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: Constraints on the $γ$-parameter for the vacuum solution of Cotton gravity with geodesics and shadows.







	 ednaldobarrosjr@gmail.com
Faculdade de Física, Universidade Federal do Pará, Campus Universitário de Tucuruí, CEP: 68464-000, Tucuruí, Pará, Brazil

     
    tarcisojunior17@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill

	 fslobo@ciencias.ulisboa.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal
Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     esialg@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill
Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará, Campus Universitário de Abaetetuba, 68440-000, Abaetetuba, Pará, Brazil

   drubiera@ucm.es
Departamento de Física Téorica and IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain

      
        fc53497@alunos.fc.ul.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     henriquefisica2017@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill



	





§ ABSTRACT 

We consider a recently introduced extension of General Relativity dubbed as Cotton gravity (CG), based on the use of the Cotton tensor, to estimate the size of a new constant γ appearing within a spherically symmetric, vacuum solution of the theory. Taking into account its non-asymptotically flat character, we use the inferred size of the central brightness depression of the supermassive object at the heart of the Milky Way galaxy (Sgr A*) by the Event Horizon Telescope to constrain at 2σ the CG parameter as γ M ≈ 3.5 × 10^-12. We study the potential observational consequences from the smallness of such a value using exact and numerical expressions for the deflection angle, optical images from optically and geometrically thin accretion disks, isoradials, and instability scales (Lyapunov index) of nearly bound geodesics associated to photon rings. Our results point towards the impossibility to distinguish between these two geometries using current and foreseeable techniques in the field of interferometric detection of optical sources.






Constraints on the γ-parameter for the vacuum solution of Cotton gravity with geodesics and shadows

    Henrique A. Vieira
    August 5, 2024
====================================================================================================







§ INTRODUCTION



Despite the undeniable success of Einstein's General Relativity (GR) in explaining a large number of gravitational phenomena in different regimes <cit.>, there are several drawbacks of the theory. On the one hand, the accelerated expansion of the Universe, as found by Type-Ia supernovae observations  <cit.>, can only be accounted for, in the frame of GR, via a hypothetical fluid known as dark energy <cit.>, whose properties can be assimilated to that of a cosmological constant term <cit.> yet there is no direct proof of its existence. Similarly, the explanation of galactic rotation curves demands more than 90% of the total mass of most galaxies to be dominated by a non-baryonic form of matter dubbed as dark matter <cit.>, which we have been unable to detect in any terrestrial observatory yet. On more theoretical grounds, the problems with the unavoidable development of space-time singularities inside black holes and in the early cosmological evolution and the loss of predictability they bring about <cit.>, the Hawking process and the associated information loss paradox <cit.>, and the development of a framework unifying gravity and quantum mechanics (quantum gravity <cit.>), are challenges that still haunt the gravitational community.

In response to these challenges, a large avenue of research has emerged in the last few decades arranged around alternative approaches to GR, generally dubbed as modified theories of gravity (see e.g. <cit.> for some reviews). Within this context, recently Harada presented a theory called “Cotton Gravity" (CG) <cit.>. In this proposal, the effects describing gravity are attributed to the Cotton tensor, and the corresponding field equations contain third-order derivatives. The main interest in this proposal comes from the fact that any solution of the Einstein equations, with or without a cosmological constant term, also satisfies the field equations of CG. Furthermore, the cosmological constant itself emerges as an integration constant out of the field equations. In addition, in <cit.> Harada used the post-Newtonian limit of CG to solve the field equations numerically, interpreting the rotation curves of several galaxies without employing dark matter sources. Note that the possibility that dark matter may be geometric effects of modified theories of gravity has been explored in the literature <cit.>.


Regarding new solutions within CG, the first non-trivial one with static, spherical symmetry was found in <cit.>, which extends the Schwarzschild solution via a new parameter γ. Other non-trivial solutions were found in <cit.>. On the other hand, in <cit.> it was emphasized that the Codazzi tensor can be derived from the equations of CG, which implies that Harada's formulation can be understood as a modification of the GR equations due to the presence of this residual tensor. This way, with the introduction of the Codazzi tensor, it became possible to express CG through a new formulation, which allowed for the representation  of conformally flat cosmological solutions without a vacuum in CG <cit.>. 

However, there is ongoing debate about the viability of CG as a potential extension of GR. Some researchers see CG as promising for addressing GR's limitations, while others doubt its theoretical consistency <cit.>. Further research is needed to determine whether CG can replace or complement GR, or if it will remain a theoretical curiosity.


In order to set constraints in the new parameter γ of CG, one can resort to the gravitational lensing of light. This is the deflection of light trajectories as they pass close to massive bodies (stars, black holes, galaxies, etc) and can act like a natural telescope by amplifying the source brightness and even creating multiple images. 
Recent developments related to this effect can be divided into two groups:  microlensing (i.e. the measurement of the collective magnification of various images) and strong lensing <cit.>. For the sake of this work we are (mostly) interested in the strong field regime, given the fact that it offers the possibility of testing GR beyond the weak-field limit tests carried out so far. In such a limit,  Bozza <cit.> proposed a formalism to calculate the deflection angle for every spherically symmetric metric in terms of a logarithmic expansion. Such a formalism has been very influential in advancing analytical methods to compare theoretical predictions with real images <cit.>, including alternative spherically symmetric metrics <cit.>.


The black hole shadow and photon ring heuristics are two such examples of visual phenomena caused by strong gravitational lensing <cit.>. The former consists in the dark silhouette of a black hole that is projected in the celestial plane, with a critical boundary separating backwards-traced geodesics that are scattered towards infinity from those that are captured by a black hole <cit.>. The latter appears as an infinite series of discrete and stacked, exponentially-damped in luminosity, lensed images of the emission surrounding a black hole, under the assumption of an optically thin and non-spherical accretion disk <cit.>. The recent progress in high-frequency very-long-baseline interferometry (VLBI) provides a direct way of comparing such heuristics with actual images, by the Event Horizon Telescope (EHT) Collaboration, of the super-heated overflow of the plasma around the supermassive central objects of M87* <cit.> and Sgr A* <cit.>. In turn, this provides an unprecedented opportunity to perform tests of new gravitational Physics, which has been put to good use very recently, see e.g.  <cit.>.

The main aim of this work is therefore to make use of this opportunity to study the optical appearances of the CG black hole above. In particular, we shall employ the reported bounds on Sgr A*'s shadow by the EHT Collaboration to constrain the CG black hole's parameter space. Next, we generate images of this geometry within this restricted parameter space when illuminated by an optically and geometrically thin accretion disk, modeled by three semi-analytic emission profiles developed by Gralla-Lupsasca-Marrone (GLM) <cit.>, and compare its optical appearance with that of a Schwarzschild black hole. Furthermore, we study the Lyapunov exponents of nearly bound orbits to elaborate on the differences between both geometries. In this work we consider units G=c=1. 

This paper is outlined in the following manner: In Sec. <ref>, we briefly present the field equations of CG and the static, spherically symmetric metric used throughout this work. In Sec. <ref>, we obtain the null geodesic equation and the effective potential of the CG black hole, using the above-mentioned spherically symmetric metric.
Section <ref> addresses gravitational lensing in CG, where we first derive the exact analytical expression for the deflection angle using elliptic integrals, followed by an approximate expression for the weak field regime.
In Sec. <ref>, we generate the images of the Cotton black hole using the restricted γ-space parameter obtained in the previous sections,  and qualitatively compare its appearance to that of a Schwarzschild black hole of the same mass parameter.
In Sec. <ref>, we perform an analysis of isoradial curves in the Cotton and Schwarzschild space-times, to complement the black hole images presented in the previous section. In Sec. <ref>, we explore the Lyapunov exponent of nearly bound orbits, which is a quantity that measures the instability scale of these orbits.
Finally, in Sec. <ref>, we summarize our results and conclude.
 



§ A SPHERICALLY SYMMETRIC SOLUTION OF COTTON GRAVITY




Recently, as mentioned above, a generalization of the gravitational field equations of GR, dubbed as “Cotton Gravity" (CG) <cit.>, was proposed. The field equations read as follows

    C_αμν=κ^2∇_βΘ_βαμν^β,

where κ^2=8π, is  Newton's constant in suitable units, ∇_β is the usual covariant derivative  built with the Christoffel symbols of the metric, C_αμν  is the Cotton tensor defined as

    C_αμν≡∇_μR_να-∇_νR_αμ-1/6(g_αν∂_μ-g_αμ∂_ν)R,

where R_μν stands for the Ricci tensor and R  is the Ricci scalar, and finally the energy-momentum tensor  Θ_βαμν is defined as

    Θ_βαμν   ≡   1/2(g_βμΘ_αν-g_αμΘ_βν-g_βνΘ_αμ+g_ανΘ_βμ) 
        
          -1/6(g_βμg_αν-g_αμg_βν)Θ ,

with Θ_βμ=g^ανΘ_βαμν and Θ=g^βμΘ_βμ. 

The most remarkable feature of CG gravity is that every solution of the Einstein equations with a cosmological constant term G_μν+Λ g_μν=κ^2Θ_μν satisfies Eq. (<ref>) <cit.>, while the opposite is not true. In contrast to GR, however, the cosmological constant in CG is determined by integration of the field equations. Furthermore, the conservation law (∇_βΘ_μν^β=0) is a natural consequence of the theory, just like in GR, as can be verified by contracting Eq. (<ref>) by g^αν to yield

    g^ανC_αμν=κ^2∇_βΘ_μν^β=0.


For the sake of this work we use the exact vacuum solution obtained by Harada in <cit.>. Similarly to GR, where R_μν=0, this solution is obtained in Cotton Gravity when C_αμν=0 is considered. Such a solution belongs to the static, spherically symmetric class of metrics with line element

    ds^2=-A(r)dt^2+dr^2/A(r)+C(r)(dθ^2+sin^2θ dϕ^2).

whose metric function reads as <cit.>

    A(r)=1-2M/r-Λ/3r^2+γ r  ,       C(r)=r^2  .

This is the first non-trivial solution found in CG theory, which contains a new parameter γ, not present in the Schwarzschild case of the GR, and which breaks the asymptotically flat character of the space-time. Moreover, the cosmological constant is obtained naturally as an integration constant no matter the value of the cosmological constant Λ, which we take to be vanishing in this work. We draw attention to the fact that solutions containing a linear γ r term have been obtained in other theoretical contexts (e.g. <cit.>).






§ GEODESICS AND EFFECTIVE POTENTIAL





 §.§ Gravitational lensing


In this section, we obtain the null geodesic equation and the effective potential of the CG black hole, using the spherically symmetric line element (<ref>). To this end, we consider the point-particle Lagrangian density given by

    L=1/2g_μνẋ^μẋ^ν=ϵ,

where ẋ^μ is the four-velocity of the photon,  ϵ=(-1,0,1) indicates the time-like, null, or space-like character of the geodesic, respectively, and an overdot denotes the derivative with respect to the affine parameter λ. Taking into account the metric (<ref>) yields the following form for the Lagrangian

    L=1/2[-A(r)ṫ^2+ṙ^2/A(r)+C(r)θ̇^2+C(r) sin^2 θϕ̇^2]=ϵ .



Given  the spherically symmetric character of the space-time, we can consider the motion in the equatorial plane, θ=π/2, without any loss of generality. In addition, since the Lagrangian (<ref>) does not directly depend on t and ϕ, the Euler-Lagrange equations imply the conservation of two quantities:


    E   =A(r)ṫ,
     
    L   =C(r)ϕ̇.,


where E is the energy of the particle and L the angular momentum, respectively (per unit mass). Since we are interested in null geodesics, ϵ=0, Eq. (<ref>) with the conserved quantities (<ref>) and (<ref>), leads to

    ṙ^2=1/b^2-V_ eff(r).

where we have conveniently re-absorbed a factor L in the affine parameter λ, and introduced the impact parameter b ≡ L/E. In this equation, the effective potential reads as

    V_ eff(r)=A(r)/C(r).

which for the  metric function considered in this work, Eq. (<ref>), reads

    V_ eff(r)=1/r^2-2 M/r^3+γ/r.

The maximum of the effective potential determines the region of unstable orbits, dubbed as the photon sphere, whose radius is denoted by r=r_p, and the corresponding impact parameter as the critical impact parameter, dubbed by b_c. The latter separates those orbits that are captured by the black hole, b<b_c, from those that are scattered back to asymptotic infinity, b>b_c. Those that have b=b_c may linger indefinitely at the photon sphere, turning several times around the black hole. To determine these quantities, we note that circular light orbits obey ṙ=0 and r̈=0 simultaneously. Under these conditions, r_p and b_c are expressed for the metric function (<ref>) as


    r_p   =√(1+6 γ  M)-1/γ,
    
        b_c   =3√(6)  M/√((1+ √(1+6 γ   M)+6 γ   M √(1+6 γ  M)+9 γ  M)),


which for γ=0 it recovers the critical b_c=3√(3)M and the photon sphere radius r_p=3M of the Schwarzschild geometry.

With the expressions (<ref>) and (<ref>) we can develop the equation of motion which describes the trajectory of the photon as

    (dr/dϕ)^2=C^2(r)(1/b^2-V_ eff).

These are the general expressions for a given spherically symmetric metric. 





 §.§ Shadow's radius and constraints to the γ parameter



The shadow of a black hole is a silhouette, a  region marked by a strong decrease in the observed luminosity, projected onto the celestial plane of a distant observer. This projection arises from photons originating from a background source, which are absorbed due to its intense gravity. To calculate the angular size and the shadow of the black hole, we adopt the formalism described by <cit.>. In this approach, they consider a static observer located at (r_0, θ_0=π/2), projecting an angle θ relative to the radial coordinate r, as given by

    tanθ=√(C(r)/A^-1(r))dϕ/dr|_r=r_0.

The shadow of the black hole, understood as the region which borders (on the observer's plane image) the critical curve, is thus constructed by the light rays emitted from the observer back to the source.

To proceed with our derivation, we conveniently re-write Eq.(<ref>) into

    (dr/dϕ)^2=C(r)/A^-1(r)(C(r)/A(r)E^2/L^2-1).

A light ray with impact parameter b>b_c will be scattered off the black hole at a radial distance r_0>r_m, while  the photon sphere (i.e. the locus of unstable bound geodesics, r_p) can be found via the condition dr/dϕ|_r_p=0  which allows to express the impact parameter b as

    1/b^2=A(r_p)/C(r_p).

Now, by introducing the definition

    h^2(r)=C(r)/A(r),

we can rewrite Eq. (<ref>) as

    (dr/dϕ)^2=C(r)/A^-1(r)(h^2(r)/h^2(r_p)-1).


Substituting Eq.(<ref>) into Eq.(<ref>), and after proper manipulations, allows to find the expression

    sin^2θ=h^2(r_p)/h^2(r_0),

which yields the angular size of the black hole's shadow θ_s as

    sin^2θ_s=b_c^2/h^2(r_0)=b_c^2A(r_0)/C(r_0).

Using the expressions worked out above, for CG the angular shadow's size reads as 

    sin^2θ_s   =   2/r_0^2γ^2 (γ +8γ^2M)(-1+6M√((1+6γ M))
       +   √((1+6γ M))-9γ M )(1-2M/r_0+γ r_0).








Similarly, we can obtain an expression for the shadow radius as seen by a far-away observer, a regime in which we approximate sin(θ) ∼θ and θ_s=b_c/r_0
which gives us the following relation (assuming θ=π/2)

    r_s=r_p√(g_tt(r_0)/g_tt(r_p)).

For CG this reads as

    r_s=(√(a)-1)/γ√((√(a)-1)[r_0(γ r_0+1)-2M]/r_0(1-√(a)+4γ M)) ,

where a=1+6γ M. In Fig. <ref> we depict the behaviour of the shadow's radius in CG with the parameter γ, considering the black hole distance and mass estimates reported in Table <ref>.




We now aim to restrict the parameter space of γ by assessing the compatibility of the CG geometry's corresponding shadow radius r_s with the EHT's inferred shadow size of Sgr A* using the data above. This is so because, as argued in <cit.>, it is reported that the observed shadow size of Sgr A* can be inferred by utilizing the size of the bright emission ring as its proxy, subject to a calibration factor that multiplicatively accounts for the theoretical and observational uncertainties. Such a correlation becomes possible thanks to priors on Sgr A*'s mass-to-distance ratio obtained via orbital dynamics measurements of the stars closest to the galactic center (i.e. dubbed S-stars), and which are reported in Table <ref>.  These measurements allow the EHT collaboration to estimate the fractional deviation δ between Sgr A*'s inferred shadow size r_sh and that of a Schwarzschild black hole r_sh,Sch= 3 √(3) M, with angular radius ω_g = M/D  via the formula

    r_sh/M=(δ +1) 3 √(3) ,

which, under the assumption of a normal distribution of the estimation uncertainties, yields the following shadow size bounds <cit.>

    4.55 ≲ r_sh/M ≲ 5.22  ,

at 1σ deviation and

    4.21 ≲ r_sh/M ≲ 5.56  ,

at 2σ deviation.



We note that, to quantify the theoretical biases in the calibration factor utilized in the inference of the shadow's size of Sgr* A, the EHT collaboration performs simulations of black hole images to study a wide range of effects concerning the accretion flow properties and the underlying space-time. In this regard, the collaboration assumes the Kerr metric by default, although they also cover black hole metrics that either deviate from the Kerr space-time or describe alternative geometries, such as wormholes or boson stars. Because their analysis does not include the Cotton geometry, by performing tests with the above constraints we introduce a theoretical bias in our own analysis, since we assume these results to be valid for the CG spacetime. The extent to which the Cotton geometry could modify the above fractional deviation, and the constraints stemming from it, could be evaluated by performing a similar analysis to that by the EHT collaboration. However, such an effort lies well beyond the purpose of this work. 

To perform tests on the CG metric with its shadow radius, we compare the evolution of r_s as a function of the logarithm of γ M, in Fig. <ref>. We restrict our analysis to positive values of γ, which are required to describe the large-distance gravitational acceleration. 
Because we are dealing with a non-asymptotically flat space-time, one must take into account that its asymptotic Arnowitt-Deser-Misner (ADM) mass <cit.> is undefined for an observer at radial infinity and cannot be associated with the mass parameter appearing in the metric.
To circumvent this issue, we shall take a similar approach to <cit.> and report the the shadow radius in units of the local mass M, based on the reasoning that the stellar orbital dynamics of the S group of stars probe this quantity, since they are influenced by the local gravity of Sgr A*. 
Additionally, since the CG black hole's shadow radius depends explicitly on the observer's radial position, recall Eq.(<ref>), we compute the shadow radius considering an observer located at r_O ∼ 8000  kpc <cit.>. 




Upon inspection of Fig. <ref>, we verify that r_sh starts to increase around values of γ M ∼ 10^-12. Below this value, the CG black hole shadow radius coincides with that of a Schwarzschild black hole. We note that this is qualitatively the same behavior that is evidenced in <cit.> for the Rindler metric. Actually, the contribution of γ to the shadow radius becomes significant when the term γ r ≫ 1, while the upper 2σ bound in Eq.(<ref>) becomes saturated at a value of approximately γ M ≃ 3.525 × 10^-12.
For the values of γ M predicted by galaxy radial acceleration rotation curves in <cit.>, or the recent constraints obtained by analyzing CG's influence on CMB power spectrum and large scale structures <cit.> (i.e. both estimating γ M ∼ 10^-17), the shadow radius of the Cotton black hole lies well within the constraints imposed by Sgr A*'s shadow bounds, though there is no appreciable difference between it and its Schwarzschild counterpart. This result is not surprising, given that γ encodes large-distance gravitational effects within the CG framework. 
We have also found that the Cotton black hole shadow radius remains within the 2σ bounds for γ M ∼ 10^20. 
Such values describe extremely large and unphysical gravitational acceleration effects, exceeding any other current estimates by several orders of magnitude. For this reason, we have opted for excluding them from the γ parameter range in Fig. <ref>.










§ GRAVITATIONAL LENSING



This section deals with the phenomenon of gravitational lensing for CG. First, we will obtain the exact analytical expression for the deflection angle in terms of elliptic integrals. Then we will calculate an approximate expression valid for the weak field regime.





 §.§ Exact deflection angle 


To proceed with our derivation, it is convenient to make the following variable change in the radial coordinate u = 1/r in Eq. (<ref>). This allows us to re-write the photon equation of motion as

    ( du/d ϕ)^2 = 2 M G(u),

where the function G(u) reads, for CG, as

    G(u) =u^3 - u^2/2M  - u γ/2M + 1/2Mb^2.


The general expression for the deflection angle at a finite distance has been worked out in <cit.> as

    α = √(1/2M)( ∫_u_R^u_0du/√(G(u))+∫_u_S^u_0du/√(G(u))) + Ψ_R - Ψ_S,

where u_0 = 1/r_0 is the inverse of the distance of closest approach, while u_R = 1/r_R and u_S are the inverse of the radius of the observer and the source, respectively, and the indices refer to the position of the observer (R) and the source (S). 
The function Ψ encodes a correction term accounting for the finite distance, and is given by

    Ψ = arcsin( bu √(A(u))) .

Note that if we take the limit of  γ→ 0, and consider also that observer and source are located at infinity we will have

    α_SC = √(2/M)∫_0^u_0du/√(G(u)) - π,

which is the well-known deflection angle formula for the Schwarzschild solution.  This change in the definition of the deflection angle is necessary because the Cotton metric is not asymptotically flat. Moving on, the integrals of Eq. (<ref>) can be solved by rewriting the function G(u) as

    G(u) = (u-u_1)(u-u_2)(u-u_3),

where u_1,u_2 and u_3 are the roots of the cubic polynomial. We can check that u_0 = 1/r_0 is one of the roots of this polynomial. We have chosen (for reasons that will become clear later) u_2 = 1/r_0. Now we can write

    u^3 -u^2(u_1 +u_3 +1/r_0) +u (u_1 u_3 +u_1/r_0 + u_3/r_0) 
       
    
         - u_1 u_3/r_0  = u_3 - u^2/2M - u γ/2M + 1/2Mb^2.

After comparing the coefficients in Eq. (<ref>), we get

    u_1 +u_3 + 1/r_0 =   1/2M, 
        u_1u_3 + u_1/r_0 + u_3/r_0 =  
            - γ/2M.

Solving the above system we find that u_1 and u_3 are  given by

    u_1 = -2 M
       +r_0 ∓√(-(12 M^2 -8 γ  M r_0^2-4 M r_0-r_0^2))/4 M r_0 , 
       u_3 = -2 M+r_0 ±√(- (12 M^2-8 γ  M r_0^2-4 M
       r_0-r_0^2))/4 M r_0,

respectively. Note that we have two different solutions, one of which is u_1> u_3 and another that u_3> u_1.
Since our goal is that these roots fulfill the condition u_3> u_2> u_1, we choose the negative sign in the expression u_1 and the positive sign in the expression u_3. Now both integrals of Eq. (<ref>) can be put into the form

    I_1(u^') =  ∫_u^'^u_2du/√((u-u_1)(u-u_2)(u-u_3))

with u_3>u_2>u^'>u_3 and u^' being any finite distance. The result of the above integral is 

    I(u^') = 2/√(u_3-u_1) F(δ,k^2),

where F(δ,k^2) is an incomplete first-order elliptic integral of the parameters

    δ   =   arcsin√((u_3-u_1)(u_2-u^')/(u_2-u_1)(u_3-u^')), 
    
        k^2    =   u_2-u_1/u_3-u_1.



 To get the result of the integrals given in (<ref>) we just need evaluate I(u_R)+I(u_S). The complete result will also depend on the correction given by the functions Ψ(u_R) - Ψ(u_S). 
For the Cotton metric, the function Ψ is 

    Ψ(u) = sin ^-1(b u √(1-2 M u+γ/u)),

while we have for the parameters

    δ(u^') = sin ^-1{ 2 √(2)[ 
        (M r_0 (1-u^' r_0) √(r_0^2 c)/√(r_0^2 c)  +6 M  r_0- r_0^2) ×
    ×(√( r_0^2 c)-2 M
        r_0 (2 R  r_0+1)+ r_0^2)^-1]^1/2},












where c=-12 M^2+4 M r_0 (2 γ  r_0+1)+r_0^2

    k^2= √( r_0^2-12 M^2+4 M r_0 (2 γ 
      r_0+1))+6 M -r_0/2√(r_0^2 -12 M^2+4 M
       r_0 (2 γ  r_0+1)).

which closes all the expressions we needed to characterize the deflection angle in CG. It is also interesting to re-write these expressions in terms of the impact parameter by inverting its definition, in CG, as 

    r_0    =   b^2 γ/3+2/3 b √(b^2 γ ^2+3)×
          ×cos[1/3cos^-1((2 b^4 γ ^3 +9 b^2 γ
       -54 M)/2 b (b^2 γ^2+3)^3/2)].









For the sake of the graphical representation of the deflection angle we prefer to use the variable b^' = 1-b_c/b. This variable is limited to a value between 0 ≤ b^'≤ 1, where zero represents the strong field limit and one the weak field one. In Figs. <ref> and <ref> we show the deflection angle as a function of the variable b^' for values of γ in the range from 10^-16 to 10^-9. We see that α decreases with increasing γ and that there is no significant change in the values of α from γ <10^-13. If we increase the value of the parameter γ beyond the order of magnitude of 10^-9, the deflection angle begins to give negative values. Observationally, this value diverges for b^' =0 and tends to zero for b^' =1. We expect our theoretical result to reproduce this behaviour.
In other words, for the phenomenon of light deflection, Cotton's parameter is limited to values smaller than γ = 10^-9.










§ IMAGES OF BLACK HOLES WITH THIN ACCRETION DISKS



Having restricted the γ-space parameter, we now proceed to generate the images of the Cotton black hole (by pushing the value saturating the upper 2σ bound of the Sgr A* shadow) and qualitatively compare its appearance to that of a Schwarzschild black hole of the same mass parameter. We consider the setting of a black hole illuminated by an optically (i.e. fully transparent to its own radiation) and geometrically thin accretion disk, when observed at both a face-on inclination (i.e. θ_o = 0^∘) and at a nearly edge-on inclination (i.e. θ_o = 80^∘).  

We begin our considerations from Eq. (<ref>), which we conveniently re-write in the form

    dϕ/dr=±b/C(r)√(1/1-b^2V_ eff),

that describes the variation in a photon's azimuthal angle as a function of its radial coordinate. By numerically ray-tracing the previous equation for a bunch of trajectories within some range of the impact parameter b, we compute the null geodesics from an observer's screen backwards towards their point of origin. Figure <ref> displays the geodesics in the Schwarzschild space-time (left panel) for M=1, and in the CG space-time (right panel), the latter with a value of γ M = 3.525 × 10^-12. Upon close observation of the figure, one highlights the absence of any significant qualitative differences between the near-horizon lensing properties of the two geometries, under the considered parameter values. In addition to larger distances, the effects of γ become more noticeable at much higher orders of magnitude than those considered here, as is discussed in Section <ref> below. 





To simulate the emission from an accretion disk we resort to the semi-analytic models provided by Gralla, Lupsasca and Marrone in Ref. <cit.>, which describe the radial emission profile of an accretion disk based on a Standard Unbound (SU) Johnson distribution function given by

    I(r;γ,μ,σ)=e^-1/2[γ+arcsinh(r-μ/σ)]^2/√((r-μ)^2+σ^2),

where the γ, μ and σ parameters modify the skewness, peak location and kurtosis of the distribution, respectively. Such models are designed to reproduce the most prominent image features of time-averaged General Relativistic Magneto-Hydrodynamic (GRMHD) simulations of a thick accretion flow around a Kerr black hole, providing a less computationally taxing alternative towards the generation of black hole images <cit.>.
These models describe an infinitesimally thin, axisymmetric, radially dependent emission profile, assumed to be monochromatic in the disk's frame of reference and optically thin at the emitting frequency. In the absence of absorption, the relativistic radiative transfer equation <cit.> implies that the specific intensity I_ν/ν^3 is conserved along a light ray such that I_ν_o=g^3 I_ν_e, where the subscripts o and e denote the observer and emitter frames, respectively. The quantity g is defined as the redshift factor g=ν_o/ν_e = √(A(r)/A(r_∞)). Since the disk is geometrically thin, a light ray gains additional brightness for each time it intersects the accretion disk while on its way to a distant observer (i.e. every n half-orbit). Thus, considering a monochromatic emitter I_ν_e≡ I(r), the integrated (bolometric) intensity I=∫ I_ν_o d ν_o in the observer's plane is computed according to

    I_o(r)=∑_n(A(r)/A(r_o))^2.I(r)|_r=r_n.



For the purpose of generating black hole images we restrict our analysis up to the n=2 photon ring, as higher-order photon rings will provide a negligible contribution to the net luminosity (as shall be explained in the next section). We shall suitably adapt the three models introduced in <cit.> to a spherically symmetric scenario, to simulate the accretion flow around the Schwarzschild and Cotton black holes, with the parameters

    GLM1   :   γ=-3/2  ,   μ=0   ,      σ=M/2 , 
    GLM2   :   γ=0   ,     μ=0   ,      σ=M/2 , 
    GLM3   :   γ=-2  ,    μ=17M/3  ,  σ=M/4 .

whose corresponding emission profile is depicted in Fig. <ref>.



The GLM1 and GLM2 models are utilized to simulate an accretion disk with an emission profile extending all the way towards the event horizon, being distinguished by the radial emission decay near this region. In contrast, the GLM3 model represents an accretion disk with its emission peak located near the radius of the innermost stable circular orbits (r_ISCO), which is 6M for a Schwarzschild black hole. The latter allows us to separate the individual luminosity contributions of the n=1 and n=2 rings from the direct emission, thereby facilitating the comparison of these features between the Schwarzschild and Cotton black holes, whereas in the former models the photon rings appear as a discrete set of boosts of luminosity superimposed on top of the disk's direct emission <cit.>. In our simulations, the observed luminosity is normalized with respect to its total value, to ensure both geometries are compared in identical settings. We resort to the Geodesic Rays and Visualization of IntensiTY profiles (GRAVITYp) ray-tracing code, developed by some of us.



By inspection of Fig. <ref>, we report no appreciable visual differences between a Schwarzschild and a Cotton black hole of the same mass, with γ M ≃ 3.525 × 10^-12, regardless of the emission conditions. At the considered distance, the Cotton black hole's corresponding event horizon radius, photon sphere radius and critical impact parameter, are nearly indistinguishable from those of a Schwarzschild black hole of the same mass M,  differing only by a factor of the same order of magnitude of γ. Since for a non-asymptotically flat geometry the shadow radius does not coincide with the critical impact parameter, the geometries appear identical. This similarity persists onto the photon ring(s), where the gravitational lensing features play a significant role. In the literature, however,  various black holes with a similar critical impact parameter have been found to be distinguished by the properties of their corresponding photon rings <cit.>. These differences occur when, within the parameter space compatibility with Sgr A* shadow radius, the additional parameter in a given geometry modifies significantly the effective potential (i.e. its shape, location, concavity) seen by photons on their approach to a black hole, from the canonical Schwarzschild case. For some geometries, however, the constraints to the parameter space are too restrictive, to the point that there are virtually no changes to the effective potential, which is our case. Finally, we point out that although Eq. (<ref>) suggests the observed intensity depends on the observer's radial coordinate, for a non-asymptotically flat space-time, there are no visual differences between the Cotton and Schwarzschild black holes since the intensities are normalized with respect to their total value. Without normalizing the intensities, the observed intensities of the accretion disk emission in the Cotton spacetime would decrease, as in this case A(r) →∞ for an observer at infinity.





§ ISORADIALS



In this section, we perform an analysis of isoradial curves in the Cotton and Schwarzschild space-times, to complement the black hole images presented in the previous section. We solve the isoradial equatorial curves corresponding to the n=0 emission, following the analytical approximation provided by <cit.>, which states that the arrival point of a light ray may be mapped onto an observer's plane with Cartesian coordinates (X',Y') as 


    [ X'; Y' ]
        = R/√(A(R))√(1-(1-x)^2)[  cos(φ); sin(φ). ]

Here, R denotes a given radial distance from the origin, while φ corresponds to the polar angle in the observer's plane[For a better understanding of this coordinate representation, the reader is referred to Fig. 7 of <cit.>.]. The value of x may be computed via the following approximation


    x    =    A(R)y {1
     -e/100(1-A(R))y[ ln(1-y/2) +y/2] 
     
           +(315 I_1^2 -135 I_2 + 180 A^2(R) +120 I_1 +32)/720y^2
        
                   + (I_1/2 + 1/6)y
     },









with y being expressed by the trigonometric relation between the polar angle in the observer's plane and the observer's inclination angle θ_o via

    y=1+sin(φ)/√(sin^2(φ)+cot^2(θ_o)),

and where the quantities I_1 and I_2 are calculated as

    I_1=∫_0^1 P(u)   du  , 
    
     I_2=∫_0^1 P^2(u)   du,

with P(u)=-4u^2A(u) + A(R), and where the variable change r=R/u was considered to obtain A(u).

