Paper Title: Test of the Equivalence Principle for Superconductors.




Center for Experimental Nuclear Physics and Astrophysics, University of Washington, Seattle, Washington 98195, USA

Department of Physics and Astronomy,
 Western Washington University, Bellingham, Washington 98225, USA

Department of Physics and Astronomy,
 Carleton College, Northfield, Minnesota 98133, USA

Department of Physics and Astronomy,
 Western Washington University, Bellingham, Washington 98225, USA

Department of Physics,
 Columbia University, New York, NY 10027, USA







Center for Experimental Nuclear Physics and Astrophysics, University of Washington, Seattle, Washington 98195, USA

§ ABSTRACT 
We searched for violations of the weak equivalence principle using a cryogenic torsion balance with a pendulum comprised of superconducting niobium and copper. We constrain the Eötvös parameter with 95%-confidence to η_Nb*-Cu ≤ 2.0×10^-9 and η_CP-ee≤6.9×10^-3 for superconducting niobium and Cooper pairs, respectively.


Test of the Equivalence Principle for Superconductors
     J.H. Gundlach
    
=====================================================



The weak equivalence principle (WEP) underpins our geometric description of gravity (i.e. general relativity). Possible violations of the WEP are traditionally parameterized by the Eötvös parameter <cit.>: 

    η=2(a_1-a_2)/a_1+a_2

where a_1,2 are the gravitational accelerations for two test bodies made from different materials. The WEP states that this parameter vanishes for all choices of material. This has been precisely tested for a variety of materials. <cit.>.

It has been argued <cit.> that the non-local quantum-mechanical properties of Cooper pairs should lead to violations of the WEP, with sometimes remarkable effects. For example, it has been suggested that superconductors strongly reflect gravitational waves <cit.>, which would imply a major violation of the WEP.

To our knowledge, only two previous experiments <cit.> have probed the gravitational properties of superconductors. In particular, <cit.> compared the change in the inertial and gravitational masses of niobium in the normal and superconducting states and obtained η_Nb*-Cu < 4.0× 10^-7.

Here, we describe a sensitive test of the WEP comparing superconducting niobium with normally-conducting copper. The experiment was made possible by the development of a unique cryogenic, torsion balance apparatus.  

Apparatus - Very few torsion balances have operated at cryogenic temperatures <cit.>. Our cryogenic balance (described in detail in <cit.>) is shown in Figure <ref>. 

The balance consisted of a torsion pendulum with a niobium-copper composition dipole suspended from a 30-µm thick, 16.5-cm long tungsten fiber which was attached to a passively isolated “swing" damper. The pendulum was housed in a vacuum chamber with a 0.4 mPa helium environment and surrounded by two layers of thermal shielding. The innermost layer of shielding and the suspension point of the fiber were cooled to ∼ 5 K by a two-stage helium pulse-tube cooler. An outer shield was cooled to ∼ 50 K. The exterior of the inner shield was wrapped in a cylindrical layer of lead to provide superconducting magnetic shielding when at cryogenic temperatures. Additionally, a traditional magnetic shield (Co-NETIC AA) surrounded the outer thermal shield.

The pendulum had eight test bodies each with a mass of 9.5 g, four made of OFHC copper and four of niobium (99.9% purity), in a dipole arrangement. The parameters that describe the torsion balance are displayed in Table <ref>. When the pendulum was cooled below the critical temperature of niobium, 9 K <cit.>, the test bodies formed a superconductor-normal-conductor dipole. The superconductivity of the niobium was verified by demonstrating the Meissner effect with an auxiliary coil placed near the pendulum. The angular position of the pendulum was measured by a multi-slit autocollimator <cit.>. 



Environmental sensors were placed throughout the instrument. These included an orthogonal set of tiltmeters attached to the top flange of the vacuum chamber, a pair of external magnetometers, and an array of temperature sensors placed on the top flange, first and second stage of the cold head, and inner and outer thermal shields.



Superconductivity of Niobium Test Bodies - A direct measurement of the pendulum temperature was not practical without spoiling the performance of the torsion balance. Although the inner shield is at ∼ 5 K, the equilibrium temperature of the pendulum at low pressures (<0.1 mPa) is above the critical temperature of niobium. This is due to the heat loads from the autocollimator light and residual thermal radiation from room temperature surfaces visible to the pendulum through the optical access window.

We achieved superconductivity by injecting helium exchange gas into the chamber to a final pressure of 0.4 mPa. This pressure allowed sufficient thermal coupling between the pendulum and the inner shield to bring the pendulum below the critical temperature of niobium.

We verified that the test bodies were superconducting by observing the pendulum's response to an injected sinusoidal magnetic field with coils placed inside and outside the chamber. The torque acting on the pendulum due to an external magnetic field, B⃗, has three terms: one due to the intrinsic magnetic moment of the pendulum, m⃗_0, a second due to an effective magnetic moment induced by pinned flux, m⃗_flux, and a third due to the expulsion of the magnetic field. These are described by:

    τ⃗=(m⃗_0+m⃗_flux)×B⃗-r⃗×(∫B^2/2 μ_0 dA⃗).


An applied sinusoidal magnetic field, B⃗=B⃗_0 sinω t, produces three distinct ω-dependent torques:


    τ⃗=   τ⃗_0 sinω t +τ⃗_fluxsinω t 
       +τ⃗_Meissner(cos 2ω t)

where τ⃗_0, τ⃗_flux, τ⃗_Meissner are the torque amplitudes due to the magnetic moment of the pendulum, the pinned flux, and the Meissner effect, respectively. 


A response at the drive frequency, ω, is expected when the test bodies are not superconducting as both the pinned flux and Meissner terms do not contribute. However, when the test bodies are superconducting, an increase in the response at the drive frequency and a large response at twice the drive frequency, 2ω, are expected.

Figure <ref> shows the measured torque spectra at room temperature as well as at low temperature, with and without exchange gas, while magnetic fields were injected via exterior and interior coils. With the addition of the exchange gas, the pendulum has a large 2ω response to the interior coil due to the Meissner effect and an increased 1ω response due to pinned flux, confirming that the niobium test bodies are in a superconducting state. Additionally, the response to the exterior field vanishes due to the lead shield becoming superconducting during cryogenic operations.







Analysis - The experiment was operated from March 26, 2024 to April 22, 2024 with occasional disruptions, yielding 22 days of usable data. We searched for a violation of the equivalence principle towards the sun that would appear as a daily modulated torque. Our analysis began by filtering the observed torque with a second-order low-pass Butterworth filter with a corner frequency of 0.1 mHz. This filter removes high-frequency noise caused by the angular readout as well as the free torsional oscillation. 

The raw torque showed a clear correlation with the temperature of the inner shield. We interpret this coupling as thermally induced torques of the pendulum's suspension point. This correlation was removed by subtracting the inner shield temperature scaled by a constant coupling factor. This coupling factor was determined to be 4.6 × 10^-11 N m/K by correlating the raw torque and inner shield temperature across the entire data run. Additionally, coupling to the tilt of the apparatus was subtracted from the measured torque. This subtraction did not significantly change the observed torque. The tilt coupling was determined by deliberately tilting the chamber to be -5.5 × 10^-8 N m/rad and 3.5 × 10^-8 N m/rad respectively for the two horizontal directions. An upper limit of 10^-11 N m/Gauss was placed on coupling to external magnetic fields during the magnetic injections. This indicates a negligible influence (<0.1 fN m) of external magnetic fields on the observed torque. None of the environmental couplings changed appreciably throughout the data run.

We periodically rotated the pendulum by 180^∘, effectively exchanging the copper and niobium test bodies, to minimize the environmental effects that could mimic an EP violation. An EP-violating torque would have the opposite sign when the pendulum is flipped whereas many environmental signals (daily tilts, etc.) would not change. This allows EP-violation-mimicking environmental effects to be subtracted from the results. The pendulum requires damping and a thermalization period after each rotation. As such, the data was recorded in roughly week-long sets. A third-order polynomial was fit and subtracted from each set to account for long term drifts.

A series of day-long cuts of the corrected torque were extracted and analyzed independently. Each cut was linear least-squares fitted to the sum of two functions of the sun's location and an offset term: 

    τ (t)=    τ_in a_in(t)+ τ_out a_out(t)+ C_offset

where τ corresponds to the observed torque, a_in is the “in-phase" horizontal projection of the sun's location perpendicular to the pendulum's composition dipole, a_out is the “out-of-phase" orthogonally-oriented horizontal projection, t is time, τ_in and τ_out are the respective torque fit coefficients, and C_offset is the fit offset. 

The sun's location was obtained using the Astropy <cit.> libraries. Figure <ref> shows the observed and fit torque along with the daily modulated in-phase function and the residual of the fits. An EP violating signal would appear as a non-zero in-phase fit coefficient. 

Figure <ref> shows the collection of extracted in-phase coefficients. We take the standard deviation of the residual torque for each cut as a conservative estimate of the uncertainty of the corresponding torque fit coefficient. The collection of in-phase coefficients were averaged with inverse-variance weighting to yield a measurement of the in-phase torque. We find no evidence of a EP-violation with a measured in-phase torque amplitude of:

    τ_in=-1.3±5.2 fN m


We also measured no significant out-of-phase component, τ_out=-2.1±5.2 fN m. We extract limits on the Eötvös parameter from our torque measurements via:

    η=τ_in/m r a_g

where a_g is the gravitational acceleration towards the sun. Our results yield a 95%-confidence limit on the Eötvös parameter for superconducting niobium versus copper of 

    η_Nb*-Cu ≤ 2.0 × 10^-9.


It has been suggested that the material of interest is not the bulk superconductor but only the Cooper pairs <cit.>. Additionally, parameterizing in terms of Cooper pair number allows for comparison between different superconducting materials. 

The total mass of Cooper pairs within a test body is:

    m_CP≈m_e/m_nN_e/N_nN_v/N_eN_CP/N_vm

where m_e and m_p are respectively the mass of the electron and proton, N_e, N_n, N_v, and N_CP are the number of electrons, nucleons, valence electrons, and electrons in Cooper pairs, respectively. The ratio N_CP/N_v was estimated via <cit.>:

    N_CP/N_v=m_e a^3/2 μ_0 e^2 λ_L^2

where a is the lattice constant, μ_0 is the vacuum permeability, e is the charge of the electron, and λ_L is the London penetration depth. 

 To estimate λ_L, we use a conservative upper limit of the temperature of the pendulum of 8 K which corresponds to a λ_L(8 K)=1 µ m for niobium <cit.>. This corresponds to a Cooper pair fraction of N_CP/N_v=0.05.

Using Equation <ref>, we convert the measurement for superconducting niobium to a 95%-confidence limit on the Eötvös parameter for Cooper pairs versus unpaired electrons of:

    η_CP-ee≤ 6.9× 10^-3.


Conclusion - We used a cryogenic torsion pendulum with superconducting-niobium and copper test bodies to constrain deviations from the WEP for superconducting niobium and Cooper pairs with 95%-confidence to η_Nb*-Cu ≤ 2.0×10^-9 and η_CP-ee≤6.9×10^-3, respectively. This improves upon the result reported by <cit.> by more than two orders of magnitude. Our experiment places strong constraints on any enhanced gravitational effects of superconductors.

Data Availability - Code to generate the results shown here can be found at <https://github.com/EotWash/SuperEP>. Data avaible upon request.

Acknowledgements - Participation from the University of Washington, Seattle, was supported by funding from the NSF under Awards PHY-1607385, PHY-1607391, PHY-1912380, and PHY-1912514.

unsrtnat




 Paper Title: Stochastic gravitational wave background from cosmological neutrino-dominated accretion flows.





    Tong Liutongliu@xmu.edu.cn0000-0002-9130-2586]Yun-Feng WeiDepartment of Astronomy, Xiamen University, Xiamen, Fujian 361005, China0000-0001-8678-6291]Tong LiuDepartment of Astronomy, Xiamen University, Xiamen, Fujian 361005, China§ ABSTRACT 
    We investigate the stochastic gravitational wave background (SGWB) from neutrino-dominated accretion flows (NDAFs) based on the results of our fallback core-collapse supernova (CCSN) simulations. We find that the predicted SGWB is mainly determined by the typical CCSN initial explosion energy and progenitor metallicity. For the optimistic cases in which the typical initial explosion energy is low, the SGWB from NDAFs without disk outflows might be detected by next-generation space-based interferometers such as Decihertz Interferometer Gravitational wave Observatory (DECIGO) and Big Bang Observer (BBO). In the low-frequency regime ∼10^-3-10^-1 Hz, this background is comparable to that expected from standard inflationary models. Therefore, the SGWB from NDAFs may become a foreground for searches of the SGWB generated in the inflationary epoch. Combining the diffuse NDAF neutrino background and SGWB from NDAFs, one may constrain the properties of the CCSNe and NDAFs.
    
    
    § INTRODUCTION
    
    
    The stochastic gravitational wave background (SGWB) is expected to be created by the superposition of gravitational waves (GWs) from many uncorrelated sources. Such a background could be produced in many processes during cosmological and astrophysical evolutions. A cosmological SGWB may be produced by inflation <cit.>, cosmic strings <cit.>, alternative cosmologies <cit.>, and a variety of other phenomena. Astrophysical SGWB could have various possible sources (see <cit.> for a review ), including inspiral and coalescence of compact binaries <cit.>, white dwarf binaries <cit.>, rotating neutron stars <cit.>, core-collapse supernovae <cit.>, gamma-ray bursts <cit.>, Population III (Pop III) stars <cit.>, stellar core collapse <cit.>, and so on.
    
    The next-generation space GW detectors, such as Decihertz Interferometer Gravitational wave Observatory <cit.> and Big Bang Observer <cit.>, targeting 0.1-1 Hz, might be possible to detect SGWB from both cosmological and astrophysical origins. Notably, these experiments are expected to detect the primordial SGWB that arises in the very early universe during the inflationary epoch <cit.>. Such SGWB may serve as a powerful tool for studying the extremely early universe. However, astrophysical foreground sources could be a significant problem for searches of the inflationary SGWB. The SGWB from CCSNe <cit.> and Population III stars <cit.> are expected to be comparable to or mask the inflationary SGWB in some range of frequencies. Thus, a good understanding of astrophysical foregrounds around the deci-Hertz band is essential for DECIGO and BBO detecting the inflationary SGWB. In this paper, we focus on the low-frequency GW generated by anisotropic neutrino emission of neutrino-dominated accretion flows (NDAFs) and estimate the SGWB spectrum from NDAFs.
    
    NDAFs around rotating stellar-mass black holes (BHs) are one of the plausible candidates of GRB central engines in massive collapsars and compact object mergers. In the collapsar model <cit.> for long-duration GRBs, the core collapse of a massive star can produce a BH hyperaccretion system via the fallback process. For the very high accretion rate, the disk might be in a state of NDAFs. In the inner region of the disk, photons are trapped, and only neutrinos are emitted from the disk surface. These neutrinos annihilate in the space out of the disk and then form the primordial fireball to power a GRB. The properties of NDAFs have been widely studied in recent decades <cit.>, and for a review see <cit.>. In particular, the GW signals generated by the anisotropic neutrino emission from NDAFs have been investigated by some previous works <cit.>. The neutrino-induced GWs are detectable for ∼10 Mpc by DECIGO/BBO <cit.>. Of course, the BZ mechanism <cit.> could coexist even be dominated in the BH hyperaccretion process, but it has no GW radiation.
    
    An earlier estimation of the SGWB produced by NDAFs was given by <cit.>. They adopted some typical GW energy spectra of NDAFs to calculate the SGWB. Meanwhile, they used the GRB formation history to estimate the event rate of NDAFs. The results show that the SGWB from NDAFs is below the detection limit of DECIGO/BBO. However, the effects of progenitor properties and the initial explosion energy on the GW spectrum of NDAF are not included in their work. Besides, the event rate of NDAFs might be underestimated. In the collapsar model, an observable GRB is only triggered when jets launched by the central engine can break out from the envelope and circumstellar medium in the prompt emission phase <cit.>. If the jet is chocked in the envelope or the observer is not in the jet's line of sight, no GRB would be detected. Therefore, the event rate of GRBs should be much less than that of NDAFs. Of course, the GWs from NDAFs are partly anisotropic. In this work, we estimate the SGWB from NDAFs based on fallback CCSN simulations. In <cit.>, we revealed how the GW emission of NDAFs depends on the initial explosion energies, masses, and metallicities of the progenitor stars. Here, we aim to improve the estimation of the SGWB from NDAFs by including those dependencies.
    
    In this work, we mainly focus on the SGWB from NDAFs around BHs. Note that NDAFs around NSs may also operate in some GRBs and CCSNe <cit.>. Actually, for the newborn NSs in the center of massive collapsars, their strong magnetic fields <cit.> will restrain the accretion process and destroy the inner region of the disk. Besides, since the rapidly decreasing fallback accretion rate, NDAFs might only exist for the very initial stage of the NS hyperaccretion process. Therefore, the contribution of NDAFs around NSs to the SGWB is not considered here.
    
    The paper is organized as follows. In Section 2, we describe the setup of our CCSN simulations and discuss the effects of the initial explosion energies and the masses and metallicities of the progenitor stars on the GW emission of NDAFs. In Section 3, we estimate the amplitude of the SGWB from NDAFs. We discuss how the SGWB depends on the initial explosion energy and progenitor metallicity. We also explore the effect of initial mass function (IMF) on the SGWB from NDAFs. The conclusions and discussion are presented in Section 4.
    
