Paper Title: Nanohertz gravitational waves from a quasar-based supermassive black hole binary population model as dark sirens.











Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China


Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China


School of Physics and Optoelectronic Engineering, Hainan University, Haikou 570228, China



Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China


College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China
Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China






Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China

Corresponding author
zhangxin@mail.neu.edu.cn
Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
Key Laboratory of Data Analytics and Optimization for Smart Industry (Ministry of Education), Northeastern University, Shenyang 110819, China
National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang 110819, China





§ ABSTRACT 

Recently, several pulsar timing array (PTA) projects have detected evidence of the existence of a stochastic gravitational wave background (SGWB) in the nanohertz frequency band, providing confidence in detecting individual supermassive black hole binaries (SMBHBs) in the future. Nanohertz GWs emitted by inspiraling SMBHBs encode the luminosity distances of SMBHBs. They can serve as dark sirens to explore the cosmic expansion history via a statistical method to obtain the redshift information of GW sources' host galaxies using galaxy catalogs. The theoretical analysis of the dark siren method relies on the modeling of the population of SMBHBs. Using a population model consistent with the latest SGWB observations is essential, as the SGWB provides significant information about the distribution of SMBHBs. In this work, we employ a quasar-based model, which can self-consistently account for the SGWB amplitude, to estimate the population of SMBHBs. We constrain the Hubble constant using the mock GW data from different detection cases of PTAs in the future. 
Our results show that a PTA consisting of 100 pulsars with a white noise level of 20 ns could measure the Hubble constant with a precision close to 1% over a 10-year observation period, and a PTA with 200 pulsars may achieve this goal over a 5-year observation period. The results indicate that modeling the SMBHB population significantly influences the analysis of dark sirens, and SMBHB dark sirens have the potential to be developed as a valuable cosmological probe.




















Nanohertz gravitational waves from a quasar-based supermassive black hole binary population model as dark sirens
    Xin Zhang
    
================================================================================================================





§ INTRODUCTION

The study of cosmology has entered the era of precision cosmology <cit.>. 
Five of the six basic parameters of the ΛCDM model have been precisely constrained by the Planck cosmic microwave background (CMB) data with constraint precisions better than 1% <cit.>.


However, as the precision of cosmological parameter measurements improves, some puzzling tensins have come to light. The Hubble constant (H_0) is a crucial cosmological parameter describing the universe's current expansion rate. Its values inferred from the Planck CMB observation (assuming the ΛCDM model) <cit.> and the model-independent local-universe distance ladder measurement <cit.> are in more than 5σ tension, which has become a critical crisis in cosmology <cit.> (see Refs. <cit.> for related discussions). 
While the Hubble tension indicates the potential for new physics beyond the ΛCDM model, there remains a lack of consensus on a valid extended cosmological model to address this tension. Additionally, it is imperative to explore new cosmological probes capable of independently measuring H_0.


Compact binary coalescence (CBC) events, a category of gravitational wave (GW) sources, have been discovered by the LIGO-Virgo-KAGRA collaboration in recent years, with approximately a hundred events detected.
The luminosity distances of CBC events can be determined through the analysis of GW waveforms, and these events are commonly referred to as “standard sirens” <cit.>. 
Standard sirens can provide absolute distance measurements, avoiding poorly understood calibration processes inherent in the distance ladder method. If the redshifts of GW sources are also determined, the distance-redshift relation could be established to study the expansion history of the universe (see Refs. <cit.> for related discussions).  
The binary neutron star coalescence event, GW170817, gives H_0=70^+12_-8  km s^-1 Mpc^-1 with a precision of 14% <cit.>. Its redshift was determined by directly identifying the electromagnetic (EM) signals produced when the binary merged <cit.>. Such standard sirens are known as bright sirens, with only one case detected to date. 


For the standard sirens without EM counterparts (usually referred to as dark sirens), we can cross-match them with galaxy catalogs and statistically infer their redshifts (see Refs. <cit.> for related works). Recently, using 46 dark sirens with signal-to-noise ratios (SNRs) over 11 from GWTC-3 and the GLADE+ K-band galaxy catalog <cit.>, researchers achieved a ∼19% constraint on H_0. Combined with the bright siren GW170817, the constraint precision of H_0 improves to ∼10% <cit.>. 

The frequencies of GWs generated by the GW sources with varying masses are hugely different, which requires GW detectors sensitive to diverse frequency bands. The current LIGO-Virgo-KAGRA detector network <cit.> can detect only stellar-mass CBCs at around hundreds of hertz. In the future, precise measurements of GWs in various bands will become possible. In the frequency band from a few hertz to several thousand hertz, the third-generation ground-based GW detectors, e.g., the Cosmic Explorer <cit.> and the Einstein Telescope <cit.>, will significantly enhance the detection capabilities for stellar-mass CBCs. In the decihertz band, the Decihertz Interferometer Gravitational-Wave Observatory <cit.> and lunar-based detectors <cit.> could detect intermediate-mass black hole binaries and binary white dwarfs. In the millihertz band, GWs from massive black hole binaries and extreme mass ratio inspirals could be detected by space-borne GW detectors such as the Laser Interferometer Space Antenna <cit.>, Taiji <cit.>, and TianQin <cit.>. In the nanohertz (nHz) band, the pulsar timing arrays (PTAs) could detect continuous GWs from the inspiraling supermassive black hole binaries (SMBHBs) resulting from galaxy mergers.

The pulsars constructing PTAs are millisecond pulsars (MSPs) and have highly stable rotation periods. Their radio signals can be periodically detected by radio telescopes on the Earth with extremely high precision. GWs passing between the MSPs and the Earth will alter the proper distances between them. This leads to a discrepancy between the actual and expected arriving times of radio signals, which is known as the timing residuals. By analyzing the time residuals of PTAs, we can extract information about nHz GWs.
Recently, North American Nanohertz Observatory for Gravitational Waves <cit.>, European PTA + Indian PTA <cit.>, Parkes PTA <cit.>, and Chinese PTA <cit.> have reported substantial evidence for the existence of a stochastic GW background (SGWB) <cit.>. 

Such evidence boosts confidence in making more advances in detecting nHz GWs. For instance, detecting individual SMBHBs has become one of the next important missions of PTA observations, which has been widely discussed in Refs. <cit.>. Although extracting an individual GW signal from the superimposed GW background is challenging, this goal may be achievable with the addition of more advanced radio telescopes, especially in the era of the Square Kilometre Array (SKA).

Recently, Refs. <cit.> indicated that utilizing SMBHBs as standard sirens could help to improve the measurement precisions of cosmological parameters. 
The authors of Ref. <cit.> pointed out that the EM counterparts of SMBHBs can be observed during its inspiraling stage and forecast the constraint on the equation of state of dark energy with 154 SMBHB bright sirens observed by SKA-era PTAs.

The authors of Ref. <cit.> investigated the dominant factors affecting the detection capability of PTAs for SMBHBs and found that the root mean square (RMS) of white noise, σ_t, is more dominant than the number of pulsars in affecting the detections of SMBHBs. They also proposed that detecting nHz GWs from individual SMBHBs has significant implications for exploring the properties of dark energy and precisely determining the Hubble constant.

In the aforementioned papers, the SMBHB standard siren mock data are simulated based on the galactic major merger model proposed by Mingarelli et al. <cit.> (hereafter the M17 model).
Recently, Casey-Clyde et al. <cit.> indicated that most major merger models predicted characteristic amplitudes of SGWB lower than that derived from the NANOGrav 12.5-yr data. In this context, they proposed a quasar-based SMBHB population model (hereafter the C21 model), considering that galaxy mergers could trigger quasars, and SMBHBs arising from galaxy mergers might be linked to quasar populations. This model can self-consistently predict the local number density of SMBHBs given the amplitude of the SGWB.


Based on the C21 model and the common-process signal in the NANOGrav 12.5-yr dataset, the number of SMBHBs in the local universe is roughly five times higher than that predicted by the M17 model. 
It is evident that there is a significant difference in the number of SMBHBs predicted by these two models. Notably, the number of SMBHB dark sirens significantly influences the constraints on cosmological parameters, especially when the localization of dark sirens is poor <cit.>. Nevertheless, previous studies have not forecast the capability of SMBHB dark sirens of constraining cosmological parameters based on the C21 model. Consequently, a crucial question arises: Based on the C21 model that is more consistent with SGWB observations, what role can SMBHB dark sirens actually play in measuring cosmological parameters?

To answer this question, we simulate the mock SMBHB dark siren data based on the C21 quasar-based model and the SGWB signal inferred from the NANOGrav 15-yr dataset <cit.>.
We analyze the ability of SKA-era PTAs to detect the local-universe SMBHBs based on various PTA detection cases, and then combine the mock dark siren data with the 2 Micron All Sky Survey (2MASS) Extended Source Catalog <cit.> to perform cosmological parameter estimations. 
In this work, we only focus on the ability of SMBHB dark sirens to measure H_0. This is because our aim is to investigate how SMBHB dark sirens can contribute to resolving the Hubble tension, and we primarily mock SMBHB dark sirens in the local universe, where the distance-redshift relationship is only sensitive to H_0. Therefore, we use the SMBHBs dark sirens to infer H_0 and fix all other cosmological parameters to the Planck 2018 values. 

This paper is organized as follows. Section <ref> introduces the methodology used in this work. In Section <ref>, we report our constraint results and make relevant discussions. The conclusion is given in Section <ref>. Unless otherwise stated, we adopt the system of units in which G = c = 1 throughout this work.



§ METHODOLOGY




 §.§ Subcatalog of 2MASS galaxy catalog

Massive galaxies are hypothesized to have a higher probability of hosting SMBHBs <cit.>. To select massive galaxies in the local universe, we first estimate their K-band absolute magnitudes by M_K=m_K-5 ×log _10(d_ L)-25-0.11 × A_v, where d_ L represents the luminosity distance to the galaxy. Ref. <cit.> considered the impact of peculiar motions on redshift measurements, allowing for precise calculations of distances for all galaxies in the 2MASS Redshift Survey (2MRS) catalog <cit.>. The apparent magnitude m_K and the extragalactic extinction A_v can be obtained from the 2MRS catalog. We implement a K-band cutoff with M_K≤ -25 for selecting galaxies with stellar mass M_⋆≥ 10^11 M_⊙. 
 


Based on this criterion, we select 13172 galaxies from the 2MRS catalog as potential host galaxies for SMBHBs. These candidate host galaxies constitute a new catalog, referred to as the 2MASS catalog in the following text.


Since the 2MASS catalog is incomplete at higher redshifts, we further consider its completeness.
The completeness fraction, P_com(𝒮), can be expressed as

    P_com(𝒮) ≡N_cat(𝒮)/n̅_gal V_ c(𝒮),

where 𝒮 = 𝒮(z, Ω̂; Δ z, ΔΩ) represents a volume that is a portion of a cone. This cone is defined by the central redshift z with a range from z - Δ z to z + Δ z, and the central angle Ω̂ with a solid angle ΔΩ. 
n̅_gal represents the density of galaxies per comoving volume.
V_ c is the comoving volume of 𝒮, and N_cat is the number of galaxies in the catalog within 𝒮.
We assume that the completeness of the 2MASS catalog is isotropic and count the number of galaxies in each redshift bin. The redshift range from 0 to 0.1 is equally divided into 30 bins. Additionally, we assume that galaxies are uniformly distributed within the comoving volume and consider the 2MASS catalog complete at z < 0.04. Therefore, we can derive the completeness of the 2MASS catalog as a function of redshift using Eq. (<ref>), which is shown in Fig. <ref>.







 §.§ Population models of SMBHBs

The differential comoving number density of SMBHBs can be described as d^3Φ_BHB/d M_BH, 1d z d q, where M_ BH,1 is the mass of the larger SMBHB, and q is the mass ratio defined as q=M_ BHB/M_ BH,1-1 with M_BHB being the total binary mass. q is assumed to follow a log-normal distribution, p(q), centered at log_10q = 0 with a standard deviation of σ_ log_10q = 0.5dex, within the range [0.25, 1.0], similar to the “hiq” model described in Sesana et al. <cit.>.


While the galactic major merger models can effectively estimate the distribution of SMBHBs, these models are based on the assumptions about the evolution processes of the galaxies and black holes, the galaxy merger rates, and the relationship between the galaxy mass and the black hole mass, etc. However, these assumptions vary greatly among different major merger models, leading to significantly different SMBHB number densities in simulations. It should be noted that the SGWB characteristic amplitude predicted by the M17 model was lower than that of NANOGrav 12.5-yr data. This issue must be addressed if the SGWB signal is considered to be contributed by SMBHBs. 
We note that the C21 model <cit.> can self-consistently incorporate the SGWB characteristic amplitude as a parameter to infer the distribution of SMBHBs, and the number of nearby SMBHBs inferred from the NANOGrav 12.5-yr data is more than five times the number estimated by the M17 major merger model. Therefore, we choose the C21 model to estimate the distribution of SMBHBs and compare it with the M17 major merger model <cit.>.




  §.§.§ Major merger model

We convert M_K to the stellar masses for early-type galaxies using the empirical equation, log_10M_⋆ = 10.58-0.44(M_K + 23) <cit.>. The masses of SMBHBs in these galaxies are determined via the M_ BHB-M_ bulge relationship from Ref. <cit.>, where the bulge mass, M_ bulge, is approximated to the stellar mass M_⋆. The mass ratio q of the binary is drawn from a log-normal distribution, p(q), centered at log_10 q = 0 and with the standard deviation of σ_ log_10q = 0.5 dex, within the range [0.25, 1.0]. Then, the probability of the j-th galaxy hosting an SMBHB in the PTA band, p_j, is given by


    p_j(M_⋆, z)=t_ c,j/T_ life∫_μ_⋆≥ 0.25 dμ_⋆d N/ d t(M_⋆, μ_⋆, z') T_ life,

where t_ c, j = (5/256) (π f_ low)^-8/3 (GM_ c/c^3)^-5/3 is the time to coalescence of the SMBHB in the j-th galaxy, with f_ low=1 nHz being the lower limit of the PTA band. T_ life = T_ df + T_ sh is the lifetime of the SMBHB, where T_ df and T_ sh are the dynamical friction timescale <cit.> and the stellar hardening timescale <cit.>, respectively. 
d N / d t(M_⋆, μ_⋆, z') represents the galaxy merger rate derived from the Illustris cosmological simulation project <cit.>, where M_⋆ denotes the stellar mass of the galaxies, μ_⋆ denotes the progenitor stellar mass ratio, and z' denotes the redshift at which the galaxies merge, calculated at T_z with Planck cosmological parameters <cit.>.

The number of SMBHBs in the galaxy catalog is N_ SMBHB = ∑_j p_j for one simulation. We perform numerous simulations to obtain a distribution of N_ SMBHB. From this distribution, we select the simulation that yields the N_ SMBHB value closest to the maximum likelihood. We find that there are 213 sources emitting GWs in the PTA band in the 2MASS catalog. Then, we randomly select 213 galaxies from the 2MASS catalog as the host galaxies of SMBHBs according to the probability p_j.




  §.§.§ Quasar-based model

The characteristic amplitude h_ c of the GW signal generated by the population of merging SMBHBs can be expressed as <cit.>

    h_ c^2(f)=4/π f^2∭   d M_BH, 1d z d q 
       ×d^3Φ_BHB/d M_BH, 1d z d q1/1+zd E_gw/dln f_r,

where d^3Φ_BHB/d M_BH,1 dz dq represents the differential comoving number density of SMBHBs, dE_gw/ dln f_ r is the GW energy emitted per unit logarithmic frequency change, and f_ r = f(1 + z) is the GW frequency in the rest frame of the source. For a population of SMBHBs in circular orbits, the GW energy per unit logarithmic frequency change is <cit.>

    d E_gw/ dln f_r    = (π f_ r)^2 / 3/3M_ c^5 / 3,

where the SMBHB chirp mass is given by M_ c^5/3 = q/(1+q)^2M^5/3_BHB.
Therefore, the differential comoving number density of SMBHBs is the main parameter influencing the value of h_ c. 


Since some quasars could be driven by the mergers of black holes, the populations of quasars and black holes might be correlated. Therefore, the C21 model makes a simplified assumption that the ratio of the differential comoving number density of quasars to that of black holes remains constant:

    d^3Φ_BHB/d M_BH, 1d z d q∝d^2Φ_Q/d M_BHd z p(q).

That is to say, the differential comoving number density of SMBHBs is proportional to the differential comoving number density of quasars, d^2Φ_Q/d M_BH d z, and an assumed mass ratio distribution p(q). The number density of quasars is derived from the quasar luminosity function and Eddington ratio distribution of the lifetime of the quasar from the galaxy merger simulations in Ref. <cit.>. 
To estimate the number density of quasars, we convert the differential quasar triggering rate <cit.>, d^2Φ_Q/d M_BHd t_ r, into a redshift distribution using the ΛCDM model with WMAP9 data,

    d^2Φ_Q/d M_BHd z=d^2Φ_Q/d M_BHd t_ rd t_ r/d z.


According to Eqs. (<ref>) and (<ref>), we can renormalize the SMBHB population with both the SGWB characteristic amplitude and the quasar number density distribution,

    d^3Φ_BHB/d M_BH, 1d z d q=Φ_BHB, 0/Φ_Q, 0d^2Φ_Q/d M_BHd t_ rd t_ r/d zp(q)/∫_q ≥ 0.25d q p(q),

with


    Φ_BHB, 0 =    . ∬_M_BH, 1≥ M_BH, det, 
     q ≥ 0.25d M_BHd q 
    d^3Φ_BHB/d M_BH, 1d z d q|_z=0,



    Φ_Q, 0=.∫_M_BH≥ M_BH, detd M_BHd^2Φ_Q/d M_BHd t_ rd t_ r/d z|_z=0.


The best-fit value of the SGWB characteristic amplitude from the NANOGrav 15-yr data set is h_ c = 2.4 × 10^-15 <cit.>. Assuming that this SGWB signal is contributed by numerous SMBHB inspirals, according to our calculations using Eqs. (<ref>)–(<ref>), there are approximately 688 SMBHBs in the redshift range (0, 0.1). 
Considering the completeness of the 2MASS catalog, we use Eq. (<ref>) to calculate the total completeness fraction of the 688 SMBHBs. This completeness fraction actually refers to the proportion of the host galaxies included in the 2MASS catalog relative to the total host galaxies. We find that it is approximately equal to 490. Therefore, we randomly select 490 galaxies based on their P_ com values from the 2MASS catalog as SMBHBs’ host galaxies. We assume that the host galaxies of the other 198 SMBHBs are not included in the 2MASS catalog and ignore their contributions to the subsequent calculations for dark sirens.







 §.§ GW detection and Fisher analysis


The timing residuals induced by a single GW source located in the direction represented by the unit vector Ω̂ and measured at time t on Earth can be expressed as

    s(t, Ω̂)=F^+(Ω̂) Δ A_+(t)+F^×(Ω̂) Δ A_×(t).

Here, F^+(Ω̂) and F^×(Ω̂) are the antenna pattern functions <cit.>, which can be expressed as

    F^+(Ω̂)=1/4(1-cosθ){(1+sin ^2δ) cos ^2δ_pcos[2(α-α_p)]. 
       .  -sin 2 δsin 2 δ_pcos(α-α_p)+cos ^2δ(2-3 cos ^2δ_p)}, 
       F^×(Ω̂)=1/2(1-cosθ){cosδsin 2 δ_psin(α-α_p). 
       .  -sinδcos ^2δ_psin[2(α-α_p)]},

where (α, δ) and (α_ p, δ_ p) represent the right ascension and declination of the GW source and the pulsar, respectively. θ represents the opening angle between the GW source and the pulsar,

    cosθ=cosδcosδ_pcos(α-α_p)+sinδsinδ_p.

In Eq. (<ref>), Δ A_{+, ×} (t)=A_{+, ×}(t)-A_{+, ×}(t_p) <cit.> represents the difference between the Earth term A_{+, ×}(t) and the pulsar term A_{+, ×}(t_p).
t_p is the time when the GW passes the pulsar and can be expressed as t_p=t-d_p(1-cosθ)/c, with d_p representing the pulsar distance.

For evolving SMBHBs in circular orbits, we have

    A_+(t)=     h(t)/2 π f(t){(1+cos ^2ι) cos 2 ψsin [ϕ(t)]
       +2 cosιsin 2 ψcos[ϕ(t)]},


    A_×(t)=     h(t)/2 π f(t){(1+cos ^2ι) sin 2 ψsin [ϕ(t)] 
       -2 cosιcos 2 ψcos[ϕ(t)]}.

Here, ψ is the GW polarization angle, ϕ is the phase of the GW, ι is the inclination angle of the GW, and the GW strain amplitude h(t) can be formulated as

    h(t) = 2 (G M_c)^5 / 3/c^4(π f(t))^2 / 3/d_L,

where the frequency of GW signals is calculated by the following formula <cit.>,

    f(t) = π^-1(GM_ c/c^3)^-5/8[256/5(t_ c-t)]^-3/8.

Here, t_ c is uniformly sampled from [100 yr, 26 Myr]. This range represents the time to coalescence of a SMBHB with M_ BH,1 = 10^9 M_⊙ and q = 1 from 1 nHz to 100 nHz.
h_0 and f_0 represent the values of h(t) and f(t) when t=0, respectively.
Fig. <ref> shows the distribution of mock SMBHBs on the f_0-h_0, f_0-M_ BHB, z-M_ BHB planes.

The inclination angle, the polarization angle, and the initial phase are randomly sampled in the ranges cosι∈ [-1, 1], ψ∈ [0, 2π], and ϕ_0 ∈ [0, 2π], respectively.
The GW phase, ϕ(t), can be expressed as

    ϕ(t)=ϕ_0 + 1/16(G M_c/c^3)^-5 / 3{(π f_0)^-5 / 3-[π f(t)]^-5 / 3}.

The square of SNR of a GW signal detected by a PTA is defined as

    ρ^2=∑_j=1^N_p∑_i=1^N[s_j(t_i)/σ_t, j]^2,

where N is the total number of the data points of timing residuals for each pulsar, N_ p is the number of pulsars, s_j(t_i) is the timing residual induced by the GW signal in the j-th pulsar at time t_i, and σ_t,j is the RMS of the white noise of the j-th pulsar. The SNR threshold is set to 10 in the simulation.

It should be noted that in this work, we did not consider red noise, nor did we take an unresolved SGWB as red noise into account. We assume that the pulsar noise is primarily dominated by white noise and that individual GW sources can be disentangled from the SGWB through specialized methodologies. 

We utilize the Fisher information matrix for parameter estimations. For a PTA comprising N_ p independent pulsars, the Fisher information matrix can be expressed as

    Γ_a b=∑_j=1^N_ p∑_i=1^N∂ s(t_i)/σ_t, j∂θ_a∂ s(t_i)/σ_t, j∂θ_b,

where θ denotes the free parameters to be estimated.

For each GW source, the timing residuals depend on N_p+8  system parameters, including eight parameters of the GW source (i.e., M_ c, α,δ, ι, ψ, ϕ_0, f_0, d_ L) and the distances of pulsars d_p,j ( j = 1, 2,⋯, N_ p). With additional EM information priors, the estimation precision of parameter can be improved by adding to the appropriate diagonal element of the Fisher information matrix, i.e., Γ_a b→Γ_a b-⟨∂^2ln P(θ_i)/∂θ_a∂θ_b⟩. Here, P(θ_i) is the prior distribution of the parameter θ_i <cit.>. Since the disk direction is randomly distributed in a 4π solid angel, the prior distribution of the inclination angle ι is given by P(ι) ∝sin (ι). Consequently, we can modify the Fisher information matrix element Γ_i i to Γ_i i→Γ_i i+1/sin ^2ι with θ_i = ι.  


The 1σ absolute error of the parameter θ_a can be estimated by 

    Δθ_a=(Γ^-1)_a a^1 / 2.


The angular resolution ΔΩ can be calculated according to the uncertainties of the parameters,

    ΔΩ=2 π√((cosδΔδΔα)^2-(cosδ(Γ^-1)_δα/ΔδΔα)^2).

For the parameter d_ L, we first estimate its instrumental error by Eq. (<ref>). Besides, we also consider the error arising from weak lensing, which is expressed by the fitting formula from Refs. <cit.>,

    Δd_L^lens(z)     = d_L(z) × 0.066[1-(1+z)^-0.25/0.25]^1.8.

The total error of d_ L is expressed as

    Δd_L    = √((Δd_L^ inst)^2+(Δd_L^ lens)^2).

It should be added that we assume the distance to each pulsar has been precisely determined. Under this assumption, the derivatives of GW frequencies can be obtained, and the chirp masses of the SMBHBs can be further measured. This enables us to break the degeneracy between M_ c and d_ L, and therefore precisely infer d_ L from GW amplitudes. 
The distributions of the d_ L relative errors and the angular resolutions of simulated SMBHBs versus SNR based on 20, 100, and 200 MSPs with σ_t = 20 ns and 10-year observation are displayed in Fig. <ref>. The sky locations of MSPs are randomly selected from the Australia Telescope National Facility pulsar catalog <cit.>.








 §.§ Inference of the Hubble constant


In this work, we adopt the flat ΛCDM model to make the cosmological analysis. The Hubble parameter, which quantifies the rate of expansion of the universe at redshift 
z, can be expressed as

    H(z) = H_0√(Ω_ m(1+z)^3+1-Ω_ m),

where H_0 is the Hubble constant and Ω_ m is the current matter density parameter. The luminosity distance at redshift z is given by

    d_ L(z) = c(1+z) ∫_0^z1/H(z^')d z^',

where c is the speed of light in vacuum. Throughout this work, 
we adopt the flat ΛCDM model to generate the dark siren data with the best-fit values of Ω_ m=0.3111 and H_0=67.66  km s^-1 Mpc^-1 given by the Planck 2018 TT, TE, EE+lowE+lensing+BAO data <cit.>. Based on the flat ΛCDM model, we utilize the GW signals from SMBHBs as dark sirens to infer H_0, fixing the other cosmological parameters to the Planck 2018 results.

Using the Bayesian method to infer the cosmological parameters, the posterior probability distribution of H_0 can be expressed as

    p(H_0|𝒟_GW, 𝒟_EM) 
       = ∏_i=1^N_GW p(𝒟_GW, i, 𝒟_EM, i|H_0)p(H_0)/γ(H_0),

where 𝒟_GW and 𝒟_EM represent the detected GW data and the EM data, respectively. p(H_0) is the prior distribution of H_0, and N_GW is the number of GW events. p(𝒟_GW,i, 𝒟_EM,i|H_0) is the likelihood function of a single GW event, which can be written as

    p(𝒟_GW, i, 𝒟_EM, i| H_0) 
       = ∫ p(𝒟_GW, i, 𝒟_EM, i, d_L, α, δ, z | H_0) dd_Ldαdδd z 
       = ∫ p(𝒟_GW, i|d_L, α, δ) p(𝒟_EM, i| z, α, δ) p(d_L| z, H_0) 
           × p_0(z, α, δ| H_0) dd_Ldαdδd z 
       = ∫ p(𝒟_GW, i|d_L, α, δ) p(𝒟_EM, i| z, α, δ) 
           ×δ.(d_L-d̂_L(z, H_0).) p_0(z, α, δ| H_0) d d_Ldαdδd z 
       = ∫ p.(𝒟_GW, i|d̂_ L(z, H_0), α, δ.) p(𝒟_EM, i| z, α, δ) 
           × p_0(z, α, δ| H_0) dαdδd z.


p.(𝒟_GW, i|d̂_L(z, H_0), α, δ.) is the likelihood of the GW data, which can be expressed as

    p.(𝒟_GW, i|d̂_L(z, H_0), α, δ.) ∝ e^-χ^2 / 2,

with χ^2 = (x-x_ gw)^T𝒞^-1(x-x_ gw), where 𝒞 is the 3 × 3 covariance matrix only relevant to (d_L, α, δ), obtained from the Fisher information matrix. x = (d̂_L(z,H_0), α, δ) represents the three-dimensional (3D) position in the sky.
x_ gw=(d_ L,i, α_ i, δ_i) represents the 3D position of the GW source. 

Considering the bias between the true value and the central value of the posterior distribution from the actual observation, we use the Fisher matrix to calculate the Gaussian distribution of x_ gw, and randomly select the value of x_ gw according to this Gaussian distribution.

We establish the boundary of the GW source’s localization volume based on the 3D measurement errors of x_ gw. The galaxies whose positions satisfying χ^2 ≤ 11.34  (corresponding to 99% confidence) can be considered as the potential host galaxies of the GW source. The number of potential host galaxies for a GW source is defined as N_ in, which is depend on the prior values of H_0. We display N_ in as a function of SNR in Fig. <ref>, fixing H_0=67.66  km s^-1  Mpc^-1. Due to the lack of EM signals in dark sirens, p(𝒟_EM,i| z, α, δ) can be set to a constant value <cit.>.

For the potential candidate host galaxies, p_0(z, α, δ| H_0) in Eq. (<ref>) can be expressed as

    p_0(z, α, δ| H_0) =     f_ com p_cat(z, α, δ)+(1-f_ com) 
       × p_miss(z, α, δ| H_0),

where f_ com is the completeness fraction of the catalog, which can be expressed as

    f_ com = N_cat/n̅_gal V_ max,

where N_cat is the number of galaxies in the galaxy catalog, and V_ max is the entire comoving volume bounded by the upper limit of redshift, i.e., z_ max= 0.1. p_cat(z, α, δ) is the observed galaxy distribution function provided by the galaxy catalog, which is expressed as 

    p_cat(z, α, δ)=1/N_in∑_j=1^N_in   𝒩(z_j, σ_z, j) δ(α-α_j)
       ×δ(δ-δ_j),

where 𝒩 represents the Gaussian distribution with mean z_j and the standard deviation σ_z,j = (1+z) √(⟨v_ p^2⟩)/c <cit.>. 
√(⟨v_ p^2⟩) is the standard deviation of the radial peculiar velocity of a galaxy, set to 500  km s^-1 <cit.>.   
And p_ miss is given by 

    p_miss(z, α, δ| H_0) ∝ (1 - p_com(z, α, δ)) d V_ c/d z dαdδ.



The normalization term γ(H_0) in Eq. (<ref>) can be written as 

    γ(H_0)= ∫   p_det^GW(d_L(z, H_0), α, δ) p_det^EM(z, α, δ) 
       × p_0(z, α, δ) d z dαdδ,

where p_det^GW and p_det^EM represent the GW selection effect and the EM selection effect, respectively. The GW selection effect can be expressed as

    p_det^GW(d_L(z, H_0), α, δ)=
       ∫_𝒟_GW>𝒟_GW^th p(𝒟_GW|d_L(z, H_0), α, δ) d𝒟_GW,

where 𝒟^ th_ GW denotes the SNR threshold for GW detections and is set to 10. The EM selection effect can be expressed as p_det^EM(z, α, δ) ∝ℋ(z_max-z), where ℋ is the Heaviside step function and z_max=0.1.




§ RESULTS AND DISCUSSIONS








In this section, we compare the two SMBHB population models, discuss the GW parameter estimations, and present the constraint results for H_0 from various PTA detection cases.

As shown in Fig. <ref>, the h_0, z, M_ BHB, and f_0 distributions of SMBHBs simulated from the two population models are essentially similar. Nevertheless, the numbers of simulated SMBHBs for the two population models are 490 and 213, respectively, showing that the C21 model has a noticeably larger number of SMBHBs than the M17 model. We consider the 10-year observations based on the C21 model as the representative example of PTA observations for the following discussions.











 
In Fig. <ref>, we present the distributions of ρ, ΔΩ, and Δ d_ L/d_ L based on the 20, 100, and 200 MSPs with σ_t = 20 ns. Note that the calculations are based on 213 (M17 model) and 490 (C21 model) simulated SMBHBs at z < 0.1.
As shown in the left panel of Fig. <ref>, fewer available pulsars result in lower SNRs, thus showing worse localization abilities. We note that the localization capability with 100 MSPs is much better than that of 20 MSPs, but 200 MSPs are slightly better than 100 MSPs. This indicates that 100 MSPs seem to be sufficient and more economical, as further increasing the number of pulsars does not significantly enhance the localization ability.









For the mock SMBHBs from the two population models, we consider PTA cases of monitoring 20, 100, and 200 MSPs with σ_t = 20 ns to detect them. The observation time spans are T_ span = 5, 10, 15, 20 years, respectively, with a cadence of 2 weeks that is the typical cadence used in current PTAs. The detected SMBHBs are used as dark sirens to constrain H_0. 


Figure <ref> illustrates the relative errors for H_0 as a function of T_ span. Table <ref> shows the relative error of H_0, defined as ε(H_0)=σ(H_0)/H_0. We observe that the PTAs with N_ p=20 give the poorest measurement precision for H_0. Based on the C21 model, the constraint precision of H_0 is 2.44% for a 5-year observation, improving to 1.71% over a 20-year observation. These values are worse than the constraint precision obtained from current distance ladder measurements. Increasing the number of MSPs to 100 significantly enhances the precision of H_0 compared to using 20 MSPs. With a 5-year observation, the constraint precision of H_0 is 1.33% and can be improved to 0.85% with a 20-year observation; both values are close to or better than 1%. Furthermore, monitoring 200 MSPs provides even better constraints on H_0, achieving precisions ranging from 1.07% to 0.71% over 5 to 20 years of observations.




Figure <ref> presents the constraints on H_0 obtained from individual GW events as well as the combined results. The colored lines represent the constraints from individual events and the black line represents the combined result of all events.

There is a notable discrepancy between the two SMBHB population models in terms of the H_0 posterior distributions given by the individual events. In the M17 model, there are several events whose SNRs far exceed the others, but in the C21 model, there is no significant difference in the SNRs given by all the events. This discrepancy can also be seen in Fig. <ref>: Several SMBHBs (represented by the orange circles) in the M17 model have significantly higher masses and GW strains than the others. This is due to the fact that the mechanisms of the two models in predicting the SMBHB populations are substantially different.
The M17 model predicts the masses of SMBHBs based on the apparent magnitude of the galaxies in the galaxy catalog. 
A few of these galaxies have significantly larger masses, and the SMBHBs inhabiting them could produce the GW
signals with notably higher SNRs than the other SMBHBs. However, for the C21 model, its SMBHB distribution is derived from the SGWB signal, which is the overall superposition of many GW events and does not reflect the information of individual sources. As a result, the SMBHB distribution predicted by the C21 model is statistical rather than individualistic. Therefore, in the C21 model, no event exhibits a SNR that is significantly higher than other events.







Compared with the dark siren analysis in Ref. <cit.>, under the same detection conditions and population models, the constraint precision of H_0 is improved in this paper. This is mainly due to the fact that we simulate SMBHBs over a broader range of redshift (0<z<0.1), resulting in a larger number of GW sources available as dark sirens. We further consider the effect of incompleteness for galaxies at z>0.04.
The authors of Ref. <cit.> also indicated some advantages of nHz GW standard sirens over the standard sirens in other frequency bands. Here, we want to emphasize another advantage: Since SMBHBs are all hosted in supermassive galaxies, we can directly exclude those galaxies with low masses when using the dark siren method, and only need to consider the supermassive galaxies within the localization volume of the GW source. Compared with the stellar-mass dark sirens, it can greatly reduces the computational costs and the uncertainties caused by spurious host galaxies in Bayesian inference.




 










§ CONCLUSION

Several PTA projects have reported substantial evidence for the existence of a SGWB in the nHz band. In addition to SGWB, the GW signals from individual SMBHBs may also be observed in the future. Such individual GW events can be used as dark sirens to constrain cosmological parameters. In the previous studies on dark sirens, the simulations of SMBHBs were based on the M17 model, which cannot fit the observed SGWB signal well. In this paper, we use the C21 model, which can self-consistently predict the local number density of SMBHBs given the amplitude of the SGWB, to investigate the potential of SMBHB dark sirens in measuring the Hubble constant. According to our results, the following conclusions can be drawn:

(i) In the C21 model, a PTA with 100 MSPs over a 10-year observation period could measure H_0 with a precision close to 1%. If the number of MSPs increases from 100 to 200, a 5-year observation could measure H_0 with a precision close to 1%.

(ii) The C21 model predicts a larger number of SMBHBs than the M17 model. The numbers of SMBHBs derived from the M17 and C21 models are 490 and 213, respectively.  



(iii) The SMBHB distribution in the C21 model is derived from the SGWB signal, which is the overall superposition of the GW signals from individual events. Therefore, the mass distribution of SMBHBs predicted by the C21 model is more statistical than those predicted by the M17 model.

Our conclusions indicate that modeling the SMBHB population has a great impact on the analysis of SMBHB dark sirens. In the future, more accurate measurements of the SGWB signal will not only give us a better understanding of the distribution and evolution history of SMBHBs, but also have important implications for measuring cosmological parameters via the dark siren method.






















We thank Shang-Jie Jin for helpful discussions.
This work was supported by the National SKA Program of China (Grants Nos. 2022SKA0110200 and 2022SKA0110203), the National Natural Science Foundation of China (Grants Nos. 11975072, 11875102, 11835009, 12305069, and 12305058), the National 111 Project (Grant No. B16009), the Program of the Education Department of Liaoning Province (Grant No. JYTMS20231695), and the Natural Science Foundation of Hainan Province (Grant No. 4240N215).






 Paper Title: Neutron Star Collapse From Accretion: a Probe of Massive Dark Matter Particles.







liuning@njnu.edu.cn
Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 230021, China



arvindm215@gmail.com
Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 230021, China


		

	

§ ABSTRACT 


We explore the multi-scatter capturing of the massive dark matter (DM) particle inside the neutron star via a momentum-dependent dark matter-nucleon scattering cross-section. We find that the capturing enhanced for the positive velocity and momentum transfer dependent DM-nucleon scattering in comparison with the constant cross-section case. Further, a large capture of the DM particles can be thermalized and lead to black hole formation and, therefore, destroy the neutron star. Using the observation of the old neutron star in the DM-dominated region, we obtain strong constraints on massive DM parameters.  

		

		
		
		Neutron Star Collapse From Accretion: a Probe of Massive Dark Matter Particles
    
	Arvind Kumar Mishra 
	
	https://orcid.org/0000-0001-8158-6602
    < g r a p h i c s > 

    August 5, 2024
=============================================================================================



	


§ INTRODUCTION


The various observations suggest that dark matter (DM) exists from its gravitational interaction with the baryons  <cit.>). The ongoing DM searches experiment assumes some form of the DM-baryon interactions; however, irrespective of current technological progress, the micro-physical properties of the DM particles remain an enigma <cit.>. 
The direct DM-detection experiments are sensitive for 𝒪(1-10^4) GeV DM masses. However, if the DM particles are massive, i.e., m_χ> 10^6 GeV, or lighter, m_χ< GeV, then during DM-nucleon collision, the energy transfer to the nucleus becomes small; therefore the detection experiment becomes insensitive to these masses.


In this work, we aim to explore the constraints on the massive dark matter, m_χ> 10^6 GeV, from the neutron star observations. The neutron star provides a lucrative place to study the DM-nucleon interaction, thanks to their strong gravity and density, see a recent review <cit.>. NS situated in the DM-dominated regions can be used to investigate the constraints in two ways: First, after capturing, DM particles become thermalized in the NS (via DM-nucleon scattering or DM-annihilation to the visible particles), which leads to heating of the NS and, therefore, observation of isolated of NS with high temperature can constraint the DM parameters <cit.>. Second, after thermalization, the DM particles can form a self-gravitating core, which can collapse, form a black hole, and ultimately destroy the NS; therefore, observation of the old neutron star in a DM-dominated region (central region of galaxies) can used to probe the DM microphysics <cit.>. However, most of the previous work has been done using a constant cross-section, without form factor, and the rest frame of the DM halo. Furthermore, incorporating the above effects into account, we obtained a strong constraint on the low-mass DM particles from the survival of NS <cit.>.


The constraint on the massive DM has been explored from heating for both a single and multiscatter capture formalism <cit.>. In previous works, <cit.>, the single scatter capture formalism has been applied to put constraints on the massive DM particles from the survival of NS, specifically using constant DM-nucleon cross-section without form factor. However, for actual estimation, one needs to consider the multiscatter capture formalism for massive DM particles <cit.>.  For other methods to explore massive DM particles, please see Refs <cit.>. 


In this work, we explore the constraint on the massive DM particles from the neutron star survival. To explore the capture of massive DM particles on the neutron star, we adopt from the multiscatter capture approach. Here, we focus on the velocity-dependent and momentum-transfer dependent DM-nucleon cross-section, which, to our knowledge, was not explored for massive DM particles in the context of black hole formation (see, for low mass DM constraint in our previous work <cit.>). In addition, we have also considered the general relativistic effect, the finite size effect of the nucleon, and the relative motion of the neutron star and galaxy frame in the capture rate. 


Further, considering the power law parametrization of momentum and velocity dependent DM-nucleon cross-section, we found that the capture rate is enhanced with a positive power of velocity and momentum. We obtained that pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel, where v_rel is the relative velocity of DM and nucleon.
Using this model, pulsar observation strongly constrain bosonic DM mass range, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and fermionic DM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. We also report that although the DM accretion is higher for σ∝ v^4_rel model, a large thermalization time puts weak constraints on the DM parameters. Interestingly, a cross-section model with a large momentum dependency, i.e., σ∝ q^4 (q represent the momentum transfer), provides no constraint on the DM parameters. 


The arrangement of our work is as follows: In section <ref>, we present a brief discussion on the DM particle capture rate using a multiscatter capture process. In Section <ref>, we discuss the black hole formation from DM accretion. Further, we present DM-Nucleon cross-section and estimate the captured particles <ref>. In Section <ref>, we present and discuss our main results. Finally, in Section <ref>, we conclude our work.



§ MULTISCATTER CAPTURE OF DM PARTICLES ON NEUTRON STAR 



We assume that the neutron star is situated in a dark matter-rich environment and explore the capture of the massive DM particles to the neutron star. In the case of massive DM particles, an energy transfer via a single interaction between the DM particles and nucleons will be insufficient to capture the DM particles. Thus, a multiple scattering will be required <cit.>. Inside the NS, the number of times by which DM scatter with the nucleons can be obtained as the ratio of the typical diameter of the NS, 2R_NS with the mean free path of the DM particles, λ_MFP, given by  N∼ 2R_NS/λ_MFP, where R_NS is neutron star radius. Assuming λ_MFP∼ 1/(n_Nσ), one can obtain, N∼ 2n_Nσ R_NS, where σ is DM-nucleon scattering cross-section, and n_N is the nucleon density <cit.>. 


Further, the probability of DM particles, p_N(τ) participating in N collisions with an optical depth, τ  can be defined via Poisson distribution (τ,N) and given by <cit.>

    p_N(τ)=2∫^1_0 y e^-yτ(yτ)^N/N! dy  ,

where 
 y is related to the incidence angle subtended by DM particles while entering the star.
The optical depth is defined as τ=3σ/2σ_sat, where σ is the DM-nucleon cross-section and σ_sat is the saturation cross-section allowed during the scattering between the DM-nucleon interaction. In the context of NS, a saturation limit is dictated via geometrical cross-section, given by  σ_geo=π R^2_NS/N_B where, R_NS is the radius of NS, and N_B is total number of the baryons, respectively. For NS, considering, R_NS=10.6 km and N_B=1.7× 10^57, we obtain σ_sat=2× 10^-45 cm^2, which we assumed to remains a constant with DM mass    [Although we have assumed the saturation limit to be a constant quantity throughout the DM mass range, in general, it depends on the DM mass, see Refs. <cit.>. Therefore our estimation are somewhat conservative.].





In the multiscatter capture process, the capture rate for N scatter, C_N is given by <cit.> [The derivation has been done  is the assumption of multi-capture with v_esc (r)≃ v_esc (R_NS) and number density of the baryons are constant inside the neutron star, n_B(r)=n_B see Ref. <cit.>]

    C_N=  π R^2_NS p_N(τ)∫^∞_v_escf(u)/u^2  w^3 g_N(w) dw  .

where g_N(w) corresponds to the probability of DM particles reducing their velocity lower than the escape velocity, v_esc after N collisions, and f(u) is the DM velocity distribution function.  In Eq. (<ref>), u is the DM velocity at very large distance, and w=√(u^2+v^2_esc). 
Therefore, the total capture rate after the N collision is obtained as

    C_total=∑^∞_N=1 C_N .

However, summation can be cut off at some value N_max corresponding to condition, p_N_max(τ)≈ 0. In our calculation, 
we approximate, g_N(w)=Θ[v_esc(1-⟨ z_i⟩β_+/2)^-N/2-w], where, β_+=4m_N m_χ/(m_N+m_χ)^2 in which, m_χ and m_N, represents the mass of the DM particles and nucleons, respectively <cit.>. Here we assume, ⟨ z_i⟩=1/2, <cit.> however for more general estimation of g_N(w), see Ref. <cit.>. 
Furthermore, the DM particle velocity distribution function is considered to be Maxwellian, given in the neutron star frame as <cit.> 

    f_0(u) = (3/2)^3/24/√(π)ρ_χ/m_χu^2/u^3_0e^-3/2u^2/u^2_0 ,

 where ρ_χ, and u_0 represent energy density, mass, and average velocity of the DM particles, respectively.


Furthermore, as on the surface of the neutron star, the DM velocity is high, so during the interaction, DM imparts large energy to the nucleons, see Ref.<cit.>. Therefore, in this analysis, we consider the finite-size effect of the nucleon during the DM-Nucleon interaction. Here, we adopt a conservative approach and assume the dipole form factor for DM-nucleon interaction (considered in the next subsection), given by <cit.>

    F(q^2)=Λ^4/(q^2+Λ^2)^2

where q≃√(2)(m_χm_N/m_χ+m_N)v_esc is momentum transfer during the DM-nucleon collision and m_N is the nucleon mass. Further, we assume Λ≃ 0.25 GeV throughout our analysis <cit.>. Furthermore, in our computation, we use the general relativistic (GR) correction, which modifies the capture rate and escapes velocity. So, using the GR correction, the capture rate and escape velocity is modifies via <cit.>

    C_N→C_N/(1-v^2_esc)  and    v_esc→√(2[ 1-(1-v^2_esc)^1/2]).

 





§ BLACK HOLE FORMATION FROM DM ACCRETION



After the accretion, the DM particles start interacting with the neutron star material and may get thermalized. In the context of the multi-capture process, the interaction of the DM particles with the neutron star is complex because, at each step, the DM particle transfers less momentum, and hence, the estimation of the thermalization time is complex.  Now, we consider the simplified approach and assume the form of the thermalization time scale as <cit.>

    t_th≃   10700μ/(1+μ)^2(10^5K/T_NS)^2(10^-45cm^2/σ)   years.

where, μ=m_χ/m_N, and σ is DM-nucleon scattering cross-section. When the thermalization time scale is smaller than the lifetime of the NS (t_th<t_NS), DM particles will be thermalized inside the NS and lie inside the  thermal radius given by 

    r_th=
    ( 9T_NS/8π G_Nρ_B m_χ)^1/2  .

where G_N is the Newton's gravitational constant,  and ρ_B is the baryon energy density. For a given density and temperature of the NS, r_th∝√(1/ m_χ), which implies that the thermal radius is smaller for massive DM particles and vice versa. 
Further, DM particles start self-gravitating whenever

    ρ_χ>ρ_B  for    r ≤ r_th.

Therefore, with the help of the above condition, we define the DM particles dictated via N_self, necessary for the self-gravitation of the DM particles inside the NS as

    N_self≃  4.8× 10^41( 1.4× 10^14gm/cm^3/ρ_B)^1/2( 100 GeV/m_χ)^5/2( T_NS/10^5K)^3/2  .

So the condition for DM self-gravitation suggest, N_χ≥ N_Self.



Furthermore, when the number of DM particles becomes higher than those defined by the Chandrashekhar limit, then self-gravitating DM particles collapse and form a black hole. In the case of fermionic dark matter (FDM) particles, gravitational collapse requires gravity to surpass the fermi pressure. For non-interacting fermionic DM particles, the Chandrashekhar  limit is obtained as <cit.>

    N^Fermions_Chandra≃ 1.8× 10^51( 100 GeV/m_χ)^3 .

From the above equation, it is important to emphasize that the massive DM particles form low-mass BH and vice versa.
In this case, the BH formation occurs when DM particles start self-gravitating, i.e., N_χ≥ N_Self, and captured DM particles are higher than the Chandrashekhar limit,i.e.,N_χ>N^Fermions_Chandra. The mass of formed BH is obtained as M_BH∼ m_χN^Fermions_Chandra, and given by

    M_BH≃    1.8× 10^53( 100 GeV/m_χ)^2GeV.

From the above equation, it is important to emphasize that the massive DM particles form low-mass BH and vice versa. 
Furthermore, to form a BH in the bosonic dark matter (BDM) particle case, gravity must overcome pressure originating from the uncertainty relation. For non-interacting bosons, the Chandrashekhar limit is given by <cit.>

    N^Bosons_Chandra≃ 1.5× 10^34( 100 GeV/m_χ)^2 .
 
In bosonic DM case, BH forms as soon as DM starts self-gravitating, 

    N_χ≥ N_Self,  for  m_χ≤ 10^21( ρ_B/1.4× 10^14gm/cm^3)  ( 10^5K/T_NS)^3 GeV  .

In this case, the mass of formed BH is obtained as M_BH∼ m_χN_Self, and explicitly given by

    M_BH≃ 4.8× 10^43( 1.4× 10^14gm/cm^3/ρ_B)^1/2( 100 GeV/m_χ)^1/2( T_NS/10^5K)^3/2  GeV.

From the above equation, we find that the massive DM particles and low neutron star temperature form low-mass BH and vice versa. 


After its formation, BH starts growing via accreting the surrounding material and also reducing its mass via Hawking evaporation. The BH mass evolves  as <cit.>

    dM_BH/dt≃ 4πλ_s( G_NM_BH/v^2_s)^2ρ_B v_s+(dM_BH/dt)_DM -1/1560π G^2_N M^2_BH  ,

where λ_s and v_s are the accretion eigenvalue and sound speed. In the r.h.s, the first and second term corresponds to the accretion rate obtained from baryon (Bondi-Hoyle accretion rate) and DM particles, and the last term represents Hawking evaporation. Assuming the DM accretion contribution is small, Eq. (<ref>) suggests an initial critical BH mass given as M^crit_BH=1.2× 10^37 GeV. Whenever formed BH with initial mass,M^ini_BH is larger than the critical mass, i.e., M^ini_BH>M^crit_BH, BH will grow via accretion and destroy the host NS. However, in the opposite case, i.e., M^ini_BH<M^crit_BH, black hole will evaporate and decrease its mass with the time. 