By solving the above equations for all polar angles, for a light ray with its emission point located at a constant r=R, one obtains the isoradial curves of the n=0 emission. In Fig. <ref>, we display isoradial curves for a Schwarzschild black hole and for a Cotton black hole with γ M ≃ 3.525 × 10^-12, as seen by an observer with inclination angle θ_o = 80^∘. 



This result emphasizes the similarity in the horizon-scale lensing phenomena between these two geometries, as seen by the complete overlap of the isoradial curve of both space-times; something which is in agreement with the analysis in the previous section. 




§ THEORETICAL LYAPUNOV EXPONENT




In this section, we explore the Lyapunov exponent of nearly bound orbits, a quantity that measures the instability scale of such orbits. Since these instabilities take place near the photon sphere, the Lyapunov exponent tracks a local effect and is therefore independent of the observer's distance to the black hole. 

We begin our analysis by considering the effect of a small radial perturbation near the photon sphere (i.e. r=δ r + r_p  , δ r ≪ r_p) and apply it to Eq. (<ref>). We perform a second-order Taylor series expansion of the effective potential V_ eff around r_p

    V_ eff(r)≃ V_ eff(r_p)+V'_ eff(r_p)δ r+1/2V”_ eff(r_p)(δ r)^2,

where the first term of the expansion corresponds to the critical impact parameter b_c and the second term disappears, since unstable circular null geodesics correspond to critical maxima of the effective potential. The prime notation above denotes a derivative with respect to the radial coordinate. Substituting this result into Eq. (<ref>), leads to the following expression 

    (d δ r/dϕ)^2 = -1/2C^2(r_p)V”_ eff(r_p)(δ r)^2.

By writing the differential equation in the form

    πd δ r/dϕ = λδ r,

whose solutions are of an exponential form, δ r_n=δ r_0e^λ n, we obtain the Lyapunov exponent λ which quantifies the instability scale of a small perturbation. Computing the second derivative of the effective potential and utilizing the critical curve equation C'(r_p)A(r_p)-C(r_p)A'(r_p)=0 <cit.>, yields the following expression for the theoretical Lyapunov exponent

    λ=π(AC”-A”C/2)^1/2.


For the Schwarzschild geometry γ = π, indicating that an initial perturbation grows (in radial distance) by a factor e^γ with each half-orbit around the black hole. For the Cotton geometry, the Lyapunov exponent reads

    λ=π(1+γ r_p)^1/2.

This result supports the qualitative analysis done in the previous sections. The λ exponent of the CG black hole, differs from the Schwarzschild black hole by an order of magnitude proportional to γ.  Consequently, despite the exponential behaviour driven by the Lyapunov exponent, the deviations between these two geometries are captured by photon rings of a similarly order, requiring many half-orbits around the black hole before becoming significantly different. The gravitational lensing causing the n=1 and n=2 photon rings is far from being sensitive to the contribution of γ. Thus, one cannot hope to be able to distinguish between the Schwarzschild and CG geometries via their n=1 and n=2 ring features (i.e. relative intensities, relative widths), which are the ones that could be within reach of future interferometric detectors <cit.>.





§ CONCLUSION


In this work we investigate the phenomenon of gravitational lensing and its associated observables within an extension of GR recently proposed in the literature and dubbed as Cotton gravity. We estimated the size of the new parameter γ of Cotton gravity and which arises with a newly found spherically symmetric vacuum (Schwarzschild-like) solution, by making a numerical comparison with bound on the shadow's size of the supermassive black hole at the center of our galaxy (Sgr A*) reported by the Event Horizon Telescope Collaboration. Such a bound  is saturated, for CG, for values as small as γ M = 3.525 × 10^-12 (at 2σ in such bounds), where we took into account some subtleties associated to the non-asymptotic flatness of CG geometry to identifying the mass parameter of it with the one of the Schwarzschild case. 

To study the impact of this result in geodesic effects, we first computed the exact expression for the deflection angle in the form of elliptic integrals. There are two important considerations about this result. Firstly, in order to obtain the result of the Schwarzschild solution, one must first assume γ=0 and additionally u_R and u_S=0 (which corresponds to the consideration of infinite distances). This is because the result for Schwarzschild is usually determined using the definition for infinite distances, which is not possible in the case of Cotton gravity given its non-asymptotically flat character. Secondly, the deflection angle can be written as a function of the impact parameter b, though under an explicit cumbersome form. Furthermore, we computed the deflection angle using a Taylor series approximation to solve the integrals, which yields a result which is valid for the weak field regime where M/b ≪ 1. We have considered a linear approximation for the parameter γ in addition to the series with respect to the mass (we expect that γ has in reality a much smaller order of magnitude than mass and distances), and checked the consistence of the analytical approximation with the numerical integration of the deflection angle. Overall, we find that the deflection angle remains unchanged for values of the CG parameter γ < 10^-14.


The smallness of such a parameter of CG drove us to consider the chances of distinguishing between Schwarzschild and CG black holes using observables associated to gravitational lensing. To this end, we simulated the appearance of a CG black hole, with such a value of the γ parameter, surrounded by a thin accretion disk, modelled by three semi-analytic radial intensity profiles. We found the constraints placed on γ by Sgr A*'s shadow bounds to be too tight for any significant visual differences to appear. In this regard, we note that although in CG the observed intensity depends on the observer's radial distance, our images do not show any difference in this regard due to having normalized intensity profiles. 

In looking for further settings to search for the presence of any differences, we studied the isoradial curves of a CG black hole against those of a Schwarzschild black hole, utilizing a generalization of the Poutanen formula. We found that there is a full overlap of the isoradial curves between the CG and Schwarzschild geometries, confirming the absence of any visual differences in this regard. We also considered the theoretical Lyapunov exponent for a Cotton gravity black hole, associated to the instability scale of photon rings. We derived its exact expression by perturbing the photon equation of motion about the photon sphere radius. The CG theoretical Lyapunov exponent was found to differ from Schwarzschild's by a factor proportional to CG's γ parameter, and thus photon rings do not help in discriminating between both geometries, at least for those rings (n=1 and n=2) that could be expected to be measured in the last decades via interferometric means.

To conclude, the analysis carried out in this work points that the CG space-time recently found in the literature is very unlikely to be distinguishable from the expectations of the canonical Schwarzschild black holes with the devices we have at this moment and in the foreseeable near future.



MER thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Brazil, for partial financial support. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
FSNL acknowledges support from the Fundação para a Ciência e a Tecnologia (FCT) Scientific Employment Stimulus contract with reference CEECINST/00032/2018, and funding through the research grants UIDB/04434/2020, UIDP/04434/2020 and PTDC/FIS-AST/0054/2021.
DRG is supported by the Spanish Agencia Estatal de Investigación Grant No. PID2022-138607NB-I00, funded by MCIN/AEI/10.13039/501100011033, UE (European Union), ERDF (European Regional  Development Fund) A way of making Europe.


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 Paper Title: Measuring the Spin of the Galactic Center Supermassive Black Hole with Two Pulsars.








Department of Astronomy, School of Physics, Peking University, 
Beijing 100871, China
Kavli Institute for Astronomy and Astrophysics, Peking University, 
Beijing 100871, China

lshao@pku.edu.cn
Kavli Institute for Astronomy and Astrophysics, Peking University, 
Beijing 100871, China
National Astronomical Observatories, Chinese Academy of Sciences, 
Beijing 100012, China

§ ABSTRACT 
As a key science project of the Square Kilometre Array (SKA), the discovery and
timing observations of radio pulsars in the Galactic Center would provide
high-precision measurements of the spacetime around the supermassive black hole,
Sagittarius A* (Sgr A*), and initiate novel tests of general relativity. The
spin of Sgr A* could be measured with a relative error of ≲ 1% by
timing one pulsar with timing precision that is achievable for the SKA. However,
the real measurements depend on the discovery of a pulsar in a very compact
orbit, P_b≲0.5  yr. Here for the first time we propose and
investigate the possibility of probing the spin of Sgr A* with two or more
pulsars that are in orbits with larger orbital periods, P_b∼ 2- 5 
yr, which represents a more realistic situation from population estimates. We
develop a novel method for directly determining the spin of Sgr A* from the
timing observables of two pulsars and it can be readily extended for combining
more pulsars. With extensive mock data simulations, we show that combining a
second pulsar improves the spin measurement by 2-3 orders of magnitude in some
situations, which is comparable to timing a pulsar in a very tight orbit.


Measuring the Spin of the Galactic Center Supermassive Black Hole with Two Pulsars
    Lijing Shao
    August 5, 2024
==================================================================================



Introduction.—Black holes (BHs) are the most intriguing objects for
testing general relativity (GR). While the detection of gravitational waves from
stellar-mass BH mergers <cit.> provided abundant knowledge of the
highly dynamical, strong-field regime of gravity, the supermassive BHs (SMBHs)
existing at the center of most massive galaxies <cit.> represent another ideal laboratory for BH physics.  The SMBH
dwelling in our Galactic Center (GC) <cit.>, Sagittarius A* (Sgr A*), with the largest mass
to distance ratio among the known BHs, is the most promising target for
precision observations and unique tests of strong-field
gravity <cit.>.

Previous studies have shown that timing observations of radio pulsars in compact
orbits around Sgr A* would allow us to probe the spacetime around this SMBH to 
unprecedented accuracy and provide us with unique tests of the no-hair theorem 
for Kerr BHs <cit.>. Regular timing
of a normal pulsar in a highly eccentric orbit (e≳ 0.8) tightly around
Sgr A* (P_b≲ 0.5  yr) will measure the spin and quadrupole moment
of Sgr A* to a relative precision of about 1% within five years for a timing
precision of 1  ms, which is achievable for future radio
telescopes <cit.>.
However, the existence and detectability of such pulsars in very close orbits
around Sgr A* is still unclear. The large dispersion measures and the scattering
caused by highly turbulent interstellar medium in the GC make observations favor
high radio frequencies <cit.>, which then render pulsars
flux-limited because of the steep spectra of emission. Nevertheless, theoretical
models suggest that there can be 10^2-10^3 pulsars orbiting around Sgr A* with
P_b≲ 100  yr <cit.>, and present
observations including the discoveries of a magnetar <cit.>, a millisecond pulsar <cit.> and several normal
pulsars <cit.> in that region all suggest that a
large population of pulsars are awaiting for discoveries in the GC for future
high-frequency surveys.

Measuring the BH spin is crucial for gravity tests.  Determining the spin of
Sgr A* based on the timing of a single pulsar suffers from parameter degeneracy
at the leading order <cit.>.  The measurable secular
changes of the pulsar orbit in a short time, caused by the frame-dragging
effect <cit.>, are the linear rate of the periastron precession,
ω̇, and the linear change of the projected semimajor axis of the
pulsar orbit, ẋ. However, these two quantities cannot fully determine
the BH spin which has three components. To break the degeneracy, one needs at
least one accurate measurement of  higher-order time derivatives such as 
ω̈ or ẍ <cit.>, which in general requires a
much longer observation time span or a pulsar in an orbit with very short 
orbital period whose relativistic effects are large. 

There are still two possible ways to break the leading-order degeneracy and 
provide a better spin determination without requiring a pulsar in a rare orbital
situation. One is to combine the measurement of the proper motion of the pulsar
from astrometric observations <cit.>.  One can infer the secular
change in the longitude of the ascending node, Ω̇—which is not
observable in timing observation—from the proper motion of the pulsar and
provide an additional constraint on the spin of the central BH. However, an
expected astrometric accuracy in the order of 10 μ
as <cit.> is still not enough and contributes little when 
combining with the timing data <cit.>. The other possibility is to
combine the observation of another pulsar with a different orbital inclination
i, as the leading-order degeneracy mainly depends on i <cit.>.
Considering a potentially large pulsar population in the GC, finding two or more
pulsars orbiting around Sgr A* in the future may provide us a more likely way in
measuring the spin of Sgr A*.

In this Letter, we develop a novel method for determining the spin of
Sgr A* by combining  observations of two or more pulsars. We demonstrate that
for a pulsar with orbital period P_b∼ 2-5  yr, combining a second
pulsar can improve the spin measurement significantly. While previous studies
focused on timing one pulsar in a very tight orbit (P_b≲ 0.5 
yr), we show that finding two or more pulsars with larger orbital periods are
more likely to be the case for future observations.

Combining two pulsars.—When the changes in the Keplerian parameters
of a pulsar orbit are slow over the observation time span, the direct way to
characterize  secular effects is to fit the timing data with timing parameters
varying linearly in time, such as  ω=ω_0+ω̇ t and 
x=x_0+ẋ t <cit.>. In a pulsar-SMBH system, the
leading-order derivatives will be measured soon after the start of observation
and have a rather high measurement precision. Here we use the two combinations
introduced by <cit.> for the leading-order derivatives and other
timing observables, which are 𝒳≡ -ẋs_i^2
(xΩ̂)^-1 and 𝒲≡(ω̇-
ω̇_M)s_i^2Ω̂^-1, where s_i=sin i,
Ω̂=4πβ_O^3/P_b(1-e^2)^3/2 and 
ω̇_M=6πβ_O^2/P_b(1-e^2), with β_O=(2π GM/c^3P_b
)^1/3 the orbital velocity parameter introduced by <cit.>.
Note that s_i and M in these combinations can be measured precisely from the
Shapiro delay <cit.> that is large enough here even for face-on
orbits <cit.>. 

To combine the timing results of two pulsars and determine the spin of the 
central BH, one has to translate the measurement constraints of both pulsars in
a common parameter space. In this case, the natural choice is the
χ-χ_λ plane, where χ_λ is the projection of the
dimensionless spin χ of the BH in the line of sight direction. For each
pulsar, timing observations give one constraint represented by a hyperbola,


    𝒳^2/s_i^2(1-s_i^2)+𝒲^2/s_i^2(1+3s_i^2)=
        χ^2-1+3s_i^2/(1-3s_i^2)^2(χ_λ+3
        𝒲c_i/1+3s_i^2)^2 ,


where c_i=cos i.  If one  observes two or more pulsars, all these curves in
the χ-χ_λ plane should intersect at one point that gives χ
and χ_λ. 





Figure <ref> shows an illustration of using the timing of two pulsars to
determine the spin of Sgr A* in the χ-χ_λ plane. The system 
parameters are randomly chosen to avoid the accidental degeneracy which is
discussed later.  For each pulsar, the measurements of ( X,  W)
give a hyperbola shown by the solid curve, and the i↔π-i
ambiguity gives another hyperbola shown by the dashed line. In general, there
can exist at most eight different solutions for (χ, χ_λ).  A rough
measurement of any second-order derivatives, for example, the ω̈ 
for one of the pulsar, as shown by the vertical red line, or timing of a third 
pulsar can reduce this degeneracy into two solutions whose χ has the same
value and χ_λ has opposite signs.

The methodology described above can be easily extended for combining more
pulsars.  For each additional pulsar, it provides an independent test of GR as
all these curves should intersect at one point within their measurement
uncertainties. On the other hand, if one assumes that GR is correct, timing more
pulsars can be a unique probe for us to explore and model the (stellar and gas)
mass distribution near the GC <cit.>. As
only the leading-order time derivatives and a rough value of a higher-derivative
parameter are used, one is expected to obtain a high-precision measurement after
a short time observing two pulsars.

Numerical simulation.—To further explore the method introduced above
and give a quantitative description of the expected measurement precision in
measuring the spin of Sgr A* by timing two pulsars,  we perform mock data
simulations based on the numerical timing model for pulsar-SMBH systems
developed earlier <cit.>. In this timing model, we numerically
integrate the pulsar orbital motion based on the post-Newtonian (PN) equation of
motion, r̈=-GMr̂/r^2+r̈_ 1PN
+r̈_ SO+r̈_ Q, where r is the relative
separation pointing from the SMBH to the pulsar, r̈_ 1PN is
the first PN correction, r̈_ SO and r̈_ Q
are the spin and quadrupole contribution from the central SMBH
respectively <cit.>, M is the mass of the SMBH,
r≡|r|, and r̂≡r/r. Taking advantage of the PN
expansion, we  treat the dimensionless spin χ and the dimensionless
quadrupole moment q of the SMBH as independent parameters, which renders a
test of the no-hair theorem q=-χ^2 for Sgr A* possible <cit.>.
As the mass ratio m_ PSR/M≲ 10^-6, we treat the pulsar as a test
particle in the BH's spacetime. We take into account the Römer delay, the
leading-order Shapiro delay and the Einstein delay in the timing
model <cit.> and ignore the effects
caused by the proper motion of Sgr A* <cit.>. More
details can be found in Ref. <cit.>.

The full parameter set, Θ=Θ_ BH∪Θ_
PSR, in the timing model  for a single pulsar includes the parameters of the
SMBH, Θ_ BH={M,χ,q,λ,η}, where λ and
η describe the direction of the spin, and the parameters of the pulsar,
Θ_ PSR={P_b,e,f,i,ω,N_0,ν,ν̇}, where f is the
initial orbital phase and {N_0,ν,ν̇} relate the pulsar's rotation
number N and pulsar's proper time T via N(T)=N_0+ν T+ν̇T^2/2.  We
have set the longitude of the ascending node of the pulsar orbit, Ω, to
be zero as it is not an observable for timing observation. When combining the
timing of two or more pulsars, one should have one set of Θ_ PSR for
each pulsar while Θ_ BH is common for different pulsars. In
addition, the difference between the longitudes of the ascending node, ΔΩ=Ω_2-Ω_1, should also be included in the timing
model <cit.>. As we can have an overall rotation of the system
around the line of sight direction, we fix Ω_1=0 in the following
discussions and we denote the pulsar with a smaller orbital period as PSR1 or
the inner pulsar.

We use the Fisher matrix method for parameter estimation <cit.>. 
The log-likelihood function ℒ in the covariance matrix C_αβ= (∂^2ℒ/ ∂Θ^α∂Θ^β)^-1 is simply the sum from two pulsars,
ℒ=ℒ_1+ℒ_2, and 


    ℒ_i=1/2ν_i^2∑_a=1^N^(i)_ TOA[N^(i)_a(Θ)-N^(i)_a(Θ̅)]^2/(σ^(i)_ TOA)^2 ,  i=1,2


where N^(i)_a(Θ)=N (Θ;t_a^(i)  TOA) is the pulsar
rotation number calculated by the timing model corresponding to the a-th pulse
arrived at t_a^(i)  TOA. Θ̅ denotes the true system
parameters and σ_ TOA^(i) is the timing precision for pulsar i. 
As we do not consider the interaction between the two pulsars, the properties of
the Fisher matrix method and the timing model allow one to construct the full
Fisher matrix F≡ C^-1 though the combination of the Fisher matrices
F_i of each single pulsar. This simplification is important for combining more
pulsars.



We assume a timing precision σ_ TOA=1  ms and an observation 
time span T_ obs=5  yr for both pulsars, which are realistic for 
future instruments such as the Square Kilometre Array (SKA) and the 
next-generation Very Large Array (ngVLA) <cit.>.
In the upper panel of Fig. <ref> we show the measurement precision of the
spin of Sgr A* from combining the timing of two pulsars (solid curves) compared 
to the results obtained from the inner pulsar only (dashed line). Here we fix 
the orbital period of the inner pulsar to be P_b1=2.3  yr and orbital 
eccentricity e_1=0.8 while varying the orbital period P_b2 and 
eccentricity e_2 of the second pulsar. The lower panel of this figure shows 
the improvement factor F_ imp=δχ_ single / δχ_
combine. For this case, the inner pulsar with an orbital period P_b1=2.3
  yr can only constrain δχ / χ to ∼ 15%. Combining a 
second pulsar even with a large orbital period can provide an improvement factor
of ∼ 10 because of the breaking of the leading-order degeneracy.  The
improvement factor from combining a second pulsar will be smaller if the inner
pulsar has a tighter orbit for which itself can break the leading order
degeneracy. Another case study of P_b1=0.5  yr shows an improvement
factor of 2∼ 3 when combined with a second pulsar whose orbital period
P_b2≲ 3  yr.



As the second pulsar has a larger orbital period than the inner one, the 
constraint of the spin from it is in general looser than that from the inner
pulsar <cit.>.  Thus the
improvement of the spin measurement by combining two pulsars is mainly caused by
the breaking of the leading-order degeneracy.  Figure <ref> shows an
example of the dependence of the degeneracy on the most relevant orbital
parameter, the orbital inclination i. In this example we set P_b1=2.3 
yr, e_1=0.8, P_b2=4.2  yr, and e_2=0.6. The orbital
inclination i_1 of the inner pulsar is fixed to be 60^∘ and we vary the
orbital inclination of the second pulsar, i_2.  One can clearly see that the
improvement factor varies over two orders of magnitude and it is not dominated
by  the second pulsar's measurement precision. 

The simulation results are consistent with the prediction from the analysis we
presented before.  From Eq. (<ref>), it is obvious that the shape of the
hyperbola of each pulsar is controlled by their inclination angles, which
determine the leading-order degeneracy. Moreover, from Eq. (<ref>) one
can also predict the position of the three degeneracy peaks for the combined
results in Fig. <ref>, where the improvement from combining a second
pulsar is the smallest. The central peak at i=90^∘ originates from the 
denominator of the first term in Eq. (<ref>), while the other two peaks 
are attributed to the situation where two hyperbolas are tangent to each other
at the point they intersect. We show the predicted peak positions from
Eq. (<ref>) in Fig. <ref> with vertical lines.

Pulsars in the GC.—We have shown the large potential of measuring the
spin of Sgr A* with two or more pulsars. Compared to the measurement with a
single pulsar, the key advantage of the approach we proposed here is that it
does not require a pulsar in an extremely compact orbit around Sgr A*. Although
several pulsars including a magnetar and a millisecond pulsar have been found in
the GC region <cit.>, none of them are close enough
to the Sgr A* for doing the measurement described above. Here we give an
estimation of the probability of finding suitable pulsars in future
high-frequency radio surveys.

We define two possibilities P_1 and P_2 as follows. P_1 represents the 
probability of finding the innermost pulsar with an orbital period P_b≤
0.5  yr  via future instruments like SKA, which then can be used to
measure the spin of Sgr A* by timing itself alone, while P_2 represents the
probability of finding two innermost  pulsars both with orbital period P_b≤
5  yr. By combining the timing observations of these two pulsars, one may
constraint the spin of Sgr A* to ∼ 1% within five years according to our
simulations.  We assume a simple power-law distribution of pulsars near the
Sgr A*, p(r) ∝ r^-α, where p(r) is the probability density of
finding a pulsar in a unit volume at a distance r from the Sgr A*. 
Considering the large uncertainties in the GC pulsar population, the power-law
index α can vary largely depending on the detailed
models <cit.>. 

Under these assumptions, one can calculate P_1 and P_2 via

    P_1   =   1-exp[-N_4(r_0.5  yr/R_4)^3-α]
         ,
    
        P_2   =   1-[1+N_4(r_5  yr/R_4)^3-α]exp[-N_4(r_5  yr/R_4)^3-α] ,

where r_0.5  yr∼ 100  AU and r_5  yr∼ 475 
AU are the semimajor axes of the pulsar orbits with P_b = 0.5   yr and
P_b = 5  yr respectively. N_4 is the expected total number of pulsars
that can be observed within R_4=4000  AU around Sgr A*.  P_1 and P_2
for different α and N_4, motivated from current observations and
simulations, are listed in Table <ref> <cit.>. In the reasonable parameter space of α and N_4, one
always has P_1≲ P_2,  indicating that it is more likely to find two
pulsars that are suitable for measuring the spin of Sgr A*.




Discussions.—In this Letter, we proposed to measure the spin of
Sgr A* with timing observations of two or more pulsars, which does not require a
pulsar with an extremely small orbital period but can still provide similar
measurement precision by breaking the leading-order degeneracy of spin
parameters. Although the GC pulsar population has a large uncertainty at
present, it is expected to find a lot of new pulsars in future high-frequency
radio surveys towards the GC and we give a rough estimation of the probability
of finding proper systems.  It seems that we are more likely to find the systems
that are suitable for the procedure proposed here. Even if we can find a pulsar
with an orbital period P_b≲ 0.5  yr, combining a proper second
pulsar can still provide an improvement factor  ∼ 2 for the SMBH spin
measurement.

In this work we have ignored the complex environments near the Sgr A*, which
will complicate the measurement <cit.>. In general, pulsars with larger orbital periods
will suffer from more perturbations caused by the mass distributions around
Sgr A*, which may even spoil the measurement. It is suggested that one may only
use the timing data around the periastron passages and treat each periastron
passage incoherently <cit.>. This will  enlarge the measurement
uncertainty of the spin from timing a single pulsar, where second-order time
derivatives are required but hard to  measure during the short periastron
passage. However, as only the leading-order time derivatives are required in the
method we proposed here, it is expected to have a good measurement even only the
timing data around the periastron passages are used. On the other hand, by
combing the timing observations of more pulsars, one will have the ability to
model the mass distribution around Sgr A* that will be of vastly useful for
studies of the GC region.


We thank Norbert Wex, Ziri Younsi, and Fupeng Zhang for helpful discussions.
This work was supported by the National SKA Program of China (2020SKA0120300),
the National Natural Science Foundation of China (11991053), the Beijing Natural
Science Foundation (1242018), the Max Planck Partner Group Program funded by the
Max Planck Society, and the High-performance Computing Platform of Peking
University. 






 Paper Title: Black bounces in Cotton gravity.






	 ednaldobarrosjr@gmail.com
Faculdade de Física, Universidade Federal do Pará, Campus Universitário de Tucuruí, CEP: 68464-000, Tucuruí, Pará, Brazil

     
    tarcisojunior17@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill

	 fslobo@ciencias.ulisboa.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal
Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     esialg@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill
Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará, Campus Universitário de Abaetetuba, 68440-000, Abaetetuba, Pará, Brazil

   drubiera@ucm.es
Departamento de Física Téorica and IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain

      
        fc53497@alunos.fc.ul.pt
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016 Lisbon, Portugal

     henriquefisica2017@gmail.com
Faculdade de Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, Pará, Brazill





§ ABSTRACT 


Recently, J. Harada proposed a theory relating gravity to the Cotton tensor, dubbed as “Cotton gravity" (CG). This is an extension of General Relativity such that every solution of the latter turns out to be a solution of the former (but the converse is not true) and, furthermore, it is possible to derive the cosmological constant as an integration constant within it. In this work we investigate CG by coupling it to both non-linear electrodynamics (NLED) and scalar fields.  We study static and spherically symmetric solutions implementing a bouncing behaviour in the radial function so as to avoid the development of singularities, inspired by the Simpson-Visser black bounce and the Bardeen model, both interpreted as magnetic monopoles. We identify the  NLED Lagrangian density and  the scalar field potential generating such solutions, and investigate the corresponding gravitational configurations in terms of horizons, behaviour of the metric functions, and regularity of the Kretchsman curvature scalar. Our analysis extends the class of non-singular geometries found in the literature and paves the ground for further analysis of black holes in CG.



04.50.Kd,04.70.Bw



Black bounces in Cotton gravity
    Henrique A. Vieira
    August 5, 2024
===============================


HMS










 








§ INTRODUCTION




Recently, Harada <cit.> proposed a theory that generalizes General Relativity (GR), known as “Cotton gravity" (CG). It is based on the use of the Cotton tensor, assumed to encapsulate the effects describing gravity, and leading to third-order field equations in the derivatives of the metric tensor  <cit.>. Part of the interest in this theory comes from the fact that  every solution of  Einstein's equations of GR (including a cosmological constant term) is a solution of CG gravity via a particular Cotton tensor. Indeed, the cosmological constant itself arises as an integration constant of the theory. Furthermore, in <cit.> it was shown that a particular new solution of CG gravity not present within GR could be able to explain the galactic rotation curves without the need for dark matter, further raising the interest in the theory. As a consequence, different applications of this theory have been studied very recently in the literature <cit.>. In fact, there is a current debate on the theory as a viable extension of GR <cit.>, which highlights the need for further work to determine the potential of Cotton gravity to supersede GR.

Among the many gravitational configurations explored in the literature beyond the canonical (Kerr) black hole of GR, those aimed to solve the singularity problem are of particular interest. Such a problem is related to the unavoidably development of space-time singularities (as given by the incompleteness of at least one geodesic trajectory) inside black holes within the GR framework (see <cit.> for a discussion of this topic), and which typically correlates with the presence of infinities in some sets of curvature scalars. A canonical strategy in the field at this regard is to imbue the radial function with a non-trivial bouncing behaviour in order to prevent the focusing of geodesics appearing in the singularity theorems, this way allowing geodesic congruences to re-expand at a minimum radial distance. Examples of this are the well known Bardeen model <cit.> and, more recently, the class of black bounces introduced by Simpson and Visser in <cit.>. Typically, models of this kind introduce one (or more) parameters associated to a minimum length scale, which is responsible for the bouncing behaviour. Furthermore, the bounce can be hidden behind event horizons thus representing regular black holes, or be naked instead, interpreted as a kind of traversable wormhole allowing free communication between far-away distances <cit.>. 

The various forms of the black bounce proposal have led to the development of several additional classes of solutions <cit.> and the analysis of their properties. Among them, we can underline regularity, causal structure and energy conditions <cit.>, gravitational lensing  <cit.>, and observational  signatures <cit.>, besides generalizations to the rotating case <cit.>. Typically, such bouncing configurations are supported by a combination of a non-linear electrodynamics (NED) and a scalar field   <cit.> working under a reverse procedure: the geometrical line element incorporating the bouncing behaviour is set first, and then the field equations are driven back to find the explicit matter sources treading the geometry.

The main aim of this work is to implement suitable generalizations of both Simpson-Visser and Bardeen solutions within CG gravity according to the new features of the theory. In particular, we shall generalize the exact vacuum solution of CG of <cit.>, characterized by a new parameter γ, towards bouncing geometries supported by NLED and scalar fields. The geometries obtained this way depend on the mass of the black hole M, on the magnetic charge q, and on the CG parameter γ, besides the cosmological constant itself Λ. Given the four-parametric nature of these CG solutions, we shall take different choices for them in both the Simpson-Visser and Bardeen solutions, hopefully covering the most relevant conceptual classes of configurations in terms of types of horizons and global shape of the corresponding metric functions. We shall then obtain the functional shapes of the NLED Lagrangian density and the scalar field potential. 









This work is organized as follows. In Sec. <ref>, we briefly present the field equations of CG coupled to NLED and scalar fields, introduce the class of geometries we are interested in, and find the equations allowing to reconstruct the matter (NLED supported by magnetic charge) and scalar field potential. In Sec. <ref>, we  generalize the black bounce geometries and study in detail their properties, and in Sec. <ref> we do the same with the Bardeen one. In Sec.  <ref>, we summarize and discuss our results. We work in units G=c=1.







§ COTTON GRAVITY COUPLED TO NON-LINEAR ELECTRODYNAMICS AND SCALAR FIELDS






 §.§ Field equations of Cotton gravity



In CG the gravitational field equations are given by  <cit.> 
    C_αμν=κ^2∇_βΘ^β_βαμν,

where κ^2=8π, while the tensors C_αμν and Θ_αμν are defined as

    C_αμν   ≡   ∇_μR_αν-∇_νR_αμ
          -1/6(g_να∇_μR-g_μα∇_νR),
    	 
    ∇_βΘ_μαμν^β   ≡   1/2(∇_μΘ_αν-∇_νΘ_αμ)
          -1/6(g_να∇_μ-g_μα∇_ν)Θ .

respectively, where C_μν stands for the Cotton tensor, R_μν for the Ricci tensor and Θ_μν  for the energy-momentum tensor, and their traces are represented by R and Θ, respectively. Moreover, Einstein's equations of GR are solutions of Eq. (<ref>), and therefore, GR can be seen as a particular case of CG. Note that if we contract the equation of motion  (<ref>) with g^αν, the conservation law g^ανC_αμν=κ^2∇_βT_βμ^β=0 is naturally  satisfied.

For the matter sector of the theory, in this work we are interested in space-times sourced by NLEDs and scalar fields, whose energy-momentum tensors can be written as

    ∇_βΘ_μαμν^β=∇_βFΘ_μαμν^β+∇_βφΘ_μαμν^β.