    
    
    § MODEL
    
    
     §.§ CCSN simulations
    
    
    Here, we briefly review the setup of our CCSN simulations. In this paper, we adopt the presupernova (pre-SN) models with initial mass in the range of 20-40M_⊙ as progenitor models <cit.>. For those models, the models with zero metallicity (Z/Z_⊙ = 0) and solar metallicity (Z/Z_⊙ = 1) are referenced from <cit.> and <cit.>, respectively, as well as the ones with metallicity Z/Z_⊙ =0.01 are provided by Prof. Alexander Heger in private communication, where Z_⊙ represents the metallicity of the Sun. The stellar collapse and explosion simulations <cit.> are performed in a series of 1D simulations with the Athena++ code <cit.>. We use the piston approach <cit.> to carry out spherically symmetric explosion simulations. For each progenitor star, a piston was initially located at the outer edge of the iron core. When the star collapses, the piston firstly moves inward for 0.45 s and then moves outward with an initial high velocity and decelerates smoothly until coming to rest at 10^9 cm. We follow this approach to determine the initial explosion condition at the inner boundary.
    
    The simulation was divided into two steps to reflect the initial collapse and the subsequent explosion. In the first step, the numerical grid has an inner boundary at 10^9 cm, and a unidirectional outflowing inner boundary condition was used at the inner boundary to mimic the suction effect resulting from the hypothetical piston moving inwards. The outer boundary is set at the surface of the progenitor star. In this step, the grid has 10^4 cells with a logarithmic uniform interval for the radial direction. The simulation is run to 0.45 s, and then the piston turns outwards, corresponding to the outward propagation of the blast.
    
    In the second step, we use the same outflowing inner boundary condition, which is set at 10^9 cm. The outer boundary is set at 10^16 cm, and the medium outside the star is maintained in a constant state with a pressure and density three orders of magnitude lower than the corresponding ones on the star surface. In this step, the grid has 2,000 logarithmic cells. At the beginning of this step, we map the results of the first step to the new grid for the second step and inject energy into the innermost cell adjacent to the inner boundary to mimic the outward blast passing through the inner boundary. Here, the injected energy is the setting energy, which is assumed to have three values for each case, i.e., 2, 4, and 8 B (1 B=10^51  erg). All simulations were run until the remnant growth ceased. For more details of simulations, see <cit.> and <cit.>.
    
    For each simulation, we record the evolution of the fallback mass supply rate at the inner boundary. Ignoring the disk outflows, we roughly consider the mass supply rate as the mass accretion rate of the disk. If the mass accretion is high, the hyperaccretion disk would be in the state of NDAFs. Meanwhile, the mass and spin of the BH will violently evolve within tens of seconds. According to the conversion of the energy and angular momentum, the evolution equations of a spinning BH can be expressed as <cit.>,
    dM_BH/dt=Ṁ/√(3x_ms) ( 4-3a_*/√(x_ms) ),
    
    and
    da_*/dt=2√(3)Ṁ/M_BH ( 1- a_*/√(x_ms) )^2,
    
    where M_BH and Ṁ are the mass of the BH and the mass accretion rate, respectively, a_* is the dimensionless spin parameter of the BH, and x_ms=3+Z_2-√((3-Z_1)(3+Z_1+2Z_2)) is the dimensionless radius of the marginally stable orbit <cit.>, where Z_1=1+(1-a_*^2)^1/3[(1+a_*)^1/3+(1-a_*)^1/3] and Z_2=√(3a_*^2+Z_1^2) for 0< a_*< 1.
    
    In this work, the initial spin parameter is set as a_*=0.9, and the starting time is set at the time when the initial BH mass (core mass) reaches 2.3 M_⊙.
    
    
    
     §.§ GWs from NDAFs
    
    
    Neutrino emission of NDAFs is mainly determined by the mass accretion rate and the properties of central BH <cit.>. In the relativistic global solutions of NDAFs  <cit.>, the detailed neutrino physics, the nuclear statistical equilibrium, and the conditions of the ignition and neutrino trapping are all considered. For the case of the viscosity parameter α = 0.1, we can derive the fitting formula for the neutrino luminosity as a function of the BH mass and spin and the mass accretion rate, which is expressed as
    
        log L_ν (erg s^-1) =    52.80-0.03m_BH+1.01a_*
           +1.08logṁ,
    
    where m_BH=M_BH/M_⊙ and ṁ=Ṁ/M_⊙  s^-1 are the dimensionless BH mass and accretion rate, respectively. It should be noticed that the ignition and trapping accretion rates are roughly ∼ 0.001 and ∼ 5  M_⊙  s^-1, respectively, which also be significantly affected by the BH mass and spin and disk viscosity parameter <cit.>. But ∼ 0.01 M_⊙  s^-1 can be considered as the minimum accretion rate for detectable neutrino radiation from BH-NDAF systems in the center of massive collapsars <cit.>.
    
    
    
    Here, we estimate the GWs from anisotropic neutrino emission, which is firstly proposed by <cit.>. We follow the formalism developed by
    previous works <cit.> to calculate the neutrino-induced GW signals. For a geometrically thin disk model for NDAFs, the local energy flux of GWs can be written as (for details, see <cit.>)
    dE_GW/dA   =G/36π c^5D^2(1+2cosθ)^2tan^4(θ/2)
       ×∫_-∞^∞dt L_ν(t)^2,
    
    where dA=D^2dΩ is the surface element, D is the distance between the observer and the source, θ is the viewing angle. Here, θ=π/2 corresponds to the case that the observer is located in the equatorial plane. Then, integrating over a sphere surrounding the source, one can calculate the total energy E_GW. For θ=π/2, the E_GW is calculated as
    
    E_GW=β G/9c^5∫_-∞^∞dt L_ν(t)^2,
    
    where β∼ 0.47039. In order to calculate GW spectrum, one can write L_ν(t) in terms of the inverse Fourier transform as
    
    L_ν(t)=∫_-∞^+∞L̃_ν(f)e^-2π iftdf;
    
    then, the GW energy spectrum can be deduced as
    dE_GW(f)/df=2β G/9c^5 |  L̃_ν(f) |^2.
    
    In this work, we assume the orientation of disks is random and calculate the angle average GW energy spectrum per NDAF as
    
    <dE_GW(f)/df>=∑_i∫_△θ _i dθ/∫_0^π/2 dθdE_GW,i(f)/df
    where △θ_i is the angle range of angle bin i, and dE_GW,i(f)/df is the observed GW energy spectrum at the corresponding viewing angle. In the subsequent calculations, we all adopt angle-averaged GW energy spectra.
    
    
    
    The effects of the progenitor mass and metallicity on the GW energy spectrum of NDAFs are displayed in Figure 1. Panels (a), (b), and (c) correspond to the cases of progenitors with metallicity of Z/Z_⊙=0, 0.01, and 1, respectively. The initial explosion energy is 2 B. In Panels (a) and (c), we only show GW energy spectra of NDAFs from progenitors with a mass greater than 30 M_⊙. In our simulations, for progenitors with metallicities of Z/Z_⊙=0 and 1, the final remnant of progenitors with a mass less than 30M_⊙ are NSs rather than BHs. The GW amplitude is mainly determined by the neutrino luminosity, which is related to the mass accretion rate. Most of the fallback and neutrino emission come from the core of the star. Therefore, the GW emission of the NDAF is determined by the compactness of the pre-SN star core. Some previous works <cit.> studied the dependences of progenitor masses and metallicities on the structural characteristics of pre-SN stars and found a non-monotonic behavior for compactness as a function of progenitor mass and metallicity. As a result, for the same initial explosion energy, the GW emission of NDAFs is not strictly dependent on progenitor mass and metallicity. Besides, previous stellar evolutionary studies <cit.> found that the core compactness parameters of solar metallicity stars are commonly smaller than those of low metallicity stars. Thus, solar metallicity is unfavorable for neutrino emission and GW emission of NDAFs. Moreover, although the accretion rate is still large for massive progenitors, the central BH mass is more massive which will influence the neutrino luminosity of NDAFs <cit.>.
    
    Figure 2 shows the effects of the initial explosion energy on the GW energy spectra of NDAFs. The metallicity is set to Z/Z_⊙=0.01. The black, red, and blue curves correspond to the initial explosion energies of 2, 4, and 8 B, respectively. For different progenitor stars, the amplitudes of the spectral lines increase with decreasing initial explosion energy because the weaker explosion energy corresponds to a more powerful fallback and neutrino emission. In Figures 2(a) and 2(b), the case of 8 B is not shown because the final remnants of these progenitors are both NSs.
    
    
    
    § SGWB FROM NDAFS
    
    
    A prediction of the SGWB requires a good understanding of the average GW energy spectrum and the event rate of NDAFs. The properties of progenitors and the initial explosion energy would determine whether the fallback process produces NDAFs and GW emission of NDAFs. Here, we investigate the effects of metallicities and initial explosion energies of progenitors on SGWB from NDAFs.
    
    
    
     §.§ Cosmic NDAF history
    
    
    The progenitor stars of CCSNe have relatively short lifetimes (≲ 10^8 years) compared to cosmic timescales <cit.>. Thus, the rate of NDAFs can be calculated using the star formation rate and IMF. The cosmic NDAF rate at a redshift of z is calculated as
    
    R_NDAF (z)=R_SFR(z)∫_M_min^M_maxΨ (M)dM/∫_0.1^125 MΨ (M)dM,
    
    where R_SFR(z) is the cosmic star formation rate in units of Mpc^-3  yr^-1, which can be deduced from observations <cit.>. Here, we adopt the continuous broken power law description by <cit.>,
    
    R_SFR (z)= ρ̇ _0 [ (1+z)^αη +(1+z/C )^βη + (1+z/D )^γη ]^1/η,
    
    where α =3.4, β =-0.3, γ =-2, η =-10, C≃ 5100, D≃ 14, and ρ̇ _0=0.014  Mpc^-3  yr^-1. Here, we adopt a Salpeter IMF <cit.> with Ψ (M)∝ M^ζ with ζ  =-2.35 in the mass range of 0.1-125 M_⊙, but explore a liberal range from -2.15 to -2.45<cit.> in our final calculations. M_max and M_min are the maximum and minimum masses of progenitors that produce BH-NDAF systems, respectively. Due to the influence of the metallicity of the progenitor and initial explosion energy on the fallback process of CCSNe, not all massive stars can produce NDAFs. Especially, M_min depends on the metallicity of progenitor and initial explosion energy <cit.>. According to the results of our simulations, we gave M_min for different metallicity and initial explosion energy, and the results are listed in Table <ref>. Of course, we also checked that the fallback accretion rates of these progenitors are suitable for NDAFs until they decrease lower than the ignition accretion rates. The SGWB prediction depends weakly on the M_max and we set M_max∼ 50 M_⊙, which is the approximately upper limit of the progenitor mass for NDAFs because the new-born BH mass is too larger to effectively ignite NDAFs <cit.>.
    
    
    
     §.§ IMF-weighted average GW energy spectra
    
    
    The average GW energy spectrum for a population of progenitors is computed by weighting each progenitor by
    dE(f)/df =∑_i∫_△ M _iΨ (M)dM/∫_M_min^M_maxΨ (M)dM <dE_GW(f)/df>_i,
    
    where Ψ (M) is once again the IMF, △ M _i is the mass range of mass bin i, and <dE_GW(f)/df>_i is the angle-averaged GW energy spectrum of the NDAF with progenitor mass of M_i.
    
    
    
     §.§ Estimations of SGWB
    
    
    It is usually to characterize the SGWB by the energy density parameter <cit.>, i.e.,
    Ω_GW(f)=1/ρ_cdρ _GW/d ln f,
    
    where ρ_GW is the gravitational energy density and ρ_c=3H_0^2/8π G is the critical energy density needed to close
    the universe. Then, following <cit.>, the sum of the energy densities radiated by a large number of independent NDAFs at each redshift is given as
    Ω_GW(f)   =1/ρ_cc^2
       ×∫_0^∞dzR_NDAF(z)/1+z| dt/dz|f(z)dE(f_z)/df,
    
    where f_z≡ f(1+z) and |dt/dz  |=[H_0(1+z)√(Ω_Λ+Ω _m(1+z)^3)]^-1. Here, we ignore the anisotropy of GWs from NDAFs and adopt the cosmological parameters  Ω _m=0.3, Ω_Λ=0.7, and H_0= 100 h_0 km s^-1 Mpc^-1 with h_0=0.72.
    
    Figure 3 shows the effect of the metallicity of the progenitor on the spectrum of SGWB from NDAFs. The initial explosion energy of all progenitors is set as 2 B. The black, red, and blue curves correspond to Z/Z_⊙=0, 0.01, and 1, respectively. The gray dashed lines represent the sensitivity curves of DECIGO <cit.>, BBO <cit.>, and the ultimate-DECIGO <cit.>. Solar metallicity is not beneficial for the detection of the SGWB from NDAFs. This is because the solar metallicity is unfavorable for neutrino emission and GW emission of NDAFs. Moreover, if the solar metallicity is universal, the event rate of NDAFs would decrease. The horizontal orange dashed line shows the SGWB generated in the inflationary epoch. In low-frequency region (∼10^-3-10^-1 Hz), our estimate for the SGWB from NDAFs can be comparable to the most optimistic SGWB from slow-roll inflation.
    
    In Figure 4, we display the effects of the initial explosion energy on the spectrum of SGWB. We assume all progenitors have the same metallicity, Z/Z_⊙=0.01. The black, red, and blue curves correspond to the initial explosion energy of 2, 4, and 8 B, respectively. The weaker explosion energy corresponds to a more powerful fallback, which enhances the GW emission and the event rate of NDAFs. As a result, the initial explosion energy will significantly affect the detection of the SGWB from NDAFs.
    
    
    
    In Figure 5, we explore the effect of IMF on the spectrum of SGWB from NDAFs. The black curve corresponds to the Salpeter IMF with ζ =-2.35. The red and blue curves correspond to ζ =-2.15 and ζ =-2.45, respectively. A shallower IMF (i.e., ζ =-2.15) is beneficial for the detection of SGWB from NDAFs. The SGWB differs by almost a factor of 3 between the case of ζ =-2.15 and the case of ζ =-2.45. As shown in Figure 1, for the same progentior metallicity and initial explosion energy, relatively higher mass stars correspond to the higher accretion rate, then the more powerful GW emission of NDAFs until the effects of the central BH mass dominate the ignition of NDAFs. Shallower IMFs have more high-mass stars which contribute to the SGWB.
    
    
    
    § CONCLUSIONS AND DISCUSSION
    
    
    In this paper, we estimate the SGWB produced by NDAFs based on fallback CCSN simulations. The effects of progenitor properties and initial explosion energies on SGWB from NDAFs are studied. These factors can affect the GW energy spectra and event rates of NDAFs. The GW emission of NDAFs is determined by the fallback process in CCSNe. Lower initial explosion energy is beneficial for producing a more powerful fallback, resulting in stronger neutrino emission and GW emission of NDAFs. Moreover, low initial explosion energy may enhance the event rates of NDAFs. Therefore, if low initial explosion energy is universal CCSNe, even for failed CCSNe, the SGWB from NDAFs may be detected by interferometers such as DECIGO and BBO. The influence of metallicity on SGWB from NDAFs is not monotonic because the compactness of the pre-SN star core is not strictly dependent on metallicity. However, solar metallicity is not beneficial for the detection of SGWB. Another uncertainty in our models is the IMF. A shallower IMF would increase the amplitude of SGWB from NDAFs. Therefore, the SGWB could be a valuable tool to investigate the IMF.
    
    In NDAF models, we adopt α=0.1 as a typical viscosity parameter. Actually, a disk with low α is denser and has a higher neutrino luminosity <cit.>. Thus, the GW emission of NDAFs with low α might be more powerful. Besides, in our calculations, we do not consider the disk outflows, which plays an important role in critical accretion systems <cit.>. The neutrino luminosity of NDAFs with strong disk outflows would be at least one order of magnitude lower than that of NDAFs without outflows <cit.>. As a result, powerful disk outflow will decrease the GW emission of NDAFs. If powerful disk outflows are universal in NDAFs, the amplitude of SGWB from NDAFs would decrease.
    
    In summary, the uncertainties of the SGWB from NDAFs are large. The detection of the corresponding neutrino background will lead to an improved prediction of the SGWB from NDAFs. In <cit.>, we investigated the diffuse NDAF neutrino background (DNNB) in detail. Metallicity and initial explosion energy have similar effects on both DNNB and SGWB from NDAFs. For the optimistic cases where the typical initial explosion energy is low, the DNNB might be detected by the upcoming larger neutrino detectors such as Hyper-Kamiokande, Jiangmen Underground Neutrino Observatory (JUNO), and Deep Underground Neutrino Experiment (DUNE). The detection of DNNB would help constrain the average neutrino emission and event rate of NDAFs. By combining the DNNB and SGWB, we would better understand CCSNe and NDAFs.
    
    It is noted that the SGWB from NDAFs might mask the SGWB generated in the inflationary epoch at low frequency. Besides, the SGWB from cosmological CCSNe and Pop III stars are expected to mask the inflationary GWs in some range of frequencies. These astrophysical foreground sources could be a significant problem for searches of the inflationary SGWB. It is necessary to disentangle these foreground sources from the inflationary SGWB in future detection.
    
    
    We thank Prof. Alexander Heger for providing us pre-SN data. We also thank Xing-Jiang Zhu for helpful discussion. This work was supported by the National Key R&D Program of China (Grant No. 2023YFA1607902), the National Natural Science Foundation of China (Grant Nos. 12173031, 12221003, and 12303049), the Postdoctoral Fellowship Program of China Postdoctoral Science Foundation (Grant No. GZC20231424), and the science research grants from the China Manned Space Project.
    