Furthermore, the value of the critical mass of the formed BH can be applied to estimate an upper limit on the mass of DM particles, which can be probed from the BH formation in the core of NS. The initial formed BH mass formed by fermionic DM particles,M_BH∼ m_χN^Fermions_Chandra, and bosonic DM particles via M_BH∼ m_χN_Self. Therefore assuming that BH grows via accreting the NS materials, we obtain the m_χ<2.5× 10^8 GeV (for T_NS=10^7 Kelvin) for BDM particles, and m_χ<10^10 GeV for the FDM particles. We emphasize that these constraints are obtained using the assumption that formed BH is growing via accreting the NS material and not considered DM accretion. Therefore, when DM accretion is taken into account (or NS temperature is changed for the BDM case), the constraint will be modified.    


 

§ DM ACCRETION FROM NON-CONSTANT DM-NEUCLEON ELASTIC SCATTERING

 

In this work, we explore the DM-Nucleon interaction, which is dependent on relative velocity and momentum transfer during the collision <cit.>, which are investigated in contest of the DM capturing on sun <cit.>, white dwarfs <cit.> and neutron stars <cit.>. 
 In this work, we parameterize the relative velocity-dependent cross-section (VDCS) as a power law form, given by <cit.>

    σ^vd_χN(v_rel)=σ_χN(v_rel/u_0)^2α≃σ_χN(w/u_0)^2α,

where, v_rel is the DM-nucleon relative velocity, and u_0 represent the normalized velocity. Here α depends on the DM models, and we specifically focus on the positive velocity dependence cases, i.e., α=1, and 2.  This kind of interaction is called p-wave (α=1), and d-wave (α=2) can be originated for the cases when initial state particles have relative angular momentum of 1 or 2 units, see Refs. <cit.>,<cit.>.  Here we consider the typical value, u_0=220 km/sec <cit.>. 
  

Further, we parameterize the momentum-dependent cross-section (MDCS) in the power law form as<cit.>

    σ^md_χN(q)=σ_χN(q/q_0)^2β,

where q is the momentum transfer and  q_0 is a normalized momentum. The momentum dependent cross-section can originate from various mechanisms such as dipole interaction between the DM and baryons, parity violating DM-baryon interaction, see Refs. <cit.>. Further, for calculation purposes, we assume, q_0=40 MeV <cit.>). 



Having been equipped with the form of the DM-baryon interaction, we will now calculate the probability and DM capture rate using the multicapture process.  This can be obtained by using Eq.(<ref>) and Eq.(<ref>). We have estimated the multiplescatter capture rate using constant, MDCS and VDCS in Appendices <ref>,<ref>, and <ref>, respectively. Below, we will briefly summarize the  results:


Capture rate in constant cross-section case: For constant cross-section case, the expression of probability is obtained after using Eq.(<ref>) as

    P^cont_N(τ_0)=2/τ^2_0 N![Γ(2+N)-Γ(2+N,τ_0)]

where, τ_0=3σ_χN F(q^2)/2σ_sat, and Γ(a,b) represent the incomplete Gamma function. Therefore using the above probability, C_N can be obtained as
 
    C^cont_N(τ_0)=   R^2/1-v^2_escn_χ/6u_0P^cont_N(τ_0)f(v_N,η)Θ(v_N-v_esc)

where, v_N=v_esc(1-β_+/2)^-N/2 is the DM velocity after N scattering <cit.>. Further, the total capture rate after N collision is given by C^cont_tot(τ_0)=∑^∞_N=1C^cont_N(τ_0). In this case, an analytic expression of the total capture rate is obtained using various analytic approximations, see Refs. <cit.>. Further, to calculate the total collected DM particles, we considered DM particles to be asymmetric (neglecting DM annihilation). In this assumption, the number of collected DM particles by neutron star throughout its lifetime is given by N_χ(t_NS)=C^cont_tot(τ_0) t_NS. Here, we will not apply the analytical approximations and calculate N_χ(t_NS) numerically.  


Capture rate in MDCS case: For MDCS, the optical depth is given by τ_md=τ_0(q/q_0)^2β. So, the probability after N collision is obtained as

    P^MDCS_N(τ_0,q)=2/τ^2_0 N!( q/q_0)^-4β[Γ(2+N)-Γ(2+N,τ_0 ( q/q_0)^2β)] .

The form of C_N in the MDCS case is given by 

    C^MDCS_N(τ_0,q)= R^2/1-v^2_escn_χ/6u_0P^MDCS_N(τ_0,q)f(v_N,η)Θ(v_N-v_esc)

  Further, the total capture rate after N collision is given by C^MDCS_tot(τ_0,q)=∑^∞_N=1C^MDCS_N(τ_0,q). Further, assuming the DM particles to be asymmetric, the total captured DM particles in the MDCS case is given by N_χ(t_NS)=C^MDCS_tot(τ_0,q) t_NS. In this case, we will calculate N_χ(t_NS) numerically.  

Capture rate in VDCS case: For VDCS, the optical depth is given by τ_vd=τ_0(w/u_0)^2α. So, the probability after N collision is obtained as

    P^VDCS_N(τ_0,w)=2/τ^2_0 N!( w/u_0)^-4α[Γ(2+N)-Γ(2+N,τ_0 ( w/u_0)^2α)] .

Using the above probability after simplification (see appendix <ref>), we obtain an analytic expression for the capture rate as
  
    C^VDCS_N(τ_0,w_0)= R^2/1-v^2_escn_χ/6u_0P^VDCS_N(τ_0,w_0)f(v_N,η)Θ(v_N-v_esc)

In this case, the total capture rate after N collision is obtained as C^VDCS_tot(τ_0,w_0)=∑^∞_N=1C^VDCS_N(τ_0,w_0). Further, in the assumption that the DM particles are asymmetric, the total captured DM particles in this case is given by N_χ(t_NS)=C^VDCS_tot(τ_0,w_0) t_NS. In this case, we will calculate N_χ(t_NS) numerically. 






Furthermore, the estimation of total accreted DM particles using the general expression of capture rate and probability is tedious. Therefore, we try to restrict our calculation up to a certain accuracy as follows:  We calculate the total sum up to N=N_max, where N_max≤τ <cit.>. Further, we use a simplified expression of probability in some extreme cases of optical depth. For a single scatter (τ≪ 1) and multiscatter (τ≫ 1), the probability is approximated as  <cit.>

P_N (τ)≈ 
2τ^N/N!(N+2)+O(τ^N+1),   if   τ≪1

2/τ^2(N+1) Θ(τ-N),   if   τ≫1 

Therefore, using the capture rate expression estimated above, we calculate the accreted DM particles for constant, MDCS, and VDCS cases numerically.





§ RESULTS AND DISCUSSIONS



In this Section, we will present our results. For calculation purposes, we assume the NS parameters as M_NS=1.44 M_⊙, R_NS=10.6 km, N_B=1.7× 10^57, and v_esc=1.8× 10^5 km/sec <cit.>, otherwise specified explicitly. 



 §.§ Effect of DM-Nucleon interactions on the multiscatter probability


In Fig. <ref>, we plot the probability of the DM multiscatter capture as a function of a number of scattering, N, (left panel) and cross-section (right panel). It can be seen that the probability decreases as the number of scattering N increases and becomes zero for some value of N, say N_max, i.e., p_N_max(τ)≈ 0. Further, fixing the DM parameters, m_χ=10^6 GeV,ρ_χ=0.3GeV/cm^3 and N_max takes values 1 and 2, and 3 corresponding to σ_χN=10^-47cm^2, σ_χN=10^-46cm^2, and σ_χN=10^-45cm^2, respectively. 
 Also, increments in the cross-section lead to an enhanced probability and, therefore, enlarge N_max. 

Further, the right panel of Fig <ref> suggests that
depending on the N, the probability becomes max at some σ_crit and vanishes on low DM-Nucleon scattering cross-section, σ_min such that σ_min<σ_crit . Nevertheless, increasing the number of scattering will shift the peak probability position and σ_min on a comparably higher cross-section.  Furthermore, we also find that increasing the DM masses decreases the probability.






In Fig. <ref>, we plot the probability of the DM capture using MDCS, σ∝ q^2β (Left panel) and VDCS, σ∝ v^2α_rel (Right panel) cases.  From the Left panel of Fig. <ref>, it can be seen that the probability decreases as the number of scattering N increases and becomes zero for some value N_max as discussed previously. However, β=2 shows an interesting feature that probability increases first and decreases sharply after a large scattering. For our parametrization and fixing the other DM parameters, m_χ,ρ_χ and σ_χN, N_max takes values 6,30 and 1000 corresponding to β= 0,1, and 2, respectively.   Further, in the VDCS case (Left panel of Fig. <ref>), DM capturing inside the NS requires more scattering, N_max. For example, keeping DM parameters fixed, m_χ=10^6 GeV, ρ_χ=0.3GeV/cm^3 and σ_χN=10^-45cm^2, DM capturing in α=2 case required ∼ 10^9 scattering. Further, for constant cross-section, the probability decreases promptly, leading to a smaller N_max value in comparison with the MDCS and VDCS cases. It is important to emphasize that the value of N_max depends on the DM parameters; therefore, changing it will modify the N_max value.






 §.§ Accreted DM particles via multi-capture process

 Fig. <ref> shows the accreted DM particles via multi-scatter capture for MDCS (Left panel), and VDCS (Right panel) cases. We found that for both cases, the number of the accreted DM particles, N_χ, enhanced with the DM scattering, but decreased with the DM mass. For the MDCS case, N_χ increases with the positive power of cross-section, β>0 and number of scattering, N. Here we see that for β=1, N_χ is same for N=1 and N=10. This is because for such a small DM-neutron scattering and also parameters used in the calculation (see Left panel of Fig. <ref>), DM capturing happens through single scattering, and therefore  N_χ remains constant for N>1. For the β=2 case, DM capturing becomes possible via multiscattering, and therefore, N_χ increases with N until it gets saturated at N_max=1000.  Furthermore, in the VDCS case (Right panel of Fig. <ref>) and for α=1, the number of captured DM particles is equal for N=1 and N=1000 (same reason as previously). For the strong VDCS case, α=2, DM particles get captured via multiscattering, and therefore, N_χ increases till its saturation at N_max=5000. From  Fig. <ref>, we also point out that for positive MDCS and VDCS cases, the number of captured DM particles is higher than the constant cross-section case.








 §.§ Thermalization time scale


To estimate a DM thermalization time for non-constant DM-nucleon interaction, we assume as follows: In VDCS and MDCS case, we replace σ via Eq.(<ref>), and Eq.(<ref>), and the relative velocity is assumed to be the thermal velocity inside the NS, i.e., v_rel≃ v_th=√(3T_NS/m_χ). With this simplification, we plot the thermalization time scale for our various parameterizations in Fig. <ref>. We find that for constant and σ∝ v^2_rel, σ∝ q^2, t_th is remains constant but increases linearly with mass, i.e., t_th∝ m_χ. This behavior is evident from Eq.(<ref>), and Eq.(<ref>). It can be seen that for a few cases, i.e. 
σ∝ v^2_rel,σ∝ v^4_rel and σ∝ q^2, DM quickly thermalized inside the NS, and therefore, these cases are interesting in the context of the BH formation. However, for the σ∝ q^4 case, the thermalization scale is quite large; therefore, DM does not thermalize in the typical lifetime of the NS. 






 §.§ Constraining massive DM from NS collapse





Now, we will constrain the DM-nucleon parameter space represented via σ_χN -m_χ using the survival of the neutron star situated in the DM-dominated region. We consider the cold and old neutron stars whose distance from the galaxy center, central temperature, and age are estimated accurately. For this purpose, we consider pulsar J2124-3858 situated at a distance 270 pc with surface temperature, T_e<4.6× 10^5K (central temperature, T_c=2.5× 10^7K [The central temperature can be estimated using the surface data from the analytic formula discussed in Ref. <cit.> .]), t_age∼ 7.81×10^9 years. As this pulsar is close to the sun, we may approximate u_s≃ u_⊙= 230 km/sec <cit.>, and ρ_χ=0.3GeV  cm^-3, where u_⊙ represent the sun's speed relative to galactic halo. 



Fig <ref> shows a constraint on the DM-nucleon scattering cross-section for a constant cross-section using the pulsar data J2124-3858. Here, orange and red colors are excluded from the BH formation via bosonic and fermionic DM particles. The olive and brown color regions represent the BH evaporation regions for bosonic and fermionic DM particles, and in these regions, the constraint is relaxed. Further, sky-blue regions are excluded via non-thermalization of the captured DM particles, i.e., t_th>t_NS. The black dashed line is the saturation limit of the DM-Nucleon scattering cross-section, and above this line, the constraints are not valid.   

 
From Fig <ref>, we see that for BDM, the number of the accreted DM particles is high enough to form the BH across all the DM mass ranges. The higher DM particles form low mass BH, so for sufficiently large DM masses, formed BH masses become smaller than the critical mass, i.e., M^ini_BH>M^crit_BH, so BH start evaporating. Due to evaporation, constraints are relaxed; hence, the collapse of the neutron star may not constrain quite massive DM particles, i.e., m_χ>6× 10^8 GeV for bosonic DM. 
Therefore, most of the DM parameter space constraints are relaxed from the evaporation constraint. The BH formation from BDM  constrains very limited parameter spaces 10^-45 cm^2≤σ_χN≤ 4× 10^-46 cm^2 of BDM masses 5× 10^7 GeV≤ m_χ≤ 5× 10^8 GeV. Further, we find that for FDM particles, BH formation happens on high DM masses, i.e. m_χ>10^9 GeV. This is expected because overcoming the degeneracy pressure of the FDM requires more mass. Furthermore, BH formation in FDM does not constrain the DM microphysics as all the parameter space is relaxed from the saturation limit and evaporation constraints.       


 Fig. <ref> shows the constraint on the DM-nucleon scattering cross-section for the MDCS, case from the pulsar observation. The Left and Right panel corresponds to σ∝ q^2 (N_max=10) and  σ∝ q^4 (N_max=1000) cases, respectively. In this case, the DM accretion is enhanced (in comparison to the constant cross-section); therefore, the pulsar data constrain the massive DM particles. For σ∝ q^2, the pulsar data constraint  on cross-section 10^-48 cm^2≤σ_χN≤ 10^-45 cm^2 for BDM masses 10^6 GeV≤ m_χ≤ 6× 10^8 GeV and 
 2× 10^-47 cm^2≤σ_χN≤ 10^-45 cm^2 for FDM masses 10^8 GeV≤ m_χ≤ 10^10 GeV. 
 Although the pulsar data also constrains more massive DM particles, those parameter spaces are relaxed by Hawking evaporation. Furthermore, for σ∝ q^4 case, DM accretion is even higher than  σ∝ q^2 case.
 However, in this model, the thermalization time is higher than the lifetime of the pulsar, which makes the BH formation highly unlikely, and therefore, this model does not constrain the DM parameters.  






Furthermore, we constrain the velocity dependent DM-Nucleon scattering cross-section from the pulsar observation in Fig. <ref>. The Left and Right panel corresponds to σ∝ v^2_rel (N_max=1000) and  σ∝ v^4_rel (N_max=5000) cases, respectively.  For σ∝ v^2_rel, the captured DM particles are higher than the constant case; therefore, it constrains the DM parameter space more strongly. This model constrain 10^-52 cm^2≤σ_χN≤ 10^-50 cm^2 for BDM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 2× 10^-47 cm^2 for FDM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. Furthermore, due to large positive power, the number of accreted DM particles is highest for model σ∝ v^4_rel (among all models discussed previously).  Therefore we expect that this model will put the tightest constraint. However, as seen in Fig. <ref>, the thermalization constraint strongly restricts the DM parameter space and limits the cross-section, 10^-50 cm^2≤σ_χN≤ 10^-47 cm^2 for the BDM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 3× 10^-46 cm^2 for the FDM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. 

From the above discussion, we find that the pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel. 






§ CONCLUSION



 A neutron star situated in a DM-dominated environment can efficiently capture the dark matter particles, and for large capturing, dark matter forms a black hole and destroys the host neutron star. In this work, we constrain the massive dark matter particles from the survival of the neutron star.

 To estimate the DM capture rate of massive dark matter particles m_χ>10^6 GeV, we adapted the multiscatter capture formalism to a non-constant DM-nucleon scattering cross-section. Adapting a model-independent approach, we assumed a power law parametrization of the velocity dependent and momentum dependent DM-Nucleon scattering cross-section, σ^vd_χN(v_rel)=σ_χN(v_rel/u_0)^2α and σ^md_χN(q)=σ_χN(q/q_0)^2β.
  Further, using the form factor and incorporating the general relativistic correction, we obtain analytic expressions for the capture rate in the neutron frame.
 Our estimation shows that the capture rate enhanced for the positive velocity and momentum dependent cross-section, and a number of the DM-Neucleon scattering, which leads to a black hole formation in a typical lifetime of the neutron star.    

 Further, applying the observation of pulsar data J2124-3858, we explored the constraint on the dark matter parameter space, i.e.,  σ_χN -m_χ for non-constant cross-section. We obtained that pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel. 
This model constrain 10^-52 cm^2≤σ_χN≤ 10^-50 cm^2 for bosonic DM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 2× 10^-47 cm^2 for fermionic DM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. Interestingly, we also found that although for model σ∝ v^4_rel, the DM accretion is higher, but due to a large thermalization time, it put weak constraints on the DM parameters. Furthermore, a large momentum dependent cross-section model, i.e., σ∝ q^4 provides no constraint on the DM parameters. 

To the best of our knowledge, this is the first constraint on the massive DM particles from the neutron star survival using the velocity and momentum dependent cross-section (in the multiscatter capture formalism). Nevertheless, we also point out that our calculation of capture rate has been done under certain assumptions, such as non-relativistic scattering of DM-nucleon, saturation cross-section is constant with DM mass (however, varies for massive DM particle), nucleon density  (also escape velocity) is constant. After inclusion, these effects will change the constraint, and we leave these interesting aspects for the future. 




§ ACKNOWLEDGEMENT


This work is supported by the National Natural Science Foundation of China (NNSFC) under grants No. 12275134, No. 12335005, and No. 12147228. AKM would like to thank Prof. Lei Wu and Prof. Chih-Ting Lu for the useful discussions. 




§ CALCULATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA CONSTANT CROSS-SECTION



After having equipped with the probability expression (see Section), we now estimate the multiscatter capture rate using Eq.(<ref>). The derivation of Eq.(<ref>) has been done is the assumption of multi-capture with v_esc (r)≃ v_esc (R_NS) and number density of the baryons are constant inside the neutron star, n_B(r)=n_B see Ref. <cit.>. To estimate the capture rate, we apply the Maxwell distribution function in the neutron star frame, Eq. (<ref>).


In a constant cross-section case, the probability, P^cont_N(τ_0) is independent of the DM velocity. So, applying Eq.(<ref>), Eq.(<ref>), and Eq.(<ref>) the analytic expression of C_N is obtained as
 
    C^cont_N(τ)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^cont_N(τ_0)f(v_N,η)Θ(v_N-v_esc)

 where 

    f(v_N,η)= 1/η[e^-3(v^2_N-v^2_esc)/2u^2_0+2√(v^2_N-v^2_esc) η/u_0 +η^2]
    ×[-√(6π)u_0( -1+2√(v^2_N-v^2_esc)η/u_0)√(v^2_N-v^2_esc)
       (1+e^√((v^2_N-v^2_esc))η/u_0-2e^-3/2u^2_0[(v^2_N-v^2_esc)+2√((v^2_N-v^2_esc)u_0η])u_0η 
    +π(3v^2_esc +u^2_0(1+3η^2))e^3/2u^2_0[v^2_N-v^2_esc+2√(v^2_N-v^2_esc)u_0η+u^2_0η^2]
       { ( 2 Erf√(3/2η) + Erf(√(3/2)(√(v^2_N-v^2_esc)-u_0η)/u_0) 
     + Erf(√(3/2)(√(v^2_N-v^2_esc)+u_0η)/u_0)}] ,

 where, v_N=v_esc(1-β_+/2)^-N/2 is the DM velocity after N scattering, and erf(z)=2/√(2)∫^z_0e^-y^2dy represent the error function.
In the above equation, ρ_χ is the DM energy density surrounding the NS, and N_B is the total number of baryons inside the NS. We checked that in the rest frame of DM halo, i.e., η→ 0, and point-like structure of nucleon (neglecting the form factor), our expression of C^const_χN matches with the Ref. <cit.>.  





§ DERIVATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA MDCS



In this case, σ^md_χN∝ q^2β, where q≃√(2)(m_χm_N/m_χ+m_N)v_esc.  So, P^MDCS_N(τ_0,q) is independent of the initial DM velocity. Using this, we obtain the analytic form of the capture rate after N scattering for the arbitrary momentum dependence cross section. The form of C_N in the MDCS case is given by

    C^MDCS_N(τ_0,q)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^MDCS_N(τ_0,q)f(v_N,η)Θ(v_N-v_esc)





§ DERIVATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA VDCS



In this case, σ^md_χN∝ w^2β, where w=√(u^2+v^2_esc). Therefore the probability P^VDCS_N(τ_0,w) depends on the DM velocity, see Eq. (<ref>). Using this probability expression, no reliable analytic (or numerical) solution was obtained. Therefore, to calculate the capture rate, we follow a conservative approach and approximate  w≃ w_0=√(u^2_0+v^2_esc) in the probability expression. After applying this approximation, we can easily integrate and obtain an analytic expression for the capture rate as

    C^VDCS_N(τ_0,w_0)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^VDCS_N(τ_0,w_0)f(v_N,η)Θ(v_N-v_esc)



utphys

 


 Paper Title: Catalog-based pseudo-$C_\ell$s.


Quantum Simulation via Stochastic Combination of Unitaries
    Prineha Narang
    August 5, 2024
==========================================================





§ INTRODUCTION

  In cosmology and astrophysics, a plethora of observable quantities can be described by projected fields on the sky. These fields may be observed and mapped in a continuous manner, as is the case of Cosmic Microwave Background (CMB) observations, or may be observed only at the discrete positions of a finite number of sources. Depending on the physical nature of the quantity at hand, these quantities will transform differently under rotations of our angular coordinates, leading to spin-0 fields (i.e., scalars, such as the CMB temperature anisotropies, the lensing convergence, or the galaxy overdensity), spin-1 fields (i.e., vectors such as transverse peculiar motions or lensing displacements), spin-2 fields (i.e., tensors such as the CMB polarization anisotropies or weak lensing shear), and higher-spin fields (e.g. the spin-4 polarization of gravitational waves). The angular power spectrum, corresponding to the covariance between angular fluctuations in two different fields at a given multipole scale ℓ, is one of the most used summary statistics in cosmology and astrophysics and, in particular, encodes all available information in the case of Gaussian fields. For this reason, being able to accurately estimate the angular power spectrum is a key requirement for many cosmological probes. In the case of galaxy weak lensing, for instance, a precise and accurate reconstruction of the cosmic shear power spectrum on sub-arcminute scales is of vital importance to obtain meaningful constraints on the gravity-dominated, nonlinear regime of cosmic structure formation, scrutinizing the Universe’s matter content and nature of Dark Energy <cit.>. This is a key goal for current and next-generation galaxy imaging surveys, such as the Kilo Degree Survey (KiDS)[<http://kids.strw.leidenuniv.nl>], the Dark Energy Survey (DES)[<https://www.darkenergysurvey.org>], the Hyper Suprime-Cam[<https://www.naoj.org/Projects/HSC/>], Euclid[<https://www.euclid-ec.org>], and the Rubin Observatory Legacy Survey of Space and Time (LSST)[<https://www.lsst.org>]. Valuable complementary insight may be gained from the angular power spectrum of other discretely sampled tracers of the matter density, such as the overdensity of galaxies <cit.>, fluctuations in their sizes <cit.>, fluctuations in the luminosity distance of standard sirens <cit.>, transverse peculiar motions <cit.>, or the dispersion measure of extragalactic Fast Radio Bursts (FRBs) <cit.>.

  There is a large literature of unbiased power spectrum estimators that are optimal, have well-understood statistical properties, and robustly meet common observational challenges such as partial sky coverage, linear sky contamination, and inhomogeneous noise <cit.>. For small angular scales, optimal quadratic estimators are usually computationally prohibitive, and the most common alternative is their simpler incarnation, the so-called “pseudo-C_ℓ” estimator <cit.>. The pseudo-C_ℓ estimator effectively assumes a diagonal covariance matrix for the data, and is therefore close to optimal for spatially uncorrelated or white-noise-dominated data. As a consequence of this assumption, the mode-coupling induced by the spatial variation of the resulting inverse-variance weights (i.e., the “mask”) can be estimated using fast, analytical methods. Variations of the method allow it to account for the impact of linear contaminant deprojection, and for the effects of E- and B-mode purification in non-scalar fields <cit.>.

  In its standard implementation, the pseudo-C_ℓ method takes, as input, pixelized maps of the fields in question. When the original observations are in the form of discrete measurements at arbitrary sky positions, as is the case for any catalog-based observable, this results in two fundamental problems. First, generating a pixelized map from the discrete sources induces a bias at scales comparable to, and smaller than, the pixel size. The form of this bias depends on a combination of pixel smoothing and aliasing of small-scale power, and is non-trivial to characterize in general <cit.>. Second, when the source number density is low, the mask is dominated by Poisson noise, also known as shot noise, which, as we will see, leads to statistical couplings between distant angular multipoles. If not treated carefully, this can lead to strong numerical instabilities and catastrophic biases in the estimator. The issue of small-scale power in the mask has been traditionally handled via apodization and smoothing of the mask edges <cit.>, but this approach is highly impractical in the context of catalog-based observations for sparse samples.

  This work addresses the issue of how to define an unbiased power spectrum estimator for fields of arbitrary spin discretely sampled on the sphere. Building on the recent work on this topic by <cit.>, we forgo the pixelization step completely and directly estimate spherical harmonic transforms of the field and its measurement weights at the discrete source positions. This allows us to build a pixel-independent version of the pseudo-C_ℓ estimator, addressing the first problem described above. We address the second problem (shot noise instabilities) by introducing a simple modification to the estimator that accounts for uncorrelated noise, both in the mask and the field, and simultaneously eliminates both the associated noise bias and the numerical instabilities of the mask-induced mode coupling. More specifically, the contribution to mode coupling from the mask shot noise is estimated analytically and subtracted from the measured pseudo-C_ℓ, together with a naïve, data-based estimate of the noise bias. As we will show, the resulting estimator is then immune to any bias from uncorrelated noise, and benefits from a numerically stable mode-coupling matrix, independently of the number density of sources in the catalog. We implement this methodology in the publicly available package[<https://github.com/LSSTDESC/NaMaster/>]. As we were preparing this manuscript, we learned of related work presented in Tessore et al. 2024 <cit.>. The results there agree with those presented here.

  This paper is structured as follows. In Section <ref>, we lay out the formalism that defines our unbiased estimator. In Section <ref>, we validate it on simulated cosmic shear catalogs with a realistic source distribution, based on real data from the Kilo Degree Survey, also demonstrating its validity on significantly lower number densities. In Section <ref>, we present several example applications of the methodology, including state-of-the-art cosmic shear data from the Dark Energy Survey, transverse proper motion surveys, and FRB-based dispersion measure maps. We conclude in Section <ref>.




§ METHODS

  This section presents the formalism to estimate the angular power spectrum from sparse observations of a given field at a discrete set of points on the sphere using the pseudo-C_ℓ formalism. Section <ref> reviews the main concepts behind the standard pseudo-C_ℓ method, and the application to discrete tracers is described in Section <ref>. We first present the formalism in the simpler case of scalar fields, and then, in Section <ref> present the generalization to spin-s quantities.

  

 §.§ Pseudo-C_ℓs

    Let f() be a scalar field on the sphere, and let w() be a sky mask (often also called weights maps, or sky window function). The mask may be as simple as a binary map (0 for unobserved regions, 1 for observed regions) but, more usefully, it should represent the precision with which f has been measured at different points (e.g. an inverse-variance map will be optimal for noise-dominated observations). Because of this, w() is often referred to as a “window function” or “weights map”, but we will stick to the simpler “mask” in what follows.

    The field is related to its harmonic coefficients f_ℓ m through a spherical harmonic transform (and its inverse):
    
    f()=∑_ℓ m Y_ℓ m() f_ℓ m↔f_ℓ m=∫ d f() Y^*_ℓ m(),

    where Y_ℓ m() are the spherical harmonic functions. The spherical harmonic coefficients of the mask, or any other sky map, are defined in the same way.

    Let f^w()≡ f()w() be the masked field. The “pseudo-C_ℓ” of f is simply the naïve estimate of the masked field's power spectrum ignoring the fact that it's masked:
    
    C̃^f_ℓ≡1/2ℓ+1∑_m=-ℓ^ℓ f^w_ℓ mf^w*_ℓ m.

    We commonly assume the field to receive signal and noise contributions: f=s+n, where the signal s is a statistically isotropic field, such that
    
    ⟨ s_ℓ ms^*_ℓ' m'⟩=δ_ℓℓ'δ_mm'S_ℓ,

    here S_ℓ is the signal power spectrum, and is the quantity we are ultimately after.

    The key result of the pseudo-C_ℓ estimator is the relation between the ensemble average of C̃^f_ℓ and S_ℓ:
    
    ⟨C̃^f_ℓ⟩=∑_ℓ'M^w_ℓℓ'S_ℓ'+Ñ_ℓ,

    where M^w_ℓℓ' is the mode-coupling matrix:
    
    M^w_ℓℓ'=2ℓ'+1/4π∑_ℓ”C̃^w_ℓ” (2ℓ”+1)ℓℓ'ℓ”000^2,

    here C̃^w_ℓ≡∑_m |w_ℓ m|^2/(2ℓ+1) is the pseudo-C_ℓ of the mask. Ñ_ℓ is the “noise bias”, given by the ensemble average of the pseudo-C_ℓ of the masked noise:
    
    Ñ_ℓ≡⟨1/2ℓ+1∑_mn^w_ℓ mn^w*_ℓ m⟩.

    Depending on the type of observations, and our understanding of the noise properties, Ñ_ℓ may be estimated analytically, via Monte-Carlo simulations, or avoided by using cross-correlations between observations with the same signal but different noise realizations (e.g. CMB maps constructed from disjoint sets of timestreams). As we will see, in the case of discretely-sampled fields, the noise bias can be entirely avoided by construction under the assumption that the noise contributions from different points are uncorrelated. The expressions above may be generalized to the case of cross-correlations between different fields, by simply replacing C̃^w_ℓ with the cross-spectrum between the masks of both fields.

  

 §.§ The case of discretely-sampled fields

    Consider now observations of the field f only at a discrete set of N points on the sphere. Most commonly, these will be the sky positions of sources in a catalog. Let w_i be the weight associated to the i-th point, with sky position _i, and f_i≡ f(_i). To ease notation, in what follows, the subscript _i applied to any function defined on the sphere will be shorthand for the value of that function at _i. The mask and masked field can, in this case, be written as a sum over delta functions:
    
    w()=∑_i w_i δ^D(,_i),f^w()≡ w() f()=∑_i w_if_i δ^D(,_i),

    where δ^D(_1,_2) is the Dirac delta function on the sphere[That is, δ^D(_1,_2)≡δ(φ_1-φ_2)δ(cosθ_1-cosθ_2), where (θ_n,φ_n) are the spherical coordinates of the unit vector _n.]. The harmonic coefficients of w and f^w are then sums over the harmonic functions sampled at the positions of the discrete sources
    
    w_ℓ m=∑_i w_i Y_ℓ m,i^*,f^w_ℓ m=∑_i w_if_i Y_ℓ m,i^*.

    An efficient evaluation of such discrete harmonic transform over a non-regular set of points is, in general, computationally more expensive than standard approaches making use of regular pixelization schemes <cit.>. Nevertheless, approaches for dealing with this problem have been recently developed <cit.>, and we will use the implementation of such “general” spherical harmonic transforms built in the ducc library[<https://mtr.pages.mpcdf.de/ducc>], as described in <cit.>. This paper will not discuss the details of the numerical implementation of these methods and instead will deal with the implementational details of the pseudo-C_ℓ algorithm itself, once these discrete transforms are available. We refer readers to <cit.> further details.

    The pseudo-C_ℓ of both mask and masked field is given by:
    
    C̃^w_ℓ=∑_ijw_iw_j 1/2ℓ+1∑_mY_ℓ m,i^*Y_ℓ m,j,
       C̃^f_ℓ=∑_ijw_iw_jf_if_j 1/2ℓ+1∑_mY_ℓ m,i^*Y_ℓ m,j.

    These equations may be further simplified making use of the addition theorem for spherical harmonics (Eq. <ref>). However, it will be more useful to leave them as is for now.

    

  §.§.§ Noise bias from uncorrelated measurements

      Let us now consider our measurements of the field f=s+n to contain some common signal s and noise component n, which we assume to be uncorrelated among different sources. Under this assumption, the noise contribution to C̃^f_ℓ reads
      
    ⟨ n_i n_j⟩ = δ_ijσ_N,i^2,

      where σ_N,i^2 is the noise variance in the i-th point. In this case, the noise bias only receives contributions from the variance at the same source (i.e., the zero-lag correlation function):
      
    Ñ_ℓ=∑_ijw_iw_j⟨ n_in_j⟩1/2ℓ+1∑_mY^*_ℓ m,iY_ℓ m,j=1/4π∑_i w_i^2σ_N,i^2,

      where, in the last equality, we have made use of the addition theorem for spherical harmonics (Eq. <ref>) for the special case i=j. The noise bias is therefore “white” (i.e., scale-independent), and non-zero at all ℓs.

      Since the standard pseudo-C_ℓ estimator requires us to estimate Ñ_ℓ and subtract it from C̃^f_ℓ, in order to obtain an unbiased estimator of the signal power spectrum S_ℓ, it is important for us to consider practical ways to estimate Ñ_ℓ. Assuming our measurements of f to be dominated by noise at any single point, one potentially simple way to do this would be to estimate Ñ_ℓ directly from the variance of the data itself. This methodology has been commonly applied, in the past, when estimating cosmic shear power spectra, where measurements are indeed dominated by noise <cit.>. 
      In this situation, and looking at Eq. <ref>, we can thus propose the following estimator for Ñ_ℓ:
      
    Ñ^f≡1/4π∑_i w_i^2f_i^2.

      Expanding the field measurements into signal and noise components, we can thus see that Ñ^f is a biased estimate of Ñ_ℓ due to the presence of signal in our measurements:
      
    ⟨Ñ^f⟩=Ñ_ℓ+1/4π∑_i w_i^2⟨ s_i^2⟩.

      For a statistically isotropic field, such as s, the field variance is related to its power spectrum via
      
    σ_S^2≡⟨ s_i^2⟩=∑_ℓ2ℓ+1/4πS_ℓ,

      and thus we can write the bias of Ñ^f as:
      
    ⟨Ñ^f⟩=Ñ_ℓ+Ñ^w σ_S^2,

      where we have defined the combination
      
    Ñ^w≡1/4π∑_iw_i^2.


    

  §.§.§ Mask noise

      Drawing intuition from our discussion of uncorrelated noise in the field, we can see that the pseudo-C_ℓ of the mask C̃^w_ℓ will also receive a white-noise-like contribution from the terms i=j in Eq. <ref>. By inspection, we can see that this is none other than the quantity Ñ^w we just calculated. We will thus refer to Ñ^w as the “mask shot noise”. In terms of it, we can thus write the mask power spectrum as
      
    C̃^w_ℓ=S̃^w_ℓ+Ñ^w,

      where the “signal” component is, mathematically:
      
    S̃^w_ℓ=1/4π∑_i≠ jw_iw_j P_ℓ(μ_ij),

      with μ_ij≡_i·_j, and P_ℓ(μ) the Legendre polynomials. In practice, the fastest way to estimate S̃^w_ℓ is not to compute the double sum above, but to compute C̃^w_ℓ from the harmonic transform of w() (Eq. <ref>), and then subtract Ñ^w from it, calculated using Eq. <ref>.

      The presence of Ñ^w is a feature exclusive of the discrete origin of our observations. It sets a floor below which C̃^w_ℓ cannot deviate significantly and, for most common point distributions of interest in astrophysics, C̃^w_ℓ will, in fact, tend to Ñ^w at high ℓ. This is contrary to the usual scenario for power spectrum estimation over incomplete skies. In more common cases, in which the mask is a smooth function, and has support over a significant fraction of the sky (compared to the angular scales of interest), C̃^w_ℓ decays with ℓ. This causes the sum over ℓ” in Eq. <ref> to converge quickly, and the resulting mode-coupling matrix to be highly concentrated along the diagonal. It is therefore important to understand what the contribution from Ñ^w would be to M^w_ℓℓ' and, in general to the ensemble average of C̃^f_ℓ. Calling these contributions M^w,N_ℓℓ', and (ΔC̃^f_ℓ)_N, we can calculate them by replacing C̃^w_ℓ with Ñ^w in Eq. <ref>, and then substituting the result into Eq. <ref>. The resulting contribution to the mode-coupling matrix is:
      
    M^w,N_ℓℓ'=2ℓ'+1/4πÑ^w,

      where we have made use of one of the orthogonality relations of the Wigner-3j coefficients (Eq. <ref>). We can thus see that the mask shot noise leads to a contribution that couples all multipole ℓs with all other multipoles ℓ' (in principle up to ℓ'=∞). Although this does not imply that the mode-coupling matrix is necessarily singular, it does mean that an accurate implementation of the pseudo-C_ℓ estimator might involve (depending on the size of Ñ^w), estimating M^w_ℓℓ' up to very large values of both ℓ and ℓ' in order to account for mode coupling. This is problematic, since estimating M^w_ℓℓ' is the most expensive step of the pseudo-C_ℓ estimator.

      An alternative to avoid this, is to estimate the contribution to ⟨C̃^f_ℓ⟩ from the mask shot noise term, and to bring it to the left hand side of Eq. <ref>. From our result above, this contribution is, simply:
      
    (ΔC̃^f_ℓ)_N=∑_ℓ'M^w,N_ℓℓ'S_ℓ'=Ñ^wσ_S^2.

      Thus, we find that the contribution to the masked field's pseudo-C_ℓ coming from the mask shot noise is precisely equivalent to the unwanted contribution to our naive estimate of the noise bias in Eq. <ref>. This result allows us to construct a fully unbiased estimator for the power spectrum of a discretely-sampled field with uncorrelated noise, which we describe below.

    

  §.§.§ An unbiased estimator

      Gathering our results from the last two sections, the algorithm presented in this paper is as follows:
      
        
  * Estimate the spherical harmonic coefficients of the masked field, and of the mask (Eq. <ref>).
        
  * From these, compute the pseudo-C_ℓ of the mask (C̃^w_ℓ) and of the masked field (C̃^f_ℓ).
        
  * Estimate the “noise” components of both power spectra, Ñ^w, Ñ^f directly from the measurements, using Eqs. <ref> and <ref>, respectively.
        
  * Subtract both noise contributions to form the “signal” pseudo-C_ℓs of the mask and the masked field[As we discussed, S̃^f_ℓ is actually a biased estimator of the signal component of C̃^f_ℓ, since we are subtracting a contribution sourced by the signal variance. Nevertheless, having corrected C̃^w_ℓ in the same way, the resulting estimator of S_ℓ will be unbiased.] S̃^w_ℓ≡C̃^w_ℓ-Ñ^w, S̃^f_ℓ≡C̃^f_ℓ-Ñ^f.
        
  * Apply the standard pseudo-C_ℓ estimator to these “signal-only” spectra. This means, compute the mode-coupling matrix using S̃^w_ℓ instead of C̃^w_ℓ, and apply its inverse[The usual caveats about power spectrum binning and invertible mode-coupling matrices should be borne in mind, however. See <cit.>.] to S̃^f_ℓ to obtain an unbiased estimate of the signal power spectrum.
      
      To reiterate, subtracting the noise-like contribution from the mask pseudo-C_ℓ has two desirable consequences: first, it automatically allows us to account for the bias induced by correcting for a noise power spectrum computed from our own data. Secondly, the resulting mode-coupling matrix is now computed from the noise-debiased mask spectrum which, by construction, will decay with ℓ (at least for the usual point distributions that make up most cosmological observations). Hence, the sum over ℓ” in <ref> will have better convergence properties, and the mode-coupling matrix will be less numerically unstable.

    

  §.§.§ Noise hardening

      The result presented in Section <ref> may seem slightly disconcerting: by correcting for the noise-like component of the mask pseudo-C_ℓ, which we can estimate entirely from the data, we are able to exactly correct for the bias induced by the contribution to our estimate of the noise power spectrum coming from the signal variance. However, this contribution in principle depends on the true signal power spectrum, which we do not know a priori. For this to be possible, it must be that our correction of the mask power spectrum automatically makes the estimator immune to any noise-like (i.e., constant additive) contribution to the power spectrum. In other words, the corrected mode-coupling matrix M^w,S_ℓℓ'≡ M^w_ℓℓ'-M^w,N_ℓℓ' must satisfy that ∑_ℓ' M^w,S_ℓℓ'S_ℓ'=0 if S_ℓ= constant for all ℓ. In linear algebra terms, M^w,S_ℓℓ' has a zero eigenvalue with eigenvector v_ℓ=1 ∀ℓ. We can show this explicitly:
      
    ∑_ℓ'M^w,S_ℓℓ'   =1/4π∑_ℓ'ℓ”(2ℓ'+1)(2ℓ”+1)ℓℓ'ℓ”000^2[C̃^w_ℓ”-Ñ^w],
       =∑_ℓ”2ℓ”+1/4π[C̃^w_ℓ”-Ñ^w],
       =∑_i≠ jw_iw_j/4π∑_ℓ” m”Y^*_ℓ” m”,iY_ℓ” m”,j,
       =∑_i≠ jw_iw_j/4πδ^D(_i,_j)=0,

      where, in the second line, we have used the orthogonality relation Eq. <ref>, in the third line we have expanded C̃^w_ℓ and Ñ^w using Eqs. <ref> and <ref>, and in the last line we have used the completeness of the spherical harmonic functions (Eq. <ref>).

      In practice, what this means is that this power spectrum estimator is not able to recover any completely flat component of the true power spectrum or, in other words, that it has infinite variance for these flat modes. This is not a strong shortcoming, since the amplitude of such modes can be reconstructed from the zero-lag correlation function (i.e., the variance of the field across all sources), assuming a model for the noise. This also means that, to avoid numerical instabilities, it will be useful to employ singular value decomposition methods before inverting the noise-corrected mode-coupling matrix. The crucial advantage is the fact that we do not need to know which fraction of the total field variance is attributed to uncorrelated noise and which fraction to the signal autocorrelation: changes in either of them will only affect, and are exactly captured by, Ñ^f. This fundamentally establishes the unbiasedness of our estimator.

    

  §.§.§ Cross-correlations

      Cross-correlations between two different fields measured on two disjoint sets of points, or cross-correlations between standard maps and discretely-sampled fields, can be estimated by making use of the standard pseudo-C_ℓ algorithm, taking advantage of discrete harmonic transform methods in the case of the sampled fields. No additional care must be taken in this case about subtracting noise components in the mask or field power spectra, since these are zero by construction. Care must be taken, however, if the two different fields to be correlated have been sampled at the same discrete positions. A good example is the cross-correlation between the cosmic shear signal of a set of sources and the PSF ellipticity at their position, as a diagnostic for potential systematics. In the most general case, different weights may be applied to the sources when mapping either field. In this case, the mask and masked-field pseudo-C_ℓs must be corrected for noise bias.

      Let e and f be the two fields being correlated, and let v_i and w_i, respectively, be the weights applied to each source when mapping either field. The noise-like contributions to the mask pseudo-C_ℓ and to the masked-field pseudo-C_ℓ are simple generalizations of those presented in the case of auto-correlations:
      
    Ñ^vw≡1/4π∑_i v_i w_i,Ñ^ef≡1/4π∑_i v_i w_i e_i f_i.


  

 §.§ Spin-s fields

    Let f^a(), with a∈{1,2} be a spin-s_f field, and let f^α_ℓ m be its spherical harmonic coefficients (with α∈{E,B}, corresponding to E- and B-modes). The two are related via a spin-s_f spherical harmonic transform (and its inverse):
    
    f^a()=∑_ℓ m _s_fY^aα_ℓ m()f^α_ℓ m,↔ f_ℓ m^α=∫ d f^a() _s_fY^aα*_ℓ m(),

    where the generalized spin-s spherical harmonic functions _sY^aα_ℓ m() are defined in Appendix <ref>, and we implicitly sum over all repeated indices (e.g. a and α above).

    The pseudo-C_ℓ of the masked field is defined in the same way as in the scalar case, except now we have to keep track of E/B-mode indices:
    
    C̃^f,αβ_ℓ≡1/2ℓ+1∑_m f_ℓ m^α f_ℓ m^β*.

    The generalization of the scalar pseudo-C_ℓ estimator from Eq. <ref> to spin-s fields is:
    
    ⟨C̃^f,αβ_ℓ⟩=∑_ℓ'( M^w_ℓℓ')^αβ_α'β' S^α'β'_ℓ'+Ñ^αβ_ℓ,

    where the mode-coupling matrix is <cit.>:
    
    M_ℓℓ'^w=2ℓ'+1/4π∑_ℓ”C̃^w_ℓ” (2ℓ”+1)ℓℓ'ℓ”s_f-s_f0^2  D_ℓ+ℓ'+ℓ”,

    and the matrix D_n is:
    
    ( D_n)^EE_EE=( D_n)^EB_EB=( D_n)^BE_BE=( D_n)^BB_BB=1+(-1)^n/2,
       ( D_n)^EE_BB=-( D_n)^EB_BE=-( D_n)^BE_EB=( D_n)^BB_EE=1-(-1)^n/2.


    Following the same logic used in Section <ref>, in the case of discretely-sampled fields, the spherical harmonic transform of a sampled and masked spin-s_f field is
    
    f^w,α_ℓ m=∑_iw_if_i^a _sY^aα*_ℓ m,i,

    and its pseudo-C_ℓ is
    
    C̃^f,αβ_ℓ=∑_ijw_iw_jf_i^af_j^b1/2ℓ+1∑_m _sY^aα*_ℓ m,i _sY^bβ_ℓ m, j.


    Now, let us separate the field into signal and noise components f^a=s^a+n^a, where the signal is statistically isotropic (⟨ s^α_ℓ ms^β*_ℓ'm'⟩=δ_ℓℓ'δ_mm'S^αβ_ℓ), and the noise is uncorrelated between different sampling points, which in this case means ⟨ n_i^an_j^b⟩=δ^abδ_ijσ^2_N,i. Inserting this in the previous equation, and using Eq. <ref>, we obtain the noise bias:
    
    Ñ^αβ_ℓ=∑_iw_i^2σ_N,i^21/2ℓ+1∑_m _sY^aα*_ℓ m,i _sY^aβ_ℓ m,i=δ^αβ/4π∑_iw_i^2σ_N,i^2.

    Based on this result, a potentially useful estimate of the noise bias, using only our measurements of f, would be:
    
    Ñ^f,αβ≡δ^αβ/4π∑_i w_i^2 1/2f_i^af_i^bδ_ab.

    As in the spin-0 case, this estimator will be biased due to the contribution from the signal component of f. To calculate this bias, we take the expectation value of the previous equation, and find:
    
    ⟨Ñ^f,αβ⟩   =Ñ_ℓ^αβ+δ^αβ/4π∑_iw_i^2δ_ab/2⟨ s^a_is^b_i⟩
       =Ñ_ℓ^αβ+δ^αβ/4π∑_iw_i^21/2∑_ℓS^α'β'_ℓ∑_m _sY^aα'*_ℓ m, i _sY^aβ'_ℓ m, i
       =Ñ_ℓ^αβ+δ^αβÑ^w∑_ℓ2ℓ+1/4πS_ℓ^EE+S_ℓ^BB/2
       ≡Ñ_ℓ^αβ+δ^αβÑ^wσ_S^2,

    in analogy with Eq. <ref>, after generalising the expression for the field variance σ_S^2 in Eq. <ref>. In the second line above, we have simply expanded s_i^a in its spherical harmonic coefficients, and used their statistical isotropy. We have then made use of Eq. <ref> in the third line.

    The final step in our derivation shows that the contribution from Ñ^w to Eq. <ref> cancels out the term ∝σ_S^2 in the last equation, thus yielding and unbiased power spectrum estimator. To begin with, let us replace C̃^w_ℓ with N̅^w in Eq. <ref> to calculate the mask noise contribution to the mode-coupling matrix, M^w,N_ℓℓ':
    
    M^w,N_ℓℓ'=Ñ^w 2ℓ'+1/4π∑_ℓ”(2ℓ”+1)ℓℓ'ℓ”s_f-s_f0^2  D_ℓ+ℓ'+ℓ”.

    To simplify this further, note that all elements of D_ℓ+ℓ'+ℓ” either are zero, or take the form [1±(-1)^ℓ+ℓ'+ℓ”]/2 (see Eq. <ref>). We can use the term ∝(-1)^ℓ+ℓ'+ℓ” to swap the sign of the lower spin indices of one of the two Wigner 3j symbols, using Eq. <ref>. The resulting sum over ℓ” for this term is then zero by virtue of Eq. <ref>, and we are left only with the contribution from the first term in the D matrix elements, which is always 1/2 (or zero). Again, using Eq. <ref> on this term, we obtain the simplified result:
    
    M^w,N_ℓℓ'=Ñ^w2ℓ'+1/4π  d/2,

    where the matrix d has elements:
    
    ( d)^EE_EE=( d)^EB_EB=( d)^BE_BE=( d)^BB_BB=( d)^EE_BB=( d)^BB_EE=1,  ( d)^EB_BE=( d)^BE_EB=-1,

    and zero otherwise. Substituting this in Eq. <ref>, and after a bit of algebra[This can be done by writing out d as a 4×4 matrix, and multiplying it by the vector (S_ℓ^EE,S_ℓ^EB,S_ℓ^BE,S_ℓ^BB), with S_ℓ^BE=S_ℓ^EB.], we find that the resulting contribution to ⟨C̃^f,αβ_ℓ⟩ is
    
    (ΔC̃^f,αβ_ℓ)_N=δ^αβÑ^wσ_S^2.