Next, we will explicitly show the forms of these matter components described by the energy-momentum tensor:

    ∇_βFΘ_μαμν^β   ≡   1/2(∇_μFT_αν-∇_νFT_αμ)
          -1/6(g_να∇_μ-g_μα∇_ν)FT,
    ∇_βφΘ_μαμν^β   ≡   1/2(∇_μφT_αν-∇_νφT_αμ)
          	-1/6(g_να∇_μ-g_μα∇_ν)φT ,

respectively, with

    FT_μν   =g_μν L_ NLED(F)- L_FF_μαF_ν^να,
    FT   =4 L_ NLED(F)-4 L_FF,
    φT_μν   =2 ϵ ∂_μφ∂_νφ-ϵ 
    g_μν∂^σφ∂_σφ+g_μνV(φ),
    φT   =-2 ϵ ∂^νφ∂_νφ+4V(φ),

where  ϵ=+1 is the scalar field, while ϵ=-1 represents the phantom scalar field, φ is the scalar field, while V(φ) denotes the scalar potential,  L_ NLED(F) is the Lagrangian density describing a NLED that depends on the electromagnetic scalar

    F=1/4F^μνF_μν,

 and the  Maxwell-Faraday antisymmetric tensor is defined by F_μν = ∂_μ A_ν -∂_ν A_μ,

  

 where A_α is the vector potential. 
 
 The field equations for these two matter contributions can be found by the usual variational procedure as 

    ∇_μ ( L_F F^μν)   =1/√(-g)∂_μ (√(-g) L_F F^μν)=0,
    
         2∇_ν∇^μφ   =-dV(φ)/dφ ,

respectively, where g is the determinant of the metric and we have denoted  L_F=∂ L _ NLED(F)/∂ F.





 §.§ Spherically symmetric solutions



For the sake of our considerations we shall work in the static, spherically symmetric, line element 

    ds^2=A(r)dt^2-B(r)dr^2-Σ^2(r)   dΩ^2,

where A(r), B(r) and Σ(r) are functions of the radial coordinate r and dΩ^2≡ dθ^2+sin^2(θ)dϕ^2



is the line element on the two-spheres. We point out that in this line element we have included a non-trivial radial function Σ^2(r) given the fact that we are interested in geometries implementing a bouncing behaviour in the radial sector, for which Σ^2(r) is non-monotonic. On the other hand, it is always possible to perform a change of variables of the form B(r)dr^2=dx^2/A(r) to write the line element (<ref>) under a more canonical form with g_tt=g^xx, meaning that one of these two metric functions is redundant. Nonetheless, for the sake of generality, we shall work in the line element above and later fix one of the metric functions. 

In Ref. <cit.>, Harada found the following exact vacuum solution of the CG field equations  (<ref>) as

    A(r)=1/B(r)=1 - 2M/r - Λ/3r^2 + γ r.

where M is the standard mass term of the Schwarzschild solution (to which the solution above reduces when Λ=0 and the parameter γ resulting from the theory goes to zero, γ→ 0). Furthermore,  Λ arises as an integration constant, and given the fact that the term -Λ r^2/3 denotes a de Sitter term makes Λ naturally interpreted as a cosmological constant term. 

Our aim is to generalize the above solution in CG to implement bouncing behaviours in the radial function. To this end we shall take the canonical choice, due to Ellis <cit.>, as

    Σ(r)=√(L_0^2+r^2) ,

where the parameter L_0 ∈ is a regularization parameter and possesses the dimension of a length. This constant can be consistently interpreted as a magnetic charge q, provided that we source the line element with a NLED and scalar fields. This means that the coordinate r extends over the whole real line, r ∈ (-∞,+∞), while the function Σ(r) which gives the area of the two-spheres as S=4πΣ^2(r) takes a minimum value Σ(r)=q at r=0.

The invariants associated to such fields are

    F_23=q sinθ ,

for the electromagnetic field, and 

    F=q^2/2 Σ^4(r).

for the scalar one. In addition, to assess the consistency of our solutions we shall also use the following useful relationship

    L_F=∂ L _ NLED/∂ r(∂ F/∂ r)^-1 .


The field equations (<ref>) are third-order coupled gravitational equations for the metric, which makes it very difficult to obtain a solution by direct integration when the matter content is included. To overcome this difficulty and find new solutions for these equations, we use the following approach in our work: we propose the spherical symmetry for the metric as well as a functional form that already contains the new characteristic term of the vacuum solution in CG while at the same time implementing the bouncing behaviour given by Eq. (<ref>), and then integrate the equations of motion to determine the Lagrangian of the NLED as well as its derivative in order to check the consistency equation (<ref>). In this way, we will be able to generalize some known bouncing solutions found within GR (namely, the Simpson-Visser and Bardeen solutions), and obtain the specific shapes of the matter (electromagnetic and scalar),

To run this procedure we first solve the (0,0,1) and (2,1,2) components of the CG field equations (<ref>). This provides cumbersome yet analytic expressions for both the NLED Lagrangian density and its derivative as


    L_ NLED(r)= f_0+2f_1q^2/√(q^2+r^2)+∫1/2κ^2q^2A(r)^3B(r)^3(q^2+r^2)^5/2{2B(r)^2(q^2+r^2)^3A'(r)^3
          
    +(q^2+r^2)^2A(r)B(r)A'(r)[A'(r)(2(q^2+r^2)B'(r)-rB(r))-4B(r)(q^2+r^2)A”(r)]
    
          	
    +2A(r)^3[-2rB(r)^3(q^2+r^2)-2r(q^2+r^2)^2B'(r)^2+2rB(r)^2(κ^2ϵ(q^2+r^2)^2φ'(r)^2+3q^2+r^2)
    	.
    
          
    +B(r)(q^2+r^2)(r(q^2+r^2)B”(r)+3q^2B'(r))]
    -A(r)^2(q^2+r^2)[-2(q^2+r^2)^2A'(r)B'(r)^2
          	
    +B(r)(q^2+r^2)(3(q^2+r^2)A”(r)B'(r)+A'(r)((q^2+r^2)B”(r)+rB'(r)))
    
          
    +2B(r)^2(A'(r)(κ^2ϵ(q^2+r^2)^2φ'(r)^2+q^2+2r^2)-(q^2+r^2)(A^(3)(r)(q^2+r^2)+2rA”(r)))] } dr
          +∫1/κ^2A(r)^3B(r)^3(q^2+r^2)^3{-2B(r)^2(q^2+r^2)^3A'(r)^3
          	-(q^2+r^2)^2A(r)B(r)A'(r)[A'(r)(2(q^2+r^2)B'(r)-rB(r))-4B(r)(q^2+r^2)A”(r)]
    	
          	-A(r)^3[-4r(q^2+r^2)^2B'(r)^2+B(r)^3(q^2+r^2)(κ^2(q^2+r^2)^2V'(r)-4r)
    	.
          	+B(r)(q^2+r^2)(2r(q^2+r^2)B”(r)+B'(r)(κ^2ϵ(q^2+r^2)^2φ'(r)^2+6q^2))
    	
          .+2B(r)^2(2r(3q^2+r^2)-κ^2ϵ(q^2+r^2)^3φ'(r)φ”(r))]
    	
          	+A(r)^2[-2(q^2+r^2)^2A'(r)B'(r)^2-2B(r)^2(q^2+r^2)(A^(3)(r)(q^2+r^2)+2rA”(r))
          	+B(r)(q^2+r^2)(3(q^2+r^2)A”(r)B'(r)+A'(r)((q^2+r^2)B”(r)+rB'(r)))
    	
          	+B(r)^2A'(r)(3κ^2ϵ(q^2+r^2)^2φ'(r)^2+2(q^2+2r^2))](q^2+r^2)]} dr,

and 
 
    L_F(r) = (q^2+r^2)^3/2{f_0+∫1/2κ^2q^2A(r)^3B(r)^3(q^2+r^2)^5/2{2B(r)^2(q^2+r^2)^3A'(r)^3
          
    +(q^2+r^2)^2A(r)B(r)A'(r)[A'(r)(2(q^2+r^2)B'(r)-rB(r))-4B(r)(q^2+r^2)A”(r)]
    
          	
    +2A(r)^3[-2rB(r)^3(q^2+r^2)-2r(q^2+r^2)^2B'(r)^2+2rB(r)^2(κ^2ϵ(q^2+r^2)^2φ'(r)^2+3q^2+r^2)
    	.
    
          
    +B(r)(q^2+r^2)(r(q^2+r^2)B”(r)+3q^2B'(r))]
    -A(r)^2(q^2+r^2)[-2(q^2+r^2)^2A'(r)B'(r)^2
          	
    +B(r)(q^2+r^2)(3(q^2+r^2)A”(r)B'(r)+A'(r)((q^2+r^2)B”(r)+rB'(r)))
    
          
    +2B(r)^2(A'(r)(κ^2ϵ(q^2+r^2)^2φ'(r)^2+q^2+2r^2)-(q^2+r^2)(A^(3)(r)(q^2+r^2)+2rA”(r)))] } dr}.


respectively, where f_0 and f_1 are integration constants.

As for the scalar field φ, we choose the corresponding expression of the GR case which is given by <cit.>

    φ(r)=tan ^-1(r/q)/√(κ ^2 (-ϵ )).

After we substitute Eq. (<ref>) into Eqs. (<ref>) and (<ref>) and then into the condition (<ref>) with the choice (<ref>) we arrive at the equation

    (q^2+r^2)/2r[-2A'(r)/κ^2-(q^2+r^2)^2V'(r)/q^2]=0 ,

which yields the following solution for the potential:

    V(r)= V_0 + ∫ -2  q^2 A'(r)/κ ^2(q^2+r^2)^2 dr.


This completes our framework to finding new solutions in CG gravity. In order to achieve it so we just need to set a functional form for the metric coefficient A(r) (since as mentioned above we can always choose B=A(r)^-1 without any loss of generality). In this sense, in the following sections  <ref> and <ref> we present the bouncing solutions for the generalizations of Simpson-Visser and Bardeen solutions. In this sense, we use the set of equations   (<ref>) and (<ref>)  for the NLED Lagrangian density and its derivative, and (<ref>) and (<ref>)  for the scalar field potential, in order to reconstruct the functional form of the matter field Lagrangians.





§ SIMPSON-VISSER TYPE SOLUTION






 §.§ Metric and solutions



A natural generalization of the solution (<ref>) is to imbue it with the choice (<ref>) for the radial function, that is, to propose the metric coefficient

    A(r)= 1-2M/(q^2+r^2)^1/2-1/3Λ(q^2+r^2)+γ√(q^2+r^2).

This is the counterpart of the Simpson-Visser black bounce introduced in <cit.>, which is recovered when the CG parameter γ=0 (and for vanishing cosmological parameter). The black bounce proposal via the choice (<ref>) has become canonical in  the literature given its ability to regularize curvature singularities in any spherically symmetric space of interest, like ours. Furthermore, it has been shown that it is generically supported by combinations of NLED and scalar fields, see e.g. <cit.>. A similar approach is possible in our case, as shall be shown later in this section. 


In order to analyze the properties of the metric function (<ref>) we shall characterize the presence of horizons via the condition g_rr^-1(r_H)=0 (which coincides with the Killing horizon g_tt=0), that is:

    A(r_H)=0,

where the radius r_H denotes the presence of the horizon. In order to analyze the number and nature of such horizons, we add the condition 

    d A(r_H)/dr|_r=r_H=0,

which is capable to detect the extreme black holes (characterized by a degenerate horizon). Such a relation will numerically provide the critical values of the parameters of the model, namely, M, q, Λ or γ for which such extreme black holes exist. Elaborating from them we shall be able to characterize the remaining solutions within each choice of the parameters.  Given  the fact that there are four free parameters, in our approach we set three of them and take the remaining one (either M, q, or γ)  as a free parameter, and investigate the corresponding solutions for both positive and negative values of the cosmological constant Λ. This leads to six classes of solutions in order to explore all the possible geometrical structures within this setting.  For each of them we numerically determine the value of the critical values of the free parameter (either M, q, or γ) from the simultaneous resolution of Eqs. (<ref>) and (<ref>).








(i) Type-I solution: {q = 0.5, Λ = -0.2, γ=0.01}

For these values the critical mass, found by solving simultaneously Eqs.(<ref>) and (<ref>),  takes the value M_c = 0.255. Fig. <ref> shows the behavior of the metric function (<ref>) in terms of the radial coordinate r, for three mass scenarios: M > M_c, M = M_c, and M < M_c. When the mass exceeds the critical value, M > M_c, we observe the presence of two event horizons. This is possible given the fact that at r=0 we have a bounce from the region r > 0 to the region r < 0, and conversely, due to the symmetry with respect to r, a bounce from the region r < 0 to the region r > 0 is also possible. In the case where the mass is equal to the critical mass, M = M_c, we observe a degenerate double horizon at r=0, and a bounce is still present at r=0. Finally, when the mass is less than the critical mass, M < M_c, there is no horizon formation, and the geometry we have is a wormhole with a traversable throat at r=0. 



























(ii) Type-II solution: {q = 0.5, Λ = 0.2, γ=0.01}

In this case the critical mass is given by M_c=0.770. In the case where M < M_c, we see in Fig. <ref> the formation of four horizons throughout the entire region r∈(-∞,+∞), two on each side of the bounce. The innermost horizons on each side correspond to the event horizons, and the outermost are the cosmological horizons, denoted here by r_Λ. This is due to the fact that outside the cosmological horizons the metric function A(r) is negative with signature for the metric (-,+,-,-). The bounce thus connects two black hole space-times both surrounded by a cosmological horizon. In the case where the mass is equal to the critical mass, M = M_c, we observe the existence of degenerate horizons on each side of the bounce, made up of the event and cosmological horizons. When the mass exceeds the critical value, M > M_c, there is no horizon of any kind, and the metric function remains negative throughout the entire r region, which can thus be interpreted as a traversable wormhole embedded in a de Sitter space-time.







(iii) Type-III solution: {γ=0.01,M=2.0,Λ=-0.2}

Now the combined resolution of Eqs.(<ref>) and (<ref>) provides instead a critical value for the charge as  q_c=2.665. The behavior of A(r) is shown in Fig. <ref> for the scenarios of q > q_c, q=q_c and q<q_c.
Now, when q>q_c, there is no event horizon, and the space-time is described by a traversable wormhole with its throat at r=0. In the configuration where q=q_c, we have a case similar to Fig. <ref> for M=M_c with a degenerate horizon at r=0. And when q<q_c, we find a situation similar to the dashed green curve represented in Fig. <ref>, which corresponds to the scenario in which M > M_c for which there are event horizons on both sides of r=0. 







(iv) Type-IV solution: {γ=0.5,M=2.0,Λ=0.2}

The critical charge for these conditions reads q_c=2.240. The behavior of A(r) as a function of r is shown in Fig. <ref>, where the charge takes  the values q > q_c, q=q_c, and q < q_c. When q>q_c, we observe the presence of two event horizons and two cosmological horizons; for q=q_c event horizons and cosmological horizons merge to form a single degenerate horizon; while for q<q_c then we have a traversable wormhole. This scenario is thus analogous to the case represented in Fig. <ref> with M<M_c. 







(v) Type-V solution: {q=0.75,M=1.0,Λ=-0.2}

In this case the joined resolution of Eqs.(<ref>) and (<ref>) provides a critical value for the constant γ_c=2.172. The behavior of A(r) is shown in  Fig. <ref> for the scenarios γ < γ_c, γ=γ_c and γ>γ_c. Now, when γ>γ_c, there is no event horizon and a traversable wormhole arises, similarly as the cases of M<M_c and q>q_c described in Figs. <ref> and <ref>, respectively.  
In the configuration where γ=γ_c, we have a case similar to Fig. <ref> for M=M_c and also to the case of q=q_c described in  Fig. <ref>  (middle left). And when γ<γ_c, we find a situation similar to the curve represented in Fig. <ref>, which corresponds to the scenarios of M > M_c and also to the case of q<q_c described in  Fig. <ref>.







(vi) Type-VI solution: {q=0.52,M=1.2,Λ=0.2}

Under these conditions, we obtain γ_c=0.133. The behavior of A(r) as a function of r is shown in  Fig. <ref>,  where the parameter takes the values γ> γ_c, γ=γ_c and γ < γ_c. In the case that γ>γ_c, we observe a similar behaviour as in  Fig. <ref>,  for M<M_c and also for the case that q < q_c, as described in  Fig. <ref>. In the case that γ=γ_c, we have a similar scenario as in  Fig. <ref>,  for M= M_c. Finally, for γ < γ_c, we have a wormhole geometry, analogous to the case in  Fig. <ref>, but with M>M_c. 





 §.§ Kretschmann scalar



On the other hand we can study the regularity properties of our model by computing the  Kretschmannn scalar as


    K=	
    1/3(q^2+r^2)^5{36 M^2 (3 q^4-4 q^2 r^2+4 r^4)+	12 M q^2[q^2 (r^2 (4 Λ√(q^2+r^2)-9 γ)-8 √(q^2+r^2))
    	
       +8 r^2 √(q^2+r^2)-2 r^4 (γ -Λ√(q^2+r^2))+q^4 (2 Λ√(q^2+r^2)-7 γ)]
    	
       +(q^2+r^2)^[4Λ^2q^8+8r^6(3γ^2-3γΛ√(q^2+r^2)+Λ^2r^2)+q^6(27γ^2-20γΛ√(q^2+r^2)+16Λ(Λ r^2-1))
    	. 
       ..4q^2r^4(12γ^2-13γΛ√(q^2+r^2)+2Λ(3Λ r^2-1))+q^4(-48γ√(q^2+r^2)(Λ r^2-1)+51γ^2r^2+4(7Λ^2r^4-6Λ r^2+9))]} 	.


which, at asymptotic infinity takes the value R(r →∞) →8 Λ ^2/3, while being regular at the bouncing region r=0, as is transparent from Eq. (<ref>).





 §.§ Matter Lagrangian reconstruction



The above solutions can be shown to be supported by a NLED plus a scalar field matter field. To obtain the explicit shapes of their Lagrangian densities, we use Eq. (<ref>)  to obtain L_ NLED as

    L_ NLED(r)    =  f_0+2f_1q^2/√(q^2+r^2)-2q^2/6κ^2√(q^2+r^2)(q^2+r^2)^2×
       ×[-45M-3γ(q^2+r^2)+4√(q^2+r^2)]+
    
       2q^2/κ^2(-6M/(q^2+r^2)^5/2-γ/3(q^2+r^2)^3/2+1/2(q^2+r^2)^2),

and its derivative L_F as

    L_F(r)    =   (q^2+r^2)^3/2[f_1-(-45M-3γ(q^2+r^2)
    	..
          .. +4√(q^2+r^2))/6κ^2(q^2+r^2)^2],

while for the scalar field  φ(r) we use Eq.  (<ref>) to find the potential V(r) as

    V(r)= 2 q^2 [6 M+5 (q^2+r^2) (γ -Λ√(q^2+r^2))]/15 κ ^2 (q^2+r^2)^5/2 .


On the other hand, inverting numerically the expressions above to obtain  r(F) and r(φ), we determine

    L _ NLED (F)     =   	f_0+2√(2)f_1√(F)q^3/2-2F/3κ^2
           +6√(2)F^5/4M/κ^2√(q)+2^3/4γ F^3/4√(q)/3κ^2,
   
and

    V(φ)    =   2(3Mcos(2√(κ^2(-ϵ)))+3M)/15κ^2(q^2^2(√(κ^2(-ϵ))))^3/2
          +2q^2(γ-Λ√(q^2^2(√(κ^2(-ϵ)))))/3κ^2(q^2^2(√(κ^2(-ϵ))))^3/2.
  
Considering f_0=0, f_1=0, γ=0 and Λ=0 in the Lagrangian (<ref>) we obtain the same expression as the one of GR as found in Ref. <cit.>. On the other hand, in expression (<ref>) the scalar field must necessarily be a phantom field, i.e. 
ϵ=-1, as is transparent in Eq. (<ref>). Note that in the limit F≫ 1, the NLED Lagrangian behaves as

    L _ NLED (F)∼    f_0+2√(2)f_1q^3/2F^1/4+6√(2)F^5/4M/κ^2√(q)
       +2^3/4γ q^1/2F^3/4/3κ^2-2F/3κ^2.

which is a non-Maxwellian behavior in the central region of the field. The bottom line of this analysis is that, similarly as in the case of GR <cit.>, the Simpson-Visser solution can be modelled as a solution of a NLED and a scalar field, though with a more general functional form.

In Fig. <ref> we depict the behaviour of the NLED Lagrangian density given by Eq. (<ref>) using the blue dashed curve for f_0=0 and f_1=0.2 and for the case similar to GR with the red dotted-dashed curve with the values f_0=0 and f_1=0. Similarly, in Fig. <ref>. we illustrate the behaviour of the scalar field potential in Eq. (<ref>) for some choices of the model parameters. Note that the amplitude of this potential changes depending on the charge values. We illustrate this for three different values of the charge: q = 0.2, q = 0.3 and q = 0.4.














§ BARDEEN-TYPE SOLUTION






 §.§ Metric and solutions



Consider now the following metric function:

    A(r)= 1-2Mr^2/(q^2+r^2)^3/2-1/3Λ(q^2+r^2)+γ√(q^2+r^2).

This is a generalization, within CG, of the so-called Bardeen magnetic monopole <cit.> via the CG parameter γ, using a similar trail of thought as in the Simpson-Visser solution of the previous section with the replacement of the radial function with the choice (<ref>). Bardeen's monopole avoids the presence of a singularity via its replacement by a de Sitter core, a canonical procedure within GR to achieve this end <cit.>, here generalized to display a bouncing behaviour on its radial sector. In this section we shall study the properties of this metric function using again to this end Eqs. (<ref>) and (<ref>) for the radius of the horizon and the critical parameters of the solution, and split the discussion also into six sub-cases.




























(i) Type-I solution: {q = 0.5, Λ = -0.2,γ=0.01}

In this case the critical mass reads M_c = 0.686. The behaviour of the metric function (<ref>) is shown in Fig. <ref> for three mass scenarios: M > M_c, M = M_c and M < M_c. When the mass exceeds the critical mass, M > M_c, we observe a total of four horizons in the entire r region, the innermost horizons being the Cauchy horizons and the outermost being the two event horizons.  At r=0 we have the throat of the wormhole, which is hidden behind these horizons. For the case where M=M_c, we have two degenerate horizons, one at r > 0 and the other at r < 0,  connected by the throat at r=0. In the case where M < M_c, we find the geometry of a traversable wormhole, a scenario similar to that in Fig. <ref> for M < M_c, Fig. <ref> for q > q_c and Fig. <ref> for γ>γ_c. 







(ii) Type-II solution: {q=0.5,Λ=0.2,γ=0.01}

In this case we find the critical mass value M_c=0.811. The scenarios M > M_c, M = M_c and M < M_c are illustrated in Fig. <ref>. When  M > M_c, two event horizons are formed, one in each r region, connected at r=0 by a bounce in the radial function. When M=M_c, we observe the formation of four horizons, the innermost being Cauchy horizons, while the outermost ones are degenerate horizons that arise from the union of cosmological and event horizons. And when  M < M_c, we observe the formation of six horizons: the two innermost ones (on each side) are Cauchy horizons, the  two intermediate ones are the event horizons, and the two outermost ones are the cosmological horizons.
 






(iii) Type-III solution: {γ=0.01,M=2.0,Λ=-0.2 }

In this case we find the  critical charge value of q_c=1.204. The behavior of A(r) is shown in Fig. <ref> for the cases q > q_c, q=q_c, and q < q_c. When q > q_c, we observe a similar behavior to the curve described in Fig. <ref> for the case where M < M_c, finding a traversable wormhole geometry. Similarly, when q=q_c we find the same behaviour as that of M=M_c in Fig. <ref>. Finally, when  q < q_c the metric function exhibits the same behavior as the one of Fig. <ref> with M > M_c. 







(iv) Type-IV solution: {γ = 0.01,M = 0.5,Λ = 0.12}

Now the critical charge reads q_c = 0.389. The behavior of A(r) is shown in Fig. <ref> for the cases q > q_c, q = q_c, and q < q_c. When q > q_c we observe a similar behavior as in Fig. <ref> for the case q > q_c; for q=q_c we note the presence of degenerate horizons and two cosmological horizons, one each in the regions r>0 and r<0; and when  q < q_c, it shows the same behavior as in Fig. <ref>, but with M < M_c.







(v) Type-V solution: {q=0.75,M=2.0,Λ=-0.2}

Now we obtain the critical value of γ_c=0.792. The behavior of A(r) is shown in Fig. <ref> for the scenarios γ < γ_c, γ=γ_c and γ>γ_c. When γ>γ_c we have a similar behavior to the cases of M<M_c and q>q_c as described in Fig. <ref> and <ref>; when γ=γ_c a similar behaviour as Fig. <ref> for M=M_c and Fig. <ref> for q=q_c; and when γ<γ_c a similar situation as in Fig. <ref> when  M > M_c.







(vi) Type-VI solution: {q=0.75,M=1.0,Λ=0.2 }

Under these conditions we obtain γ_c=0.107. The behavior of A(r) as a function of r is shown in Fig. <ref> for the values γ> γ_c, γ=γ_c and γ < γ_c. When  γ>γ_c we observe a similar behavior as in Fig. <ref> for the case of q > q_c; when γ=γ_c we also have a similar scenario as in Fig. <ref> for q= q_c; and for γ < γ_c a similar behaviour as is Fig. <ref> for q<q_c.





 §.§ Kretschmann scalar



As for the regularity properties of this family of solutions, we compute the Kretschmann scalar as


    K=	
    1/3(q^2+r^2)^7{ 12M^2(4q^8-44q^6r^2+169q^4r^4-68q^2r^6+12r^8)+4Mq^2(q^2+r^2).	
    	
       +r^6(2Λ√(q^2+r^2)-30γ)-3q^2(16r^2√(q^2+r^2)+15γ r^4)+24r^4√(q^2+r^2)
        +q^6(4Λ√(q^2+r^2)-6γ) +q^4r^2(2Λ√(q^2+r^2)-21γ)	
       +(q^2+r^2)^3[4Λ^2q^8+8r^6(3γ^2-3γΛ√(q^2+r^2)+Λ^2r^2)+q^6(27γ^2-20γΛ√(q^2+r^2)+16Λ(Λ r^2-1))+.	
       ..4q^2r^4(12γ^2-13γΛ√(q^2+r^2)+2Λ(3Λ r^2-1))+q^4(-48γ√(q^2+r^2)(Λ r^2-1)+51γ^2r^2+4(7Λ^2r^4-6Λ r^2+9))]} 	.


which provides a finite value at r→0 and, therefore, likewise in the previous case the geometry is everywhere regular.





 §.§ Matter Lagrangian reconstruction



Using Eq. (<ref>), we obtain the NLED Lagrangian density as

    L_ NLED(r)    =    f_0+2f_1q^2/√(q^2+r^2)+q^2/21κ^2(q^2+r^2)^7/2×
          ×[M(41q^2+231r^2)+7(q^2+r^2)×
          ×(γ(q^2+r^2)-√(q^2+r^2))],

its derivative as

    L_F(r)   =   (q^2+r^2)^3/2{f_1-[190Mq^2-3γ(q^2+r^2)^2
          -165M(q^2+r^2)+4(q^2+r^2)^3/2]/[6κ^2(q^2+r^2)^3]} ,
 
and the scalar field potential as







    V(r)= 2 q^2 /3 κ ^2(6 M (7 r^2-8 q^2)/35 (q^2+r^2)^7/2+γ/(q^2+r^2)^3/2-Λ/q^2+r^2),


respectively. Expressing r(F) and r(φ), we determine

    L _ NLED (F)  = 	f_0+2 √(2) f_1 √(F) q √(q/√(F))-2 F/3 κ ^2
        -380 2^3/4 F^2 M √(q/√(F))/21 κ ^2+2^3/4γ  F √(q/√(F))/3 κ ^2
        +22 √(2) F^3/2 M √(q/√(F))/κ ^2 q.
  
and

    V(φ)    =    -2 γcos ^4(φ√(κ ^2 (-ϵ ))) √(q^2  ^2(φ√(κ ^2 (-ϵ ))))/3 κ ^2 q^2
           +4 M sin ^2(φ√(κ ^2 (-ϵ ))) cos ^6(φ√(κ ^2 (-ϵ ))) /5 κ ^2 q^4×
          ×√(q^2  ^2(φ√(κ ^2 (-ϵ )))) -2 Λcos ^2(φ√(κ ^2 (-ϵ )))/3 κ ^2
            -32 M cos ^8(φ√(κ ^2 (-ϵ ))) √(q^2  ^2(φ√(κ ^2 (-ϵ ))))/35 κ ^2 q^4.
 

Considering again f_0=0, f_1=0, λ=0 and Λ=0 in the Lagrangian of equation (<ref>), we obtain the same expression as the Lagrangian of GR, found in Ref. <cit.>. The central limit, F≫1, of the corresponding NLED Lagrangian reads as 

    L _ NLED (F)  ∼  f_0+2√(2)f_1q^3/2√(F)+22√(2)F^5/4M/κ^2√(q)
    
    -2F/3κ^2   
       -380M2^3/4√(q)F^7/4/21κ^2+2^3/4γ√(q)F^3/4/3κ^2.

which is again a non-Maxwellian behaviour there.

Fig. <ref> shows the behavior of the NLED Lagrangian density using Eq. (<ref>) for the values {f_0=0,f_1=0.2} (dashed blue curve), and {f_0=0,f_1=0} (dotted-dashed red curve). Similarly, in Fig. <ref> we depict the behaviour of the scalar field potential following  Eq. (<ref>).













§ SUMMARY AND CONCLUSION





In this paper we have studied spherically symmetric geometries implementing a bouncing in the radial function within a recently proposed theory of gravity known as Cotton Gravity (CG). Such geometries are inspired by a recently found vacuum solution of CG which generalizes the Schwarzschild solution of GR via a new parameter γ, and are further extended towards the black bounce mechanism to avoid the focusing of geodesics ascribed to the singularity theorems.  We have investigated such solutions by coupling the field equations of CG to a combination of a NLED and a scalar field. Solving the corresponding field equations we have found the free functions L_ NLED (r) and L_F(r) that characterize the NLED sector, and the scalar field potential V(r) associated to a specific scalar field found within the context of GR. 

Subsequently, we have studied two generalizations of bouncing solutions found within GR, namely the  Simpson-Visser and Bardeen type solutions, both understood to be supported by magnetic NLED plus scalar fields. Such geometries differ from their GR counterparts  via an additional term depending on the parameter γ of CG, and further supplemented by a cosmological constant term given the fact that the latter naturally arises as an integration constant in CG. Given the presence of four constants in our model, {M,q,Λ,γ}, we used an approach in which we set three of them and vary the remaining one (keeping always fixed the cosmological constant term Λ to both positive and negative values), thus leading to six types of solutions on each case. 


For the first model we used a generalized version of the bouncing metric of the Simpson-Visser type and simultaneously solved the equations (<ref>) and (<ref>) to find the critical parameters for {M_c,q_c,γ_c}. For Λ < 0, in the case of the critical mass M_c,  when M > M_c we found an event horizon on both sides of the bounce r=0, which for M = M_c merge into a degenerate horizon at r=0, while if M < M_c a traversable wormhole is formed. In the case of Λ > 0, no horizons are formed for M > M_c, while degenerate horizons are observed for M = M_c, and M < M_c, two horizons are observed on each side of the bounce, the outermost ones being cosmological horizons and the two innermost ones event horizons. On the other hand, for Λ < 0, in the case of critical charge, we observe that there is no horizon for q > q_c, when q = q_c we have degenerate horizons, whereas for q < q_c we observe two horizons. In the scenario where Λ > 0, we identify the presence of two cosmological horizons in all cases, whether q > q_c, q = q_c, or q < q_c.  We also analyzed the Kretschmann scalar of this model finding its regularity everywhere and, in particular, at the bounce location r=0. Finally, we verified via direct computation of their specific shapes that such bouncing geometries are supported by a specific combination of a NLED Lagrangian and a scalar field potential. Our setting recovers the GR one when we choose f_0=f_1=γ=0.