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 Paper Title: Cosmologies in $f(R,\mathcal{L}_m)$ theory with non-minimal coupling between geometry and matter.























theoremTheorem[section]
lemma[theorem]Lemma
proposition[theorem]Proposition
corollary[theorem]Corollary
remark[theorem]Remark
proof[1][Proof.]
      
  #1
     definition[1][Definition]
     
  #1
     example[1][Example]
     
  #1
abbrvnat

































sergiobravom@javeriana.edu.co

Departamento de Física,
 Pontificia Universidad Javeriana, Cra.7
No.40-62, Bogotá, Colombia


marek.nowakowski@ictp-saifr.org

 ICTP-South American Institute for Fundamental Research,
Rua Dr. Bento Teobaldo Ferraz 271, 01140-070 São Paulo, SP Brazil


contact@rvlobato.com

  ICRANet, Piazza della Repubblica 10, Pescara, 65122, Italy



davide.batic@ku.ac.ae


Department of Mathematics, 

Khalifa University of Science and Technology,
Sas Al Nakhl Campus,
P.O. Box 2533 Abu Dhabi,
United Arab Emirates



             

§ ABSTRACT 
Among the recent extensions to standard General Relativity, f(R,ℒ_m) gravity has risen an interest given the possibility of coupling between geometry and matter.
We examine the simplest model with non-minimal coupling in the context
of cosmology. We pay special attention to the question of how far this
model could reproduce the observational fact of our universe.



 
                             

                              
Cosmologies in f(R,ℒ_m) theory with non-minimal coupling between geometry and matter
    Davide Batic
    August 5, 2024
====================================================================================





§ INTRODUCTION

Since the discovery of the accelerated expansion of the universe, cosmologists have developed the Λ CDM model, a standard cosmological model rooted in Einstein's general relativity <cit.>. This model incorporates a positive cosmological constant Λ <cit.>, the Friedmann-Robertson-Walker metric <cit.>, and the concept of Dark Matter (DM) <cit.>. The otherwise robust model is not completely without problems.
To start with, despite numerous efforts, a candidate for DM has not yet been found. The
second problem is coined as Hubble tension, careful phrasing of the fact that different measurements of the Hubble constant yield different results <cit.>. In the future, as we seek to explain these issues, we may require a new theoretical model. This could involve modifying Einstein's gravity <cit.>, replacing the cosmological constant with different models of Dark Energy (DE) <cit.>, and exploring alternative candidates for DM <cit.>. The number of extensions of general relativity as well as the number of DE models is, of course, quite large <cit.>.

Despite the remarkable success of the Λ Cold Dark Matter (ΛCDM) model in explaining a wide array of cosmological observations, from the cosmic microwave background (CMB) anisotropies to the large-scale structure of the universe, several significant challenges remain unresolved. The nature of dark energy, epitomized by the cosmological constant Λ, poses a profound theoretical conundrum, known as the "cosmological constant problem" <cit.>. The astonishingly small value of Λ, as required to explain the observed acceleration of the universe, stands in stark contrast to theoretical predictions from quantum field theory, typically larger by many orders of magnitude <cit.>.

Moreover, the ΛCDM model does not escape from the Hubble tension, a growing discrepancy between the values of the Hubble constant, H_0, measured directly from local astronomical observations and those inferred from the CMB under the ΛCDM framework <cit.>. These inconsistencies point to potential physics beyond the standard model, either in the form of new particles or fields, or through modifications to the theory of general relativity itself.

In response to these challenges, a plethora of modified gravity theories have been proposed as alternatives to general relativity, aiming to provide a more comprehensive theoretical framework that can naturally incorporate the phenomena attributed to dark energy and dark matter. Theories such as f(R) gravity, where the Ricci scalar R in the Einstein-Hilbert action is replaced by a function f(R), offer promising ways to explain cosmic acceleration without the need for a cosmological constant <cit.>.

Expanding further, f(R,T) theories introduce a coupling between matter, represented by the trace of the energy-momentum tensor T, and geometry, providing a framework to explore the effects of such couplings on the dynamics of the universe <cit.>. The f(R,ℒ_m) models we focus on in this paper extend this idea by incorporating the matter Lagrangian ℒ_m directly into the gravitational action, potentially offering new insights into the interaction between dark matter and dark energy, and their impact on the evolution of cosmic structures <cit.>.

Our motivation for selecting f(R,ℒ_m) gravity stems from its ability to unite the geometrical and material sectors of the universe in a single, coherent theoretical framework. This approach not only allows for the exploration of the cosmic acceleration and dark matter problems from a new angle but also provides a platform for testing the limits of Einstein's general relativity on cosmological scales. By investigating the cosmological implications of f(R,ℒ_m) gravity, we aim to contribute to the ongoing discussion on viable alternatives to ΛCDM, exploring whether these theories can offer a more satisfactory explanation of observational phenomena without some of the fine-tuning issues that plague the standard model.

It is therefore a priori not clear
which class of models is theoretically preferred over the others.
We think that simplicity of the modification or, in other words,
Occam's razor, could serve us here as guiding principle.
Within a range of models, this possibility seems quite likely. Among the wide class of models based on a Lagrangian function of the type f(R), where R is the Ricci scalar <cit.>, one could choose to focus on R^2 gravity <cit.> and the resulting cosmological models <cit.>. Recently, another class has been examined, based on
f(R, ℒ_m) where ℒ_m <cit.>
represents the matter
Lagrangian, often taken to be the energy density ρ, pressure p, or the trace of the
energy-momentum tensor T-with all these quantities being diffeomorphically invariant. In this class, an additional term in the Einstein-Hilbert Lagrangian of the form σ Rℒ_m would be considered a mild extension of the previous model. Moreover, it provides a straightforward coupling of the geometry encoded in R with matter represented
astrophysically by ℒ_m.  The
astrophysical and other implications of such a theory have been examined in
<cit.> with consequences for Neutron star and
White Dwarf physics. Some cosmological aspects with specific choices
of ℒ_m have been considered in <cit.>.
Except for <cit.>, the choice of the function
f(R, L_m) is different from ours. The
f(R,ℒ_m) theories have their own theoretical
peculiarities that require careful examination. One notable aspect is
that choosing
ℒ_m=±ρ, ± p, T does not
lead, through standard metric variation to the energy-momentum tensor of a perfect fluid T_μν <cit.>. On the other hand, a constrained variation
based on the Lagrangian with constraints can result in
the prefect fluid energy-momentum tensor. However, this comes at the cost of  T_μν no longer being conserved.  We show that explicitly using
the new Friedmann equation in f(R, ℒ_m).


In solving the new Friedmann equations, we try to make the model resemble our present universe. While it seems possible to achieve this by choosing adequate initial values, the evolution into the past and future can bring some surprises, as will be shown below.  One of the simplest and model independent feature of our
universe is the lower limit on its age as given for example by the
redshift of the oldest galaxies <cit.>.
A second constraint independent of the model comes from uranium decays  <cit.>.
If a cosmological model, as
appealing as it may be, cannot
reproduce these facts, it is then certainly not a good candidate to
describe our universe. We will pay a special attention to these
questions while examining the details of the model under discussion.

The paper is organized as follows. In Section II, we outline the basics
of f(R,ℒ_m) theory with minimal coupling, denoted as σ
Rℒ_m. In this section, we also present the new Friedmann equations with a cosmological constant in a dimensionless form suitable for numerical integration.
In Section III, we present the generalization of one of the
Friedmann equations. In its standard form, it reads
Ω_Λ,0 +Ω_m,0=1. However, in the context of f(R,ℒ_m) theory, it generalizes to
Ω_Λ,0 +Ω_m,0+ Ω_σ, 0=1. In the same section, we discuss the possible initial values, a necessary undertaking, since the new Friedmann equation contains the second derivative of the density. We believe it is necessary to provide a brief overview of the standard cosmological model in its analytical form (see Section IV). This is important because in Section V, we will present the numerical results and compare the new model with the standard one. In Section VI, we will draw our conclusions.





§ THE F(R,ℒ_M) THEORY

As a generalization of the standard Einstein-Hilbert Lagrangian in General Relativity, i.e.

    ℒ_GR=1/2κR+ℒ_m,

where κ=8π G_N, a new Lagrangian has been proposed by
making use of the invariants R and ℒ_m.  It makes use
of a general function, f(R,ℒ_m), and reads simply
with an action <cit.>

    S=∫ d^4x √(-g) f(R,ℒ_m).

From the formal variation of the action (δ_gS=0) with respect
to the metric, the following field equations are obtained <cit.>

    f_RR_μν + (g_μν-∇_μ∇_ν)f_R-1/2(f-f_L_mℒ_m)g_μν=1/2f_L_mT_μν,

where f_R=∂ f/∂ R, f_L_m=∂
  f/∂ L_m and the energy-momentum tensor T_μν is
given by

    T_μν=-2/√(-g)δ(√(-g)ℒ_m)/δ
      g^μν.

An alternative form of these field equations may be obtained by taking the trace of (<ref>) and solving for f_R, namely

    f_R=1/6f_L_mT+2/3(f-f_L_mL_m)-1/3f_RR.

When replacing it back into (<ref>), we obtain the equation

    f_RG_μν+1/6(f+f_RR - f_L_mL_m)g_μν-1/2f_L_m( T_μν-1/3T g_μν)-∇_μ∇_νf_R=0.

Here, G_μν=R_μν-1/2g_μνR represents the Einstein
tensor. As mentioned in the Introduction, the particular form of
f(R,ℒ_m) we wish to work with is the same as the one presented in <cit.>

    f(R,ℒ_m)=R/2κ+ℒ_m+σ Rℒ_m,

where σ is the parameter which determines the coupling between
matter and geometry.

With respect to the choice of the matter Lagrangian ℒ_m
that reproduces the perfect fluid Energy-Momentum tensor there is a
current debate <cit.>. While some authors make the
choice ℒ_m=±ρ, others suggest ℒ_m=± P <cit.> with particular values for the polytropic index. With the choice in <cit.>, i.e. ℒ_m=-P, the field equations read

    (1/2κ-σ P)R_μν-σ[g_μν-∇_μ∇_ν]P-1/4κR g_μν=1/2( 1+σ R)T_μν,

or in the alternative form












    (1-2κσ P) G_μν+1/3Rg_μν-κσ P/3Rg_μν-κ (1+σ R) (T_μν-1/3Tg_μν)+2κσ∇_μ∇_νP=0,

which is consistent with the equations displayed above.  The conservation
of the energy-momentum tensor is given in general by <cit.>

    ∇^μT_μν=2∇^μln[f_L_m(R,ℒ_m) ]∂ℒ_m/∂ g^μν.

This suggests that choosing the matter Lagrangian to be proportional to
ρ or P leads to the conservation of the energy-momentum
tensor. However, this is only the case when relying directly on
equation (<ref>). Unfortunately, given the choices of the
matter Lagrangian as mentioned above, it does not reproduce the
perfect fluid energy-momentum tensor. To illustrate this point, if we
start with ℒ_m=±ρ, the energy-momentum tensor
defined in (<ref>) will not contain the pressure term.
On the other hand, by using a constrained variation in the metric <cit.>  or alternatively, a
constrained Lagrangian formalism, we can explicitly show that, for example, choosing the energy density as the matter Lagrangian yields a perfect fluid energy-momentum tensor. However, its conservation no longer holds, as we introduce the metric through the constraints.

There is a certain confusion on this issue in the literature
<cit.>. This is a point of divergence between our approach and others. Therefore, we explicitly demonstrate the non-conservation of T_μν. The
equations of motion, as presented in
(<ref>), (<ref>), and (<ref>), remain valid when using the
constrained formalism. In this formalism, T_μν is the
energy-momentum tensor of perfect fluid, which, in general, is not conserved and is defined as

    T_μν=(ρ+P)u_μu_ν-Pg_μν.

As mentioned earlier, the model under discussion displays a
non-conservation of the energy-momentum tensor, specifically ∇_μT^μν≠ 0 (at least in general). This
non-conservation is also a characteristic of other gravity models such as Rastall Gravity <cit.>, and has been advocated in modular gravity models <cit.>.
From the field equations (<ref>) we can
obtain the modified Friedmann equations in the flat (k=0)
Friedmann–Lemaitre–Robertson–Walker metric

    ds^2=g_μνdx^μdx^ν=dt^2-a^2(t)(dr^2+r^2dθ^2+r^2sin^2θ dϕ^2).

By taking the 0-0 components of the new field equations, one obtains <cit.>

    3H^2f_R+1/2(f-f_RR-f_L_mℒ_m)+3Hḟ_R=1/2f_L_mρ,

where H is the Hubble parameter given by H=ȧ/a.
The space components (i.e. μ=i, ν=j) give

    Ḣf_R+3H^2f_R-f̈_R-3Hḟ_R+1/2(f_L_mℒ_m-f)=1/2f_L_mP,

where the f̈_R term comes from the double derivatives in the field equations ( and ∇_μ∇_ν). For the model we have chosen, we have

    f_R=1/2κ+σℒ_m,      f_L_m=1+σ R,     ḟ_R=σρ̇,     f̈_R=σρ̈.

Together with the choice ℒ_m=ρ, the new Friedmann equations read







    H^2   =   κρ/3+σκ[ 4Ḣρ+6H^2ρ-2Hρ̇],
    ä/a   =   -κ/6(ρ + 3P )-σκ [ Ḣ(3P+ρ)+6H^2P +ρ̈+2Hρ̇].

In general relativity, the two Friedmann equations lead to the
conservation law ρ̇+3H(ρ+P)= 0. It is known that the
same conservation law follows from ∇^μT_μν=0
<cit.>.  It is straightforward to derive from (<ref>)
and (<ref>) the following relation

    ρ̇+3H(ρ+P)=-6σ[HḢ(3P+11ρ)+ρ̇(3H^2+Ḣ)+6H^3(P+ρ) +2Ḧρ],

which clearly gives back the conservation law (and therefore, the
conservation of the energy-momentum tensor) in the case
σ=0. The nature of the non-conservation of the energy-momentum
tensor is explicitly proved here. It should have been addressed also in
<cit.> where the authors
consider cosmological models of f(R, L_m) theories.

Commonly, equations (<ref>) and (<ref>) are known as (modified)
Friedmann equations. In some papers <cit.> equation (<ref>) is also
called Raychaudhuri equation. Both equations are related by (<ref>).

In rewriting the Friedmann equations in a dimensionless form, specific choices were made to align the theoretical framework with observational benchmarks and simplify the numerical analysis. By introducing χ = ρ/ρ_ crit, we scale the density to the critical density, which is pivotal in determining the curvature of the universe. Similarly, scaling the Hubble parameter by its current value H_0 through h^2 = H^2/H_0^2 allows us to examine the evolution of the expansion rate relative to its present value. The introduction of σ̃ = H_0^2 σ helps in assessing the impact of the coupling constant σ within the typical energy scales of the current universe. Lastly, by using η = H_0 t, time derivatives are normalized to the current expansion rate, making them dimensionless and easier to handle numerically. These transformations are not only mathematically convenient but also provide a direct link between theoretical predictions and observable quantities. To this end, we introduce

    χ≡ρ/ρ_ crit =κρ/3H_0^2,      h^2≡H^2/H_0^2,
        σ̃≡ H_0^2σ,    η=H_0 t,
        ρ_crit=3H_0^2/κ,

where ρ_0 is a reference value of ρ at the present epoch.
Similar comment applies to the Hubble parameter H_0. In addition, we
write the time derivatives in terms of derivatives with respect to the
dimensionless parameter η by means of

    ∘A≡dA/dη=dA/dtdt/dη=Ȧ1/H_0.


In the context of our model, the barotropic index γ
characterizes the equation of state (EOS) of the cosmological fluid,
namely P = (γ - 1) ρ. This parameter is crucial for
defining the thermodynamic properties of the universe's contents,
where γ = 1 corresponds to a universe dominated by
non-relativistic matter (the choice we make here for the calculations), γ = 4/3 to one dominated by radiation, and γ = 0 to the cosmological constant scenario with vacuum energy. Each value of γ distinctly affects the evolution dynamics of the universe, as reflected in the modified Friedmann equations. By means of the aforementioned (EOS), equations
(<ref>), (<ref>) and (<ref>) take the form

    h^2   =   χ + σ̃[ 18h^2χ + 12∘hχ-6h∘χ],
    
    h^2+∘h   =   -1/2(3γ-2)χ-σ̃[3(3γ-2)∘hχ+18(γ-1)h^2χ+3∘∘χ+6h∘χ],
    
    3∘χ+9γ hχ   =    -6σ̃[(9γ+24)h∘hχ+3∘χ(3h^2+∘h)+18γ h^3χ+6∘∘hχ].

Since we want to compare the modified cosmological model with standard
cosmology  we should also include the cosmological constant
Λ. This can be done without much effort by considering the action

    S_f(R,ℒ_m)+Λ=∫ d^4x √(-g)(f_Λ(R,ℒ_m)-Λ/κ)

with

    f_Λ(R,ℒ_m)=R/2κ+ℒ_m+σ Rℒ_m.

This gives the new Einstein equations

    f_RR_μν+(g_μν -∇_μ∇_ν)f_R-1/2(f-f_L_mL_m)g_μν+Λ/2κg_μν=1/2f_L_mT_μν

as well as the generalized Friedmann equations

    3H^2f_R+1/2(f-f_L_mℒ_m-f_RR)+3Hḟ_R-Λ/2κ   =   1/2f_L_mρ,
    Ḣf_R+3H^2f_R-f̈_R-3Hḟ_R+1/2(f_L_mℒ_m-f)+Λ/2κ   =   1/2f_L_mP.