    Thus, the recipe for an unbiased estimator described in Section <ref> can be directly applied to spin-s fields, simply replacing the scalar noise bias Ñ^f by its spin generalization Ñ^f,αβ in Eq. <ref>.

    

  §.§.§ Different spins

      One important generalization of the formalism we just described is the calculation of cross-power spectra between two fields of different spin sampled at the same source positions. A good example of this would be measurements of both the shape (spin-2) and size (spin-0) of galaxies, as proxies for weak lensing shear and convergence. The formalism we just described, in which the mode-coupling matrix is estimated from the noise-debiased mask pseudo-C_ℓ, can be applied directly to these situations without any modifications. In fact, the orthogonality relation Eq. <ref> guarantees that, for fields of different spin, the contribution from Ñ^w to the mode-coupling matrix vanishes exactly. Likewise, by virtue of the addition theorem for spin-weighted spherical harmonics (Eq. <ref>), the noise bias to the pseudo-C_ℓ of the masked fields also vanishes in the limit of uncorrelated noise. Thus, in these cases, one may even forego the calculation of Ñ^w and Ñ^f altogether.

  

 §.§ Clustering

    A conceptually different type of discretely sampled field is the case of galaxy clustering, in which the overdensity of points in the discrete source catalog is itself the field. This case has been discussed in detail in <cit.>, but we repeat part of the discussion here for completeness. In this case, the number density of objects is
    
    n_d()=∑_i=1^N w^d_i δ^D(,_i),

    where w^d_i is the weight of the i-th galaxy (which may be used to account for observational conditions, instrumental effects, such as fiber collisions, and other systematics <cit.>).

    Critical in the study of galaxy clustering is our ability to account for spatial variations in the expected mean number density of sources n̅(), caused by depth variations across the footprint, contamination by galactic or instrumental systematics, and the complex geometry of the observed footprint itself. This has been commonly done through the use of “random catalogs”: synthetic catalogs containing discrete points that trace the underlying expected density of sources, but that are otherwise not clustered. The mean number density of sources in this case is then
    
    n̅()=α∑_i=1^N_r w_i^r δ^D(,_i),

    where w_i^r is the weight of the i-th random point, N_r is the total number of random points, and the normalization factor α accounts for the potentially different number of objects in the data and random catalogs:
    
    α≡∑_i=1^Nw_i^d/∑_i=1^N_rw_i^r.

    Ideally, the field whose power spectrum we wish to estimate is the galaxy overdensity f()≡δ_g()≡ n_d()/n̅()-1, although, obviously, the discretized nature of n̅() complicates the construction of such a map. We can proceed, however, under the following consideration: a close to the optimal choice of mask for the standard pseudo-C_ℓ method is one that is proportional to the inverse noise variance of the field under study. Assuming n_d to be a Poisson realization of n̅(1+δ), a reasonable choice of mask is therefore w()=n̅(). With this choice, the masked field and the associated mask are simply
    
    f^w()=n_d()-n̅()=∑_i=1^Nw_i^d δ^D(,_i)-α∑_i=1^N_rw_i^r δ^D(,_i),
       w()=α∑_i=1^N_r w_i^rδ^D(,_i).


    The methodology described in Section <ref> can therefore be extended directly to the case of galaxy clustering with the following changes:
    
      
  * The SHTs of the masked field and the mask are
      
    f^w_ℓ m=∑_i=1^Nw_i^dY^*_ℓ m,i-w_ℓ m, w_ℓ m=α∑_i=1^N_rw_i^rY^*_ℓ m,i,

      
  * The shot noise components to field and mask are:
      
    Ñ^f=1/4π∑_i=1^N(w_i^d)^2+Ñ^w,
            Ñ^w=1/4π∑_i=1^N_r(α w_i^r)^2.

    

    Although the use of randoms to characterize survey homogeneity has been common, particularly in the analysis of spectroscopic galaxy surveys, it is also possible to map spatial variations in survey homogeneity at every point in the celestial sphere. It is therefore important to also consider the case in which a “completeness map” W is provided, such that W()∝n̅(). In this case, the method must be adapted as follows:
    
      
  * The sky mask is generated by normalizing the completeness map, to make it correspond to the expected number density of galaxies:
      
    w()=α W(),α=∑_i=1^Nw_i^d/∫ d W().

      
  * The SHT of the masked field is still given by the first equality in Eq. <ref>, with the mask SHT w_ℓ m estimated from the continuous w() as usual.
      
  * Since the mask is now a map, rather than a collection of sources, its shot noise is Ñ^w=0. The shot-noise component of the field's pseudo-C_ℓ, in turn, is Ñ^f=∑_i=1^N (w_i^d)^2/(4π).
    



§ VALIDATION
 
  Having presented the mathematical formalism of an unbiased power spectrum estimator for discretely sampled fields, we now proceed to validate it against simulations. We focus on what is likely the most relevant probe for which this formalism is useful: cosmic shear catalogs. The results presented here will therefore correspond to spin-2 fields, but other spin cases do not change the problem conceptually[Except perhaps in the case s=0, where the separation between  E- and B-modes is meaningless.]. We showcase examples of the spin-1 and spin-0 cases in Sections <ref> and <ref>, respectively.

  Cosmic shear measures the apparent distortion of a galaxy's shape due to the gravitational lensing effect caused by the foreground matter distribution integrated along the line of sight. This effect, while weak and indistinguishable from the galaxy's intrinsic ellipticity when considering single objects, represents a powerful probe of the cosmic large-scale structure when studied in a statistical way on large catalogs of galaxies. In the weak lensing regime, the shear γ_a,i that the i-th galaxy's image has undergone, provides a linear contribution to its measured ellipticity e_a,i (estimated from the traceless component of the projected inertia tensor, see e.g. <cit.>):
  
    e_a,i = γ_a,i + n_a,i,

  where a=1,2 labels the two components of ellipticity and shear, and the noise component n_a,i includes both the intrinsic (i.e., unlensed) galaxy ellipticity and any additional measurement noise. Crucially, it is common to work under the assumption that all contributions to n_a,i are uncorrelated between different sources.

  Section <ref> below describes the setup used to validate the catalog-based methodology described in the previous section, comparing it against the results of the standard map-based approach. The results of this validation, as a function of the number density of catalog sources, are then presented in Section <ref>.

  

 §.§ Validation setup

    To validate our estimator, we generate 500 synthetic cosmic shear catalogs. Each catalog is created by first generating a sky realization of the spin-2 cosmic shear field as a Gaussian random field from a theoretical E-mode power spectrum. This realization is generated using the pixelization scheme with a resolution parameter N^ hires_ side=4096, corresponding to a pixel size of ∼0.86'. The native resolution is chosen to be significantly higher than the angular scales we will try to extract using either the map-based or catalog-based methods, to ensure that the finite-resolution effects inherent to these ground-truth maps are negligible compared to the typical statistical error. From this map, we then produce a mock observed shear catalog, by sampling it at the positions of a discrete set of sources. Specifically, to each source we assign the value of the shear field corresponding to the pixel it falls in[Unfortunately, more sophisticated interpolation schemes are highly computationally intensive for .]. We do not add any shape noise to the catalog, in order to reduce the statistical uncertainties of the recovered power spectra, thereby maximizing our sensitivity to any potential bias.

    Once a mock shear catalog has been generated, we estimate its angular power spectrum using the catalog-based method described above, and the standard map-based method. For the map-based case, we follow the procedure outlined in <cit.>: a map of the average shear in each pixel is constructed at a resolution N_ side=1024. In each pixel, the local shear field is calculated as the weighted average of the source ellipticities in it. As our simulated catalogs do not contain any additional shape noise, we do not subtract any data-based noise bias in the standard map-based approach (although we do subtract Ñ^f and Ñ^w in the catalog-based method, as described in Section <ref>). The pixelization suppresses the small-scale information in the shear map, reducing the map RMS by a pixel window function, which needs to be accounted for when computing pseudo-C_ℓs <cit.>. Assuming that the pixelization is coarse enough so that every pixel contains at least one galaxy, we can approximate the pixel window function in real space by a Gaussian smoothing kernel with an FWHM of ∼ 41.7 deg/N_ side size <cit.>. We do not correct for this bias in the simulated data, but account for it when comparing with the bandpower-convolved theory power spectrum.

    In order to produce mock catalogs with realistic source distributions, we use the angular positions and shape weights of real data from the public KiDS-1000 shear catalog <cit.>. In particular, we use all sources from the fourth photometric bin, which contains 4,544,395 sources, observed in two different fields, covering an area of approximately 800  deg^2 after masking. The ground-truth maps were generated using the theoretical prediction for this redshift bin based on the best-fit ΛCDM parameters from <cit.>, assuming no intrinsic alignments. We note that, by choosing to use real KiDS-1000 sources, we ensure that the method is validated on catalogs with not only a realistic number density, but also a spatial distribution that reproduces both the expected clustering of sources, and their inhomogeneity caused by depth variations across the observed footprint. In order to quantify the performance of the catalog- and map-based methods as a function of source number density, we repeat our analysis on down-sampled versions of these catalogs, in which we select a fraction of the KiDS-1000 sources. Specifically, we will study the results of selecting only 1% and 0.01% of the sources. We choose a multipole binning of constant width Δℓ=30 up to a maximum of ℓ_ max=2N_ side=2048.

  

 §.§ Validation results


    

    Figure <ref> shows the main results of our validation exercise. The upper right panel shows the results in the fiducial setup, in which all the KiDS-1000 sources are used to measure the power spectrum. The power spectrum averaged over all simulations is shown as orange and blue circles for the catalog- and map-based approaches, respectively. The theoretical predictions, estimated by convolving the input power spectrum with the bandpower window functions corresponding to the mode-coupling matrix derived with either method, are shown as dashed red and solid grey lines (the latter barely visible behind the red dashed lines). We multipliy the theory spectra by the squared pixel window function (N_ side^ hires=4096 for the catalog-based estimator, N_ side=1024 for the map-based one). The narrow subpanel below shows the residual of each method with respect to its corresponding prediction normalized by the error on the mean of the 500 simulations (i.e., the scatter across simulations divided by √(500)). The horizontal grey bands show 3-σ confidence intervals.
    
    The catalog-based method is able to produce an unbiased estimate of the power spectrum, while at ℓ≳ 500, the map-based approach deviates significantly due to pixelization effects at the map level and shot-noise-related bias in the estimator itself. Note that, in a realistic scenario, the statistical uncertainties would be at least √(500)∼22 times larger, not accounting for the impact of shape noise, and hence these deviations would likely be subdominant. We discuss the various sources of this bias in Appendix <ref>. For the catalog-based estimator, we must account for the pixelization of our ground truth shear map, and can do so accurately by means of the window function.[The simulated catalog contains field values that are, by design, constant within every pixel of the ground truth map. This makes the catalog-based field effectively band-limited, with no power at scales below the pixel scale, and we can account for this by means of the analytic pixel window function.] Ideally, we could choose an even higher N_ side^ hires to avoid this effect, but given the higher numerical cost, we prefer to correct the theory accordingly. With the map-based approach, the estimate is biased both for the uncorrected case (not shown in the figure) and the case where the theory is convolved with the pixel window function (shown). As explained in Appendix <ref>, the two main reasons are the impact of aliasing, and the numerical instability of the pixel-based mode-coupling matrix due to mask shot noise. Note that this coupling matrix is truncated to multipoles ℓ<3N_ side, and thus fails to capture the full correlation between scales induced by the mask for the sparsest catalogs.

    The bottom panels of Fig. <ref> show the results of repeating this analysis on catalogs containing 100 times and 10,000 times fewer sources (left and right panels, respectively). We find that, in spite of the substantial drop in number density, the catalog-based method is able to recover unbiased power spectrum measurements in both cases within the range of scales of interest. The map-based approach, however, recovers significantly biased measurements, which cannot be corrected for by our analytical pixel window function. This bias becomes gradually larger on smaller scales for the 1% catalog, reaching very significant levels after ℓ∼100. For the 0.01% catalog, in turn, the map-based method fails catastrophically, due to significant shot noise in the field pseudo-C_ℓ estimator, as well as numerical instabilities in the estimated mode-coupling matrix. These lead to the method misestimating the measured power spectrum at all scales (bottom right panel). In contrast, the catalog-based approach remains unbiased even for this extremely low number density (the shear field is sampled at the positions of just 454 sources). For the thinned catalogs, the catalog-based estimator is again fully unbiased even at scales ℓ>1024 after accounting for pixelization-related bias mentioned above, which becomes less important for smaller source densities due to higher estimator noise (∼2σ and ≲1 σ at ℓ=1024 for 1% and 0.01% thinning, respectively).
    
    
    The upper left panel of Fig. <ref> shows the mask pseudo-C_ℓ in the catalog-based case, C̃^w_ℓ, for the fiducial catalog, and the 1% and 0.01% catalogs (from top to bottom). In each case, the mask shot noise level Ñ^w is marked by horizontal black dashed lines, and the raw mask pseudo-C_ℓs are shown as the solid lines. We see that, for the original KiDS source density, the mask shot noise is low enough to not affect the mode-coupling matrix significantly, except above ℓ∼2000. In turn, the 1% catalog becomes shot-noise dominated at ℓ∼200 (compared to ℓ∼100 where the decoupled power spectrum estimate starts deviating significantly from the input). Finally, the 0.01% catalog is shot-noise dominated at almost all scales, leading to severe numerical instabilities in the standard pixel-based algorithm, which does not account for the impact of mask shot noise.

    

    Figure <ref> shows the mode coupling matrices computed from mock catalogs, and compares the result to the map-based analog. We show this for the full KiDS-1000 catalog (blue) and the two thinned versions (orange, green) but focus on the former for now. The left panels show specific columns of the ^EE_EE component of the catalog-based corrected mode coupling matrix M_ℓℓ'^w,S (solid lines, see Sect. <ref>) and the standard pixel-based mode coupling (dashed lines), normalized to one at ℓ=ℓ'. The right panels show, in the same units, the relative difference between the columns of the map-based mode coupling matrix (<ref>) and the catalog-based one[For a direct comparison between the catalog-based and the map-based mode coupling matrix elements, we need to divide the latter by the squared pixel area, (4π/N_ pix)^2.]. From top to bottom, we display columns with ℓ=(30,  100,  800,   2000). We find that, in the case of the full catalog, both matrices differ by a factor of less than 10^-4, indicating that the mask shot noise correction is subdominant. The matrix columns peak at ℓ=ℓ' as expected, rapidly decline to percent level at |ℓ-ℓ'|∼ 200, and converge to a nearly constant value at the 0.1 percent level. 

    For the 1%-thinned catalog, both methods start differing at the percent level. The elements of the catalog-based matrix match on average those for the full catalog, but are much noisier for couplings between distant multipoles, while the pixel-based coupling matrix includes a now very significant contribution due to mask shot noise, with the expected scaling ∝ (2ℓ+1), see Eq. (<ref>). For the highly shot-noise dominated case with 454 sources, the catalog-based mode coupling becomes more noisy, with fluctuations reaching percent level, but its mean stays compatible with the one for the full catalog. On the other hand, the pixel-based coupling matrix is almost completely dominated by the shot noise component, and differs from the catalog-based version by more than 100% at ℓ≳500. This shows that even with an extremely low source number density, the catalog-based approach recovers a stable mode coupling matrix that allows us to estimate unbiased power spectra from sparsely sampled sources on the sky. 
    
    

    To understand what causes the bias in the pixel-based estimator, we compare both approaches in terms of the coupled pseudo-C_ℓ estimators. Figure <ref> shows, in the same color scheme as in Fig. <ref>, the coupled shear EE pseudo-C_ℓs (see Eq. <ref>) computed using the catalog-based and pixel-based approach, and compared to the input theory, multiplied by the square of the respective pixel window, and coupled with the respective mode coupling matrix. The field noise level N_f is shown as a horizontal black dashed line for comparison. The left and the right panel show the cases of 1% and 0.01% thinning, respectively, and the lower panels show the bias with respect to the mode-coupled theory prediction. The pixel- and catalog-based coupled estimators differ at similar scales as their decoupled versions shown in Fig. <ref>, but without being affected by the numerical instability of the inverse binnned mode-coupling matrix.
    
    In contrast, the coupled theory bandpowers do depend on the mode coupling matrix. We see that the catalog-based theory prediction becomes noisy for small scales, which is inherited from the noisy mode-coupling matrix seen in Fig. <ref>. For both thinned catalogs, the catalog-based estimator is unbiased, while the map-based estimator receives a bias at intermediate and small scales. In summary, Fig. <ref> illustrates that for low source number density, the pixel-based coupled pseudo power spectrum is shot-noise dominated. Analogously, the mask shot noise dominates the pixel-based mode coupling at small scales and leads to a severe misestimation of the mode-coupled theory theory prediction. The decoupled pixel-based estimator inherits this field noise bias, and suffers from numerical instabilities from applying the inverse mode coupling matrix. In contrast, the catalog-based estimator does not deviate significantly from theory predictions, even when considering the statistical power of 500 noiseless simulations combined.




§ EXAMPLE APPLICATIONS
 
  

 §.§ Spin-2 example: cosmic shear from DES-Y3
 
  As a first real-data example, we apply our estimator to the highest redshift bin of the galaxy shear catalog from DES-Y3 <cit.>. This catalog contains 𝒪(10) more galaxies in a four times larger survey footprint than the fourth redshift bin of KiDS-1000 used in Sect. <ref>. Cosmic shear is a common example of a spin-2 field sampled at discrete positions, and one of the most powerful probes of the cosmic LSS. It measures the lensing-induced distortion of galaxy shapes due to the foreground matter distribution along the line of sight. In the Limber approximation <cit.>, the cosmic shear angular power spectrum is related to the 3D matter power spectrum P(k,   z(χ)), and can be modeled as
  
    C_ℓ = G_ℓ^2 ∫χ/χ^2  q^2(χ)   P( k=ℓ + 1/2/χ,   z(χ) )   ,

  where k is the modulus of the Fourier wave vector, z is the redshift, and χ is the comoving radial distance. The lensing kernel q(χ) quantifies the radial contribution of foreground matter to the angular galaxy lensing power spectrum, and depends on the redshift distribution p(z) of galaxies that may act as lensed images. In the case of a flat universe in which general relativity is valid and c=1, the lensing kernel is given by
  
    q(χ) = 3/2 H_0^2 Ω_m χ/a(χ)∫_z(χ)^∞ z’   p(z’)χ(z’) - χ/χ(z’)  ,

  where H_0 is the Hubble constant, Ω_m is the matter density parameter, and a=1/(1+z) is the scale factor. Transitioning from the 3D Laplacian of the gravitational potential to the angular Hessian of the associated lensing potential requires the correction factor <cit.>
  
    G_ℓ≡√((ℓ+2)!/(ℓ-2)!)1/(ℓ + 1/2)^2.

  Galaxy shear data are notoriously biased by galaxy shape noise, which has to be corrected for explicitly in the pixel-based approach <cit.>, but is removed at the catalog level using our estimator.

  We use the fourth redshift bin of the DES-Y3 data, containing 25,091,297 catalog galaxies between redshifts ∼0.4 and ∼1.5. We correct the catalog galaxy ellipticities for linear and multiplicative bias as described in <cit.>. The survey footprint covers an effective sky area of 4143 deg^2 as shown in the left panel of Figure <ref>. The weighted source number density of DES-Y3 is n_ eff = 5.59  gal/ arcmin^2, compared to 6.17  gal/ arcmin^2 for KiDS-1000, yielding a similar mask shot noise level.

  Figure <ref> in the right panel shows the results in terms of the decoupled power spectrum, comparing our catalog-based approach with recent pixel-based measurements using the pixel-based approach published in <cit.>. Both estimators use the same set of multipole bins up to ℓ=8192, corresponding to 2 N_ side of the map used with the pixel-based estimator, or an angular scale of ∼26 arcsec. We corrected the map-based estimate by the analytical pixel window function. Both results use 1-sigma error bars from <cit.>, derived from an empirical power spectrum covariance that includes a Gaussian and a non-Gaussian super-sample contribution. The lower panel shows the relative difference between both methods in units of sigma. We find that both methods agree within 1 sigma within the shown multipole range.

  


  

 §.§ Spin-1 example: proper motion surveys

    
    Although most quantities of cosmological interest are either scalars or spin-2 objects (e.g. cosmic shear, CMB polarization), there are examples of particularly useful spin-1 fields, especially in terms of catalog-based observations. Examples of these are lensing displacements, galaxy angular momenta, and transverse peculiar velocities <cit.>. Astrometric surveys, such as the satellite, are able to track the angular position of sources over time with high precision. Although the main use of such observations is reconstructing the dynamics of stars in the Milky Way, it has been argued that they may be useful for cosmology when applied to extra-galactic sources, placing constraints on the local rate of expansion <cit.>, detecting the Solar System's proper motion dipole <cit.>, and even constraining structure growth <cit.>.

    As an example application, we generate simulated observations of quasar proper motions for a -like sample. In particular, we use the positions of quasars in the public quasar sample <cit.>. We select 342,971 quasars with spectro-photometric redshifts in the range 1<z_ sp<1.5. The spatial and redshift distribution of this sample is shown in Fig. <ref>. Next, we generate the theoretical prediction for the angular power spectrum of the expected cosmological proper motion signal. Following the formalism in <cit.>, the projected proper motion map Ȧ is a spin-1 field given by
    
    Ȧ()=∫dz/(1+z)χp(z) (1-^T)· v(χ),

    where p(z) is the redshift distribution of the sample, v is the peculiar velocity, projected on the plane transverse to the line of sight via the projector 1-^T. At linear order in perturbation theory, Ȧ is a pure E-mode field, since it can be written as the covariant derivative of a projected velocity potential <cit.>. Thus, in what follows, we study only the power spectrum of its E-mode component.

    We compute a theoretical prediction for the signal power spectrum using the redshift distribution of the sample and assuming standard ΛCDM cosmological parameters. We then generate Gaussian realizations of this field at resolution N_ side=1024, and sample them at the positions of the quasars. Since the peculiar motion signal is most relevant at large scales, and given the relative sparsity of the sample, we will only attempt to reconstruct the power spectrum up to scales ℓ=360, and thus this resolution is sufficient. We generate 100 such simulated catalogs, estimate their power spectrum using the catalog-based method presented here and the standard pixel-based approach, using maps of resolution N_ side=128 in the latter case.
    

    The result of this exercise is shown in Fig. <ref>. The upper panel shows the E-mode power spectrum bandpowers measured and averaged over 100 simulations, as well as the associated theoretical prediction, given by the input power spectrum convolved with the bandpower window functions. The lower panel shows the relative residuals between simulations and theory prediction in both cases. We find that, while the catalog-based approach is able to obtain an unbiased estimate of the power spectrum at all scales, the pixel-based method under-predicts it severely on small scales (ℓ≳70).

  

 §.§ Spin-0 example: FRB-based dispersion measure maps
 
    As an example of a scalar (spin-0) quantity sampled at discrete positions, we study the case of maps of the projected electron density distribution constructed from the dispersion measure of fast radio bursts (FRBs). FRBs are powerful millisecond-long radio pulses emitted by (most likely) extra-galactic sources of unknown nature. As they propagate through the interstellar and intergalactic plasma, the group velocity of these pulses changes with frequency. This leads to a time delay between pulses at different frequencies ν that scales as Δ t=t_0  DM/ν^2, where t_0≡ 4.15×10^-3   s  GHz^2  cm^3  pc^-1, and the dispersion measure DM is
    
    DM()≡∫_0^χ_sdχ/an_e(χ),

    with χ_s the comoving radial distance to the source, a the scale factor, and n_e( x) the comoving electron density. As in the case of cosmic shear and peculiar transverse velocities, one can thus make use of FRB catalogs to construct maps of the projected electron density. For a catalog with a redshift distribution p(z), this is given by <cit.>
    
    DM()=∫dz/H(z)(1+z) P(>z) n_e(χ(z))  , P(>z)≡∫_z^∞ dz' p(z').

    where P(>z) is the cumulative redshift distribution.
    

    Current FRB catalogs are significantly sparser than the transverse velocity and cosmic shear examples from previous sections, with existing FRB samples containing a few hundred sources (although this number will grow significantly in the future <cit.>). Another distinctive feature of FRB catalogs is the fact that DM measurements are highly accurate. Typically, the DM can be measured with a precision of ∼0.1  cm^-3 pc, while extragalactic DM values are orders of magnitude higher (e.g. DM_z∼0.8∼1000  cm^-3  pc), and the contributions from the Milky Way are also comparatively small (DM_ MW∼ 50  cm^-3 pc). Typical host contributions are likely of the order DM_ host∼100  cm^-3  pc (although the distribution of host DM may have significant tails <cit.>). Thus, in this case, the per-source measurements of the field are signal-dominated, unlike the previous examples.

    To showcase the ability of the method presented here to recover accurate power spectrum measurements in this regime, we use the latest public catalog of FRBs made available by the CHIME collaboration <cit.>. After removing duplicates, the catalog contains 492 unique sources, with DM values in the range DM≲2000. The left panel of Fig. <ref> shows the spatial distribution of this sample and their measured dispersion measures. We assign a nominal redshift to each source from their measured DM using the background relation between DM and redshift:
    
    DM(z)- DM_0=∫_0^zdz'/H(z') (1+z') n̅_e,n̅_e=3H_0^2Ω_b/8π G m_px_e(1+x_H)/2,

    where m_p is the proton mass, H_0 is the expansion rate today, Ω_b is the fractional density of baryons (which we set to Ω_b=0.05), x_e is the ionization fraction (which we set to x_e=1), and x_H is the hydrogen mass fraction (which we set to x_H=0.75). We fix the offset DM_0=100  cm^-3  pc to account for the typical host / Milky Way contribution (although our results do not depend on this choice). The corresponding redshift distribution of the sample is shown in the right panel of Fig. <ref>, and can be approximated as p(z)∝ z^2exp(-z/0.15) (solid line in the same figure).
    

    We use this redshift distribution to generate a theoretical prediction for the power spectrum of the DM field in Eq. <ref>, assuming a linear bias for the electron overdensity of b_e=0.8 <cit.>. We generate a suite of 1000 simulated Gaussian realizations of DM() at a resolution of N_ side=512, which we then sample at the positions of the CHIME catalog sources. Given the sparsity of the sample, we only attempt to reconstruct the power spectrum up to scales ℓ=180, and hence this reduced resolution is sufficient. The left panel of Fig. <ref> shows, in orange error bars, the reconstructed power spectrum averaged over all simulations, together with its theoretical prediction (i.e., the bandpower-convolved power spectrum used to generate the simulated maps) in solid black lines. In blue error bars, we show also the result of using the standard, pixelized version of the estimator, using a resolution N_ side=64. The numerical instability of the mode-coupling matrix, and aliasing due to limited pixel resolution, causes the standard estimator to fail catastrophically in this case.
    
    The right panel of the same figure shows rows of the normalized mode-coupling matrix M_ℓℓ' for four different values of ℓ. Results are shown for the catalog-based and pixel-based approaches as solid and dotted lines, respectively. In the catalog-based case, M_ℓℓ' displays significant coupling between multipoles separated by up to Δℓ∼20, with a more non-trivial structure at larger multipole separations. In turn, the pixel-based approach, which does not correct for the effect of mask shot noise, shows significant correlation extending across the whole range of multipoles, growing like (2ℓ'+1) (see Eq. <ref>). The mode-coupling matrix in this case is thus significantly more numerically unstable. Together with the impact of aliasing (see Appendix <ref>), this leads to the failure observed in the right panel.

  

 §.§ Clustering example

    We end this section showcasing the applicability of the discrete pseudo-C_ℓ approach to the case of source clustering, in which the density distribution of sources is itself the field under study. The formalism for this case was first presented in <cit.>, and we described it within the general framework presented here in Section <ref>. To validate our implementation of the formalism in , we generate fast simulations of a galaxy clustering sample as follows:
    
      
  * We calculate an initial angular power spectrum corresponding to the projected ovedensity of sources with a linear galaxy bias b=1 and a normal redshift distribution centered at z=0.6 with a standard deviation σ_z=0.1.
      
  * We create a completeness map c() to simulate the observational survey inhomogeneities that the random catalogs described in Section <ref> aim to trace. To do this, we draw 200 random unit vectors and select a spherical cap with a 10^∘ radius centered at each of them. For each cap, we then draw a random number between 0.97 and 1 to represent the sky completeness in that patch (i.e., up to 3% incompleteness). We multiply the resulting map by a sky mask that removes regions of high dust extinction and star contamination, constructed as described in <cit.>.
      
  * We generate a Gaussian overdensity field, δ_G() from this power spectrum, and perform a lognormal transformation to turn it into a positive-definite angular density field:
      
    n^ true_g()=n̅_g exp(δ_G()-σ_G^2/2),

      where σ_G is the standard deviation of δ_G, and n̅_g is the desired average number of sources per steradian[The -σ_G^2/2 counterterm ensures that the mean of the resulting field is indeed n̅_g, assuming δ_G is Gaussian.]. The observed mean source density is then constructed by multiplying this field by the completeness map described above: n^ obs_g()=c() n^ true_g().
      
  * We Poisson-sample this field, assigning each source a random position within its pixel, to generate the “data” catalog. The “random” catalog is generated by drawing random positions on the sphere, and discarding them with a probability given by the local value of the completeness map. The random catalog thus constructed has 50 times more sources than the data.
    
    

    Keeping the completeness map and associated random catalog fixed, we generate 100 data simulations at N_ side=1024 with a number density n̅_g=0.01  arcmin^-2, and estimate the power spectrum of each simulation using the procedure described in Section <ref>. The use of a lognormal transformation to generate the galaxy density field prevents us from using the initial angular power spectrum as a theoretical prediction to compare against. Although the effect of a lognormal transformation can be propagated to the power spectrum analytically (see <cit.>) we resort to a more direct approach that avoids potential numerical instabilities of Hankel transforms: for each simulation, we calculate the power spectrum of the lognormal density field (δ_ LN=exp(δ_G-σ_G^2/2)-1) on the full sky (i.e., in the absence of completeness, masking, and discrete-source effects). The “true” power spectrum against which we compare the result of our simulations is then given by the average of this power spectrum over simulations corrected for the impact of the pixel window function used in these simulations.

    The left panel of Fig. <ref> shows the completeness map used in this exercise. The right panel of the same figure shows the mean power spectrum recovered from our simulation (blue points with error bars), the true C_ℓ (orange line), and the relative residual between them (bottom subpanel). The catalog method implemented here is able to recover the true underlying power spectrum at very high accuracy (the error bars shown represent the statistical uncertainties in the mean, corresponding to 1/10 of the error of a single realization).



§ CONCLUSION
 
  In this work, we present a simple extension to the standard pseudo-C_ℓ estimator to tackle the problem of fields sampled at the positions of a discrete and finite catalog of sources. By treating the problem directly at the catalog level, without resorting to an intermediate pixelization of the sources onto a regular grid, we demonstrate that the method is able to tackle three prevalent sources of systematic bias in the standard pixel-based approach: resolution and aliasing effects due to the finite pixel area, shot noise in the field estimate due to the discrete nature of the field tracers, and numerically unstable mode-coupling matrices due to shot noise in the pixelized mask constructed from the sources. We interpret the catalog data as representing a masked field, with a mask given by a set of delta functions at the position of each source, and take advantage of new methods developed for the calculation of spherical harmonic transforms on irregular grids. We show that a simple correction of the shot-noise contributions to the power spectra of the masked field and of the mask itself, using fast and model-independent analytical expressions, is able to significantly improve the numerical stability of the mode-coupling matrix, while simultaneously rendering the estimator immune to any uncorrelated noise bias, irrespective of the size of the signal contribution. The method is a generalization of the technique presented in <cit.> in the context of galaxy clustering. The most evident use of this method is in the analysis of cosmic shear data, although it is broadly applicable to any datasets described by observed properties of discrete sources.
  
  We validate our implementation of this method on a set 500 simulated KiDS-1000-like cosmic shear catalogs, with sources sampled at their real sky positions. We show that the resulting estimator is unbiased even for unrealistically small samples (100 and 10,000 times less dense than the real KiDS-1000 sample), where mode-coupling instabilities due to shot noise render standard pixel-based methods unusable. Even at realistic sample sizes, including the larger densities achieved by deeper samples, we show that pixel-based methods encounter difficulties to accurately characterize the effects of pixelization on the small-scale power spectrum. While these can be circumvented through the use of higher pixel resolutions, this is at a significantly higher cost in computational time and complexity, which the catalog-based method presented here avoids altogether.

  To demonstrate the practical use of our estimator on other suitable astrophysical datasets, with varying sample sizes and signal-to-noise ratios, we apply it in four additional distinct scenarios. First, we demonstrate its feasibility in the analysis of significantly larger catalogs, reproducing the analysis of the 3-year DES cosmic shear sample, comparing the results to those presented in <cit.>. We then study the case of a spin-1 field: the peculiar transverse motion of extragalactic objects as traced by an astrometric survey, using the quasar sample. Besides the different spin, this type of dataset features a significantly lower source density and signal-to-noise ratio than cosmic shear. We also consider measurements of the spin-0 projected electron density in the form of Fast Radio Burst catalogs. These correspond to samples with extremely low number densities (10^2-10^3 objects) but a very high signal-to-noise ratio. Finally, we showcase the applicability of our formalism to the analysis of galaxy clustering, in which the density of catalog sources is itself the field of interest.  Our results from all four examples confirm our previous findings that our estimator is unbiased and can overcome the numerical challenge of source sparseness when characterising mode coupling.

  We expect the method presented to be useful in the analysis of upcoming large-scale structure datasets, particularly in the context of cosmic shear, as well as novel probes such as FRBs, radial and transverse peculiar motion surveys, weak lensing of gravitational waves, and any other catalog-based observations. Our implementation is made available with the latest release of the public power spectrum package [<https://github.com/LSSTDESC/NaMaster/>]. 



§ ACKNOWLEDGMENTS

  When the work presented here was at an advanced stage, we learnt of similar work being developed by Tessore et al. 2024 <cit.>. We thank the authors of this work, particularly Nicolas Tessore and Benjamin Joachimi, for sharing their draft paper and for useful discussions. We also thank Anton Baleato, Elyas Farah, Robert Reischke, and Martin White, for useful comments and feedback. KW acknowledges support from the Science and Technology Facilities Council (STFC) under grant ST/X006344/1. DA acknowledges support from the Beecroft trust. We made extensive use of computational resources at the University of Oxford Department of Physics, funded by the John Fell Oxford University Press Research Fund.



§ GENERALITIES OF SPHERICAL HARMONIC FUNCTIONS

  We use the following definition for the generalized spin-s spherical harmonic functions[The -1 sign preceding these definitions is a mostly inconvenient convention that is dropped for the case of spin-0 fields, so that _0Y^1E_ℓ m= _0Y^2B_ℓ m=Y_ℓ m, and _0Y^2E_ℓ m= _0Y^1B_ℓ m=0.]:
  
    _sY^1E_ℓ m=-1/2[ _sY_ℓ m+(-1)^s _-sY_ℓ m],
        _sY^1B_ℓ m=-i/2[ _sY_ℓ m-(-1)^s _-sY_ℓ m],
    
        _sY^2E_ℓ m=i/2[ _sY_ℓ m+(-1)^s _-sY_ℓ m],
        _sY^1B_ℓ m=-1/2[ _sY_ℓ m-(-1)^s _-sY_ℓ m],

  where _sY_ℓ m() are the spin-weighted spherical harmonic functions of spin s.
  These functions satisfy the following properties:
  
    ∫ d _sY_ℓ m^aα*()_sY_ℓ' m'^aβ()=δ_αβδ_ℓℓ'δ_mm',
       ∫ d _sY_ℓ m^aα*()_sY_ℓ' m'^aβ() Y_ℓ”m”()=(-1)^s+m√((2ℓ+1)(2ℓ'+1)(2ℓ”+1)/4π)
       ℓℓ'ℓ”-mm'm”ℓℓ'ℓ”s-s0 d^αβ_ℓ+ℓ'+ℓ”,
       d^EE_n=d^BB_n=[1+(-1)^n]/2,d^EB_n=-d^BE_n=-i[1-(-1)^n]/2.
       ∑_m _sY_ℓ m^aα*() _s'Y_ℓ m^aβ()=2ℓ+1/4πδ_αβδ_ss'.

  The last relation can be derived from the addition theorem for spin-weighted spherical harmonics <cit.>. For scalar harmonics, it reads simply
  
    ∑_m Y^*_ℓ m(_1)Y_ℓ m(_2)=2ℓ+1/4π P_ℓ(_1·_2)   ,

  which, for _1 = _2, simply evaluates to (2ℓ+1)/(4π). Furthermore, the spherical harmonic functions are a complete basis on the sphere, so that:
  
    ∑_ℓ mY^*_ℓ m(_1)Y_ℓ m(_2)=δ^D(_1,_2).

  We will also employ the following properties of the Wigner 3j symbols:
  
    ∑_mm'ℓℓ'ℓ”mm'm”ℓℓ'ℓ”'mm'm”'=δ_ℓ”ℓ”'δ_m”m”'/2ℓ”+1,
       ∑_ℓ”(2ℓ”+1)ℓℓ'ℓ”s-s0ℓℓ'ℓ”s'-s'0=δ_ss',
       ℓℓ'ℓ”-m-m'-m”=(-1)^ℓ+ℓ'+ℓ”ℓℓ'ℓ”mm'm”.




§ PIXELIZATION EFFECTS IN CATALOG-BASED OBSERVABLES

  The main text has emphasized the impact of the numerical instability of the mode-coupling matrix due to the presence of shot noise in the mask power spectrum for catalog-based observables that are first pixelized before estimating their angular power spectrum. This appendix explores the two other sources of potential bias that affect pixelized catalog-based data: the impact of pixel window functions, and aliasing.
  

  For this exercise we generate two types of simulated realizations of a  scalar Gaussian random field using a base pixelization of N_ side=1024:
  
      
  * Band-limited realizations with a power spectrum of the form
      
    C_ℓ=exp[-(ℓΘ)^2]/ℓ+50.

      The exponential cutoff, with Θ=0.3^∘ sharply reduces any structure on scales ℓ≳200.
      
  * Non-band-limited realizations with a power spectrum of the form
      
    C_ℓ=1/ℓ+50,

      which is rather flat, decaying significantly more slowly at high ℓ than the previous case.
  

  First, to isolate the effects of pixel averaging and aliasing from the practicalities of fields sampled at arbitrary discrete positions, we construct maps at lower resolution, N_ side=256 and N_ side=64 by simply averaging the N_ side=1024 maps onto larger pixels. We calculate the angular power spectrum of each of these maps, and estimate the effective pixel window function of the measured power spectrum as:
  
    b^ eff_ℓ≡√(Ĉ_ℓ/C_ℓ^ in),

  where Ĉ_ℓ is the estimated power spectrum, and C_ℓ^ in is the input power spectrum (given by Eq. <ref> or Eq. <ref>). We repeat this for 100 simulations and compute the mean and scatter of the effective pixel window function.

  The top left and top right panels of Fig. <ref> shows the result of this exercise for the band-limited and non-band-limited realizations, respectively. We find that, for the band-limited case, the impact of pixel averaging is captured by the pixel window function with relatively good accuracy[The slight difference for the N_ side=256 maps is due to the fact that averaging over only 4×4 sub-pixels is less representative of a true area averaging than the 16×16 pixels for N_ side=64.]. In the case of non-band-limited simulations, however, we see very significant differences in the effective pixel window function. These differences are caused by aliasing of the small-scale power in the base maps onto the larger pixels via averaging. Interestingly, the resulting effective pixel window function has a similar shape both in the N_ side=256 and the N_ side=64 case.

  We now repeat the exercise with a key modification: we sample the base maps at the discrete positions of 10^6 sources drawn at random on the sphere, and we then construct maps of the resulting field using the standard pixel-based approach at N_ side=256 and N_ side=64 (corresponding to ∼30 and ∼1.3 sources per pixel on average, respectively). When calculating the power spectra of theses maps we use the density of said sources as their corresponding masks. Finally, to factor out the effects caused by shot noise in the mask, we subtract the analytical shot noise contribution to the mask and field pseudo-C_ℓ (Ñ^w and Ñ^f), mimicking the procedure described in the main text to avoid these effects. The bottom left and right panels of Fig. <ref> show the results of this exercise for the band-limited and non-band-limited maps, respectively. The conclusions are analogous to those found in the previous exercise: regardless of the average number of sources per pixel, the most relevant effect is aliasing in the case of non-band-limited data. The effects of pixelization on band-limited data are well captured by the pixel window function.





 Paper Title: Probing beyond local-type non-Gaussianity with kSZ tomography.






adshead@illinois.edu



atishue@illinois.edu



















§ ABSTRACT 
Non-Gaussianity of the primordial curvature perturbations may arise from a variety of well motivated early Universe scenarios. In particular, inflationary theories with additional light degrees of freedom can generate a bispectrum that is peaked in the squeezed limit. While the presence of an additional massless scalar  can produce local-type primordial non-Gaussianity, in general the squeezed limit of the bispectrum depends on the mass of the new degree of freedom, and can deviate from the local shape. The resulting bispectrum leaves a distinct imprint on the amplitude and scale-dependence of the galaxy bias, which for massive fields can differ from the k^-2 scaling from local type non-Gaussianity, providing an observational window into the physics of the early Universe.  
In this work we demonstrate that kinematic Sunyaev-Zeldovich tomography with next generation cosmological surveys will offer significant additional constraining power for both the shape and amplitude of scale-dependent bias arising from primordial non-Gaussianity beyond the local type. We show that this improved constraining power is robust to various obstacles such as the optical depth degeneracy, photometric redshift errors, and uncertainty in the galaxy bias model. With CMB S4 and the Large Synoptic Survey Telescope, we forecast that compared to the galaxy survey alone the addition of kSZ tomography will offer a roughly factor of two reduction in the measurement uncertainty of the amplitude  of primordial non-Gaussianity well beyond the local (massless) limit. We find that kSZ tomography extends the range of masses for which order unity constraints on  are achievable, as well as extending the range of masses for which the late time probes of the matter power spectrum outperform the sensitivity of the CMB itself.  


Probing beyond local-type non-Gaussianity with kSZ tomography
    Avery J. Tishue
    Received XXX; accepted YYY
=============================================================






§ INTRODUCTION 

Probing non-Gaussianity of the primordial density fluctuations is a key objective for upcoming next-generation cosmic microwave background (CMB) experiments, such as the Simons Observatory <cit.> and CMB-S4 <cit.>, and large scale structure (LSS) surveys, such as SPHEREx <cit.>. Detailed measurements of the distribution of structures in the Universe  have the potential to provide critical information about the mechanism responsible for setting the initial state of the Universe. On the one hand, existing observations are consistent with an early period of accelerated expansion—inflation.  A Gaussian, red-tilted and adiabatic spectrum of initial density fluctuations, consistent with models for single-field, slow-roll inflation <cit.>, accounts for all observations to date.  On the other hand, primordial non-Gaussianity holds the potential to provide insights beyond the single-field slow-roll inflationary paradigm, and probe the microphysics of the very early Universe <cit.>.  



One primary goal for upcoming searches is to improve measurement sensitivity to the amplitude of local-type non-Gaussianity <cit.>. Here the non-Gaussian primordial curvature perturbation ζ(𝐱) is generated by the fluctuations in a Gaussian field ζ_g(𝐱) via

    ζ(𝐱) = ζ_g(𝐱) +(ζ_g(𝐱)^2 - ⟨ζ_g(𝐱)^2⟩).

The corresponding momentum space bispectrum 

    ⟨ζ(𝐤_1)ζ(𝐤_2) ζ(𝐤_3) ⟩' =    P_ζ(k_1) P_ζ(k_2) + perms.,

is peaked in the squeezed limit, k_2,k_3 ≫ k_1. The prime, ⟨…⟩', denotes that we have suppressed the momentum conserving delta function (2π)^3 δ(𝐤_1 + 𝐤_2 + 𝐤_3 ), which restricts the momenta to form a closed triangle. Local-type non-Gaussianity leads to mode coupling between short- and long-wavelength fluctuations. The short wavelength modes control the local density of collapsed objects like galaxies, which are then non-trivially correlated across long distances due to this type of non-Gaussianity.

The squeezed limit of the bispectrum is particularly important because single-field slow-roll inflation, or more generally single-clock inflation, predicts that the observable bispectrum should vanish in this limit <cit.>. Consistency relations for large-scale structure <cit.> then ensure that properties of correlation functions of galaxy overdensities are determined by local physics <cit.>. These relations prohibit the development of the long range correlations predicted by local-type non-Gaussianity, leaving any such correlation as a clean signature of primordial non-Gaussianity. An important benchmark, then, is the limit σ () ≲ 1  <cit.>. Reaching this limit would test the single-clock inflationary paradigm; detection of ≠ 0 would rule out all single clock  inflationary models and indicate the presence of additional light degrees of freedom during inflation <cit.>. 



While the importance of testing local primordial non-Gaussianity is manifest, in general  the details of the primordial non-Gaussianity can be complicated and need not be described by a squeezed bispectrum with amplitude . Additional light degrees of freedom during inflation <cit.>, particle production during inflation <cit.>, models with higher derivative interactions <cit.>, excited initial states <cit.>, curvaton models <cit.>, inhomogeneous reheating <cit.>, transient violations of slow-roll  <cit.> and bounce cosmologies <cit.> are a few key examples of early Universe physics that can produce different types of primordial non-Gaussianity, for example predicting signals dominated by equilateral (k_1 ≈ k_2 ≈ k_3) and folded (k_1 ≈ 2k_2 ≈ 2k_3) bispectra configurations. Tests of the details of primordial non-Gaussianity are therefore tests of the details of early Universe physics, and thus have to potential to shed light on the fundamental interactions that seeded the initial conditions of the hot Big Bang. Although Planck <cit.> already constrains local and equilateral primordial non-Gaussianity to = -0.9 ± 5.1 and f_NL^equil. = -26 ± 47 respectively, the theoretical target of order unity constraints remains to be reached. Novel and improved probes of primordial non-Gaussianity therefore remain an acutely important avenue to testing fundamental physics. 

To date, information from the CMB has provided the most stringent constraints on non-Gaussianity <cit.>. However, due to the lack of remaining modes to be measured in the  primary CMB, future information will come from other probes, such as large scale structure <cit.>, including line intensity mapping <cit.>, and the secondary CMB <cit.>. Extracting information about primordial correlation functions from these measurements remains an ongoing challenge. One promising approach is to capitalize on the wealth of information stored in the correlations induced between the CMB anisotropies and LSS observables as photons from the surface of last scattering interact with matter in the late Universe. Cosmological inference with these correlations will become increasingly powerful in the near future as next generation CMB and LSS experiments probe small scale fluctuations with unprecedented sensitivity. This includes measurements of weak lensing (see e.g. Ref. <cit.>), the moving-lens effect <cit.>, and the thermal and kinematic Sunyaev-Zeldovich (kSZ) effects <cit.>. 



In this work, we focus on the kSZ effect—a secondary CMB temperature anisotropy due to inverse scattering of photons off electrons with bulk line-of-sight velocity. The imprint of the kSZ effect on the CMB, combined with a LSS survey on the same portion of the sky, enables the reconstruction of the radial velocity field, v_r. In particular, the cross correlation of the radial velocity field with the galaxy field is projected to be measured with a high-signal-to-noise ratio on large scales, and can be described by an estimator that is unbiased by other secondary CMB anisotropies <cit.>. The velocity field contains information about the density fluctuation and the growth rate. Reconstructing v_r with the kSZ effect therefore provides an additional measurement of the large scale matter power spectrum that, crucially, has a different bias than reconstruction of the density field with galaxy surveys. This also facilitates sample variance cancellation, where the comparison of the density field reconstruction with two different tracers — here, galaxies and velocities —  enables improved measurement of the respective biases and the underlying matter power spectrum on large scales, which otherwise would hinder either probe in isolation due to survey volume limitations. Altogether, this makes kSZ tomography a powerful probe of cosmology and any new physics imprinted on the matter power spectrum, or the respective biases, on large scales.  As we focus on here, one such imprint is scale-dependent galaxy bias arising from primordial non-Gaussianity <cit.>. The scale-dependence and amplitude of this bias depends on the details of early Universe physics.  Importantly, for a variety of models, the scale-dependent bias has the most significant effect on the largest scales, which is precisely the regime where kSZ tomography is projected to have the greatest signal to  noise. 



The potential of sample variance cancellation <cit.>, as well as the additional constraining power on large scales motivates the study of kSZ tomography as a probe of primordial non-Gaussianity. Indeed, in Ref. <cit.> it was shown that kSZ tomography can be a powerful probe of local non-Gaussianity in the form of constraints on the bispectrum amplitude , improving on constraints from galaxies alone by a factor of a few. Similarly, kSZ tomography may also improve constraints on local primordial non-Gaussianity in terms of the trispectrum amplitude τ_NL <cit.>. Furthermore, current and upcoming galaxy surveys can constrain primordial non-Gaussianty beyond the local type via measurement of scale-dependent bias on linear and quasi-linear scales <cit.>. 

In this paper, we carry out forecasts to show that kSZ tomography can be leveraged similarly to improve constraints on primordial non-Gaussianity beyond the local type due to its imprint as a scale-dependent galaxy bias on large cosmological scales.  The improved constraining power from kSZ tomography is due to it providing an additional measurement of the matter power spectrum, which facilitates sample variance cancellation and helps mitigate various degeneracies between the non-Gaussianity parameters and astrophysical and cosmological parameters. We also demonstrate that the additional constraining power provided by kSZ in this context is robust when including various realistic effects including photometric redshift (photo-z) errors and uncertainty in details of the galaxy bias, galaxy-electron correlation, and ΛCDM parameters. We find kSZ tomography will have significant practical application for constraining primordial non-Gaussianity beyond the local type, both in terms of outperforming the CMB and for achieving the theoretical target of order unity constraints.  In our analysis we show this practical application applies not only to pushing down upper limits on the amplitude of primordial non-Gaussianity, but also to improving the measurement significance of non-zero f_NL, should it be detected in future data. 



This paper is organized as follows. In Sec. <ref> we provide a brief overview of the primordial non-Gaussianity scenario that we consider and the corresponding signal in the galaxy bias. In Sec. <ref> we briefly review  galaxy and kSZ tomography, and in Sec. <ref> we describe the setup for our forecasts. Results for our baseline forecasts are presented in Sec. <ref>, and the sensitivity of these results to various model and survey details is presented in Sec. <ref>. We conclude with a discussion of these results and directions for future work in Sec. <ref>.