For the second model, we adopted Bardeen's regularized metric function and also determined the solutions for the horizons numerically. In the case of critical mass, for Λ < 0, when M > M_c we observe four horizons, where the innermost horizons are Cauchy horizons on each side of the bounce; when M = M_c, degenerate horizons are observed; and if M < M_c we find a traversable wormhole. For Λ > 0, if M > M_c we observe two horizons, when M = M_c we find  four horizons, the innermost of which are the Cauchy horizons. In the scenario where M < M_c, we discover six horizons, the innermost of which is the Cauchy horizon and the outermost being the cosmological horizons. For the critical charge in which Λ < 0, if q > q_c there is no horizon, if q = q_c we observe a degenerate horizons on each side of the bounce, while if q < q_c we have two horizons on each side (Cauchy and event). For Λ > 0, we also observe that there are cosmological horizons in all scenarios (q > q_c, q = q_c, q < q_c) and, in particular, for q < q_c a total of three horizons on each side are observed. With the critical parameter γ for Λ<0, if γ > γ_c, we have a wormhole, for γ =γ_c, degenerate horizons occur, and if γ < γ_c we have four horizons in total, with the innermost ones being Cauchy horizons. In the case where Λ > 0, two horizons are formed for γ > γ_c, degenerate horizons are formed for γ = γ_c, and for γ < γ_c we have six horizons. Similarly as in the Simpson-Visser geometry we verified the regularity of the  Kretschmann scalar and found the explicit shapes of the NLED Lagrangian density and the scalar potential.

The bottom line of our analysis is that CG naturally allows for suitable generalizations of bouncing geometries of GR in terms of the new CG parameter γ, extracted out of the class of vacuum solutions found within this theory. The geometries built this way display promising features to be studied within the context of black hole thermodynamics, the analysis 
 the stability of solutions through the analysis of perturbations, gravitational lensing and black hole shadows, topics in which we hope to report soon. 
 







MER thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Brazil, for partial financial support. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
FSNL acknowledges support from the Fundação para a Ciência e a Tecnologia (FCT) Scientific Employment Stimulus contract with reference CEECINST/00032/2018, and funding through the research grants UIDB/04434/2020, UIDP/04434/2020 and PTDC/FIS-AST/0054/2021.
DRG is supported by the Spanish Agencia Estatal de Investigación Grant No. PID2022-138607NB-I00, funded by MCIN/AEI/10.13039/501100011033, UE, and ERDF A way of making Europe.





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 Paper Title: Nonsingular universe through zero-point length cosmology.






asheykhi@shirazu.ac.ir
Department of Physics, College of
Science, Shiraz University, Shiraz 71454, Iran
Biruni Observatory, College of Science, Shiraz University,
Shiraz 71454, Iran

Department of Physics, College of
Science, Shiraz University, Shiraz 71454, Iran

kimet.jusufi@unite.edu.mk
Physics Department,
State University of Tetovo, Ilinden Street nn, 1200, Tetovo, North
Macedonia





§ ABSTRACT 
Taking into account the zero-point length correction to the
gravitational potential and using the entropic force scenario, we
derive a modified expression for entropy that includes zero-point
length correction terms. We then apply the thermodynamics-gravity
conjecture and the emergent gravity scenario to derive the
modified Friedmann equations. These equations describe the
evolution of the universe with zero-point length corrections
included in the entropy expression. Further, we examine the time
evolution of the total entropy, including the entropy of the
apparent horizon and the matter field entropy inside the horizon
and find that the generalized second law of thermodynamics is
satisfied. Besides, we investigate cosmological implications of
the modified cosmology through zero-point length. We study several
cases, including radiation dominated era, matter dominated era and
universe filled with matter and cosmological constant. We observe
that the zero-point length correction does not change the general
profile of the universe evolution, however, it shifts the time of
the phase transition in a universe filled with matter and
cosmological constant. We explore the age of the universe for our
model and observe that the predicted age of the universe becomes
larger compared to the standard cosmology. By calculating the
explicit form of Ricci and Kretchmann invariants, we confirm that
in our model, the initial singularity of the universe is removed.
This is an expected result, because the main motivation for
considering zero-point length correction in the gravitational
potential is to remove singularity at the origin.

Nonsingular universe through zero-point length cosmology
    Kimet Jusufi
    August 5, 2024
========================================================




§ INTRODUCTION

The discovery of black holes thermodynamics in 1970's, prompted
physicists to speculate a profound connection between gravity and
thermodynamics. Jacobson was the first who argued that the
Einstein field equations of general relativity is just an equation
of state for the spacetime <cit.>. The relationship
between gravitational field equations and the first law of
thermodynamics was generalized to other theories, beyond Einstein
gravity, such f(R) gravity <cit.>, Gauss-Bonnet gravity,
scalar-tensor gravity, and more general Lovelock gravity
<cit.>. In the cosmological setups, it was shown that
the first law of thermodynamics on the apparent horizon,
dE=TdS+WdV, can be translated as the first Friedmann equation of
the Friedmann-Robertson-Walker (FRW) universe
<cit.>. Here, T and S are,
respectively, the temperature and entropy associated with the
apparent horizon, dE is the change in the total energy inside
the apparent horizon due to the volume change dV of the
universe, while W is the work density. The connection between
thermodynamics and gravity reveals that the field equations of
gravity are just equation of state for the spacetime.

Another approach for understanding the nature of gravity proposed
by Verlinde <cit.>, who argued that gravity is not a
fundamental force and can be regarded as an entropic force which
appears due to the change in the information associated with the
system.  Two main assumptions of Verlinde' proposal are the
holographic principle and the equipartition law of energy for the
degrees of freedom of system. Using these two principles, Verlinde
derived the Newton's law of gravity, the Poisson equation and in
the relativistic regime the field equations of general relativity.
The studies on the entropic force scenario were generalized to
cosmology, where it has been shown that Friedmann equations can be
derived by using Verlinde's proposal <cit.>.


Although Jacobson and Verlinde arguments reveal thermodynamic
nature of gravity from thermodynamics and statistical mechanics
point of view, however, they assumes spacetime as a pre-exist
geometry. At a deeper level, Padmanabhan <cit.> argued that
the spatial expansion of our Universe can be understood as a
consequence of the emergence of space, and the cosmic space is
emergent as the cosmic time progress.  By equating the difference
between the number of degrees of freedom in the bulk and on the
boundary, Padmanabhan derived the Friedmann equation that
describes the evolution of the universe <cit.>. The idea that
gravity is an emergent phenomenon not only checked in Einstein
gravity, but also has been generalized to Gauss-Bonnet gravity,
Lovelock gravity, and braneworld scenarios
<cit.>. In all these studies, the aim is to
extract the dynamical field equations in the background of FRW
universe, by calculating the number of degrees of freedom on the
boundary and in the bulk.

Despite the thermodynamic nature of gravity, general relativity
suffers some basic problems such as singularity at the center of
the black hole solutions and initial singularity at the origin of
the universe. The singularity also exist in Newtonian gravity for
a point particle with mass M, where the gravitational potential,
ϕ=-GM/r, diverges at r=0. Many attempts have been done to
remove singularities in the solutions of general relativity. For
example, regular charged black holes have been proposed which are
nonsingular local solutions of the field equations in the presence
of nonlinear electrodynamics <cit.>. There is also
an independent way for removing singularity in the black hole
theory, which is based on the concept of T-duality in string
theory. According to T-duality the momentum space propagator for
the massless particles is given
by <cit.>

    G(k)=-l_0 /√(k^2) K_1(l_0
    √(k^2)).

where k=p/ħ, l_0 represents the zero-point length of
spacetime, and K_1(x) is a modified Bessel function of the
second kind. Based on this, the static Newtonian potential
corresponding to G(k) at distance r get modified as
 <cit.>

    V(r)=-M /√(r^2+l_0^2),

where M is the mass of the source of the potential and we have
set G=ħ=1, for simplicity. Using the above potential, the
metric of the Schwarzschild black hole is obtained as
<cit.>

    g_tt(r)=g^-1_rr(r)=1-2Mr^2/(r^2+l_0^2)^3/2,

which is regular at r=0. The main difference between this
solution inspired by T-duality, with those obtained for regular
black holes in <cit.> is that here we do not need
nonlinear electrodynamics for removing singularity at r=0.

In the present work, we would like to examine the cosmological
consequences of zero-point length during the history of the
universe. Recently, the cosmological viability of the modified
Friedmann equations through zero-point length was explored in
<cit.>. They also explored several cosmological aspects of
the theory, including the inflationary paradigm under the
slow-roll condition, density perturbations in the linear regime,
power-law spectrum as well as gravitational waves spectrum
<cit.>. It has been shown that this model is compatible with
observational data <cit.>. Our work differs from
<cit.> in that we obtain the modified Friedmann equations by
applying the emergence scenario of gravity and check the validity
of the generalized second law of thermodynamics. Besides, we
explore some novel cosmological aspects of this theory which were
not addressed in <cit.>. In particular, we shall calculate
the age of the universe for several cases and confirm that this
model is capable to remove the initial cosmic singularity and
reproduce a nonsingular universe.


This paper is organized as follows. In the next section, we use
the entropic gravity and obtain the entropy expression with
zero-point length correction terms. Then, by applying the first
law of thermodynamics on the apparent horizon, we derive the
modified Friedmann equations inspired by zero-point length. In
section III, we employ the emergence scenario for the cosmic space
and derive the modified Friedmann equations which is coincides
with the result of section II. In section IV, we check the
validity of the generalized second law of thermodynamics. In
section V, we explore cosmological consequences of the model for
various cases. We finish with closing remarks in the last section.



§ MODIFIED FRIEDMANN EQUATIONS THROUGH ZERO-POINT LENGTH 

As we mentioned in the introduction, the gravitational potential
get modified due to the zero-point length effects. The modified
gravitational potential ϕ(r) of spherical mass M at
distance r is given by <cit.>

    ϕ(r)=-M /√(r^2+l_0^2),

where l_0 is the zero-point length and its value is expected to
be of Planck length <cit.>.
If a point particle with mass
m is placed at distance r=R in this gravitational potential,
it will feel the following force

    F⃗=-m ∇⃗ϕ(r)|_r=R=-Mm/R^2[1+l_0^2/R^2]^-3/2r̂.

Taking the entropic force proposal into account, the correction
term to the horizon entropy is given by <cit.>

    1+(1/2 π
    R)ds(A)/dR=(1+l_0^2/R^2)^-3/2.

The total entropy of the horizon is given by

    S_h=A/4+s(A).

where the correction terms is denoted by s(A). Note that A=4π
R^2 is the area of the horizon which is the boundary of the
volume V=4π R^3/3. Combining Eqs. (<ref>) and (<ref>), we
find

    dS_h=2π R (1+l_0^2/R^2)^-3/2 dR

The explicit form of the total entropy associated with the
horizon, can be obtained by integrating the above equations. We
find

    S_h   =   π R^2 (1+l_0^2/R^2)^-1/2+3π l_0^2
    (1+l_0^2/R^2)^-1/2
          -3 π
    l_0^2 ln(R+√(R^2+l_0^2)),

In the limiting case where l_0^2→ 0, the area law is
recovered. We assume our Universe is described by the line
elements

    ds^2=h_μνdx^μ dx^ν+R^2(dθ^2+sin^2θ
    dϕ^2),

where R=a(t)r, x^0=t, x^1=r, and h_μν=diag (-1,
a^2/(1-kr^2)) stands for the two dimensional metric. The spatial
curvature of the universe is described by k = -1,0, 1. The
radius of the apparent horizon is  <cit.>

    R=1/√(H^2+k/a^2).

The temperature associated with with the apparent horizon is
obtained as <cit.>

    T_h=-1/2 π R(1-Ṙ/2H R).

We further assume total energy momentum tensor is in the form of
the perfect fluid,

    T_μν=(ρ+p)u_μu_ν+pg_μν,

where ρ and p are, respectively, the energy density and
pressure of the matter field inside the universe. The continuity
equation, which shows the conservation of energy in the background
of FRW universe, is given as usual

    ρ̇+3H(ρ+p)=0,

where H=ȧ/a is the Hubble parameter. In an expanding
universe, the work is done due to the volume change. The
corresponding work density is given by  <cit.>

    W=-1/2 T^μνh_μν=1/2(ρ-p).

The total energy inside a 3-sphere is E=ρ (4π R^3/3),
where its differential form is given by

    dE=4πr̃_A^2ρ dr̃_A-4π H r̃_A^3(ρ+p) dt,

where we have used the continuity equation (<ref>). Combining
Eqs. (<ref>), (<ref>), (<ref>), (<ref>) with the
first law of thermodynamics on the apparent horizon, dE = T_h
dS_h + WdV, we arrive at

    4π H R^3 (ρ+p)dt=(1+l_0^2/R^2)^-3/2 dR.

The above equation can be rewritten as

    -2/R^3(1+l_0^2/R^2)^-3/2 dR=
     8π/3dρ.

where we have again used Eq. (<ref>). Integrating and then
expanding, for small value of l_0^2/R^2, we reach

    -2/l_0^2[ 1-
    l_0^2/2R^2+3/8l_0^4/R^4+...]+C=8π/3ρ.

where C is a constant of integration. When l_0^2/R^2
→ 0, we expect the above relation to restore the
standard Friedmann equation. This implies that

    C≡2/l_0^2-Λ/3,

where Λ is a constant which can be interpreted as the
cosmological constant. Note that one may also define C= 2/l_0^2
by absorbing/neglecting the contribution from vacuum energy to the
total energy density ρ.

The final form of the modified Friedmann equation, due to the zero
point length correction terms, is obtained as

    H^2+k/a^2-α(H^2+k/a^2)^2=8π/3(ρ+ρ_Λ),

where α=3l_0^2/4, ρ_Λ=Λ/8π and we have
used the relation (<ref>). Since l_0^2 is very small, we can
neglect the higher order term of α in Eq (<ref>).
Solving Eq. (<ref>) yields

    H^2+k/a^2=8π/3ρ (1+Γρ),

where Γ=2π l_0^2 and we have st Λ=0, for
simplicity. As an additional point of information, we note that
the modified Friedmann equation given by the last equation has a
form similar to those found in Brane-cosmology models (see, in
particular, Eqs. (656) and (657) in <cit.>). In
such a case, the factor Γ can be linked to to the brane
tension σ, by the inverse relation Γ∼
1/(2σ). Another important result can be found from Eq.
(<ref>), namely if we set ȧ=0 (H=0) and
Λ=0; and we further solve this equation using a=a_
min, we get two branches

    k/a_ min^2=3 ±√(9-96 απρ)/6 α,

which implies a critical density to exist with the condition,
i.e., 9-96 απρ≥ 0|_ρ=ρ_ c, which
gives ρ_ c≤ 1/(8 π l_0^2). If we set
ρ=ρ_ c and we solve for the minimal scalar factor we
get

    a_ min=√(6 k)/2 l_0.

This equation indicates the existence of a minimum length for the
scale factor and suggests a possible bouncing scenario for a
closed universe, given the additional condition Ḣ>0. On
the other hand, for a flat universe (k=0), Eq. (<ref>)
reads

    H^2-α H^4=8π/3(ρ+ρ_Λ).

In the absence of the cosmological constant (Λ=0), we have

    H^2-α H^4=8π/3ρ.

where ρ is the total energy density. In terms of the density
parameter, we can rewrite the above equation as

    1-α H^2=Ω.

where Ω=8πρ/(3H^2)=Σ_iΩ_i is the total
density parameter and Ω_i=8πρ_i/(3H^2). It is
clear that in this case Ω<1.

It is instructive to take a glance at Eq. (<ref>) in the
high energy limit (early universe) where Γρ^2≫ρ.
For flat universe (k=0) we have

    H^2≈8π/3Γρ ^2.

 Using p=ρω in the continuity equation (<ref>), the following expression is obtained.
 
    ρ(t)= ρ_0   a^-3(1+ω)

In the radiation dominated era where ω=1/3, we have
ρ_r=ρ_r,0 a^-4. In this case Eq. (<ref>)
admits a solution of the form a(t)=C_1 t^1/4, with
C_1≡[ 4 √(8πΓ/3)ρ_r,0]^1/4. Thus in this case the scale factor evolves as
a(t) ∼ t^1/4. Comparing to the radiation dominated era in
standard cosmology (a(t) ∼ t^1/2), we see that the rate of
the universe expansion is slower in this case. In a matter
dominated universe where ω=0, we get
 ρ_m=ρ_m,0 a^-3. Thus, Eq. (<ref>)
admits the solution a(t)=C_2 t^1/3, where in this case
C_2≡[3 √(8πΓ/3)ρ_m,0]^1/3. Therefore, the scale factor evolves as a(t) ∼
t^1/3, which compared to the matter dominated era in standard
cosmology (a(t) ∼ t^2/3), we observe that the rate of our
Universe expansion is slower when the zero point length is taking
into account in the cosmological field equations.



§ EMERGENCE OF THE COSMIC SPACE

The notion that gravity is an emergence phonomania and the cosmic
expansion can be understood by calculating the difference between
degrees of freedom on the horizon and in the bulk was pointed out
by Padmanabhan <cit.>. According to Padmanabhan the increase
dV of the cosmic volume, is given by <cit.>

    dV/dt=G(N_sur-N_bulk),

where N_sur and N_bulk stand for the
number of degrees of freedom on the boundary and in the bulk,
respectively. The studies on the emergence gravity were
generalized to Gauss-Bonnet and Lovelock gravity
<cit.>, and braneworld scenarios <cit.>,
where it has been shown that the corresponding Friedmann equations
can be derived by properly modifying proposal (<ref>). In this
work, following <cit.> we assume (G=1)

    dV/dt=RH (N_sur-N_bulk).

We propose the temperature associated with the apparent horizon is
given by

    T=1/2π R,

In order to write down the number of degrees of freedom on the
horizon, we first expand the modified entropy (<ref>) in terms
of l_0 ^2/R^2. We find


    S_h   =   π R^2 +5l_0^2/2 -9π/4
    l_0^4/R^2+....

Neglecting the term of order l_0 ^4, and taking into account the
fact that N_sur∼4 S_h, we can recognize

    N_sur=4π R^2+10 π l_0 ^2,

where the second term comes from the leading order term in the
entropy expansion and reflects the influences of the zero-point
length corrections in the entropy expression.

The Komar energy inside the sphere with volume V=4 π R^3/3 is given by

    E_Komar=|(ρ +3p)|V.

From the equipartition law of energy, the number of degrees of
freedom of the matter field in the bulk is given by

    N_bulk=2|E_Komar|/T.

Therefore, the number of degrees of freedom in the bulk is
obtained as

    N_ bulk=-16 π^2/3  R^4 (ρ+3p),

where we have assumed ρ+3p<0 in an expanding universe.
Combining relations (<ref>) and (<ref>) with Eq.
(<ref>), after simplifying, we get

    2 R^-3Ṙ/H-2R^-2-5 l_0 ^2 R^-4=8π/3(ρ+3p).

Multiplying both side of Eq. (<ref>) by factor ȧa,
after using the continuity equation (<ref>), we arrive at

    d/dt( a^2 R^-2)+5/2l_0
    ^2/R^4d/dt (a^2)=8 π/3d/dt(ρ
    a^2).



Using the fact that R=a r in the second term, we can integrate
the above equation. We find

    H^2+k/a^2-γ(H^2+k/a^2)^2=8π/3(ρ+ρ_Λ),

where γ=5 l^2_0/2 and we have used relation
(<ref>) in the last step.  This result coincides with the
one obtained from the first law of thermodynamics in Eq.
(<ref>). This further supports the viability of the
Padmanabhan's perspective of emergence gravity.



§ GENERALIZED SECOND LAW OF THERMODYNAMICS
 Now we are
going to check whether the obtained entropy associated with the
apparent horizon, given in Eq. (<ref>) can satisfy the
generalized second law of thermodynamics. For our Universe, which
is currently experiencing a phase of accelerated expansion, the
generalized second law of thermodynamics have been investigated in
<cit.>.

Using Eq. (<ref>) as well as the continuity equation, we
get

    Ṙ= 4π R^3 H (ρ+p)
    (1+l_0^2/R^2)^3/2.

It is a matter of calculations to show that

    T_hṠ_̇ḣ   =   4π H R^3 (ρ+p)(1-Ṙ/2HR).

Since in an accelerated universe, one may have ρ+p<0, which
implies the second law of thermodynamics may break down,
Ṡ_̇ḣ<0. Therefore we consider the generalized second law
of thermodynamics. From the Gibbs equation we have <cit.>

    T_m dS_m=d(ρ V)+pdV=V dρ+(ρ+p)dV,

where T_m and S_m stand for the temperature and entropy of
the matter fields in the bulk, respectively. Assuming the thermal
equilibrium hypothesis, the temperature of the matter field in the
bulk and the horizon temperature are the same, namely T_m≈
T_h <cit.>. This assumption is necessary to prevent the
flow of energy between the bulk and the boundary. The Gibbs
equation (<ref>) can be easily written as

    T_hṠ_m =4πR^2Ṙ(ρ+p)-4πR^3H(ρ+p).

Adding Eqs. (<ref>) and (<ref>), after using Eq.
(<ref>), we reach

    T_h( Ṡ_̇ḣ+Ṡ_̇ṁ)=8 π^2 H R^5(ρ+p)^2
    (1+l_0^2/R^2)^3/2.

From this equation we always have Ṡ_̇ḣ+Ṡ_̇ṁ≥ 0,
which guaranties the validity of the generalized second law of
thermodynamics.



§ COSMOLOGICAL IMPLICATIONS 

In this section, we would like to study cosmological consequences
of the modified Friedmann equations. For simplicity, we consider a
flat universe, although one can generalize the arguments to a
nonflat universe.


 §.§ Matter dominated era

Let us begin with the case where our Universe is dominated with
pressureless matter and the cosmological constant is equal to
zero. From the continuity equation ρ̇_m+3H ρ_m=0, we
have ρ_m=ρ_m,0 a^-3. Dividing both side of Eq.
(<ref>) by H_0^2, we find

    -β(H/H_0)^4+(H/H_0)^2-Ω_m,0
    a^-3=0,

where we have used definition Ω_m,0=8πρ_m,0/(3
H_0^2 ) and β≡α H_0 ^2=3l_0^2 H_0 ^2/4≪ 1. Note
that at the present time (H_0=H) and (a=a_0=1), Eq.
(<ref>), can be written as

    Ω_m,0+β=1,

Solving Eq. (<ref>), we find

    (H/H_0)^2=-1/2β[-1+
    √(1-4βΩ_m,0 a^-3)].

Since β is very small, we can expand the RHS of the above
equation up to the linear term in β. The result is

    H^2/H_0^2=Ω_m,0 a^-3(1+βΩ_m,0
    a^-3)+𝒪(β^2).

If we define, as usual, the redshift parameter a^-1=1+z, we
can rewrite Eq. (<ref>) as

    H^2(z)/H_0^2=Ω_m,0 (1+z)^3{1+βΩ_m,0 (1+z)^3}+𝒪(β^2).

Let us calculate Eq.(<ref>) at the present time (z=0).
Using Eq. (<ref>), we have

    (H/H_0)^2_z=0   =   Ω_m,0+βΩ^2_m,0+𝒪(β^2)
       =   (1-β)+β(1-β)^2≃1+𝒪(β^2),

where we have neglected the terms of 𝒪(β^2)  and
higher order of β. We can also rewrite Eq. (<ref>) as

    da/a √(Ω_m,0 a^-3(1+βΩ_m,0
    a^-3))=H_0dt.

where we have used H=(da/dt)/a. By integrating the aforementioned
relation, we arrive at

    2a^3/2/3√(Ω _m,0)+β/3√(Ω _m,0/a^3)=H_0t,

Finally, we can obtain the explicit form of the scale factor as a
function of time. We find

    a(t)={9/4Ω_m,0(H_0 ^2 t^2-4
    β/9)}^1/3,

where the integration constant is assumed to be zero. In the
limiting case where β=0, one recover a(t)∼ t^2/3,
which is the well-known solution in standard cosmology for the
matter only Friedmann model.

In Fig. <ref>, we explore the behavior of the scale factor
a(t) for different values of β. We see that in zero-point
length cosmology (β≠0), the scale factor lies below the
scale factor of standard cosmology where β=0. In addition,
we observe that for a given value of H_0 t, the scale factor
decreases as β increases. Note that for β≠0, there
is a lower bound on H_0 t in which H_0 t > 2√(β)/3
which leads to t> l_0/√(3). Note that c=1 in our
discussion. Clearly, for H_0 t< 2√(β)/3 one needs to
consider the effects of quantum gravity and the classical general
relativity no longer valid.

Taking the time derivative of Eq. (<ref>), after some
algebra, we obtain the deceleration parameter as

    q(z)=-1-Ḣ/H^2=1/2+ 3/2×βΩ_m,0 (1+z)^3 /1+βΩ_m,0 (1+z)^3,

Eq. (<ref>) indicates that in zero-point length cosmology and
for a matter dominated universe we have always a decelerated
universe where q>1/2.

In Fig. <ref>, we plot the deceleration parameter q versus
redshift parameter z. We see that q(z) is always greater than
1/2 during the history of the universe.

We can also estimate the age of the Universe for the matter only
dominated Universe, when the zero-point length is taken into
account. The age of the Universe is defined as

    t_0=∫_0^∞dz/(1+z)H(z).

Inserting H(z) from (<ref>), we have

    t_0=1/H_0∫_0^∞dz(1+z)^-1/√(Ω_m,0 (1+z)^3[1+βΩ_m,0
    (1+z)^3]).

If we change the variable as (1+z)^-1=x, the above expression
can be written as

    t_0   =   1/H_0∫_0^1x
    dx/√(Ω_m,0x[1+βΩ_m,0
    x^-3])
       =   2/3 H_0 √(Ω_m,0)(√(1+βΩ_m,0)-√(βΩ_m,0)).

Using the fact that β=1-Ω_m,0, we can rewrite the
final form of the age as

    t_0=2/3 H_0(√(1+Ω_m,0^-1-Ω_m,0)-√(1-Ω_m,0)).

For β=0, we have Ω_m,0=1 and one recover t_0=2/(3
H_0), which is well-known result in standard cosmology for the
matter dominated universe.


In Table <ref>, we present the age of the universe for different values of β within the framework of zero-point length cosmology in a matter-dominated universe. Compared to standard cosmology, we observe that the age of the universe increases.

Substituting the scale factor from (<ref>) into ρ_m(t)=
ρ_m,0   a^-3, we obtain
  
    ρ_m(t)/ρ_m,0=4/9(1-β) 1/(H_0 ^2 t^2-4
     β/9),

where we have used β=1-Ω_m,0.


Fig. <ref> illustrates the behavior of the energy density
ρ(t) for different values of β. We observe that in
early epoch of the universe as β increases, the energy
density increases too. Since ρ _m>0, we must have H_0
t>2√(β)/3. Thus H_0 t cannot go to zero due to the
presence of zero-point length.



 §.§ Radiation dominated era

It is generally believed that, following inflation, our universe underwent a radiation-dominated era during which most particles behaved relativistically. In this case the continuity equation implies
ρ_r=ρ_r,0 a^-4 and the Friedmann equation
(<ref>) can be rewritten as

    -β(H/H_0)^4+(H/H_0)^2-Ω_r,0
    a^-4=0.

Solving the above equation, and keeping the terms up to
𝒪(β), we find

    H^2/H_0^2= Ω_r,0 a^-4(1+βΩ_r,0
    a^-4)+𝒪(β^2).

In terms of the redshift parameter, we have

    H^2(z)/H_0^2= Ω_r,0 (1+z)^4{1+βΩ_r,0 (1+z)^4}+𝒪(β^2).

We can also rewrite Eq. (<ref>) as

    da/a √(Ω_r,0 a^-4(1+βΩ_r,0.
    a^-4))=H_0dt.

Integrating the aforementioned relation, we arrive at

    a^2/2√(Ω _r,0)+β/4√(Ω _r,0/a^4)=H_0t,

Solving for the scale factor, we find

    a(t)={4 Ω_r,0(H_0^2 t^2 -β/4
    )}^1/4,

where the integration constant is set to zero. When β=0, one
recovers a(t)∼ t^1/2, which is an expected result in
standard cosmology for the radiation dominated epoch.

In Fig. <ref>, we plot the behavior of the scale factor
a(t), for different values of β. We observe that at each
time, as β increases the scale factor decreases. This
indicates that the radius of the universe is smaller compared to
the standard cosmology. Besides for zero-point length cosmology
the scale factor does not go to zero for H_0 t> √(β)/2.
This may show removing the initial singularity of the Universe. To
become more confidence, we calculate Ricci scalar and Kretchmann
invariant which are given by R= g^μν R_μν, and
K=R_μναρ R^μναρ, respectively.
For FRW metric they are given by

    R   =6( a ä+ȧ^2)/a^2,
    
    K   =12(  ä^2 a^2+ȧ^4)/a^4,

where the "dot" denotes the derivative with respect to time. Substituting a(t) from (<ref>) into the above expressions, we obtain

    R   =-12  β  H_0^2/(4H_0^2  t^2 - β)^2,
    
    K   =48 (8 H_0^4  t^4 + 4 H_0^2  β  t^2 + β^2)
    H_0^4/(4 H_0^2  t^2 - β)^4.

For the domain H_0 t> √(β)/2, the Ricci scalar and the Kretschmann invariant both have finite values and do not diverge. On the other hand, in the limit t→ 0 we obtain the finite values R=-12 H_0^2/β and K=48 H_0^4/β^2 for β≠0. This implies the avoidance of an initial singularity in the cosmological model due to the zero-point length. Such a result is to be expected, since the zero-point length removes the singularity in the potential.

Further one can study the deceleration parameter which is given by

    q(z)=-1-Ḣ/H^2=1+ βΩ_r,0 (1+z)^4
    /1+βΩ_r,0 (1+z)^4.

This quantity is shown to be always positive, indicating that our Universe was undergoing a decelerated phase during the radiation-dominated era.


Let us estimate the age of the Universe for a radiation-only dominated era, taking into account the zero-point length. To do so, we can substitute H(z) from (<ref>) in (<ref>), yielding

    t_0=1/H_0∫_0^∞dz(1+z)^-1/√(Ω_r,0 (1+z)^4[1+βΩ_r,0
    (1+z)^4]).

Further by considering (1+z)^-1=x, the above expression
can be written as

    t_0   =   1/H_0∫_0^1x
    dx/√(Ω_r,0[1+βΩ_r,0
    x^-4])
       =   1/2 H_0 √(Ω_r,0)(√(1+βΩ_r,0)-√(βΩ_r,0)).

Using the fact that β=1-Ω_r,0, we can write the expression for the age of a radiation-only dominated Universe as follows

    t_0=1/2 H_0(√(1+Ω_r,0^-1-Ω_r,0)-√(1-Ω_r,0)).

In particular if we set β=0, we have Ω_r,0=1, then one can recover t_0=1/(2
H_0). This is an expected result in standard cosmology for the
radiation dominated Universe.

In Table <ref>, we present the age of the Universe for different values of β. It can be observed that the age problem in standard cosmology can be mitigated.

By means of  Eq. (<ref>) and the relation ρ_r(t)= ρ_r,0   a^-4, we arrive at

    ρ_r(t)/ρ_r,0=1/4 1/Ω_r,0(H_0^2 t^2 -β/4),

where Ω_r,0=1-β. It follows that

    ρ_r(t)/ρ_r,0=1/4 1/(1-β) (H_0^2 t^2 -β/4)


 Fig. <ref> illustrates the behavior of the energy density ρ_r(t)  for
 different values of β in a radiation-dominated universe.  We observe that, in the early epoch, as β increases, the energy density of the Universe also increases. According to the equation H_0 t>√(β)/2, we have ρ _r>0, which is depicted in the figure.



 §.§ Multiple component universe

Let us now focus on the general case using the modified Friedmann equation, which can be written as

    -β(H/H_0)^4+(H/H_0)^2=Ω_m,0
    a^-3+Ω_r,0 a^-4+Ω_Λ,0.

If we neglect the contribution from radiation, which is small compared to matter and vacuum energy, we have the following relation

    -β(H/H_0)^4+(H/H_0)^2-(Ω_m,0
    a^-3+Ω_Λ,0)=0.

Now, by solving the above equation for small β, we obtain

    H^2/H_0^2=(Ω_m,0
    a^-3+Ω_Λ,0)[1+β (Ω_m,0
    a^-3+Ω_Λ,0)]+𝒪(β^2).

Specifically, in terms of the redshift parameter, we have

    H^2(z)/H_0^2   =   
    [Ω_m,0(1+z)^3+Ω_Λ,0]
          ×[1+β (Ω_m,0
    (1+z)^3+Ω_Λ,0)].

On the other hand, we can also rewrite Eq. (<ref>) as

    da/a √((Ω_m,0 a^-3+Ω_Λ,0)+β
    (Ω_m,0 a^-3+Ω_Λ,0)^2)=H_0dt.



Fig. <ref> illustrates the behavior of the scale factor
a(t) for different values of β in a multiple-component
Universe. We observe that as β increases, the predicted age
of the Universe also increases compared to the standard cosmology.
Moreover, it is indicated that at each point in time, the scale
factor decreases as  β increases. Consequently, the radius
of the Universe is smaller compared to that in standard cosmology.
The plot shows that as time tends to zero, the scale factor does
not tend to zero.