With the explicit choice of f_Λ the modified Friedmann equations
simplify to

    H^2   =   κρ/3+σκ[ 4Ḣρ+6H^2ρ-2Hρ̇] + Λ/3,
    ä/a   =   -κ/6(ρ + 3P )-σκ [ Ḣ(3P+ρ)+6H^2P +ρ̈+2Hρ̇]+Λ/3.

Their dimensionless form can be written by using the dimensionless parameter Ω_Λ=ρ_ vac/ρ_ crit as follows

    h^2   =   χ + σ̃[ 18h^2χ + 12∘hχ-6h∘χ]+Ω_Λ,
    
    h^2+∘h   =   -1/2(3γ-2)χ-σ̃[3(3γ-2)∘hχ+18(γ-1)h^2χ+3∘∘χ+6h∘χ]+Ω_Λ.

These equations are now suitable for numerical integration.

Normally, the inclusion of Λ in a modified gravity theory
might not be well justified as the modification itself can
account
for Dark Energy. But it appears that this is a too global statement
and the role of Λ can vary from model to model. In the
section on numerical results we will demonstrate that the cosmological
constant does not affect the characteristic feature of the model,
i.e., its shortcoming to produce a reasonable lifetime of the universe.



§ THE INITIAL VALUES AND THE ∑_I Ω_I=1 RELATION

The Friedman equations are now first order in h and second order in
χ. Therefore, we need three initial values, which we provide at the time t_0 corresponding to the present epoch. We have

    h_0≡ h(t_0)=H_0/H_0=1,   χ_0≡χ (t_0)=κρ_0/3H_0^2,   ∘h_0≡∘h(t_0)=Ḣ_0/H^2_0,   ∘χ_0≡∘χ(t_0)=κρ̇_0/3 H_0^3.

We look for model independent measurements for the density of the Universe (ρ_0), the Hubble parameter (H_0) and its first and second derivatives (Ḣ_0, Ḧ_0). For Ḣ we use the deceleration parameter q, given by <cit.>

    q≡-äa/ȧ^2,   →  Ḣ/H^2=-(1+q)

and thus,

    Ḣ_0=-H_0^2(1+q_0),

where q_0=q(t=t_0) has been obtained in a model independent way
<cit.>. The jerk parameter will be useful for
Ḧ_0 appearing in the second modified Friedmann equation. This is given by <cit.>

    j=⃛a/aH^3→ j=Ḧ/H^3+3Ḣ/H^2+1

and therefore,

    Ḧ_0=(j_0-1)H_0^3-3Ḣ_0H_0=(j_0+q_0+2)H_0^3.

To ensure that our exploration of the modified f(R, ℒ_m) gravity model remains grounded in observational reality, we adopt standard cosmological parameters that are widely accepted within the community. This choice allows us to rigorously test whether the modifications introduced by the theoretical framework can provide a viable alternative to the ΛCDM model, particularly in light of recent tensions and discrepancies such as those in measurements of the Hubble constant. By using these parameters, the model's predictions can be directly compared with those derived from both local and cosmological scales, providing a comprehensive evaluation of its empirical adequacy. For the numerical values, it is important to note that due to recent
discrepancies in measurements of the Hubble constant, known as the
Hubble Tension <cit.>, there are several values for H_0. For example we have
H^+_0= 73,52±1,62 km/s/Mpc and H^-_0= 67,4 ±0,5
km/s/Mpc, respectively. As for the current energy density
of the Universe and the deceleration parameter, we use values found in
<cit.>:
χ_0=ρ_0/ρ_ crit=0.285±0.012, q_0 =
-0.545 ± 0.107, j_0=1.30± 0.37 with the value for
the Hubble parameter taken as H_0=71.34±1.74 km s^-1
Mpc^-1. It is worth noting that these are specific parameter
choices, but other sets of parameters are available in
<cit.>. Possible values for these parameters are
displayed in Table <ref>. This particular choice of parameters that as of today we are as close as possible to the standard cosmology.

For a standard reference value of ∘χ_0, we consider the standard cosmological model and use the continuity equation (<ref>) with σ̃=0, even in the presence of Λ≠0. This leads to

    3∘χ+9γ h χ=0.

For the initial value we then obtain

    ∘χ_0=-3γ h_0χ_0=-3γχ_0.

Since h_0=1, in the case of dust γ=1, we have

    ∘χ_0=-3χ_0.

In mathematics, when solving differential equations, one typically
assumes that all parameters and functions entering the equation, as
well as the initial values are
known and given. However, in physics, the situation can become more
complex because these parameters and initial values may not be
precisely known. When considering a differential equation at the point
where the initial values are applied, it leads to a relationship
between
these initial values and
the parameters of the equation. The Friedmann equations provide a notable example of this phenomenon.

The first modified Friedmann equation can be expressed in terms of the density
parameters Ω_i. To this purpose, let us start by defining the Ω_m parameter as

    Ω_m=ρ/ρ_ crit=κρ/3H^2.

This, in turn, implies that χ=Ω_m and ρ=Ω_mρ_ crit=Ω3H^2/κ. Thus, our first Friedmann equation can be reformulates as follows

    H^2=Ω_m H^2+Ω_m H^2(18σ H^2)+Ω_m
    H^2 (12σḢ)-σκ H ρ̇ +Λ/3.

Recognizing this equation as

    Ω_m + Ω_Λ+ σ̃[18 Ω_m H^2+12 Ω_mḢ-κρ̇/H]=1,

we can express it in a more compact form. Finally, defining the term proportional to σ as Ω_σ, we arrive at

    Ω_m,0 +Ω_Λ, 0+Ω_σ,0=1,

where we have taken the values at t=t_0. Making use of (<ref>) and
 (<ref>) the above equation reduces to

    Ω_m,0 + Ω_Λ,0+σ̃Ω_m,0 (15-12 q_0)=1.

Equation (<ref>) establishes a relationship between initial
values and parameters within the modified Friedmann equation. Moreover, it
introduces the deceleration parameter q_0, which is a measured
quantity,
though not strictly
necessary for solving the first order modified Friedmann equation in H.
Consequently, there are generally two approaches to solve the new
Friedmann equations.  The first one involves specifying h_0=1, χ_0,
∘χ_0, and Ω_Λ while varying σ̃. Equation (<ref>) can then be used to determine q_0. The second approach relies on the measured value of q_0 and fixes the other parameters
as previously described. In this case, equation (<ref>) provides
the parameter σ̃. In particular, we note that in the case without coupling between R and ℒ_m, the relative densities today read

    Ω_m,0+Ω_Λ,0=1.


Numerically, however, there is an uncertainty in the obtained values,
reflected in the relation Ω_m,0+Ω_Λ,0=1±Δ, where Δ represents this uncertainty. We can exploit it in our model by setting, as shown in equation (<ref>),

    σ̃=±Δ/Ω_m,0(15-12q_0).














§ THE STANDARD COSMOLOGY

Before embarking on a discussion of the numerical solutions it is
convenient to briefly outline the analytical picture of the standard
cosmology. This makes sense since we will be comparing the two scenarios.


The dimensionless solution for h_Λ+ for the standard
cosmological model can be obtained by taking σ̃=0
in equations (<ref>) and (<ref>) with γ=1, namely

    h^2   =   χ +Ω_Λ,
    
    h^2+∘h   =   -1/2χ +Ω_Λ.

Combining the above equations, we arrive at a Riccati equation of the form

    ∘h=-3/2 h^2 + 3/2Ω_Λ.

We can solve the differential equation above by setting

    ∫_h'=1^h'=hdh'/Ω_Λ-(h')^2=3/2∫_η'= η_0^η'=η dη',

where η_0=H_0t_0 is the time at which the
Hubble parameter is H_0, which, in turn, is related to the present values of the parameters used. To solve the integral above, we redefine x=h'/√(Ω_Λ) and write

    1/√(Ω_Λ)∫_x=1/√(Ω_Λ)^x=h/√(Ω_Λ)dx/1-x^2=3/2(η-η_0).

One possible solution for the integral on the left-hand side is

    ∫dx/1-x^2=1/2ln(1+x/1-x).

However, such a solution is valid for -1<x<1. This would imply that h<√(Ω_Λ) and thus, χ=h^2-Ω_Λ<0, which would suggest an unphysical negative density. Another solution is

    ∫dx/1-x^2=1/2ln(x+1/x-1).

It is valid whenever |x|>1. Moreover, it implies h^2>Ω_Λ and, consequently χ=h^2-Ω_Λ>0, ensuring a positive and physically meaningful density. If we also use the fact that

    ^-1x=1/2ln(x+1/x-1)

and solve for the integral (<ref>), we end up with the result

    1/√(Ω_Λ)[^-1(h/√(Ω_Λ))-^-1(1/√(Ω_Λ)) ]=3/2(η-η_0).

Solving for h in the γ=1 case leads to

    h_Λ+(η)=√(Ω_Λ)[3/2√(Ω_Λ)(η-η_0)+^-1(1/√(Ω_Λ)) ].

For the general γ case, the equation above reads

    h_Λ+(η)=√(Ω_Λ)[3γ/2√(Ω_Λ)(η-η_0)+^-1(1/√(Ω_Λ)) ].

In the standard notation of the scale factor a_Λ(t) and the Hubble parameter H_Λ(t), we get

    a_Λ+(t)   =   (1-Ω_Λ/Ω_Λ)^1/3γsinh^2/3γ(μ t+ν),
    
    H_Λ+(t)   =   √(Λ/3)(μ t+ν),

where

    μ=γ√(3Λ)/2,  ν=1/2ln(1+√(Ω_Λ)/1-√(Ω_Λ))-μ t_0

for a positive cosmological constant Λ. For completeness, we
also give the solution for Λ<0 which reads

    a(t)_Λ-   =   a_0[cos(α-β t)/cos(α-β t_0)]^2/3γ,
    
    H_Λ-(t)   =   √(-Λ/3)tan(α-β t)

with

    α=β t_0+tan^-1(√(-3/Λ)H_0),  β=γ√(-3Λ)/2.


Without loss of generality we can set t_0=0. Then from (<ref>)
we can infer that 0 <α < π/2 and hence cos(α)>0. To
ensure that the scale factor is positive it suffices to choose t ∈
[α -π/2/β, α +π/2/β]
  corresponding to Big Bang and collapse.

We will explore later how this simple solutions evolves when we introduce a non-zero σ.

Two important model-independent restrictions on a realistic
cosmological model relate to the universe lifetime. In the
standard cosmological model, we can calculate the lifetime (neglecting
the short radiation period) by setting the argument of the Hubble
function to zero, which results in a singularity. Choosing η_0=0,
we obtain

    |η_univ|=
      1/3√(Ω_Λ)ln(1+√(Ω_Λ)/1- √(Ω_Λ)),

we get |η_univ| ≃ 0.96 for Ω≃ 0.7. This
lifetime is not in conflict with model-independent estimates,
such as those from the oldest galaxies <cit.>
(which indicate the existence of galaxies some 400 million years after
the Big Bang) and <cit.>,
which points to a lower limit of 12.5Gy.







§ NUMERICAL RESULTS

Assuming σ≠ 0, we have converted the system under discussion
into a first order system which we solved by standard numerical
methods. As an independent check, we used MAPLE18 routines and found that the
results coincide.

When solving the modified Friedmann equations numerically, the question arises:
What values should we take for σ̃? The standard
cosmology has two constants, the Newtonian one G_N and the
cosmological constant. Interestingly, the dimension of σ is the same as that of the Newtonian constant.  If we take σ as
nG_N with n some number of order 1, σ̃ would turn out extremely small. Therefore, it makes sense to initially try more moderate values for σ̃. In the subsequent figures, the blue curve represents the standard cosmological model prediction.


In Figure <ref>, we have chosen the initial values as discussed
in Section III, with Ω_Λ=0.7. The chosen values of
σ̃ are relatively small to illustrate that the
dimensionless Hubble function closely follows the one of the standard
cosmological model around η close to zero.
Several notable features can be observed in the plot, which are likely to persist for other values of σ̃. First of all, the Hubble function reaches the singularity much earlier than in the standard model. This poses a challenge if we aim to adapt the model under discussion as a viable cosmological model of our universe. As mentioned earlier, one of the constraints on any cosmological model is its ability to be roughly compatible with a lower limit on the lifetime of the universe. This constraint arises from evidence such as the existence of galaxies just 400 million years after the Big Bang <cit.>. Therefore, any cosmological model must strive to be in agreement with the estimated lifetime of the universe as predicted by the standard cosmological model.
Secondly, we notice another interesting feature. The Hubble function becomes negative which implies a contraction of the universe. Therefore, the dimensionless density χ starts increasing
again. This is known as the Big Crunch scenario which happens also
in f(R, L_m) models with the simple choice L_m=ρ. The Big Crunch here is a result of the model and cannot be avoided.




In Figure <ref>, we have plotted the same functions for two
values of σ̃ appearing also in figure
<ref>, but over a larger scale of time. One can appreciate the
steep rise of h at a relatively modest negative values of η.




In Figure <ref>, we have increased the values of
σ̃ but there is  no visible effect on the time of the
Big-Bang. However, for larger values of the parameter, the density becomes unphysically negative.




This is confirmed in Figure <ref> which shows the same functions,
but over a larger timescale. The differences between the models
become then visible at large η.





Increasing the values of σ̃ does not bring any new
insight, but for the first time there appears a cusp for
σ̃=5. MAPLE18 interpretes this as singularities. This is
demonstrated in Figure <ref>.



In Figure <ref>, we return to relative small values of
σ̃ to show that the cusp behavior is quite common in the
model. This is interesting from a purely theoretical point of view, but
leaves doubts about the viability of the model over large time scales.



Finally, we turn our attention the the behaviour of h and χ for
negative values of σ̃ which is depicted in Figures 
<ref>, <ref>, and <ref>.



In Figure <ref>, we encounter a universe whose contraction
slows down to turn into a expanding universe at η around
zero. Again many cusps accompany this behavior which could well mean
that universe undergoes a singularity there.



For small and very small values of negative σ̃ we have
again negative densities.

We have also explored a second approach for selecting the parameter
σ̃. More precisely, we utilized (<ref>)
and considered the error bars associated with measured values in standard cosmology.
Figure <ref> provides an example where we calculated
σ̃ with high precision, staying within the range of
uncertainty. This allowed us to examine how sensitive the model is to the choice of parameters. The lifetime of the universe comes out
approximately
|η_univ| ≃ 0.4, which is too small to account for the
existence of the first galaxies in our universe. The Big
Crunch appears to be unavoidable again.




In the third approach, we used equation (<ref>) while keeping
the standard value for Ω_m,0, but varying Ω_Λ
and q_0. We then solved the equation to find σ̃. An
example is illustrated in Figure <ref>.

This variation led to a slight increase in the universe lifetime. Motivated by this observation, we
explored scenarios with a negative cosmological constant.  Two such
examples are noteworthy. Setting q_0=-0.5 and Ω_Λ=-50 resulted in σ̃≃ 10 and a lifetime of |η_univ| ≃ 0.51. Alternatively, when choosing a  deceleration parameter of q_0=-1, we obtained  σ̃≃ 7.8 and a lifetime of  0.56. However, it appears challenging to significantly extend the model lifetime beyond these values.

In summary, this model represents a mild extension of
    Einstein gravity, particularly when viewed from the perspective of
    its Lagrangian. Surprisingly, the cosmology given by this model for a
spatially flat, homogenous universe is quite different from the
standard case based on Einstein equations.





§ CONCLUSIONS

Proposing a new gravity theory which goes beyond Einstein leads inevitably
to the examination of the new model in a cosmological context. In this context, theoretical advancements revealing new phenomena are equally important as empirical efforts to evaluate the model potential to supersede the standard cosmological framework.
We have emphasized both aspects studying the extension f(R, 
  L_m) theory with an explicit coupling between geometry and matter
in the form σ R ρ with ρ being the energy density. This
appears as a natural choice used also in astrophysical context. We
explored the model over a wide range of the coupling constant σ
with initial data mimicking our present universe.  Given that the model predicts a universe lifespan that is drastically shorter than the observed age of our universe (approximately 13.8 billion years), it fails to meet a fundamental criterion for a viable cosmological model. This discrepancy leads us to conclude that the model, in its current form, cannot adequately describe the observed universe. As such, the potential occurrence of a Big Crunch, while theoretically interesting, becomes a secondary concern because the model is already ruled out based on its inability to account for the observed age of the universe.

All in all, the findings highlight
challenges in adapting the model to realistically represent the
universe, particularly regarding the universe lifetime and the issue
of negative densities in some scenarios.

Before comparing our work with existent literature, let us point out
that once we derive the field equations from the Lagrangian principle,
the Lagrangian should contain a matter part L_m from which the
perfect energy-momentum tensor can be derived. The latter seems only
possible via a constrained variation with respect to the metric. The
conservation of the energy-momentum tensor in theories of the type
f(R, L_m) is not guaranteed and requires special attention. In
<cit.> the choice of the Lagrangian is f(R, L_m)=R/2+L_m
+σ RL_m with L_m=-p.  It is then difficult to study the dust
scenario with p=0 as it implies L_m=0. Furthermore, the
energy-momentum tensor is not conserved in this choice in contrast to
the authors' claim.  In <cit.> the choice is f(R,L_m)=R/2
+L_m^α with α being some number. It is not clear how we
can derive from that a hydrodynamical energy-momentum tensor and, as
explained above, whether we obtain a conservation law.  The choice of
the Lagrangian in <cit.> is also quite different from ours
as it lacks the direct coupling between geometry (represented by R)
and matter (represented by L_m). This shows that such coupling has
special consequences. In <cit.> the Lagrangian reads
f(R,L_m)=R/2 +α L_m^n - β with α and β constants. Remarks similar
to the case of <cit.> would apply also here.