§ NON-GAUSSIANITY AND GALAXY BIAS 

For simplicity, and to demonstrate the efficacy of kSZ tomography, we focus on a scenario in which the primordial non-Gaussianity arises from additional light degrees of freedom during inflation, for example in models such as quasi-single-field inflation <cit.>. When extra, light degrees of freedom are present,  the three-point function of the curvature perturbation ζ can receive contributions in the squeezed limit, which take the form

    lim_𝐤_1 → 0⟨ζ(𝐤_1)ζ(𝐤_2) ζ(𝐤_3) ⟩' =   ( k_1/k_2)^Δ P_ζ(k_1) P_ζ(k_2)

where the amplitude is controlled by the parameter , and the shape is controlled by the scaling exponent Δ_± = 3/2 ±√(9/4 - m^2/H^2), where m is the mass of the light field. This scenario has been explored in significant detail, see, for example, Refs. <cit.>.  The squeezed limit is dominated by Δ_-, so that a massless field leads to local primordial non-Gaussianity, Δ = 0. Further, Δ smoothly approaches 3/2 as m → 3H/2. Larger values of Δ are possible, for example a very massive spectator can yield a correction to the bispectrum with Δ=2 <cit.>. Further, in single-field inflationary scenarios the leading physical term in squeezed bispectrum has a factor ∼ (k_1 / k_2)^2  <cit.> corresponding to Δ = 2 (with an amplitude of the order of 𝒪(ϵ_H)) <cit.>. 
In what follows, we  focus our analysis on the range Δ∈ [0,2].





The bispectrum in Eq. (<ref>) yields a correlation between the variance σ_R of curvature perturbations on some scale k and the primordial curvature perturbation on some scale k^', which in turn induces a correlation between the variance and the late time total matter overdensity δ,

    ⟨σ_R(𝐤) δ(𝐤^') ⟩' ∝ (kR)^Δ/k^2T(k) P_mm(k),

where T(k) is the transfer function (normalized such that T(k) → 1 as k→ 0). Hence the galaxy overdensity on a scale k is correlated with the matter overdensity on a scale k^'. The upshot is that this results in an additional, scale-dependent term in the galaxy bias, , that takes the form <cit.>[Note the factor of 3 gives a normalization  3 f_NL^(Δ = 0) = f_NL^loc. ]

    (k,z) = 3 b_ϕ(z)/k^2 𝒯(k,z) (k R_⋆)^Δ.

Here, 𝒯(k,z) = 2T(k)D(z)/3H_0^2 Ω_m, where the growth function D(z) is normalized such that D(z) ∼ (1+z)^-1 during matter domination. The parameter R_⋆ is the comoving Lagrangian radius of the halos under consideration, determined by the minimum halo mass. We take the minimum halo mass to be 10^13 M_⊙ corresponding to R_⋆ = 2.66 h^-1Mpc <cit.>. The non-Gaussian bias parameter b_ϕ(z) can be non-trivial in general, and in principle introduces a perfect degeneracy with the non-Gaussianity amplitude f_NL^(Δ) that we hope to measure. Following Ref. <cit.>, we assume b_ϕ is given by universality conditions so b_ϕ(z) = 2δ_c (b_1(z) - 1) where δ_c = 1.42 is the critical overdensity of collapse and b_1(z) is the standard linear galaxy bias. We note that uncertainty in b_ϕ, and in the universality conditions that relate b_ϕ to the linear bias b_1, may degrade constraints on  <cit.>, although it has been shown that multitracer analyses may help circumvent this obstacle <cit.>. 



§ GALAXY AND KSZ TOMOGRAPHY
 




In linear theory, which is valid on large scales, the continuity equation relates the velocity mode to the density mode  via 

    v(𝐤) = i faH/kδ_m(𝐤),

where 

    D(k,a) = (P_mm(k,a)/P_mm(k,a=1))^1/2

and

    f(k,a) = d ln(D(k,a))/d ln(a)

are the growth function and the growth rate, respectively. Reconstruction of the velocity field therefore provides a measurement of the underlying density field. In particular, if the signal to noise of this reconstruction is sufficiently high, the factor of k^-1 in Eq. (<ref>) indicates that velocity tomography can potentially probe the matter power spectrum on the largest scales more precisely than a `direct' density tracer such as galaxies. 

One way to reconstruct these large-scale velocity flows is with the kSZ effect, whereby CMB photons scatter with electrons present in the gas in galaxies and galaxy clusters. Electrons with bulk peculiar-velocity 𝐯 upscatter CMB photons via inverse Compton scattering, inducing correlations between the CMB anisotropies and the intervening large scale structure.  In the so-called `snapshot geometry' <cit.>, in which the Universe is taken  to be a 3D box of comoving side length L taken at a `snapshot' time t corresponding to a redshift z and comoving distance χ(z), a minimum variance quadratic estimator for the reconstructed large-scale (radial) velocity field, v̂_r, can be written as an integral over small scales containing the observed power spectrum of the density tracer, e.g. galaxies, the observed CMB temperature power spectrum, and the galaxy-electron cross spectrum P_ge <cit.>. The reconstruction noise power spectrum for the radial velocity estimator is given by

    1/ N_v̂_r v̂_r(k_L,μ, z) =    K^2(z)/χ^2(z)×
       ∫dk_Sk_S/2πP_ge^2(k_S,z)/P_gg^obs (k_S,k_L,μ,z) C^TT,obs_ℓ = χ(z) k_S,

where μ = k̂·r̂, k_L is the amplitude of the three-dimensional Fourier wavevector corresponding to the reconstructed large scale mode,
and the integral over small scales k_S is dominated roughly by the range k_S ∈ [0.1, 10] Mpc^-1 for the surveys we consider in this work. Here 

    K(z) = σ_T n_e,0 x_e(z) e^-τ(z) (1+z)^2,

is the kSZ radial weight function, where σ_T is the Thompson scattering cross section, n_e,0 is the electron number density, x_e(z) is the free electron fraction, and τ(z) is the optical depth to redshift z. The velocity mode and its radial component are related via v̂ = μ^-1v̂_r, and hence N_v̂v̂ = μ^-2 N_v̂_r v̂_r. 

The observed galaxy power spectrum is 

    P_gg^obs(k_S,k_L,μ,z) = P_gg(k_S,z) + N_gg(k_L,μ,z),

where the noise,

    N_gg(k_L,μ,z) = 1/n_g(z) W^2(k_L,μ,z),

is determined by the shot-noise via the galaxy number density n_g(z) (in Mpc^-1) and by photometric redshift errors described by the  photo-z kernel function,

    W(k_L,μ,z) = exp(-μ ^2 k_L^2 σ_z^2/2H^2(z)).
 

With the galaxy field and the reconstructed radial velocity field, we have access to the galaxy and velocity power spectra, P_gg and P_v̂_r v̂_r, and the cross-spectra P_g v̂_r, defined in the usual way via ⟨ X_𝐤_1 Y_𝐤_2⟩ = (2π)^3 δ( 𝐤_2 + 𝐤_2) P_XY,

    P_gg(k,z)    = b_g^2(z,k)  P_mm(k,z) , 
     
        P_g v̂_r(k,z)    = b_v(z) μf(k)aH/kb_g(k,z) P_mm(k,z)  , 
     
        P_v̂_rv̂_r(k,z)    = b_v^2(z) μ^2 (f(k)aH/k)^2 P_mm(k,z) .

Here, b_g is the galaxy bias and b_v is the velocity reconstruction bias. The velocity reconstruction bias arises if the fiducial model for the small scale galaxy-electron cross correlation P_ge(k_S) does not match the true underlying one, in which case the estimator for the reconstructed velocity field is biased with respect to the true field, v̂_r = b_v v_r, where the bias b_v is given by <cit.>

    b_v   = ∫ dk_S F(k_S) P^true_ge(k_S)/∫ dk_S  F(k_S) P^fid_ge(k_S) ,

where 

    F(k_S)    = k_S P^fid_g e(k_S)/P^obs_g g(k_S) C_ℓ = k_S χ^TT, obs  .

This is important because the small scale galaxy-electron cross correlation P_ge(k_S) is determined largely by astrophysics and is a source of theoretical uncertainty. A precise prediction of P_ge(k_S) requires precise knowledge of the mean optical depth of the measured galaxies <cit.>, as well as the spatial distribution of electrons <cit.>. This presents an obstacle to using kSZ tomography for cosmological inference: for example, when attempting to use kSZ tomography to measure the large-scale galaxy-velocity cross correlation, the underlying signal is a squeezed bispectrum of two galaxy modes and one CMB mode <cit.>, obeying ⟨δ_g δ_g T ⟩∝ P_ge(k_S) P_gv(k_L). Hence the large scale galaxy-velocity cross-spectrum we hope to measure suffers a perfect overall amplitude degeneracy with the unknown small-scale galaxy-electron cross-spectrum. This is the so called `optical depth degeneracy.' In forecasts, the optical depth degeneracy can be fully accounted for by marginalizing over b_v. The situation is analogous to marginalizing over the galaxy bias when attempting to use the galaxy power spectrum to constrain cosmology. Importantly, the bias b_v is scale-independent, in contrast to the scale-dependent bias  induced by primordial non-Gaussianity which we aim to constrain, and hence the optical depth degeneracy in principle does not introduce a fatal degeneracy for our purposes.


The total galaxy bias is 

    b_g(k,z) =      b_1(z) + (k,z) + b_rsd(z) f μ^2 
       + b_k^2(z) (kR_⋆)^2 + b_k^4(z) (kR_⋆)^4,

where we have included the contributions from the standard linear bias b_1(z); the contribution from non-Gaussianty, (k,z), given by Eq. (<ref>); the Kaiser effect  RSD term <cit.> f μ^2 with a bias b_rsd that accounts for anisotropic assembly bias from selection effects <cit.>[In subsamples of the Baryon Oscillation Spectroscopic Survey Data Release 12 <cit.>, |b_rsd-1| was shown to be roughly 0.1 to 0.3.]; and  gradient biases b_k^2 and b_k^4 that account for non-linear clustering on small scales. 

This model for the galaxy bias as well as the model for the observed matter power spectrum could be further extended, for example by considering a general bias expansion in the effective field theory of large scale structure (from which the gradient terms originate). However, such extensions should not change the main conclusions of this work because kSZ tomography adds constraining power on the largest cosmological scales, but additional terms arising from the bias expansion become important on small scales compared to the regime where kSZ adds appreciable cosmological information. Including the gradient terms allows us to still broadly capture the general impact that small scale contributions to the total galaxy power spectrum have on our results.





In Fig. <ref>, we show the effect that (k) from primordial non-Gaussianity has on the galaxy power spectrum for a range of scaling exponents and amplitudes. For smaller Δ values, the galaxy power spectrum is enhanced on large scales where the kSZ tomography signal-to-noise is greatest. As Δ→ 2, the effect on large scales diminishes and the enhancement shifts to small scales. Therefore, we naively expect kSZ to assist more in reducing σ() for smaller scaling exponents.



§ FORECASTS 

We forecast results for a galaxy survey with specifications matching those proposed for the Large Synoptic Survey Telescope (LSST) Y10 <cit.> and an S4-like CMB survey <cit.>. We consider five redshift bins z ∈ [0.2, 0.7, 1.3, 1.9, 2.6], with corresponding fiducial values for the  linear galaxy bias b_1(z) ∈ [1.05,1.37,1.79, 2.22, 2.74 ] (see Secs. 3.7, 3.8, and 13.5 in Ref. <cit.>). For all redshift bins we take the fiducial values b_rsd(z) =1, b_k^2(z) = b_k^4(z) = 0, and b_v(z) = 1. 

To carry out our forecasts, we construct a Fisher information matrix, 

    F_ab = V/2∫^1_-1 dμ ∫k^2 dk/4π^2 Tr[ C^-1dS/dπ_a C^-1dS/dπ_b].

Here V is the survey volume at each redshift bin,

    V = [5.2, 43.6, 75.9, 89.3, 119.9] × 10^9 Mpc^3,

C_ij=S_ij+N_ij is the covariance matrix of the signals S_ij and noises N_ij where the indices label the observables (i.e. S_ij corresponds to galaxy and velocity power and cross spectra P_ij in Eqs. (<ref>)-(<ref>)), π_c is a vector of the model parameters, and the trace runs over the observable indices ij. For notational brevity we have suppressed the (k,μ,z) dependence and i,j indices of C and dS in Eq. (<ref>). 
The diagonal noise matrix is 

    N_ij(k,μ,z) = diag[N_gg(z),N_v̂_rv̂_r(k,μ,z)],

and the galaxy shot noise n_g(z) in our redshift bins is taken to be <cit.>

    n_g=[5,2,0.6,0.15,0.03 ] × 10^-2 Mpc^-3.

For the model of the total observed CMB C_ℓ^TT,obs used to compute the velocity reconstruction noise, we follow Ref. <cit.>, including the moving lens effect, late-time and reionization kSZ contributions, the cosmic infrared background, the thermal SZ effect, radio sources, and weak lensing. We also include a white noise contribution

    N_ℓ^white = Δ_T^2  exp[ ℓ (ℓ+1) θ_FWHM^2 /8 ln(2)],

where Δ_T is the white noise RMS and θ_FWHM is the beam full-width half-maximum. We assume a CMB experiment with five frequency bins, [39,93,145,225,280]  GHz, with corresponding beam widths [5.1,2.2,1.4,1.0,0.9]' and noise RMS values [12.4,2,2,6.9,16.6]  μK'. We assume ℓ_max=6500, which is a reasonable estimate for the maximum CMB multipole for which foreground modeling is expected to be under control. Later, in Sec. <ref>, we further comment on how our results depend on the details of the CMB experiment.  


In our forecasts, the parameters we consider are the bias prefactors appearing in Eq. (<ref>), where in  this corresponds to marginalizing over  and, for some of our analysis, Δ. In our baseline forecasts, we also marginalize over a six parameter  ΛCDM model,

    π^ΛCDM = [A_s, n_s, H_0 [ km s^-1Mpc^-1], Ω_b h^2, Ω_c h^2, τ]

with corresponding fiducial values  

    π^ΛCDM_fid=  [2.15 × 10^-9, 0.9625, 0.67 , 0.022,0.12,0.066 ]

Later in Sec. <ref>, we  examine the impact that uncertainty in the ΛCDM parameters has on the forecasted constraints on primordial non-Gaussianity. 

To compute the Fisher information matrix in Eq. (<ref>), the integral over wavenumber  is an integral over the large scale modes of the galaxy and velocity spectra. This integral runs from k_min = π / V^1/3, determined by the box size of the snapshot geometry, to a k_max value jointly determined by (a) the smallest scale assumed accessible by the galaxy survey, and (b) the minimum of the k_S values over which we integrate to reconstruct the large scale velocity modes. In practice, we take this to be k_max =0.1 Mpc^-1 for our baseline forecasts.  

Following the approach taken in Ref. <cit.>, we consider two cases for the non-Gaussianity parameters  and Δ in our forecasts. Current data are consistent with vanishing , so in the first case  we marginalize over , assume a fiducial value = 0, and take the scaling exponent to be fixed to its fiducial value, , in the range [0,2]. This strategy is reasonable as long as the data comfortably permit =0. The reason we do not marginalize over Δ in this case is that there is no information about the shape of b_NG(k,z) if its fiducial amplitude is assumed to be vanishing. This is immediately clear if one considers how the signals P_ij change as Δ is varied, for which dP_ij/dΔ = 0 for the fiducial choice = 0. Hence, instead we ask how much can kSZ improve constraints around = 0 as the scaling exponent Δ is fixed at different values.

We also consider a second case in which we assume a non-vanishing fidicual value for  and also marginalize over the scaling exponent. This case allows us to consider how well galaxy and kSZ tomography can simultaneously measure the shape and amplitude of the scale-dependent galaxy bias if future data were to detect non-zero . This approach may not prove necessary in data analysis in the immediate future, but should hints of primordial non-Gaussianity be revealed in next-generation surveys, the analysis here elucidates some of the possibilities and challenges for simultaneous inference of the shape and amplitude of scale-dependent bias from primordial non-Gaussianity.





§ RESULTS 




 §.§ Baseline forecasts 

We begin by presenting results for our baseline forecasts described in the previous section. In Fig. <ref> we demonstrate how 1σ constraints on  improve with the inclusion of kSZ tomography in the case that f_NL^(Δ),  fid =0 and the scaling exponent is fixed to a value Δ∈ [0,2]. We first note that we find excellent agreement between our results and other work for local-type non-Gaussianity, =0 <cit.>.[To compare with Ref. <cit.>, we repeated the analysis only marginalizing over , b_1, and b_v, taking the other bias parameters and the ΛCDM parameters fixed to their fiducial values. After accounting for the factor of 3 normalization in Eq. <ref>, we find σ()^gg = 1.39, agreeing with Ref. <cit.> to within 3%. We find σ()^gg+gv+vv = 0.68, and Ref. <cit.> finds σ()^gg+gv+vv= 0.45. This difference is a consequence of our more conservative model for the CMB spectrum which yields a larger velocity reconstruction noise. When implementing a similar CMB spectrum as in Ref. <cit.>, we recover their value of σ() = 0.45.]  For local-type non-Guassianity, we find the inclusion of kSZ tomography yields σ() = 0.68, more than two times smaller than that achievable with galaxies alone, which would permit a roughly 1.5σ exclusion of the theoretical single-field target of ≲𝒪(1). 

Importantly, we see that kSZ tomography also has the ability to significantly improve constraints on the amplitude of primordial non-Gaussianity beyond Δ=0, compared to galaxies in isolation. In the range ≲ 1.3, the improvement is over a factor of 2 and is weakly dependent on . Importantly, the projected measurement uncertainty  σ() remains less than order one for ≲ 0.3, suggesting promising prospects for constraining primordial non-Gaussianity beyond the local type. For galaxies alone, the range for order one constraints on  is smaller, roughly ≲ 0.1. For larger scaling exponents, roughly ≳ 1.3, the improvement in σ() from kSZ tomography rapidly decreases, already dropping below ≈ 50% by =1.5. When the scaling exponent is assumed to be perfectly known, kSZ tomography offers very little additional constraining power to galaxy tomography as Δ→ 2. The basic reason for this is that kSZ tomography has poor signal to noise as k approaches smaller (quasilinear) scales, which is where larger scaling exponents cause greater modification to the galaxy bias. Near =2 the 1σ constraints on  are 𝒪(10^3), which is over an order of magnitude larger than current constraints on f_NL^equil. from Planck 2018 <cit.>. We can also compare with searches for quasi-single-field shapes in Planck data <cit.>, the most recent of which give much weaker constraints than galaxy plus kSZ tomography at Δ=0,   σ(f_NL^Δ=0) =12, but a much stronger constraint for Δ≳ 0.85 (or ≳ 0.65 when compared to galaxies alone), for example yielding σ(f_NL^Δ=3/2) =26. By way of further comparison, forecasted constraints from the CMB alone, with CMB-S4 and low-ℓ Planck, predict σ() = 1.8 and σ(f_NL^equil.) = 21.2 <cit.>. These constraints are about a factor of two improvement over Planck alone, and so to first approximation we may expect the quasi-single field constraints to improve similarly, in which case galaxy and kSZ tomography would outperform the CMB alone for Δ≳ 0.7 (or ≈ 0.5 without kSZ). In short, this suggests galaxy and kSZ tomography will play an important role in constraining non-Gaussianity beyond the local type, achieving the target of order unity constraints for quasi-single field shapes with scaling exponents close to the local type, roughly for ≲ 0.3, and extending the range of scaling exponents for which the probes of the late time power spectrum outperform the CMB alone. Conversely, neither galaxy nor velocity tomography will likely play a significant role in constraining larger scaling exponents in next generation surveys. 





We now turn to the second case, in which we consider how well the  amplitude and scaling exponent of the bispectrum can be inferred simultaneously if a non-zero  were detected. We follow a similar approach as in Ref. <cit.>, setting the fiducial value of  to the value consistent with a 5σ detection, f_NL^(Δ), fid = 5σ(). However, for simplicity, and to maintain consistency with present bounds from the CMB, we impose this relation using the σ() values from Ref. <cit.>. We find that using the fiducial  values from this procedure yields forecasted measurement errors from galaxy and kSZ tomography that would give a < 5σ detection in those observables internally when marginalizing over Δ. In other words, quasi-single-field amplitudes  presently consistent with the CMB to within 5σ are not large enough to separately yield a ≥ 5 σ detection with galaxy and kSZ tomography if the scaling exponent is simultaneously marginalized. A minor exception occurs at very small scaling exponents, ≲ 0.1, for which the forecasted measurement errors with galaxy and kSZ tomography are small enough to yield a ≲ 5.9 σ detection.

We show our results for this analysis in Figs. <ref> and Figs. <ref>, which show the forecast constraints on the amplitude and scaling exponent, respectively. Focusing on Fig. <ref>, we find that the 1σ measurement error on  will be significantly larger in this case than what we found in Fig. <ref> where the scaling exponent was fixed. This is most pronounced for local-type non-Gaussianity, Δ=0, where the measurement error is roughly a factor of 50 larger, while at the effect becomes less pronounced as Δ approaches larger values, however, the constraints there are too weak to yield a significant detection with galaxy and kSZ tomography anyway. Regardless, we find that the addition of kSZ tomography can significantly reduce the measurement error on  from galaxies alone — by over a factor of 2 — for a range of scaling exponents. As a concrete example, for = 0.2, galaxies alone would have sensitivity for a roughly 1.5σ hint, while with the addition of kSZ tomography, this would jump to a 4σ hint. Similarly,  for =0, galaxies alone would yield just over a 2σ hint, while the addition of kSZ tomography would raise this to nearly 6σ detection. This highlights that kSZ tomography can play a practical role in reinforcing measurements from the CMB of primordial non-Gaussianity beyond the local type, even where the constraining power from galaxy surveys alone is limited. 

Another promising picture emerges in Fig. <ref>, where we show the forecasted measurement error on the scaling exponent from galaxy and kSZ tomography. Under the assumptions we have made here, we find that kSZ tomography will significantly improve the ability to infer the shape of the scale-dependent bias originating from light fields. Once again, this is especially relevant for smaller scaling exponents, where kSZ reduces the measurement error by over a factor of 2; this will make 𝒪(10^-1) constraints on the scaling exponent possible up to ≈ 0.5, while such constraining power with galaxies alone is essentially only possible for local type non-Gaussianity.















 §.§ Sensitivity to model and survey details 


In this section we examine how robust the kSZ-driven improvement in constraints on primordial non-Gaussianity is to various biases, degeneracies, and survey details. We focus our analysis on the case where the scaling exponent is fixed and constraints are forecasted around f_NL^(Δ),fid = 0. We omit an extended discussion of b_rsd, but we note that we found fixing b_rsd had insignificant effect on the results; doing so improved σ() by at most 𝒪(10%) for galaxy tomography alone, and only 1-3% when kSZ tomography is included. The minimal impact of b_rsd is expected, given that it only enters the total galaxy bias with very mild scale-dependence via f(k), which is very distinct from the scaling in (k). Therefore, while kSZ tomography does help mitigate the degeneracy between b_rsd and , in practice uncertainty in b_rsd  has very little impact on constraints on primordial non-Gaussianity.








  §.§.§ Optical depth degeneracy 

In Fig. <ref>, we demonstrate the impact that the optical depth degeneracy, e.g. uncertainty in the velocity reconstruction bias parameter b_v, has on the results we have presented here. In the top panel, we compare the 1σ   constraints from the combination of galaxy and kSZ tomography when marginalizing over b_v to assuming a fixed value b_v =1. The bottom panel shows the fractional difference between these two curves, indicating that taking b_v to be perfectly known leads to constraints on  that are overoptimistic by less than ≈ 2%. Hence, the optical depth degeneracy does not significantly diminish the ability of kSZ tomography to improve constraints on primordial non-Gaussianity via measurement of scale-dependent bias.












  §.§.§ Linear galaxy bias 

In Fig. <ref> we  demonstrate the impact of uncertainty in the linear bias by considering the effect that various b_1 priors have on constraints on . For simplicity we have chosen to implement the same priors across the five redshift bins. For illustrative purposes, we show a 2% and a 0.1% prior; the 2% prior is roughly the precision needed to see significant degeneracy breaking in the feature at ≈ 1.4, while the 0.1% is a rough `upper limit' beyond which a stronger prior does not impact the results. In next-generation surveys, achieving external, redshift-independent b_1 priors of this magnitude is in all likelihood unrealistic (although potentially some progress can be made, for example, by  cross correlating lensing with the same galaxy sample). However, these simplifying choices allow us to easily demonstrate the  effect that uncertainty in the linear galaxy bias has on σ() over the range of scaling exponents. 


We find that uncertainty in b_1 significantly hinders constraints on  for scaling exponents ≈ 1.5, with the most significant effect occurring for slightly larger scaling exponents when including kSZ tomography.  Near this value of , the two priors on b_1 have the potential to reduce σ() from galaxies alone by 50% and over a factor of 3 respectively. KSZ tomography helps to reduce the sensitivity to uncertainty in b_1, in part breaking the degeneracy between the linear bias and the non-Gaussianity amplitude, and hence the priors yield a smaller reduction in σ() from the combination of galaxiy and kSZ tomography. Correspondingly, the improvement in σ() from including kSZ tomography is generally less as the b_1 prior gets stronger. 





Even so, setting a strong enough prior such that b_1 is effectively fixed does not push down the measurement error enough to make galaxy and kSZ tomography competitive with the CMB at large scaling exponents. As we pointed out in the previous section, the practical application of kSZ tomography is likely to be for smaller scaling exponents,  ≲ 0.7 - 0.85, and here the impact of uncertainty in b_1 is very small, as evidenced by minimal response to the strong priors for those scaling exponents. In this range, for galaxies alone, the stronger prior reduces the measurement error on  by at most ≈ 30%, while with the addition of kSZ tomography, this figure drops to roughly 5%. Overall, these results suggest that the practical application of kSZ tomography to probing non-Gaussianity beyond the local type is robust to uncertainty in standard linear galaxy bias.






  §.§.§ Gradient biases 

Our baseline forecasts have marginalized over the gradient biases b_k^2,b_k^4 which modify the galaxy bias in a scale dependent fashion, roughly on a characteristic scale k_* = 1/R_* ≈ 0.25 Mpc^-1 (depending on the precise value of b_k^2n). On one hand, this is a much smaller scale than the regime where we expect kSZ tomography to add appreciable constraining power. On the other hand, for larger values of the scaling exponent, the impact of (k) shifts from larger to smaller scales and becomes more blue-tilted, and so heuristically we may expect the degeneracy between the gradient biases and (k) to be greater for larger . We quantify the impact that uncertainty in the gradient biases has on constraints on  in Fig. <ref>, where we show forecasted 1σ constraints on  with and without b_k^2n priors. As we did with the linear bias, we take the simplifying approach of imposing the same prior for all redshift bins.[Achieving observational priors on small scale galaxy bias is very likely impractical in the near future. However, it is possible that theory priors, e.g. from analytic models or simulations, may provide some guidance <cit.>, and implicit in the gradient expansion is the assumption the b_k^2n are order one parameters <cit.>.] As expected, priors on the the small scale gradient biases make a more significant impact on σ() at larger scaling exponents, for which the effect of (k) is more heavily weighted toward smaller scales. For scaling exponents near =2, constraints on  from both galaxy and kSZ tomography improve significantly (by over an order of magnitude) for strong priors on the gradient biases, confirming significant degeneracy in that regime. As the lower panel of Fig. <ref> indicates, the inclusion of kSZ tomography reduces the sensitivity to uncertainty in the gradient bias over most of the range  ∈ [0,2], although this behavior is somewhat nontrivial. In short, the inclusion of kSZ tomography helps to mildly break the degeneracy between the gradient bias and the non-Gaussianity amplitude.  

From a practical point of view, the impact of uncertainty in the gradient bias is very similar to that of the linear bias. Even effectively fixing the gradient biases does not reduce the measurement error enough at large scaling exponents to bring galaxy and kSZ tomography into competition with the CMB, while at scaling exponents (≲ 0.85) for which galaxy and kSZ tomography are likely to outperform the CMB, uncertainty in b_k^2n is not very significant. Similar to the case with the linear bias, the addition of kSZ tomography helps to effectively eliminate what moderate sensitivity the galaxy surveys have to uncertainty in b_k^2n: fixing b_k^2n would reduce the measurement error by up to 30% for galaxies alone, but the inclusion of kSZ tomography reduces this figure to under 3%. In summary, these results suggests that the near term, practical application of kSZ tomography for improving constraints on primordial non-Gaussianity will be largely robust to  uncertainty in the gradient bias, or likely any biases that only become important on non-linear scales. 













 



  §.§.§ Cosmological parameters 


In our baseline forecasts we have marginalized over a six-parameter ΛCDM model with fiducial values given in Eq. (<ref>). Previous work has shown that, for the scaling exponent held fixed to Δ=0, fixing the cosmology does not lead to significantly overoptimistic results because the degeneracy between  and the six ΛCDM parameters is sufficiently weak in the matter power spectrum <cit.>. In Fig. <ref>  we demonstrate how a fixed cosmology affects constraints on . In agreement with previous work, we find that for fixed Δ=0, fixing the cosmology only underestimates σ()  from galaxies alone by about 15%, and the inclusion of kSZ tomography negates this effect. Hence in this case, the improvement in σ() gained from including kSZ tomography does not change very significantly when marginalizing over the six ΛCDM parameters. 

However, this conclusion changes if the scaling exponent deviates from Δ=0. Fixing the cosmology can underestimate σ() by up to a factor of 2, depending on the fiducial value of the scaling exponent, mostly due to degeneracy between  and n_s, Ω_b h^2, and Ω_c h^2. The reduction in σ() from fixing the ΛCDM parameters is much greater for galaxies alone. That is, kSZ tomography significantly reduces the impact that uncertainty in the ΛCDM parameters has on constraints on . For example, for ≲ 0.85, the measurement error from galaxies alone is up to 60% larger with π^ΛCDM marginalized, but less than 10% larger with the inclusion of kSZ tomography. 




For simplicity, and to compare with previous work, we have made the comparison with a fixed cosmology, but in practice the cosmological parameters will receive strong priors from, for example, the contemporaneous CMB experiment used in the kSZ velocity reconstruction. However, we find only 𝒪(10%) difference in our results between fixing the ΛCDM parameters and using Planck priors <cit.>, with no qualitative difference in the conclusions. Hence in Fig. <ref>, the gray dotted curve in the bottom panel is likely a more realistic representation of the improvement in σ() one can expect from kSZ tomography. Likewise, in the upper panel of Fig. <ref>, the dotted lines are more realistic predictions for the total 1σ constraints on . This highlights that precise measurement of the underlying cosmology can be important for extracting constraints on primordial non-Gaussianity with scale-dependent galaxy bias. The upshot is that for the scaling exponents of practical interest, kSZ tomography will enable a roughly 2-fold reduction in σ() over galaxies alone.






  §.§.§ Survey specifications 

Our forecasts with galaxy and kSZ tomography have assumed an S4-like CMB experiment and an LSST-like galaxy survey. Here we consider how our primary conclusions are affected by some of the specifications of the experimental setup. 


First, we consider the consequences of a galaxy survey with access to fewer large scale modes. This is important for two reasons: first, for the linear galaxy power spectrum, the largest wavelength modes are measured with the less accuracy than modes close to the baryon acoustic oscillation scale or the peak of the matter power spectrum; secondly, velocity tomography is projected to be a powerful probe on the largest scales where it benefits from the k^2 signal-to-noise scaling discussed above in Sec. <ref>. In Fig. <ref>, we show the impact of varying the smallest scale accessible to the survey, k_min, on σ(), For simplicity, we vary k_min in the box formalism we work in by fixing n_min>1, where the nth mode is k_n = nπ/(V^1/3), and the baseline analysis sets k_min via n_min=1. More realistically, these largest scale modes will be present but with significant noise, so our analysis here provides a simple upper bound on increase in measurement error. As the bottom panel of Fig. <ref> indicates, the overall constraints on  suffer significantly from the absence of the largest scale modes. Moving from n_min=1 to n_min=2 can increase the measurement error σ() by up to  ≈ 55% (70%) for galaxy (kSZ and galaxy) tomography, and moving from n_min=2 to n_min=3 causes a moderately smaller, but still significant, further increase. 
KSZ tomography suffers proportionally more from the loss of the largest modes, which is also reflected in how
the kSZ-driven improvement in the constraints suffers from the loss in large scale information, as can be seen  in the middle panel of Fig. <ref>. For scaling exponents ≲ 1.4, the improvement in σ() from kSZ is reduced by roughly 15% for each removed mode. The impact is much less severe for larger values of the scaling exponent because the effect of (k) shifts to smaller scales, and so the loss of large scale information is less detrimental. 




The galaxy number density n_g is also crucial for obtaining strong constraints on  from scale dependent galaxy bias. For the case of , the improvement in σ() from kSZ tomography can be significantly better for larger galaxy number densities <cit.>. We do not carry out a similar test here, as the result for cases with scaling exponents close to Δ = 0 should be very similar to the result for . For  significantly larger scaling exponents, where the effects of (k) move to smaller scales, a larger n_g should not significantly improve constraints from kSZ tomography for two reasons. First, for an S4-like CMB experiment, the velocity reconstruction noise is approaching saturation with respect to n_g <cit.>. Second, on smaller scales (where there is more information about larger scaling exponents), the velocity reconstruction noise is totally dominated by the exponential photo-z error, rendering even significant increases in n_g irrelevant. Indeed, as Fig. <ref> already demonstrates, changes in signal-to-noise at large scales has a diminishing effect as we go to larger scaling exponents. 



KSZ tomography is also sensitive to the details of the CMB experiment, and in particular the velocity reconstruction benefits from a high precision CMB experiment that can accurately measure the temperature anisotropies at high multipole ℓ. In Fig. <ref> we examine the impact of increasing the maximum multipole in the CMB survey for our baseline S4+LSST setup. We also compare these results to a futuristic scenario with a CMB experiment with specifications matching that of CMB HD <cit.>, specifically with a ∼ 20^'' beam and white noise Δ_T = 0.1  μK^'. We find that the reduction in σ() from and S4-like survey with a more optimistic maximum multipole, ℓ_max=9000, yields a nearly 15% smaller σ() for ≲ 1.3. A futuristic, HD-like CMB experiment would offer significantly stronger constraints. For example, with ℓ_max =15000, the improvement over our baseline S4 results is ≳ 75%. At large , this improvement is not enough to bring kSZ and galaxy tomography into contention with the constraining power of the primary CMB itself, but constraints on , for example, would be much stronger, and such a setup would modestly increase the range of order unity constraints on  to ≲ 0.4. The additional constraining power comes from a reduction in the kSZ reconstruction noise that, on the largest scales, and at higher redshifts, is over an order of magnitude. Note that in this comparison, we have only changed the CMB specifications and not the details of the galaxy survey. This allows us to clearly isolate the impact that a higher precision CMB experiment has on constraining primordial non-Gaussianity with kSZ tomography. However, in practice, CMB HD would likely be accompanied by a contemporaneous galaxy survey, such as MegaMapper <cit.>, and such a futuristic galaxy survey with improved number density and/or lower (or no) photo-z error would benefit both galaxy and kSZ tomography, which may change the quantitative conclusions about how much improvement kSZ tomography offers in a futuristic setting. Our point here is simply to demonstrate that constraints on  from kSZ tomography can significantly benefit from a higher precision CMB experiment. 







§ DISCUSSION 


In this paper we have explored the possibility of using kSZ tomography to improve constraints on primordial non-Gaussianity beyond the local type. We have focused on a scenario where the light degrees of freedom present during inflation lead to a bispectrum scaling as k^Δ in the squeezed limit. This leaves a distinct imprint on the scale dependence of the galaxy bias, which can differ from the characteristic large scale k^-2 behavior associated with local-type primordial non-Gaussianity. 
The primary results of this paper, encapsulated in Figs. <ref>-<ref>, indicate that with next-generation CMB and galaxy surveys, kSZ tomography will significantly improve the measurement of the  scale-dependent galaxy bias on large scales for a range of scaling exponents Δ, thereby tightening constraints on primordial non-Gaussianity beyond the local type. 

The main conclusion from this work is that the practical application of kSZ tomography for probing primordial non-Gaussianity is not limited to constraining . Most importantly, our results show that kSZ tomography will significantly improve constraints around =0 compared to galaxy surveys alone, enabling late-time probes of the matter power spectrum to outperform the CMB for a wider range of scaling exponents beyond Δ=0. Similarly, kSZ tomography will extend the range of scaling exponent for which the thoeretical target of order unity constraints on  are achievable, opening a wider window into the physics of the early Universe. Furthermore, we have demonstrated that if a non-vanishing  were to be detected, kSZ tomography will improve the synergy between the CMB and late-time probes of primordial non-Gaussianity, enhancing the detection significance of  and significantly reducing the measurement error on the scaling exponent Δ.  

In our analysis we have explored how improved constraints on primordial non-Gaussianity from kSZ tomography can quantitatively change in response to a range of potential biases, degeneracies, experimental parameters, and modelling details. However, we have demonstrated that our primary conclusions are robust to these various uncertainties. The optical depth degeneracy, which can potentially hinder cosmological inference with kSZ tomography, is essentially a non-factor in the context of the present work. This is a simple consequence of two factors. First, the key measurement is that of the large-scale galaxy bias, and second, combining galaxy and kSZ tomography is effectively a multi-tracer approach to measuring the total matter power spectrum. Combined, these mean that the main role of kSZ tomography in this context is to facilitate sample variance cancellation, circumventing cosmic variance limitations that would otherwise hinder the galaxy or velocity power spectrum in isolation. The combination of galaxy and kSZ tomography, therefore, enables the unambiguous measurement of the galaxy bias on the largest cosmological scales. In a similar vein, it was shown in Ref. <cit.> that, for an LSST-like survey, the constraints on  may significantly improve with explicitly multi-tracer galaxy tomography  (for beyond local-type primordial non-Gaussianity), while in Ref. <cit.> it was demonstrated that the additional constraining power for  offered by kSZ tomography depends on the multi-tracer nature of the galaxy survey. Hence, one important future direction is to explore the role kSZ tomography would play in constraining  if the galaxy surveys benefit from some internal sample variance cancellation.

We have also examined how uncertainty in the details of galaxy bias may impact constraints on primordial non-Gaussianity in this scenario. In particular, we have shown that uncertainty in both the standard linear galaxy bias b_1, as well as in the small scale biasing details (here studied via gradient biases b_k^2n) may significantly worsen constraints on , but that kSZ tomography generally helps to mitigate this effect. Furthermore, the degeneracies between the non-Gaussianity amplitude and the other contributions to galaxy bias are weakest for smaller scaling exponents, for which kSZ tomography holds the most promise. Therefore, in practice, we anticipate the uncertainty in the model for the galaxy bias will play little role where kSZ (and galaxy) tomography has any practical application. 

Beyond the details of biasing, our results show that the inclusion of kSZ tomography can significantly reduce the sensitivity of the measurement error on   to uncertainties in the ΛCDM parameters. This reduction is significant for non-Gaussianity beyond the local limit, particularly for scaling exponents Δ≈ 1.  As a corollary, this means marginalizing over the cosmological model mildly overestimates the improvement in constraining power from kSZ tomography compared to galaxies in isolation. In real analysis, additional information about the cosmological model will be furnished by the CMB. In that case, a good rule of thumb is that, for scaling exponents for which the late-time power spectrum probes compete with the sensitivity of the CMB, kSZ tomography will reduce σ() from galaxies alone by about a factor of 2. These results indicate that the assistance granted by kSZ tomography in this context is not only to facilitate sample variance cancellation, but also to better pin down the cosmology that controls the underlying matter power spectrum, and that these cosmological parameters do introduce non-negligible degeneracies that diminish  constraints on the amplitude of primordial non-Gaussianity. This also suggests that future studies should examine, for example, how extended cosmological models, e.g. dynamical dark energy, impact constraints on primordial non-Gaussianity from measurements of the galaxy and velocity fields.

While we have focused on galaxy surveys and kSZ tomography, measurements of the CMB anisotropies place independent constraints on primordial non-Gaussianity. Although quasi-single-field shapes with scaling exponents Δ≳ 0.85, will very likely remain better probed by the CMB itself, our results reinforce findings that galaxy and kSZ tomography will play an important role in providing competitive, and possibly stronger, constraints on non-Gaussianity of the local type and beyond.  Fully characterizing how constraints from LSS will faithfully compare with future CMB analyses of quasi-single-field scenarios requires a more dedicated study, which we leave for future work.


While we have focused on probing primordial non-Gaussianity, the present work also highlights the broader capability of kSZ tomography as a test of any physics imprinted in scale-dependent bias on the largest cosmological scales. For example, simulations suggest that scale-dependent bias on large scales is present even in standard cosmological settings without primordial non-Gaussianity <cit.>.  
Probing primordial non-Gaussianity, of local type and beyond, is one demonstration of the myriad new opportunities for cosmological inference that will be made possible by the greatly improved small scale sensitivity of next generation surveys. These surveys will open new avenues for probes of fundamental physics beyond the cosmic variance limit and robust to various astrophysical and cosmological uncertainties.

























§ ACKNOWLEDGEMENTS

We are grateful to Gil Holder, Selim Hotinli, and Srinivasan Raghunathan for useful conversations. The authors are supported in part by the
United States Department of Energy, DE-SC0015655.






 Paper Title: Quasilocal Newtonian limit of general relativity and galactic dynamics.


apsrev4-1



marco.galoppo@pg.canterbury.ac.nz
School of Physical & Chemical Sciences, University of Canterbury, 
 Private Bag 4800, Christchurch 8140, New Zealand


federico.re@unimib.it
Dipartimento di Fisica Giuseppe Occhialini, Università di Milano Bicocca,
 Piazza dell'Ateneo Nuovo 1, 20126, Milano, Italy
 &
INFN, sezione di Milano, Via Celoria 16, 20133, Milano, Italy


david.wiltshire@canterbury.ac.nz
School of Physical & Chemical Sciences, University of Canterbury, 
 Private Bag 4800, Christchurch 8140, New Zealand

§ ABSTRACT 
A new Newtonian limit of general relativity is established for stationary axisymmetric gravitationally bound differentially rotating matter distributions with internal pressure. The self-consistent coupling of quasilocal gravitational energy and angular momentum leads to a modified Poisson equation. The coupled equations of motion of the effective fluid elements are also modified, with quasilocal angular momentum and frame–dragging leading to novel dynamics. The solutions of the full system reproduce the phenonomenology of collisionless dark matter for disc galaxies, offering an explanation for their observed rotation curves. Halos of abundant cold dark matter particles are not required. 


Quasilocal Newtonian limit of general relativity and galactic dynamics
    David L. Wiltshire
    August 5, 2024
======================================================================



The Newtonian limit of general relativity is crucially important to both fundamental physics and astrophysical phenomenology. In cosmology two key features consistent with observation are: (i) far from any localised sources the Universe is not empty; (ii) the Universe is not globally rotating on the largest scales. 

Conventionally, one invokes an asymptotic Minkowski space, with respect to which additive Newtonian potentials are defined, despite observation (i). Moreover, even though known rotating asymptotically flat solutions of Einstein's equations—the Kerr and Kerr-Newman geometries—possess global non-zero angular momenta at spatial infinity, it is implicitly assumed that superposition of the effectively quasilocal angular momenta of such pointlike sources can be reconciled with observation (ii).

In this Letter we show that such implicit assumptions are misguided, even at nonrelativistic orbital speeds. The conventional limit neglects an important coupling of the quasilocal energy and angular momentum defined by the regional time–averaged motion of matter sources. We set out a new, self-consistent quasilocal Newtonian limit of general relativity, which does not assume a Minkwoskian spacetime background a priori. It introduces novel, first-order features that fundamentally modify the dynamics considered. A direct consequence is the elimination of the need for abundant cold dark matter particles in the modelling of galactic dynamics.

The prescription of a fixed, global spacetime background is not a requirement of general relativity. Indeed, the nonlinearity of Einstein's field equations (EFE), which arises from the self–interaction of matter and geometry, results in dynamical, regional backgrounds in which spacetime itself carries its own energy and angular momentum <cit.>. Moreover, the highly nonlinear nature of general relativity gives rise to a noncommutativity of averaging and limiting procedures in spacetime <cit.>. This has fundamental consequences for the dynamics of non-pointlike sources of large spatial extent, such as disc galaxies <cit.>.

The crucial ingredient of the differentially rotating models <cit.>, and in particular of the exact solutions of Ref. <cit.>, is that essential nonlinearity is retained before any low energy limit is considered. No global background is assumed. This contrasts with: (a) approaches that investigate frame–dragging on asymptotically flat <cit.> or asymptotically Friedmann <cit.> backgrounds, necessitating dark matter halos; (b) dark matter alternatives which modify the laws of gravitation, e.g., modified Newtonian dynamics (MOND) <cit.>.

In this Letter we demonstrate how insights from the conventional Newtonian framework may nonetheless be embedded in a quasilocal framework with an effective fluid. Furthermore, Einstein was never fully satisfied by the extent to which the EFE embody Mach's principle, viz., “Local inertial frames are determined through the distributions of energy and momentum in the universe by some weighted average of the apparent motions” <cit.>. The quasilocal Newtonian limit exhibits such a suitable weighted average, which we apply to disc galaxies showing that the EFE may actually contain an understanding of inertia consistent with MOND phenomenology.

The spacetime metrics can be written in the generalized Lewis–Papapetrou–Weyl form <cit.>,

    
    s^2     =-c^2 e^2Φ(r, z)/c^2( t+A(r, z)  ϕ)^2 +e^-2Φ(r, z)/c^2[W(r,z)^2 ϕ^2 +e^2k(r, z)/c^2( r^2+ z^2)]  ,



The energy-momentum tensor takes the form

    T^μν=((r, z)+p(r,z)/c^2)   U^μU^ν  + p(r,z) g^μν,

where (r,z) is the local matter density, p(r,z) is the effective pressure, and each element of the fluid possesses a 4–velocity U^μ, given by

    U^μ∂_μ=(-H(r,z))^-1/2(∂_t +Ω(r,z)  ∂_ϕ)  .

Here dϕ/dt=Ω(r, z) uniquely defines the angular speed of rotation at any point, and H(r, z) is a normalization factor. Since U^μU_μ=-c^2, it follows that

    H=-e^2Φ/c^2(1+A  Ω)^2+e^-2Φ/c^2W^2   Ω^2/c^2  .

Relevant physical velocities have to be identified to correctly implement any low-velocity limit. In the present case, we define the kinetic and dragging velocities

    := r Ω  , 
        := r χ  ,

where χ := -g_tϕ/g_ϕϕ is the frame–dragging term. For systems in the low-energy regime—i.e., nonrelativistic relative local velocities v ≪ c, weak pseudo-Newtonian potential Φ v^2, nonrelativistic frame–dragging, and small pressure p  v^2—then to leading order  coincides with the special relativistic interpretation of the redshift, whilst  follows by analogy <cit.>.

Let us then write the EFE as

    R_μν = 8 π G/c^4(T_μν-1/2T g_μν)  ,

where T=T^μ_μ = 3p- c^2. With (<ref>)–(<ref>) this yields the partial differential equations (PDEs) <cit.>

   
    Φ^,a_,a + W^,aΦ_,a/W + c^4A^,aA_,a/2W^2e^4Φ/c^2 = 4 π G e^2(k-2Φ)/c^2[(+p/c^2)(1+AΩ)^2e^2Φ/c^2+c^-2W^2Ω^2e^-2Φ/c^2/-H+2p/c^2]   , 
       A^,a_,a - W^,aA_,a/W + 4/c^2Φ^,aA_,a = 16 π G/c^4 W^2Ω1+AΩ/H(+p/c^2)e^2(k-2Φ)/c^2  , 
       W^,a_,a = 16 π G/c^4p   , 
        W_,rr-W_,zz+2/c^2(k_,zW_,z-k_,rW_,r)+2W/c^4(Φ_,r^2 - Φ_,z^2) + c^2/2We^4Φ/c^2(A_,z^2 - A_,r^2) = 0   , 
       c^2/2We^4Φ/c^2A_,z A_,r-W_,rz-2W/c^4Φ_,rΦ_,z+1/c^2(k_,zW_,r-k_,rW_,z) = 0   , 
       1/c^2(Φ^,a_,a-k^,a_,a)-1/2WW^,a_,a + 1/c^2W^,aΦ_,a/W - 1/c^4Φ^,aΦ_,a+ c^2/4W^2e^4Φ/c^2(A_,z^2 + A_,r^2) = 4 π G/c^2e^2(k-2Φ)/c^2(-p/c^2)  ,
 

where a∈{r,z}. Eq. (<ref>) and (<ref>) are the tt and rzcomponents of (<ref>) respectively, (<ref>) and (<ref>) are obtained from the R_zz± R_rr equations, (<ref>) is found by taking the combination W^-1c^-2(g_ϕϕR_tt-2 g_tϕR_tϕ+g_ttR_ϕϕ), and finally (<ref>) follows from the combination 2c^-2(R_tϕ-A R_tt). 

The system of PDEs is completed by the perfect fluid elements' equations of motion, which give 

    
    Hp_,a/e^2(k-Φ)/c^2=[Φ_,a+(c^2 A_,a+2AΦ_,a)Ω+(c^2AA_,a+A^2Φ_,a)Ω^2]e^2Φ/c^2+Ω^2(W^2Φ_,a/c^2-WW_,a)e^-2Φ/c^2  ,


We now take an expansion in powers of v/c to implement the nonrelativistic limit, where v is any relevant local velocity. We apply the self-consistent ansatz 

    W(r,z) = r + 𝒪(v^4/c^4)   ,

to solve (<ref>) up to fourth order. From (<ref>), and the definition of χ, we find A = r  /c^2+𝒪(v^3/c^3). Thus, by (<ref>) it follows that 

    H = -1 + ^2/c^2 - 2/c/c + 2 Φ/c^2 + 𝒪(v^4/c^4) .