Finally, the deceleration parameter is given by the expression

    q(z) =-1+3 Ω_m,0 (1+z)^3  /2[Ω_m,0
    (1+z)^3+Ω_Λ,0] 
    ×[1+2 β (Ω_m,0
    (1+z)^3+Ω_Λ,0)]/[1+β (Ω_m,0
    (1+z)^3+Ω_Λ,0)],


Fig. <ref>, shows the relationship between q and z. Our
Universe experiences a phase transition from deceleration  (q>0)
to acceleration (q<0) during its history, and this transition
affects the formation and development of cosmic structures. We
observe that at lower redshifts, the transition from deceleration
to acceleration occurs. Moreover, it is evident that the onset of
cosmic acceleration happens later compared to standard cosmology.

To calculate the age of the universe for a multi-component
universe by considering zero-point length, we substitute H(z)
from (<ref>) in (<ref>), we have

    t_0=1/H_0∫_0^∞dz(1+z)^-1/√(X(1+β X)).


    X=Ω_m,0
    (1+z)^3+Ω_Λ,0.

By changing the variable as (1+z)^-1=x, the above expression
can be written as

    t_0=1/H_0∫_0^1x
    dx/√((Ω_m,0x+Ω_Λ,0x^4)[1+β( Ω_m,0
    x^-3+Ω_Λ,0)])
    
    = 1/H_0( 2  tanh^-1( √(Ω_Λ,0)/√(Ω_m,0 + Ω_Λ,0))/3 √(Ω_Λ,0).
     + β√(Ω_m,0 + Ω_Λ,0)/3
    
     - .β/3√(Ω_Λ,0) sinh^-1( √(Ω_Λ,0)/√(Ω_m,0))/√(Ω_m,0)√(1 + Ω_Λ,0/Ω_m,0))

Note that for the general case  Ω _m,0=1-Ω
_Λ ,0-β. For β=0, we have  Ω _m,0=1-Ω
_Λ ,0. Thus, we can recover the age of multiple-component Universe in standard cosmology. Therefore, the above expression can be written as

    t_0 = 1/H_0(2  tanh^-1  ( √(Ω_Λ,0))/3 √(Ω_Λ,0)).



In Table <ref>, we present the age of the universe for
different values of β for a multi-component universe, where
the contribution of radiation is negligible compared to other
components. It is observed that the age problem in standard
cosmology can be mitigated.

In general, the equation of state parameter is defined as

    ω_tot=p_m+p_Λ/ρ_m+ρ_Λ,

where p_m=0. By dividing both the numerator and denominator by ρ_Λ and using the relation p = ρ , ω, we obtain

    ω_tot=ω_Λ Ω_Λ/Ω_m+Ω_Λ.

By using the fact that Ω_m+Ω_Λ=1-β, we obtain

    ω_tot=ω_Λ Ω_Λ/1-β.

Using the definition Ω_Λ= 8π/3H^2ρ_Λ and ρ_Λ=Λ/8 π, we find

    Ω_Λ= Λ/3H^2.

Substitute H(z) from (<ref>), we obtain

    Ω_Λ=Λ/3H_0^2 
    ×[1/(Ω_m,0(1+z)^3+Ω_Λ,0)(1+β (Ω_m,0
    (1+z)^3+Ω_Λ,0))]

 Combining relations (<ref>) and  (<ref>) and using ω_Λ=-1, we find

    ω_tot=-Λ/3H_0^2 (1-β)
    ×[1/(Ω_m,0(1+z)^3+Ω_Λ,0)(1+β (Ω_m,0
    (1+z)^3+Ω_Λ,0))]



In Table <ref>, we present the total equation of state
parameter for different values of β in a multi-component
universe. It is observed that as β increases, ω_tot
decreases as well.

A de Sitter universe is a solution to the Einstein field equations
of general relativity, named after Willem de Sitter. It models the
universe as spatially flat while neglecting ordinary matter,
resulting in dynamics dominated by the cosmological constant.
Therefore, by ignoring the contributions from radiation and matter
in Eq. (<ref>), we obtain

    -β(H/H_0)^4+(H/H_0)^2-Ω_Λ,0
    =0.

Solving the above equation and retaining terms up to the linear term in β, the result is

    H^2/H_0^2=Ω_Λ,0 +β Ω_Λ,0^2+𝒪(β^2).

We can can rewrite the above equation as

    da/a √(Ω_Λ,0 +β Ω_Λ,0^2)=H_0dt.

Finally, we can obtain the explicit form of the scale factor as a function of time in a de Sitter universe. This results in

    a(t)=exp[  √(Ω_Λ,0)√(1+β Ω_Λ,0)  H_0t ]

Using Eq(<ref>), the final form of the scale factor can be rewritten as

    a(t)=exp[  √(Λ/3)√(1+β Ω_Λ,0) t ]

In the limiting case when β=0, we recover the expression
a(t)∼exp( √(Λ/3) t), which is
the well-known solution in standard cosmology for the de Sitter
Universe.

In Fig. <ref>, we observe the behavior of the scale factor
a(t) for different values of β in a de Sitter universe. As
β increases, the predicted age of the Universe also
increases compared to standard cosmology. Additionally, at each
point in time, the scale factor decreases as β increases.
Consequently, the radius of the universe decreases compared to
standard cosmology.

Finally, for the deceleration parameter we have

    q(z)=-1-Ḣ/H^2=-1

This value is always negative, indicating that the de Sitter
universe was undergoing an accelerated phase. In the presence of a
cosmological constant, this leads to accelerated expansion.



§ CLOSING REMARKS 

In order to address fundamental questions in modern cosmology,
exploring modified cosmology can enhance our understanding of the
universe. In this paper, by incorporating zero-point length
corrections to the gravitational potential and applying the
entropic force scenario, we obtained corrections to the entropy
associated with the horizon. In particular, we started from the
first law of thermodynamics, dE=T_hdS_h+WdV, on the apparent
horizon of the FRW universe, and obtained the modified Friedmann
equations. The implications of these corrections and their impact
on the generalized second law of thermodynamics are discussed.

We also derived the same modified Friedmann equations from a
different perspective, specifically through the concept of
emergent gravity. By calculating the number of degrees of freedom
in the bulk and on the boundary of the universe, we reached the
same result.

In the following sections of the paper, we analyzed in details the
cosmological implications of the modified Friedmann equations.
Specifically, we focused on single-component flat universes
dominated by matter, radiation, and de Sitter space, as well as
multi-component flat universes composed of pressureless matter and
a cosmological constant or dark energy. It is argued that in
modified cosmology inspired by zero-point length, the predicted
age of the universe is greater compared to standard cosmology.
Consequently, the age problem can be mitigated. Additionally, it
is observed that the scale factor decreases compared to standard
cosmology, resulting in a smaller radius of the universe. Our
findings also indicate that the transition from a decelerated
universe to an accelerated one occurred at lower redshifts
compared to standard cosmology, with the onset of cosmic
acceleration happening later.

Focusing on modified cosmology with only pressureless matter or a
radiation-dominated era, and in the absence of any dark energy,
results in a decelerating universe. This model cannot account for
an accelerated expansion without dark energy. However, it is shown
that as t → 0 both Ricci Scalar and Kretchmann
invariant have finite values and do not diverge for β≠0.
Thus, the initial singularity of the universe can be mitigated by
the zero-point length.

As a final remark, we calculated the parameter of the equation of
state and the energy density of our universe and examined their
behavior for different values of  β. We observed that the
effects of β are significant in the early universe but have
little impact on current times.

These findings not only demonstrate the influence of modified
entropy into the geometric component of field equations, but also
highlight the significant implications that arise from such
modifications. Such results challenge our current understanding of
the universe's evolution and provide valuable insights into its
dynamics.


Acknowledgements: We thank A. Dehyadeghari for useful
discussion and valuable comments. The work of A. Sheykhi is based
upon research funded by Iran National Science Foundation (INSF)
under project No. 4022705.


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 Paper Title: Spurious Solar-Wind Effects on Acceleration Noise in LISA Pathfinder.





Spurious Solar-Wind Effects on Acceleration Noise in LISA Pathfinder]Spurious Solar-Wind Effects on Acceleration Noise in LISA Pathfinder



^1Institute for Computing in Research, 145 Washington Ave, Santa Fe, NM 87501
^2California Institute of Technology, 1200 East California Boulevard Pasadena, California 91125, United States
^3Los Alamos National Laboratory, Los Alamos National Laboratory P.O. Box 1663 Los Alamos, NM 87545
gdmeadors@lanl.gov




  August 5, 2024

§ ABSTRACT 
Spurious solar-wind effects are a potential noise source in the measurements of the future Laser Interferometer Space Antenna (LISA). Comparative models are used to predict the possible impact of this noise factor and estimate spurious solar-wind effects on acceleration noise in LISA Pathfinder (LPF). Data from NASA’s Advanced Composition Explorer (ACE), situated at the L1 Lagrange point, served as a reliable source of solar-wind data. The data sets were compared over the 114-day time period from March 1, 2016 to June 23, 2016. To evaluate these effects, the data from both satellites were formatted, gap-filled, and adapted for comparison, and a coherence plot comparing the results of the Fast Fourier Transformations. The coherence plot suggested that solar-wind had a minuscule effect on the LPF, and higher frequency coherence (LISA’s main observing band) can be attributed to random chance correlation. This result indicates that measurable correlation due to solar-wind noise over 3-month timescales can be ruled out as a noise source. This is encouraging, although another source of noise from the sun, solar irradiance pressure, is estimated to have a more significant effect and has yet to be analyzed.








Keywords: Space weather, solar wind, gravitational waves, LISA















§ INTRODUCTION

The future Laser Interferometer Space Antenna (LISA) will be the first space-based gravitational wave detector on a heliocentric orbit <cit.>. The LISA project expands upon the work of the ground-based Laser Interferometer Gravitational-wave Observatory (LIGO), which has made significant contributions to our understanding of the universe <cit.>. LISA will measure frequencies ranging from approximately 0.1 millihertz to 100 millihertz <cit.>.  LISA will consist of three spacecraft in an equilateral triangle with arms 2.5 million km long. The antenna will follow 20 degrees behind the Earth. Each spacecraft will contain two centered, free-falling test masses. Deviations from the geodesic motion of the test masses are detected with time-delay interferometry (TDI) <cit.>, using six laser links that connect each of the three spacecraft bidirectionally. TDI constructs a set of time series data, sampled every few seconds, from the combination of these laser links that represents that stretching of space, known as (dimensionless) strain. LISA's TDI output is, in effect, three virtual, correlated space-based interferometer channels sensitive to gravitational waves in the millihertz band.



To confirm the viability of the LISA concept, the European Space Agency (ESA) developed and launched the LISA Pathfinder (LPF). LPF consists of a test version of a LISA module - two free acceleration test masses held in the module's center. The LPF was launched on December 3, 2015, and traveled 1.5 million km from Earth to orbit the L1 Lagrange point; see Figure 1 for positioning in relation to the Sun and Earth. The goal of LPF is to show that a drag-free control measurement technique is viable for a long baseline gravitational wave observatory. LPF was sent into space with 2 test masses, the second one located 38 cm from the first, to avoid common noise sources: solar radiation pressure, solar-wind, magnetic environment, and particle impacts. In addition, spacecraft acceleration was measured to observe external disturbances on a single-test mass signal. One of these sources, solar-wind, has been generally understood to be a very modest noise source <cit.>, below other factors - actuation fluctuations, Brownian noise, micrometeoroids, and more.

Theoretical extrapolation from solar wind and solar irradiance data to LISA <cit.> has set the stage for the analysis in this manuscript, which considers whether correlations are measurable between solar wind and LPF acceleration data. Analyzing LPF's data can be proof that solar-wind noise impact is minimal or demonstrate that in the future, LISA might act as a serendipitous solar-wind sensor at an unusual orbit, similar to STEREO <cit.>. Constant solar wind, sunlight radiation pressure, or other static forces are not expected to have an impact on LISA. However, a time-varying force could accelerate the LISA spacecraft, particularly if the three spacecraft experience the differential forces due to orbital position and orientation. Studies of solar wind variability <cit.> demonstrate that the solar wind can vary from 350 km/s to 700 km/s in the span of a week, doubling the solar-wind force on a timescale of less than 1 Mega-second, motivating an investigation of the micro-Hertz to milli-Hertz frequencies even though they are below the typical LISA science band.

The idea was tested by comparing the LPF spacecraft's Z and X axis data with the corresponding ACE solar-wind data from March 1, 2016, to June 23, 2016 <cit.>. The methods used can find direct, linear couplings at the measured timescales, but nonlinear “upconversion" of solar-wind noise to higher frequencies cannot be ruled out. The Advanced Composition Explorer (ACE), orbiting the L1 point as the LPF did, is used to measure solar-wind data, such as proton speed, proton density, proton to alpha particle ratio, x, y, z components of proton velocity, and Coordinated Universal Time (UTC) <cit.>. This data can be modeled onto LPF and compared with the data from LPF to analyze the effects of solar-wind.



 §.§ Related Works

Space weather has been identified as a key noise factor in possible future LISA measurements. Past work that assessed the impact of solar wind involved using data from the Solar and Heliospheric Observatory’s (SOHO) Virgo Solar irradiance and the ACE Solar-wind to model possible impacts on the future LISA mission <cit.>. Another major source of noise identified for LPF is micrometeoroids. Through various models, 54 impact candidates were identified across a 4348-hour time period. They were shown to be similar to those resulting from Jupiter Family Comets (JFC), Oort cloud comets, Halley-type comets, and asteroids <cit.>. Additionally, the development of LISA is renewing the interest in the environment of space <cit.><cit.><cit.>. This environment includes solar energetic particles, galactic cosmic ray fluxes, and the effects of solar neutrons and interplanetary electrons <cit.>.




§ METHODS

 Figure 2 shows the steps to plot and compare the ACE and LPF data sets.
 We first formatted both data sets to include only data present is within the time frame and for the axes being studied. Next, we filter and gap-fill the ACE data set because the data set is in time-series format. An Inverse Fast Fourier Transform (IFFT) is applied to the LPF data set because the set is in the frequency domain <cit.>. Once the IFFT is performed, the LPF data is filtered and gap-filled. The ACE data set must be modeled onto LPF. Lastly, both data sets are plotted and compared.




 §.§ Process for Creating Plottable ACE Data

Solar-wind data was obtained from the Advanced Composition Explorer (ACE). ACE data included information on solar-wind velocity components (x, y, z), alpha particle to proton ratios, proton densities, and proton speeds at 64-second intervals measured in Coordinated Universal Time (UTC) <cit.>. The area of the solar array was modeled as a 2.9 meter diameter flat circular array, based off of the concept S/C design <cit.>. ACE's UTC measurements were converted to GPS time because the LPF data is in GPS time. Bad data values from the ACE data were also removed (the ACE data header specifies that bad data is indicated by the value -9999.9) <cit.>. The gaps created by removing the bad data were split into three categories based on the process described in Modeling Spurious Forces on the LISA Spacecraft Across a Full Solar Cycle <cit.>. Type A gaps are missing one data point and are filled using linear interpolation. Type B gaps are missing two to twenty-four data points and are filled using linear interpolation with Gaussian Noise. Eq. 1 derives Gaussian Noise (σ) where σ_1 and σ_2 are the standard deviations found by sampling data of equal length to the gap on either side of the ACE data gap. In the equation, for the ith entry in a gap of length l such that 1 ≤ i ≤ l, a point was generated based on σ_1 and σ_2 <cit.>. 



    σ=l-(i-1)/l+1σ_1+i/l+1σ_2


Type C gaps are missing twenty-five or more data points; they are filled using a Hann window with a slow fall-off to zero, as defined by Eq. 2 <cit.>. Twenty-five data points on either side of the gap are used to calculate simulated interpolation data ω(n) where N=51 and 0<n<N-1, creating the data to fill a type C gap.


    ω(n)=0.5[1-cos(2π n/N-1)]




Following previous methods <cit.>, the force exerted on the satellite by solar-wind particles was calculated using ACE data.

Eq. 3 calculates the number of protons (N_p) and alpha particles (N_α) hitting the satellite per unit of time,


    N=n v Acos(ϕ)

with n being particle density and v being wind speed. The surface area of the LPF solar array is represented by A. R is the fraction of particles reflected. ϕ represents the angle between the normal of the array and the orbital plane <cit.>. LPF's eta sensor data of the second test mass showed ϕ was less than ±1 × 10^-7 for all values, making it reasonable to assume ϕ in that case to be 0. The eta sensor data was stated to be within a 2-degree margin of error. Therefore, both a 0-degree and 2-degree value for the angle were tested. Equations 4,5,6 were used to calculate the final forces for the x and z axis. The y-axis equation was disregarded as the amount of force occurring on that axis was negligible. <cit.>.

The following equations are reproduced verbatim from previous LISA analysis <cit.>.

    F_x    = (N_pm_p+N_αm_α)[(1+Rcos(2ϕ))v_x+Rsin(2ϕ)v_z], 
    
    F_y    = (N_pm_p+N_αm_α)[(1-R)v_y], 
    
    F_z    = (N_pm_p+N_αm_α)[(1+Rcos(2ϕ))v_z+Rsin(2ϕ)v_x]


N_p, the proton collision rate, and N_α, the alpha particle collision rate, were found using Eq. 3. The proton and alpha-particle masses from the ACE solar-wind data are represented by m_p and m_α. The x, y, z components of solar-wind velocity are represented by v_x,v_y, and v_z. Assuming the worst-case scenario (that all the particles reflect), R will be defined as 1. Then Equations 4,5,6 can be simplified to the following <cit.>:

    F_x    = (N_pm_p+N_αm_α)[(1+cos(2ϕ))v_x+sin(2ϕ)v_z], 
    
    F_y    = 0, 
    
    F_z    = (N_pm_p+N_αm_α)[(1+cos(2ϕ))v_z+sin(2ϕ)v_x].



The ACE solar-wind force values derived using Equations 7,8,9 are now able to be compared to the processed LPF data by being plotted and transformed as described in Section III.



 §.§ Process for Creating Plottable LPF Data

Data from the LPF was used to predict the effect of solar-wind on the future LISA data. The LPF data obtained from the NASA Goddard Space Flight Center combines information from various sensors to estimate the free-body motion of LPF. The data spans 114 days and is organized into 370 files, each 16384 seconds long. The data was taken at 2.5-second intervals from the second sensor on the x-axis and the second sensor on the z-axis. The data for LPF was in the frequency domain; thus, an IFFT was applied. The IFFT added the complex conjugates of negative values and zeros to the positive, non-zero values, returning the data to the original time series format. To calculate force, the acceleration from the IFFT was multiplied by the mass 422 kg <cit.>. The 31 gaps in both the x and z axes were filled using the Hann Window (Type C) method (Table. 1) <cit.>.




 §.§ Plotting Methods

A simple time domain comparison and a Fourier Transform comparison were used in Figures 3 and 4, respectively, to see if there were any apparent overlaps when the data were graphed over each other. The time domain data was zero-meaned to highlight the differences in the force data collected between LPF and ACE. The  function is used to determine coherence between LPF and ACE data and the averages of different lengths of correlation values <cit.>. As described in the legends of Figures 5, 6, 7, and 8, the coherence comparison used increasing segment lengths multiplied by factors of 4 from 1 to 4^5.




§ RESULTS


Figure 3 depicts a time series that presents a direct comparison between the LPF time domain and the ACE force time domain, with both datasets having been zero-meaned. The ACE force data was calculated using the force equations and a 2-degree estimate for angle ϕ. The time series do not show any direct correlation between ACE and LISA data. Long constant segments in LPF indicate data gaps. 



Figure 4 illustrates some overlap between the ACE and LPF data but nothing significant enough to be seen as a solid correlation between the two data sets. FFT was done with a 1/N normalization coefficient and a sampling time interval of dt = 2.5 seconds and dt = 64 seconds for LPF and ACE respectively.


From the coherence plot (Figure 5), it was observed that the coherence between the two data sets was minimal, and any correlation is probably due to random chance. In particular, spectral lines in LPF, due to instrumental artifacts, the line above 10 millihertz in ACE, do not show coherence on closer inspection.


Figure 5 shows six different lengths of coherence value averages. Each decrease in the values averaged increases the accuracy of the line. As the accuracy increases, the coherence decreases. The low coherence indicates that solar-wind will have little to no direct effects on the future LISA. Although with an estimated normal angle of 0, there should be no effect on the x-axis, the 2-degree margin of error of the angle was tested for coherence in the x-axis in Figure 6.




§ TEST VERIFICATION

To ensure the accuracy of the previous tests and to create a visual of what would happen if there was a finding, the correlation methods were tested with simulated data. A measured known excitation of the sinusoidal form was added to both the LPF and ACE data sets after they had been filtered, gap-filled, and post-processed.


    F(t) = A sin(2π f t+ϕ)


In Eq. 10, F(t) represents the Force excitation added to the data where A is amplitude, f is frequency, t is time, and ϕ is phase. The phase was set to 0 because it does not affect the coherence measurement. Amplitude and frequency were varied as follows. Adding this excitation in both sets of data simulates a specific frequency of the wave that should then be picked up by the coherence function, thus showing an excitation in that frequency on the coherence plot.

In each test, a single sinusoidal frequency was excited. Multiple frequencies were excited in independent tests. The test frequencies 1 × 10^-2 Hz, 1 × 10^-3 Hz, 2 × 10^-3 Hz, 5 × 10^-3 Hz, 1 × 10^-4 Hz, 5 × 10^-4 Hz were measured. All frequencies displayed distinct excitation at the injected frequency in the coherence spectrum. They were also tested for the least excitation needed for at least 50% correlation when 1024 points were used per segment.
To illustrate this, Figure 7 shows the frequency 2 × 10^-3 Hz at 99.5% coherence for 1024 segments, and Figure 8 shows the frequency 5 × 10^-3 Hz at 68.36% coherence for 1024 segments per average with only 0.2 nanoNewtons of force. Figure 9 shows the frequency 5 × 10^-3 Hz again, but with the same 5 nanoNewtons of force as Figure 7 and displays a similar 99.9% correlation. Thus showing that the coherence function works accurately and can properly compare the 2 datasets.







The excitation frequencies of 2 and 5 millihertz in this paper represent the low-to-mid range of LISA, relevant to both the galactic white-dwarf binary population and black-hole mergers around a million solar masses. At these frequencies, prior work <cit.> has estimated the force coupling of solar wind pressure into LISA: Figure 3 of that paper shows forces in the year 2000, near solar maximum, resulting in a predicted force at 2 millihertz of roughly 20 picoNewtons, and less at 5 millHertz. While a comprehensive statistical sampling from that work’s force spectrum is reserved for future research, this paper shows that near 200 picoNewtons, magnitude-squared coherence levels do reach above 0.68, or about two-thirds, at 5 millihertz. Comparing the two frequencies, the coherence would be expected to be lower at 2 millihertz, consistent with the noise spectrum.




§ CONCLUSION

The effects of spurious solar-wind on acceleration noise of LISA Pathfinder (LPF) data from the x and z axes were thoroughly analyzed. As shown in Figures 5 and 6, the coherence between the Advanced Composition Explorer (ACE) and LPF data was sporadic for the given frequency range, and the average correlation for each data section was minuscule. The coherence test has verified that there is a measurable correlation for a simulated excitation of known amplitude, as seen in Figures 7, 8 and 9. The analysis shows that the correlation seen in the real, unexcited data is indistinguishable from chance given the data available and no relationship was found between ACE solar-wind data and LPF data. The models and plots of the x and z axes found a negligible relation between the ACE solar-wind data set and the LPF.

This result suggests that there is a reasonable safety margin between solar wind forces and measurable acceleration noise in LISA. More detailed analysis is warranted, as solar variability could reach levels even higher than in 2000, but comparison with solar irradiance <cit.> suggests that those forces may pose a larger concern. To the benefit of LISA, albeit a loss to the space weather community, it currently appears difficult, with such low coherence at expected force levels, to solve the inverse problem of finding solar wind force from LISA acceleration, as proposed previously <cit.>.

These findings are encouraging for the future of the LISA as less noise will allow LISA to see much more of the universe. However, solar irradiance data, another noise factor with likely greater effects on the LPF, has yet to be tested <cit.>, and the limited 3-month time frame of data that were analyzed means that correlations over longer timescales are possible as well. Future analyses should also focus on working with individual subsets of the data and plotting the relationships between those to eliminate the possible confounding effects created when using gap-fill methods. 




§ ACKNOWLEDGMENTS

Thank you to the Institute for Computing in Research: Mark Galassi and Rhonda Crespo. Thank you I. Thorpe and J. Slutsky at the NASA Goddard Space Flight Center for the data and helpful comments. Additionally, thank you to B.M. Frank, B. Piotrzkowsky, B. Bolen, M. Cavaglià, and S.L. Larson for their methods and helpful comments. Indie Desiderio-Sloane and Arnold Yang were supported by the Institute for Computing in Research. GDM was supported by multiple Los Alamos National Laboratory internal funding sources, including the Intelligence and Space Research Early Career (ISR-EC) 2022 Pitch Day award XB4F00 and the Center for Space and Earth Science project development award W88600. This document has Los Alamos Unlimited Release (LA-UR) number LA-UR-23-30801. This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).




§ SUPPLEMENTARY DATA

The source code and procedures to run it are saved at codeberg.org/Poarthan/solarWind- DataAnalysisCode. The ACE and LPF data used can be found respectively at zenodo.org/records/6955182 and zenodo.org/records/6954044.



§ REFERENCES


iopart-num







 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: Electric Penrose process and the accretion disk around a 4D charged Einstein-Gauss-Bonnet black hole.









malloqulov@gmail.com
Institute of Fundamental and Applied Research, National Research University TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan 
University of Tashkent for Applied Sciences, Str. Gavhar 1, Tashkent 100149, Uzbekistan


sanjar@astrin.uz
Institute for Theoretical Physics and Cosmology,
Zhejiang University of Technology, Hangzhou 310023, China
Institute of Fundamental and Applied Research, National Research University TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan
University of Tashkent for Applied Sciences, Str. Gavhar 1, Tashkent 100149, Uzbekistan








§ ABSTRACT 

In this paper, we aim to examine the electric Penrose process (PP) around a charged black hole in 4D Einstein–Gauss–Bonnet (EGB) gravity and bring out the effect of the Gauss–Bonnet (GB) coupling parameter α and black hole charge on the efficiency of energy extraction from the black hole. This research is motivated by the fact that electrostatic interactions significantly influence the behavior of charged particles in the vicinity of a charged static black hole. Under this interaction, decaying charged particles can have negative energies, causing energy to be released from black holes with no ergosphere. We show that the GB coupling parameter has a significant impact on the energy efficiency of the electric PP, but the efficiency can be strongly enhanced by the black hole charge due to the Coulomb force. Finally, we consider the accretion disk around the black hole and investigate in detail its radiation properties, such as the electromagnetic radiation flux, the temperature, and the differential luminosity. We show that the GB coupling parameter can have a significant impact on the radiation parameters, causing them to increase in the accretion disk in the vicinity of the black hole. Interestingly, it is found that the 4D EGB charged black hole is more efficient and favorable for the accretion disk radiation compared to a charged black hole in Einstein gravity.   




Electric Penrose process and the accretion disk around a 4D charged Einstein-Gauss-Bonnet black hole
    Sanjar Shaymatov
    August 5, 2024
====================================================================================================





§ INTRODUCTION


In general relativity (GR), black holes have been known as a generic result of Einstein's gravity, as their geometric properties are described by simple mathematical equations.  Among other properties the occurrence of singularity makes black holes very exciting and fascinating objects, even though this marks the limits of Einstein's theory.  However, exploring black hole's unknown properties and the limits of the theory's applicability has been extremely important. In this context general relativity is known as an incomplete theory and thus for its validity and applicability higher-order theories have been regarded as possible extensions of general relativity <cit.>. We know that Einstein's gravity is constructed from linear order Riemann curvature, while Gauss-Bonnet (GB) gravity continues to the quadratic order with higher order invariants, thus referring to the Lovelock theory as a generalization of Einstein's theory <cit.>. Note that GB gravity with the quadratic order gives a contribution to the gravitational dynamics only in D>4, and hence this is known as the Lovelock theory. However, it was recently suggested that a 4-dimensional Einstein-Gauss-Bonnet (4D EGB) theory could exist and that Lovelock's theorem could be bypassed by a suitable redefinition of the GB coupling constant <cit.>. The new 4D EGB is presently under scrutiny on the basis of two main arguments. One involving the ill-posedness of the action for the theory <cit.> and the other regarding the validity of the rescaling of the GB constant, which may be possible only for systems with certain symmetries <cit.>. Both objections, if valid, may invalidate the 4D EGB theory as an alternative to Einstein's theory. However, the fact remains that solutions with high symmetries exist, they have a clear physical interpretation that mirrors the corresponding solutions in GR, and may be regarded as coming from an effective prescription for the lower dimensional limit of GB gravity. For this reason, the investigation of the properties of such solutions has attracted great interest (see, e.g.  <cit.>). There have also been several investigations on these lines  <cit.> addressing the dynamics of spinning particle motion, the impact of the GB term  on the superradiance process <cit.>, the plasma effect on weak gravitational lensing <cit.>, the question of destroying the black hole horizon in strong and weak forms <cit.>, and the Bondi-Hoyle accretion process around 4D EGB black hole <cit.>. It is worth noting that the 4D EGB theory was also extended to obtain black hole solutions with electric charge and rotation <cit.>. Later, it was also extended to the 3D BTZ black hole solution in EGB theory <cit.>. Additionally, interesting aspects that pertain to the GB black hole in higher dimensions (i.e., D>4 ) have also been investigated in Refs. <cit.>. 

The Penrose process (PP) <cit.> is a mechanism proposed to extract the rotational energy of rapidly rotating black holes, usually referred to as a potential explanation for highly energetic astrophysical phenomena. It utilizes the ergosphere, which appears in the region between the horizon and the static limit bounded from the outer surface. In this process, a falling particle is divided into two parts, with one part falling into the black hole and the other escaping to infinity with more energy than the incident particle. This allows the energy of the escaping particle to be extracted from the black hole, leading to a slowdown in the black hole's rotation. The PP has since been applied in various contexts <cit.>. It is to be emphasized that the PP has also been extended to higher dimensional rotating black holes <cit.> and Buchdahl stars <cit.>. Bardeen et al. <cit.> and Wald <cit.> demonstrated that extracting more energy from the black hole would be difficult unless the incident particle is relativistic. Subsequently, the PP was reformulated as a new mechanism, known as the magnetic Penrose process <cit.>. In this mechanism, the influence of the magnetic field on an escaping particle enables it to surpass the constraint velocity and become relativistic, significantly improving the efficiency of energy extraction from a rotating Kerr black hole. This mechanism has since been extended to various scenarios  <cit.>) addressing the impacts of a purely magnetic field on the energy extraction process. Hence, it has been accepted that the energy extraction process can only occur inside the ergoregion of rotating black holes acting as an engine for the higher energetic astrophysical processes mentioned above. Despite this fact, the PP can be applied even in non-rotating black holes, referred to as the electric Penrose process, which can also serve as a high-energy emission event <cit.>. Therefore, it is also important to examine the electric PP to gain a deeper explanation of high-energy astrophysical phenomena. 


In this paper, we study a 4D charged EGB black hole and explore the efficiency of energy extraction using the electric PP. This is our main focus for investigation, with further analysis of the accretion disk properties around the black hole. Additionally, we examine test particle dynamics and the innermost stable circular orbit (ISCO) around the black hole, along with the accretion disk's radiative energy efficiency in 4D EGB theory. 

It is commonly accepted that the radiation of the accretion disk and recent observations pertaining to particle outflows  <cit.> have been tested to examine the astrophysical aspects of black holes. Therefore, to gain a deeper understanding of gravity and to test it in the strong field regime it is worth analyzing the rich observational phenomenology of electromagnetic radiation with expected thermal spectra through a thin accretion disk <cit.>. In this sense, it becomes important to explore the electromagnetic radiation by a thin accretion disk in the close vicinity of black holes. In the current paper, we investigate the radiative properties of the aforementioned thin accretion disk using the geometrically thin and optically thick Novikov-Thorn disk model in the strong field regime of 4D charged EGB black hole. We also provide a precise comparison with the standard Reissner-Nordström black hole in Einstein gravity. 

The paper is organized as follows. We briefly review a charged black hole in 4D EGB theory in Sec. <ref>. We consider the electric Penrose process with a test particle dynamics in Sec. <ref>. We study the geometrically thin Novikov-Thorne model for the accretion disk around the black hole in 4D EGB theory in and further discuss the flux of the radiant energy over the accretion disk, accretions disk's radiative efficiency, temperature profile and differential luminosity in Sec. <ref>. We end up with conclusion in Sec. <ref>. Throughout this work, we use the signature (–, +, +, +) for the spacetime metric and system of units G=c=1.