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 Paper Title: General Relativistic Magneto-Hydrodynamic Simulations with BAM: Implementation and Code Comparison.
















^1 Institut für Physik und Astronomie, Universität Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany
^2 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany
^3 Theoretical Physics Institute, University of Jena, 07743 Jena, Germany
^4 Dipartimento di Fisica G. Occhialini, Università di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy

^5 INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
^6 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy
^7 CFisUC, Department of Physics, University of Coimbra, Rua Larga P-3004-516, Coimbra, Portugal
^8 GRAPPA, Anton Pannekoek Institute for Astronomy, Institute of High-Energy Physics, and Insitute of Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
^9 Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan



§ ABSTRACT 
Binary neutron star mergers are among the most energetic events in our Universe, with magnetic fields significantly impacting their dynamics, particularly after the merger. While numerical-relativity simulations that correctly describe the physics are essential to model their rich phenomenology, the inclusion of magnetic fields is crucial for realistic simulations. For this reason, we have extended the  code to enable general relativistic magneto-hydrodynamic (GRMHD) simulations employing a hyperbolic `divergence cleaning' scheme. We present a large set of standard GRMHD tests and compare the  code to other GRMHD codes, , , and , which employ different schemes for the evolution of the magnetic fields.  Overall, we find that the  code shows a good performance in simple special-relativistic tests. In addition, we find good agreement and consistent results when comparing GRMHD simulation results between  and . 


General Relativistic Magneto-Hydrodynamic Simulations with BAM: Implementation and Code Comparison
    Masaru Shibata^2,9
    August 5, 2024
==================================================================================================





§ INTRODUCTION



Magnetic fields play a crucial role in various astrophysical high-energy scenarios. In particular, they are important in binary neutron star (BNS) mergers because the magnetohydrodynamical effects can influence the lifetime of the merger remnant and shape the matter outflow in the post-merger phase. In the multi-messenger event GW170817 <cit.>, the first observation of a BNS system by gravitational waves (GWs) accompanied by electromagnetic (EM) counterparts including the gamma-ray burst (GRB) GRB170817A <cit.>, strong evidence has been found for BNS merger to be sources of GRBs triggered by a relativistic jet, e.g., <cit.>. The launch of such a relativistic jet is most likely powered by magneto-hydrodynamic processes, e.g., <cit.>. 

A key in the study of BNS systems are numerical-relativity (NR) simulations that solve Einstein's field equations together with magneto-hydrodynamics equations. Due to the high complexity and dynamics of the merger process, an accurate description of the spacetime and hydrodynamics is crucial to allow for realistic predictions and to enable us to study details of the physical processes. Furthermore, we can extract the GW signal and the matter outflow from NR simulations to inform GW models as well as models for the EM counterparts, which are fundamental for the correct interpretation of multi-messenger observations. Among others, this allows for studies on the behavior of matter at supranuclear densities, e.g., <cit.>, the expansion rate of the Universe, e.g., <cit.>, and the synthesis of heavy elements, e.g., <cit.>. In the last decade, there has been rapid progress towards more realistic representations of microphysics in NR simulations, such as using tabulated equations of state (EOS) based on nuclear physics calculations to describe the interior of neutron stars (NSs), including neutrino emission and transport, as well as including magnetic fields.

In this work, we focus on the latter, i.e., the implementation of general relativistic magneto-hydrodynamics (GRMHD) routines in our NR code  <cit.>, but also refer to previous works concentrating on employing tabulated EOSs <cit.> and on the implementation of an advanced multipolar first-order (M1) neutrino transfer scheme <cit.> to highlight recent progress in performing more accurate simulations with . Magnetic fields, although negligible for the inspiral, are amplified during and after the merger due to the Kelvin-Helmholz instability (KHI), e.g., <cit.>, the Rayleigh-Taylor instability, e.g., <cit.>, and the magneto-rotational instability (MRI), e.g., <cit.>. Accordingly, they influence in particular the evolution of the remnant system and the outflow of matter in the post-merger phase due to magnetically driven winds and jet formation, making their incorporation crucial for correct ejecta models and predictions of EM signals, e.g., <cit.>. At the same time, they significantly increase the complexity of the simulations and pose new numerical challenges. 

One challenge in GRMHD is to fulfill the divergence-free condition for the magnetic field. There are various approaches in the literature to ensure that no magnetic monopoles are formed. Among the most popular schemes used in the NR community nowadays are the constraint transport scheme as initially developed by <cit.>, e.g., <cit.>, vector potential methods, e.g. <cit.>, and divergence cleaning approaches, e.g.,<cit.>. Constraint transport uses the induction equation and Stokes theorem to integrate the magnetic field (see <cit.> for a detailed discussion of constraint transport methods). In vector potential methods, one exploits the fact that the magnetic field can be expressed by a vector potential, which is then evolved instead. For both methods, the divergence of the magnetic field is by construction zero at round-off accuracy. However, both methods require a staggered grid, as the magnetic field components are usually defined on the surface of a grid cell, which complicates implementation in codes using mesh refinement. Also, vector potential methods do not guarantee magnetic flux conservation across the refinement boundary when mesh refinement is employed. 

We apply in  the hyperbolic divergence cleaning approach following <cit.>. In this approach, a new field variable is introduced to damp and advect the divergence of the magnetic field. Hyperbolic divergence cleaning of this type originated in <cit.> and was introduced to NR in <cit.> for magnetized rigidly rotating NSs. <cit.> showed very promising results for standard tests of special relativistic MHD, while divergence cleaning encountered some preliminary issues for GRMHD, so their focus was on constraint transport for GRMHD. 

Divergence cleaning offers the advantage of a relatively simple implementation, but has limitations in maintaining an exact zero divergence of the magnetic field. In fact, we expect to find an order of 10^-3 for the divergence of the magnetic field normalized to the magnetic field strength <cit.>. On the other hand, divergence cleaning is directly compatible with high-order schemes for fluxes, and thus, high-resolution shock capturing schemes can be applied directly without large code changes. In codes with constraint transport or vector potential, the magnetic field needs to be interpolated from the cell surfaces to the cell center, which is usually done linearly. 

In the present work, the goal is to analyse how well divergence cleaning actually performs compared to the other methods. For this reason, we conduct a comprehensive comparative analysis of our implementation with other, well-established GRMHD codes using different treatments for the divergence-free constraint: the  code <cit.>, the sacra variant of <cit.> henceforth called , and the  code <cit.>. The latter uses a vector potential method, while the other two codes use constraint transport. Such direct code comparisons are still rare, but are crucial to identify how dependent results are on the method used and how serious errors due to small divergences of the magnetic field are in the evolution also in relation to other numerical inaccuracies. In <cit.>, a comparison for different open-source GRMHD codes showed discrepancies in BNS merger scenarios, particularly in the merger times and remnant lifetimes. 

Our code comparison includes relativistic shock tests in one, two and three dimensions as well as BNS merger simulations. While the basic structure of the codes is relatively similar, we have to consider in the comparison not only differences in the magnetic field treatment but also in several other numerical implementations, such as high-resolution shock capturing (HRSC) schemes or algorithms for flux conservation at refinement boundaries. By using higher and lower orders for the reconstruction methods of fluid variables and different grid setups, we can estimate their effects and try to separate them from differences caused by the divergence-free constraint treatments. In fact, the tests show that the latter are negligible compared to the effects caused by using different reconstruction methods.
For the BNS merger simulation, we run the setup also at three different resolutions to carefully analyze the resolution effects and to determine whether the differences become larger or smaller at higher resolutions. 

The article is structured as follows: Section <ref> summarizes the relevant GRMHD evolutions equations, describes the divergence-free constraint treatment, and new numerical methods implemented in the  code. In Sec. <ref>, we present a series of relativistic tests performed with  and compare them with results of the GRMHD codes , , and . We then discuss results of BNS merger simulations performed with  and , and compare them in Sec. <ref>. A summary of our main results follows in Sec. <ref>. In this article, we apply a metric with (-,+,+,+) signature and geometric units with G=c=M_⊙=1, unless otherwise specified. 



§ RELEVANT EQUATIONS AND NUMERICAL METHODS



We summarize in this section our new GRMHD implementation in the  code <cit.>. Below, we briefly outline the framework of the NR code, including spacetime evolution and grid structure. We then discuss the relevant evolution equations and applied divergence-free constraint treatment. Further, we highlight new numerical techniques that we implemented in  to address the increased complexity of the systems due to magnetic fields. 



 §.§ Spacetime Evolution and Grid Structure


 evolves the gravitational field in time using the methods-of-line approach and employing finite difference stencils for spatial discretization. For this, Einstein's field equations are solved in a 3+1 decomposition, in which the line element is:

    ds^2 = -α^2dt^2 +  γ_ij(dx^i + β^i dt)(dx^j + β^j dt),

with α as lapse function, β^i as shift vector, and γ_ij as spatial part of the metric tensor g_μν induced on a three-dimensional, spatial hypersurface. The latter is given by:

    γ_αβ = g_αβ + n_αn_β,

where n^α is the timelike, normal vector to the three-dimensional hypersurface defined by: 

    n^μ = 1/α(1, -β^i) ,  n_μ = (-α,0,0,0).


In this work, we apply a fourth-order Runge-Kutta integration and a Courant-Friedrichs-Lewy (CFL) coefficient of 0.25 in all  simulations. We use the BSSN reformulation <cit.> for the tests presented in Sec. <ref> and the Z4c reformulation with constraint damping terms <cit.> for the BNS simulations presented in Sec. <ref>, combined with 1+log slicing <cit.> and gamma-driver shift conditions <cit.>. 


The infrastructure of  includes adaptive mesh refinement (AMR). The grid consists of a hierarchy of L refinement levels labeled by l=0,...,L-1 with one or more Cartesian boxes. Using a 2:1 refinement strategy, the grid spacing on each level is given by h_l = h_0/2^l. For inner refinement levels with l ≥ l_ mv, the Cartesian boxes can move and adjust dynamically during the evolution to track the compact objects. Thereby, each box consists of n grid points per direction or respectively n_ mv grid points per direction for inner, moving boxes. The points are cell-centered and staggered to prevent division by zero at the origin. As a result, the grid points of the successive levels l and l+1 do not coincide. 

Furthermore, we apply in  the Berger-Oliger scheme <cit.> for local-time stepping (see <cit.>) and the Berger-Colella scheme <cit.> to ensures flux conservation across refinement boundaries (see <cit.>).



 §.§ General Relativistic Magneto-Hydrodynamics



We use the Valencia formulation of the GRMHD equations <cit.>. We note that we apply the ideal GRMHD approximation assuming infinite conductivity and zero resistivity. Although a resistive approach would be more accurate, it leads to more complications and stiff source terms in the final evolution equations, which is why it is rarely used in the literature; see <cit.> for some examples. 


In GRMHD, the full electromagnetic field is described by the Faraday tensor:

    F^μν = n^μ E^ν - n^ν E^μ + ϵ^μνκλ n_κ B_λ,

where ϵ^μνκλ is the four-dimensional Levi-Civita symbol and where E^ν and B^ν are, respectively, the electric and magnetic field as measured by an Eulerian observer. Accordingly, we can obtain the electric and magnetic field by the Faraday tensor and its dual ^* F^μν = 1/2ϵ^μνκλ F_κλ via:

    E^μ = F^μν n_ν B^μ = ^*F^μν n_ν.


In the case of ideal GRMHD with infinite conductivity, there is no charge separation and the electric field vanishes in the fluids rest-frame. Thus, ideal GRMHD corresponds to imposing u_μ F^μν=0 where u_μ is the fluid four-velocity. Following this condition, the electric field can be expressed by:

    E^ν = -1/Wϵ^μνκλ n_κ B_λ u_μ,

where W = α u^0 = 1/√(1-v^2) is the Lorentz factor. It is therefore sufficient to evolve the magnetic field only, which greatly simplifies the final evolution equations. 

We introduce the magnetic four-vector b^μ = u_μ^* F^μν describing the magnetic field for a comoving observer. The components are given by:

    b^0 = W B^i v_i/α,  b^i = B^i/W + b^0 (α v^i - β^i),

with v^i = u^i / W + β^i / α as the spatial components of the fluid velocity measured by an Eulerian observer. 

The full stress-energy tensor for a perfect fluid is then defined by:

    T^μν = ( ρ h + b^2 ) u^μ u^ν + (p + b^2/2) g^μν - b^μ b^ν,

with h = 1 + ϵ + p/ρ being the specific enthalpy, ρ the rest-mass density, p the pressure, and ϵ the internal energy density. b^2 = b^μ b_μ represents twice the magnetic pressure with p_ mag = b^2/2. We note that in our definition of the magnetic field a factor of 4 π is absorbed. 


The evolution equations for ideal GRMHD are derived from the conservation laws of baryon number and energy-momentum:

    ∇_μ(ρ u^μ) =0, ∇_μ T^μν = 0,

as well as from the Maxwell's equations:

    ∇_μ^*F^μν = 0,

with ∇_μ being the covariant derivative. Furthermore, an EOS is required to close the evolution system. In order to write the evolution system in the form of a balance law with:

    ∂q/∂ t + ∂F^i/∂ x^i = s,

we define the primitive variables w=(ρ, v^i, ϵ, p, B^i ) and the following conservative variables q:

    D    =   √(γ)ρ W, 
    
        S_j  =   √(γ)((ρ h +b^2) W^2 v_j- α b^0 b_j), 
    τ =   √(γ)((ρ h +b^2) W^2 - (p+p_ mag). 
       .- α^2 (b^0)^2 )- D, 
    B̂^k  =   √(γ) B^k ,
     
where γ is the determinant of γ_ij.
Finally, we obtain the following evolution equations in the form of Eq. (<ref>):

    q   = [D, S_j, τ, B̂^k], 
    F^i    = α[                                       D ṽ^i; S_j ṽ^i + √(γ)(p+p_ mag) δ_j^i - b_j B̂^i/W;   τṽ^i + √(γ)(p+p_ mag) v^i - α b^0 B̂^i/ W;                         B̂^k ṽ^i - B̂^i ṽ^k ], 
    s   = α√(γ)[                                    0; T^μν( ∂ g_ν j/∂ x^μ - Γ^λ_ μν g_λ j);    α(T^μ 0∂ln (α)/∂ x^μ-T^μνΓ_ μν^0);                                  0^k ],
 
with ṽ^i = v^i -β^i/α and the Christoffel symbols Γ_ μν^λ.



 §.§ Divergence-Free Constraint Treatment



While the system of equations derived above is generally sufficient to perform simulations and evolve the magnetic field in time, there is no guarantee that the time component of Maxwell's equations (<ref>) is satisfied. Even when starting from constraint-satisfying initial data, numerical errors can accumulate leading to magnetic monopoles and thus to violations of Eq. (<ref>). Approaches typically used to address this issue include constraint transport schemes as implemented in  and , vector potential methods as implemented in , and divergence cleaning techniques. We refer to <cit.> and <cit.> for implementations of constraint transport and vector potential methods in , , and , respectively. 

Our GRMHD implementation follows the hyperbolic divergence cleaning scheme from <cit.>. In particular, a new field variable ζ is introduced to damp and advect divergences of the magnetic field. We use a modification of the Maxwell's equations:

    ∇_μ( ^*F^μν + g^μνζ) = κ n^νζ,

with the damping rate κ. We set κ = 1 in this work. For ζ→ 0, the equation reduces again to the Maxwell's equations (<ref>). From the time component of Eq. (<ref>), we obtain an evolution equation for ζ:

    ∂_t ζ   + ∂_i ( α/√(γ)B̂^i - ζβ^i ) 
       = ζ (-κα - ∂_i β^i) + B̂^i ∂_i (α/√(γ)).


The spatial part of Eq. (<ref>) leads to the following modification for the evolution equation of the magnetic field:

    ∂_t B̂^j    + ∂_i [ (α v^i - β^i ) B̂^j - α v^j B̂^i + α√(γ)γ^ijζ] 
       = - B̂^i ∂_i β^j + ζ∂_i ( α√(γ)γ^ij).


Although divergences of the magnetic field are damped, this approach does not prevent the occurrence of constraint violations, unlike the constraint transport or vector potential methods. This is an obvious disadvantage of the method, leading to finite errors. Nonetheless, these errors should converge to zero in a reliable and controlled manner with the scheme employed, similar to the convergence of other constraints, e.g., the Hamiltonian constraint. On the other hand, the implementation is much simpler as there is no staggering of the magnetic field necessary and available high-order HRSC schemes can be directly employed. While  already uses cell-centered fields for both the metric and fluid variables, additional work would be required since the magnetic field is defined at the cell faces and not in the cell centers. In particular, the AMR restriction and prolongation algorithms would have to be changed and would become more complex. For a magnetic field defined at the cell surface, it would first have to be interpolated to the cell centers. In most codes using constraint transport and vector potential methods, this is often done linearly and, thus, the magnetic field relies on second-order methods. For a cell-centered magnetic field, however, we can use the same methods as for other variables, typically using fourth-order schemes in our implementation. Apart from questions of complexity and convergence rate, a key question is how a given physical system depends on numerical errors in the divergence constraint, which we investigate in the numerical examples that follow.



 §.§ Numerical Schemes




  §.§.§ Conversion to Primitive Variables


In order to write the evolution equations in form of a balance law, we use the conservative variables q as defined in Eqs. (<ref>)–(<ref>) which are analytical functions of the primitive variables w. Nevertheless, we need to know the primitive variables, for instance, to compute the flux terms F^i at each time step. However, the inversion of Eqs. (<ref>)–(<ref>) to obtain the primitive variables is non-trivial and must be solved numerically. We refer to <cit.> for a summary and discussion of different approaches commonly used in NR codes. The scheme we use follows the approach implemented in the RePrimAnd library <cit.>. More precisely, we use its master-function (see Eq. (44) of Ref. <cit.>) and use a Brent-Dekker scheme to find its root. In the numerical tests as well as in the BNS merger simulations, which we discuss in Secs. <ref> and <ref>, this method has proven to be robust and provides reliable results.