Hence, by direct substitution of (<ref>) into (<ref>)–(<ref>) we find

    ΔΦ + 1/2r^2||∇⃗Ł_D||^2 = 4 π G  + 𝒪(v^2/c^2)  , 
       Δ̂Ł_D = 0 + 𝒪(v^2/c^2)   , 
        k_,r = 1/4r(Ł_D,z^2-Ł_D,r^2)+ 𝒪(v^2/c^2)    , 
        k_,z = -1/2r(Ł_D,z Ł_D,r) + 𝒪(v^2/c^2)   ,

where Ł_D := r is the quasilocal angular momentum per unit of mass associated with spacetime rotation, Δ is the standard laplacian and Δ̂ := ∂^2_r -(1/r)∂_r + ∂^2_z, is the Grad–Shafranov laplacian <cit.>. The number of equations reduces by one, as (<ref>) is equivalent to (<ref>) to the order considered. Furthermore, we see that the fluid effective pressure does not enter the EFE as a source term at this order of approximation. 

Eq. (<ref>) is a general relativistic generalization of the Poisson equation, and coincides with it in the absence of spacetime rotation. The extra term in (<ref>) can be interpreted as the rotational energy associated with the quasilocal angular momentum of the averaged background spacetime. This term is absent in the conventional Newtonian approximation in which the background is taken to be Minkowski space a priori. Furthermore, when moved to the r.h.s. of (<ref>), it can be identified as an effective density, ρ𝒬=-||∇⃗Ł_D||^2/(8π G  r^2).  This represents a negative gravitational binding energy relative to the background <cit.>.

Even for nonrelativistic dragging velocities spacetime rotation gives first-order corrections to the field equations. For fixed , Eq. (<ref>) shows that the strength of the pseudo-Newtonian potential needed to sustain equilibrium decreases as the local rotational energy of the background geometry increases. Furthermore, the pseudo-Newtonian potential in (<ref>) is essential to accommodate nonrigid rotation of the perfect fluid source. Indeed, if we set Φ=0 then (<ref>)–(<ref>) are the same equations that characterize a system of rigidly rotating dust <cit.>. As such this choice is unphysical, requiring an identification of  purely with rotational energy.

At the same order, equations (<ref>) reduce to 

    -p_,r/=Φ_,r+Ω Ł_D,r - ^2/r + 𝒪(v^2/c^2)   , 
       -p_,z/=Φ_,z+ΩŁ_D,z+𝒪(v^2/c^2)  .

Hence, the effective pressure of the fluid still plays a crucial role in determining the physics of the system, as expected. In Eq. (<ref>) and (<ref>), we recognize the gradient of the pseudo-Newtonian potential and the centrifugal acceleration in the term ^2/r. The conventional Newtonian equations follow in the case of negligible dragging (≪). However, more generally Eqs. (<ref>), (<ref>) once again show the crucial role that the quasilocal angular momentum of the regional background plays in determining the dynamics. For fixed  and p, the magnitude of the pseudo-Newtonian potential needed to match the l.h.s. of (<ref>) and (<ref>) decreases linearly with the quasilocal angular momentum of the underlying averaged geometry.

The quasilocal Newtonian limit defined here is thus realized by Eqs. (<ref>)-(<ref>). As a first application of the quasilocal Newtonian limit, we derive the dragging velocity profile for a disc galaxy rotation curve supported exclusively by baryonic matter. The baryonic mass distribution in a disc galaxy can be written as =  +, where  and  are the densities of the bulge and the disc, respectively. 

We assume a spherical bulge component given by the Plummer density profile <cit.>

    (R) = 3^2/4π(R^2+^2)^5/2 ,

where R = √(r^2+z^2),  is the scale parameter of the bulge and  is the total bulge mass. The resulting circular velocity in the conventional Newtonian limit is

    (R) = √(G R^2(R^2+^2)^-3/2)  .


The baryonic disc matter distribution is usually taken as (r, z)=Σ(r)Z(z). The surface density of the disc is well approximated by the exponential profile

    Σ(r)=/2π^2e^-r/  ,

where  is the total disc mass and  is the scale length in the radial direction for the galactic disc. The distribution along z, with ∫ Z(z)dz=1, is assumed to be

    Z(z)=1/2e^-|z|/  ,

where  is the thickness scale length and ≪. When the thickness  tends to zero, we apply the conventional infinitely thin disc approximation, i.e., Z(z)=δ(z)<cit.>. The Newtonian circular velocity field in the thin disc approximation, , is given by <cit.>

    
    (r) = √(G/2^3r^2[I_0(r/2)K_0(r/2) -I_1(r/2)K_1(r/2)])  ,


where I_{0,1} and K_{0,1} are modified Bessel functions of the first and second kind.

To explain the observed rotation curves in the conventional Newtonian framework, a spherical cold dark matter (DM) halo must be included, so that the matter density is =  +, where  is typically given by the isothermal profile <cit.>

    (R)=ρDM0/1+(R/)^2  .

Here  is the halo scale parameter, and ρDM0 is the maximum DM density. The DM halo contribution to the rotation curve, , is then <cit.>

    (R)^2=4π G ^2 ρDM0(1- arctan(R/)/R/)   .

Thus the circular velocity of the matter in Newtonian modelling of the disc galaxy, , on the equatorial plane, is given by

    (r,0)=√((r,0)^2+(r)^2+(r,0)^2)  .

In Fig. <ref>, we consider a Milky Way-like galaxy with = 0.8 · 10^10M_⊙, = 0.8 kpc, = 8.1 · 10^10M_⊙, = 2.1 kpc, = 5.69 kpc and ρDM0= 1.88·10^-21 kg/m^3. We plot the predicted rotation curve (r,0), and the respective (r,0), (r) and (r,0). 


To apply the general relativistic solution, let us now assume that =, i.e., that the inferred Newtonian rotation curve corresponds to that of a disc galaxy, and that =. We will demonstrate that the quasilocal energy and angular momentum of the galaxy produce observations equivalent to those of a conventional cold dark matter halo.

We begin by writing the pseudo-Newtonian potential as Φ=-, so that  is given by the Newtonian solution for the chosen , while  is defined as the solution of the Poisson equation sourced purely by the quasilocal rotational energy of the spacetime 

    Δ=||∇⃗Ł_D||^2/2r^2  .


We now solve (<ref>), (<ref>), and (<ref>) for Ł_D by noting that in the thin disc approximation the pressure term is exactly the same for both the Newtonian and the general relativistic cases. Moreover, a similar ansatz can be applied to the spherical bulge, where dark matter is not required to explain the observed dynamics, i.e., it is well captured by purely Newtonian models with purely baryonic matter. Thus we can take p(r,z):=p N (r,z) for the whole system, where p N (r,z) is the expected effective Newtonian pressure. We then use Eqs. (<ref>),(<ref>) to deduce

    = ΩŁ_D,r -(^2-^2)/r   , 
        = ΩŁ_D,z ,

where := √(^2+^2) is the Newtonian contribution to the observed rotation curve from the baryonic mass.

Substituting (<ref>), (<ref>) in (<ref>) it follows that

    Δ =Ω_,rŁ_D,r +2ΩŁ_D,r/r +Ω_,zŁ_D,z -(^2-^2)_,r/r  ,

where we have also used the quasilocal Newtonian limit of the (tϕ) Einstein Eq. (<ref>). Eqs. (<ref>), (<ref>) then give a first-order nonlinear PDE for Ł_D

    ∇⃗Ł_D·[∇⃗Ł_D - 2∇⃗(r)] + 2r(^2-^2)_,r = 0  .

We evaluate Eq. (<ref>) on the galactic plane, z=0, where thanks to the plane symmetry we have Ł_D,z(r,0)=vK,z(r,0) = 0. Thus, the nonlinear PDE (<ref>) is reduced to an algebraic equation for Ł_D,r on the galactic plane, whose solutions are given by 

    Ł_D,r=(r)_,r±√([(r)_,r]^2 - 2r(^2-^2)_,r)  .


We choose the minus sign in (<ref>) so that it reduces to the expected Newtonian case for =. Furthermore, given the baryonic density distribution and the conventional Newtonian profile for (r,0), we can numerically solve (<ref>) for (r,0). In Fig. <ref> we show the resulting frame–dragging velocity profile generating the simulated galaxy rotation curve, . The frame–dragging speed reaches a maximum 42.2km/s at r=24.4kpc, well within our nonrelativistic approximation. Furthermore, as we would expect: (i)  decreases with increasing radius, tending to zero at infinity; and (ii)  is negligible in the bulge, showing the self-consistency of the pressure ansatz used in the calculations. 


The new quasilocal Newtonian limit applies to stationary axisymmetric matter distributions with geometry (<ref>). These symmetry assumptions mean that for a Milky Way-like disc galaxy the approximations here, along with the exact solutions <cit.> hold on time scales, 10^7 t10^9yr. Eqs. (<ref>)–(<ref>) will not apply during galaxy formation, and therefore do not explain how disc galaxies form in a cold dark matter free environment. While such complex problems must be left to future work, the precise definition of finite infinity <cit.> provides a mathematical framework for further exploring the quasilocal Newtonian limit, in settings from the early Universe through to highly dynamical astrophysical systems. The interplay of quasilocal gravitational energy and angular momentum must play a key crucial role.

In summary, we have shown that via a novel self-consistent quasilocal Newtonian limit, general relativity can explain the observed rotation curves of disc galaxies without the need for halos of cold dark matter particles. The phenonomenology of collisionless dark matter is reproduced by the regional gravitational energy and angular momentum of the average spacetime geometry.

This discovery has far-reaching consequences for cosmology, astrophysics and particle physics. Indeed, many interesting questions are now opened up. Within the given realization of the quasilocal Newtonian limit, disc galaxy stability, gravitational lensing and fits to observational datasets must be addressed to definitively test the new paradigm. Comparisons with MOND phenomenology potentially opens the prospect not only of placing MOND within the theoretical framework of general relativity, but also of providing insights into the development of this important new physical limit of Einstein's theory. Finally, in light of our results, the phenomenology of dark matter in the early universe, in the dynamics of galaxy clusters, and in other astrophysical environments should now be reconsidered and further investigated.

Acknowledgments  DLW is supported by Marsden Fund grant M1271 administered by the Royal Society of New Zealand, Te Apārangi. We thank Davide Astesiano, Luca Ciotti, Sergio Cacciatori, John Forbes, Vittorio Gorini, Christopher Harvey-Hawes, Frederic Hessman, Morag Hills, Emma Johnson, Zachary Lane, Ryan Ridden-Harper, Antonia Seifert, Chris Stevens, Shreyas Tiruvaskar and Michael Williams for useful discussions. 






 Paper Title: Tip of the Red Giant Branch Distances with JWST. II. I-band Measurements in a Sample of Hosts of 9 SN Ia Match HST Cepheids.






0000-0002-8623-1082]Siyang Li
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA

0000-0002-5259-2314]Gagandeep S. Anand
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA

0000-0002-6124-1196]Adam G. Riess
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA


Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA

0000-0001-9420-6525]Wenlong Yuan
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA

0000-0003-3889-7709]Louise Breuval
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA

0000-0002-1775-4859]Lucas M. Macri
NSF's NOIRLab, 950 N Cherry Ave, Tucson, AZ 85719, USA

0000-0002-4934-5849]Daniel Scolnic
Department of Physics, Duke University, Durham, NC 27708, USA

0000-0002-1691-8217]Rachael Beaton
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA

0000-0001-8089-4419]Richard I. Anderson
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL),
 Observatoire de Sauverny, 1290 Versoix, Switzerland

§ ABSTRACT 

The Hubble Tension, a >5σ discrepancy between direct and indirect measurements of the Hubble constant (H_0), has persisted for a decade and motivated intense scrutiny of the paths used to infer H_0. Comparing independently-derived distances for a set of galaxies with different standard candles, such as the tip of the red giant branch (TRGB) and Cepheid variables, can test for systematics in the middle rung of the distance ladder. The I band is the preferred filter for measuring the TRGB due to constancy with color, a result of low sensitivity to population differences in age and metallicity supported by stellar models. We use James Webb Space Telescope (JWST) observations with the maser host NGC4258 as our geometric anchor to measure I-band (F090W vs F090W-F150W) TRGB distances to 7 hosts of 9 Type Ia supernovae (SNe Ia) within 27 Mpc: NGC1448, NGC1559, NGC2525, NGC3370, NGC3447, NGC5584, and NGC5643. We compare these with Hubble Space Telescope (HST) Cepheid-based relative distance moduli for the same galaxies and anchor. We find no evidence of a difference between their weighted means, 0.01 ± 0.04 (stat) ± 0.04 (sys) mag. We produce fourteen variants of the TRGB analysis, altering the smoothing level and color range used to measure the tips to explore their impact.  For some hosts, this changes the identification of the strongest peak, but this causes little change to the sample mean difference producing a full range of 0.01 to 0.03 mag, all consistent at 1σ with no difference. The result matches past comparisons of I-band TRGB and Cepheids when both use HST.  SNe and anchor samples observed with JWST are too small to yield a measure of H_0 that is competitive with the HST sample of 42 SNe Ia and 4 anchors; however, they already provide a vital systematic crosscheck to HST measurements of the distance ladder.






§ INTRODUCTION
 

The `Hubble Tension' refers to a 5-6 σ discrepancy between direct measurements and cosmological inference of the present expansion rate of the Universe, parameterized as the Hubble constant (H_0). Currently, the “gold standard” method to directly measure H_0 utilizes a three-rung ladder to obtain distances to galaxies in the Hubble flow. This method uses geometric distances derived from parallaxes (such as those from ), water masers <cit.>, and detached eclipsing binaries <cit.> to calibrate the luminosities of primary distance indicators such as Cepheid variables (first rung). This calibration is in turn used to measure distances to galaxies that host both Cepheids and Type Ia supernovae to calibrate the luminosities of the latter (second rung), which are then finally used to measure distances to galaxies in the Hubble flow to measure H_0 (third rung). The Supernovae, H_0, for the Equation of State (SH0ES) team, for instance, has used this method to measure H_0 = 73.17 ± 0.86 km s^−1 Mpc^−1 <cit.>. The cosmological inference uses the cosmic microwave background together with a cosmological model such as ΛCDM to infer the present expansion rate. <cit.> used this approach to derive H_0= 67.36 ± 0.54 km s^−1 Mpc^−1, a >5σ difference from the value obtained by <cit.>. Several independent teams and measurements using a wide range of distance indicators have also yielded measurements in tension with those from Planck. No precise direct measurement of H_0 has yielded a value that is lower than that from Planck (see compilations in ). This tension suggests a possible need to revise the standard ΛCDM model, though efforts continue to search for unidentified systematics in the measurements at any level[For an extensive list of studies that have studied systematics related to the tension, we refer the reader to <https://djbrout.github.io/SH0ESrefs.html>.]. 

Independent distance measurements to the same galaxies using multiple standard candles offers one of the best ways to ensure systematics are well accounted for and understood. The tip of the red giant branch (TRGB) refers to the maximum luminosity reached by first-ascent red giant stars before transitioning onto the horizontal branch, and has been used as a standard candle to measure distances <cit.>, and the Hubble constant <cit.>. The TRGB can thus be used as a means to crosscheck systematics of other standard candles, such as Cepheid variables.

The TRGB is commonly measured in the I band, or the Hubble Space Telescope (HST) F814W and James Webb Space Telescope (JWST) F090W equivalents, where the magnitude of the TRGB exhibits the smallest metallicity (and hence color) dependence <cit.>. While it is possible to measure the TRGB in the NIR to take advantage of brighter magnitudes, doing so requires careful calibration of the TRGB's color dependence <cit.> where it rises to ∼ one magnitude per magnitude of measured color, introducing additional systematics. In this study, we focus on measuring the TRGB in the JWST F090W filter, which is most similar to I and F814W used with HST.

The launch of JWST has opened up the possibility of performing extensive crosschecks of Cepheids, TRGB, and other standard candles in unprecedented resolution. In particular, the dual-module configuration of the JWST NIRCam instrument allows for at least three standard candles (Cepheids, TRGB, and Carbon stars) to be simultaneously observed with one set of observations - see for instance, JWST programs GO-1685 <cit.>, GO-1995 <cit.>, and GO-2875 <cit.>, among others.
In a previous study, <cit.> (Paper I), we measured the TRGB with JWST in the maser host NGC4258, which serves as one of four geometric calibrators of the distance ladder, and presented measurements in 2 SN Ia hosts (NGC1559 and NGC5584). The derived luminosity of the TRGB in JWST F090W in Paper 1 matched that independently determined by <cit.>, also in F090W, to ± 0.01 mag, indicating a reliable foundation for distance measurements with JWST in SN Ia hosts.
Here we study an expanded sample using JWST observations of 7 hosts of 9 SNe Ia from Cycle 1 program GO-1685 <cit.> and Cycle 2 GO-2785 <cit.> to measure TRGB-based distances to these galaxies and test their consistency with HST Cepheid-based distances to those same galaxies.



§ DATA
 

We retrieve JWST F090W and F150W observations of NGC1448, NGC1559, NGC5584, and NGC5643 from GO-1685 <cit.> and NGC2525, NGC3370, and NGC3447 from GO-2875 <cit.> using the Barbara A. Mikulski Archive for Space Telescopes (MAST)[<https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html>]. We list observation details in Table <ref> and show galaxy footprints in Fig. <ref>. For NGC2525, we found that the images corresponding to the fourth (and final) dither for each filter were noticeably blurred. Upon further investigation, we found that the guide star had wandered onto some bad pixels in the Fine Guidance Sensor. We thus excluded these dithers from our analysis and generated our own stage 3 image from only the first three dithers for use of mutual image alignment of the underlying stage 2 images with DOLPHOT. 






cccccccc

Galaxy     Program     Observation Number     Observation Date     Filters     Exposure Time [s]      Version
NGC1448     GO-1685     13     2023-08-02     F090W/F277W     418.7 x 4     1215 

NGC1448     GO-1685     13     2023-08-02     F150W/F277W     526.1 x 4     " 

NGC1448     GO-1685     14     2023-18-02     F090W/F277W     418.7 x 4     " 

NGC1448     GO-1685     14     2023-18-02     F150W/F277W     526.1 x 4     " 

NGC1559     GO-1685     1     2023-06-30     F090W/F277W     418.7 x 4     " 

NGC1559     GO-1685     1     2023-06-30     F150W/F277W     526.1 x 4     " 

NGC1559     GO-1685     2     2023-07-15     F090W/F277W     418.7 x 4     " 

NGC1559     GO-1685     2     2023-07-15     F150W/F277W     526.1 x 4     " 

NGC5584     GO-1685     9     2023-01-30     F090W/F277W     418.7 x 4     "  

NGC5584     GO-1685     9     2023-01-30     F150W/F277W     526.1 x 4     "  

NGC5584     GO-1685     10     2023-02-21     F090W/F277W     418.7 x 4     "  

NGC5584     GO-1685     10     2023-02-21     F150W/F277W     526.1 x 4     "  

NGC5643     GO-1685     11     2023-07-07     F090W/F277W     311.4 x 4     "  

NGC5643     GO-1685     11     2023-07-07     F150W/F277W     418.7 x 4     "  

NGC5643     GO-1685     12     2023-07-22     F090W/F277W     311.4 x 4     "  

NGC5643     GO-1685     12     2023-07-22     F150W/F277W     418.7 x 4     "  

NGC2525     GO-2875     1     2023-04-23     F090W/F277W     472.4 x 3     1225 

NGC2525     GO-2875     1     2023-04-23     F150W/F277W     526.1 x 3     " 

NGC3370     GO-2875     2     2023-06-28     F090W/F277W     472.4 x 4     1234 

NGC3370     GO-2875     2     2023-06-28     F150W/F277W     526.1 x 4     " 

NGC3447     GO-2875     12     2023-05-24     F090W/F277W     472.4 x 4     "  

NGC3447     GO-2875     12     2023-05-24     F150W/F277W     622.7 x 4     "  

Summary table for the JWST NIRCam observations of NGC1448, NGC1559, NGC2525, NGC3370, NGC3447, NGC5584, and NGC5643 used in this study. Columns from left to right are: galaxy name, program number, observation number, observation date, filters, exposure time per dither, and context () file version.





§ PHOTOMETRY
 

We follow the general procedures outlined by previous works from our team for photometric reductions <cit.>. In detail, we perform PSF photometry[The photometry for this study is available publicly upon publication in a Zenodo repository, DOI: 10.5281/zenodo.13131990.] using the DOLPHOT[<http://americano.dolphinsim.com/dolphot/>] software package <cit.>, together with the latest (February 2024) version of the JWST/NIRCam module <cit.>. We use the stage 3 F150W  files as reference frames and perform photometry directly on the stage 2  images.  We use the Vega-Vega system to remain consistent with earlier versions of DOLPHOT's JWST module, which gives a ∼0.04 mag offset (in F090W) from the Vega-Sirius system that is currently default in DOLPHOT NIRCam module. We use the “-etctime" option, which allows the package to use () for S/N calculations. We apply the following DOLPHOT quality cuts, similar to those from <cit.>:  < 0.5 (in both bands),  < 0.01 (in both bands), and  ≤ 2. In addition to these quality cuts, we also apply S/N cuts in both bands for each target. For NGC1448, NGC1559, NGC3447, NGC5584, and NGC5643, we adopt a minimum threshold of =5, whereas we choose a value of =3 for NGC2525 an NGC3370 (due to the decreased relative depth of their photometry). 

We also correct for foreground extinction using the Python package  <cit.> and galaxy coordinates from the NASA/IPAC Extragalactic Distance Database (NED)[<https://ned.ipac.caltech.edu/>]. We use the <cit.> reddening law with R_v = 3.1 and A_λ/A_V = 1.4156 and 0.6021 for F090W and F150W, respectively <cit.>. 



§ MEASUREMENT
 

Measuring the TRGB magnitude involves locating the discontinuity or inflection point of the luminosity function consisting of red giant and asymptotic giant branch stars. This can be done many ways; for instance, using a Sobel filter <cit.>, or model-based methods, with either least-squares fitters <cit.> or maximum-likelihood estimation <cit.>. In this study, we measure the TRGB using a Sobel filter using the measurement routine without spatial clipping (which we instead perform using elliptical annuli) from the Comparative Analysis of TRGBs (CATs) team <cit.>, which is publicly available on GitHub[<https://github.com/JiaxiWu1018/Unsupervised-TRGB>]. The CATs team had optimized their measurement parameters (such as smoothing value, color cuts, etc.) by minimizing field-to-field dispersion in the TRGB measurement across multiple fields. At the moment, there are a limited number of JWST fields in each host galaxy, which prevents us from performing a similar optimization. For this study, we adopt a smoothing value of 0.05 mag and unweighted, or `simple' weighting as named in the CATs algorithm (i.e. no further weighting of the output of the edge-detection measurement). We note that without the CATs optimizations in place, this methodology is equivalent to the TRGB measurements adopted by the Carnegie-Chicago Hubble Program <cit.>, but without the additional weighting of the Sobel filter output (which has been shown to cause biases in the outputs; ). We explore the effects of varying the level of smoothing of the luminosity function and the color range used to measure the luminosity function, 14 combinations in all, in the next section. TRGB uncertainties are estimated using bootstrap resampling of the luminosity function with 100 samples.  For NGC 5584, using the smaller smoothing value of s=0.05 mag produces a Sobel response that shows two peaks of nearly the same height. Following <cit.>, if there are two Sobel peaks that are of equivalent height (≤ 2%), we select the brighter peak.  

Before measuring the TRGB, we perform spatial selections to limit contamination from younger stellar populations in these star-forming galaxies. The NIRCam regions used to measure the TRGB are highlighted in Fig. <ref> - in all cases (except NGC3447, described below), only regions exterior to the B=25 mag/sq arcsec isophote (D_25; ) are selected <cit.>. For NGC1559 and NGC5584, the parallel NIRCam module contains a suitable sampling of stars, although only half of the parallel module is used (the far half of each contain too few stars to perform precise empirical PSF adjustments and aperture corrections). The same is nearly true for NGC5643, though we remove a small portion of each parallel module that is interior to D_25. In NGC1448, NGC2525 and NGC3370, a sizeable portion of the main module is outside of D_25, which we include to increase the sampling of stars and for NGC2525 and NGC3370, the parallel modules contain too few stars to perform photometry on. Lastly, NGC3447 is an actively interacting system, often described as two systems, N3447A and B. Here, we use two ellipses centered at RA, Dec = (10:53:23.998, +16:46:26.78) and (10:53:29:3934, +16:47:02.294), major axes of 140 and 50 arcsec, minor axes of 60 and 40 arcsec, and position angles of 80 and 115 degrees, respectively, following <cit.>. A color image of NGC3370 is shown in Fig. <ref>, along with the same D_25 selection region adopted for the spatial cuts. It can be seen that the outer regions of the image used for our analysis are of low stellar density, where stars are easily separated. 






The TRGB in F090W is expected to have minimal variation (<0.02 mag) with color over a range of 1.15 < F090W - F150W < 1.75 mag <cit.>. Here we adopt a color selection of 1.30 < F090W - F150W < 1.75 mag for our baseline result (see Fig. 2 in ), where the tighter range aims to limit the effects of a limited amount of supergiants which remain in our CMDs.









§ DISTANCES
 

We use the TRGB measurements shown in Fig. <ref> to measure distances to the host galaxies. We adopt a TRGB zero-point of M^F090W_TRGB = -4.372 ± 0.033 (stat) ± 0.045 (sys) mag from <cit.> - based on m_TRGB = 25.045 mag, which is the mean of the two simple Sobel measurements using a D > D_25 and outer spatial selections (either being within ± 0.005 of this mean). This calibration uses the geometric maser distance to NGC4258 <cit.>, to calibrate the TRGB in the JWST F090W system and is stable for smoothing values s=0.04 mag to s=0.12 mag due to the absence of weighting (see Fig. 6 in ). This absolute calibration is also highly consistent with the independently derived zero-point of M^F090W_TRGB = -4.36 ±  0.025 (stat) ± 0.043 (sys) mag from <cit.>, after adjusting their value from the Sirius-Vega system to the Vega-Vega zeropoint system (M. Newman, priv. communication).  We list and plot the distances to the host galaxies, compared to Cepheid-based distances from <cit.>. We compare to fit variant 10 from <cit.> for consistency in anchor selection; the distances produced by fit variant 10 are on average greater than those using all anchors (baseline) by ∼0.015 mag, consistent with their differences in H_0 of 72.5 km s^−1 Mpc^−1 for NGC4258 and 73.0 km s^−1 Mpc^−1 using all anchors.


We find a weighted mean difference between the TRGB (baseline results with 1.30 < F090W-F150W<1.75 mag and smoothing of s=0.05 mag) and HST Cepheid distances of 0.01 ± 0.04 (stat) ± 0.04 (sys) mag, which includes the systematic error due to measurement choice listed in Table <ref>, between the TRGB distance moduli measured here and Cepheid distance moduli from HST. We find no statistically significant difference between the two sets of distances measured between the same anchor and SN Ia hosts but with independent distance indicators.  We have not attempted to account for any further population-based differences intrinsic to the TRGB between the field of NGC 4258 and the mean of the SN Ia hosts, but believe these are limited due to the limited metallicity and age differences expected in F090W and our restricted color window <cit.>.


ccccccccccccc



Galaxy    
TRGB    
σ    
TRGB    
σ    
ΔTRGB    
μ_0(TRGB)    
σ    
μ_0(TRGB)    
σ    
μ_0(Cepheid)    
σ (Cepheid)    
σ (var)

    
s=0.10    
s=0.10    
s=0.05    
s=0.05    
(s=0.10-s=0.05)    
s=0.10    
s=0.10    
s=0.05    
s=0.05    
    
     

NGC1448     26.96     0.01     27.01     0.05     -0.05     31.33     0.05     31.38     0.07     31.30     0.05     0.02 

NGC1559     27.12     0.01     27.13     0.01     -0.01     31.49     0.05     31.50     0.05     31.50     0.07     0.01 

NGC2525     27.48     0.03     27.43     0.06     0.05        31.86     0.07     31.81     0.09     32.06     0.11     0.04 

NGC3370     27.83     0.02     27.84     0.08     -0.01     32.20     0.06     32.21     0.09     32.13     0.06     0.02 

NGC3447     27.55     0.03     27.54     0.08     0.01        31.93     0.06     31.92     0.09     31.95     0.05     0.01 

NGC5584     27.58     0.06     27.43     0.10     0.15        31.95     0.08     31.8      0.11     31.77     0.06     0.07 

NGC5643     26.19     0.01     26.20     0.01     -0.01     30.56     0.13     30.57     0.13     30.55     0.06     0.01 

TRGB and Cepheid distances to NGC1448, NGC1559, NGC2525, NGC3370, NGC3447, NGC5584, and NGC5643. Cepheid distances are from <cit.>, fit variant 10. Distance errors do not include an 0.032 mag uncertainty from the maser distance for either Cepheids or TRGB.  TRGB distances include a (correlated across the host galaxies) measurement error for NGC4258 of 0.045 mag from <cit.>. We adopt half the foreground extinction correction, with 0.01 mag added in quadrature to account for potential halo extinction in the host galaxy, as the uncertainty on the foreground extinction correction (as done in ). We also add in quadrature a systematic uncertainty from measurement and peak choices with values and measurement variants listed in Table <ref> and Table <ref>, respectively. Columns from left to right: 1) Galaxy, 2) TRGB measured here with a smoothing value of 0.10 mag, 3) error on that TRGB measurement, 4) TRGB measured here with a smoothing value of 0.05 mag, 5) error on that TRGB measurement, 6) difference between the TRGBs measured with s=0.05 and s=0.10 mag, 7) TRGB distance modulus measured here using a smoothing value of 0.10 mag, 8) error on that TRGB distance modulus, 9) TRGB distance modulus measured here using a smoothing value of 0.05 mag, 10) error on that TRGB distance modulus, 11) Cepheid distance <cit.>, 12) error on the Cepheid distance, 13) dispersion of variants (measurement choices), may be considered as a systematic error on an individual distance measurement <ref>.










The choice of algorithm and selection parameters can impact final TRGB measurements. One compelling approach is to optimize the measurement parameters to minimize scatter in the field-to-field variations (see ), however, the limited number of JWST fields available to measure the TRGB within our host galaxies prevent us from performing such an optimization. To investigate the effects of how different parameter choices can impact our measurements, we explore a fourteen combinations of different luminosity function smoothing, color selection, and zero-point choice. We list the different measurement variants explored in this analysis, together with the weighted mean difference between the TRGB and HST Cepheid distances for each variant, in Table <ref>. We include a thirteenth variant using the zero-point from <cit.>, adjusted for the Vega-Vega photometric system which has differences of ∼0.04 mag in F090W and ∼0.02 mag in F150W. We also add a variant where we use the median of all TRGB measurements (three smoothings, four color ranges) to compare with Cepheids.  We add the standard deviation of the measured TRGBs across the smoothing and color range variants for each galaxy, excluding the variant using the zero-point from <cit.> and median TRGB, as a systematic error to our measurements, listed in Table <ref>. We find that for all variants, the mean differences between the TRGB and Cepheid distances across all galaxies for a given variant are consistent with zero to within their statistical uncertainties with a full range of 0.01 to 0.03 mag.  

In Fig. <ref>, the error in the mean μ_TRGB-μ_Cepheid of 0.03-0.04 mag is a statistical uncertainty.  There is also a systematic uncertainty in the mean TRGB and Cepheid comparison from the measurement uncertainty of the TRGB in NGC4258 of ± 0.04 mag <cit.> which pertains to all calibrated TRGB distances and likewise for Cepheids in NGC4258 of ± 0.02 mag, or ± 0.045 mag when they are combined.  This systematic term due to the measurement of NGC4258 for both methods is the dominant term in the comparison and may only be reduced in the future with the use of additional observations of NGC4258.  The maser distance uncertainty of ± 0.032 mag should not be considered in this distance comparison as it is common to both the Cepheid and TRGB distances.

We can also compare the distances to NGC1559 and NGC5584 from <cit.>, who used the same dataset as here. <cit.> found distances of μ_0 = 31.49 ± 0.07 mag (Δ = 0.01 mag) and 31.80 ± 0.08 mag (Δ = 0.00 mag) to NGC1559 and NGC5584, respectively, when comparing their results to ours from the same smoothing scale (for NGC1559, we now have combined both visits to obtain higher S/N photometry). The TRGB distance for NGC1559 is also in agreement with that measured with Miras in <cit.> of μ_0 = 31.41 ± 0.050 ± 0.060 mag. 

An unavoidable feature of the TRGB method of edge detection is a potential ambiguity when multiple Sobel peaks of similar height are present.  For NGC5584 there are two Sobel peaks separated by 0.2-0.3 mag. Because the peaks are similar in height, different measurement choices may cause changes in which peak is higher. To account for this possibility, we provide an additional individual distance uncertainty listed in Table <ref> estimated from the variation of individual results of all the variants in Table <ref>. This uncertainty should be added for citing a single distance estimate for each host.  A similar situation involving two peaks at similar heights, but at a smaller level, 0.05 mag, is seen for NGC1448 and in the reverse direction for NGC2525. 

The Extragalactic Distance Database  <cit.> and CCHP <cit.> teams have measured HST TRGB distances to NGC1448 and NGC5643, which can provide an interesting comparison with the JWST distances found here. These distances also reflect differences in TRGB measurement methodology; the EDD team uses a model-based approach <cit.>, while the CCHP team uses a Sobel filter applied on a Gaussian-windowed, Locally Weighted Scatterplot Smoothing (GLOESS) smoothed luminosity function <cit.>. We list this comparison in Table <ref>. For NGC1448, these teams independently found distance moduli of 31.38 mag (EDD) and 31.32 mag (CCHP), compared to 31.33 mag and 31.38 mag (different levels of smoothing) found here resulting in a standard deviation of 0.03 mag. For NGC5643, these team found 30.47 mag (EDD) and 30.48 mag (CCHP), compared to 30.57 mag (which agrees well with Cepheids) found here resulting in a standard deviation of 0.04 mag. 


ccc

Galaxy     Reference     μ_0(TRGB)
NGC 1448     <cit.>     31.38  

NGC 1448     <cit.>     31.32 

NGC 1448     Here     31.33 (s=0.05), 31.38 (s=0.10)   

NGC 5643     <cit.>     30.47  

NGC 5643     <cit.>     30.48  

NGC 5643     Here     30.57  

Summary table for distances to NGC 1448 and NGC 5643 from EDD <cit.>, CCHP <cit.>, and here.

 


ccccccc

Smoothing [mag]     Color Cuts [mag]     μ_TRGB - μ_Cepheid [mag]     σ [mag]     Calibration     χ^2     Weighted Scatter [mag]
0.05     1.15, 1.65     0.03     0.04    <cit.>     1.075     0.10 

0.05     1.15, 1.75     0.03     0.04    <cit.>     1.468     0.12 

0.05     1.30, 1.75     0.01     0.04    <cit.>     0.718     0.09 

0.07     1.15, 1.65     0.02     0.04    <cit.>     0.919     0.09 

0.07     1.15, 1.75     0.02     0.04    <cit.>     1.377     0.11 

0.07     1.30, 1.75     0.02     0.04    <cit.>     1.142     0.10 

0.10     1.15, 1.65     0.02     0.03    <cit.>     0.916     0.09 

0.10     1.15, 1.75     0.02     0.03    <cit.>     1.081     0.09 

0.10     1.30, 1.75     0.02     0.04    <cit.>     0.954     0.09 

0.10     1.30, 1.75     0.01     0.04    <cit.>     0.946     0.08 

0.15     1.15, 1.65     0.01     0.03    <cit.>     0.912     0.08 

0.15     1.15, 1.75     0.02     0.03    <cit.>     0.878     0.08 

0.15     1.30, 1.75     0.02     0.03    <cit.>     0.802     0.08 

All Smoothings     All Color Ranges     0.02     0.03    <cit.>    1.063     0.09 

Summary table for the weighted mean differences between the TRGB and Cepheid distances (TRGB - Cepheid) to NGC1448, NGC1559, NGC2525, NGC3370, NGC3447, NGC5584, and NGC5643 across various measurement variants. Cepheid distances are from <cit.>, fit variant 10. The error in the mean difference, TRGB-Cepheid, does not include a systematic uncertainty of 0.03 mag for the TRGB measurement error in NGC4258 from <cit.>. Measurements use `simple' smoothing (i.e. no weighting, see ) and do not include the measurement choice error listed in Table <ref> except for the last variant that uses the median TRGB across all smoothing and color range variants. Bolded values correspond to the baseline variant that uses s= 0.05 mag smoothing with a color range of 1.3 < F090W - F150W < 1.75 mag. For the variant that uses the <cit.>, we use the errors listed in their Tables 4 and 6 added in quadrature with the TRGB measurement and foreground extinction errors. Columns from left to right are: Smoothing value, color cuts, mean difference between TRGB and Cepheid distances, error on the mean difference, chi-squared, and weighted scatter.





§ DISCUSSION
 

We measure F090W TRGB-based distances to NGC1448, NGC1559, NGC2525, NGC3370, NGC3447, NGC5584, and NGC5643 using JWST NIRCam observations and test their consistency with HST Cepheid-based distances to those same galaxies. We find excellent agreement between the two independent sets of distance measurements, which were measured between the same hosts and anchor but using data taken with different telescopes and distance indicators.

Examining the Hubble Tension necessitates close scrutiny of the consistency and systematics of distance indicators. Based on the measurements made in this study, we find no evidence of a bias or systematic difference between the Cepheid and TRGB methods that would cause a difference translatable to H_0 that can solve the Hubble Tension. This finding is in agreement with <cit.>, who find, “the excellent agreement between the published Cepheid distances in Riess et al. (2022) and TRGB distances in Freedman et al. (2021), which in the mean, agree to 0.007 mag". In addition, Fig. <ref> shows a comparison between Cepheid and TRGB distances, both measured with HST, using TRGB and Cepheid distances from <cit.> and <cit.>, respectively. The TRGB and Cepheid distances in Fig. <ref> yield a mean difference of 0.00 ± 0.02 (stat) ± 0.04 (sys) mag, also consistent with no difference even without additional consideration of populations differences in TRGB measurements <cit.>. This comparison will benefit from more JWST TRGB measurements in the future (for instance, from GO-1995; ). As the Hubble Tension corresponds to 5log_10(73/67.5) ∼ 0.17 mag, the demonstrated consistency is meaningful and is inconsistent with providing evidence for a solution to the Hubble Tension at 3σ.






 We do not yet report a value of H_0 from the JWST-calibrated SN subsamples because they are too small and the anchors too few to be competitive with the HST sample of 42 SN Ia and 4 anchors.  The uncertainty in H_0 would be 2.5 times greater with σ∼ 2-2.5 km/s/Mpc vs. 0.9  km/s/Mpc for HST, which would trivially remove the Hubble Tension simply by inflating the error in H_0.  Rather, the greater power of these presently small JWST samples comes from a direct comparison of what they can both measure in common, the distances between the same anchor and SN hosts.  

It will be important to also scrutinize the difference in direct H_0 measurements that arise from different treatments of SNe Ia. <cit.> find that the addition of peculiar flow corrections and cross-calibration of datasets which are included in the Pantheon and Pantheon+ analyses increase H_0 measurements each by 0.5 and 1.1 km/s/Mpc from the <cit.> analysis, bringing it closer in line with the value obtained from Cepheids.  An under recognized source of variations in determinations of H_0 is also the make-up of the SN Ia calibration sample which can produce fluctuations of ∼ 1-2 km/s/Mpc due to small sample statistics.

Crosschecks between standard candles can be made more comprehensive with additional, independent standard candles such as Miras and carbon stars. The carbon star method, also called the J-region Asymptotic Giant Branch (JAGB), originated in the 1980s <cit.> and has since been revived and further pioneered in the 2000s <cit.>. Although the JAGB method still requires further development and standardization (see discussions of asymmetric luminosity functions and metallicity effects in ), the great luminosities of carbon stars in the near-infrared allow the JAGB to reach farther than the I and J-band TRGB, potentially allowing for a more comprehensive crosscheck of Cepheid variables via galaxy distances.

The TRGB measurements made here can benefit from further analysis, such as a future contrast ratio calibration (see CATs, <cit.> or via modelling of the luminosity function <cit.> instead of edge-detection algorithms (Anand et al., in prep). However, even without these additions, the consistency of the TRGB-based distance measurements with those of Cepheid-based distances measurement do not show evidence of a bias or systematic offset in Cepheid distances that would resolve the Hubble Tension. Similar analyses in the future will also benefit from further studies investigating the effects of red giant diversity to improve the accuracy of TRGB distances <cit.>. Future observations of TRGB fields with HST (for instance, from GO-17520;  ), JWST, and the Roman Space Telescope will provide more opportunities to scrutinize the second rung of the distance ladder by providing a more extensive set of observations that can be used to compile a more comprehensive comparison between these distance indicators. Specifically, wide-field JWST observations within the anchor galaxy NGC 4258 will help reduce systematic uncertainties with regards to the absolute calibration of the TRGB, and deep JWST observations of galaxies with the highest levels of mismatch between HST Cepheid and HST TRGB distances (e.g. NGC 3021, NGC 4038/9; ) will help elucidate any individual discrepancies between the two techniques. Lastly, a fully independent distance ladder from the traditional Cepheid+SNe Ia route is also under construction, via the usage of the TRGB combined with Surface Brightness Fluctuations as the final rung <cit.>.



We are indebted to all of those who spent years and even decades bringing JWST to fruition. We thank Yukei Murakami for helpful conversations. SL is supported by the National Science Foundation Graduate Research Fellowship Program under grant number DGE2139757. GSA acknowledges financial support from JWST GO-1685 and JWST GO-2875. DS is supported by Department of Energy grant DE-SC0010007, the David and Lucile Packard Foundation, the Templeton Foundation and Sloan Foundation.  RLB is supported by the National Science Foundation through grant number AST-2108616. RIA is funded by the SNSF through an Eccellenza Professorial Fellowship, grant number PCEFP2_194638.

This research made use of the NASA Astrophysics Data System. This work is based on observations made with the NASA/ESA/CSA JWST. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with programs #1685  and #2875. The JWST data presented in this article were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via DOI 10.17909/556t-7522.

The Digitized Sky Surveys were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions.





JWST (NIRCam)

astropy <cit.>, 
SAOImage <cit.>, 
DOLPHOT <cit.>


aasjournal



 Paper Title: Properties of the diffuse gas component in filaments detected in the Dianoga cosmological simulations.




   

   


Faculty of Mathematics and Physics, University of Ljubljana, Jadranska cesta 19, 1000 Ljubljana, Slovenia
INAF – Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy
IFPU – Institute for Fundamental Physics of the Universe, Via Beirut 2, I-34014 Trieste, Italy
Department of Physics, Astronomy Section, University of Trieste, via G. B. Tiepolo 11, I-34131 Trieste, Italy
INFN, Instituto Nazionale di Fisica Nucleare, Via Valerio 2, I-34127, Trieste, Italy
ICSC - Italian Research Center on High Performance Computing, Big Data and Quantum Computing, via Magnanelli 2, 40033, Casalecchio di Reno, Italy
Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstr.1, 81679 München, Germany 
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany


   
   



 
  
  
        Cosmic filaments are observationally hard to detect. Hydrodynamical cosmological simulations are ideal laboratories where the evolution of the cosmic web can be studied. This allows for easier insight into the nature of the filaments.
      
        We investigate how the intrinsic properties of filaments are evolving in areas extracted from a larger cosmological simulation. We aim to identify significant trends in the properties of Warm-Hot Intergalactic Medium (WHIM) and suggest possible explanations.
      
        To study the filaments and their contents, we select a subset of regions from the Dianoga simulation. We analysed these regions that were simulated with different baryon physics, namely with and without the AGN feedback. We construct the cosmic web using the Sub-space Constrained Mean Shift (SCMS) algorithm and the Sequential Chain Algorithm for Resolving Filaments (SCARF). We examined the basic physical properties of filaments (length, shape, mass, radius) and analysed different gas phases (hot, WHIM and colder gas components) within those structures. The evolution of the global filament properties and the properties of the gas phases were studied in the redshift range 0 < z < 1.48.
      
        Within our simulations, the detected filaments have, on average, lengths below 9 Mpc. The filaments' shape correlates with their length; the longer they are, the more likely they are curved. We find that the scaling relation between mass M and length L of the filaments is well described by the power law M ∝ L^1.7. The radial density profile is widening with redshift, meaning that the radius of the filaments is getting larger over time. The fraction of gas mass in the WHIM phase does not depend on the model and is rising towards lower redshifts. However, the included baryon physics has a strong impact on the metallicity of gas in filaments, indicating that the AGN feedback impacts the metal content already at redshifts of z ∼ 2.
  
   

   

   Properties of the diffuse gas component in filaments detected in the Dianoga cosmological simulations
    Samo Ilc
          1
          Dunja Fabjan
          1,2,3
          Elena Rasia
          2,3
          Stefano Borgani
          4,2,3,6,5
          Klaus Dolag
          7,8
          
    Received  March 23, 2024; accepted July 27, 2024
================================================================================================================================================================================================





§ INTRODUCTION


    
    Clusters of galaxies lie at the intersection of the filamentary structure that forms the cosmic web <cit.>. While cluster properties have also been intensively studied in a broader cosmological context, the filamentary structure, i.e., both the large-scale cosmic filaments and bridges connecting pairs of clusters, is still poorly understood. The complex evolution of the cosmic web can be studied by describing the properties of the large-scale filaments and shorter bridges between galaxy clusters.

    In the last few decades, the search for large-scale filaments has taken a big leap forward, with a compilation of a number of filament catalogues (usually based on the galaxy distribution), e.g. within the Sloan Digital Sky Survey (SDSS) <cit.>, the two degree Field Galaxy Redshift Survey (2dF GRS) <cit.>, the Galaxy and Mass Assembly survey (GAMA) <cit.>, in the Cosmic Evolution Survey (COSMOS) field <cit.>, the Sydney-AAO Multi-Object IFS (SAMI) Galaxy Survey <cit.>, the VIMOS Public Extragalactic Redshift Survey (VIPERS) <cit.>.

    As for the gas component tracing the cosmic filaments, it is currently difficult to detect due to the low density and low emissivity that diffuse baryons have within filaments. The gas component in the filamentary structures, such as intercluster filaments and larger cosmic filaments, is studied by combining different data and using different techniques. For example, ROSAT X-ray data <cit.> and stacked Compton-y maps from Planck satellite <cit.> were used to obtain gas temperatures and overdensities for a large number of cosmic filaments at intermediate redshifts (detected previously using SDSS data) having lengths above 30 Mpc. A study of the network of filaments around the Coma cluster was performed by <cit.> combining SDSS data with the cosmic web detection algorithm DisPerSe <cit.>, obtaining information about cluster connectivity and a tentative detection of Sunyaev-Zeldovich (SZ) signal within filaments using Planck data. To analyse the signal of hot gas in extended structures <cit.> combined Compton-y maps from the Atacama Cosmology Telescope (ACT) stacked on redMaPPer cluster positions from the optical Dark Energy Survey (DES).
    
    Gas in bridges between clusters is commonly studied in X-rays <cit.> combining data from Chandra, XMM-Newton, Suzaku and recently from eRosita. Some bridges (between Abell 399 and Abell 401) were also studied using thermal SZ effect and performing a multiwavelength analysis <cit.>.
    
    Focusing on the intergalactic gas, according to numerical simulations, at present epoch ≃ 40-50% of the baryons hosted within filaments are in the form of a Warm-Hot Intergalactic Medium (WHIM) <cit.>. In the last decade, a number of observations have focused on the detection and characterization of the diffuse WHIM in the vicinity of clusters <cit.> and between pairs of clusters or cluster systems, e.g. A222/223 <cit.>, A3391/A3395 <cit.>, A2029/A2033 <cit.>, Abell 98 triple merging system <cit.>, A399/401 <cit.>. Recently, <cit.> analyzed the complex galaxy cluster system Abell 3391/95 using SRG/eROSITA data and found that it contains (among other structures) a warm-hot emission filament 15 Mpc long. <cit.> used data from the eRASS All-Sky X-ray survey combined with the SDSS optical filament catalogue, detecting WHIM by stacking around 8000 X-ray filaments. Mild positive detections of the WHIM in cosmic filaments were reported using the X-ray stacking analysis of cosmic filaments <cit.> and stacked SZ effect signal from intergalactic medium <cit.>.
 
    In simulations, the detection of filaments can be done with different methods. Usually, four main components are detected as parts of the cosmic web: knots, filaments, sheets, and voids. In the past years, a number of different methods were employed to trace the large-scale structure and its components. Some of the methods are able to detect all the objects of the cosmic web, e.g. NEXUS+ <cit.>, T-web <cit.> and V-web <cit.>. Similarly, DisPerSE <cit.> and Spineweb <cit.> detect all the cosmic web elements, but instead of knots, they map topological nodes. Some of the methods are able to obtain all but knots, e.g. MMF-2 <cit.>, while others focus only on filaments, e.g. SCMS <cit.>, MST <cit.> and Bisous <cit.>. Despite the different approaches, all of the methods give overall comparable results, with very similar structures detected. For a detailed comparison and more in-depth analysis of the main methods, see <cit.>.

    With a large number of available methods to detect the cosmic web, cosmological simulations are nowadays used to characterise the properties of filaments <cit.>, their evolution <cit.>, the properties of galaxies within filaments <cit.> and the connection between the cluster outskirts and filaments <cit.>. <cit.> studied clumps in and around galaxy clusters in non-radiative simulations. Since high-density clumps are easier to detect with X-ray telescopes, they expect them to be tracers of the filaments that compose the cosmic web. They find the filament temperature to (mildly) correlate with the mass of the main cluster. Besides studies of the properties of the cosmic web, simulations are also used to investigate the origin of the gas in the observed filamentary regions <cit.> and to predict future observational strategies <cit.>.

    In this work, we used a sample of resimulated volumes extracted from a larger cosmological simulation to investigate the intrinsic properties of the filaments within those regions. By selecting regions with different accretion histories and comparing simulations where different feedback effects are at work, we aim to study the mechanisms that impact the thermal and chemical properties of the intergalactic gas within the detected filaments, focusing on the WHIM phase.
    
    In Sect. <ref>, we describe the simulations used for this work. Sect. <ref> is devoted to the detailed description of the methods used to detect filaments with the Sub-space Constrained Mean Shift (SCMS) algorithm and how to determine their properties. In Sect. <ref>, we explore the physical properties of the filaments and their evolution in time. The last Sect. <ref> summarizes our conclusions.