§ SPACETIME GEOMETRY IN THE 4D EGB GRAVITY 


The action for EGB gravity can be defined by the following  action <cit.>


    S=1/16π∫ d^Dx√(-g)(R+α/D-4 G^2-F_αβF^αβ)  ,

where α refers to Gauss-Bonnet (GB) coupling parameter, while G to the GB invariant defined as

    G^2=R^2-4R^αβR_αβ+R^αβμνR_αβμν  ,

where R is the Ricci scalar and R_αβ and R_αβμν the Ricci and Riemann tensors. Based on the action underlined above, the line element describing the static black hole spacetime in 4D EGB theory is given by  <cit.>

    ds^2=-f(r) dt^2+dr^2/f(r)+r^2(dθ^2+sin^2θ dϕ^2) ,

with metric function 

    f(r)   =1+r^2/2α(1-√(1+4α/r^2(2M/r-Q^2/r^2)))
       =2(r^2-2Mr+Q^2+α)/r^2+2α+√(r^4+4α(2Mr-Q^2))  ,

where M is the total mass and Q the black hole electric charge. The corresponding vector potential is written as follows:

    A_α=(-Q_e/r,0,0,Q_mcosθ)  .


The parameter range of the coupling constant α can be found by imposing the minimum condition of f(r), i.e., it is given as 0≤α/M^2≤√(1-α). It is obvious that one can recover the Reissner-Nordstrom solution in the limit of α→ 0, so it is given by 

    lim_α→ 0f=1-2M/r+Q^2/r^2  .

In the case of α≪1, it does however take the following form 

    f≃ 1-2M/r+Q^2/r^2+4α M/r^3+ O(α^2) .

Interestingly, albeit the complicated form of f(r), the horizon radius takes a simple form as r_±=M+√(M^2-Q^2-α), similarly to the horizon cases of RN or Kerr black hole spacetime in Einstein gravity. It must be noted that, in the equatorial plane (i.e., θ=π/2), the Faraday tensor, F_μν=A_ν,μ-A_μ,ν, can be defined by only one independent nonzero component as 

    F_tr=-F_rt=-Qr^2  .
 







§ ELECTRIC PENROSE PROCESS AND THE EXTRACTED ENERGY 


Here we consider the electric PP to examine energy extraction from a charged black hole in 4D EGB gravity. It must be emphasized that energy extraction from rapidly rotating black holes remains one of the most important issues in astrophysics. PP was proposed as a key explanation for higher energetic astrophysical phenomena, such as the active galactic nuclei with luminosity of approximately ∼ 10^45 erg· s^-1. For the PP to be valid, there must exist an ergosphere, in which an incident particle splits into two fragments, with one falling into the black hole and the other escaping to infinity with more energy. However, this is not true for a non-rotating black hole. One can consider electrostatic interaction in the so-called generalized ergosphere around a static charged black hole, where it is assumed that decaying charged particles can have negative energies under the electrostatic interaction. This is how the electric PP can become efficient, releasing energy from charged black holes with no ergosphere.




We assume that the splitting point of an incident particle takes place in the equatorial plane, with the four-velocity u^α=u^t(1,v,0,Ω), where v=dr/dt is the radial velocity of the particle and Ω=dϕ/dt the angular velocity. We can write the following equation for the angular velocity of the splitting particles using the normalization condition u^αu_α=-k with k=0 for massless and k=1 for massive particle:

    (u^t)^2[v^2/f(r)-f(r)+Ω^2 r^2]=-k .

Based on the above equation, for a distant observer at infinity, the angular velocity of the splitting particles, Ω=dϕ/dt, is defined by 

    Ω=±1/u_tr√((u_t)^2(f(r)-f^-1(r)v^2)-kf^2(r))  .

The allowed range of Ω is then given by  

    Ω_-≤Ω≤Ω_+        Ω_±=±√(f(r))/r  ,

with the Keplerian orbits.
Further we consider the conservation laws for the incident particle, m_1, which splits into two fragments, m_2 and m_3, especially very close the black hole horizon. The conservation laws after splitting of the incident particle are then written as follows: 

    E_1 = E_2 + E_3  ,    
    L_1 = L_2 + L_3 
    q_1 = q_2 + q_3  ,

and 

    m_1ṙ_̇1̇ = m_2ṙ_̇2̇ + m_3ṙ_̇3̇,    
    m_1 ≥ m_2 + m_3  ,

where dot denotes a derivative with respect to the proper time, τ. Following to the conservation laws, we write the momentum as <cit.>

    m_1u_1^ϕ = m_2u_2^ϕ + m_3u_3^ϕ  ,

with the four-velocity u^ϕ=Ωu^t=Ωe/f(r), where i = 1,2,3 depicts the number of particles in the process. Eq. (<ref>) then yields 

    Ω_1m_1e_1=Ω_2m_2e_2+Ω_3m_3e_3  .

Eq. (<ref>) solves to give the analytic form of E_3 as

    E_3=Ω_1-Ω_2/Ω_3-Ω_2(E_1 + q_1A_t)-q_3A_t,

with the ith particle's angular velocity Ω_i=dϕ_i/dt.

For the further analysis we shall for simplicity consider neutral incident particle (i.e., q_1 = 0) in order to maximize the ionized particle's energy. We also assume E_1/m_1=1. With this in view, for the incident particle m_1, the angular velocity given in Eq. (<ref>) can be rewritten as follows:

    Ω_1^2=f(r)[1-f(r)]/r^2, 
          Ω_2=Ω_-  ,
          Ω_3=Ω_+  .

with q_2=-q_3.
The particle m_3 can be considered an ionized particle.  This particle can reach its maximum energy, depending on the highest value of this ratio (Ω_1 - Ω_2) / (Ω_3 - Ω_2) with the corresponding values of angular momenta Ω_i. So one can then find   

    Ω_1-Ω_2/Ω_3-Ω_2=1/2[1+√(1-f(r_ion))]  ,

where r_ion refers to the ionization radius.
In the limit of M=1 and Q=α=0, it reduces the Schwarzschild black hole case with the following form <cit.>

    Ω_1-Ω_2/Ω_3-Ω_2=1/2+1/√(2 r_ion)  .

Taking all into account, the ionized particle's energy take the following form 

    E_3=1/2[1+√(1-f(r_ion))] (E_1 + q_1 A_t) - q_3 A_t  .

As underlined above, we assume q_1=0 and q_2=-q_3. Within this assumption, the ionized particle's energy takes the following form at the splitting point 

    E_3 = 1/2[1+√(1-f(r_ion))] E_1 - q_3 A_t  ,

which, on the other hand, is rewritten as follows: 

    E_3/E_1 = 1/2[1+√(1-f(r_ion))]  -  q_3 A_t/E_1  .

The above equation represents the energy efficiency of the electric PP. We do further analyze the efficiency of energy extraction from a charged black hole in 4D EGB gravity. It is clear from the above equation that the time component of the electromagnetic four potential is properly proportional to the black hole charge Q. Therefore, the energy of the ionized particle becomes maximum when considering the sign for q_3 and Q as expected, resulting in the charged particle getting accelerated due to the Coulomb repulsion force acting between the black hole charge and particle charge.

The key point to note here is that the ionized particle gets accelerated only if the right-hand side of Eq. (<ref>) is greater than unity. Since it is complicated to solve analytically we explore it numerically and provide illustrative plots. We then analyze the energy efficiency as the function of black hole charge for various possible cases. In Fig. <ref>, we demonstrate the energy efficiency released from the charged black hole in 4D EGB gravity due to the incident particle splitting into fragments in the generalized ergosphere. To be more precise, the left panel depicts the impact of black hole charge on the energy efficiency at different splitting points for fixed GB coupling parameter, while the right panel shows the same behavior at the fixed position for various combinations of GB coupling parameter. As can be seen from the left panel of Fig. <ref>, the shape
of the energy efficiency (ratio) shifts up to higher efficiency when the splitting point occurs near the black hole horizon. Additionally, the energy efficiency increases with increasing black hole charge due to the increase in the Coulomb force. However, it decreases as the GB coupling parameter increases, compared to the RN black hole in Einstein gravity.      
  













§ THE ACCRETION DISK AROUND A 4D CHARGED EINSTEIN-GAUSS-BONNET BLACK HOLE AND ITS RADIATIVE EFFICIENCY


In this section, we consider the accretion disk around a charged black hole in 4D EGB gravity using the Novikov-Thorne model. According to this model, we assume that the accretion disk is optically thick and geometrically thin around the black hole. The point to note is that the accretion disk can be extended horizontally as its vertical size is considered thin enough. With this in view, the accretion disk can be characterized by two size parameters, such as the height h and the radius r of the disk. For these two parameters, h/r ≪ 1 is then satisfied well as the disk's radius is much more larger in the horizontal direction. Hence, this property of the this accretion disk can cause the vertical entropy and the pressure gradients to be negligible for a gas and dust/particle in the disk. However, the accretion disk becomes hot enough and generates heat as a result of the dynamical friction, which can be released as the thermal radiation on the surface of the accretion disk starting from its inner edge located at the innermost stable circular orbit (ISCO) around the black hole. 

Keeping the above in mind we turn to the analysis of the accretions disc. To that end, we begin to examine the accretion disk's bolometric luminosity, which is defined by  <cit.>

    ℒ_bol=ηṀc^2  ,

where Ṁ is the growth rate of matter that gets absorbed by the black hole and η the energy efficiency of the accretion disk. From an astrophysical point of view, observing bolometric luminosity is still a complicated process due to black hole parameters and its geometry. Therefore, it is valuable to measure the bolometric luminosity using theoretical analysis and models since energy efficiency can be applied for explaining important aspects of the accretion process. It is envisaged that the remaining of mass-accreting matter is converted into the electromagnetic radiation that is emitted from the black hole. This is what we further define as the disk's energy efficiency extracted from black hole due to the infalling matter. This energy efficiency can be represented by the radiation rate of the photon energy emitted from the disk surface; see details in <cit.>. We then proceed to determine the energy efficiency through the energy measured at the innermost stable circular orbits (ISCO), usually referred to as the emitted photons from the disk that travel out to infinity as radiation. This is defined by  

    η=1-ℰ_ISCO  .

From the above relation, we can determine the energy efficiency extracted from the accretion disc around the black hole. To do this, we need to determine the energy measured at the ISCO first.  







Therefore, we now focus on the particle motion around BH in 4D EGB gravity. The effective potential for radial motion of a test particle with mass m in the equatorial plane of a static black hole is written in general form <cit.> 


    V^±_ eff(r)=q A_t ±√(f(r)(1+ L^2/r^2sin^2θ))  ,

which defines the radial motion of the charged test particles around the black hole in 4D EGB gravity. It is clear that the effective potential has two parts, i.e., Column and gravitational interaction parts. Also, the effective potential consists of two different solutions and it keeps its symmetry under the transformation, such as qQ → -qQ: V^+_ eff→ -V^-_ eff and V^-_ eff→ -V^+_ eff. It is to be emphasized that we further focus on the positive one, V^+_ eff that reflects physically meaningful timelike motion. Based on the effective potential, one can easily proceed to determine the ISCO parameters, such as r_ISCO, ℰ_ISCO and ℒ_ISCO. To find these quantities, we impose the following conditions  

    V_ eff(r)=0,  V'_ eff(r)=0 V”_ eff(r)=0  .

All results associated with the ISCO radii for various combinations of the GB coupling parameter and black hole charge are tabulated in Table <ref> and <ref>. As seen in Table <ref> and <ref>, the ISCO radius decreases as α and Q increases. Here, the GB coupling parameter can be interpreted as a gravitational repulsive charge, thus resulting in the gravity weakening in the background geometry <cit.>. Therefore, it influences the radiative efficiency of the accretion disk around the black hole and it increases with increasing the GB coupling parameter. It is clear that the radiative efficiency is significantly enhanced due to the combined effects of the GB coupling α parameter and the black hole charge, compared to the the Schwarzschild and the RN black hole case in Einstein gravity <cit.>. Interestingly, it can be observed that the radiative energy efficiency of the accretion disk is higher for neutral particles compared to charged particles. This is due to the Coulomb interaction, which allows charged particles to retain their energy in the accretion disk.       



§ THE ACCRETION DISK'S RADIATIVE PROPERTIES AROUND A 4D CHARGED EINSTEIN-GAUSS-BONNET BLACK HOLE


Here, we investigate the accretion disk's radiative properties, such as the flux created in the  accretion disk around the charged black hole in 4D EGB gravity. It is worth noting that in most cases, accretion disks can be found around massive black holes. Any matter that enters the accretion disk can follow a trajectory in a spiral form within the disk, which is formed by gas and dust/particle orbiting around a black hole. Therefore, the gas and particle within the disk can rub and collide with each other as they move in a turbulent flow, causing frictional heating that is released as radiation energy. As a result of this process, the angular momentum of the gas and dust/particle decreases, causing them to drift inward towards the black hole. As they orbit closer to the black hole, their velocity increases, leading to an increase in frictional energy. Within this process, more energy is radiated away, resulting in the accretion disk becoming hot enough to emit X-rays in the close vicinity of the black hole. The main objective of this study is to investigate how the GB coupling parameter and the black hole charge can affect the accretion disk radiation. This may cause the gas and dust within the accretion disk to become highly ionized around the charged black hole, leading to the emission of exceptionally high-energy radiation such as X-rays that can be observed. Studying the accretion disk radiation around the charged black hole in 4D EGB gravity and analyzing the effects of its parameters on the disk can provide valuable insights into the properties of the disk and help us gain a deeper understanding of remarkable aspects of the black hole geometry. Following to <cit.>, we determine the flux of electromagnetic radiation via the following  equation

    ℱ(r)=-Ṁ_̇0̇4 π√(g)Ω_,r(E-Ω L)^2∫ _r_ISCO^r (E-Ω L) L_,r d r ,

where we denote g as the determinant of the three-dimensional subspace with (t, r , ϕ), i.e., √(g)=√(-g_ttg_rrg_ϕϕ). From the flux equation given above, ℱ(r) is sensitive to the disk mass accretion rate, Ṁ_̇0̇ which remains unknown. For further analysis, we shall for simplicity choose Ṁ_̇0̇. Here, the angular velocity (i.e., the Keplerian frequency) of the charged test particle is defined by <cit.>

    Ω^2=-g_tt,r/g_ϕϕ,r-2q^2/m^2g_ϕϕA^2_t,r/g_ϕϕ,r+1/g^2_ϕϕ,r[4q^4/m^4g^2_ϕϕA^4_t,r+
           + 4q^2/m^2A^2_t,rg_ϕϕ,r(g_ϕϕg_tt,r-g_ttg_ϕϕ,r)]^1/2  .

The angular velocity reduces to Ω^2_0=-g_tt,r/g_ϕϕ,r in the limit of q→ 0, referred to as the Keplerian frequency for a neutral particle. By imposing the conditions Eq. (<ref>) for the energy and angular momentum, the flux of the electromagnetic radiation, ℱ(r), can be determined explicitly. We now turn to analyze the flux numerically as deriving its analytical form is complicated. We show the radial dependence of the flux of electromagnetic radiation in Fig. <ref>. The left and the right panels depict the impact of the GB coupling parameter and the black hole charge on the electromagnetic radiation flux, respectively. As can be seen from Fig. <ref>, the shape of the electromagnetic radiation flux of the accretion disk is shifted upward towards larger values due to the impact of the GB coupling parameter α, compared to the RN black hole case in Einstein gravity. The combined effects of the GB coupling parameter and the black hole charge can also enhance the flux significantly, as seen in the right panel of Fig. <ref>. The electromagnetic radiation flux can reach its maximum value in the accretion disk as expected under the influences of α and Q. We are able to define the black body radiation flux as ℱ(r)=σ T^4, together with the Stefan-Boltzmann constant, σ. In Fig. <ref>, we show the radial profile of the accretion disk temperature as the disk energy. As seen in Fig. Fig. <ref>, the maximum of the temperature profile corresponds to high temperature in the accretion disk around the black hole. The disk temperature increases as α increases, but it decreases accordingly at larger distances from the disk. To be more representative we show the density plot in Fig. <ref>, addressing the parameter dependence on the accretion disk temperature. In the top row of Fig. <ref>, the density plots depict how the disk temperature changes with the change in the GB coupling parameter and black hole charge. As observed from the density plot, the accretion disk temperature distribution starts to become hot enough from the inner edge of the disk and extends to its outer edge, as indicated by the red regions corresponding to the maximum temperature, as shown in the bottom panel of Fig. <ref>. 

Additionally, we now study the differential luminosity, which is one of key quantities and helps to reveals the accretion disk's nature <cit.>. The differential luminosity is written as follows:

    d ℒ_∞d lnr=4 π r √(g) E ℱ(r)  .

It can be assumed that the radiation emission is regarded as the radiation of black body, thus allowing the spectral luminosity ℒ_ν,∞ to be defined by the radiation frequency variable ν at infinity <cit.>

    νℒ_ν,∞=60π^3∫_r_ISCO^∞√(g) EM_T^2(u^t y)^4exp[u^t y(M_T^2 ℱ)^1/4]-1  ,

where we have defined y=h ν /k T_⋆ with the characteristic temperature T_⋆. Here, k and h respectively refer to the Planck and Boltzmann constants, while M_T to the total mass. The characteristic temperature can be determined via the Stefan-Boltzmann law, and it is then given by 

    σ T_⋆= Ṁ_04 π M_T^2  ,

with the Stefan-Boltzmann constant σ. We examine the differential luminosity and show its radial dependence in Fig. <ref>. Similarly to what is observed in the electromagnetic flux, the differential luminosity can have a similar behavior as a consequence of the GB coupling parameter's impact. It increases in the accretion disk with the increasing of α. Taking all into account, we can deduce that the accretion disk parameters are more sensitive to the coupling parameter α in EGB gravity case, compared to the RN black hole in Einstein's case.      




§ CONCLUSIONS 


In this paper, we considered a charged black hole solution in 4D EGB gravity and investigated the efficiency of electric Penrose process and the accretion disk radiation properties around this black hole spacetime. We showed that the efficiency of the energy extraction from the black hole via the electric PP increases with the increasing black hole charge, but it is enhanced by the effect of positive values of the GB coupling parameter α for small values of black hole charge. However, the energy efficiency is strongly enhanced with increasing black hole charge that causes the Coulomb force interaction to increase. 

Accretion disks around black holes are considered the primary source of information regarding gravity and the surrounding geometry in the strong field regime. We also examined the radiative energy efficiency of the accretion disk. To be more quantitative, we estimated the radiative efficiency of the accretion disk using the ISCO energy of the charged particles for various possible cases and showed that the radiative efficiency increases as the ISCO radius becomes closer to the black hole as a consequence of an increase in α and Q; as seen in Table <ref> and <ref>. Interestingly, it was found that the radiative efficiency is higher for neutral particles compared to charged particles. This occurs because the Coulomb interaction can allow charged particles to retain their energy in the accretion disk. 

Further, we studied the accretion disk's radiative properties and examined the effects of 4D EGB gravity to provide valuable insights into the properties of the disk. We found that the electromagnetic radiation flux of the accretion disk is significantly enhanced due to the impact of the positive GB coupling parameter α, compared to the RN black hole case in Einstein gravity. It was also shown that the flux increase under the combined effects of the GB coupling parameter α and the black hole charge Q. Additionally, we examined the accretion disk temperature as a function of α and Q and showed that it starts to increase and reach its maximum due to the increase in α, but it decreases at larger distances from the disk. Based on the density plot, we also showed that the temperature distribution of the accretion disk starts to become hot in the accretion disk as a result of the combined effects of the coupling parameter and black hole charge. Finally, we studied the differential luminosity and demonstrated that it begins to increase with the rise in the value of the GB coupling parameter.

We examined the distinct future and effect of the GB coupling parameter and black hole charge on the efficiency of energy extraction and properties of the accretion disk. Based on the results, we found that the 4D EGB black hole is more efficient for the accretion disk as compared to the RN black hole in Einstein gravity case. These theoretical results may help assess the validity of 4D EGB black holes in explaining the distinct departures from Einstein gravity and astrophysical observations regarding the accretion disk radiations. 




§ ACKNOWLEDGEMENT


This work is supported by the National Natural Science Foundation of China under Grant No. 11675143 and the National Key Research and Development Program of China under Grant No. 2020YFC2201503. 




 Paper Title: Dynamical System Analysis Of Chameleon Mechanism in Brans-Dicke Scalar-Tensor Model.



Article Title]Dynamical System Analysis of Chameleon Mechanism in Brans-Dicke Scalar-Tensor Model










[1,2]Azwar Sutionoazwarsu@gmail.com

1,3]Agus Surososuroso@itb.ac.id
These authors contributed equally to this work.

1,3]Freddy Permana Zenfpzen@itb.ac.id
These authors contributed equally to this work.

*[1]Theoretical Physics Laboratory, THEPI Division Indonesia, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, Indonesia

[2]Department of Physics, Universitas Hasanuddin, Jl. Perintis Kemerdekaan KM. 10, Makassar, 90245, Indonesia

[3]Indonesia Center for Theoretical and Mathematical Physics (ICTMP), Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, Indonesia





We investigated the stability of the chameleon screening mechanism in the Brans-Dicke scalar-tensor model. We define a constraint on the Brans-Dicke parameter ω_BD^* identifying two stability groups, ω_BD>ω_BD^* and 0<ω_BD<ω_BD^*. The first group achieves stability with both appropriate eigenvalues and a density profile consistent with dark energy dominance. The second exhibits eigenvalue stability but contradicts conditions for a stable universe. We explore the impact of variations in the scalar field potential and matter coupling by analyzing different parameter sets. Each unique set of parameters results in a distinct ω_BD^*. Dynamic analysis reveals that stability is achieved when the scalar field dominates, highlighting the importance of the kinetic and potential terms while minimizing the influence of matter density. In high matter density regions, the scalar field's negligible presence aligns with standard gravitational behavior, whereas in low matter density regions, the scalar field grows exponentially, driving dark energy and cosmic acceleration.







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    [
    August 5, 2024
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§ INTRODUCTION

Modified gravity refers to theoretical modifications or alternative theories to Einstein's general theory of relativity, which describes the force of gravity as the curvature of spacetime caused by mass and energy <cit.>. While general relativity has been extremely successful in explaining various gravitational phenomena, there are some observations on cosmological and galactic scales that it doesn't fully account for. Modified gravity theories attempt to address these issues by introducing new concepts, fields, or modifications to the gravitational force. The field of modified gravity was still an active area of research, with scientists exploring various theoretical frameworks and observational tests. One of the most significant observations in cosmology is that the expansion of the universe is accelerating. This discovery, based on observations of distant supernovae, led modified gravity to the proposal of dark energy as a hypothetical form of energy that permeates space and counteracts the attractive force of gravity <cit.><cit.><cit.><cit.><cit.><cit.>. Some examples of modified gravity theories include Scalar-Tensor Theories, f(R) Gravity, Extra Dimensions and Braneworlds, and the Tensor-Vector-Scalar (TeVeS) Theory <cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.>. 

Screening mechanisms come into play within these modified gravity theories. They are mechanisms that suppress or screen the effects of the modified gravity in environments where standard gravity predictions are well-established. This screening allows the theory to be consistent with local tests of gravity (such as those conducted in our solar system) while still producing noticeable effects on cosmic scales. Common types of screening mechanisms include the chameleon mechanism, Vainshtein mechanism, and symmetron mechanism <cit.><cit.><cit.><cit.>. These mechanisms allow modified gravity theories to evade detection in regions where gravitational effects are well-understood, ensuring agreement with experimental tests conducted in our immediate cosmic neighborhood.  The background of screening mechanisms is rooted in the quest to reconcile cosmological observations, particularly the accelerated expansion of the universe, with modified gravity theories that can explain these phenomena without relying on the introduction of dark energy. These mechanisms enable consistency with local tests of gravity while allowing for deviations at cosmic scales <cit.><cit.><cit.><cit.><cit.>.

The Chameleon mechanism, embedded in the Brans-Dicke scalar-tensor model <cit.><cit.><cit.><cit.>, adeptly conceals gravitational modifications in regions of high density. This mechanism marks a progression from the conventional chameleon mechanism, intricately coupling the scalar field with matter <cit.><cit.>. As a consequence, the scalar field assumes a variable mass contingent upon the density of matter in the relevant region. In regions characterized by extremely low matter density, the scalar field within the model reverts to its role as dark energy, contributing to the universe's expansion acceleration <cit.>. 

Equally crucial, the pursuit of screening mechanism stability in modified gravity originates from the need to understand the behavior and viability of these theories in different environments, particularly on cosmological and astrophysical scales. The indispensability of stability becomes evident in ensuring the viability of modified gravity theories. An unstable mechanism could manifest observable deviations from general relativity in regions subject to experimental and observational constraints. Stability ensures that the modified gravity theory remains consistent with empirical evidence and can provide a viable alternative explanation for gravitational phenomena. The exploration of screening mechanism stability serves as a crucial means to anticipate and understand how these theories behave in diverse cosmic environments <cit.><cit.><cit.>. It offers insights into the interplay between gravity and matter at different energy scales and provides a testing ground for the compatibility of modified gravity theories with the principles of general relativity. Analogous to conventional dark energy models, the Chameleon mechanism must satisfy stringent stability conditions to establish itself as a trustworthy dark energy model. Hence, a thorough analysis of the dynamic system is conducted to assess the stability of the mechanism.

The structure of the paper is outlined as follows. Section <ref> give review of the thin-shell solution of the Brans-Dicke chameleon.  Section <ref> delves into the Brans-Dicke chameleon model, presenting the system of equations formulated as an autonomous system. Moving to Section <ref>, a comprehensive system dynamics analysis is conducted to assess the stability of the mechanism. Section <ref> elucidates the correlation between the Brans-Dicke chameleon and the universe's expansion acceleration. The final section encapsulates a summary and draws conclusions based on the findings of the study.



§ THIN-SHELL SOLUTION OF BRANS-DICKE CHAMELEON

In our previous work, we investigate the thin-shell solution of the Brans-Dicke Chameleon mechanism <cit.>. The Brans-Dicke Chameleon action is given as follows:

    S=∫ d^4x√(-g)[ϕℛ-ω_BD/ϕ(∂ϕ)^2-2V(ϕ) ] + 2S_m(A^2(ϕ)g_μν, ψ),

where (∂ϕ)^2=g^μν∂_μϕ∂_νϕ, V(ϕ) is the scalar field self-interaction potential, w_BD is the Brans-Dicke dimensionless coupling parameter, A(ϕ) is the coupling function to matter, g_μν is the metric in Jordan Frame and g̃_μν=A^2(ϕ)g_μν is the metric in Einstein Frame. The potential V(ϕ) is assumed to be of the runaway form that monotonically decreasing and satisfies: V, V_,ϕ/V, V_,ϕϕ/V_ϕ...→∞ as ϕ→∞, as well as V, V_,ϕ/V, V_,ϕϕ/V_ϕ...→∞ as ϕ→ 0 <cit.><cit.>. By appropriately selecting V(ϕ) and A(ϕ), it is feasible to create a scenario where the scalar propagates unrestrictedly and facilitates a fifth force in regions characterized by a low Newtonian potential. However, this force is deactivated in high-density areas, such as those found in the solar system. Variation of action (<ref>) with respect to ϕ give the equation of motion for ϕ,

    ∇^2ϕ=2/3+2ω_BD[ϕ V_,ϕ-2V+(A_,ϕϕ-A/2)ρ],

with ρ is density in general. In order to understand how the chameleon force is suppressed in the presence of high density region, we solve for the field profile in the presence of a massive compact object. We consider a spherically symmetric object characterized by a constant radius R and density ρ_c for simplicity. Additionally, we envision that this object exists in the presence of a homogeneous background of density ρ_b. We choose the thin-shell case that refers to a region around a massive object where the chameleon field experiences a significant change in its behavior. This thin shell arises due to the interaction between the chameleon field and matter, particularly in regions of high density. Within the thin shell, which typically extends to a certain distance from the surface of the massive object (its distance is R_c from the center of the massive object), the chameleon field transitions from its light, weakly interacting state to its heavier, strongly interacting state. This transition zone effectively shields the effects of the chameleon field from being observable at larger distances.


We find the approximate  exterior solution of the thin-shell case,
 
    ϕ_thin(r)=-ηe^-m_b(r-R)/r+ϕ_b,
 
with,

    η   =   e^-(R_c+R)√(Qβρ_c/M_pl)(M_pl-2β) /2β(e^2R_c √(Qβρ_c/M_pl)M_plR_c(-R_c+√(M_pl/Qβρ_c))( 1+R√(Qβρ_c/M_pl))/-R_c(Qβρ_c+M_pl)+M_pl(√(M_pl/Qβρ_c)- √(Qβρ_c/M_pl)).
          .+e^2R √(Qβρ_c/M_pl)R_c(R+√(M_pl/Qβρ_c))(Q R_cβρ_c+√(Qβρ_c/M_pl))/√(βρ_c/M_pl)(R_c(Qβρ_c-M_pl)+M_pl( √(M_pl/Qβρ_c)+ √(Qβρ_c/M_pl)))),

where Q=2/3+2ω_BD, M_pl, β, ϕ_b and m_b  are Planck mass, coupling constant, a minimum of the effective potential in the background, and the mass of small fluctuations about ϕ_b, respectively. Note that η is a constant that depends on Planck mass, theory parameters, and the compact object radius and density.

The solution is subsequently implemented on earth by choosing the suitable constants. As observed, the chameleon field undergoes exponential growth once it surpasses the Earth's radius, eventually stabilizing at an infinite distance. In regions with high matter density, where the chameleon field has a large effective mass, the associated fifth force becomes short-ranged. The force acts effectively only over small distances, confined to the immediate vicinity of the massive object. The short-range nature of the fifth force leads to a screening effect. On the contrary, in low-density regions far from the Earth, where a lighter scalar field operates, a long-range force is mediated, highlighting the impact of modified gravity on the cosmological background.



§ AUTONOMOUS SYSTEM OF BRANS-DICKE CHAMELEON ACTION

Considering the variation of the Chameleon Brans-Dicke action with respect to both the metric g_μν and the scalar field ϕ in the flat Friedmann–Lemaître–Robertson–Walker background, we arrive at the following Friedmann equation and dynamic equation:

    3H^2   =   -3Hϕ̇/ϕ+ω_BD/2(ϕ̇/ϕ)^2-V/ϕ+Aρ_m/2,
    
    2Ḣ+3H^2   =   -2Hϕ̇/ϕ-ω_BD/2(ϕ̇/ϕ)^2-ϕ̈/ϕ-V/ϕ,
    ϕ̈+3Hϕ̇   =   2/3+2ω_BD[ϕ V_,ϕ-2V+(A_,ϕϕ-A/2)ρ_m].

where ρ_m represents the matter density. The matter fluid is considered as pressureless dust (p_m = 0). In cosmology, pressureless dust is a good approximation for non-relativistic matter, such as cold dark matter and baryonic matter, at large scales. These components dominate the matter content of the universe in the current epoch, making pressureless dust a relevant model for studying the chameleon mechanism <cit.>. In the absence of a screening effect,  without any coupling between matter and the scalar field, the analysis reduces to the standard Brans-Dicke theory <cit.>. 
Combining equation (<ref>), (<ref>) and (<ref>), we get modified continuity equation as,

    ρ̇_m+3Hρ_m=-3/4A_,ϕϕ̇ρ_m/A
The effective equation of state of the system is ω_eff=-1-2/3Ḣ/H^2 <cit.>. In order to study the cosmological dynamics of the system, it is convenient to introduce the following dimensionless variables: 

    x=ϕ̇/Hϕ,
          y=1/H√(V/3ϕ),
          z=1/H√(ρ_mA/3ϕ).

The equation (<ref>) can be simplified and written as a constraint,

    ω_BD/6 x^2 -x - y^2 +z^2 = 1.

We also define the energy fraction of dark energy and matter:

    Ω_ϕ=ω_BD/6x^2-x - y^2;      Ω_m=1-Ω_ϕ.

From equation (<ref>) and (<ref>), we get

    Ḣ/H^2   =   -x/2-ω_BDx^2/4+3( Q-1/2)y^2+3Qz^2/4-Q/2H^2(V_,ϕ+A_,ϕρ_m)-3/2,
    ϕ̈   =   -3Hϕ̇-6Qy^2H^2ϕ-3Qz^2H^2ϕ/2+Qϕ( V_,ϕ+A_,ϕρ).