  §.§.§ Reconstruction



 contains HRSC methods for the treatment of shocks and discontinuities in the hydrodynamic variables. In order to evaluate the fluxes F at the individual grid cell interfaces and to solve the Riemann problems, the variables have to be reconstructed to the cell surfaces from either side. Since all field variables are cell-centered in our GRMHD implementation, we can use the same reconstruction schemes for the magnetic field variables and ζ that are already implemented in . These include, for example, a third-order convex-essentially-non-oscillatory (CENO3) scheme <cit.>, a fifth-order weighted-essentially-non-oscillatory (WENOZ) scheme <cit.>, or a fifth-order monotonicity preserving (MP5) scheme  <cit.>. 

While higher order schemes are generally more accurate and less dissipative, they can exhibit stronger oscillations, as we show and discuss in some relativistic shock tests in Sec. <ref>. Enhanced Oscillations can lead to unphysical values. For this reason, in the BNS merger simulations with  presented in Sec. <ref>, we apply the following procedure in lower density regimes to ensure the physical validity of reconstructed variables and, in particular, to maintain the positivity of the density and pressure:


    
  * We use a high-order reconstruction scheme to compute the GRMHD variables at the interface between grid cell i and i+1. 
    
  * If the rest-mass density ρ drops below a threshold density, we examine the oscillation of all variables by comparing the reconstructed values at the interface with the cell-centered ones of i and i+1. 
    
  * Whenever one reconstructed variable is below or above a value in the grid cells i and i+1, we change to a low-order reconstruction and recompute the values of all variables at the interface.
    
  * Finally, we examine the physical validity of the reconstructed values by demanding a positive rest-mass density and a positive pressure. If this is not given, we use a linear reconstruction method.


For high-order reconstruction schemes, we typically use WENOZ or MP5 as fifth-order methods or CENO3 as third-order method. We then fall back on a linear total variation diminishing (TVD) method with “minmod” slope limiter <cit.> as low-order reconstruction scheme.



  §.§.§ Riemann Solver



For the GRMHD simulations with  in Secs. <ref> and <ref>, we use the Harten, Lax, and van Leer (HLL) Riemann solver <cit.>. HLL uses a two-wave approximation via estimates for the fastest left- and right-moving signal speeds:

    F^ HLL   = λ_ maxF(q^+) - λ_ minF(q^-)/λ_ max - λ_ min
       - λ_ maxλ_ min/λ_ max - λ_ min(q^+-q^-),

where q^+ and q^- are the variables respectively reconstructed from the left and right side of the cell face, and F(q^+) and F(q^-) are the according flux terms. The fastest left- and right-moving signal speeds are determined by:

    λ_ max   = max(λ_i(q^+),λ_i(q^-),0), 
    λ_ min   = min(λ_i(q^+),λ_i(q^-),0),

with λ_i being the characteristic wave speeds. 

While there are three characteristic modes in pure general relativistic hydrodynamics, there are seven independent characteristic waves in ideal GRMHD: the entropy wave, two Alfven waves, and four magnetosonic waves (two slow and two fast modes). For a detailed discussion of the individual waves and derivation of the eigenvalues, we refer to <cit.>. Since the exact wave speeds require the solution of a non-trivial quartic equation for the magnetosonic waves, we use the commonly applied approximation proposed in <cit.>. We note that by using divergence cleaning, two additional modes with characteristic velocities equal to the speed of light must be taken into account. For the magnetic field variables and ζ, we therefore set λ_ max=1 and λ_ min=-1 in the Riemann solver. 


In addition to HLL, we started to implement the HLLD, a five-wave relativistic Riemann solver, following <cit.>, with minor changes to adapt to the divergence cleaning formulation. The implementation is still in test phase. For this reason, we use it only in some of the special relativistic tests presented in Sec. <ref>.



  §.§.§ Atmosphere



In grid-based NR simulations, the vacuum region surrounding the compact objects is usually modeled with an artificial atmosphere. The reason is that too low rest-mass densities are numerically very demanding and can lead to errors in the recovery of the primitive variables and in HRSC schemes. Therefore, we set in  a grid cell to atmosphere when its density falls below a density threshold ρ_ thr <cit.>. We assume a cold, static atmosphere: the density is reduced to a fraction of the star's original central density ρ_ atm = f_ atmρ_c, pressure and internal energy are determined according to the zero temperature part of the EOS, and the velocity of the fluid is set to zero. The magnetic field remains unchanged. The density threshold is defined as the fraction of the atmosphere value with ρ_ thr = f_ thrρ_ atm to avoid fluctuations around the density floor. In the BNS merger simulations presented in Sec. <ref>, we use f_ atm = 10^-11 and f_ thr = 100.



§ SPECIAL RELATIVISTIC TESTS



For validating our new implementation, we have performed a series of well-established special relativistic magneto-hydrodynamics tests in one-, two-, and three-dimensions. In particular, it is important to analyze the reliability of the code with respect to shocks. In each test, we compare our results and performance with the codes  <cit.>,  <cit.>, and  <cit.>.



 §.§ One-Dimensional Tests








Relativistic shock tube problems are simple, one-dimensional tests that can be used to demonstrate a code's ability to capture a variety of different shock wave structures. By setting different states of the fluid to the left and right of an interface, a shock is generated and evolved. Among the most popular and well-known shock tube problems in magneto-hydrodynamics are the Balsara tests <cit.>. We summarize the initial data for the Balsara tests I–V in Tab. <ref> by specifying the respective fluid variables of the left and right states. The tests are performed along the x-axis with the initial fluid discontinuity at x=0. In all Balsara tests, an ideal-gas law EOS with p=(Γ-1)ρϵ, where Γ is the adiabatic constant, is assumed: in Balsara test I with Γ = 2.0 and in Balsara test II–V with Γ = 5/3. We evolve the shock waves until t=0.4 for Balsara tests I–IV and t=0.55 for Balsara test V. 

The results are shown in Fig. <ref> for each Balsara test at three different resolutions with 1000, 2000, and 4000 grid points in x ∈[-0.5, 0.5]. We use MP5 reconstruction. For each test, we show the profiles of the density ρ and the B^y component of the magnetic field at the respective final time. Additionally, we show the exact solution of <cit.> as comparison. Overall, all tests are in good agreement. As expected, higher resolutions increase the accuracy and are better able to capture the shock fronts. This is particularly evident in the Balsara tests II, III and IV, where a high resolution is required to capture the sharp edges of the shock waves in ρ. Some oscillations are visible, e.g., in B^y of Balsara tests III and IV, that we attribute to the use of a reconstruction scheme with relatively high order. 

In order to evaluate how our GRMHD implementation performs in these tests with respect to other NR codes, we compare the results obtained with  with those obtained with , , and . Figures <ref> and <ref> show respectively the Balsara tests I and III for all four codes. The resolution is set to 2000 grid points in x ∈[-0.5, 0.5]. Although we try to choose the technical configurations rather similar, each working group using different codes has preferred numerical methods that also differ in the individual implementations. For instance, the HLLD Riemann solver as described in <cit.> is used for both tests by . This solver is more advanced than the HLL solver used by , , and . Furthermore, different methods are used to reconstruct the fluid variables: While  uses a fifth-order WENO5 scheme,  applies a third-order piecewise parabolic method (PPM).  also employs PPM reconstruction in Balsara test I. However, since Balsara test III is more demanding due to the large jump in the initial pressure,  uses here a second-order TVD method with “minmod” limiter. To assess the differences caused by different reconstruction algorithms, we show for  results obtained with a linear TVD method, labeled as `low order' in Figs. <ref> and <ref>, additionally to those with MP5 reconstruction. 

All codes demonstrate their ability to reliably capture and evolve the shock waves with small differences at the shock fronts. For Balsara test III, , , and  (with MP5) seem to be superior in modeling the exact solution, despite showing small oscillations. We explain this by the usage of higher orders for the reconstruction.  uses a second-order scheme in this test. In fact, the solution of  with linear TVD method perfectly matches the results of  when the same reconstruction method and Riemann solver are used. Also in Balsara test I, the largest differences appear for  using different reconstruction methods. This suggest that the differences in Figs. <ref> and <ref> for the individual codes are due to different HRSC algorithms rather than different methods for the magnetic field or the divergence-free constraint treatments.






 §.§ Two-Dimensional Tests




  §.§.§ Cylindrical Explosion







The first two-dimensional shock problem we perform in this test series is the cylindrical explosion, also known as cylindrical blast wave, a standard problem for GRMHD codes, e.g., <cit.>. We consider a dense medium within a cylindrical radius of r_ in = 0.8 in a less dense ambient medium outside of r_ out =1. The denser medium has an overpressure and therefore expands. The initial values for density ρ and pressure p are: ρ_ in  = 10^-2 and p_ in  = 1.0  for the inner medium, and ρ_ out  = 10^-4 and p_ out  = 3 × 10^-5 for the outer medium. For the transition region 0.8 ≤ r ≤ 1, we apply:

    ρ   = exp( (r_ out -r ) lnρ_ in + (r - r_ in) lnρ_ out/r_ out-r_ in), 
    
        p    = exp( (r_ out -r ) ln p_ in + (r - r_ in) ln p_ out/r_ out-r_ in),

with cylindrical radius r = √(x^2 + y^2) in the x-y plane. In this test, an ideal-gas EOS with adiabatic index Γ = 4/3 is used and we set a uniformly directed magnetic field along the x-axis with B^x = 0.1 and B^y = B^z = 0. We evolve this configuration until t=4. 

The resulting blast wave at the final time is presented in Fig. <ref>. We show snapshots of ρ, p, B^x, and B^y in the x-y plane of a simulation with  using CENO3 reconstruction and a grid consisting of two refinement levels with 200 × 200 grid points each, covering a range of [-6, 6] in x and y directions. A fast forward shock and a reverse shock bounding the inner region are observable. While there is no exact solution available for this tests, the structures shown in our results are consistent with those in <cit.>. 

For a more quantitative comparison of our results with , , and , we show one-dimensional slices of the final blast wave for ρ, p, and B^x along the x- and y-axes in Fig. <ref>. To avoid discrepancies due to different algorithms for flux conservation at the refinement boundaries, we use here a grid consisting of a single refinement level with grid spacing Δ x = 0.03 and 400 × 400 points. For , we apply the reconstruction as described in Sec. <ref> with CENO3 as higher order scheme and linear TVD as lower order method, examining oscillations in the fluid variables in regions with ρ < 2× 10^-4. Additionally, we run  once using a pure linear TVD method, labeled as `low order' in Fig. <ref>. Both,  and , apply here a second-order TVD scheme, while  results are shown for third-order PPM reconstruction. Lacking a well-known analytical solution for this test problem, we perform additionally one high-resolution test run with  using 8000 × 8000 grid points, i.e., a spacing of Δ x = 0.0015, using CENO3. Except for , that applies the HLLD Riemann solver, the results are obtained with HLL. The results for , , and  overlap with the ones obtained with  using the linear TVD reconstruction method. The higher-order reconstruction leads to stronger oscillations at the shock fronts, which are mainly visible in the ρ and p profiles along the y-axis. Nevertheless, all codes converge approximately to the same solution. As before in the Balsara tests, we conclude that the differences are primarily due to different HRSC schemes.



  §.§.§ Magnetic Rotor






As a second two-dimensional standard test, we consider the magnetic rotor, originally described for classical magneto-hydrodynamics <cit.> and later generalized to the relativistic case <cit.>. For this test, we initialize a dense, fast rotating fluid in the center of a non-rotating ambient medium. The central rotating fluid is set inside an area with cylindrical radius r=0.1 with ρ_ in = 10 and a uniform angular velocity of Ω = 9.95, resulting in a maximum velocity of the fluid of v_ max = 0.995. The density of the surrounding medium is ρ_ out = 1. We apply in this test an ideal-gas EOS with Γ = 5/3 and consider a uniform pressure with p=1 and magnetic field with B^x = 1, and B^y=B^z=0. The test is simulated until t=0.4. 

We perform this test with  on a grid with two refinement levels and 200 × 200 points covering the range [-0.5, 0.5 ] for x and y using CENO3 reconstruction. During the simulation, the rotation leads to a twisting of the magnetic field lines in the central region, which in turn slows down the rotor. Figure <ref> presents maps of the density ρ, the pressure p, and the magnetic field variables b^2, which can be considered as twice the magnetic pressure, and the B^x component at the final time. The results are in good agreement with those in <cit.>. 

Like for the cylindrical explosion, we compare our results with , , and  for one-dimensional slices at the final time. Figure <ref> shows profiles of ρ, p, and B^x along the x- and y-axes. As for the cylindrical explosion, we choose for the comparison a grid without mesh refinement consisting of 400 × 400 grid points with a spacing of Δ x = 0.0025. The results with  are obtained with the HLLD Riemann solver in combination with a third-order PPM reconstruction, while  and  apply HLL and a second-order TVD reconstruction. For  we show the results using the HLL Riemann solver once with CENO3 and once with linear TVD, labeled as `low order' in Fig. <ref>. Again, we perform additionally one high-resolution test run for  with HLL and CENO3 using 8000 × 8000 grid points and a spacing of Δ x = 0.000125. 



Overall, the results of the individual codes agree well. Mostly at the shock fronts, i.e., along x between -0.2 and -0.35 and along y between 0.1 and 0.25, some codes show stronger oscillations than others, which is strongly depending on the HRSC method used. The largest differences are present for  for the B^x profile. However, the differences decrease for higher resolutions. As shown in Fig. <ref>, with increasing resolution from Δ x = 0.0025 to Δ x = 0.00125 up to Δ x = 0.000625 the B^x profile for  converges against the same solution. The shock along the y-axis is well captured even at the lowest resolution, while the shock along the x-axis needs a higher resolution. Since the main difference between  and the other codes is the Riemann solver, we attribute to the HLLD solver being more suitable for different types of shock waves than for others.



  §.§.§ Kelvin Helmholz Instability




The last two-dimensional special relativistic magneto-hydrodynamics test we consider in this work addresses the KHI as proposed in <cit.>. For this test, we are focusing the comparison on the  and  codes. We use the same setup as in <cit.> and specify a tanh-shaped velocity profile with:

    v^x = -v_ shtanh(y/a).

Here, v_ sh determines the shear velocity and a the thickness of the shear layer. As in <cit.>, we set v_ sh = 0.25 and a = 0.02. Further, we initialize a uniform density of ρ = 1 and pressure of p=20, and apply an ideal-gas EOS with Γ = 4/3. The components of the magnetic field are B^x = √(2σ p) and B^y = B^z = 0. We set σ = 0.01. 

To disturb the shear layer, movement along the y direction is introduced by:

    v^y = A_0 v_ shsin(2 π x) exp(-100 y^2),

where A_0 = 10^-4 determines the strength of the perturbation. The v^z component remains zero. 

Figure <ref> presents snapshots of the density ρ for  at t=10.9 and for  at t=10.0. In both cases, the grid consists of one refinement level with 100 × 200 points covering a range for x ∈[-0.5, 0.5] and y ∈[-1, 1]. We use in  MP5 and in  PPM reconstruction. Both codes demonstrate their ability to resolve the vortex formed in the KHI test. The observable structure is very similar, although the vortex forms later for  than for . We attribute this to the usage of different Riemann solver:  applies HLL and  HLLD. A similar behavior for different solvers is also found in <cit.>. 





To be able to compare the simulations quantitatively, the perturbed velocity difference Δ v^y = 0.5 (v^y_ max-v^y_ min) is shown as a function of time in Fig. <ref>. We show the results for  additionally for using the HLLD solver that is still under development. As in <cit.>, Δ v^y grows exponentially until nonlinear saturation is reached. The growth rate is higher and nonlinear saturation is reached faster for  and  using HLLD than for  using HLL, whereas the saturation amplitude is about the same for all cases. This demonstrates that the exponential growth for the KHI test depends strongly on the Riemann solver used, while the nonlinear saturation is only weakly dependent.



 §.§ Three-Dimensional Tests




  §.§.§ Spherical Explosion





As a demanding three-dimensional shock problem for GRMHD codes, we perform the spherical explosion test with  and . We follow the description in <cit.>, which corresponds to a three-dimensional version of the cylindrical explosion test in Sec. <ref>, where the cylindrical radius is substituted for a spherical radius: A dense medium with ρ_ in  = 10^-2 and p_ in  = 1.0 is set inside a spherical radius of r_ in = 0.8. The ambient medium outside of r_ out = 1 has ρ_ out  = 10^-4 and p_ out  = 3 × 10^-5. In the transition region between r_ in = 0.8 and r_ out=1, we set:


    ρ   = exp( (r_ out - r ) lnρ_ in + (r - r_ in) lnρ_ out/r_ out-r_ in), 
    
        p    = exp( (r_ out - r ) ln p_ in + (r - r_ in) ln p_ out/r_ out-r_ in).

We apply again an ideal-gas EOS with Γ = 4/3 and initialize a uniform magnetic field with B^x = B^y = 0.0 and B^z = 0.1. The results obtained with  are shown in Fig. <ref> at the final time t=4. For this test, we apply CENO3 reconstruction and use a grid with two refinement levels and 200 × 200 × 200 points ranging from [-6,6] in each direction. The grid spacing is accordingly Δ x = 0.06 on l=0 and Δ x = 0.03 on l=1. 