§ DATA



    

 §.§ The simulation code

    
    
    In this work, we focused on a subset of Lagrangian regions that are centred around massive galaxy clusters and are part of the Dianoga set of simulations. These zoomed-in regions were extracted from a parent dark matter-only cosmological simulation (of h^-3 1 Gpc^3) originally described in <cit.> and later re-simulated with an improved resolution with the addition of the baryonic component. The set of simulated regions adopted for this work was performed with GADGET-3, which is a modern version of the Tree-PM Gadget code <cit.> and accounts for a new description of the smooth particle hydrodynamics that includes higher order interpolation kernels (Wendland C^4 kernel with 200 neighbours) and advanced formulations for artificial viscosity and thermal diffusion <cit.>. The full set of simulations was described initially in <cit.> and extensively studied in a number of later works <cit.>.
    
    
    
    Simulations presented here are based on a ΛCDM model, with cosmological parameters consistent with 7-year WMAP measurements <cit.>: Ω_m = 0.24 and Ω_b = 0.04 for the density parameters of matter and baryons, H_0 = h_0   100 km s^-1 Mpc^-1 and h_0=0.72 for the present day Hubble parameter, n_s = 0.96 for the primordial spectral index and σ_8 = 0.8 for the amplitude of the power spectrum of the density fluctuations. The selected Lagrangian regions are extracted and re-simulated with the zoomed-initial technique (ZIC) described in <cit.> by increasing the resolution in mass. In the high-resolution region, gravity is calculated with a Plummer-equivalent softening length of ε = 2 h^-1 kpc for stars and black holes and ε = 3.75 h^-1 kpc for DM and gas particles. The softening is fixed to comoving coordinates for all except DM particles, to which below z = 2 is given in physical units. The highest mass resolution of DM particles in this set of simulations is m_ DM = 8.47 × 10^8 h^-1 M_⊙, while the initial mass of a gas particle is m_ gas = 1.53 × 10^8 h^-1 M_⊙. 
    
    The set of Lagrangian regions was re-simulated with two different baryon physics models. The main difference between the two is the presence (or absence) of feedback from Active Galactic Nuclei (AGN). In particular:
    
        
  * CSF (Cooling and Star Formation) model - This model accounts for radiative cooling and subsequent formation of star particles. Star particles that describe the evolution of a stellar population include in the subgrid model also the feedback from supernovae (SN) and the metal enrichment from different stars (SN Ia, SN II and asymptotic giant branch AGB stars) <cit.>. Stellar evolution and metal enrichment models are described in <cit.> and allow to follow the distribution of fifteen chemical species (H, He, C, Ca, O, N, Ne, Mg, S, Si, Fe, Na, Al, Ar and Ni). These elements also contribute to the cooling. Models by <cit.> and <cit.> account for rates of metal-dependent radiative cooling and for the effects of the UV/X-ray background emission. Prescriptions for star formation are based on the work by <cit.>, where the velocity of galactic winds v_w originated by SN driven outflows, which are fixed at 350 km/s.
        
  * AGN (Active Galactic Nuclei) - These simulations rely on the same prescriptions described in the CSF case, and, in addition, they include the feedback effect from AGNs. The AGN feedback is based on a subgrid model of accretion on supermassive black holes (SMBH), where the mechanical and radiative outflows are accounted for as thermal feedback in both cases. Kinetic feedback from jets would be required to resolve sub-kpc scales and is not modelled explicitly <cit.>. The gas accretion rate (Eddington-limited) and the SMBH mass are both included in the efficiencies of the two outflows and provide a smooth transition between the radio and quasar mode. The model can account for hot and cold accretion, but in Dianoga simulations <cit.>, only cold gas accretion is considered. We refer to <cit.> for a detailed description of the model and its performance in general.
    
    
    

 §.§ The cluster sample

    Table <ref> contains the main characteristics of the high-density regions selected for the analysis. Three of the regions (namely D1, D6 and D22) contain a central cluster with mass M_ 200 > 1.7 · 10^15 M_⊙   h^-1, while the main clusters in the other two regions (D5 and D9) have central clusters with ∼ 10 times smaller masses. The latter regions were selected as "isolated regions", and their present-day volume is three times smaller than that of the largest and denser regions. We also checked the merging histories of the central clusters. In all but one case, the last major merger[Major merger is here defined as a merging event, in which the less massive progenitor is at least one-fourth of the more massive one.] happened at lookback time of about 6.5 Gyr or greater (corresponding to z > 0.7). The only exception is region D6, where the last major merger happened 3.77 Gyrs ago (at z ≃ 0.3).

    Inside each Lagrangian region, the code identifies the main halos using a Friend-of-Friend (FoF) algorithm, while substructures within haloes are detected with the Subfind algorithm <cit.>. The identification of self-bound substructures inside FoF halos was defined in its original form for DM-only simulations and extended to simulations with baryon physics by <cit.>. The sample of main haloes is obtained with a FoF algorithm with a linking length equal to 0.16 times the mean dark matter particle separation. The centre of each halo is defined by the most bound particle (the particle with the lowest gravitational potential). Radii and masses at different overdensities are then calculated around each cluster centre, while substructures are identified within the cluster virial radius. Filaments are detected within each region using the centres of substructures.



§ DETERMINING THE FILAMENTS AND THEIR PROPERTIES





 §.§ The Sub-space Constrained Mean Shift (SCMS) algorithm



    The Subspace Constrained Mean Shift (SCMS) algorithm, described by <cit.>, was modified and used as a method for filament detection by <cit.>. This method models filaments as ridges of the galaxy probability density function. The same algorithm has been successfully used for addressing different problems: reconstructing real cosmic filaments using SDSS <cit.>, studying galaxy-filament alignment <cit.>, filaments in the hydrodynamic simulation MassiveBlack-II[See https://sites.google.com/site/yenchicr/homeCosmic Web Reconstruction page for references and filament catalogues obtained in the MassiveBlack-II simulation.] <cit.>, weak lensing maps <cit.> and studying velocity around stellar filaments <cit.>. 
    
    SCMS is a multiple-step algorithm that determines filaments based on the density function p(x), where x is the spatial coordinate. The density function can be calculated with the standard kernel density estimator (KDE)
    
    p(x) = 1/nh^d∑_i=1^n K(||x-X_i||/h),


    where n is the number of tracers with coordinates X_i and ||x-X_i|| is the Euclidean distance between the i tracer and the location where KDE is evaluated; d is number of spatial dimensions (e.g. d=2 or d=3), h is the smoothing bandwidth, K is the smoothing (e.g. Gaussian) kernel. The details of the algorithm can be found in Appendix <ref> and in <cit.> while here below, we specify the input values that we chose for this work:

    
         
  * Tracer coordinates: as previously said, we aim at identifying the filaments from the galaxy probability function to follow an observational-like approach; thus, our tracers are the substructures identified with Subfind. Since we select them by applying a cut in total mass, they can be associated with galaxies (there will be a subtle difference in the selection function). Specifically, we consider all substructures with total masses > 10^11  h^-1M_⊙, which is a good compromise between the resolution of our simulations and the need for large statistics of tracers.
 
        
  * Smoothing bandwidth h: this parameter controls the smoothing of the distribution of tracers and thus depends on the local property of the density field. We utilize the expression provided by <cit.>:
        
    h = A_0 (1/d+21/n)^1/d+4σ_min,

    
        where n and d have the same meaning as before, σ_min is the minimal value of the three standard deviations associated with the spatial coordinate, and A_0 is the only parameter that does not depend on the local environment and needs to be calibrated according to the analyzed simulation. In our case, we chose A_0=0.5 after visually inspecting the overlap of the resulting skeleton and the tracer density field (see Appendix <ref>). The usual value of the smoothing bandwidth h is in the range 1-1.6 Mpc.

    
        
  * Threshold parameter τ: when defining a filament in a selected region, we can a priori avoid the computation in lowest-dense volumes. This translates to discarding all x locations whose density function is below a certain threshold,  p(x)<τ. Applying this selection avoids false filament detections, reduces overall noise, and speeds up the process. The value of the threshold parameter again depends on the local properties of the environment. Namely, it is defined as the difference between the local, p̂, and the mean density, p̅:
        
    τ = σ(p̂) ≡(∫_𝕂 (p̂(x) - p̅(𝕂))^2 dx )^1/2∼p̂ - p̅,

    
        where 𝕂 is the region to which we are applying the algorithm, and the local density is computed in 1 cubic cell with a size equal to the size of 𝕂 divided by 100 (see definition of ℳ in Appendix <ref>).  
   
    

    The SCMS algorithm works on a mesh of points that are shifted towards the density ridges defined by the tracers (see Appendix <ref> for a description). The final output of the algorithm is a collection of points. In Fig. <ref>, we show one example of the output of the SCMS algorithm in 2D space. The skeleton is extracted from a thin slice (∼ 1.2 Mpc thick) in region D6 at redshift z=0. The black dots represent the skeleton of the cosmic web. They are overimposed on the colour-coded KDE and the tracing substructures, which are marked with green dots. A thin cyan line separates the regions with p(x) > τ, where τ=9.78·10^-6. The blue circle represents the virial radius of the massive clusters in the region. Note that some skeleton points lie inside their virial radius. A large part of the skeleton follows the densest regions and the tracer distribution. We notice, though, that there are some filaments that, while in the dense region, do not appear to be close to tracers. Most of them are very short (some are even single points), and since they are the result of the noise, at later stages, they will be removed.

    



 §.§ Sequential Chain Algorithm for Resolving Filaments

    
    As shown and presented above, the outcome of the SCMS algorithm is a skeleton consisting of an ensemble of points that need to be automatically sorted and organized into filaments. For this purpose, we developed a Sequential Chain Algorithm for Resolving Filaments (SCARF).

    The SCARF algorithm is a two-step process: the initial step involves combining neighbouring points into a singular chain (see Sect. <ref>), while the second step breaks down the single chain into separate filaments (see Sect. <ref>). A chain is a sequential arrangement of points where each point is next to its closest neighbour, forming a connected series like a linked chain. This two-step approach allows SCARF to organize and sort points of the skeleton into filaments in an effective and automated manner.

    

  §.§.§ Constructing a single chain

    
    
    To facilitate the explanation, we start by presenting the main concepts used in the procedure, which are also displayed in Fig. <ref>. Considering the entire collection of points forming the skeleton of the cosmic web or, in other words, the outcome of SCMS, the first step will be to identify the closest neighbour to the point of reference. This neighbour point is always referred to as g. To identify the chain, we will use the letter c. The final result of SCARF will be one unique chain for all points in the skeleton. However, at the intermediate steps (one of them is shown in Fig. <ref>), more chains can be built, and these will be distinguished with a superscript indicating their numberings, e.g. c^0 and c^1 will be the first and the second chain that is created and so on. In each step, the initial and final points of the i-th chain are respectively called c^i_0 and c^i_-1. These points will change through time as the chain becomes longer. In some instances, the chain could be constituted of only one point and then c^i_0≡ c^i_-1.

    The procedure starts by selecting a random point, which by definition is associated with the chain c^0. Subsequently, the closest companion, the point g, is identified, and its distance with respect to c^0 is computed. The closest companion to g is searched, and if it is farther away than the distance between g and c^0, then g will be associated with the first chain c^0, which now constitutes of two points, and the search for another neighbour will continue. Otherwise, the point g will not be connected to c^0, but it will be considered the first point of a second chain, c^1.

    Once two separate chains are identified, the procedure focuses on the closest neighbour of c^1, called again g. The program then compares the distance between the new g and c^1 with the distance between c^1 and the end point of the chain c^0. If the latter is smaller than the former, then c^1 will be connected to c^0, and the two separate chains become one, named c^0. In the other case, g will be attached to c^1 only if it does not have any other closest neighbours. In this circumstance, indeed, it will be the starting point of a third chain c^2.

    Generalizing the concept and the notation, whenever a new neighbour g is found, the program computes its distance with its closest chain c^i as the minimum distance between g and the chain final points:
    
    d_g, c = min{||𝐠 - 𝐜^i_0||, ||𝐠 - 𝐜^i_-1||}.

    
    This distance will be compared with the distance between two previously identified subsequent chains, defined as the minimum distance between their final points:
    
    d_c, c = min{||𝐜^i_0 - 𝐜^i-1_0||, ||𝐜^i_0 - 𝐜^i-1_-1||, ||𝐜^i_-1 - 𝐜^i-1_0||, ||𝐜^i_-1 - 𝐜^i-1_-1||}.

    
    If d_c,c is smaller, then the two chains are fused into one. Otherwise, the program checks whether g has any other companion at a distance smaller than d_g,c. If it does not, then g is associated with c^i; otherwise, g will be the starting point of the new chain c^i+1.
    
    The procedure continues until all points from G are connected in one single chain.

    

    

  §.§.§ Resolving the filaments

    

    Once the entire skeleton is regrouped into a single chain, the second step of SCARF fragments the chain into individual filaments. The procedure starts by computing the vectors between two subsequent points: ξ_i = c_i+1 - c_i. Each vector has two main properties: length  ||ξ|| and direction, from which we can evaluate the change of direction of two neighbouring vectors as the angle
    
    θ_i = arccos(ξ_i·ξ_i-1).

    
    The criteria used to break the chain into filaments are that either ||ξ||>1 Mpc or θ_i<π/6. We select the first threshold to be of the order of the cluster radii, while the second limit is the minimum angle that still allows for smooth transitions of directions whenever the angle is below 30 degrees.

    In Fig. <ref>, we show the final results of the SCARF algorithm in 2D space where each filament is shown with a different colour. The points that lie inside the virial radius of the clusters with mass M>10^13 M_⊙ (in this case, two) are removed. A low-opacity line represents a chain that connects all the skeleton points detected by the SCSM algorithm. Filaments arising from the noise are marked with small, low-opacity points. Because of their shortness, they are easily removed. In this particular 2D case, SCARF detected 21 filaments. The choices limiting the vector length and direction change angle are applied to all 3D skeletons for all the regions analyzed.
 
    
    
    

 §.§ Mapping the filament

    
    
    A filament consists of N points in successive order f_0, f_1, ..., f_N. We now presume that the filament is shaped like a bent cylinder. We denote a particle's position as 𝐩. The longitudinal distance l for a particle in 𝐩 (e.g. gas or dark matter particle) is defined as the distance between the filament's starting point f_0 and the particle's projected point on the filament's spine and is calculated as 
    
    l = ∑_m=0^n ||f_m+1 - f_m|| +  (𝐩 - f_n) · (f_n+1 - f_n)/||f_n+1 - f_n||,


    where n is the sequence number of the point in the filament. Points f_n and f_n+1 are the closest filaments' points to the projected point 𝐩. The radial distance  r is the shortest distance between the particle and the spine of the filament and is calculated as
    
    r = ||(𝐩 - f_n) × (f_n+1 - f_n)||/||f_n+1 - f_n||.

    


 §.§ Length and radius of the filament



    We define the length of the filament as the sum of distances between neighbouring points:
    
    L = ∑_n=0^N ||f_n+1 - f_n||,

    where N is the total number of points in the filament's spine. This differs from the usual definition of length, where filaments are connecting the nodes. SCMS does not define nodes, and this is the reason why we also remove points that lie inside the virial radii of groups and clusters, as described in Sect. <ref>. Consequently, the filaments may be slightly shorter than expected compared to other studies. Nonetheless, this definition still offers a reliable estimate of the filaments' geometrical properties.

    To estimate the filaments' radius, we first map the gas, dark matter and star particles surrounding the filament, as discussed in Sect. <ref>, up to 5 Mpc from the filament spine. We obtained the particle distribution around the filament in l and r coordinate system. Our goal is to determine the filament's radius based on its overdensity, defined as
    
    δ = ρ/ρ_crit - 1,


    where ρ_ crit is the critical density[Critical density is computed using the cosmological parameters of the simulations as ρ_crit (z) = 3H_0^2/8π G(Ω_m(1+z)^3+Ω_Λ).] at the given redshift. This allows us to normalize the overdensity over all different regions, making them comparable. 

    We create an overdensity map that is described using the longitudinal and radial coordinates, l and r. The map is divided into cells with a height and width of approximately 0.1 Mpc. We map the particles at the same radial distance r from the spine in the cell at the correspondent longitudinal distance l. We applied the Gaussian filter with σ = 0.2 Mpc, equivalent to the size of 2 neighbouring cells, to smooth the map and damp sharp irregularities. We then categorize the cells as overdense if their overdensity is δ > 0 and underdense otherwise.

    Ideally, the filament would include most of the overdense cells close to the spine relative to the entire map. We can take two approaches: either determine a constant radius using the cells of the entire map or calculate a variable radius R(l) using cells associated only with that l. In practice, since R(l) can change rapidly, we applied a Gaussian filter with σ = 1 Mpc, allowing for a more gradual change in the filament's radius.

    The computation of both radii is shown in the following example. The main panel of Fig. <ref> illustrates an example of the colour-coded overdensity map around a single filament, with length L≈ 6.2 Mpc, in the D6 region (AGN simulation). The purple colour represents the underdense cells. The solid blue line shows the variable radius R(l). We also evaluated a constant radius, indicated by the dashed blue line, in the following way. In the right panel, for each radius r we computed the frequency of overdense cells for the upper area (red, above r) and lower area (green, below r). The constant radius is shown as the radius where the difference in the frequency of overdense cells is maximum (yellow line). In the lower panel, the calculated variable radius is represented by a solid line, while the dashed line illustrates the radius after applying the Gaussian filter. This example shows that the constant radius cuts off part of the overdense region, while the filament's variable radius traces better the filament's radial boundaries.

    However, issues can arise when no overdense cells are present at l, thus getting R(l)=0, or when objects, such as galaxies or small groups, are located outside the filament (red clumps in the overdensity map in Fig. <ref>). In that case, we can encounter a lone overdense cell (or a few) located far away from the filament's spine; because of that, the radius will extend beyond the actual filament. This can be mitigated by imposing a hard limit, where if the first few cells from the filament's spine are underdense, then the radius at that point is R(l) = 0. This overcomes, to some extent, the issue. In the cases where the average overdensity of the filament is below the threshold δ < 0, we choose to remove the filaments from further analysis. On average, we had to remove 2 additional filaments per snapshot.
    
    

    

 §.§ Physical properties of the filament

    

    To see if the filaments are the fair containers of cosmic baryons, we computed the gas, stellar and baryon depletion factors. They are calculated as in <cit.>:
    
    Y_x = M_x/M_tot/Ω_b/Ω_m,


    where x can be gas, stars or baryons (gas and stars combined), M_x is the total mass of the x component, M_tot total mass of the filament, and Ω_b and Ω_m the density parameters used in the simulations.
        
    We analysed the properties of the gas included in a larger region around the spine of the detected filaments. For the purpose of this work, we studied the gas components, dividing them into three phases based on temperature and hydrogen number density n_H, as done, for example, in <cit.>. The hot gas phase corresponds to gas having temperatures above T > 10^7 K and any number density. This phase is common in the potential wells of massive clusters, but as we will see, we can find this phase in filaments as well, especially near the filament ends (if they are near a cluster). For the WHIM gas phase, we used the range 10^5 K < T < 10^7 K for temperature and n_H > 10^-4(1+z) cm^-3 for the hydrogen number density. We expect most of the gas to be found in this phase. The remaining gas is treated separately and includes the colder gas associated with the ISM of galaxies and cold IGM, as well as the warm CGM created by shock heating and feedback processes near galaxies. In Figure <ref> we plotted the phase diagram of hydrogen number density n_H and temperature T for all gas particles in all regions combined at redshift z=0 in AGN and CSF simulations. Dashed lines separate the different gas phases that we have defined previously: hot, WHIM and other. Here, we can also note the difference between AGN and CSF simulations. The biggest difference is seen at the tail-end of the diagram, where high-density and low-temperature gas is located. This is the star-forming gas, which is more abundant in CSF simulations, while it is efficiently removed from the star-forming phase by AGN feedback.

    
    
    To show the different behaviour of each gas component inside the filamentary structures, we calculated the mass fraction for each gas phase as M_ x/M_gas, where x is the mass fraction of the desired gas phase and M_gas the total gas mass inside the filament.

    To explore the metallicity content of filaments, we obtained the iron metallicity Z_Fe for each gas phase in the filament as
    
    Z_Fe = ∑_i m_Fe, i/∑_i m_H, i,

    where the sum is done over all the particles contained in the region of the filament and where the solar abundance is Z_Fe, ⊙ = 1.77 · 10^-3 as in <cit.>. To be consistent with the observed temperatures, we also used the definition of spectroscopic-like temperature T_SL, as described in <cit.> <cit.>
    
    T_SL = ∑_i ρ_i  T_i^0.25 m_i/∑_i ρ_i T_i^-0.75  m_i,

    where T_i is the temperature of the gas particle, ρ_i its density and m_i its mass.
    
    

§ RESULTS

    

    First, we applied the SCMS algorithm on the selected set of the Dianoga regions, obtained with two different physics (AGN and CSF), as described in Sect. <ref>. To study the evolution of gas properties, we selected redshifts z = 0, 0.25, 0.51, 0.76, 1, 1.26 and 1.48. We removed all the points of the skeleton that fall inside the virial radii of the groups and clusters with masses above 10^13 M_⊙. After that, we ran the SCARF algorithm on the remaining skeleton. The final output is one catalogue of filaments per region, redshift and physical model.

    We calculated the physical length L of all the filaments as specified in Sect. <ref>. We only keep filaments with length L>2 Mpc for further analysis. Shorter filaments could be, at best, considered galaxy bridges and are entirely disregarded. With this, we also remove the false filaments, as discussed in Sect. <ref>. Almost all of the detected filaments are inside the high-resolution regions entirely, as they are either connecting two massive halos within this denser region or are situated in between filaments. A few of the filaments do not have a cluster or group at one of the ends. We still consider these filaments, as our main focus is the study of diffuse gas.
    


 §.§ Geometrical properties


    Before analysing the geometrical properties of filaments, we provide in Table <ref> the number of filaments with L>2 Mpc found per region, redshift and simulation type. We do expect that the number of filaments is related to the volume of the region. However, we see a steady increase in time of the filament absolute number in all regions, the numbers growing by ∼ 60% for the smallest ("isolated") regions, D5 and D9, and ∼ 40% for the other three largest regions. This is expected since the density contrast increases with time, especially in isolated regions <cit.>. We checked that, in general, filaments become longer with lower redshift in proper coordinates, though not considerably. There is no significant difference in the evolution of the number of filaments between AGN and CSF simulations. We do observe slight deviations between AGN and CSF simulations, but the difference is, in most cases, below 5%. Since the formation of the cosmic web is mostly determined by gravity, a similar number of filaments is expected as physical models used in AGN and CSF simulations mostly affect the gas component.
    
    
    
    Along with the filaments' length, we can also observe their shape (whether the filament is straight or curved). Therefore, we computed the distance D between the filament's first and last point and compared it with the filament length L.

    Since, in this case, we are not interested in environmental, model or redshift dependency, we combined all filaments with L≥2 Mpc from all regions and redshifts. In Fig. <ref>, the 2D histogram between D and L is plotted, with dashed lines representing the values of D/L and solid black line representing the median values of D at given L. On the side are the histograms for L and D. The ratio D/L represents the shape of the filament; the lower the value, the more curved the filament. As expected, since we are working with a low-volume simulation box, most of the filaments have lengths below 9 Mpc, with number counts falling with length. Similarly, this happens with distance D since this quantity is correlated with the length of the filament. According to multiple works <cit.>, filament populations can be divided into short and long filaments. Though there is no universally set limit, in all of the works, long filaments have a length of at least 20 Mpc. This means that in this work, we exclusively study short filaments. The median line (solid black line in Fig.<ref>) indicates that the longer the filament, the more likely it is to be curved. The median value is close to D/L=0.9 for short filaments. With larger lengths, the median value is moving towards the lines with lower values for D/L, indicating that the longer filaments are more likely to be curved since they do interact tidally with clusters located close to the filaments <cit.>. Even so, there are a number of longer filaments that are straight and smaller filaments that are curved.
    
    

    We explored how the shape of the filaments changes with redshifts. Shown in Fig. <ref> are median values plotted against redshift z for AFN and CSF simulations. The median values were calculated at each redshift combined for all the filaments in all regions. We can observe the filaments' shape is consistently rising with lower redshift, which means that the filaments become straighter with time. A similar conclusion was reached in <cit.>, where the filament's shape was similarly defined. As expected, the evolution to straighter filaments is similar for both AGN and CSF simulations. 

    
    


 §.§ Global properties



    We calculated the radial profile for each filament and then averaged all of them into a single radial profile at a given redshift, the same way as in <cit.>. The results are shown in Fig. <ref>, where we have plotted the dark matter radial profiles for all filaments from regions D1, D6 and D22 (left panel) and all filaments from regions D5 and D9 (right panel), separately. These radial profiles are in agreement with the results from other works <cit.>. For all of the regions, as the redshift decreases, we observe a decline in physical density. It is also clear that the filaments around the more isolated objects have a higher density close to their spine. Still, they fall quicker to values similar to the other three regions at larger radii. In the same figure, we also observe the widening of radial profiles with decreasing redshift. This indicates that, on average, the radius of filaments increases in time. This aligns with the findings in <cit.>, where they examined the radii of short, medium, and long filaments in the MilleniumTNG simulations. For short filaments (median value 7.56 Mpc), which most of the filaments in our study are, the radius decreased until redshift z=2 and then began to increase. Similar trends were observed for medium and long filaments, with the turning point occurring at redshift z=1. However, for long filaments, the radius remained relatively unchanged after this redshift. 
    
    

    The following computations are performed for all identified filaments across five regions at seven distinct redshifts. We begin by calculating for each filament the mean radius, denoted as ⟨ R ⟩, by taking the mean of the variable radii. We verified that, similarly to what is shown in Fig.<ref>, ⟨ R ⟩ increases with redshift. Subsequently, we gather all particles located within the filaments' variable radius and collect their masses to obtain the total mass of the filament. Given that we know the filament's mass and volume, we can compute its average density ρ and its average overdensity δ. 
    
    
    
    In Fig. <ref>, the top-left panel displays the distribution of total filament mass against its length for filaments at redshift z=0 in AGN simulations. The results for CSF simulations are similar. The graph includes median mass values within specific length ranges, revealing a consistent increase in mass. A power-law fit indicates a relation of M ∝ L^1.7. <cit.> determined a similar power-law relation, ∝ L^2.2, based on filaments in Millennium simulations detected with a different algorithm, NEXUS+. Notably, their simulations featured longer and, consequently, more massive filaments, a larger sample size, and exclusively dark matter-filled filaments. The existence of this correlation spanning a different range of lengths and masses implies that the longest filaments are not a fortunate connection of short filaments but rather clearly defined and possibly persistent structures. The bottom-left panel illustrates median lines for seven redshifts. Filaments consistently adhere to the same mass-length relation but exhibit decreasing mass with lower redshifts. 

    Our results might seem to go against the expectation that filaments become thinner as time passes, becoming more massive and diminishing radii with redshift. However, we should keep in mind that we are not studying large-scale filaments from cosmological boxes but relatively shorter filaments in denser regions populated with clusters. This aspect is likely influencing the evolution of filaments. Massive clusters will grow thanks to the mass flow from filaments, but we expect the filaments will not accrete mass from surroundings at the same rate. This would require a deeper investigation that is out of the scope of this paper. Moreover, as noted also in <cit.>, the correlation between the width and density of filaments depends on evolutionary processes. They found that thin filaments are usually found in underdense regions (areas where large-scale filaments form and persist), while thick ones are in overdense areas. The area we are studying is rather dense and populated with a large number of clusters that act as gravitational attractors.
    
    A similar plot is presented for mass versus mean radius in the middle panels of Fig. <ref>. Most of the mean radius are below 1 Mpc, which is a standard radius for the filaments, most recently determined by <cit.>. Still, the trend indicates that larger mean radii correspond to more massive filaments, which also correspond to longer filaments. Additionally, the radius of the filament appears to increase with redshift at the same mass, which is consistent with Fig. <ref>. The panels on the right depict mass versus overdensity. Initially, the mass appears constant at lower overdensity but rises steadily with overdensity until it becomes nearly constant again. Note that, as seen in the upper right panel, most of the filaments have overdensity larger than δ > 2.
 


 §.§ WHIM and hot gas phase properties

    To understand the broader evolution of the mass in filaments, we computed the fractions of the main matter components: dark matter, gas and stars within the boundary of the variable radius R(l). We noticed that the filaments located at the borders of the high-resolution regions have a very low number of gas particles. Therefore, we removed those with M_gas/M_tot < 0.05 from further analysis. We then calculated the depletion factor for gas, stars and baryons using Eq. <ref> for both AGN and CSF simulations. The results at redshift z=0 are shown in Table <ref>, where we can notice that filaments are not fair containers of the baryons in both simulations since Y_baryons is around ∼ 0.79 in AGN and ∼ 0.83 in CSF simulations, lower than unity. This is broadly consistent with the result by <cit.>, where they observed short filaments in IllustrisTNG simulations having Y_baryons∼ 0.9. They concluded that there is a baryon deficiency at r<0.7 Mpc, which coincides with the typical radius of the filaments we study. The gas in the filaments within AGN simulations has a comparable fraction with that of the CSF simulations. In turn, Y_ stars in CSF filaments is almost twice the value of that in AGN simulations, having both a large error. This is expected since CSF simulations are characterized by overcooling and missing the AGN feedback needed to regulate the star formation (e.g. AGN feedback), and overdense regions have higher production of stars. This increases the abundance of stars in the filaments in CSF simulations while simultaneously reducing their gas abundance. On the other hand, the reduced star formation in the AGN simulations makes more baryons available to be part of the diffuse medium, but such extra baryons are, in fact, displaced by the same AGN feedback. As a result, while Y_stars in AGN simulations decreases, there is almost a compensation of the two effects in determining the value of Y_gas. We can notice that at the end the overall baryon content of filaments is slightly reduced in AGN simulations.

    
    
    We checked that the mean mass fractions of baryons and dark matter do not change much in time (around 5% in AGN and % 10 in CSF simulations with respect to the present day value). The most notable rise of the mass fraction is for the stars in the CSF simulations, which is expected due to the high star-formation rate. Because of the relatively small mass fraction in stars, this does not visibly impact the gas and dark matter mass fraction. We then calculated the mass fractions of the three gas phases, defined in Sect. <ref> and calculated the median mass fraction for each gas phase and the 16^th and 84^th percentiles for AGN and CSF simulations at each redshift. The results are shown in Fig. <ref>. We immediately notice that the gas phases follow slightly different evolutions depending on the simulation. This is broadly consistent with what was found by <cit.>, where slightly different simulations were analysed in a broader context of the WHIM content inside smaller cosmological boxes. In AGN simulation, the WHIM gas phase consistently stands out as the most abundant across different redshifts, while the hot gas phase remains at very low levels. We recall that the filaments in this work are outside the virial radii of groups and clusters. Notably, the WHIM gas phase becomes even more abundant as redshift decreases. At redshift z = 0, its median mass fraction reaches almost 0.6-0.7 in the two physical models. In similar works, this value is between 0.5-0.7 <cit.>. Meanwhile, the presence of other gas phases gradually decreases. Though the abundance of the hot gas phase is barely noticeable, it does increase slightly at lower redshift. This indicates that the gas temperature in the filaments is slowly rising with lower redshift, likely because of the shock heating <cit.>. Due to that, the filaments would possibly be easier to detect with X-ray telescopes at lower redshifts. The two models show very similar trends with redshift. However, the fractions of the WHIM and the colder gas reach equality earlier for the AGN simulation.  

    
    

    

    One of the most striking differences between the two models can be seen in the metal content of the filaments. Furthermore, metallicity will be important in the observational detection of filaments. Therefore, we calculate the metallicity Z_Fe for each gas phase in every filament, as well as the median values at each redshift. The results are shown in Fig. <ref> for AGN and CSF simulations, respectively. 
    
    In AGN simulations, the hot gas phase exhibits the highest metallicity among the three considered phases, with the other gas phases ranking second and the WHIM displaying the lowest metallicity. Over time, the metallicity of the hot gas gradually decreases, while both the WHIM and other gas phases experience a slow but steady rise in metallicity. At redshift z=0, the metallicity of the hot gas phase falls below the value of other gas phases. These results align with those of <cit.>, who found that the decrease in metallicity at lower redshifts is attributed to increasing temperatures caused by shock heating. This results in the transition of the metal-poor WHIM phase (as well as the warm CGM phase) to the hot phase. Since hot gas is not abundant in the filaments, as shown in Fig. <ref>, the addition of metal-poor gas decreases the median metallicity.

    Conversely, the hot gas phase in CSF simulations shows much lower metallicity than in AGN simulations. It has barely larger values than WHIM, which also has lower metallicity than in AGN simulations. Other gas phases, though, exhibit similar values at all redshifts in the two physical models. Notably, across all three gas phases in CSF simulations, there is a collective increase in metallicity over time, marking a distinctive difference from the trend observed in AGN simulations, likely due to a higher star-formation rate, resulting in larger metal production. The overall metallicity values are higher in AGN simulations compared to CSF simulations, highlighting the effect of the AGN feedback model in simulations. This is consistent with the previous results shown in <cit.> where the AGN feedback, especially active at z=2, ejected metal-enriched gas from the small potential well of high-redshift galaxies. It is, therefore, expected that in AGN simulations, filaments retain gas with high metallicity, funnelling it toward clusters.

    

    Focusing on the AGN model, although the hot gas inside filaments is not a relevant part of the gas mass, the filaments seem to be enriched with high metallicity hot gas. Gas in the WHIM phase is instead reaching the highest fraction of gas mass at redshift z=0 while having an almost constant metallicity with time. At the redshift of ∼ 1.5, WHIM is already enriched. Its gas mass then grows by encompassing gas that gradually reaches the temperatures typical of this phase while being enriched as well. The same is not true for CSF simulations, where both the mass fraction of WHIM and its metallicity are growing in time. In this case, the WHIM phase is gradually enriched by the accumulation of gas with higher metallicities.

    To better explore the relation between the metal content and the temperature of the gas, we show in Fig. <ref> the metallicity Z_Fe versus spectroscopic-like temperature T_SL for the WHIM and the hot gas phase separately. The spectroscopic-like temperature was calculated according to Eq. <ref>, where only appropriate gas particles that fall into the designated temperature and density range are included, i.e. only WHIM particles are used for the calculation of the T_SL for WHIM phase.  
    
    On the upper left panel of Fig. <ref> (for AGN simulations), we see that the WHIM undergoes a swift rise in metallicity from low temperatures up to approximately T_SL≈ 3· 10^5 K. The metallicity stabilizes for larger temperatures, resulting in a relatively constant metallicity. Both trends can be attributed to shock heating. According to <cit.>, the only gas phase with metallicity lower than the WHIM is the diffuse IGM, which has a similar density to the WHIM but with temperatures below 10^5 K. As filaments undergo shock heating, their temperature increases, causing the metal-poor diffuse IGM to transition into the WHIM phase. This transition is most evident at lower temperatures but diminishes rapidly at slightly higher temperatures. Since shock heating does not add metals to the gas, the metallicity is expected to remain constant as the temperature rises. The metallicity of the WHIM is constant across all redshifts, consistent with results in Fig. <ref>. The hot gas phase in AGN simulations (upper right panel) displays variable median values at different redshifts. At higher redshifts, we observe a slight rise in metallicity with temperature and a gradual decline with respect to temperature at lower redshifts. Note that the majority of the filaments have a spectroscopic-like temperature below 2· 10^7 K. Similar to the discussion above, the metallicity of the hot gas is decreasing due to the shock heating. This is especially apparent at z=0, where we notice that the median metallicity is only slightly higher than that of WHIM and is constant at all temperatures.   
    
    On the lower panel of Fig. <ref> (CSF simulations), the metallicity of the WHIM gas phase shows a similar trend as in AGN simulations. The metallicity sharply rises until the temperature is T_SL≈ 2· 10^5 K. At higher temperatures, the metallicity rises very slowly with temperature. In time, the metallicity steadily rises at all temperatures. For the hot gas phase, the metallicity rises with higher temperatures but seemingly shows less dependence on redshift.
    
    For the WHIM phase, we found a significant diversity between the AGN and CSF simulations. In AGN simulations, the iron abundance within filaments remains unaffected by redshift, while CSF simulations show a slow rise of the values in time. This difference can be attributed to the role of AGN feedback, which was explored in the same simulations by <cit.>, accordingly to what was discussed for Fig. <ref> and <ref>. The AGN feedback is able to pre-enrich gas at z>2, and consequently, the metallicity level is already high in the AGN model for the WHIM phase.
    


§ CONCLUSIONS



    In this work, we studied the filaments in a set of high-resolution regions belonging to the Dianoga simulations that were centred around massive clusters. The set is composed of five different regions extracted from a parent cosmological simulation. Two of the regions were selected as "isolated", with a less massive cluster in their centre, while three of them are centred on massive clusters and densely populated with satellite clusters.
        
    We used The Sub-space Constrained Mean Shift (SCMS) algorithm <cit.> and the Sequential Chain Algorithm for Resolving Filaments (SCARF) algorithms to extract the single filaments. From there, we were able to quantify several geometrical and physical properties: length, shape, mass content for dark matter, gas and stars, mean radius, and mean density. We focused on the diffuse components (WHIM and hot gas) in the filaments, extracting their mass fractions and determining the mass-weighted metallicity and spectroscopic-like temperature.
    
    We summarize hereafter our main findings.
    
        
  * On average, the filaments inside the selected regions become longer in time. We found that longer filaments are more likely to be curved. With lower redshift, the filaments get straighter on average. 
        
  * The dark matter radial profiles in filaments have different central densities for regions that evolved as isolated with respect to the densest populated ones. Moreover, we noticed that the mean radius of filaments increases during evolution, consistent with the recent results <cit.>. This behaviour is related to the fact that our regions are regions around massive galaxy clusters instead of cosmological boxes covering a fair sample of environments.
        
  * We found that the mass of filaments is correlated with their length and their size. Filaments' mass increases with its length according to the power law M ∝ L^1.7, similarly to what was found by <cit.>. This indicates that the longest filaments are clearly defined and not a fortunate connection of shorter filaments. The mass also increases with a larger mean radius ⟨ R ⟩. Both correlations show a redshift dependence: at a fixed length or radius, filament masses decrease in time.
        
  * We explored the evolution of the gas found in hot, WHIM and colder gas phases inside the detected filaments. The predominant gas phase in filaments is the WHIM phase. Looking at the evolution of the WHIM mass fraction, it is steadily rising towards lower redshifts. We found that the physical models do not impact the amount of baryon mass in the WHIM phase. 
        
  * In general, we find that the metallicities of the WHIM and hot gas phases are, on average, higher in the AGN model than in the CSF one. 
        
  * The evolution of the Iron abundance in filaments depends on the physical model used in simulations. In the case of AGN simulations, the WHIM metallicity remains constant with redshift. This is connected with the AGN feedback model, which is able to displace metals at very high redshifts <cit.>. The already enriched gas is populating the WHIM gas phase, resulting in an almost constant metallicity. The opposite is true for CSF simulations, where enriched gas is not circulated by powerful outflows, increasing Iron abundance over time.  
        
  * In the case of the WHIM, metallicity is positively correlated to gas temperature, up to T_ sl∼ 3 · 10^5 K. At higher temperatures, the metallicity remains constant.
    

    
    In this work, we showed that in denser regions around massive galaxy clusters, a number of filaments can be traced. The different baryon processes included in the Dianoga simulation can be used to understand the impact of AGN feedback. The trends reported in this work, especially about the gas phases and their evolution, should be investigated in more detail in future studies. To better understand the evolution of filament properties, we plan to select single filaments that persist with redshift. These filaments, possibly connected on both ends to large virialized structures, will give us a deeper insight into the regions that connect cluster outskirts and the large-scale structure. Moreover, detecting filaments near galaxy clusters in simulations will give us the opportunity to compare them with different observational data (e.g. X-ray and/or SZ observations of cluster bridges) that are currently becoming available.

 

We thank the anonymous referee for the constructive comments that helped improving the paper overall, especially the presentation of the method and of the results. SI acknowledges the support from the Slovenian national research agency ARRS through grant MR-53649. SI and DF acknowledge financial support from the Slovenian Research Agency (research core funding no. P1-0188). SB is supported by: the Italian Research Center on High Performance Computing Big Data and Quantum Computing (ICSC), project funded by European Union - NextGenerationEU - and National Recovery and Resilience Plan (NRRP) - Mission 4 Component 2, within the activities of Spoke 3, Astrophysics and Cosmos Observations; by the PRIN 2022 PNRR project (202259YAF) "Space-based cosmology with Euclid: the role of High-Performance Computing". SB acknowledges partial financial support from the INFN Indark Grant. KD acknowledges support by the COMPLEX project from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679 as well as support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311.



aa






§ THE SCMS ALGORITHM STEPS



    Based on the density function p(x) defined by Eq. <ref>, the SCMS algorithm shifts points to the closest density ridge, converging towards them, forming a skeleton. Here, we summarize the following steps of the SCMS algorithm <cit.>:
    
    
        
  * Compute the kernel density estimator p̂(x) via Eq. <ref>.
        
  * Select a mesh ℳ of points that the SCMS will shift towards density ridges. To fully cover the box volume, we used the uniformly distributed grid, with a total of 10^6 points.
        
  * Remove x∈ℳ if p(x) < τ, where τ is a thresholding parameter. This is called thresholding and is used to remove points in low-density regions. This significantly reduces the clutter noise and number of falsely detected filaments.
        
  * For each remaining x, perform the SCMS:
        
            
  * Compute the Hessian matrix H(x) with
                
    H(x)=1/n∑_i=1^nc_i(μ_iμ_i^T-1/h^2𝕀),

            where 
                
    μ_i=x-X_i/h^2,      c_i=K(||x-X_i||/h).

            
  * Perform spectral decomposition on H(x). 
            
  * Obtain eigenvectors corresponding to the smallest d-1 eigenvalues, to form V(x)=(v_2(x),...,v_d(x)).
            
  * Update x ⟵ V(x)V(x)^Tm(x)+x until convergence, where 
                
    m(x) = ∑_i=1^nc_iX_i/∑_i=1^nc_i-x

    
                is the mean shift vector.
            
        
  * The final output is a collection of points that form the skeleton of the cosmic web.
    




§ SELECTING A_0



In this appendix, we show the calibration of the value of the free parameter A_0=0.5 and, subsequently, the value of smoothing bandwidth h. Smoothing bandwidth h controls how much we smooth the distribution of tracers when calculating p(x). We are following the guidelines as written in the Appendix of <cit.>.

We illustrate this by using a thin slice (around 1.2 Mpc thick) of the D6 region in AGN simulation at redshift z=0 (same slice as used in Fig. <ref>), centred around the largest halo in the region. The number of tracers (substructures) in this slice is n=351. In Fig. <ref>, we show the resulting skeletons of the SCMS for different values of free parameter A_0. In the top left, top right and bottom left panels are the skeleton for A_0 = 1, 0.5 and 0.25, respectively. These values correspond to a smoothing bandwidth of about 2.38 h^-1 Mpc, 1.19 h^-1Mpc and 0.59 h^-1Mpc, respectively. The contour lines encircle the areas where p(x) > τ. In the bottom right panel is the direct comparison between all three skeletons. The resulting skeletons show several distinctive differences.

Using a lower value for A_0 better follows the denser regions and more reliably detects filaments in lower-density areas. Still, using a value that is too low will detect more false filaments and possibly oversaturate points at the isolated tracers. We find that a good compromise is reached by adopting the value of A_0=0.5. Using values close to the A_0=0.5 does not change the skeleton significantly. 







 Paper Title: The mass-metallicity relation as a ruler for galaxy evolution: insights from the James Webb Space Telescope.











































                                       
                                       
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    > ∼< ∼.5ex.5ex∝∼.5ex.5ex.5exAlthæaz &NatureNew AstronomyPhys. Rev. APhys. Rev. BPhys. Rev. BPhys. Rev. DMNRASApJApJLApJSA&AA&A Supp.A&A Rev.ARA&AAJQJRASPhys. Rep.Ap&SSP&SSPubl. Astr. Soc. AustraliaPubl. Astr. Soc. Pac.J. Chem. Phys.Pub. Astron. Soc. JapanSpace Science ReviewsGeophysical Research LettersJournal of Geophysical ResearchNew Astronomy ReviewsJCAPFundamentals of Cosmic Physicsplots_png/,plots/t_ dA_100σ_ SFRσ_ZThe MZR as a ruler for galaxy evolution: insights from the JWSTPallottini et al.
        Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
         Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA
         INAF-Osservatorio di Astrofisica e Scienza dello Spazio, via Gobetti 93/3, I-40129, Bologna, Italy
        
        Galaxy evolution emerges from the balance between cosmic gas accretion, fueling star formation, and supernova feedback regulating metal enrichment of the interstellar medium. Hence, the relation between stellar mass (M_⋆) and gas metallicity (Z_g) is fundamental to understand the physics of galaxies. High-quality spectroscopic JWST data enable accurate measurements of both M_⋆ and Z_g up to redshift z≃ 10.
        
        Our aims are to understand (i) the nature of the observed mass-metallicity relation (MZR), (ii) its connection with the star formation rate (SFR), (iii) the role played by SFR stochasticity (flickering), and (iv) how it is regulated by stellar feedback.
        
        We compare the MZR obtained by the JADES, CEERS, and UNCOVER surveys, which comprise about 180 galaxies at z=≃ 3-10 with 10^6  M_⋆ 10^10, with ≃ 200 simulated galaxies in the same mass range from the serra high-resolution (≃ 20) suite of cosmological radiation-hydrodynamic simulations.
        
        To interpret the MZR, we develop a minimal, physically motivated, model for galaxy evolution that includes: cosmic accretion, possibly modulated with an amplitude  on 100 timescales; a time delay  between SFR and supernova feedback; SN-driven outflows with a varying mass loading factor ϵ_ SN, that is normalized to the FIRE simulations predictions for ϵ_ SN=1.
        
        Using our minimal model, we find the observed mean MZR is reproduced for relatively inefficient outflows (ϵ_ SN=1/4), in line with findings from JADES.
        
        Matching the observed MZR dispersion across the full stellar mass range requires a delay time =20, in addition to a significant modulation (= 1/3) of the accretion rate.
        
        Successful models are characterized by a relatively low flickering (≃ 0.2), corresponding to a metallicity dispersion of ≃ 0.2. Such values are close but slightly lower than predicted from serra (≃ 0.24, ≃ 0.3), clarifying why simulations show no clear trend for the MZR, and show some tension with the observations especially at M_⋆≃ 10^10.
        
        As the MZR is very sensitive to SFR stochasticity, models predicting high r.m.s. values (≃ 0.5) result in a chemical chaos (i.e. ≃ 1.4), virtually destroying the observed MZR. As a consequence, invoking a highly stochastic SFR (≃ 0.8) to explain the overabundance of bright, super-early galaxies leads to inconsistencies with the observed MZR. 
        The mass-metallicity relation as a ruler for galaxy evolution:
         insights from the James Webb Space Telescope
            
        A. Pallottini 0000-0002-7129-57611mailto:andrea.pallottini@sns.itandrea.pallottini@sns.it A. Ferrara 0000-0002-9400-73121 S. Gallerani 0000-0002-7200-82931 L. Sommovigo 0000-0002-2906-22002 S. Carniani 0000-0002-6719-380X1 L. Vallini 0000-0002-3258-36723 M. Kohandel 0000-0003-1041-78651 G. Venturi 0000-0001-8349-30551
            Received July 31, 2024; accepted XX XX, XXXX
        ====================================================================================================================================================================================================================================================================================================================================
        
        
        
        § INTRODUCTION
        
        
        
        
        The baryon cycle plays a pivotal role in regulating the galaxy formation and evolution process through cosmic time. As cosmic gas accretes in the dark matter halo potential well, it can cool and eventually form stars; massive stars explode as supernovae, enriching the surrounding interstellar medium with metals, that can potentially be ejected from the galaxy. Therefore key information regarding the star formation and enrichment history is encoded in the relation between stellar mass (M_⋆) and gas metallicity (Z_g), the so called mass-metallicity relation (MZR), making it effectively a standard ruler for galaxy evolution <cit.>.
        
        
        
        Over the past 20 years, it has become possible to measure the MZR from z≃ 0<cit.> up to the beginning of cosmic noon <cit.>, mostly by using the oxygen to hydrogen abundance ratio [O/H] as a proxy for Z_g.
        
        Locally, these observations have shown that Z_g increases with stellar mass up to M_⋆≃ 10^10 and beyond that point it flattens.
        
        
        This can be understood as due to the fact that low mass galaxies are prone to metal ejection due to their shallower potential well, while massive galaxies can retain most of the metals they have produced <cit.>. In this view, galaxy evolution is guided by a bathtub equilibrium <cit.> which determines the metallicity by balancing infall and outflows <cit.>.
        
        Further, observations have highlighted that Z_g and M_⋆ are tightly connected with the star formation rate (SFR) in the so-called fundamental mass-metallicity relation (FMR, ), which seems to be redshift-independent up to z≃ 3<cit.>; other factors as galaxy size <cit.> and molecular gas content <cit.> might also play some secondary role in connecting Z_g to M_⋆.
        
        
        The MZR evolution at z4 has been predicted via semi-analytical models <cit.> or cosmological simulations <cit.>. Most models predict almost no evolution or a very weak redshift dependence for z>3<cit.>; some others suggest a break down of the relation at z 9-12<cit.>.
        
        
        Thanks to the exquisite spectroscopic capabilities of JWST, we can accurately infer Z_g for galaxies well within the Epoch of Reionization (EoR, z>6). At present, surveys like JADES <cit.> and CEERS <cit.>, have provided metallicity measurements at z≃ 3-10<cit.> and z≃ 4-10<cit.>, for a combined sample of about 170 galaxies in the 5× 10^6  M_⋆ 10^10 stellar mass range.
        
        These JWST observations suggest that the MZR is already in place in the EoR, albeit downshifted by 0.5 dex with respect to the local one. The evolution in the z≃ 3-10 range is very mild, as relatively low mass (M_⋆≃ 10^9-10) galaxies can be chemically mature (Z_g≃ 0.3) already at z≃ 6. The MZR shows a r.m.s. dispersion σ_Z = 0.3 dex, and possibly a hint of flattening at z 6<cit.>.
        
        Furthermore, multiple line detections in the same target can be used to constrain abundance patterns of individual elements <cit.>. Resolved observations are beginning to probe the presence of metallicity gradients up to z≃ 8<cit.>, which should provide deeper insights into the formation process of the galaxy and the history of the stellar mass buildup <cit.>.
        
        Observations targeting lensed fields <cit.> are uncovering the metallicities of even the fainter galaxies <cit.>, which is key in order to explore the infall/outflow interplay due to stellar feedback in the less massive objects.
        