By choosing V(ϕ)=V_0ϕ^-n and A(ϕ)=A_0ϕ^α, with n and α are positive dimensionless constant, then differentiating each dimensionless variable in equation (<ref>) with respect to the number of e-foldings N= ln a, obtain the autonomous equations


    dx/dN   =   ω_BDx^3/4-3x^2/2-3x/2-3(n+2)Qy^2+3( α-1/2)Qz^2
          -3[(n/2+1)Q-1/2]xy^2+3/2( α-1/2)Qxz^2,
    dy/dN   =   -3[(n/2+1)Q-1/2]y^3+3y/2 +ω_BDx^2y/4-(n/2+1 )xy
          +3/2( α-1/2)Qyz^2,
    dz/dN   =   3/2(α-1/2)Qz^3 +ω_BDx^2z/4-3(n/2+1 )Qy^2z+3y^2z/2
          +(α/8-1)xz,

and effective equation of state,

    ω_eff   =   -1-2/3[-x/2-ω_BDx^2/4+3( Q-1/2)y^2+3Qz^2/4. 
           .+3Q/2(ny^2 -α z^2 )-3/2],

with Q=2/3+2ω_BD and d/dN=1/Hd/dt. By setting dx/dN=dy/dN=dz/dN=0, fixed points can be obtained. Solving the above equations we find the fixed points that some explicitly depend on ω_BD, n, and α as illustrated in Table <ref>. The dynamics of the autonomous system can be analyzed by finding the exact constant fixed points of the system and examining the stability around these points <cit.>.





§ STABILITY ANALYSIS OF THE BRANS-DICKE CHAMELEON


The stability of a fixed point in a dynamical system is characterized by how the system behaves in the vicinity of that point. Specifically, stability analysis involves understanding the response of the system to small perturbations or deviations from the fixed point. In the context of autonomous systems, the stability of fixed points is often determined by analyzing the eigenvalues of the Jacobian matrix evaluated at the fixed point <cit.><cit.>. The eigenvalues provide information about the behavior of perturbations around the fixed point. Consider a system of three autonomous differential equations:

    dx/dN=f(x,y,z),    dy/dN=g(x,y,z),   dz/dN=h(x,y,z),

with f, g, and h are functions describing the dynamics of the system. We consider linear perturbations (δ x,  δ y, δ z) as follows 

    x=x_c+δ x,       y=y_c+δ y,      z=z_c+δ z.

Linearizing the autonomous equation above leads to the first-order differential equations

    d/dN[ δ x; δ y; δ z;     ] = M  [ δ x; δ y; δ z;     ],

where M is 3× 3  Jacobian matrix composed of partial derivatives of  f, g, h with respect to 
x, y, z.  It is given by

    M=
     [  ∂ f∂ x  ∂ f∂ y  ∂ f∂ z;  ∂ g∂ x ∂ g ∂ y ∂ g ∂ z;  ∂ h∂ x ∂ h ∂ y ∂ h ∂ z;         ]_(x=x_c,   y=y_c, z=z_c),


The stability of the fixed point is determined by the real parts of the eigenvalues. Depending on the signs of the real parts: 1) Stable node, if all eigenvalues are real and have negative real parts. 2) Unstable node, if all eigenvalues are real and have positive real parts. 3) Saddle point, if there is a mix of positive and negative real parts among the eigenvalues. 4) Stable spiral, If the real parts of the eigenvalues are negative and at least some of the eigenvalues have non-zero imaginary parts. 5) Unstable spiral, if the real parts of the eigenvalues are positive and at least some of the eigenvalues have nonzero imaginary parts <cit.><cit.><cit.>.

In order to obtain the eigenvalues of each fixed point, we need the explicit form of the potential and the matter coupling function. The choice of potential in dark energy models is crucial because it directly affects the dynamics of the scalar field responsible for the accelerated expansion of the universe. One crucial condition that must be fulfilled is the slow-roll condition <cit.>. The slow-roll conditions are more easily satisfied with smaller values of n. This choice balances simplicity and the ability to produce a slow-roll regime that can match the observed accelerated expansion of the universe, making it a suitable and popular choice in dark energy models. As a starting step, we choose n=2, and for simplification, we also choose the same value for α. Then note that there is a special case regarding the fixed points listed in Table  <ref> for α=2. The number of critical points will be reduced to 9 points. Specifically, point P2 coincides with point P3, point P4 coincides with point P5, and points P8 and P9 coincide with point P1. We can classify the stability of a fixed point based on the limitations on the value of the parameter ω_BD. The eigenvalues of each fixed point for case n=α=2 can be seen in Table <ref>.


Based on the eigenvalues, we can determine the characteristics of each fixed point based on the limits of the parameter ω_BD. Point P1, P8 and P9 will always be a saddle point because it does not depend on the parameter ω_BD, regardless of any choice of constant values n and α. Points P2-P7 will be will be always saddle points for ω_BD>23/6. Points P10 and P11 will be stable under condition ω_BD>23/6 and will be saddle point for ω_BD<23/6. Points P12 and P13 will always be unstable when ω_BD>0 and will never be stable.

There is one conditions that permit the existence of stable fixed points. To assess the stability of our mechanism, we introduce ω_BD^* as the threshold value of the Brans-Dicke parameter. We will review one condition, focusing on the limitations of the Brans-Dicke parameter for ω_BD>ω_BD^*, with ω_BD^*=23/6  for the specific case we have chosen. The stability characteristics of each fixed point for the case ω_BD>ω_BD^* can be seen in Table <ref> for ω_BD=4. Based on the eigenvalue analysis, there are nine saddle points, namely points P1-P9, two stable points, namely points P10 and P11, and two unstable points, namely points P12 and P13. 



Points P1, P8, and P9  represent the matter-dominated era with Ω_m=1. These points indicate the early universe was dominated by matter. Points P2 to P7 are saddle points but have non-physical density parameters and do not correspond to any physical era. The density parameter values for both the scalar field and matter must not be greater than one and must not be negative. These points highlight non-physical solutions, emphasizing constraints on the model. Points P10 and P11 represent the era of dark energy domination, with Ω_ϕ=1. The stability of these points suggests that the universe will eventually evolve into a phase dominated by dark energy, leading to accelerated expansion. Points P12 and P13 can be considered as kinetic points of the scalar field with density parameter values approaching 1. The instability indicates that these states are not sustainable and the system will evolve away from these points. 





We can also identify the phase in which a fixed point is situated by analyzing the ω_eff parameter. For the universe to be in an accelerating phase, the effective equation of state parameter ω_eff must be less than -1/3. It is evident that points P2-P7 and P10, P11 satisfy the conditions for being in the accelerated phase of the universe. This is further supported by the dominance of the scalar field in these points, as indicated by their density parameter values, particularly for the stable points P10 and P11.

To gain a deeper understanding of the stability characteristics, we can analyze the trajectory of each fixed point in the phase space. Visualizations were performed using Mathematica <cit.>. Figure <ref> on the left shows the trajectory paths around points P10 and P11 in 3D phase space. To further confirm the attractor properties of the fixed point, we can set one of the critical point components as constant and visualize it in a 2D plane. Figure <ref> on the right shows the trajectory around point P10 by setting y=-1.128i, demonstrating the point's attractor nature in xz-plane. A similar plot would be observed for point P11 with y=1.128i. In Figure <ref>, we can see the trajectory paths at points P1 (and also P8,P9), P12, and P13. In the xy-plane with set z=0, the characteristics of each point are clearly visible, with P1 being a saddle point and P12 and P13 being unstable points. Points P2-P7 are not visualized due to their non-physical nature. These points may represent theoretical solutions that do not manifest in real-world scenarios or under specific model constraints, thus not contributing to the stability analysis of the system.

As a comparison, we will examine the case where n≠ 2
so that all fixed points exist with different values. Table <ref> shows the eigenvalues of each critical point for the case n=3. Point P1 still be a saddle point. Points P2, P3, P4, and P5 will be stable under the condition 1/192(107+27√(417))≤ω_BD<25/6, stable spiral for 0<ω_BD<1/192(107+27√(417)) and will be saddle points for ω_BD>25/6. Points P6 and P7 will always be saddle points under the condition ω_BD>19/6. Points P8 and P9 will be always saddle points when ω_BD<-53/24 or ω_BD>-7/4 and can be stable under certain conditions, specifically when -53/24<ω_BD<-7/4. Points P10 and P11 will be stable under condition ω_BD>25/6 and will be saddle point for ω_BD<25/6. Points P12 and P13 still be unstable when ω_BD>0 and will never be stable.




There are three conditions that permit the existence of stable fixed points, but not all of them will be examined. In the Brans-Dicke theory, the scalar field ϕ contributes to gravitational interactions that the kinetic term of the scalar field in action has the correct sign, corresponding to positive kinetic energy. Negative kinetic energy would lead to instabilities, such as ghost fields, which are unphysical. A crucial step in achieving this is by ensuring that the parameter ω_BD is positive. We will review only two conditions, focusing on the limitations of the Brans-Dicke parameter for ω_BD>ω_BD^* and 0<ω_BD<ω_BD^*, with ω_BD^*=25/6  for the specific case we have chosen.



 §.§ Case ω_BD>ω_BD^*


The stability characteristics of each fixed point for the case ω_BD>ω_BD^* can be seen in Table <ref> for ω_BD=5. Based on the eigenvalue analysis, there are nine saddle points, namely points P1-P9, two stable points, namely points P10 and P11, and two unstable points, namely points P12 and P13. 



Point P1 represents the matter-dominated era with Ω_m=1. These points indicate the early universe was dominated by matter. Points P2-P7 have non-physical density parameters and do not correspond to any physical era. Points P8 and P9 represent an era where the scalar field starts moving but matter still dominates, with Ω_m=0.922. These points suggest transitional eras where both matter and scalar fields are significant. Points P10 and P11 represent the era of dark energy domination, with Ω_ϕ=1. The stability of these points implies that the universe will eventually transition into a dark energy-dominated phase, resulting in accelerated expansion. Points P12 and P13 can be considered as kinetic points of the scalar field with density parameter values approaching 1. The instability shows that these states are not sustainable, and the system will move away from these points over time. 



We can also determine the phase in which a fixed point resides by examining the ω_eff parameter. For the universe to be in an accelerating phase, the effective equation of state parameter ω_eff must be less than -1/3. It is clear that points P2-P7 and P10, P11 meet the criteria for being in the accelerated phase of the universe. This conclusion is reinforced by the dominance of the scalar field at these points, as reflected in their density parameter values, especially for the stable points P10 and P11.



To better comprehend the stability properties, we can analyze the behavior of each fixed point in the phase space. Figure <ref> on the left shows the trajectory paths around points P10 and P11 in 3D phase space. To further confirm the attractor properties of the fixed point, we can set one of the critical point components as constant and visualize it in a 2D plane. Figure <ref> on the right shows the trajectory around point P10 by setting y=-1.109i, demonstrating the point's attractor nature in xz-plane. A similar plot would be observed for point P11 with y=1.109i. Figure <ref> shows the phase space plot of points P8 and P9, with figures on the left providing 3D visual representations. The saddle nature of point P8 is further confirmed in figures on the right when the system is observed from an alternative viewpoint in the xy-plane with set z= -0.303i. Figure <ref> shows the trajectory paths at points P1, P12, and P13. In the xy-plane with set z=0, the characteristics of each point are clearly visible, with P1 being a saddle point and P12 and P13 being unstable points. Points P2-P7 are not visualized due to their non-physical nature. These points might be theoretical solutions that fail to appear in real-world scenarios or under certain model constraints, thereby not contributing to the stability analysis of the system.





 §.§ Case 0<ω_BD<ω_BD^*


We have chosen the value ω_BD=4 which falls within the range 0<ω_BD<ω_BD^*, to examine the characteristics of each fixed point in the context of the stability of the dark energy model. According to the eigenvalues listed in Table <ref>, there are 4 stable points, 2 unstable points, and the remainder are saddle points. Interestingly, some fixed points that were initially stable have become saddle points, and vice versa. Points P1, P6-P9, P12, and P13 retain the same characteristics as they did with the previous selection of ω_BD.  Points P2-P5, previously identified as saddle points, are now stable with all negative eigenvalues and remain in the accelerated phase based on their ω_eff value. However, their density parameter values do not align with the expected properties of stable points, specifically the absolute dominance of the scalar field. Although the scalar field remains dominant with Ω_ϕ=0.93, there is still a non-zero matter density, typically a characteristic of saddle points. Conversely, points P10 and P11, previously stable, have transitioned to saddle points based on their eigenvalues, yet they still remain in the accelerated phase. Despite this, their density parameter values indicate the total dominance of the scalar field with Ω_ϕ=1 which is usually a feature of stable fixed points.  





 §.§ Variation of ω_BD


To gain a clearer understanding of the stability changes in two groups of fixed points that can potentially be stable, namely the groups of points (P2-P5) and (P10, P11), we can examine various conditions with different values of ω_BD values within the two ranges mentioned above. Table <ref> shows the stability of the fixed points for ω_BD=2 and ω_BD=3, which fall within the range 0<ω_BD<ω_BD^*, and for ω_BD=10, ω_BD=10^3 and ω_BD=10^5 which fall within the range ω_BD>ω_BD^*. In the range 0<ω_BD<ω_BD^*, points P2-P5 consistently show stability, as stable spiral points, while points P10 and P11 consistently exhibit saddle behavior. However, this stability is not supported by their density parameter values. In fact, as ω_BD decreases, the matter density becomes more dominant, leading to a contradiction between stability based on eigenvalues and stability based on the density parameters of dark energy and matter. In the range ω_BD>ω_BD^* points P2-P5 consistently exhibit saddle behavior, and points P10 and P11 consistently show stability. This is corroborated by their density parameter values, where the scalar field dominates stable conditions. This analysis sheds light on how the stability properties of fixed points in the dark energy model change with different values of ω_BD, providing insight into the complex dynamics and stability conditions influenced by the Brans-Dicke parameter.



 §.§ Variation of n and α

In the previous work, we reviewed the stability of fixed points for specific values of n and α, which led to certain constraints on the Brans-Dicke parameter. In this section, we aim to vary the values of  n and α, exploring different forms of the potential and the matter coupling function. Table <ref> presents several combinations, specifically for  n=2, n=3, n=4 and α=4, α=5, α=6. Each combination of n and α results in a unique value of ω_BD^*. As the values of both of them increase, the value of ω_BD^* also increases.  This implies that each choice of the potential and matter coupling function imposes different constraints on the Brans-Dicke parameter ω_BD. These constraints are crucial in ensuring the stability of each fixed point in the dark energy model under consideration.






§ BRANS-DICKE CHAMELEON AND THE ACCELERATION OF THE UNIVERSE'S EXPANSION

In the case examined above, we identified several stable fixed points for specific values of ω_BD. For the special case when α=2, stability is achieved when ω_BD>ω_BD^*. For the case α≠ 2, and as an example α=3 as mentioned above, the stable fixed points are divided into two groups: one where ω_BD>ω_BD^* and another where 0<ω_BD<ω_BD^*, with ω_BD^*=23/6. Both groups of fixed points are in the accelerated expansion phase of the universe with ω_eff<-1/3. In the first group, we found fixed points that are stable based on their eigenvalues and are supported by density parameter values for dark energy and matter that match the actual conditions of the universe. In the second group, we also found stable fixed points based on their eigenvalues, which contradict the density parameter values. Therefore, we will focus on the first group, which can be considered as physically stable points.
It is established that the variable x denotes the kinetic term of the scalar field, y represents the potential term of the scalar field, and z signifies the coupling term between the matter density and the scalar field. The obtained results lead to the inference that the stability of the Chameleon mechanism in the Brans-Dicke model manifests when the scalar field is dominant especially the potential terms, accompanied by the minimum coupling of matter and scalar field.

In the discourse surrounding modified gravity, specific models posit the scalar field as a potential contender for dark energy, as seen in Quintessence, Gauss-Bonnet, and similar frameworks <cit.>. Essentially, the attractor solutions derived earlier suggest that stability is achieved when the scalar field takes on a dominant role, signifying the prevalence of dark energy. Upon examining other dark energy models that also put forth the scalar field as a candidate, it becomes apparent that the attractor solutions in those models align with the dominance of the scalar field, consistent with the aforementioned results.

The dominance of the scalar field’s kinetic and potential terms in the universe’s acceleration is a concept rooted in the dynamics of specific cosmological models, particularly those involving scalar fields. The kinetic term represents the rate of change of the scalar field for a time. It is associated with the field’s motion or velocity. When the kinetic term dominates, it implies that the scalar field is dynamically evolving, and its motion contributes to the overall energy density of the universe. The potential term represents the energy associated with the scalar field at a given point in space. It characterizes the field’s potential to influence cosmic evolution. When the potential term dominates, it suggests that the scalar field has settled into a particular energy state, potentially leading to the acceleration of the universe's expansion. 

In the specific context of the Chameleon mechanism within the Brans-Dicke model and similar scenarios, the dominance of the scalar field signifies stability. It indicates that the scalar field’s dynamics are primarily influenced by its motion and energy state, with minimal interaction with matter. This dominance contributes to the stability of the Chameleon mechanism in regions of low matter density. Fundamentally, the role of the scalar field in propelling the universe’s acceleration stems from their collaborative capacity to produce an effective energy density characterized by negative pressure, resembling the attributes of dark energy. When appropriately balanced, these terms can lead to the observed accelerated expansion, as seen in certain cosmological models.

A prior investigation identified thin-shell solutions for the Chameleon Brans-Dicke field <cit.>. The findings indicated that the scalar field maintains a negligible value in proximity to regions with high matter density (near the matter source). It then experiences an exponential increase, ultimately converging to a stable value in areas of low matter density situated far from the matter source (r→∞). In locales with high matter density, the Chameleon mechanism conceals the gravitational modification effects. Conversely, in regions with low density, the scalar field reassumes its role as dark energy, instigating the acceleration of the universe's expansion. Crucially, the scalar field reverts to functioning as dark energy when it attains a very large, stable, and constant value, coinciding with very low matter density. Implicitly, these conditions lend support to the attributes of the attractor solutions outlined in the dynamic system analysis discussed earlier.




§ CONCLUSION


The results of this study provide crucial support for the stability of the Chameleon mechanism, especially within the Brans-Dicke model. By examining the eigenvalues obtained, we define a constraint on the Brans-Dicke parameter, referred to as ω_BD^*. This constraint divides the stability points into two distinct groups: one for ω_BD>ω_BD^* and another 0<ω_BD<ω_BD^*, except for the special case of α=2 which only satisfies the conditions of the first group. In the first group, the fixed point is stable and is supported by suitable eigenvalues and a density parameter that represents the era of dark energy dominance. In the second group, the fixed point is stable according to its eigenvalues, but the density parameter does not correspond to the expected conditions of a stable universe. We then explore variations in the explicit forms of the scalar field potential and the matter coupling function by selecting different values of n and α. It is shown that each unique constant value results in a distinct ω_BD^*. This demonstrates that the stability of the dark energy model is sensitive to the specific values of these parameters. The exploration reveals that the stability conditions for dark energy dominance are closely linked to the chosen values of n and α. These parameters influence the behavior of the scalar field potential and the matter coupling function, thereby affecting the overall dynamics of the universe. By identifying the unique ω_BD^* for different parameter values, we gain insights into the stability characteristics of various dark energy models, which is crucial for understanding the long-term evolution of the universe under the influence of dark energy.

From the dynamic analysis performed, it was also discovered that stability is achieved when the scalar field assumes a dominant role, with a particular emphasis on the significance of the kinetic and potential terms. This dominance is highlighted while simultaneously minimizing the influence of matter density in the cosmic environment. In regions of high matter density, the scalar field’s negligible presence effectively conceals the gravitational modification effects, aligning with standard gravitational behavior. However, in areas of low matter density, far from massive objects, the scalar field experiences exponential growth. This growth contributes to the manifestation of dark energy, ultimately leading to the acceleration of the universe. These findings not only affirm the stability of the Chameleon mechanism in the Brans-Dicke model but also provide a deeper understanding of how this mechanism operates in different cosmic environments. The interplay between the scalar field, matter density, and the terms in the gravitational equations contributes valuable insights into the intricate dynamics of the universe on both large and small scales.





§ DECLARATIONS





  * Funding

This work was supported by Riset Unggulan ITB 2023

  * Competing interests 

The authors have no relevant financial or non-financial interests to disclose.

  * Author contribution

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Azwar Sutiono, Agus Suroso and Freddy Permana Zen. The first draft of the manuscript was written by Azwar Sutiono and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript




*








 Paper Title: Non-Noetherian Conformal Cheshire Effect.






eloy.ayon-beato-at-cinvestav.mx 
Departamento de Física, CINVESTAV-IPN, A.P. 14740, C.P. 07360, Ciudad de México, México


hassaine-at-inst-mat.utalca.cl 
Instituto de Matemática, Universidad de Talca, Casilla 747, Talca, Chile


pedro.sanchez.s-at-cinvestav.mx 
Departamento de Física, CINVESTAV-IPN, A.P. 14740, C.P. 07360, Ciudad de México, México

§ ABSTRACT 
The gravitational Cheshire effect refers to the possibility of turning off the gravitational field while still leaving an imprint of the nonminimal coupling of matter to gravity. This allows nontrivial solutions in flat spacetime for which no backreaction is possible. The effect was originally shown to manifest itself for standard nonminimal couplings, such as those allowing conventional conformally invariant scalar fields. Recently, the most general scalar field action yielding a conformally invariant second-order equation was constructed, and entails a more involved nonminimal coupling explicitly breaking the conformal invariance of the action without spoiling it in the equation. We have succeeded in fully describing the spherically symmetric stealth solutions on flat spacetime supporting the Cheshire effect within this general non-Noetherian conformal theory. The allowed configurations are divided into two branches: The first one essentially corresponds to an extension of the solutions already known for the standard Noetherian conformal theory. The second branch is only possible due to the non-Noetherian conformal contribution of the action. The complete characterization of this branch is expressed by a nonlinear first-order partial differential equation. We have found the general solution of this equation using both seemingly new and well-established mathematical tools.


Non-Noetherian conformal Cheshire effect
    Pedro A. Sánchez 0009-0008-7437-0561
    August 5, 2024
========================================






§ INTRODUCTION 
 


It is evident that nonminimal couplings to gravity, arising due to the explicit presence of curvature in matter Lagrangians, vanish in the flat spacetime limit. However, it is less expected that after taking such a limit some nonminimal contributions to the variation persist, leaving an imprint on the energy-momentum tensor. This phenomenon has been dubbed the gravitational Cheshire effect <cit.>, making a reference to the Cheshire cat in Lewis Carroll's Alice's Adventures in Wonderland, which disappears leaving behind only its grin. The effect is manifested through the existence of nontrivial solutions where the nonminimal and minimal contributions cancel each other, leading to a vanishing energy-momentum tensor in flat spacetime. In fact, these are examples of gravitational stealth configurations whose definition extends beyond flat spacetime, being genuine nontrivial solutions to Einstein's equations by simultaneously canceling both their gravitational and matter contributions <cit.>. 

The Cheshire effect has been exhibited in particular for standard conformal scalar fields <cit.> whose action is 
 
    S_s[Φ,g] = - 1/2∫d^4 x √(-g)(∇_μΦ∇^μΦ + 1/6 R Φ^2 + 2λΦ^4 ),

yielding the scalar field equation 

    Φ-1/6RΦ-4λΦ^3=0,

and energy-momentum tensor 

    T_μν^s =   ∇_μΦ∇_νΦ - ( 1/2∇_αΦ∇^αΦ + λΦ^4 ) g_μν
        + 1/6( g_μν-∇_μ∇_ν+G_μν )Φ^2.


Here, λ is the coupling constant of the conformal self-interaction and the symbols , R, and G_μν stand for the d'Alembertian operator, the scalar curvature, and the Einstein tensor, respectively. The peculiarity of this system is that action (<ref>) is invariant under the conformal transformations 

    g_μν   ↦Ω^2(x) g_μν,    Φ   ↦Ω^-1(x)Φ.

In general, satisfying such property in the action will imply that the energy-momentum tensor is traceless modulo the equation of motion. To connect with our previous discussion, the last terms related to Φ^2 in (<ref>) are due to the nonminimal coupling in the action, and it is clear that in the flat limit g_μν→η_μν these terms do not vanish. In fact, they compensate with the previous minimal contributions in order to allow the existence of flat spacetime stealths that are the direct manifestation of the conformal Cheshire effect, as shown in <cit.>.

To make our terminology clear, the stealths are defined as the nontrivial extrema of any matter action not only with respect to the matter content but also with respect to the gravitational background. The first such example was shown to arise on the static BTZ black hole in three dimensions for a scalar field with the simplest nonminimal coupling to gravity, which contains the conformal coupling (<ref>) as a particular case <cit.>.  It was later on generalized for the same theory on (A)dS spacetimes of arbitrary dimension <cit.> or even on more exotic spacetimes, but likewise holographically relevant at non-relativistic limits <cit.>. Applications of stealth configurations in the context of cosmology were also reported in Refs. <cit.>.

Scalar-tensor theories, particularly those within the Horndeski class <cit.>, have proven to be an excellent laboratory for exhibiting stealth configurations. These examples are not limited to zero mass gravitational states but also include configurations defined on black holes, such as the Schwarzschild metric <cit.>, relying on having a constant kinetic term, see also Refs. <cit.>. More recently, it has been shown that in so-called beyond Horndeski theories <cit.>, there exist stealth solutions with a non-constant kinetic term <cit.>; for a detailed discussion on stealth solutions in the context of such theories, see <cit.>. The existence of such black hole stealths, which violate certain assumptions of the standard “no-hair” theorems valid in General Relativity, could prove to be of great use to distinguish such theory from scalar-tensor ones by means of future observations. In this perspective, a stealth on the top of the Kerr black hole was constructed in <cit.>, and it turns out that its disformal extension leads to another rotating black hole solution with different physical properties from those characterizing the unique prediction of General Relativity <cit.>.

In this article, we extend the known conformal realizations of the gravitational Cheshire effect provided in <cit.> by considering the most general action principle for a scalar field whose equation of motion is of second order and conformally invariant. This theory was identified in <cit.>, inspired by seminal results of <cit.>. The most innovative contribution to the conformally invariant equation is a nonminimal coupling, initially identified in <cit.>, whose inclusion breaks the conformal symmetry of the action but not of the equation; such symmetry was dubbed non-Noetherian conformal symmetry in <cit.>. Furthermore, it stands out that this new contribution makes the scalar field equation of motion fully non-linear, in contrast with the quasilinear standard conformal equation (<ref>).
 
To begin our exploration, it is natural to ask whether the standard conformal stealth configurations on flat spacetime <cit.>, retain its stealth behavior after introducing non-Noetherian conformal contributions. This task is straightforward even without imposing any symmetry on the scalar field because these Noetherian conformal configurations obey specific separability properties, as shown in <cit.>, which are also effective for the full theory. Conversely, examining the broader stealth spectrum of the full conformal theory beyond this separability presents a major challenge. The new stealth constraints are entirely nonlinear, making the quasilinearity of the standard case exploited in <cit.> inapplicable. To advance in this new study of the conformal gravitational Cheshire effect, we are focusing on spherically symmetric configurations that are still time-dependent. Unexpectedly, this simple assumption leads to the mutual cancellation of all second-order nonlinear contributions in the relevant independent constraints. This opens the door to a new branch of the gravitational Cheshire effect beyond separability, encoded in a single nonlinear first-order PDE. Remarkably, this branch exists only for a family of nonvanishing values of the non-Noetherian conformal coupling constant.

The full characterization of the new branch demands the use of sophisticated PDE tools. Indeed, finding the general solution of a fully nonlinear first-order PDE is not as straightforward as it is for its linear and quasilinear counterparts, where the method of characteristics is enough. The standard approach in the fully nonlinear case, originally formulated by Lagrange (see <cit.> and references therein), involves finding the envelope of a so-called complete integral, subject to the condition that one of its parameters is a general function of the others. This results in an implicit representation of the general solution even in the simplest nonlinear examples.

In contrast to the sophistication of Lagrange's methodology, we introduce a seemingly novel elementary strategy here, leveraging the fact that the involved nonlinear first-order PDE can be cast into a standard Hamilton-Jacobi equation. Despite being a well-known equation in mechanics, finding its general solution is not common since a complete solution suffices to obtain the general integral of the equations of motion of the mechanical system <cit.>. In this work, we address this by exploiting the quadratic dependence on the derivatives in the obtained Hamilton-Jacobi equation, allowing its identification with the definition of an hyperbola. Imposing integrability conditions on the hyperbola parameterization gives rise to a quasilinear first-order PDE for the local parameter. Naturally, this equation can be integrated using the more conventional method of characteristics.

The paper is organized as follows. In Sec. <ref>, we briefly present the most general conformally invariant second-order equation arising from an action principle and the related theory. In Sec. <ref>, we continue by reviewing the previously known separable stealth solutions defined on flat spacetime for the standard conformal case (<ref>). Then, we analyze how these solutions prevail when the most general non-Noetherian conformal theory is considered. In Sec. <ref>, we will show that turning on the non-Noetherian conformal coupling opens the possibility of obtaining radically new spherically symmetric conformal stealths beyond separability. We manage to prove that this new branch only exists for a precise family of nonvanishing values of the related coupling constant, and is exclusively determined by a single nonlinear first-order PDE. The general solution of the latter will be found using both the seemingly new and also the well-established PDE methods outlined above. Hence, we will provide several representations that can be useful for different applications and limits. Concretely, using the aforementioned quasilinearization we found three different representations for the general solution in Subsec. <ref>, two of them are implicit as expected. Notably, coordinate freedom also provides an explicit representation in a special system adapted to the problem, which is unfortunately awkward for expressing the Minkowski metric. We also apply the standard PDE approach relying on the Lagrange technique. When used on the complete integral derived from the so-called Lagrange-Charpit method, one of the representations obtained by quasilinearization is recovered in Subsec. <ref>. Since separable solutions are also a source of complete integrals we address them in Subsec. <ref>. There, we also get the unexpected result that the previously mentioned extension of the separable Noetherian conformal stealths of <cit.> can be also used as seed in the Lagrange approach to derive the most general non-Noetherian conformal stealths. Supplementary to this, in Subsec. <ref>,  we also present the so-called singular solution not contained in the general one, and show how several particular cases emerge from the general solution. The last section is devoted to a discussion concerning our results. To be as instructive as possible, we briefly review in App. <ref> all the PDE definitions and methods in which the main results are founded. Explicit expressions for relevant conformally invariant scalars are provided in App. <ref>.




§ NON-NOETHERIAN CONFORMAL SCALAR FIELDS

 


The most general second-order scalar field equation invariant upon a conformal transformation (<ref>) and arising from an action principle in four dimensions can be compactly written as <cit.>

    Φ-1/6RΦ-Φ^3(4λ + α/2𝒢̃
    -g̃^αβ∂
    f/∂g̃^αβ +2f)=0.

Here, we denote by

    g̃_μν=Φ^2g_μν,

an auxiliary metric which is conformally invariant by construction (<ref>). Furthermore, α is a coupling constant mediating a new nonminimal coupling to gravity expressed through 𝒢̃=R̃^2-4R̃_αβR̃^αβ +R̃_αβμνR̃^αβμν, which is nothing but the Gauss-Bonnet scalar built from the auxiliary metric (<ref>), and whose explicit expression is given in Eq. (<ref>). Additionally,  f=f(g̃_μν,C̃^α_ βμν) is an arbitrary function of the auxiliary Weyl tensor, only subject to the condition

    f(g̃,0) = 0.

Since the new ingredients in Eq. (<ref>) are all constructed from the auxiliary metric g̃_μν, it is clear that this equation is again conformally invariant. We remark that (<ref>) is a fully nonlinear equation since 𝒢̃ is quadratic on the second-order highest derivatives of Φ, see (<ref>), and only for α=0 this equation becomes quasilinear by involving exclusively linear contributions of the derivatives of Φ.

As was also shown in <cit.>, the
action yielding to  (<ref>) reads
 
    S[Φ,g] =    S_s[Φ,g] + S_nN[Φ,g] 
       - 1/2∫ d^4 x √(-g) Φ^4f(Φ^2g_μν,C^α_ βμν),

where S_s [Φ,g] was given in (<ref>) and <cit.> 
    S_nN[Φ,g]=   -α/2∫d^4 x √(-g)(ln(Φ)𝒢 
    -4/Φ^2G^μν∇_μΦ∇_νΦ
       -4/Φ^3(∇_μΦ∇^μΦ)Φ
    + 2/Φ^4(∇_μΦ∇^μΦ)^2).