We compare in Fig. <ref> our results for one-dimensional profiles along the x- and z-axes at the final time for ρ, p, and B^x. In this case, we focus on the comparison with  and choose a grid with two refinement levels as already described above. We include in Fig. <ref> lines for the profiles on level l=0 and l=1. Overall, the differences between  and  are small: In the center, differences are of the order 10^-8 for the rest-mass density, 10^-7 for the pressure, and 10^-5 for the magnetic field component B^x. The highest differences are found at the shock front of the p profile along the x-axis.  shows here larger values than  by a factor of almost ten. However, these larger differences are only outside the inner refinement level, where the resolution is quite low.

This test also enables us to access the conservation of the magnetic flux across refinement levels. By integrating the magnetic flux over the refinement boundary on levels l=1 and l=0, we can compare the magnetic flux violations. Figure <ref> shows the time evolution of the magnetic flux conservation across the positive z refinement boundary, i.e., the area with z=3, x ∈ [-3,3] and  y ∈ [-3,3], and presents the relative difference between ∫_z+ B_l_0 d A and ∫_z+ B_l_1 d A. We present results for , , and additionally for  using a lower resolution with Δ x = 0.12 on l=0 and Δ x = 0.06 on l=1 (labeled as `low res.'). While both codes use methods to correct fluxes across refinement boundaries, it is evident that the magnetic flux is better conserved in the simulation with  than in the simulations with . For , the violations are of the order of 10^-14, i.e., at round-off accuracy. For , the difference between the fluxes is initially zero, but increases significantly once matter crosses the refinement boundary at t ≈ 1.5, reaching orders of 10^-5. Using divergence cleaning in , the evolution equation for the magnetic field contains additional source terms. Although  incorporates algorithms to correct the fluxes across refinement boundaries, the source terms are not matched and may marginally differ, resulting in slightly different values. Figure <ref> shows small improvements for increased resolution.





§ BINARY NEUTRON STAR MERGER SIMULATIONS







 §.§ Configurations


We perform simulations of BNS merger with magnetic fields and compare the simulation results with . The publicly-available, spectral code fuka <cit.> is used to construct initial data for an equal-mass BNS system with gravitational masses m_A = m_B = 1.35 M_⊙ and baryonic masses m_b,A = m_b,B = 1.49 M_⊙. As  has an implemented fuka reader, the same initial data is used to directly compare the evolution and results of the BNS simulations. We set an initial coordinate separation of ∼ 28.4 km to ensure a short inspiral, focusing the simulation on the merger and post-merger phases. The initial ADM mass is M_ ADM =2.673 M_⊙. For the EOS, a piece-wise polytropic fit of SLy <cit.> is applied following <cit.> with four pieces, three for the NS core and one for the crust. In order to include thermal effects, the zero-temperature EOS is extended by a thermal pressure P_ th = (Γ_ th - 1) ρϵ_ th <cit.>, where ϵ_ th is the thermal part of the specific internal energy and we set Γ_ th = 1.75. We note that a slightly different value is used for the simulations with , namely Γ_ th = 1.8. 

We initialise a poloidal magnetic field for each star by a vector potential:

    A_x    = -y A_b max(p - p_ cut,0)^2, 
    
        A_y    =  x A_b max(p - p_ cut,0)^2, 
    
        A_z    =  0,

with x = x - x_ NS and y = y - y_ NS as distances from the center of the NS along x and y, respectively. We set A_b = 1000 and p_ cut=0.004 × p_ max, where p_ max is the maximum pressure in the NS at t=0 ms. In this way, we obtain a maximum magnetic field strength of the order ∼ 10^15 G inside the stars. 


For the grid, we use in  eight refinement levels where the three outermost are non-moving and the five innermost are moving levels. The simulations are performed on three different resolutions: R1 with n = 128 and n_ mv = 64 grid points, R2 with n = 256 and n_ mv = 128 grid points, and R3 with n = 512 and n_ mv = 256 grid points in each direction. The grid spacing on the finest box covering the NS are Δ x_ min≈ 368.6 m for R1, Δ x_ min≈ 184.3 m for R2, and Δ x_ min≈ 92.1 m for R3. We use the HLL Riemann solver as described in Sec. <ref> and the reconstruction method as described in Sec. <ref> with MP5 as high-order reconstruction and CENO3 as low-order reconstruction, i.e., we examine oscillations and fall back to CENO3 or further to linear reconstruction if necessary. We employ this only for low density regions: in the R1 and R2 simulations for ρ < 100 ×ρ_ atm and in the R3 simulation for ρ < 10^6 ×ρ_ atm[At the highest resolution, we encountered issues when using MP5 due to larger oscillations leading to unphysical behavior of the matter outflow along the grid axes. This problem could be solved with a higher threshold for the rest-mass density using our scheme to detect enhanced oscillations and maintain the positivity of the reconstructed density and pressure.].

 uses a similar grid configuration with ten refinement levels and the same resolutions R1, R2 and R3 as in . However, the box sizes of the refinement levels are larger as  uses a fixed mesh refinement (FMR). The half size of the finest level is set to L_ fin =35.5  km. For the inner part of the finest refinement level, a constant atmosphere is set with ρ_ atm = 10^3  g/cm^3. Outside a power-law profile is applied with ρ_ atm = 10^3 (L_ fin/r)^3   g/cm^3.  introduces a restriction for high magnetic-pressure regions with b^2/ρ≥ 10^3 by artificially suppressing the specific momentum. The reason is that the numerical accuracy is not good in those regions and, as a result, the primitive recovery procedure often fails. However, in the current work, in which the post-merger evolution is not followed over a long timescale, we do not find such a highly magnetized region. As in Sec. <ref>,  applies the HLLD solver with third-order PPM reconstruction.



 §.§ Evolution and Magnetic Field Amplification






The evolved BNS system merges after about four orbits and forms a hypermassive neutron star (HMNS), collapsing into a BH within a few tens of milliseconds. The lifetime of the HMNS varies for different resolutions because the HMNS just prior to the collapse is marginally stable, and hence, small perturbations including numerical errors can trigger the collapse. In Fig. <ref>, we present snapshots of the rest-mass density and the magnetic field lines for the simulation with the highest resolution. The snapshots show the two NSs at the beginning of the simulation, at the merger (at about 9.6  ms), and the remnant before and after the collapse. The initial magnetic dipole field inside the NSs is clearly visible, as well as the winding of the magnetic field lines during the merger, forming a helicoidal structure. 

We show the time evolution for the magnetic energy and maximum magnetic field strength in Fig. <ref> for the R1, R2, and R3 simulations with . The magnetic energy is defined by:

    E_ mag = 1/2∫ u^0 √(-g) b^2 d^3x.

Additionally, the toroidal and poloidal contribution are shown. We note that they are defined with respect to the coordinate center, which is only approximately correct after the merger assuming the remnant stays in the center. During the merger, the magnetic field is amplified. We observe here a strong resolution dependency: For R1, there is almost no amplification during the merger itself. Only on later time scales of about 10  ms after the merger the magnetic energy and field strength rise. For R2, the maximum magnetic-field strength doubles, and for R3, it increases tenfold, leading to energies up to 10^49  erg. The amplification is triggered by a variety of instabilities. For example, the shear layer between the two NS induces a KHI during the merger, which forms vortices that twist the magnetic field and lead to exponential growth, e.g., <cit.>. Another mechanism is driven by large-scale differential rotation in the post-merger, which winds up the magnetic field lines. While the magnetic winding can already be resolved with lower resolutions, capturing the KHI and also the MRI requires higher resolutions. In order to fully capture these instabilities and the associated magneto-hydrodynamic turbulence, even higher resolutions are required as illustrated in <cit.> (see also <cit.>). 

Although the magnetic field is initially purely poloidal, the toroidal component becomes dominant after the merger as shown in Fig. <ref>. For a qualitative analysis of the magnetic field in the remnant system, Fig. <ref> presents snapshots in the x-z plane after the merger. We show the magnetic field strength and the ratio between the toroidal and poloidal component before and after the collapse for the R3 simulation. Before the collapse, the magnetic field in the polar region is already predominantly poloidal. This becomes even clearer after the collapse. The disk of the remnant has rather toroidal magnetic fields. After the collapse, the magnetic field in the polar region decreases from an order of magnitude of 10^12  G to an order of magnitude of 10^11  G. The magnetic field is strongest inside the disk with up to 10^16  G. 

Additionally, we show in Fig. <ref> the divergence of the magnetic field. As already mentioned, a limitation of the implemented divergence cleaning method is that the divergence of the magnetic field is not exactly zero and cannot prevent the formation of magnetic monopoles. To assess how large the divergence actually is, we present maps of the remnant after the merger. It is largest in the interior of the disk, where the magnetic field is also strongest, with values in the order of 10^8  G / cm and smallest in the polar regions with values in the order of 10^3  G / cm. 

One important point is to compare errors introduced by the magnetic field divergence violation with other uncertainties, e.g., due to finite resolution, flux computation, or shock treatments. Following Gauss's law, we can translate the divergence into an error on the magnetic field within a grid cell by multiplying with its volume dV_i and dividing by its surface area dS_i. With E_ mag∝ B^2, we obtain an estimate of the relative error on the magnetic energy by normalizing with the magnetic field strength and squaring the quantity. In Fig. <ref>, we compare this relative error of the simulations using R1, R2, and R3 with the relative difference of the magnetic energy between resolution R2 and R3. The error in the magnetic energy due to the magnetic field divergence increases during the merger. Overall, the relative error is between 10^-8 and 10^-4 and decreases with higher resolution. The relative difference between the magnetic energy at R2 and R3 resolution is about 10^-2 and increases to 10 after the merger. We note that, in agreement with other GRMHD simulations shown in previous studies, even our highest resolution is well outside the convergence regime, explaining the large differences here. Nevertheless, this indicates that the errors due to the divergence of the magnetic field when using the divergence cleaning scheme are minor if not negligible compared to the errors caused by other resolution-dependent effects. 





 §.§ Comparison



Finally, we compare the results of the BNS simulations performed with  and  in Figs. <ref> and <ref>. The time evolution of the total baryonic mass, ejecta mass, and the central density inside the NSs is shown in Fig. <ref> and the magnetic energy and maximum field in Fig. <ref> for both codes at resolution R1 to R3. We use the geodesic criterion for the ejecta. Accordingly, matter is considered unbound if the condition u_t < -1 with a positive radial velocity v_r >0 is satisfied. We find that the merger time in the simulations with  is about 1.4  ms earlier than in the simulations with . This is due to the different gauge conditions used in . We discuss this in more details in Appendix <ref>. 



The ejecta mass is considerably larger in the simulations performed with  than in the ones performed with , i.e., by a factor of ∼ 1.9 for resolution R1, ∼ 4.1 for resolution R2, and ∼ 2.4 for resolution R3. Thereby, the variation for different resolutions is greater for  ranging between 0.015 M_⊙ and 0.022 M_⊙ than for  ranging between 0.0055 M_⊙ and 0.0064 M_⊙. A possible reason for the larger variations could be attributed to using different reconstruction schemes with different orders in  and . Another reason that could explain the strong differences in ejecta mass is the treatment of the vacuum region. As described in Sec. <ref>,  uses an artificial atmosphere with f_ atm = 10^-11, resulting in atmosphere densities of the order of 10^4  g/cm^3, while  uses lower atmospheric values of at most 10^3  g/cm^3, which further decreases with increasing radius. This can lead to differences in the ejecta as the outward flowing matter travels more freely. The lower atmosphere densities can therefore explain the higher ejecta masses in the simulations with . 





The values for the final ejecta mass, disk mass, remnant BH mass, and energy radiated by GWs evaluated at the end of the simulations at t ≈ 50  ms are listed in Tab. <ref>. Since resolution R1 is too low to give reliable results after the BH formation, we stopped the simulation for  after the collapse. For , there was no BH formation within the simulation period for R1. Results for R1 simulations are therefore not listed in Tab. <ref>. Indeed, the remnant BH masses and disk masses are more similar for both codes. Comparing the values for R3 resolution,  predicts a slightly larger BH mass of ∼ 2.57 M_⊙ than  with ∼ 2.54 M_⊙. On the other hand, the disk masses in the simulations with  are greater than the one with  by about ∼ 0.005 M_⊙ for R2 and ∼ 0.002 M_⊙ for R3. In order to assess the error size in the conservation of the energy, we compute Δ E:=M_ ADM - M_ BH - M_ disk - M_ eje-E_ GW. For R2 and R3 in , we get respectively Δ E ≈ 0.0504  M_⊙ and 0.0188  M_⊙, while for R2 and R3 in , we get Δ E ≈ 0.0121  M_⊙ and 0.0025  M_⊙. The values of Δ E for R3 are much smaller than in R2 for both codes, showing good convergent behavior. The error size for R3 in  is slightly smaller than that in , which we suggest comes from the different settings of the artificial atmosphere influencing the measured M_ eje.



Considering the violation of the divergence constraint for the magnetic field, we find larger violations for  than for the  simulations, caused by the employed evolution schemes. The norm of the magnetic field divergences normalized to the norm of the magnetic field strength lies for  between 10^-9  cm^-1 and 5 × 10^-8  cm^-1 and for  between 10^-19  cm^-1 and 2 × 10^-18  cm^-1. 

In addition, we analyze the conservation of the magnetic flux across refinement boundaries. For this purpose, we integrate the magnetic field across the refinement boundaries at level l and its adjacent coarser level l-1, and compute the relative difference of the magnetic flux similarly as in the spherical explosion test. We show the results along the positive z direction for  at the refinement boundary of level l=10 and for  at level l=6. The comparison here is not straightforward since the refinement boxes for 's FMR are much larger than for . The refinement boundary of the finest level of  lies between the refinement boundaries of l=6 and l=5 of , whereby the boxes move dynamically within . Nevertheless, it is visible that the relative differences for  are orders of magnitude larger than for : for  at orders of  10^-2 and 10^-1, whereas for  at orders of 10^-15 to 10^-14, i.e., machine accuracy. 

However, for comparing the magnetic energy and the maximum field strength in Fig. <ref>, both codes predict overall similar values for the respective resolutions. In the R1 simulations, there is almost no amplification during the merger in both cases. The magnetic field only increases after the merger, which happens slightly earlier for  than for , but with a similar slope in the magnetic energy. With R2, both codes show an amplification of the magnetic field during the merger up to 10^48  erg. Despite the different merger time, the collapse time coincides at this resolution for both codes, and the lines for the magnetic energy almost overlap between 17  ms and 25  ms. Thereafter,  predicts slightly higher energies than . In the highest-resolution simulation, both codes reach a maximum magnetic field strength of ∼ 10^16  G shortly after the merger. The simulation performed with  reaches higher magnetic energies than  by a factor of two, which could be caused by the higher-order reconstruction in  that possibly resolves small-scale effects slightly better. The collapse time is a few milliseconds earlier with  than with . In general, both codes predict similar magnetic energies and field strengths at the end of the simulation.



§ CONCLUSIONS



We successfully extended the infrastructure of the  code for ideal GRMHD simulations employing a hyperbolic divergence cleaning scheme. Additionally, we compared results for well-known special-relativistic tests with established GRMHD codes: , , and . 
Overall, the tests showed good agreement between all codes. In the one-dimensional Balsara test, we observed only minor differences at the shock fronts, which can be attributed to the fact that the reconstruction methods used in the individual codes are either more diffuse or more oscillatory.
Similarly, we found minor differences in the two-dimensional cylindrical explosion and magnetic rotor tests, which we attribute to different reconstruction schemes for the fluid variables. The largest differences occur for , whereby higher resolution converges to the same results. The  code is the only one using the HLLD Riemann solver for these tests. The other codes use the HLL solver, which could explain the larger differences here.
We also compare results for the Kelvin-Helmholz instability test between  and . Both codes show that they are able to capture the vortex. In , the vortex forms earlier than in  and the perturbation grows faster. However, we show that we achieve the same growth rate in  when using the same Riemann solver as in .
In the three-dimensional spherical explosion test, we compare the conservation of magnetic flux over a refinement boundary for  and . Both codes apply schemes to conserve the flux. Still,  proves to be superior in magnetic flux conservation. Applying divergence cleaning in  leads to additional source terms in the evolution equation for the magnetic field, which are not matched and could explain the worse performance. 

We performed first BNS simulations with our new  implementation and compare our results with simulations performed with  using the same initial data. Although we obtain large differences in ejecta mass, with  having more than twice the amount of ejecta than , both codes predict similar values for the magnetic field. The amplification of the magnetic field during the merger is stronger with increasing resolution, as capturing KHI and MRI that trigger this amplification requires high resolution. Note that none of the simulations performed are in the convergent regime, because resolving the KHI requires an extremely high grid resolution, which is not feasible for full three-dimensional GRMHD simulations. 

We demonstrate that our divergence cleaning implementation in  is able to perform reliable simulations, including the magnetic field. The scheme is simpler than other divergence-free treatments, but it does not prevent the formation of magnetic monopoles, even though this corresponds to a constraint violation near zero. However, the finite resolution error associated with other variables is larger than the errors due to divergence cleaning in all cases considered here, and all these errors should converge to zero. Our results show that uncertainties stemming from different methods for the fluxes, shock capturing methods, or Riemann solvers are more severe. 