        Importantly, using observations of lensed targets at z≃ 3-9, <cit.> suggest that the 0.5 dex shift of normalization and the higher scattering (0.3 dex) w.r.t. the local MZR can be caused by the high level of burstiness of the SFR expected in high-z galaxies <cit.>.
        
        
        Indeed, as we move from the local Universe to high z<cit.>, galaxies are expected to become more bursty, as a consequence of the increase of specific SFR <cit.> combined with a decrease in size <cit.>, which can make the stellar feedback more effective <cit.> in maintaining a higher level of turbulence <cit.>.
        
        
        With the recent availability of JWST observations <cit.>, the time varying stochastic SFR behavior (in short, flickering or burstiness), has been invoked as a possible mechanism <cit.> to explain the overabundance of bright galaxies at z10<cit.>.
        
        However, some works predict that r.m.s. amplitude of the SFR variability falls short of explaining the said overabundance <cit.>. Thus, it is unclear if flickering alone can explain the phenomenon, particularly since the required high level of variability <cit.> is apparently not seen in the data <cit.>.
        
        
        Studies of the SFR flickering have experienced a recent surge, as stochastic SFR variations can have an impact on our ability to observe distant galaxies <cit.> and affect the possibility of detecting Population III stars <cit.>. In addition, the flickering can modify the UV spectral slopes of high-z galaxies and the escape fraction of ionizing photons <cit.>, ultimately controlling the cosmic reionization history <cit.>.
        
        Analyzing the amplitude and temporal variation of SFR flickering provides unique insights into the various feedback processes that regulate the formation and evolution of early galaxies <cit.>.
        
        On the one hand, high-amplitude and high-frequency flickering, on timescales shorter than the delay between SFR and SN explosions (≃ 20-40) are invoked <cit.> to explain the observed abrupt quenching of SFR in ≃ 10^9 galaxies already at z≃ 7<cit.>.
        
        On the other hand, the presence of disks observed in M_⋆≃ 10^9-10 galaxies up to z≃ 7<cit.> implies a limited level of stochasticity, as otherwise the disk would be disrupted <cit.>.
        
        In the present work, we aim to clarify the effects of SFR flickering and feedback regulation of high-z galaxies by using the MZR as a standard ruler.
        
        The paper is structured as follows; in Sec. <ref> we present the serra simulation suite and compare it with MZR observations from JADES, CEERS, and UNCOVER; in Sec. <ref> we develop a minimal model to give a physical interpretation to the MZR; in Sec. <ref> we discuss the discrepancies found between simulations and data in the light of the analytical model; conclusions are given in Sec. <ref>.
        
        
        
        § THE MASS-METALLICITY RELATION AT HIGH REDSHIFT
        
        
        To study the MZR at high-z, firstly we adopt the data from SERRA, a suite of cosmological zoom-in simulations that follows the evolution of z 6 galaxies <cit.>.
        
        In SERRA we use the code RAMSES<cit.> to evolve DM, gas, and stars. We enable the module RAMSES-RT<cit.> to track radiation on-the-fly by coupling it <cit.> with KROME<cit.>, in order to account for non-equilibrium chemistry up to H_2 formation <cit.>.
        
        Stars are formed using a <cit.>-<cit.> like relation based on H_2. Once formed, stars produce UV radiation that ionizes the gas, photo-dissociates H_2, and builds a radiation pressure in the gas. Massive stars generate stellar winds and explode as SNe. Depending on the kind of feedback, energy injection in the gas can be of thermal (subject to radiative losses) and/or turbulent (later dissipated) nature <cit.>.
        
        Adopting a ≃ 1.2× 10^4 (≃ 20  pc) gas mass (spatial) resolution, with music<cit.> we initialize cosmological initial conditions[We adopt a ΛCDM model with vacuum, matter, and baryon densities in units of the critical density Ω_Λ= 0.692, Ω_m= 0.308, Ω_b= 0.0481, normalized Hubble constant h =  H_0/(100  km  s^-1  Mpc^-1) = 0.678, spectral index n=0.967, and σ_8=0.826<cit.>.] generated at z=100, and follow the evolution of galaxies down to z=6 (and z=4 for a sub-sample, see ).
        
        For this work, we select ≃ 200 central serra galaxies at z=7.7 with stellar mass 10^6 M_⋆ 10^10; we compute the gas metallicity as Z_g = M_g^Z/M_g by considering the total gas (M_g) and metal (M_Z) mass contained within 3 stellar effective radii from the galaxy center, which is close to the gas half-mass radius[Different assumptions on the integration radius produce only small changes in the inferred metallicity.].
        
        Note that, for a more direct comparison with observation, we should compute line and continuum emission from serra galaxies; then, we could infer M_⋆ from synthetic SED fitting and combine line emission to prepare metallicity calibrators. Such an approach would reduce potential mismatches due to uncertainties in using different calibrators <cit.>, e.g. to relax the single zone model assumption <cit.> and correct for biases in the comparison <cit.>, also avoiding potential issues in the M_⋆ determinations <cit.>.
        
        While this kind of forward modeling approach is very powerful, also allowing to prepare novel observational strategies <cit.>, here it hinders the possibility to compare with most of the other models, as the MZR is usually computed integrating metal, gas, and stellar masses; thus we avoid it in the present work.
        
        The prediction for serra are plotted in Fig. <ref>, along with JWST data from <cit.>, <cit.>, and <cit.>, for a total of about 180 galaxies at z≃ 3-10 with 10^6  M_⋆ 10^10.
        
        The bulk of serra galaxies is between 10^7 M_⋆ 10^9; these galaxies have 10^-1.5 Z_g  10^-0.5, which is broadly consistent with the observed JWST galaxies.
        
        However, at 10^7 M_⋆ 10^8 some serra galaxies show Z_g ≃ 0.25 dex higher than the observed vales; moreover, for M_⋆≃ 10^10, Z_g is on average lower than observed, up to extreme cases of 0.5 dex. In general, serra galaxies show no clear MZR trend and their dispersion appears to be somewhat larger than the observed one.
        
        Interestingly, with respect to <cit.> and <cit.>, the lensed galaxies from <cit.> seem to indicate lower values of Z_g at M_⋆ 10^7; the lensed data seems consistent with SERRA, but only a handful of simulated galaxies and observations are in that mass range.
        
        As noted by <cit.>, the lack of a clear trend and the large scatter in the SERRA relation resemble the behavior of their 6<z<10 sample. However clear differences are present, such as the relatively metal rich galaxies at M_⋆≃ 10^8 and low metallicity galaxies at M_⋆≃ 10^10.
        
        Further, the lack of a clear trend might be partially due galaxy sample selection, as most of the SERRA data has 10^8 M_⋆ 10^9 and only a small fraction of galaxies has M_⋆≃ 10^7. Thus, the observed galaxy mass range is not uniformly covered by serra galaxies, whose sample is drawn from a collection of zoom-in simulations focusing on M_⋆≃ 10^10 galaxies and their close environment. Indeed, some of the M_⋆ 10^9 mass galaxies form in a highly pre-enriched environment, which explains their relatively high metallicities (Z_g≃ 10^-0.5, see ). Note that considering different redshift intervals[We have checked the SERRA data at z=7 ,8,  and  9.] yields qualitatively similar results.
        
        
        Thus, in summary, the normalization of the MZR of serra is roughly similar to the observed galaxies, but no trend is clearly visible and some discrepancies are present, especially in the M_⋆≃ 10^10 range.
        
        Similar (or larger) tensions are present also when a comparison with other models is performed. These can be noted, e.g. from the fit obtained by the fire-2 simulations <cit.>. fire-2 data are close to the observed Z_g at M_⋆≃ 10^9.5, but the predicted slope of the MZR is different, thus the tension with <cit.> increase with decreasing M_⋆. Finally, fire-2 galaxies present a much higher scatter, that increases with decreasing stellar mass up to ≃ 1 dex at M_⋆≃ 5× 10^7.
        
        An in depth comparison between observations and simulations is presented in <cit.>, which, in addition to serra, considers the zoom-in simulations from firstlight<cit.> and FIRE<cit.>, the IllustrisTNG cosmological simulations <cit.>, and the Astraeus semi-analytical models <cit.>; as detailed in <cit.>, most simulations seem to reasonably match the Z_g data at ≃ 10^9 but have steeper slopes and underpredict the metallicities at lower M_⋆.
        
        Interestingly, some of the simulations seem to be consistent with what observed by <cit.>, which however have only eight targets, thus it is unclear if there is a change of slope of the MZR for M_⋆ 10^7.
        
        
        
        
        § A MINIMAL PHYSICAL MODEL TO EXPLAIN THE MZR
        
        
        
        Given the very contrived match between simulations and observations, it seems worthwhile to step back and try to understand the trend and dispersion of the MZR using basic physical models. The advantage of the approach consists in simplifying the complexity of numerical simulation while retaining the most important physical processes shaping the MZR. 
        
        To this aim we devise a minimal, physical model describing the evolution of DM (M_dm), gas (M_g), star (M_⋆), and gas-phase metal (M_g^Z) mass of a galaxy, following <cit.>.
        
        As in <cit.>, we introduce an explicit feedback delay resulting in a modulation of the star formation history (SFH).
        
        
        
        
         §.§ Model setup
        
        
        
        We assume that the DM halo increases as a result of cosmological accretion, which on average can be written as in <cit.>
            Ṁ_dm = 71.6 (M_dm/10^12) (0.7/h)  E(z)  ,
        
        where E(z) = [-0.24 + 0.75 (1+z)]√(Ω_m(1+z)^3 + Ω_Λ).
        
        We allow stars to form (eq. <ref>) and include stellar feedback via SNe, exploding after a delay time . SN cause mass outflows with a rate Ṁ_out (eq. <ref>), enrich the gas of metals with a yield y, and have a return fraction R.
        
        Thus, the stellar mass increases because of SFR and decreases after  because of SN explosions:
        
            Ṁ_⋆ =  SFR(t) - R  SFR(t-) ,
        
        which for t_d=0 (no-SN-delay) would give an instantaneous recycling approximation as in <cit.>.
        
        The gas mass increases because of cosmic accretion and return from stars, and it decreases because of the SFR process and SN driven outflows:
        
            Ṁ_g = f_bṀ_dm(t) + R  SFR(t-) -  SFR(t) - Ṁ_out(t-) ,
        
        where f_b = Ω_b/Ω_m is the cosmological baryion fraction.
        Finally, the gas metal content increases because of the SN yields while it decreases because of astration and outflows:
        
            Ṁ_g^Z = f_b Z_inṀ_dm(t) + y  SFR(t-) - Z_g [ SFR(t) -Ṁ_out(t-)] ,
        
        where Z_g = M_g^Z/M_g is the gas metallicity, and we include the possibility for the cosmic gas to be already enriched at Z_in, which we set to Z_in=10^-7, i.e. virtually metal free.
        
        
        
        To solve the system in eq.s <ref>, we need to specify a functional form for Ṁ_out and SFR.
        
        We take the outflow rate as in <cit.>:
        
            Ṁ_out = 3.6 ϵ_ SN  SFR(M_⋆/10^10)^-0.35 ,
        
        with ϵ_ SN an efficiency parameter that we use to calibrate the mass loading factor; ϵ_ SN=1 correspond to the results from <cit.>. Note that ϵ_ SN=1 yields a loading factor consistent with what <cit.> observes for HZ4 <cit.>. Further, using ϵ_ SN=1 in eq. <ref> would match the results from <cit.> for z≃ 5-6 ALPINE galaxies <cit.> which feature a spatially extended gas component <cit.>. However, ϵ_ SN≃ 1/3 is needed to match the loading factor inferred from z≃ 3-9 JADES galaxies <cit.>.
        
        
        We assume that the SFR is proportional to the gas mass:
        
            SFR = M_g/t_⋆ ,
        
        and we select a constant t_⋆ = 1  Gyr as our fiducial star formation time scale[In the model, galaxy evolution is self-regulated by feedback, thus the sensitivity of the results from t_⋆ is limited in our case, and null in the =0 case with constant loading factors for infall and outflow <cit.>. A posteriori, we note that the fiducial t_⋆ = 1  Gyr gives specific star formation rates sSFR =  SFR/M_⋆ which are consistent with SERRA galaxies, i.e. ≃ sSFR≃ 100-1   Gyr^-1 for galaxies with masses M_⋆=10^6 - 10^10.]. The choice of t_⋆ is roughly consistent with the upper limits of the depletion times reported by <cit.> for ALPINE galaxies, and it is discussed in more detail in Sec. <ref> (see also App. <ref>).
        
        To complete the model, we assume that halos can form stars after they reach a virial temperature for atomic cooling to be effective (10^4  K); similarly to serra, we assume return fraction and yields appropriate for a <cit.> IMF and adopting <cit.> tracks with a Z_⋆= stellar population[Changing the metallicity of the stellar population does modifies the yield and return fraction, but gives only minor modifications to the upcoming results.], which gives y≃ 0.0228 and R≃ 0.3242.
        
        
        The system (eq.s <ref> and <ref>) is solved along with z=z(t) using an embedded 5th/6th order Runge-Kutta method <cit.> with adaptive timestep, which is selected in order to have a fractional (absolute) precision of at least ≤ 10^-5 (≤ 10^2) for M_dm, M_⋆, M_g, and M_g^Z. The delayed SFR and stellar mass used to evaluate the stellar feedback are computed using a 7 points time stencil.
        
        
        
        
         §.§ Overview of the minimal physical model
        
        
        The delay between SFR and feedback  and the amplitude of the outflow (ϵ_ SN) regulate the burstiness of the galaxy in our single zone model <cit.>.
        
        To understand the impact of burstiness on galaxy evolution, it is instructive to consider a halo with mass M_dm = 3× 10^10 at z=6 and set the outflow efficiency to the standard value of ϵ_ SN=1.
        
        In Fig. <ref> we compare the evolution of the stellar, gas, and metal mass content for = 0 (no-SN-delay, dashed lines) and = 10  Myr (solid lines).
        
        By construction, the halo mass growth (eq. <ref>) is unaffected by , thus the two models follow the same DM evolution, which smoothly grows from M_dm≃ 10^7 at z ≃ 18, when the universe was t≃ 200 old.
        
        At that time, for =0 the gas and stars has ≃ 10^5 and ≃ 10^2, respectively, and both grow reaching approximately ≃ 5× 10^7 at t≃ 800, with M_⋆/M_dm≃ 5× 10^-3 broadly consistent with what expected from abundance matching models <cit.>.
        
        In the case with = 10  Myr, the secular trends are qualitatively similar, but the system experiences a series of bursts on time scales of ≃ 20  Myr=2.
        
        Let us focus on a single burst cycle around z≃ 10, when the halo has M_dm≃ 10^9 and M_⋆≃ 10^6. In the first half period of the cycle (0-), cosmic accretion quickly replenish the gas (gas infall rate is ≃ 0.5 from eq. <ref>), reaching M_g≃ 10^7 in a few Myr. Stars form at a modest rate SFR≃ 10^-2 (eq. <ref>) under feedback-free conditions <cit.>. After , the feedback kicks in, first enriching the galaxy and eventually (≃ 2) becoming powerful enough to expel both gas (gas outflow is ≃ 1 using eq. <ref>) and metals <cit.>.
        
        However, in the example the impact of SFR flickering is limited already after z≃ 15, when the galaxy has M_⋆≃ 10^5 or 10× lower if no delay is considered. At z≃ 6, the final stellar mass is M_⋆≃ 10^7 within a factor of 2 for the two models.
        
        For the gas and metals, the model with SN delay starts to converge to the no-SN-delay case at z≃ 8, when M_⋆≃ 10^7; oscillations for the gas and metals are still present up to z≃ 7, but they are milder.
        
        As in <cit.>, it is convenient to quantify the burstiness of the SFR by defining the variable δ expressing the log of the stochastic SFR variation with respect to its mean (in short: flickering)
        
            δ≡log SFR/⟨ SFR⟩ ,
        
        where log⟨ SFR/⟩ is a second order polynomial fit in the time variable t. The low order fit is needed to factor out the smooth behavior of the functions without removing time modulations <cit.>.
        
        Similarly to the flickering, we define the analog variable for the gas metallicity
        
            δ_Z≡logZ_g/⟨ Z_g⟩ ,
        
        where log⟨ Z_g /⟩ is a second order polynomial fit in the variable log M_⋆/. For both δ and δ_Z we can define the typical variation as the root mean square (r.m.s.) deviation, i.e.  and , respectively.
        
        Adopting eq.s <ref>, we find that the = 10  Myr case has a flickering ≃ 0.3 before the two models start to converge, which is slightly higher than the ≃ 0.24 obtained for SERRA galaxies <cit.>.
        
        For different  the qualitative behavior is very similar, as the model is off-balance and oscillates around the no-SN-delay case, that is similar to a bathtub solution <cit.>. For shorter (longer) ,  can be smaller (higher), as galaxy with a smaller (higher) M_⋆ can be off-balanced.
        
        The balance between infall and outflow rates is critical, since the dynamical system (eq.s <ref>) modeling the galaxy evolution give rise to exponentially increasing/decreasing trends; introducing a ≠ 0 can efficiently break the balance, causing cycles of complete gas and metal depletion.
        
        To explore the MZR variations induced by the flickering, in Fig. <ref> we plot the tracks of the selected M_dm = 3× 10^10 at z=6 with t_d from 5 to 25 Myr.
        
        Because of the longer , the SFR becomes increasingly bursty. Specifically, for the selected model the flickering ranges from ≃ 0.1 to ≃ 0.6 for =5 Myr to =25 Myr, respectively.
        
        
        The the MZR modulations induced by the flickering are highly non linear.
        
        With =5-10, the MZR dispersion is negligible-small (≃ 0.05-0.2), as the flickering can affect galaxies only up to M_⋆≃ 10^5-7.
        
        For =15, the full M_⋆ range experience flickering with a ≃ 0.5, yielding an high ≃ 0.8.
        
        Cases with >15 Myr do not show a higher stochasticity, as an increasing delay only regulates which M_⋆ can be affected by the feedback unbalance[Note that an higher/lower level of flickering can be induced by changing the infall/outflow rates, as shown later in Sec. <ref>.]. Qualitatively, the behavior is similar with higher M_dm, but the convergence to the bathtub solution is faster.
        
        In summary, a feedback delay induces a stochastic, bursty star formation behavior that becomes progressively more enhanced as t_d is increased. Such SFR flickering corresponds to analog Z_g fluctuations which act to decrease the mean metallicity as gas and metals are effectively ejected from the galaxy by outflows. 
        
        
        
        
        
         §.§ The impact of a stochastic SFR on the MZR
        
        
        We run sets of models that at z=6 have a M_dm from 10^9 to 10^13 with a logarithmic binning of 0.1 dex, in order to cover the mass range 10^5  M_⋆ 5× 10^10.
        
        We analyze different variations of the minimal models, reporting the results in different panels of Fig. <ref>, where they are compared with data from JWST observations <cit.>.
        
        
        
          §.§.§ Base models
        
        
        Let us start by discussing the base models analyzed in Sec. <ref>, i.e. with ϵ_ SN=1, consistent with fire simulations <cit.>, and selecting =20, so that most of the explored stellar mass range is expected to have a stochastic SFR.
        
        In the upper left panel of Fig. <ref> we report both the individual tracks of the base models and their M_⋆ average trends.
        
        For the base models, the median case falls significantly below the observed average MZR. Also, the dispersion is much larger than seen in JWST data, that can be barely matched by the upper envelope of the predicted curves. 
        
        The base models have a ≃0.5 flickering, that induces a 1 almost in the entire stellar mass range (M_⋆ 10^9, Fig. <ref>).
        
        The flickering is due to cycles of infall/outflow which determine an oscillatory behavior of the evolution around the =0 control case, that is largely independent of the hosting M_dm.
        
        The Z_g-M_⋆ normalization is determined by the strength of the outflows, that completely devoids the galaxy of gas and metals, not allowing the metallicity to rise to the observed level. Even for =0 the outflow seems too extreme, as only part of the lensed galaxy data can be matched <cit.>.
        
        
        In summary, the efficient feedback (ϵ_ SN=1) combined with a =20 generates a SFR flickering of ≃ 0.5, inducing MZR variations of ≃ 1.4 resulting in a chemical chaos that is not observed. Barring the option that observations are tracing only the tip of the Z_g-M_⋆ relation, in order to explain the observed MZR we modify the base models as follows.
        
        
        
          §.§.§ Weak feedback
        
        
        
        First, we reduce the impact of stellar feedback by decreasing the outflow rate from <cit.> (eq. <ref>), i.e. setting ϵ_ SN=1/4; such a loading factor is closer to what <cit.> infer for 10^7.5 M_⋆ 10^9 JADES galaxies.
        
        As shown in the upper right panel of Fig. <ref>, this modification increases the average metallicity of the modeled galaxies, matching the observed mean trends in the whole 10^6 M_⋆ 10^10 mass range. Comparing to the =0 control case, it is evident that the outflow efficiency is the main driver of the mean Z_g-M_⋆ trend.
        
        
        Note that, reducing the outflow efficiency while maintaining a fixed =20 results in ≃ 0.1 modulation of the MZR, i.e. too low with respect to the observed (≃ 0.25, see Fig. <ref>).
        
        This effect is induced by the suppression of the amplitude of the SFR flickering, especially at the high mass end.
        
        Indeed the models show a small flickering, with ≃0.15 for M_⋆ 10^8, and even slightly lower values for higher stellar masses (Fig. <ref>).
        
        As discussed in <cit.>, M_⋆ 10^9 galaxies show SFR fluctuations with characteristic periods of ≃ 20-40, which correspond to the typical evolutionary times of SNe.
        
        Such flickering timescale increases to ≃ 80-150 in more massive galaxies which are more sensitive to mergers and accretion rate modulation. The latter effects are not accounted for in our minimal model (eq. <ref>). The next step is then to heuristically incorporate them in our treatment.
        
        
        
        
          §.§.§ Weak feedback, modulated accretion
        
        
        To incorporate the extra stochastic behavior expected from variation in the cosmic accretion in a simple manner, we impose a sinusoidal modulation on a 100 timescale
        
            Ṁ_dm^mod= Ṁ_dm[1 + sin(t/100  )] ,
        
        with Ṁ_dm being the average cosmic accretion from eq. <ref> and with  scaling the amplitude of the modulation[Note that small halos should be relatively less affected by modulation on long timescales (≃ 100), since the hosted galaxies have shorter SFH. However, cosmological accretion variations can be coherently combined with the SN delay to give a higher  also in the low mass range.].
        
        
        As feedback determines the mean trend of the MZR, we keep the reduced outflow efficiency (ϵ_ SN=1/4 in eq. <ref>) and compute the evolution of a minimal model by setting the amplitude of the oscillation to =1/3, which should yield a scatter below the 0.3 dex dispersion expected for the distribution of growth rates of DM halos <cit.>.
        
        The results from these modified models are shown in the lower left panel of Fig. <ref>: both the mean trend and scatter of the observed MZR are recovered (Fig. <ref>).
        
        Such a matching set of modified models yields a roughly constant ≃ 0.2 and ≃ 0.3, satisfactorily matching the JWST data.
        
        
        
        
          §.§.§ Weak feedback, modulated accretion, no-SN-delay
        
        
        As a final check, in the lower right panel of Fig. <ref> we show the effect of removing the SN-delay from the best matching model just discussed.
        
        While the mean MZR trend is recovered, the metallicity dispersion (≃ 0.05) is much smaller than observed (≃ 0.25). Also, the median of the models fails to reproduce the relation for most of the low mass lensed galaxies <cit.>.
        
        With no-SN-delay, the SFR flickering is induced only by modulation of the cosmic accretion, which fails to unbalance the system through the ejection of a substantial fraction of the contained gas.
        
        With the selected =1/3 and =0, the model reach ≃0.1 for M_⋆ 10^8, while the flickering gets to ≃0.2 only for M_⋆≃ 5× 10^8 galaxies.
        
        
        Compared to the control case (=0), the average Z_g is slightly higher; practically, SFR flickering induced by the accretion modulation can increase Z_g since a higher gas mass can be converted into stars without efficiently ejecting the gas via outflows, i.e. the opposite outcome with respect to SFR flickering induced via SN-delay.
        
        Interestingly, despite the relatively high SFR flickering (average of ≃ 0.2), the metallicity variation is the lowest of all considered cases (≃ 0.05): only a delayed SN feedback can efficiently off-balance the system and induce modulations on both SFR and Z_g.
        
        We note that the serra cosmological simulations show a relatively modest SFR flickering <cit.>, and close to the expectations for z≃ 6 galaxies <cit.>. Yet, given the sharp sensitivity of  to SFR variations, such time variability results in a  larger than observed. 
        
        
        To summarize, our analysis shows that: (i) a weak feedback, and (ii) a relatively mild stochasticity given by (iii) SN-delay combined with a long term accretion rate modulation are critical to simultaneously match both the mean and dispersion of the observed MZR relation in the entire stellar mass range of galaxies sampled by JWST at z=3-10.
        
        Finally, we note that invoking stochasticity to explain the overabundance of bright JWST galaxies requires >0.5 (more specifically, ≃ 0.8, ≃ 0.7, and ≃ 0.6 according to , , and , respectively; see Fig. <ref>).
        
        However, such extreme flickering amplitudes are (i) higher than that recovered from SED fitting allowing for SFH variability <cit.>, and (ii) would induce a chemical chaos (1.4) that is not present in the observed MZR at z=3-10<cit.>.
        
        
        
        
        
        § DISCUSSION
        
        
        The minimal models presented here can also be useful as a tool to guide the development and compare complex cosmological simulations, particularly when focusing on relations that - like the MZR - are very sensitive to small variations of the underlying physics, that is included via sub-grid models in galaxy simulations <cit.>.
        
        
        
        
         §.§ Comparing numerical simulations
        
        
        
        
        
        In galaxy simulations, ejective feedback, such as outflows caused by mechanical/kinetic prescriptions, can effectively suppress the SFR activity as much as preventative feedback, such as turbulence/delayed cooling prescriptions <cit.>, leading to similar SFR histories when the same initial conditions are considered <cit.>.
        
        Despite similar SFR histories, other properties can differ significantly <cit.>, such as the dynamical state of a galaxy.
        
        For instance, <cit.> show that FIRE-2 galaxies <cit.> have a much lower incidence of disks compared to serra galaxies <cit.>.
        
        Additionally, SERRA galaxies tend to have higher Z_g and lower MZR scatter compared to fire-2<cit.>, especially in the M_⋆≃ 10^9 mass range. 
        
        Both effects are likely caused by the higher SFR flickering of fire-2<cit.> with respect to serra<cit.>, which is ultimately regulated by the efficiency and kind (e.g. ejective vs preventative) of included stellar feedback. Violent outflows are present in fire-2<cit.>, while the suppression of SFR due to the photo-dissociation of molecular hydrogen in serra<cit.> can more gently can modulate the SFR activity without drastically modifying the galaxy dynamics.
        
        Similar considerations hold when comparing successive improvements of simulations suites. For instance, the loading factor from fire-2<cit.> are lower by a factor of ≃ 2 with respect to fire<cit.>; this difference dictates that the MZR from fire-2<cit.> is about 0.3 dex higher than the MZR from fire<cit.>; in both cases the slope is steeper (normalization is lower) than what observed by <cit.> and is instead more consistent with <cit.>.
        
        Despite the simplicity of the minimal model presented, hints of such differences in the simulations can be highlighted and explained.
        
        
        
        
        However, it is difficult to directly quantify the differences in the simulations with the presented models.
        
        
        
        For instance, the loading factor in our simplified models (but also in relatively more complex ones as  and ) is a scalar quantity used as an input value for each galaxy, while in simulations is usually defined as an output that has a spatial dependence <cit.>. 
        
        Additionally, when dealing with simulation snapshots, quantifying both the SFR event and the consequent outflows is tricky because of the time-delay between SFR and feedback <cit.>, particularly for galaxies far from a quasi-equilibrium situation.
        
        
        
        
        Further, the simplifications done in our minimal model impose we can only capture part of the complexity present in simulations.
        
        For instance, while the SN-feedback delay is uniquely defined, simulations finds a distribution of  that depend on the underlying physical models included <cit.>; thus, trying to find the set of parameters of a minimal model that gives the best match to a simulation will only give effective parameters, a sort of summary statistic that can be used to compare different simulations.
        
        
        Apart from these difficulties, a systematic parameter fitting procedure should give a better quantification of the comparison between different simulations, and can also be used to enable an even better match between the data and the minimal models. However, such models are lacking a few ingredients that prevents reproducing the full connection between metal build up and star formation history of galaxies.
        
        
        
        
         §.§ Caveats of the minimal model
        
        
        The limitations of the presented minimal models can be appreciated by considering the fundamental mass-metallicity relation (FMR), i.e. the link between Z_g, M_⋆, and SFR that is observed to be redshift independent up to z∼3<cit.> and seems to break down at z5<cit.>.
        
        We adopt the parameterization from <cit.>
            μ≡log(M_⋆/) - 0.65 log( SFR/) ,
        
        which is obtained by minimizing the Z_g dispersion for the high sSFR sub-sample from the Sloan Digital Sky Survey (SDSS, , z 2).
        
        In Fig. <ref> we show the FMR for the minimal models matching the mean MZR and its dispersion, i.e. the weak feedback (ϵ_ SN=1/4) set with modulated accretion (= 1/3).
        
        These models are below the SDSS relation by about 0.5 dex at μ 7, with the separation that decreases with increasing μ, down to 0.1 dex at μ 9, almost connecting with the FMR trend observed at low z.
        
        As expected <cit.>, JWST galaxies are offset below the SDSS relation.
        
        The best matching models for the MZR only partially match the JWST data in the FMR plane; the minimal models are closer to the JWST data when considering the lensed galaxies from <cit.>.
        
        Further the minimal models only produce downward deviation with respect to the FMR from the SDSS.
        
        Instead, serra galaxies manage to better recover the trend for JWST galaxies around μ≃ 8, reproducing both downward and upward deviations with respect to the FMR from the SDSS; however, serra galaxies show an overabundance of low Z_g galaxies at μ≃ 8.5-9.0, which is not seen in the JWST data.
        
        
        In an equilibrium model <cit.>, the shape of the MZR is independent from the SFR, which uniquely determines the speed a galaxy can climb up the Z_g-M_⋆ curve.
        
        
        Note that the simplified model seems to deviate more from the data at low μ, i.e. a combination of relatively high SFR and low M_⋆; the adopted feedback increases in efficiency for lower M_⋆ (eq. <ref>).
        
        Thus, the observed FMR deviation might be recovered by the minimal model if we consider not only ejecting but also a preventing feedback, e.g. photodissociation of molecular hydrogen, which can suppress the star formation without affecting the gas mass (e.g. see Alyssum, a serra galaxy described in ) and can play a role in explaining some of the bright blue monsters seen by JWST <cit.>.
        
        As an alternative to keep matching the MZR and recovering the offset from the FMR, we could try to change the simplified SFR prescription (eq. <ref>); this can be done for instance by adopting as the time scale for star formation a redshift dependent depletion time <cit.>, which are typically shorter than the selected t_⋆=1  Gyr by about a factor ≃ 10, similar to what <cit.> report for z≃ 7 galaxies by using GLAM<cit.> determinations, thus promoting a faster galaxy evolution at high-z.
        
        Indeed, by adopting the depletion time from <cit.> as the time scale for SFR (see App. <ref>, in particular Fig. <ref>), the stellar mass build up is faster at M_⋆ 10^6, while later saturates to values similar to those resulting from the t_⋆=1  Gyr case as both SFHs become feedback regulated, approaching a bathtub equilibrium.
        
        
        However, on the one hand, the modifications affect the predictions of the minimal model mostly in the M_⋆ 10^6 range; on the other hand, a z average depletion time is sensitive to the population of the sample, e.g. for main sequence is ≃× 5 lower than for starburst <cit.>, and the results from <cit.> are mostly relying on the data of M_⋆ 10^8-10 galaxies from the ALPINE <cit.> and REBELS <cit.> surveys.
        
        To summarize, while trying to reconcile the FMR behavior is an interesting perspective and there are a few options, some care should be taken in extending/modifying the prescriptions for the minimal models; we leave such a possibility for a future work.
        
        
        
        
        § CONCLUSIONS
        
        
        
        
        
        
        The exquisite spectroscopic data collected by JWST allow for the first time to derive the mass-metallicity relation (MZR) in galaxies up to redshift z≃ 10.
        
        
        
        
        In this work, we considered the MZR data for a combined sample of about 180 galaxies with stellar mass 10^6  M_⋆ 10^10 at z=3-10 given <cit.>, <cit.>, and <cit.>, and obtained from the CEERS <cit.>, JADES <cit.>, and UNCOVER <cit.> surveys, respectively. 
        
        We compared the observations with the predictions from the serra simulations <cit.>, finding a broad agreement with the observed MZR. However, simulations show a lack of a clear metallicity trend, and some tension, particularly at M_⋆≃ 10^10.
        
        
        
        
        To better understand the observed high-z MZR and to clarify the behaviour of the simulations, we have devised a minimal physical model for galaxy evolution in which star formation is fueled by cosmic gas accretion <cit.> and is regulated by supernova driven outflows (eq. <ref>), with a loading factor from <cit.> that is controlled by an efficiency ϵ_ SN; additionally, we incorporated an explicit delay () between star formation and supernova feedback, inducing a stochastic SFR behavior. Some models also explore the possible modulation of cosmic accretion (eq. <ref>). s are:
        
          ∙ To recover the average trend of the MZR observed at high-z, outflows must be less efficient (ϵ_ SN = 1/4) than predicted in <cit.>. However, this evidence is consistent with the mass loading factors inferred for JADES galaxies <cit.>.
          
          
          ∙ To match the MZR dispersion in the full stellar mass range, both a delayed (=20) SN feedback, and a modulation of the cosmic accretion (= 1/3) are required.
          
          
          ∙ The r.m.s. variation of the MZR () is very sensitive to the SFR flickering (). The weak feedback necessary to reproduce the average MZR trend (ϵ_ SN = 1/4) also results in a low-amplitude SFR flickering (≃ 0.2). This prescription correctly reproduces the moderate MZR scatter (≃ 0.25) observed by JWST.
          
          
          ∙serra galaxies have slightly higher ≃ 0.24<cit.>, resulting in a metallicity scatter of ≃ 0.3, higher than observed. Differently from the analytical model, serra galaxies also overpredict the number of low metallicity (Z_g≃ 0.2) galaxies at M_⋆≃ 10^10.
          
          
          ∙ In general, any model predicting 0.5 is likely to overshoot the observed metallicity r.m.s. scatter, possibly leading to a chemical chaos (1.4) that is it not present in JWST data, for which ≃ 0.25).
          
        
        
        
        
        
        The last point also entails that simultaneously explaining the observed MZR <cit.> and the overabundance of bright galaxies observed by JWST <cit.> through high SFR flickering <cit.> is very challenging, considering that a ≃ 0.8 is required to explain z>10 observations <cit.>.
        
        We note that while both the SN-feedback delay and cosmic accretion modulation induce SFR flickering, the latter tends to give a gentler modulation of Z_g, since high feedback efficiency and a delay are needed in order to off-balance the MZR generated from a bathtub-like equilibrium <cit.>.
        
        However, variation of cosmic accretion can only go up to 0.3 dex <cit.>. Thus, alternative explanations <cit.> seem more favored to solve the overabundance problem.
        
        
        
        
        While we showed that the presented minimal model can also be used to guide our intuition in comparing complex cosmological simulations, we recall that such a simplified model does not fully capture the physical complexity of the connection between galaxy formation and metal enrichment, as highlighted by the poor match with the JWST data when analyzing the off-set from the fundamental mass-metallicity relation.
        
        
        
        Combining insights from such models with further analysis of ongoing observations, and more complex numerical simulations is crucial for understanding the galaxy formation and evolution process at high-z.
        
        
        
        We acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support from the Class B project SERRA HP10BPUZ8F (PI: Pallottini).
        
        AF acknowledges support from the ERC Advanced Grant INTERSTELLAR H2020/740120 (PI: Ferrara).
        
        SC and GV acknowledge support support from the ERC Starting Grant WINGS H2020/101040227 (PI: Carniani).
        
        Any dissemination of results must indicate that it reflects only the author's view and that the Commission is not responsible for any use that may be made of the information it contains.
        
        Partial support (AF) from the Carl Friedrich von Siemens-Forschungspreis der Alexander von Humboldt-Stiftung Research Award is kindly acknowledged.
        
        We gratefully acknowledge computational resources of the Center for High Performance Computing (CHPC) at SNS.
        
        We acknowledge usage of the Python programming language <cit.>, Astropy <cit.>, Cython <cit.>, Matplotlib <cit.>, Numba <cit.>, NumPy <cit.>, pynbody<cit.>, and SciPy <cit.>.
        stile/aa_url
        
        § MODIFICATIONS TO THE SFR TIME SCALES
        
        
        In bathtub models <cit.>, the shape of the MZR is mostly independent from the functional form of the SFR, as the equilibrium solution is determined by the balance between infall and outflow, while the SFR uniquely determines the speed a model climbs the MZR from low to high M_⋆.
        
        Instead, the SFR prescription can be important if a stochastic behavior is present in the system and when analyzing the FMR.
        
        In the fiducial model presented in the paper we assumed that the SFR is proportional to the gas mass and is regulated by a fixed time scale for star formation, i.e. t_⋆ = 1  Gyr. As a natural alternatively for a fixed t_⋆, a model can consider to adopt t_⋆ via empirical determinations <cit.> or model-based predictions <cit.> of the depletion time, which is expected to decrease with redshift, thus giving a faster star formation rate at higher z.
        
        As a test, we modify our models by changing t_⋆ in our SFR prescription (eq. <ref>) by adopting the scaling suggested by <cit.>
            t_⋆^mod   ≃8.95   Gyr/E(z)
               = 8.95   Gyr/ [-0.24 + 0.75 (1+z)]√(Ω_m(1+z)^3 + Ω_Λ) ,
        
        and setting the proportionality constant in order to match <cit.> determinations at z=0.
        
        As a reference, t_⋆≃ 20  t_⋆^mod at z≃ 10, thus a faster stellar mass build is expected for the modified model. 
        
        As a benchmark, we select the halo with M_dm=3× 10^10 at z=6 analyzed in Sec. <ref> (see Fig. <ref>), evolve it with t_⋆ = 1  Gyr and t_⋆^mod from (eq. <ref>), and plot the results in Fig. <ref> for both =10 and a reference no-SN-delay case.
        
        The largest differences between the t_⋆ = 1  Gyr and t_⋆^mod models are present for the stellar mass build up at z 10, when the galaxy has M_⋆ 10^7,
        
        The t_⋆^mod model promotes a faster build up, particularly when a SN-delay is present, as the lack of feedback combined with the short SFR timescales make he galaxy reach M_⋆ 10^6 during the first burst.
        
        As the time progresses, differences between the models tend to wash out, as the galaxy approaches M_⋆≃ 10^8, the SFH starts to become feedback-regulated, and at z≃ 6, stellar, gas, and metal masses are roughly similar in all cases. 
        
        Interestingly, while the amplitude of the flickering is similar (≃ 0.5), the modulation acts on slightly shorter time scales for the t_⋆^mod model, as a consequence of the faster SFR rise and the heavier off-balance caused by the stellar feedback.
        
        This implies that the t_⋆^mod model at z 7.5 gives an even enhanced dispersion for the MZR (from ≃ 0.7 to ≃ 1.7).
        
        Note that when no-SN-delay is present, the SFH evolution is feedback regulated from the start, and the main difference is an increased gas fraction for the t_⋆ = 1  Gyr model.
        
        Note that adopting <cit.> implies a depletion time with a ∝ z^-2.5 redshift dependence, i.e. steeper with respect to the ∝ z^-1.5 dependence from <cit.>. Thus, by assuming <cit.> instead of <cit.> to modify the SFR timescale, we expect smaller variation with respect to the fiducial t_⋆=1  Gyr model.


 Paper Title: The MillenniumTNG Project: Impact of massive neutrinos on the cosmic large-scale structure and the distribution of galaxies.



firstpage–lastpage
Curvature-induced moduli stabilization
    Flavio Tonioni
    August 5, 2024
======================================




§ ABSTRACT 
We discuss the cold dark matter plus massive neutrinos simulations of the MillenniumTNG (MTNG) project, which aim to improve understanding of how well ongoing and future large-scale galaxy surveys will measure neutrino masses. Our largest simulations, 3000  Mpc on a side, use 10240^3 particles of mass m_p = 6.66× 10^8 h^-1 M_⊙ to represent cold dark matter, and 2560^3 to represent a population of neutrinos with summed mass M_ν = 100 meV. Smaller volume runs with ∼ 630  Mpc also include cases with M_ν = 0 and  300  meV. All simulations are carried out twice using the paired-and-fixed technique for cosmic variance reduction.  We evolve the neutrino component using the particle-based δ f importance sampling method, which greatly reduces shot noise in the neutrino density field. In addition, we modify the GADGET-4 code to account both for the influence of relativistic and mildly relativistic components on the expansion rate and for non-Newtonian effects on the largest represented simulation scales. This allows us to quantify accurately the impact of neutrinos on basic statistical measures of nonlinear structure formation, such as the matter power spectrum and the halo mass function.  We use semi-analytic models of galaxy formation to predict the galaxy population and its clustering properties as a function of summed neutrino mass, finding significant (∼ 10%) impacts on the cosmic star formation rate history, the galaxy mass function, and the clustering strength. This offers the prospect of identifying combinations of summary statistics that are optimally sensitive to the neutrino mass.



cosmology: theory – large-scale structure of Universe – dark matter – galaxies: haloes – methods: numerical









§ INTRODUCTION



The advent of ever larger and more precise astronomical data sets over the past decades has led to tremendous progress in our understanding of cosmology, culminating in the establishment of a standard model, ΛCDM, whose basic premises are now subject to precision tests by ongoing and planned experiments <cit.>. In fact, the forthcoming new data will be so precise that they will not only be able to detect potential deviations from a cosmological constant, but also be sensitive to the impact of baryonic physics (for example from AGN feedback) on the matter distribution. While this can be viewed as an unwelcome complication, it of course also offers a chance for additional tests of our astrophysical understanding of the relevant aspects of galaxy formation.

In a similar vein, cosmological observations are becoming precise enough to detect deviations from the properties ascribed to purely cold dark matter. For example, dark matter may be composed of such a light particle that its thermal motions are non-negligible, causing smaller structures to be wiped out, and such warm dark matter (WDM) remains a much studied alternative to standard cold dark matter (CDM) scenarios <cit.>. The most extreme version of such a model, where the particles are all truly `hot', i.e. with large, nearly relativistic thermal velocities at high redshift, is cleared ruled out by observations because the associated top-down scenario for structure formation is inconsistent with data on multiple fronts. This is also why normal neutrinos as the dominant dark matter component were ruled out early on <cit.>.

However, at the same time we know that a significant neutrino background is left behind by big bang nucleosynthesis, and the phenomenon of neutrino flavour conversion implies a lower bound for the summed neutrino masses <cit.>. This means that neutrinos represent a hot dark matter admixture that must be present in the standard model, and which constitutes at most of order a few percent of all the dark matter according to a variety of cosmological constraints <cit.>. These are, in fact, tighter than those obtained so far from the best  experimental upper bounds on the neutrino masses which come from the KATRIN experiment, based on an examination of the  beta decay of tritium <cit.>. Their low mass limits the influence of neutrinos on structure growth to the level of a small perturbation that until recently could be safely ignored in  studies of galaxy formation. However, recent improvements in the precision of modern cosmological probes of galaxy clustering and weak lensing have started to transform this assessment. In fact, the impact of neutrinos on large-scale structure may soon become measurable with galaxy surveys, raising the prospect of actually measuring the sum of the neutrino masses and so distinguishing between the normal and inverted mass hierarchies, a fundamentally important issue in particle physics.

Computing the impact of neutrinos on structure formation in the non-linear regime faces substantial technical difficulties, because a straightforward representation  of the neutrino density field through particles is severely affected by shot noise even on large scales. This contrasts with the cold dark matter case, where shot noise is far below the intrinsic power on most scales of interest.  However, in recent years a large number of approximate simulation techniques have been developed that address this issue in a variety of ways. The recent comparative study of <cit.> gives a comprehensive overview of these methods, and compares their accuracy, albeit  over a considerably smaller spatial dynamic range than studied here, where we employ the δ f-technique originally proposed by <cit.>. This is one of the most accurate and conceptually elegant numerical techniques proposed so far. It uses a particle-based representation that is capable of tracking non-linear neutrino clustering without spatial resolution limitations while at the same time greatly reducing shot noise compared to  naive particle-based methods <cit.>.

We have used the δ f-approach in the neutrino branch of our new MillenniumTNG simulation programme. The primary introductory paper of MTNG <cit.> already briefly described some initial results from our first large neutrino run, but it deferred a more detailed discussion to later work that we now present in this study. Here we additionally include results from a second very large neutrino simulation also carried out following 1.1 trillion particles in a 3000 Mpc box, and we include calculations at the same resolution but in smaller boxes that vary the neutrino mass. Other MTNG simulations include large dark-matter-only simulations in a 740 Mpc box at different mass resolutions, as well as a high-resolution hydrodynamical model <cit.> with full galaxy formation physics for the same initial conditions, carried out with the IllustrisTNG physics model <cit.> but with a volume nearly 15 times larger than TNG300. Initial MTNG analysis papers have already looked at weak gravitational lensing <cit.>, intrinsic galaxy shape alignments <cit.>, galaxy clustering <cit.>, predictions for the high-z galaxy population <cit.>, inference of cosmological parameters <cit.>, and mock galaxy catalogue construction <cit.>.

We note that since these first MTNG papers a similarly large neutrino simulation, FLAMINGO-10k, has been added to the FLAMINGO project <cit.> and been used to study the clustering of galaxies and quasars at redshift z∼ 6 <cit.>, based on an empirical quasar model. The FLAMINGO-10k simulation has a very similar mass resolution and particle number to our large runs, but the adopted neutrino mass is different, only 60 meV, thus nicely complementing our own large simulations.  Another very large cosmological simulation of similar mass resolution, the New Worlds Simulation of <cit.>, recently accounted for neutrinos, but only at the linear level without proper treatment of non-linear neutrino clustering. <cit.> carried out large-scale cosmological simulations with massive neutrinos using a hybrid N-body/Vlasov approach solving explicitly the Boltzmann equation. On the other hand, the TianNu simulation <cit.> co-evolve nearly 3 trillion CDM and neutrino particles in a 1.2 box, making it one of the largest cosmological N-body simulations with massive neutrinos performed to date. The MassiveNuS simulations <cit.> have included the effects of radiation on the background expansion and varying the summed mass of neutrinos between 0 meV and 600 meV. Moreover, the DEMNUni <cit.> and the DUSTGRAIN-pathfinder <cit.> simulations have included the effects of massive neutrinos with dynamical dark energy and modified gravity, respectively.
Other very large cosmological simulations, such as the  AbacusSummit simulations <cit.>, have not so far included neutrinos explicitly. Also, we note that galaxy catalogues based on following the physics of galaxy formation semi-analytically, as we do here, are not yet available for these other simulations; rather, some of them typically use empirical models to represent the galaxy population, derived through halo occupation distribution modelling or similar techniques. We thus consider it one of the special and distinguishing features of the MTNG simulations that we can augment them with semi-analytic predictions computed with a novel version of the L-GALAXIES code, including the production of mock catalogues on seamless lightcone outputs <cit.>.

In this work, we will focus on introducing the neutrino methodology of our simulation set, and on highlighting the impact of neutrinos on selected clustering statistics, both for haloes and for mock galaxy catalogues. Addressing the question which combination of observational measures can most powerfully constrain neutrino masses is, however, beyond the scope of this paper.

Our paper is structured as follows. In Section <ref> we discuss the technical aspects of our neutrino simulation methods, and in Section <ref> we specify the simulations analysed here. We then present results for matter clustering in Section <ref> and for halo statistics in Section <ref>. This is followed by an analysis of galaxy clustering in Section <ref>. Finally, we present a summary of our findings and our conclusions in Section <ref>.




§ THEORETICAL AND NUMERICAL ASPECTS



While the techniques for standard cosmological N-body simulations are well understood and extensively documented in the literature <cit.>, many different approaches exist for including neutrinos <cit.>. In order to provide an unambiguous specification of our own work, we here summarise the methods we use for including massive neutrinos in the evolution both of the background cosmology and of cosmic structure, as well as for setting up the appropriate initial conditions.




 §.§ Cosmological background evolution



Here we recall some basic aspects of neutrino cosmology as needed in our numerical implementation.  To run our neutrino simulations, we have included the evolution of radiation (photons and relativistic massless neutrinos) and of a hot dark matter component in the form of massive neutrinos (which may experience a transition from relativistic to non-relativistic velocities) in the calculation of the expansion history in GADGET-4. Hence, the cosmic evolution of the density parameters of different matter and energy components in a flat Universe is given by

    Ω_r(z) + Ω_cb(z) + Ω_ν(z) + Ω_Λ = 1 ,

where Ω_r is the density parameter of the radiation components, Ω_cb is the contribution of the cold matter components (cold dark matter plus baryons), Ω_ν is the density of the non-relativistic massive neutrinos, Ω_Λ is the density parameter of dark energy in the form of the cosmological constant where Ω_Λ = 1 - [Ω_r(z) + Ω_cb(z) + Ω_ν(z)]. 








In terms of present day values, the contribution of the relativistic species, Ω_r(z) can be written as,

    Ω_r(z)    = Ω_γ(z) + Ω^ rel_ν(z)
       = Ω_γ_0[1 + N^ rel_ν7/8(T_ν/T_ CMB) ](1+z)^4(H_0/H(z))^2 ,

where the present day value of the photon density parameter is

    Ω_γ_0 = π^2/15(k_ B T_ CMB)^4/ħ^3 c^58π G/3H^2_0 ,

N^ rel_ν is the number of relativistic neutrino species, k_ B is the Boltzmann's constant, T_ν (1.9524  K) and T_ CMB (2.7255  K) are the neutrino and CMB photons temperature at the present time[The neutrinos and photons temperature are linked via the relation T_ν/T_ CMB = (4/11)^1/3, assuming instantaneous decoupling. If one wants to take into account distortions induced by non-instantaneous decoupling and neutrino flavour oscillations, one usually addresses this by a slightly modified value of N_ν^ rel.],  H_0 is the present day value of the Hubble parameter, and G is the gravitational constant. On the other hand, the cold matter density parameter, Ω_cb(z), is

    Ω_cb(z) = Ω_cdm(z) + Ω_b(z) =(Ω_cdm_0 + Ω_b_0)(1+z)^3 (H_0/H(z))^2 .