Here, C^α_ βμν and 𝒢 refer to the components of the Weyl tensor and the Gauss-Bonnet scalar of the background metric g_μν, respectively. Notice that the full action is parity invariant in the scalar field, Φ↦-Φ. For α = 0 and f = 0, the action reduces to the standard conformal action (<ref>). Interestingly, not only this term is conformally invariant but also the last one defining the Weyl coupling in the full action. In fact, both can be rewritten only in terms of the auxiliary metric up to boundary terms

    - 1/2∫d^4 x √(-g̃)(1/6R̃ + 2λ + f(g̃_μν,C̃^α_ βμν) )+b.t.,

where R̃ is the scalar curvature of the auxiliary metric g̃_μν, see Eq. (<ref>) for its explicit expression. This is compatible with the result proved in <cit.> (see their Eq. (9) and the related discussion) that any conformally invariant scalar density built from the scalar field and the metric must exclusively be expressed in terms of the auxiliary metric. However, this is no longer the case for the α-extension described by action (<ref>), which when expressed in terms of the auxiliary metric keeps a dependence on the scalar field 

    S_nN[Φ,g]=-S_nN[Φ^-1,g̃] +b.t.,

and hence, explicitly breaks the conformal invariance of the action. This was first carefully stressed by Fernandes <cit.> using an infinitesimal argument. In summary, the conformal symmetry of Eq. (<ref>) does not have a Noetherian origin, hence, such equation describes non-Noetherian conformal scalar fields <cit.>.[See Refs. <cit.> for the two-dimensional non-Noetherian analog.] 

The variation of action (<ref>) with respect to the metric gives the energy-momentum tensor whose full expression is reported in Refs. <cit.>. It is more compactly expressed if the scalar field is redefined according to its conformal weight

    σ = 1/Φ,

and in this work we only need its expression evaluated on flat spacetime that reads 
    T_μν =    T_μν^s,flat
    -α{ 4σ_μασ_ν^ α/σ^2
    -2[2σ_μν/σ
    -η_μν(σ/σ - σ_ασ^α/σ^2)
    ]
       ×(σ/σ - σ_βσ^β/σ^2)
    + η_μν((σ_ασ^α)^2/σ^4 - 2σ_αβσ^αβ/σ^2)},

where the indexes of σ denote partial differentiation, and T_μν^s,flat corresponds to (<ref>) with g_μν=η_μν. Importantly, we remark that there is no contribution from the Weyl coupling, a conclusion that can be extrapolated to any conformally flat spacetime.

Finally, it is relevant to stress that the full action (<ref>) retains a global scaling symmetry when Ω=const. in (<ref>). Although the class of second-order theories allowing such property is much broader, see <cit.> and more recently <cit.>. An additional effective conformal symmetry appears for spacetimes with vanishing Gauss-Bonnet scalar, since the energy-momentum tensor again becomes traceless modulo the scalar equation of motion <cit.>. 





§ NOETHERIAN CONFORMAL STEALTHS

 


The Cheshire effect for the standard conformal scalar field (<ref>) was first shown to exist in Ref. <cit.>. The related nonminimal conformal grin manifests through a flat spacetime stealth  parameterized as

    Φ(x^μ) = 1/ a x_μ x^μ + b_μ x^μ - c ,

where x^μ = (t, x^i) are the Cartesian coordinates of Minkowski spacetime and the integration constants a, b_μ, and c are tied in terms of the coupling constant of the conformal potential as

    λ = 2ac + 1/2b_μ b^μ.

As proved in <cit.>, this is the general solution to the standard stealth constraints T_μν^s,flat=0, defining a Noetherian conformal stealth. The above integration constants are determined up to isometries so they can be restricted further. Indeed, for non-vanishing a one can fix b_μ = 0 without any loss of generality by choosing an appropriate translation. In the same way, one can set c = 0 via a translation for vanishing a, while simultaneously aligning the vector b_μ through a rotation so that it has a single independent component.

Remarkably, the parameterization (<ref>) is also effective  for the full non-Noetherian conformal action (<ref>). This can be readily seen using the redefinition (<ref>). The fact that the scalar field (<ref>) prevails as flat spacetime stealth even when α0, relies in that the specific sum-separable form implied for σ is characterized by having a Hessian proportional to the metric, σ_μν = (σ/4) η_μν. This in turn forces the energy-momentum tensor (<ref>) to be also proportional to the metric, T_μν = (tr (T)/4) η_μν, while the condition tr(T) = 0 amounts to the constraint 

    d_∓≡1 ∓√(1-48αλ)/24α=2ac + 1/2 b_μ b^μ.

For vanishing α only the upper-sign root is well defined giving the finite limit d_- = λ, recovering the known standard constraint (<ref>). The last configurations solve then the
non-Noetherian conformally invariant equation (<ref>). Since they observe the same local dependence than those of the case with conformally invariant action, and only the constraint obeyed by the integration constants is generalized, we shall refer to them as Noetherian-like conformal stealths.

As we will show in the next section, the Noetherian-like conformal configurations above do not exhaust all the stealths allowed by the general action (<ref>). Unfortunately, the goal of finding the general solution, as done in <cit.> for the α=0 case, turns out to be a formidable task given that the stealth constraints that follow from (<ref>) are no longer quasilinear. Hence, in what follows we will restrict ourselves to the spherically symmetric configurations, Φ = Φ(t,r),  defined in flat spacetime 

    ds^2 = -dt^2+dr^2+r^2dΩ^2.

For the Noetherian-like conformal configurations (<ref>) this entails that b_i = 0, allowing only two different cases up to isometries, depending on whether a is vanishing or not. For the former, one has the spatially homogeneous solutions 

    Φ(t,r) = 1/√( -2 d_∓)  t.

While the latter gives rise to the Lorentz-invariant family

    Φ(t,r) = 1/ a ( -t^2 + r^2 ) - d_∓/(2a) ,

depending on the single integration constant a, which is a consequence of the scaling invariance of action (<ref>) discussed at the end of Sec. <ref>. For the special value of the non-Noetherian conformal coupling constant α = 1/(48λ), the two roots (<ref>) degenerate to d_∓ = 2 λ and one recovers the particular solutions first found in <cit.>, cf. their Eqs. (A8) and (A9). In fact, it was through these particular examples that they were the first to recognize that Noetherian conformal configurations can remain stealth in the presence of non-Noetherian conformal contributions. It is noteworthy that the non-Noetherian conformally supported flat universe reported in <cit.> is just a conformal transformation of the stealth (<ref>). However, the transformed configuration is no longer free of backreaction like the original, due to the lack of conformal symmetry in the action.

The Noetherian and Noetherian-like conformal solutions studied in this section rely in the sum-separability of the inverse of the scalar field σ (<ref>). Fortunately,  when such separability is not attainable it is not the end of the story, at least for spherically symmetric configurations. In the next section, we show the existence of a new possibility, only valid in the non-Noetherian conformal regime α0.




§ NON-NOETHERIAN CONFORMAL STEALTHS




In this section, we will show that spherically symmetric generalized conformal stealths are necessarily divided into two branches. One corresponds to the separable sector already characterized in the previous section. Hence, we will concentrate in this section on the study of the new branch, which is radically different and its existence is only ensured for α≠0. Concretely, assuming spherical symmetry on conformal stealths gives a single off-diagonal component for the energy-momentum tensor (<ref>)

    T_t^r=Sσ_tr/3σ^3=0,

where

    S ≡ 1+12α(σ_t^2-σ_r^2
    +2/r σσ_r).

It is then evident that Noetherian conformal stealths, given by α = 0, must be necessarily sum separable since the unique option is σ_tr=0. Interestingly, for α0 the vanishing of the quantity S opens a new possibility that we will now explore. The remaining nontrivial stealth constraints can be conveniently rewritten from the energy-momentum tensor (<ref>) as


    T_t^t    = 1-48αλ
    +16ασ/r(σ_r-rσ_rr)S-S^2 /48ασ^4=0,
    
    T_r^r    = 1-48αλ
    +16ασ/r(σ_r+rσ_tt)S-S^2 /48ασ^4=0,
    
    T_θ^θ   =T_ϕ^ϕ= (r^2σ T_r^r)_r+r^2[σ_r
    T_t^t-σ(T_t^r)_t]/2r( σ-rσ_r )=0.


The last expression is a consequence of the conservation of the energy-momentum tensor (<ref>) and the fact that it is traceless in flat spacetime, since both conditions are satisfied via the scalar equation (<ref>). In other words, it is a by-product of two symmetries of action (<ref>): general covariance and the already discussed effective conformal symmetry on flat spacetime.

One can then conclude from the previous equations that there are in fact new possibilities beyond the sum separability of the Noetherian-like conformal stealths determined by the conditions 

    S   =0,    αλ   = 1/48.

This is clearly a net departure from the standard stealths behavior, and an obvious manifestation of a Cheshire effect due to the non-Noetherian conformal contribution, since necessarily α≠0. We shall denote the resulting configurations as non-Noetherian conformal stealths. From definition (<ref>) it follows that conditions (<ref>) give rise to a nonlinear first-order PDE. In what follows we will show how to obtain its general solution using different but equivalent approaches. We start with an elementary and straightforward method to quasilinearize Hamilton-Jacobi equations that does not seem to have been considered in the PDE literature. 




 §.§ Non-Noetherian from Hamilton-Jacobi 
quasilinearization 



If one rewrites the inverse scalar field (<ref>) as

    σ = 2√(|λ|)  r u,

equations (<ref>) with definition (<ref>) yield

    ϵ S/r^2=u_t^2-u_r^2+u^2+ϵ/r^2=0,

where ϵ=±1 is the sign of the non-Noetherian conformal coupling constant α=ϵ|α|. Redefining again the involved function u as

    u=sinhΘ,    α>0, 
    coshΘ,    α<0,

we obtain in both cases the following Hamilton-Jacobi equation

    Θ_r^2-Θ_t^2=1/r^2.

This equation defines a hyperbola that can be locally parameterized by a function H=H(t,r) as

    Θ_r   =cosh H/r,    Θ_t   =sinh H/r.

Such parameterization must be subject to the integrability condition ∂_tΘ_r=∂_rΘ_t, that implies the function H obeys the following inhomogeneous quasilinear first-order PDE

    X(H)   =sinh H/r,     
    X   ≡-sinh H∂_t+cosh H∂_r.

In other words, we have managed to transform the original fully nonlinear problem defined by Eq. (<ref>) to a quasilinear first-order PDE. The general solution to the latter can be found by integrating its characteristic system, see Eq. (<ref>), which is

    -dt/sinh H=dr/cosh H=r dH/sinh H.

The equation corresponding to the second equality is separable 

    dr/r=dsinh H/sinh H,

which implies that the following function is a first integral of the system, namely

    Ω   = sinh H/r,     X(Ω)    = 0.

As is custom in the treatment of quasilinear PDE, it is advantageous to use this integral as new coordinate

    (t,r) ↦( t,Ω=sinh H(t,r)/r).

Using these coordinates, the vector field becomes

    X=X(t)∂_t+X(Ω)∂_Ω=-sinh H∂_t,

and the PDE given in (<ref>) takes the simple form

    ∂_t(cosh H)=-Ω.

This allows its straightforward integration

    cosh H(t,Ω)=-Ω t +Ω h'(Ω),

depending on an arbitrary function of the invariant Ω, appropriately chosen for later convenience. In terms of the original coordinates, this means that the most general function H(t,r) satisfying the inhomogeneous quasilinear first-order PDE (<ref>) is implicitly given in terms of an arbitrary function by

    t+r H=h'(sinh H/r).

Equivalently, the most general function Ω(t,r) satisfying the homogeneous quasilinear first-order PDE involved in (<ref>) is given by the arbitrary implicit dependence

    t+1/Ω√(1+Ω^2r^2)=h'(Ω).


Summarizing, the starting system (<ref>) is only integrable for the found dependence of the local hyperbolic parameter H, expressed either in terms of the original or the adapted coordinates. This can be corroborated by writing the differential of the function Θ in terms of the new coordinates (<ref>)

    dΘ   =sinh H/r dt + cosh H/r dr
       =Ω dt + Ωcosh H/sinh H
    d(sinh H/Ω),

or 

    dΘ - cosh^2 H/sinh HdH = Ω dt - cosh H/ΩdΩ.

Using now the found dependence (<ref>) in the right-hand side yields

    d(Θ - cosh H - lnsinh H/cosh H + 1 - Ω t + h )=0.


We are now in a position to enunciate the main result of this subsection: the general solution to the Hamilton-Jacobi equation (<ref>) is given by


    Θ(t,r)=   t/rsinh H + cosh H 
    - arcsinh( 1/sinh H) 
        - h(sinh H/r),
    
    0=   t+r H-h'(sinh H/r),


where h is an arbitrary single-argument function in terms of which the hyperbolic parameter determining the solution is implicitly defined as a local function H(t,r) from the condition (<ref>). Notice that the integration constant appearing in (<ref>) can be absorbed into the function h, which is defined modulo a constant in (<ref>). An equivalent second statement is: the general solution to the Hamilton-Jacobi equation (<ref>) can be also expressed as
 

    Θ(t,r) =   Ω t + √(1+Ω^2r^2)
    - arcsinh( 1/Ω r) 
        - h(Ω),
    
    0 =    t+√(1+Ω^2r^2)/Ω-h'(Ω),


where the solution is now determined by the function Ω(t,r), which is defined by the arbitrary implicit dependence implied from condition (<ref>). This last expression for the general solution turns out to be just the one resulting from applying the procedure devised by Lagrange to the complete integral obtained from the Lagrange-Charpit method, as we will show in the next section.

Moreover, there exists a third possibility allowing an explicit expression for the general solution by using the integral Ω as a coordinate adapted to the problem. Concretely, using (<ref>), (<ref>) and (<ref>), the general solution to the Hamilton-Jacobi equation (<ref>) also allows the explicit expression


    Θ(t,Ω) =arcsinh(1/Ω [ h'(Ω) - t ]) 
    + Ω h'(Ω) - h(Ω),

after changing the radial coordinate according to

    r = √(Ω^2[h'(Ω)-t]^2-1)/Ω.


The drawback of this explicit expression for the general solution is that in such coordinates the flat spacetime metric (<ref>) acquires a very unsuitable form.

Finally, the previous results allow to conclude that a non-Noetherian conformal Cheshire effect is possible thanks to the existence of a new α-grin whose most general time-dependent spherically symmetric stealth is determined by


    Φ   =12√(|λ|)rsinhΘ,      λ>0, 
    
    12√(|λ|)rcoshΘ,      λ<0,
          
    α   =1/48λ,


where the function Θ allows either of the two implicit representations (<ref>) and (<ref>), or even the explicit representation (<ref>).




 §.§ Non-Noetherian from Lagrange-Charpit 




It is also possible to tackle the full characterization of the spherically symmetric non-Noetherian conformal Cheshire effect by directly dealing with the PDE (<ref>). However, since this equation is no longer quasilinear as the obtained for H in the previous subsection, the integration process leading to the general solution is less straightforward. It is then needed to appeal to the Lagrange approach reviewed in App. <ref>, where the general solution is constructed starting from a complete integral. Here we will apply such procedure to the complete integral for u obtained using the Lagrange-Charpit method summarized in App. <ref>.

We concretely use the version (<ref>-<ref>) of the nonlinear first-order PDE that can be written as

    F(t,r,u,p,q) ≡ p^2-q^2+u^2+ϵ/r^2=0,

where we have defined p=u_t and q=u_r. From Eq. (<ref>), the
associated Lagrange-Charpit characteristic system reads 
    dt/2p=-dr/2q=du/2(p^2-q^2)
    =-r^2dp/2up=-r^3dq/2(ruq-u^2-ϵ).

Using the original Eq. (<ref>) in the third equality, we
obtain the separable equation

    udu/u^2+ϵ =dp/p,

whose straightforward integration yields

    H(t,r,u,p,q)≡p/√(u^2+ϵ)=c_1=const.

Isolating the first derivatives from (<ref>) and
(<ref>), we obtain

    p=c_1√(u^2+ϵ),      q=√((u^2+ϵ)
    (1+c_1^2r^2))/r,

which allows to write du=pdt+qdr as

    du/√(u^2+ϵ)=c_1dt+√(1+c_1^2r^2)dr/r.

The above Pfaff equation is integrable for the following potential 
    d[ln(√(1+c_1^2r^2)+1/c_1r(u+√(u^2+ϵ)))
    -c_1t-√(1+c_1^2r^2)]=0,

from where the complete integral can be finally isolated using the redefinitions (<ref>) with 
    Θ(t,r;c_1,c_2)=c_1t + √(1+c_1^2r^2) 
    - arcsinh(1/c_1r) - c_2.


We now apply the Lagrange procedure to derive the general solution, starting from the above complete integral obtained via the Lagrange-Charpit method. 
Following App. <ref>, the first step is to impose in the complete integral (<ref>) with (<ref>), that is c_2 = h(c_1). Differentiating the result with respect to the remaining parameter, the next envelope condition is obtained 

    ũ_1=√(u^2+ϵ)(t+√(1+c_1^2r^2)/c_1-h'(c_1))=0.

The above condition can be explicitly solved for the parameter only for precise elections of the function h, defining its coordinate dependence c_1=c_1(t,r). On the contrary, for an arbitrary function such dependence can only be defined implicitly. Consequently, according to the Lagrange procedure the general solution to the nonlinear first-order PDE (<ref>), characterizing the spherically symmetric non-Noetherian conformal Cheshire effect, allows the following parametric representation in (<ref>)  

    Θ(t,r)    = c_1t + √(1+c_1^2r^2)
    - arcsinh(1/c_1r) 
    - h(c_1),
    
    0    = t+√(1+c_1^2r^2)/c_1-h'(c_1),


where c_1(t,r) is a function implicitly determined by the second condition. This is precisely the representation (<ref>) obtained in the previous subsection following the quasilinearization procedure and parameterized by the function Ω, which is just the local dependence of c_1 in the present context. Consistently, both approaches give rise to the same representation of the general spherically symmetric non-Noetherian stealth configuration (<ref>).




 §.§ Non-Noetherian from Noetherian 




As shown at the beginning of the section, conformal stealths with spherical symmetry in flat spacetime are divided into two branches: the Noetherian-like and the non-Noetherian ones. Surprisingly enough, it happens that the two classes of solutions intersect when αλ=1/48. Indeed, it is known that other source of complete integrals for nonlinear first-order PDE are the separable solutions; in the case of (<ref>) its most general sum-separable solution is

    σ(t,r;a,τ) = a [ -( t-τ )^2 + r^2 ] - λ/a,

which is nothing other than the Noetherian-like conformal stealth (<ref>) at α=1/(48λ), before performing the time translation to make b_0 vanish in (<ref>). In other words, the Noetherian-like conformal stealths also provide a complete integral for the nonlinear first-order PDE (<ref>) describing non-Noetherian conformal stealths!

With the different complete integral (<ref>), we can apply the Lagrange procedure again to obtain another parameterization of the general solution to the PDE (<ref>). Making that the original parameter arbitrarily depends on the reinstalled time-translation one, a = a(τ), and following App. <ref> again, it is obtained that the general solution can be also represented as


    σ(t,r)    = a(τ) [ -( t-τ )^2 + r^2 ] - λ/a(τ),
    
    a'(τ)    = -2 (t-τ) a^3(τ)/λ + [ -( t-τ )^2 + r^2 ] a^2(τ),


where the function τ(t,r) is implicitly defined by the last condition.

The above representation of the general solution is linked to the one given in terms of Θ in (<ref>) in the following way. First, consider a Legendre transform of the arbitrary function of the representation (<ref>)

    Z(z)   =Ω z - h(Ω),    
    z   =h'(Ω),

where, as usual, the last relation is used to solve for Ω=Ω(z). Then the arbitrary function of the representation (<ref>) and its argument are defined by the following map 
    (z,Z) ↦(
    τ = z - tanh(Z)/Ω(z) ,
     a = √(λ) Ω(z) cosh(Z) ),

when λ > 0. For λ < 0, the same map holds if λ is replaced by its absolute value as well as sinh and cosh are interchanged in every occurrence.





 §.§ Singular and particular solutions 




For λ < 0, there exists a solution, not contained in the general one, and called singular solution, cf. App. <ref>. Solving conditions (<ref>) for the equation (<ref>), using either its version (<ref>) or (<ref>), one obtains that the singular solution is given by

    σ(r) = ± 2 √(-λ)  r.

The singular solution is not only spherically symmetric but also static, properties that are incompatible for the Noetherian-like stealths considered in Sec. <ref>.[It may seem like a linear radial dependence as (<ref>) could cause a bad behavior on the angular stealth constraints (<ref>). But this is not the case; it is merely an artifact of how those components are expressed in terms of the others.]

Let us now turn our attention to those solutions which do arise as particular cases of the general solution for spherically symmetric non-Noetherian conformal stealths. The first question to answer is if there are more static solutions. In fact, choosing a(τ)=A=const. in (<ref>) gives τ = t and the solution becomes 

    σ(r) = A r^2 - λ/A.

This is another one-parameter class of static solutions, which in contrast to the singular one is defined for any sign of λ, as long as it is related to α through (<ref>). Remarkably, the previous static stealths can be also obtained as the flat spacetime limit of the second branch of static and spherically symmetric non-Noetherian conformally dressed black holes found by Fernandes in <cit.> precisely for αλ=1/48. These configurations were originally derived in <cit.> following such approach, cf. their Eq. (A4). In contrast, the first branch of black holes given in <cit.>, which exists for the different couplings relation αλ=1/144, does not have a well-defined flat spacetime limit. That sheds some light on the relevance of the couplings special tuning (<ref>) for the manifestation of a non-Noetherian conformal Cheshire effect. The previous two static solutions turn out to be the only ones within the spherically symmetric stealths giving rise to the conformal Cheshire effect. In fact, if one considers σ = σ(r), the PDE (<ref>) becomes a quadratic first-order ordinary equation whose two roots straightforwardly yield (<ref>) and (<ref>). 

Another interesting particular solution is obtained if one chooses a(τ) = B/τ, for which (<ref>) yields

    σ(t,r) = 2 ( B t ±√( (B^2+λ) (t^2-r^2) )).

Spatially homogeneous stealths can be obtained from the previous one by setting B = ±√(-λ) for λ < 0. They coincide with the spatially-homogeneous Noetherian-like conformal stealths (<ref>) when αλ=1/48,  first obtained in <cit.>. It is straightforward to check from PDE  (<ref>) that these are the only allowed spatially homogeneous configurations. On the other hand, taking B = 0 a new Lorentz-invariant solution different from the Noetherian-like conformal stealth (<ref>) is obtained 

    σ(t,r) = ± 2√(λ( t^2-r^2 )),

valid inside or outside the light cone for positive or negative λ, respectively. By similar arguments to those used in the static case, the above solutions and  (<ref>) are the only Lorentz-invariant stealths with spherical symmetry related to the conformal Cheshire effect.





§ CONCLUSIONS 



In this work, we have explicitly shown the existence of a non-Noetherian conformal Cheshire effect for scalar fields obeying the most general second-order conformally invariant equation derived from an action principle. 

We started by showing that the configurations supporting the original conformal Cheshire effect <cit.>, characterized by a sum separable inverse of the scalar field, are also  stealth solutions for the full theory. The original restriction of the separability constants in terms of the self-interaction coupling constant λ was generalized to involve also the non-Noetherian coupling constant α. Being the latter coupling the only one responsible for breaking the conformal invariance of the action, while preserving that of the scalar equation, we have dubbed such configurations Noetherian-like conformal stealths. 

Furthermore, we have shown that the presence of the α-coupling is responsible for the manifestation of a new conformal Cheshire effect, which is strictly non-Noetherian in the sense that it only arises for α≠ 0. We arrived to that conclusion by thoroughly studying the spherically symmetric stealths of the whole theory in flat spacetime. It results that the introduction of the α-terms  allow a new branch of stealth solutions beyond the separability of the Noetherian-like conformal configurations. We managed to prove that this new sector is ruled by a single nonlinear first-order PDE. Additionally, it only occurs for the tuning αλ=1/48 between the couplings, which in particular requires both to be non-vanishing. We have called the resulting configurations non-Noetherian conformal stealths. We fully characterized these latter using different but equivalent methods, with results related at most by a combination of Legendre and point transformations.

When a coordinate system conventionally adapted to the flat spacetime metric was used, implicit representations of the general solution were obtained as it is common in the study of nonlinear first-order PDE. Unexpectedly, an explicit representation was also obtained in a coordinate system adapted to the problem, but causing the flat metric to be represented in an unconventional manner

Another interesting result is that the generic Noetherian-like solutions are just the sum-separable sector of the nonlinear first-order PDE. An outstanding lesson is that, by avoiding a premature use of isometries, these solutions have enough integration constants to serve as a seed for generating the general solution describing the full non-Noetherian sector.

We explicitly show how to recover special solutions from the general picture, including the static ones, those that are spatially homogeneous, and finally the Lorentz-invariant examples. In particular, it results that there are only two static cases, and the remaining infinite family of new conformal stealths indeed overfly Minkowski spacetime without causing even the slightest backreaction.

It was pointed out that one of the particular non-Noetherian static stealths coincides with the one obtained in <cit.> also for αλ=1/48, as the flat spacetime limit of the static and spherically symmetric conformally dressed black hole found in <cit.> for Einstein gravity coupled to the theory (<ref>) without Weyl coupling, i.e. f=0. The exhaustive analysis of this work unveils that this was just the tip of the iceberg, and that an infinite class of configurations exists for the same couplings restriction. 

Hence, the emergence of the same couplings relation both for the non-Noetherian conformal Cheshire effect and for non-Noetherian conformally dressed black hole solutions should not be considered as a mere accident, but rather due to some deeper physical reasons that deserve future exploration. For the moment, we have provided further evidence that specific nontrivial non-Noetherian conformal couplings are favored by allowing new interesting physical configurations that are inconceivable in the Noetherian conformal case we are used to.


We would like to thank D. Flores-Alfonso, U. Hernandez-Vera, Julio A. Méndez, T. Simon, D. Uribe for useful discussions and reviewing this work. This work has been partially funded by Conahcyt grant A1-S-11548 and FONDECYT grant 1210889. P.A.S. work was supported by Conahcyt “Estancias Posdoctorales por México” contract I1200/320/2022.







§ NONLINEAR FIRST-ORDER PDE 




In order to make this work self-contained, in this appendix we will briefly summarize general methods to solve nonlinear first-order PDE heavily used in the main text. They can be found in the PDE literature <cit.>, see also <cit.> and references therein for some historical remarks. 

We consider a general nonlinear first-order PDE

    F(t, r, u, p, q)=0,

where p=u_t and q=u_r. There are different types of solutions, or integrals, allowed by the PDE. A general solution is the one given in terms of an arbitrary single-argument function. Additionally, a complete integral is a solution involving two arbitrary constants which cannot be combined into a single one.[The previous definitions are generalized to the case of n independent variables in the following way: a general solution is one given in terms of an arbitrary function with n-1 arguments and a complete integral is one involving n arbitrary constants and no less.] On the contrary, a solution that involves neither an arbitrary function nor arbitrary constants is called a particular solution if it is contained as a particular case of a general or complete solution, and a singular solution otherwise. The general and singular integrals can be obtained from a complete integral using envelopes; an approach to eliminate constants and obtain other solutions. The standard procedure to find the general solution of a nonlinear first-order PDE is due to Lagrange, see <cit.>, and consists in constructing the envelope resulting from assuming that one of the integration constants of a previously found complete integral is an arbitrary function of the other constants. This is why complete integrals are relevant in the context of nonlinear first-order PDE, and an established way to build them is the Lagrange-Charpit method. In the following sub-appendices we will review these methods to obtain complete, general, and singular solutions.




 §.§ The Lagrange-Charpit method 



The Lagrange-Charpit method is a systematic approach to determining complete integrals. The method to find the general solution of a quasilinear first-order PDE was originally given by Lagrange and rests in the integration of the associated characteristic ordinary system, see (<ref>) below. Charpit introduced a generalized characteristic system for fully nonlinear first-order PDE, providing a method to find a complete integral. Such characteristic system is defined by

    dt/F_p=dr/F_q=du/pF_p+qF_q
    =-dp/pF_u+F_t=-dq/qF_u+F_r.

From this system, one needs to find a first integral independent of (<ref>), let us say

    H(t,r,u,p,q)=c_1=const.

Then, isolating the first derivatives from both equations
(<ref>) and (<ref>) should give the explicit expressions

    p=p(t,r,u;c_1),      q=q(t,r,u;c_1).

The characteristic system (<ref>) warrants that the one-form

    du-p(t,r,u;c_1)dt-q(t,r,u;c_1)dr=0,

satisfies its integrability conditions, then it is possible to straightforwardly integrate the first-order quasilinear PDE system

    u_t=p(t,r,u;c_1),      u_r=q(t,r,u;c_1),

to obtain the desired complete integral

    u=u(t,r;c_1,c_2).


In the particular case in which the original PDE (<ref>) is quasilinear, namely given by

    a_1(t,r,u) p + a_2(t,r,u) q = a_0(t,r,u),

it is enough to consider only the first two equations of the characteristic system (<ref>), since they become the following subsystem for t, r, and u

    dt/a_1 = dr/a_2 = du/a_0.

Correspondingly, the previous two equations allow for two first integrals G(t,r,u) and H(t,r,u), but the two-dimensional nature of (<ref>) implies that they are not independent. Hence, the general solution u = u(t,r) is determined from

    G = h(H),

where h is an arbitrary single-argument function. This is precisely the method used in Subsec. <ref> to find the general solution for the spherically symmetric non-Noetherian conformal stealths after quasilinearizing the involved Hamilton-Jacobi equation.




 §.§ Envelopes and singular solutions 




Let us now consider a solution involving one or more parameters

    u = u(x^μ; c_i),

and construct the following system by differentiating with respect to the parameters

    u_i ≡∂_c_i u(x^μ; c_i) = 0.

Provided the above algebraic system can be solved for the parameters as functions of the independent variables, c_i = c_i(x^μ), the envelope of the multiparametric solution is defined as the function 

    v(x^μ)=u(x^μ; c_i(x^μ)),

and is also a solution by the conditions (<ref>). 

It turns out that the envelope of a complete integral precisely yields a singular solution, when it exists. Alternatively, the singular solution can be obtained without appealing to a complete integral by eliminating p and q from the following system

    F    = 0,     F_p    = 0,     F_q    = 0.

These procedures are followed in Subsec. <ref> to identify the singular solution involved in the problem. Envelopes are also essential in the Lagrange approach to find the general solution.




 §.§ Lagrange approach to general solutions 




Lagrange conceived the following approach in order to build the general solution of a nonlinear first-order PDE. Starting from a complete integral, u = u(t, r; c_1, c_2), consider that one of its parameters is an arbitrary function of the other, let say c_2=h(c_1). Then the general solution is the envelope, as defined in the previous sub-appendix, of the function

    û (t, r; c_1) ≡ u(t, r; c_1, h(c_1)).

In terms of the resulting local dependence of the parameter, c_1 = c_1(t,r), the general solution is formally expressed as

    u_gs = u(t,r;c_1(t,r),h(c_1(t,r))),

where is manifest the dependence on an arbitrary single-argument function. However, since the function h is arbitrary, such envelope cannot be explicitly found in general. Hence, the solution is parameterized in terms of the function c_1(t,r) which is determined implicitly by the envelope condition 


    u    = u(t,r;c_1,h(c_1)),
    
    0    = u_1(t,r;c_1,h(c_1)) + u_2(t,r;c_1,h(c_1)) h'.


This is the approach we follow in Subsecs. <ref> and <ref> to reach others parameterizations of the general spherically symmetric non-Noetherian conformal stealths.




§ SECOND-ORDER CONFORMALLY INVARIANT SCALARS 




For completeness, we report here conformally invariant scalars built from the auxiliary metric (<ref>), explicitly expressed in terms of the scalar field, Φ, and the background metric, g_μν. The first corresponds to the auxiliary scalar curvature 

    R̃=1/Φ^3(R-6)Φ,

and the second to the auxiliary Gauss-Bonnet scalar, which is compactly expressed as


    𝒢̃=1/Φ^4( 𝒢+∇_μ J^μ),

where

    1/8 J^μ=   1/ΦG^μν∇_νΦ
    +1/Φ^2(∇^μΦ)Φ
    -1/Φ^2(∇_νΦ)∇^μ∇^νΦ
       1/Φ^3(∇^μΦ)∇_νΦ∇^νΦ.


Here R and 𝒢 are the analogous quantities defined from the metric g_μν, being G^μν the Einstein tensor of the latter. Notice that compact expression (<ref>) is compatible with the topological nature of the Gauss-Bonnet terms. However, it has the inconvenience that seems to incorporate third-order contributions, but these actually cancel out when considering Bianchi and Ricci identities.