The massive neutrinos contribution in Eq. (<ref>), is given by,

    Ω_ν(z) =   15/π^4(T_ν/T_ CMB)^4 N_νΩ_γ_0 (1+z)^4
       ×ℱ[M_ν/(N_ν k_ B T_ν)/1+z](H_0/H(z))^2  ,

where Ω_γ_0 is given by Eq. (<ref>), N_ν is the number of non-relativistic massive neutrinos, M_ν = ∑_i m_ν,i is the sum of the masses of the degenerate massive neutrinos and the function ℱ is defined as,

    ℱ(y) ≡∫^∞_0 x^2√(x^2 +
         y^2)/1+e^x dx .

The total matter contribution to the energy density is,

    Ω_ m(z) = Ω_ cb(z) + Ω_ν(z) ,

and the neutrino mass fraction today, f_ν, can be written as,

    f_ν(z) = Ω_ν(z)/Ω_ m(z) .

Finally, the Hubble rate throughout cosmic time in a flat Universe is given by

    H(z) = H_0[Ω_r0 (1+z)^4 + Ω_cb0(1+z)^3 + Ω_ν E^2(z) + Ω_Λ]^1/2 ,

where E(z)≡ H(z)/H_0.

The evolution of the density parameters of the different matter and energy components, Ω_ X(a), as a function of the scale factor a is shown in Figure <ref>. In the displayed case, we have considered a relativistic massless neutrino (N^ rel_ν = 1) and two degenerate massive neutrinos (N_ν = 2) with masses m_ν,i = 50 meV each, i.e., M_ν =∑_i m_ν,i = 100 meV. This also corresponds to the case we used in our largest simulations with massive neutrinos.




 §.§ Initial conditions and linear growth with massive neutrinos



Traditionally, N-body simulations have been evolved using Newtonian gravity in the context of a homogeneous Friedmann-Lemaitre background solution <cit.>, although a few authors have included relativistic effects on large scales <cit.>. This is justified because once large-scale perturbative modes are inside the horizon, a Newtonian treatment accurately recovers their linear evolution. At the same time, non-linear evolution occurs on small enough scales and involves weak enough gravitational fields that general relativistic corrections are negligible, except in the immediate vicinity of black holes and neutron stars.

However, when simulation boxes with sizes of around 1 and beyond are considered, the largest modes are actually of super-horizon size, at least at high redshift. Then ordinary Newtonian N-body codes do not deliver a correct growth rate on these scales, and questions related to gauge choices arise as well <cit.>. In simulations without neutrinos, this is usually not a significant concern, as one typically constructs the initial conditions by back-scaling the linear theory power spectrum predicted by a Boltzmann code at z=0 to the initial redshift, using a k-independent linear theory growth factor. This yields an initial power spectrum that is not fully correct at the starting redshift, but the Newtonian code will evolve it on these large scales to arrive at the correct state at low-redshift.

When massive neutrinos are included, the same trick no longer works, because the neutrinos themselves introduce a scale-dependence in the linear growth factor. Furthermore, this linear theory growth factor can no longer be computed through a closed-form integral, unlike for flat cosmologies that feature only cold matter and dark energy <cit.>. Different approaches have been proposed in the literature to deal with this complication when constructing initial conditions, including sophisticated rescaling techniques, as realised, for example, in the REPS code <cit.>. These are needed, in particular, if the linear evolution of an N-body code does not exactly reproduce the linear evolution of a Boltzmann code, even on small scales, due to the (very common) omission of a consistent accounting of relativistic components such as the background photon field.

Our strategy has been to eliminate this latter error by outfitting GADGET-4 both with a treatment of relativistic components in addition to cold matter, and with a consistent computation of the Hubble rate in the presence both of radiation components (photons and massless neutrinos), and of massive neutrinos, accounting also for the gradual transition of the latter from the relativistic to the non-relativistic regime. We can then directly use the transfer functions computed by a Boltzmann code such as CAMB <cit.> or CLASS <cit.> at the starting redshift, i.e. without requiring any back-scaling, because now the simulation code correctly accounts for the linear growth, as long as both the initial mode amplitudes and their initial growth rates are initialised correctly. The latter requires the logarithmic growth rate for every mode at the starting redshift. We obtain this from the Boltzmann code by numerically differentiating the growth factor based on two transfer function outputs spaced equidistantly around the desired starting redshift. This is done as a function of wave number  k, and for dark matter and massive neutrinos separately. 

This still leaves the problem of super-horizon modes. If we, for example, would simply initialise our simulation with the linear theory power spectrum as computed by CAMB at our starting redshift z=63, then the fundamental mode in our large box (L = 2040  = 3000  Mpc) would lag behind in growth by about 2% by redshift zero in a case without neutrinos, whereas in a M_ν=100 meV model this lag would exceed 3%. This effect is illustrated in Figure <ref>, and it arises because Newtonian gravity underpredicts the growth of these super-horizon modes in the matter dominated regime. Note that the effect gets even larger for still bigger simulation boxes, while becoming negligibly small at late times and for small boxes. In any case, we approach this by multiplying the initial power spectrum with a correction factor that boosts the linear theory input power spectrum such that  at z=0, after evolution with the Newtonian code, it will coincide with the result of the Boltzmann code <cit.>. Note that the power spectrum will also agree with the Boltzmann code at all intermediate redshifts in the presence of massive neutrinos, unlike for the standard rescaling approaches. Only at Gpc-scales and beyond, the amplitudes can differ at the percent level due to the super-horizon correction, as illustrated in Figure <ref>.

We have implemented the above corrections in the version of our initial conditions code  N-GENIC built into GADGET-4. It uses different transfer functions as computed by the  Boltzmann code CAMB for cold dark matter and massive neutrinos. In addition, we use the Boltzmann code to compute derivatives of the corresponding transfer functions with respect to the scale factor, as needed to initialise the velocities of the matter components. Finally, we apply these through 2^ nd-order Lagrangian perturbation theory to the unperturbed particle load, here modelled as two Cartesian grids for cold dark matter and neutrinos. The dark matter particles were aligned with the 10240^3 Fourier grid used to compute the displacement field, so that no interpolation to the particle positions was necessary. The unperturbed neutrino grid was shifted relative to the cold dark matter particles by half a neutrino grid spacing. Note, however, that the large streaming velocities of the neutrinos quickly decorrelates their coordinates relative to the dark matter particles anyway. As the resolution of the neutrino grid is much lower than the dark matter grid, the Fourier grid substantially over-resolves the Nyquist frequency of the neutrinos, so that interpolation errors of the displacements to the neutrino positions are likewise negligible.




 §.§ The δ f-method



Recently, <cit.> proposed a novel approach to represent massive neutrinos in cosmological simulations, which has become known as the δ f-method. We here provide our own derivation of this technique.

Let f(x⃗, v⃗, t) be the neutrino distribution function, i.e. the number density of neutrinos of mass m_ν as a function of phase space coordinates (expressed in terms of comoving coordinates and conjugate velocities) and time. Then we can obtain the neutrino mass density, 

    ρ(x⃗, t)  = m_ν∫ f(x⃗, v⃗, t) 
          dv⃗ ,

by integrating out the velocity distribution. Now let us introduce the time evolution of a spatially undisturbed neutrino background, f_0 = f_0(v⃗, t), and consider the difference between the actual neutrino density and this background:

    ρ(x⃗, t)  = m_ν∫ f_0   dv⃗  +  m_ν∫[
        f(x⃗, v⃗, t)  - f_0  ]  dv⃗.

Let us now integrate over some finite volume V (which could be a computational cell, or some region of the simulation box) in order to simplify the argument we are going to make later on. The enclosed neutrino mass inside in this volume is then

    M_ν (t)  =ρ_0 V  +  m_ν∫[
        f(x⃗, v⃗, t)  - f_0  ]  dv⃗ dx⃗,

where ρ_0 stands for the (time-dependent) average density of neutrinos. We now Monte-Carlo integrate the integral on the rhs with a sample of N points in the chosen volume. We  use a sampling function f^* proportional to f itself (which is the concept of importance sampling), with f^* normalised  as f^* = (1 / N_ν) f, where N_ν is the number of neutrino particles in the volume.
We can thus write the integral as:

    M_ν (t)  =ρ_0 V  + m_ν∫
        f - f_0  / f^*   f^*   dv⃗ dx⃗.

By construction, dp = f^* dv⃗ dx⃗ is the probability to find a sampling point in an infinitesimal phase-space volume around (dx⃗, dv⃗).
The integral can therefore be approximated as

    ∫
        f - f_0  / f^*  f^*   dv⃗ dx⃗ ≃ 1/N∑_i  f(x⃗_i, v⃗_i, t)  -
      f_0(v⃗_i, t) / f^*(x⃗_i, v⃗_i, t) .

A key step is now to argue that the sampling points need not necessarily be drawn from the current f, but instead can be identified with sampling points used to sample the initial conditions at the initial time t_0, and which were evolved along characteristics to the current time t.

At this initial time, let us assume that the sampling points have phase-space coordinates (x⃗_i^0, v⃗_i^0). The evolution of the particles between t_0 and the current time t is governed by the collisionless Boltzmann equation, which preserves the fine-grained phase-space density along individual particle orbits. We hence know that we have  f(x⃗_i, v⃗_i, t) = f(x⃗_i^0, v⃗_i^0, t_0). We therefore can write:

    M_ν (t)  =ρ_0 V  + 
    m_ν N_ν/N∑_i [ 1 -  f_0(v⃗_i, t) / f(x⃗_i^0, v⃗_i^0, t_0)].
 
The prefactor m^*_ν≡ m_ν N_ν / N is the fiducial simulation particle mass we use to represent the neutrinos. The quantity f_0(v⃗_i, t) is known and can simply be evaluated for the current velocity of the particle as the background density times the Fermi-Dirac distribution F  evaluated for v⃗_i:

    f_0(v⃗_i, t) = ρ_0 F(v⃗_i) .

A further assumption we make is that the initial phase-space distribution can be written as

    f(x⃗_i^0, v⃗_i^0, t_0) ≃ρ_0 (1+δ_i^0) F(v⃗_i^0),
 
where δ_i^0 is the initial density perturbation for the neutrino particle i. This approximation is not exact and thus will introduce some initial transients that take some time to decay away. See <cit.> for ways to improve this further, the simulations we consider here are however run with this physical approximation. Using it, the δ f-method predicts the neutrino mass in our test volume as

    M_ν (t)  =ρ_0 V  + 
    m^*_ν∑_i 
    x⃗_i ∈ V[ 1 -  F(v⃗_i) /(1 +δ_i^0)  F(v⃗_i^0)],

which is equivalent to assigning each neutrino particle a variable mass

    m̃ = m^*_ν[ 1 - 
          F(v⃗_i) /(1 +δ_i^0)  F(v⃗_i^0)],

based on its current phase-space coordinates. This mass, instead of m^*_ν, is used for computing the gravitational field with the Poisson solver implemented in the GADGET-4 code. This importance sampling greatly reduces sampling noise in the neutrino density field, a feature that is made possible by exploiting not only the current coordinate of the simulation particles, but also their current velocities. Note however that this does not change the character of each particle as representing a random neutrino orbit. All particles still follow an unchanged equation of motion, except that the force they experience is less affected by shot noise.

In our discussion so far we have ignored the fact that the neutrinos may have relativistic velocities, particularly at early times or for light neutrino masses. We address this by using the particle momentum p⃗_γ = m_νv⃗ /( 1-v^2/c^2)^1/2 in the above phase space arguments, and not the kinematic velocity v⃗ itself. Note that it is also the momentum that we obtain from the initial Fermi-Dirac distribution function. We integrate the neutrino particle momenta in the equations of motion by  approximating their temporal change with the ordinary Newtonian gravitational force, with neutrinos contributing with their rest masses to the source function of the Poisson equation. The particles are moved in space, however, with their kinematic velocity v⃗. Note that the expansion of space also redshifts p⃗_γ, and in that way automatically induces the correct transition from the relativistic to the non-relativistic regime.







As <cit.> first demonstrated, the δ f-method is indeed capable of reducing the shot noise of the neutrino particle distribution in cosmological N-body simulations. We show an example of this based on our own simulations in Fig. <ref>, where we compare the total matter power spectrum of three neutrino simulations for the M_ν = 100 meV model at the present time, using either the plain particle method or  the δ f-method. Our test simulations consist of boxes with side-length of 500 and use 540^3 CDM particles. For the particle method we addionally evolve 540^3 neutrino particles, and for the δ f-method we show two configurations with 540^3 and 116^3 extra neutrino particles, respectively[Note that 116^3 is roughly 1% of 540^3.].

When comparing the power spectra measurements for the simulations with 540^3 neutrino particles (red and orange solid lines), we can see from Fig. <ref> that the δ f-method reduces the shot-noise of the neutrino distribution by almost two orders of magnitude. In addition, we find that using only 116^3 neutrino particles with the δ f-method (green solid line) produces nearly the same level of shot-noise as using 540^3 neutrino particles with the particle method (red solid line). This is informative for selecting the relative particle numbers of neutrinos and cold dark matter in large-scale cosmological simulations with massive neutrinos. For the calculations we present later in this paper, the neutrino particle number is chosen such that the resulting shot noise level of the neutrinos at z=0 is very similar to that of the cold dark matter particles <cit.>.




§ SIMULATIONS WITH MASSIVE NEUTRINOS



We here focus on the neutrino simulations of the MillenniumTNG project <cit.> which presently consist of two large-volume cosmological simulations with box size L=2040 (3 Gpc; MTNG3000) and massive neutrinos with a summed mass of M_ν = 0.1 eV, in addition to several 430 (632.4 Mpc; MTNG630) runs with different neutrino masses. All the simulations were run with the GADGET-4 simulation code <cit.> using the δ f-method outlined above (cf. Sec. <ref>). For each of the models, we have run two realisations using the fixed-and-paired technique of <cit.>, which we denote in the following as “A” and “B” runs. The initial conditions of the runs of each pair differ in the phases of each perturbation mode by 180 degrees, or in other words, the sign of the linear density fluctuations is reversed. As <cit.> have shown, by averaging the results of such a simulation pair, certain leading higher-order deviations from linear theory cancel, effectively suppressing the variance compared to a random realisation in the same volume, and thus leading to an improved estimate of the expected cosmological mean.

Substantial computational resources, particularly memory, are needed to generate the very large MTNG3000 initial conditions.  To retain the option of being able to make relatively inexpensive zoom initial conditions for objects in these volumes we use the Panphasia method outlined in <cit.> to set the phases for the two MTNG3000 simulations. The essence of this method is to construct the Gaussian white noise field, from which the phases of the simulation are inherited, in a hierarchical way based on an octree spatial structure. The white noise field is built from piecewise polynomials localised within octree cells. Coupling this construction method with a suitable pseudo-random number generator it is possible to define extremely large realisations of the phases that cover all physical scales of interest for the original simulations and any follow up zoom simulations. The hierarchical structure of the field lends itself to the efficient creation of zoom initial conditions.

This approach, however, is not compatible with the fixing part of the fixed-and-paired method <cit.> when applied to all Fourier modes. In practice the variance suppression we need can be achieved by fixing just the poorly sampled low wavenumber Fourier modes as the sheer number of high-k modes guarantees behaviour close to the mean.  For MTNG3000, only Fourier modes with wavelengths greater or equal to 50 Mpc were fixed – a total of 452044 independent modes – while all other mode amplitudes are left free. With almost half a million fixed modes the measured linear power spectrum of the free modes is within 1% of the target for bins at least as wide as the fundamental mode of the periodic volume. 

The fixed modes are confined in the Fourier domain to large wavelengths and it must be possible to represent these fixed modes very accurately using polynomials confined to the largest octree cells. If that is the case the information in the smaller octree cells can be treated as independent allowing the creation of efficient zoom initial  conditions. This can be achieved by using higher degree polynomials than was used in the original Panphasia  field <cit.>, which was built from just degree one polynomials.  Fortunately the computational cost of going to higher degree polynomials can be offset in practice by much more efficient code for computing the Panphasia field and a six degree scheme has already been implemented in the public MONOFONIC cosmological initial conditions code <cit.>.  This version of the Panphasia field has also been used for the FLAMINGO project <cit.>. As with the original Panphasia field the phases for the MTNG3000-A volume are described by a plain text `phase descriptor', which in this case contains additional information about the degree of the polynomials and which Fourier modes are fixed. We reproduce here it for future reference[[Panph6, L19, (301309,431832,75016), S5, KK3600, CH2926231125, MillenniumTNG]].

The fiducial cosmology of our 100  meV neutrino simulations is consistent with the DES-Y3 constraints <cit.>: Ω_ cb = 0.30368, Ω_ b = 0.04870, Ω_Λ = 0.69393, h = 0.68 and σ_8 = 0.8040. In addition, two degenerate neutrino species each with 50 meV rest mass, yielding Ω_ν = 2.324× 10^-3, one massless neutrino species (Ω_ν^ rel = 1.232× 10^-5), and a standard photon background accounting for the cosmic microwave background with T_ CMB=2.7255  K were assumed (Ω_γ = 5.348× 10^-5). This means that 0.76 percent of today's mean matter density are made up of neutrinos in this cosmology – roughly half of what the stellar mass contributes.

The ICs were generated at z=63 with the method described in Sec. <ref> using the linear-theory transfer functions computed by CAMB. The MTNG3000 runs follow close to 1.1 trillion (10240^3) cold dark matter and more than 16 billion (2560^3) massive neutrino particles[Note that thanks to the δ f-method (cf. Sec. <ref>), we need only 1.5 percent of the  CDM particle number as neutrino particles in order to achieve a similar shot noise level.], leading to a cold dark matter particle mass resolution of 6.66×10^8, which is more than twice the resolution of the original Millennium simulation <cit.>, but in a much larger volume, and making them some of the largest and highest-resolution simulations with massive neutrinos run to date. The MTNG630 runs evolve 2160^3 cold dark matter and 540^3 massive neutrino resolution elements, giving the same particle mass resolution as MTNG3000.

In addition, we have run  M_ν = 0  eV (massless case) and M_ν = 0.3  eV models for the MTNG630 runs only. In these cases, we changed the Ω_ cb cosmological parameter in order to keep the value of the total matter density Ω_ m = Ω_ cb + Ω_ν = 0.306 fixed when the massive neutrino contribution is varied. We also slightly adjusted the value of Ω_Λ in order to compensate for the changing relativistic neutrino contribution and to retain an  exactly flat cosmology. For the normalisation of the initial power spectrum, we kept the CMB normalisation on the largest scales invariant, implying small changes in the linearly extrapolated small-scale σ_8 value at z=0.







In Table <ref> we summarise the cosmological parameters of the simulations, i.e. the corresponding values of the matter contributions (CDM, baryons and massive neutrinos) to the total density parameter, as well as the present-day values of the σ_8 parameter for each neutrino model. The simulations with varied neutrino mass help us to quantify the impact of massive neutrinos on galaxy observables such as the stellar mass function and the clustering of galaxies, as we will show below. The numerical specifications of our neutrino simulations are listed in Table <ref>; here we also include the specifications of the MTNG740 runs without neutrinos (full-hydro and dark matter only) in the MillenniumTNG project, for completeness.

Following the same outputting strategy as for the MTNG740 simulations <cit.>, we have measured the matter power spectra (total matter, cold matter and massive neutrinos) and produced halo/subhalo catalogues on-the-fly for 133 matching output times for all the neutrino simulations[Actually, for the MTNG3000-A run we have 13 additional output times.] between 0 < z < 63  (with more than 120 halo and subhalo catalogues  at redshifts z < 20). Halo catalogues (FoF groups) were constructed using the friends-of-friends (FoF) algorithm <cit.> at all the output times, and subhaloes were identified with the SUBFIND-HBT algorithm <cit.> as implemented in the substructure finder of GADGET-4 <cit.>. We also produced six particle lightcones for each run, keeping the same specifications and geometry as for the five lightcones of MTNG740 <cit.>, but adding a sixth lightcone in the form of an additional pencil beam.

In Figure <ref>, we show maps of the dark matter and neutrino density fields on one of the lightcones out to redshift z=2 <cit.>. One can clearly see how the clustering strength of the dark matter gradually increases from z=2 at the left and right edges of the map to the observer in the middle of the slices  at z=0, whereas the neutrino density contrast is still very small at z=2 but then starts to respond to the largest dark matter structures that have formed in the low redshift universe due to the progressive redshifting of the thermal neutrino velocities. We also produced mass-shell maps with 1.8 billion pixels for weak-lensing studies <cit.>. Furthermore, we have built merger trees from all subhaloes to allow the construction of physically motivated galaxy mock catalogues in massive neutrinos cosmologies, based on our updated version of the L-GALAXIES code for semi-analytic galaxy formation <cit.>.




We note that the data size of the MTNG3000 runs is staggeringly large. Each of the simulations contains about 1.16 × 10^9  FoF haloes at z=0 (for a minimum of 32 particles per group), and 1.46 × 10^9 gravitationally bound subhaloes. There are about ∼ 1 billion distinct merger trees[Subhaloes are stored in the same tree if they have a progenitor or descendant relation of any kind, or if they are part of the same FoF group at any time.] for each of the simulations, linking together about 1.5 × 10^11 subhaloes. Running L-GALAXIES on them produces about 1.4 billion galaxies in the volume of each of the simulations. A full particle snapshot of these simulations requires about 36 TB disc space, but thanks to the on-the-fly processing, we did not actually have to store this amount for the 133 output times (which would have led to a ballooning of the data volume to an inconvenient 4.7 PB – just for one of the two large simulations).







§ MATTER CLUSTERING



We show in Figure <ref> the clustering of the different matter components as measured through the power spectrum (separately for cold matter, P_ cb(k), and massive neutrinos, P_ν(k)) of our 100   meV simulations (MTNG3000-DM-0.1ν and MTNG630-DM-0.1ν) at redshifts z=0, 1 and 3. The vertical dotted lines indicate the fundamental mode of each box, k_ box = 2π/L_ box∼ 0.0031 and ∼ 0.0146, respectively. 

Since the 100 meV neutrino cosmological model is the same for these two boxes, we can use the comparison to quantify how box size effects limit the accuracy of our simulations. In the lower left subpanel of Fig. <ref> we can see that the cold matter power spectra of the MTNG630 run agree almost perfectly with the MTNG3000 spectra, from large-scales k∼ 0.03 down to very small-scales, k ∼ 100. Only small differences, less than 5 per cent, are seen between the small and the large box simulations near the fundamental wave number of MTNG630, reflecting only minor effects from the limited box size of this simulation. 

The box size impact on massive neutrino clustering, shown in the lower right subpanel of Fig. <ref>, is more noticeable. In this case, we find better than 2 per cent agreement between MTNG630 and MTNG3000 in the k-range 0.04 < k/[] < 0.4, which is the range over which the neutrino power spectrum is not dominated by shot noise. Interestingly, on still smaller scales – in the noise dominated regime – we observe a constant offset of ∼20 to 30 percent between MTNG630 and MTNG3000. In this regime, the power spectrum is below the nominal Poisson shot noise limit, thanks to the δ f-method, but whereas the actual Poisson shot noise for the simulations is very similar (P_ shot^ Poisson=L^3/N_ν), the much smaller effective noise level after using the δ f-method shows a noticeable box size effect. This can be attributed to the stronger neutrino clustering in the large simulation box as a result of its much better representation of massive galaxy clusters and superclusters; this diminishes the noise-damping provided by the δ f-method. The growth of such structures towards lower redshift is also the reason why the noise damping effect generally becomes weaker at late times.
In any case, these box-size tests indicate that the MTNG630 runs are still very helpful for cosmological analyses over the k-range of interest of current and future galaxy surveys such as DESI, Euclid and PFS, despite their limited box size.

Next we explore the time evolution of the clustering of the cold matter (CDM plus baryons) and massive neutrino components and compare the 100 meV and 300 meV neutrino models, using the MTNG630 simulations. The corresponding power spectra measurements are shown in Figure <ref> at redshifts z=0, 1, 3 and 7. In all panels of Fig. <ref> there is excellent agreement on large scales between the simulation measurements and the linear theory predictions (which are computed directly with CAMB for the corresponding times) for both matter components and both neutrino models. There is a monotonic evolution in the increase of power from high redshift (z=7; lower right panel of Fig. <ref>) to the present day (z=0; upper left panel of Fig. <ref>), as  expected from linear theory.

We also observe that neutrinos in the more massive case (M_ν = 300   meV; see the red dashed lines in Fig. <ref>) are more clustered than the lighter neutrinos  (M_ν = 100  meV; the red solid lines in Fig. <ref>). To compensate the higher neutrino clustering, note that the CDM+b component in the 300  meV model (blue dashed lines) ends up slightly less clustered than its 100   meV (blue solid lines) counterpart.




The impact of massive neutrinos on the matter clustering is often quantified by comparing the power spectrum with massive neutrinos to the massless (M_ν = 0   meV) case. This comparison is then expressed in terms of a suppression ratio given by

    S_ X(k) = P^ X_M_ν(k)/P^ X_ massless(k) ,

where X is here a placeholder for the component of interest, and M_ν corresponds to the total mass of the  neutrino species, i.e. 100  meV or 300  meV in our case. In the following we explore this suppression factor for the cold (cb) and total (m) matter power spectra. 

In Figure <ref> we show the suppression induced by massive neutrinos (M_ν= 100   meV and M_ν= 300   meV) in the total matter (S_ m(k), left panel) and cold matter (S_ cb(k), right panel) components with respect to the massless case at z=0.  We also include the linear theory suppression for each model, for comparison (dotted lines).

On large scales, the suppression predicted by the simulations is in good agreement with the linear theory for both contributions, P_ m(k) and P_ cb(k). In the non-linear regime (0.2<k/[]<2), however, the suppression takes a “spoon-like” shape with a minimum at ∼ 8% for the M_ν= 100  meV case and ∼ 24 % for the M_ν= 300  meV case, respectively, for the total matter power spectrum. On the other hand, the cold matter suppression is slightly weaker compared with that for the total matter. For the former we find a ∼ 7% and ∼ 20% suppression at k∼ 0.8 for the 100 meV and 300 meV models, respectively.

At very small scales, k > 2, we see an enhancement of power due to effects of nonlinear neutrino clustering. Nevertheless, the impact of massive neutrinos on small-scale structures is weak, meaning that they also are unlikely to significantly affect the abundance of low-mass dark matter haloes (cf. Sec. <ref>).



Note that the massless case does not contain a contribution from massive neutrinos, hence P_ cb(k) = P_ m(k), and the impact of massive neutrinos on the suppression, Eq. (<ref>) (see Fig. <ref>), is only driven by P^ X_M_ν(k). For this reason, we plot in Figure <ref>  the ratio between the total matter and the cold matter power spectrum measured from our 100  meV and 300  meV neutrino simulations. It is expected that this ratio, R(k) = P_ m(k)/P_ cb(k), asymptotes to the values of

    R(k) = P_ m(k)/P_ cb(k)≃
        1            for  k ≪ k_ nr
    
        (1-f_ν)^2       for  k ≫ k_ nr ,

where f_ν is the massive neutrino fraction, defined as f_ν≡Ω_ν / Ω_ m, and k_ nr is given by <cit.>,

    k_ nr≃ 0.018 Ω_ m^1/2(1  eV/M_ν) .

Our simulation measurements for R(k) are shown in Figure <ref> as solid blue and solid orange lines for our MTNG630-DM-0.1ν and MTNG630-DM-0.3ν runs at z=0, respectively. We can see that the simulations approach the value of unity at sufficiently large-scales (k ≪ k_ nr), and they agree perfectly with the asymptotic value of (1-f_ν)^2 on small-scales (k ≫ k_ nr). This also explains the ∼ 1.5% and ∼ 4.5% differences when comparing the power spectrum suppression of the total matter and cold matter of the 100  meV and 300  meV neutrino models shown in Fig. <ref>, respectively. 




§ HALO STATISTICS









The distribution of cold dark matter (sub)structures is impacted by the presence of massive neutrinos through the scale-dependent suppression of the power spectrum which we have highlighted above. On the other hand, the statistics of cold dark matter haloes, such as galaxy cluster counts, is a powerful probe to constrain cosmology <cit.>. We thus continue the analysis of the MTNG3000 and MTNG630 simulations by exploring the statistics and clustering of (cold) dark matter haloes in massive neutrino cosmologies. We measure the abundance of haloes from our group catalogues by considering each FoF group with at least one bound subhalo as a main halo, taking the particle with the minimum gravitational potential of the largest subhalo as halo centre. We characterize the halo masses using the M_200c spherical overdensity mass definition, i.e. these masses correspond to the mass contained within a radius enclosing a mean overdensity of 200 times the critical density.

To begin with, we validate the halo mass function (HMF) from our 100  meV neutrino simulations. Figure <ref> displays the redshift evolution of the halo mass functions measured as the  mean of the A and B realisations of the MTNG3000-DM-0.1ν and MTNG630-DM-0.1ν simulations, at redshifts z=3 (red lines), 2 (green lines), 1 (orange lines) and 0 (blue lines). Due to the high resolution of our simulations, we can find well-resolved haloes with masses as low as M_200c∼ 10^10.6, which corresponds to haloes with 60 CDM particles. Note that this mass limit is well below the halo mass for hosting the emission-line galaxies and Hα emitters which are the main targets of ongoing and future galaxy surveys, such as DESI and Euclid <cit.>. 

From the lower subpanels of Fig. <ref>, we observe that the MTNG630 runs predict almost the same HMF as the MTNG3000 runs in the mass range 10^11 < M_200c/[] < 10^13. The high-mass end of the MTNG630 HMF is however affected at all redshifts by the small box size of these runs. This can be seen as increasing counting fluctuations in the higher mass bins, particularly at higher redshift, corresponding to the masses of groups or clusters of galaxies with M_200c > 10^13, and in an eventual complete lack of the most massive clusters that are present in MTNG3000. However, this test confirms that the MTNG630 runs can still be very useful for cosmological analysis over a wide halo mass range, despite their limited box size.

Next we investigate the impact of varying the mass of neutrinos on the halo population. We plot the evolution of the HMF from z=3 to z=0 for our MTNG630-DM-0.1ν and MTNG630-DM-0.3ν simulations in the left and right panels of Figure <ref>, respectively. We additionally include the evolution of the massless neutrino model (MTNG630-DM-0.0ν; dashed lines) in each panel, for comparison.

As expected, the models with massive neutrinos exhibit a reduction of the HMF at the massive end. This well-known suppression of the halo mass function <cit.> is highlighted in the bottom subpanels of Fig. <ref>. We observe that the suppression is stronger for higher neutrino mass and higher redshift. For instance, at low masses (M_200c < 10^12), the suppression due to neutrinos with masses M_ν = 100  meV (left panel) is less than 10 percent, while for the more massive case (300 meV; right panel) the suppression exceeds 20 percent. On the other hand, for massive haloes with M_200c > 10^13 there is a deficit of haloes amounting to around 20% for the 100  meV case and even 50% for the 300  meV model. Naturally, this can be expected to have a substantial impact on the galaxy population as well, as we will explore below.




Before turning to this, we first investigate the impact of massive neutrinos on the halo clustering of the MTNG630 runs, but for conciseness we only focus on the present time. For definiteness we define four halo populations with number densities of n_h = 3× 10^-3 (Nh1 sample), n_h = 1× 10^-3 (Nh2 sample), n_h = 6× 10^-4 (Nh3 sample) and n_h = 3× 10^-4 (Nh4 sample), for the three neutrino models (massless, 100  meV and 300  meV). The specific number densities of our halo selected samples are motivated by the observed number densities of galaxies of recent and current surveys, such as BOSS and DESI <cit.>.

Figure <ref> shows the real-space halo two-point correlation function measured from the MTNG630-DM-0.1ν (left panel; solid lines), MTNG630-DM-0.3ν (right panel; solid lines), and MTNG630-DM-0.0ν (both panels; dashed lines) simulations at z=0, for the four halo samples described above. We measured the clustering using 20 log-spaced radial bins between 0.5 < r/[] < 50. Note that we again plot the average of the A and B realisations of each model. We clearly observe from the upper left and right panels that samples dominated by more massive haloes have a larger clustering amplitude at r > 2, independently of neutrino mass, i.e., samples with lower spatial-density are more clustered and more highly biased at larger separations r. This follows the general expectation for an increase of halo bias with halo mass <cit.>.

In order to quantify the impact of massive neutrinos on halo clustering we plot the relative difference between the massive neutrino models and the massless case in the lower subpanels of Figure <ref>. From the 100  meV model (left lower panel) we find an enhancement of 3 percent of the clustering on large scales r>20 for the Nh1, Nh2 and Nh3 halo samples. The clustering of the Nh4 sample agrees very well with its massless counterpart, with a difference of less than 1 percent for scales r>2. On the other hand, the cold dark matter haloes in the 300 meV massive neutrinos cosmology are more clustered on large-scales (r>10), producing an enhancement larger than 2 percent, independently of the halo number density selection. This indicates that the large-scale halo bias in such massive models is higher than in the plain ΛCDM (massless) model, as previously found by <cit.> and <cit.>. 

We note that the massive neutrino simulations predict a slightly higher clustering amplitude at small-scales (r<2) for the Nh2, Nh3 and Nh4 samples as well, for both massive neutrino models.  To match the number of (cold) dark matter haloes of their massless counterpart simulation, each of the corresponding samples needs to include more haloes of lower mass, which are in principle expected to be less clustered, at least on large scales. This slightly counterintuitive result could thus reflect a stronger small-scale clustering of these haloes in neutrino models, or it may also  arise due to changes in the halo bias in massive neutrino cosmologies.







§ GALAXY STATISTICS






In the following, we study the impact of massive neutrinos on the properties and the distribution of galaxies computed with a physics-based semi-analytical galaxy formation model (SAM), something that has not been done so far. We generate realistic galaxy catalogues using the updated version of the L-GALAXIES semi-analytic galaxy formation code introduced by . This code originated in work done for the Millennium simulation <cit.> and has been progressively refined over the years in many studies <cit.>. The recent improvements most relevant for the present work include the use of the more robust substructure and merger tree tracking algorithms introduced in GADGET-4, a reworked time integration scheme of the differential equations used to describe the galaxy formation physics, a more flexible storage scheme for the star formation history of galaxies, and the ability to output galaxies and their photometry directly on the past backwards lightcone of a fiducial observer.

We have run the L-GALAXIES code on the merger trees of our simulations without any modifications of the galaxy formation physics parameters (such as star formation efficiency, feedback strength. etc.), i.e. the same values for them in the parameter files and the same compilation flags are used for the three massive neutrino models (0, 100 and 300 meV), modulo the small adjustments in the cosmological parameters of the models needed to realise a massive neutrino component, which has a very small influence on the Hubble rate computation done by L-GALAXIES. We are thus not testing whether differences due to the presence of neutrinos could be partially or fully absorbed into changes of the (uncertain) galaxy formation physics parameters. Notice that throughout this section we will show results for the average of the A and B realisations, unless stated otherwise.

First, we consider the evolution of the cosmic star formation density (SFRD) for the full galaxy population as measured from the SAM galaxy catalogues of the MTNG630 simulations. In Figure <ref> we show the evolution of the star-formation rate density from z=10 to z=0. We can clearly see that the massive neutrinos models (100  meV and 300  meV) track the overall shape of the massless case, i.e., the SFRD first increases rapidly with time (decreasing redshift), reaches a peak at z∼2.5, and then decreases at later times to the present epoch as the dark matter halo population starts to become increasingly dominated by massive quenched galaxies with older stellar populations. However, the SFRD at high redshift is clearly reduced for the massive neutrino models compared to the massless case, whereas the models reach very similar star formation densities at the present time.

We directly quantify the relative impact of massive neutrinos on the SFRD in the lower panel of Fig. <ref>. The star-formation rate of the M_ν = 0.3  eV model is suppressed at almost all redshifts, but this suppression is much stronger at early times (z>5), reaching forty percent with respect to the massless case. This is due to the suppression of low-mass haloes at high redshift in this model (as shown in the right panel of Fig. <ref>; there is already a deficit of 30%  in the CDM haloes with masses M_200c∼ 10^12). While this effect is weaker for the 100  meV neutrino model, it still suppresses the star-formation density by 15% at redshifts z>5. The suppression becomes smaller for both neutrino models at late times, and at low redshift the halo population which accounts for the majority of the star formation is almost the same as in the massless neutrinos counterpart (see, e.g., the orange and blue lines in the lower subpanels of Fig. <ref>).







We now investigate the galaxy stellar mass function, which is central for describing the formation and evolution of the galaxy population over cosmic time. The galaxy stellar mass function (GSMF) measured from our MTNG630 (solid lines) and MTNG3000 (red dashed lines) simulations is shown in Figure <ref> at z=0 (left panel) and z=1 (right panel). The GSMFs are generally in good agreement with observations (blue symbols with errorbars) at both redshifts. Also, we see that the resolution of our simulations is sufficient to produce galaxies with a stellar mass of M_⋆ = 10^8, consistent with the convergence studies of <cit.>.

From the bottom subpanels of Fig. <ref>, which display the relative differences between the models, we observe that the stellar mass function measured from the MTNG630-DM-0.1ν simulation (blue solid line) is in excellent agreement with the massless case for galaxies with M_⋆ < 10^10. However, we find a suppression of the GSMF for galaxies with stellar mass M_⋆∼ 10^11 of around 10%, and this suppression is stronger for massive galaxies, reaching 30% for galaxies with M_⋆ = 10^12 at z=0, whereas at the higher redshift of  z=1 it is about 20% for galaxies with M_⋆∼ 10^11.5. 

Note that we have also included measurements of the GSMF for the MTNG3000 runs (red dashed lines) in all panels of Figure <ref>. For the stellar mass range 10^8 < M_⋆/[] < 10^10 , the suppression of the GSMF predicted by the MTNG3000-DM-0.1ν simulations is in very good agreement with that inferred from their 630 Mpc counterparts (MTNG630-DM-0.1ν; blue lines). However, for massive galaxies beyond the knee of the GSMF the models for the 3000 Mpc box size predict a higher galaxy abundance. But note that we are in this figure comparing the MTNG3000-DM-0.1ν (M_ν = 0.1 eV model)  GSMFs runs to the MTNG630-DM-0.0ν (massless) model, hence the difference at the high-mass end (M_⋆ > 10^11) is expected due to the different box sizes. In this regime, the MTNG3000 boxes produce a large number of  massive objects that are simply absent in the MTNG630 boxes, and in this sense the latter are not sufficiently representative for the universe as a whole.

We also find that the M_ν = 0.3  eV model (orange lines in the lower subpanels of Fig. <ref>) produces a suppression of 10% and 5% in the mass range 10^8 < M_⋆/[] < 10^10 at z=1 and z=0, respectively. There is a noticeable difference of around 40% at the bright end of the galaxy stellar mass function in both panels of Fig. <ref>, between the M_ν = 0.3, eV and the M_ν = 0  eV models. This is consistent with our findings for the halo mass function shown in the right panel of Fig. <ref>.

The clustering of galaxies is another powerful tool to constrain cosmological models and galaxy formation physics. For this reason we here present a preliminary analysis of the galaxy clustering from our neutrino simulations. This is primarily meant as an illustration of what is possible with our models, whereas a comprehensive analysis of the galaxy clustering properties of the MTNG630 and MTNG3000 simulations is beyond the scope of this paper and  is left for future work.




We measure the real-space galaxy two-point correlation function (2PCF) from the MTNG630 simulations at z=1 and z=0 using 30 logarithmically spaced radial bins in the range 0.1 < r/[] < 40, as presented in Fig. <ref>. We consider both stellar mass (M_⋆) and star-formation rate (SFR) selected galaxies with number densities of n_g = 3× 10^-3 (Ng1) and n_g=1× 10^-3 (Ng2). The selection by M_⋆ and SFR was chosen because it has been shown that these quantities are good proxies to define luminous-red and emission-line galaxies in the context of galaxy surveys <cit.>. The number densities of the Ng1 and Ng2 samples are similar to the expected spatial densities of galaxies that will be observed by the DESI and Euclid surveys. For definiteness, the specific values of the stellar mass and star formation cuts that we adopted to yield these number densities are listed in Table <ref>.

The galaxy 2PCF measured from the M_⋆-selected samples is shown in the upper row of Figure <ref>. From the upper right panel of Fig. <ref> we see that the three neutrino models predict almost the same clustering at z=1 at scales of r>2 for both galaxy samples,  Ng1 and Ng2. On small-scales, r<1, the massive neutrinos suppress the clustering signal by around 2% and 7%, for the 100  meV and 300  meV models, respectively. On the other hand, at z=0 (upper left panel), the clustering signal of galaxies from the 100  meV M_⋆-samples is similar to the results at z=1, because the galaxy stellar mass function is only slightly affected by the 100 meV neutrinos at z=0 and z=1 (cf. Figure <ref>). In the 300 meV model, we find a deviation of 5% below the massless case at scales r>3 at z=0. Larger effects are seen on scales r<2, where the clustering signal drops by around 15% and 10% compared to the massless model for the Ng1 and Ng2 samples, respectively. 

The clustering of SFR-selected galaxies is displayed in the bottom row of Fig. <ref>. We find that the clustering signal of SFR-selected galaxies is generally weaker than for the stellar-mass selected samples. This is because star-forming galaxies are often blue, young, low-mass galaxies that are less clustered than massive galaxies. We observe that the 2PCF of the 100   meV model at z=1 (bottom right subpanel of Fig. <ref>) is 2% larger than that of the massless case at scales r>1. The amplitude of the galaxy clustering signal of the 300  meV samples at z=1 is higher than for the massless case. This can be understood because we need to include more galaxies with lower SFR to match the desired number density.  At the present time (bottom left subpanel of Fig. <ref>), the clustering amplitude predicted by the massive neutrino models is 3% lower than for the massless case. This is because at the present time the Ng1 and Ng2 samples have almost the same SFR cut (see Table <ref>) for all models. Note that in both panels we observe that the SFR-Ng2 sample of the 100  meV model (solid purple line) produces almost the same clustering amplitude as the massless case over a broad range of scales. This is also expected based on the nearly identical SFR cut in both models. Nevertheless, there is a noticeable difference on small-scales, r<0.3, where the amplitude is decreased by 10%.

Finally, we can take advantage of the large volume of the MTNG3000-DM-0.1ν simulations to measure the large-scale baryonic acoustic oscillation (BAO) scale galaxy clustering from our SAM catalogues. One of the goals of ongoing and future galaxy surveys is to use the BAO feature as seen in different galaxy selections across cosmic time to reconstruct the cosmic expansion history, and in this way to constrain the cosmological parameters – ideally including the sum of the neutrino masses.

As an illustration of the expected effects in this approach, we measure the monopole of the redshift-space correlation function at different redshifts using our MTNG3000 runs and compare the results with current observational data. To do so, we define four galaxy samples that match the observed galaxy number densities, as follows:

    
  * DESI BGS: the desired number density is n_ BGS = 3.8× 10^-3 at the median redshift of z=0.3 <cit.>. We select galaxies ranked by their r-band magnitude from our L-GALAXIES catalogues, matching the observed n_ BGS.
    
  * CMASS: this sample has an observed spatial number density of n_ CMASS = 4.4× 10^-4 at z=0.5 <cit.>. In this case, we rank galaxies in our simulations by stellar mass in descending order, and we select the N most massive galaxies that reproduce the observational number density.
    
  * DESI LRG: similarly to the CMASS sample, we rank the SAM galaxies by stellar mass, and then select them to reproduce a number density of n_ LRG = 7.4× 10^-4 at z=0.74 <cit.>.
    
  * DESI ELG: to build this sample we just apply a simple SFR-cut in our galaxy catalogues such that we match the observed number density of n_ ELG = 1× 10^-3 at z=1 <cit.>.


The resulting clustering measurements from our simulations are shown in Figure <ref>. We observe that our simple galaxy selections based on r-band magnitude, galaxy stellar mass, and star-formation rate reproduce the amplitude and shape of the observed galaxy clustering signal quite well, which is reassuring. A potentially sizeable mismatch is seen for CMASS at scales s > 120 h^-1 Mpc, but whether or not this is really significant can only be assessed with a more careful comparison based on a detailed forward modelling approach.

To build the fully realistic mock galaxy catalogues required for this exercise, we need to apply a more sophisticated colour-magnitude selection and impose survey masks to also account for the appropriate survey geometry <cit.> and the effects this entails. However, as our goal here is merely to show that our MTNG3000 simulations have sufficiently large volume to produce galaxy catalogues that are capable of such BAO-scale analysis we defer an in-depth analysis of the match to observational galaxy surveys to a dedicated future study that implements a detailed forward modelling.




§ SUMMARY AND CONCLUSIONS



We have introduced a new set of simulations that are part of the MillenniumTNG (MTNG) project. These simulations focus on the neutrino sector and consist in each case of two realisations (A and B) for cosmic variance reduction. We include massive neutrinos with summed rest masses M_ν = 100  meV in two different volumes with box sizes L=2040 (3000 Mpc) and L=430 (∼630 Mpc), dubbed  MTNG3000-DM-0.1ν and MTNG630-DM-0.1ν, respectively. The large box runs evolve close to 1.1 trillion cold dark matter particles, and more than 16 billion neutrino particles with the δ f-technique, giving the cold dark matter a particle mass resolution of 6.66× 10^8. In addition, we have carried out two further simulations for neutrino models with masses M_ν = 300  meV and M_ν = 0  meV, using the smaller volume configuration (MTNG630-DM-0.3ν and MTNG630-DM-0.0ν).

In this paper, we have also detailed the numerical implementation of our approaches to include massive neutrinos (and other relativistic species) in cosmological simulations. This entailed changes in the GADGET-4 code to properly account for relativistic components, and the use of the so-called δ f-method <cit.> to reduce shot noise in a particle-based representation of neutrinos, as well as adjustments of the initial conditions generation to be able to directly align them with the multi-species transfer functions computed by Boltzmann codes such as CAMB, and to correct for super-horizon modes present in ∼Gpc-scale N-body simulations.

As part of a series of validation tests of the matter clustering of our M_ν = 100  meV simulations, we have found an excellent agreement between the cold matter (CDM+b) and massive neutrinos components of the measured power spectrum from MTNG3000-DM-0.1ν and  MTNG630-DM-0.1ν (see Figure <ref>). In addition, we established that the evolution of the matter clustering of the MTNG630-DM-0.1ν and MTNG630-DM-0.3ν simulations is in good agreement with linear theory predictions on the largest scales (see Fig. <ref>). Also, we observed a spoon-like suppression in the power spectrum due to presence of massive neutrinos (in Fig. <ref>). The corresponding effect is stronger for more massive neutrino models, e.g., M_ν = 300  meV, and the small-scale behaviour of the effect reflects non-linear neutrino clustering.

We have then explored the statistics and clustering signal of cold dark matter haloes in our simulations with massive neutrinos. We find excellent agreement between the halo mass functions measured from the MTNG3000-DM-0.1ν and MTNG630-DM-0.1ν simulations (see Fig. <ref>). We also quantified the impact of massive neutrinos on the halo mass function (HMF), finding a deficit of haloes that becomes larger for more massive neutrino models, and is stronger at higher redshifts (see Fig. <ref>). Massive neutrinos also affect the clustering of cold dark matter haloes, where we detect noticeable differences at large scales r>10-20, depending on the neutrino model (cf. Fig. <ref>).

Finally, one of the most important advances in the MTNG simulation project is that it can produce physically grounded galaxy catalogues with the updated version of the L-GALAXIES semi-analytical model of galaxy formation <cit.>. With such galaxy catalogues we are able to study the impact of massive neutrinos on astrophysical and cosmological observables. For instance, we showed that massive neutrinos suppress the star-formation history of galaxies across cosmic time (cf. Fig. <ref>). In addition, the galaxy stellar mass function (GSMF; Fig. <ref>) is affected  in a similar fashion to the HMF, i.e., the predicted GSMF in models with massive neutrinos is lower than that of their massless counterpart. Moreover, we established some deviations in the real-space galaxy correlation functions of stellar mass and star formation rate selected galaxies in different neutrino cosmologies (see Fig. <ref>). Furthermore, using our large boxes (MTNG3000-A/B) we were able to measure the clustering of galaxies at the BAO scale. We find that simple galaxy selection criteria that yield similar galaxy densities as in the observed samples can reproduce the shape of the large-scale galaxy clustering of current observation rather well and reproduce a clear BAO feature with very high signal to noise (as shown in Fig. <ref>).

The transformative increase of the observational volume and the number of observed galaxies ushered in by galaxy surveys such as DESI and Euclid calls for precise numerical calculations to assist the proper interpretation of this data. The MTNG simulation project is an attempt to live up to this challenge and to provide a useful basis for the generation of truly realistic galaxy mock catalogues that can improve the faithfulness of comparisons between theory and observations. While the present paper has merely scratched the surface in this regard, it established important foundations for this goal by demonstrating that simulations with massive neutrinos can be carried out with high accuracy at low computational cost, and that their effects should thus be included in future studies of precision cosmology with simulations. While the corresponding effects due to neutrinos are admittedly small, they are nevertheless large enough to offer the exciting prospect to put tighter constraints on the neutrino masses based on cosmological observations – which is addressing a truly fundamental goal of astroparticle physics.




§ ACKNOWLEDGEMENTS


CH-A acknowledges support from the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2094 – 390783311. 
VS and LH acknowledge support by the Simons Collaboration on “Learning the Universe”. 
LH is supported by NSF grant AST-1815978.  
SB is supported by the UK Research and Innovation (UKRI) Future Leaders Fellowship [grant number MR/V023381/1]. 
RK acknowledges support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant and a Discovery Launch Supplement, funding reference numbers RGPIN-2024-06222 and DGECR-2024-00144.
The authors gratefully acknowledge the Gauss Centre for Supercomputing (GCS) for providing computing time on the GCS Supercomputer SuperMUC-NG at the Leibniz Supercomputing Centre (LRZ) in Garching, Germany, under project pn34mo. 
This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility, with equipment funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. 




§ DATA AVAILABILITY


The MillenniumTNG simulations will be made fully publicly available at <https://www.mtng-project.org> in 2024. The data underlying this article will be shared upon reasonable request to the corresponding authors.







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