§ ACKNOWLEDGEMENTS


We thank I. Markin for implementing the fuka data reader in the  code and F. M. Fabbri for initiating the GRMHD implementation. KK and MS also thank K. Van Aelst for the fuka data reader in the .
TD and AN acknowledge support from the Deutsche Forschungsgemeinschaft, DFG, project number DI 2553/7. Furthermore, TD acknowledges funding from the EU Horizon under ERC Starting Grant, no. SMArt-101076369.
The simulations with  were performed on the national supercomputer HPE Apollo Hawk at the High Performance Computing (HPC) Center Stuttgart (HLRS) under the grant number GWanalysis/44189, on the GCS Supercomputer SuperMUC_NG at the Leibniz Supercomputing Centre (LRZ) [project pn29ba], and on the HPC systems Lise/Emmy of the North German Supercomputing Alliance (HLRN) [project bbp00049]. This work is in part supported by the Grant-in-Aid for Scientific Research (grant Nos. 23H01172, 23K25869 and 23H04900) of Japan MEXT/JSPS.





§ COMPARISON GENERAL RELATIVISTIC HYDRODYNAMIC SIMULATIONS





As benchmark, we run the BNS simulation with R1 once without magnetic field. From these simulations, we analyze the different merger times between  and . Additionally, we run the same set-up with the sacra code <cit.>. On the one hand,  implements an original Shibata-Nakamura version of the BSSN formulation <cit.> and employs the gauge condition based on the Nakamura variable F_i≡δ^jkγ̃_ij,k <cit.>. On the other hand, the sacra code implements the BSSN formulation employing the gauge conditions in a very similar way as . Also, the AMR implementation based on the box-in-box in sacra follows  <cit.>. Therefore, in the sacra run, we employ the same grid configuration as . In the  run, the grid configuration is a nested grid as described in the main text. 

We show the time evolution of the central rest-mass density in Fig. <ref>. The results from the corresponding simulations with magnetic fields are added as dashed lines. The different gauge condition changes the merger time in the  simulation and indeed the differences become smaller. Since the merger time is a gauge-dependent quantity, a fair comparison between our different codes with different settings is difficult. Therefore, we focus our comparison in Sec. <ref> on gauge-independent quantities such as the ejecta mass, magnetic energy, and magnetic field strength.





 Paper Title: Entropy production due to spacetime fluctuations.









thiagohenriquemoreira@ufg.br
QPequi Group, Institute of Physics, Federal University of Goiás, Goiânia, Goiás, 74.690-900, Brazil


lucas@qpequi.com
QPequi Group, Institute of Physics, Federal University of Goiás, Goiânia, Goiás, 74.690-900, Brazil

§ ABSTRACT 
Understanding the quantum nature of the gravitational field is undoubtedly one of the greatest challenges in theoretical physics. Despite significant progress, a complete and consistent theory remains elusive. However, in the weak field approximation —where curvature effects are small— we can explore some expected properties of such a theory. Particularly relevant to this study is the quantum nature of gravitational waves, which are represented as small perturbations in flat spacetime. In this framework, a quantum description of these perturbations, as a quantum field, is feasible, leading to the emergence of the graviton. Here we consider a non-relativistic quantum system interacting with such a field. We employ the consistent histories approach to quantum mechanics, which allows us to frame classical questions in a quantum context, to define a fluctuation relation for this system. As a result, thermodynamic entropy must be produced in the system due to its unavoidable interaction with the quantum fluctuations of spacetime. 


Entropy production due to spacetime fluctuations
    Lucas C. Célerihttps://orcid.org/0000-0001-5120-8176
    < g r a p h i c s >

    
=================================================================================














§ INTRODUCTION


Fluctuation theorems are strong results in nonequilibrium statistical mechanics, with applications spanning a wide variaty of fields. These range from human learning <cit.> and cosmology <cit.> to biology <cit.>, chemistry <cit.>, and quantum field theory <cit.>, to name just a few. Essentially, these theorems illustrate a strong connection between equilibrium quantities such as Helmholtz free energy and non-equilibrium ones, like energy and work. While various versions of these theorems exist, the most renowned are those formulated by Jarzynski <cit.> and Crooks <cit.>. For a historical overview of this topic, see Ref. <cit.>, and for technical details on the general theorems, consult Refs. <cit.>.

In the development of quantum field theory, one key implication is the inherent fluctuation of the quantum field energy. These fluctuations underpin various physical phenomena, such as the Casimir effect <cit.> and the Lamb shift <cit.>. When a system interacts with such fields and attempts to alter these fluctuations, it induces frictional forces, leading to dissipation and entropy production <cit.>.

In this context, a particularly intriguing aspect is the quantum nature of the gravitational field. Although a complete theory of quantum gravity remains elusive, it is possible to construct a consistent quantum field theory in the linear regime of Einstein’s theory, where curvature is small. This approach, known as the weak field approximation, introduces gravitons as the particles associated with the gravitational field <cit.>. The primary aim of this work is to examine entropy production in a system interacting with a weak quantum gravitational field. Specifically, by modeling the system as an open quantum system interacting with a bath of gravitons, we derive a fluctuation theorem for the system, the first moment of which is the entropy production. It is interesting to observe here that, contrary to other fields where energy fluctuates on a well defined spacetime, the quantum nature of the gravitational waves introduces fluctuations of the spacetime itself.

One major conceptual challenge in formulating a quantum fluctuation theorem is defining work. In classical systems, work is a well-defined quantity dependent on the trajectory and does not fluctuate, unless the system is immerse in a fluctuating environment, like a heat bath. In such cases, stochastic thermodynamics is required to describe the system and physical quantities emerge as averages over the fluctuations. In quantum systems, fluctuations are unavoidable, even in isolated systems. Moreover, the lack of the notion of a trajectory for a quantum system makes unambiguous the extension of the notion of thermodynamic quantities, like work.

For closed quantum systems, the definition of work using a two-time measurement scheme <cit.> is well-established. However, defining work for open quantum systems remains challenging. In this paper, we adopt the approach proposed in Ref. <cit.>, which defines work in a way analogous to classical systems by interpreting trajectories within quantum mechanics. To achieve this, we employ the decoherent histories formalism, primarily developed by Griffiths, Omnès, Gell-Mann, and Hartle <cit.>, with additional insights from Ref. <cit.>. This formalism was designed to describe quantum systems without relying on external observers. It employs Schrödinger's equation to help assigning probabilities to different histories, allowing us to consistently address classical questions within a quantum framework. One such question is the definition of work, which we approach by considering individual quantum histories, keeping in mind the classical thermodynamic idea of trajectory dependent quantity.

The remaining of the paper is structured as follows. First, we define our system of interest in the next section, Sec. <ref>, where the notation is also established. The decoherent histories approach to quantum mechanics is reviewed in Sec. <ref>. The main result, the fluctuation theorem, is presented in Sec. <ref>, where entropy production is directly related to quantum fluctuations of the spacetime. Finally, Sec. <ref> presents a physical discussion and our concluding remarks. We employ natural units in which ħ=c=k_B=G=1 and the metric signature (-,+,+,+).






§ LOCALIZED QUANTUM SYSTEM IN LINEARIZED QUANTUM GRAVITY



Our system of interest consists of two massive particles interacting with a gravitational wave within a flat spacetime background. Upon quantization, the fluctuations of the gravitational waves introduce a stochastic dynamic to the particles. In this section, we provide a brief overview of the approach presented in Refs. <cit.>, which derives a Langevin-like equation governing the geodesic separation between the two masses in the presence of the weak quantum gravitational field. For additional details, the reader is also referred to Ref. <cit.>.

Let us start by writing the classical action for a weak gravitational field interacting with a pair of massive particles. The total action is given by S = S_ m + S_ EH, where S_ m represents the action of the particles and their interaction with the gravitational field, and S_ EH is the Einstein–Hilbert action that describes the gravitational field itself. Denoting the coordinates of the particles as ζ^μ and ξ^μ, the action for the particles is expressed as

    S_ m   =-M∫ t √(-g_μνζ̇^μζ̇^ν)
       -m∫ t √(-g_μνξ̇^μξ̇^ν)-V(ξ).

We consider that the first particle (M) is free falling, while the second one (m) suffers the action of a potential V. Moreover, we take M≫ m such that the heavier particle is on shell with worldline ζ_0^μ(t). In this way, its dynamics becomes negligible. Furthermore, we take this particle to be at rest at the origin of our coordinate system, ζ_0^μ(t)=tδ_0^μ. Therefore, we can interpret the coordinate time t as its proper time. Since our interest lies on the lighter particle, from here on we simply refer to it as the system.

Under these assumptions it becomes appropriate to think of (t,ξ^i) as the Fermi normal coordinates defined with respect to the worldline of the heavier particle <cit.>. In these coordinates, we find <cit.>

    √(-g_μν(ξ)ξ̇^μξ̇^ν)≃1-1/2δ_ijξ̇^iξ̇^j+1/2R_i0j0(t,0)ξ^iξ^j

in the nonrelativistic limit. Here, R_μνρσ is the Riemann curvature tensor.

Now we write the spacetime metric as g_μν=η_μν+h_μν, with h_μν≪ 1, where η_μν=diag(-1,1,1,1) is the flat Minkowski metric while h_μν is a small perturbation. This is the weak field regime for the gravitational field, where the spacetime curvature is small compared with the relevant scales of the problem. Under this condition, the Einstein field equation reduces to a linear wave equation for h_μν. By taking advantage of the gauge invariance of linearized gravity, we choose to work in the transverse-traceless gauge such that the metric perturbation obeys h_0ν=0, η^μνh_μν=0 and ∂^μh_μν=0, with h_μν = h_μν - h η_μν/2 and h = h^μ_μ. In this gauge, the non-vanishing components of the Riemann curvature tensor are given by R_0i0j(t,0)=-ḧ_ij(t,0)/2, thus implying that the matter action takes the form

    S_ m=∫ t [1/2m δ_ijξ̇^iξ̇^j-V(ξ)+1/4m ḧ_ij(t,0)ξ^iξ^j],

where we have dropped non-dynamical terms.

It can be shown that the coupling with the weak gravity field leads to the total action given by <cit.>

    S   =π^2/8∫ t∫^3k ∑_s[q̇^2_s(t,k)-k^2q_s^2(t,k)] 
       +∫ t [1/2m δ_ijξ̇^iξ̇^j-V(ξ)] 
       +m/4∫ t ∫^3k ∑_sq_s(t,k) X^s(t,k),

where we have defined

    X^s(t,k)≡[2]t[ϵ_ij^s(k)ξ^iξ^j].

The gravitational field is expressed in terms of its two physical degrees of freedom, corresponding to the two polarization components by means of a Fourier transform

    h_ij(t,x)=∫^3k ∑_sϵ_ij^s(k)q_s(t,k)e^ik·x,

with ϵ_ij^s denoting the polarization tensor that satisfies the normalization ϵ_ij^s(k)ϵ^ij_s'(k)=2δ^s_s', the transversality k^iϵ_ij^s(k)=0 and the traceless δ^ijϵ_ij^s(k)=0 conditions.

Now, we assume that the initial state of the entire system is a product state, implying that initially there are no correlations between any of the components of the system —the two particles and the field. Using the Feynman-Vernon influence functional approach to open quantum systems <cit.>, we can express all (weak) quantum gravitational influences on the system through the influence functional

    e^iS_ IF[x,x'] = ∫𝒟𝒩P[𝒩]e^i∫ t𝒩^ij(t)[x_ij(t)-x_ij'(t)]
       × e^i/2∫ t t' [x_ij(t)-x_ij'(t)]D^ijkl(t,t')[x_kl(t')+x_kl'(t')],

where x_ij(t)=ξ_i(t)ξ_j(t). Here, 𝒩^ij(t) is a stochastic variable with Gaussian probability density P[𝒩] such that 𝒩^ij(t)=0 and 𝒩^ij(t)𝒩^kl(t')=N^ijkl(t,t').
The quantities D^ijkl(t,t') and N^ijkl(t,t') are called the dissipation and noise kernels, respectively.

In summary, the system is found to be described by a stochastic effective action given by <cit.>

    S_ SEA=S_ p[ξ]-S_ p[ξ']+S_ IF,

with S_ p[ξ]=∫ t [1/2m δ_ijξ̇^iξ̇^j-V(ξ)] and S_ IF being the influence action defined in Eq. (<ref>). The equation of motion is obtained by setting δ S_ SEA=0 <cit.>, yielding

    mξ_i(t)-2∫ t' D_ijkl(t,t')ξ^j(t)ξ^k(t')ξ^l(t') 
    
        =-Vξ^i+2𝒩_ij(t)ξ^j(t).

Note that the second term on the left-hand side introduces a non-linearity, making this integral differential equation very difficult to solve. It is for this reason that we choose to work on a perturbative regime, thus dropping the term involving the dissipation kernel as it was done in Ref. <cit.>. In this regime, we obtain

    mξ_i(t)=f_i(t)+2𝒩_ij(t)ξ_0^j(t),

where f_i is defined such that V(ξ)=-f_i(t)ξ^i(t) and ξ_0^i is the solution to mξ_0^i=0.

The above equation is a Langevin-like equation involving the stochastic force term 2𝒩_ij(t)ξ_0^j(t). The well-known solution for the initial conditions ξ^i(0)=ξ̇^i(0)=0 is given by

    ξ_i(t)   =1/m∫_0^t t' (t-t')f_i(t') 
       +2/m∫_0^t  t' (t-t')𝒩_ij(t')ξ_0^j(t').

This can be viewed as the classical limit for the geodesic separation between the two masses, where the stochastic behavior arises from the quantum fluctuations of the gravitational field. The key observation here is that, under certain conditions, we can employ the solutions to the Langevin equation to define work for the quantum system similarly to how it is defined for classical systems. This approach relies significantly on the concept of trajectories in space. To explore how this is feasible, we will work within the consistent (or decoherent) histories approach to quantum mechanics. This is a crucial step that allows us to properly build a fluctuation theorem.






§ DECOHERENT HISTORIES



Shortly, a quantum mechanical history is a sequence of events at successive instants of time, being characterized by a sequence of successive projection operators. Let us begin by considering a closed quantum system initially described by a density matrix ρ_0≡ρ(t_0) and we wish to consider histories consisting of n projections at times t_1<t_2<…<t_n, with t_1>t_0. In order to establish whether or not we can assign probabilities to histories (and how we do this), we need to look at the so-called decoherence functional. For a pair of histories, [α] and [β], where [α] denotes the sequence of events {α_i}_i=1^n at times { t_i}_i=1^n, and analogously for [β], the decoherence functional is defined as

    𝒟[α,β]=Tr[P_α_n… P_α_1ρ_0P_β_1… P_β_n],

Here, P_α_k is some set of projection operators compatible with event α_k (at time t_k), P_α_k=U^†(t_k,t_0)P_α_0U(t_k,t_0), where U(t_k,t_0) is the unitary time evolution operator for the system. Note that we can have different types of projections at different moments of time (although in this work we are mainly concerned with projections in position) <cit.>. The decoherence functional can be written in a more compact form as <cit.>

    𝒟[α,β]=TrT[∏_j=1^nP_β_j]T[∏_i=1^nP_α_i]ρ_0,

where T (T) stands for time (anti-time) ordering. It is straightforward to show that the decoherence functional satisfies the following properties

    
  * 𝒟[α,β]=𝒟^*[β,α],
    
  * ∑_[α],[β]𝒟[α,β]=1,
    
  * 𝒟[α,α]≥0,
    
  * ∑_[α]𝒟[α,α]=1,

where we are employing the shorthand notation ∑_[α] ≡ ∑_α_1∑_α_2…∑_α_n.

The last two properties suggest that we identify the diagonal elements 𝒟[α,α] with the probability for the history (ρ_0,t_0)→(α_1,t_1)→…→(α_n,t_n), namely p(α)=𝒟[α,α] <cit.>. However, in order for such elements to represent a genuine probability, Kolmogorov's third axiom of probability theory —the probability sum rule— must be satisfied. It is not difficult to show that

    p(α∨β)   =𝒟[α,α]+𝒟[β,β]+2Re𝒟[α,β] 
       =p(α)+p(β)+2Re𝒟[α,β].

Clearly, the sum rule is not obeyed in general due to the last term on the right-hand side. This term characterizes the quantum mechanical interference between distinct histories and it is in this sense that interference effects prevents us to give a meaningful notion of trajectory in quantum mechanics. However, this notion acquires meaning when there is strong decoherence, case in which 𝒟[α,β] vanishes for distinct trajectories. One may then write the fundamental formula for the quantum mechanics of histories as

    Re𝒟[α,β]=p(α)δ_α_1,β_1…δ_α_n,β_n.

This equation provides the necessary and sufficient condition for assigning probabilities to individual histories, as well as specifies what those probabilities are. However, for most applications of the formalism, it is often sufficient to assert that a pair of mutually exclusive histories is consistent when Re𝒟[α,β]<<𝒟[α,α], 𝒟[β,β] for [α]≠ [β] <cit.>.

From now on, let us consider projections in position basis, which is a kind of history implemented naturally in the path integral formalism. The projectors are represented by window functions P_α_k→ w_α[x(t_k)], which are equal to one if the configuration at t_k satisfies the requirement of history [α] and zero otherwise <cit.>. Equation (<ref>) can then be written as

    𝒟[α,β]=   ∫𝒟y𝒟y' e^i(S[y]-S[y'])ρ(y(0),y'(0),0) 
       ×∏_iw_α[y(t_i)]∏_jw_β[y'(t_j)].

Here S is the action for the total system described by some generic coordinate y.

Let us apply this to the problem at hand. By factoring the total action as shown in Eq. (<ref>), the integration over the gravitational variables leads to the influence functional given in Eq. (<ref>). We can then consider [α] and [β] as histories represented by two non-overlapping paths ξ^(1)(t) and ξ^(2)(t), in which case the window functions would be simply delta functions at each time instant. For this scenario, the decoherence functional was analyzed in Ref. <cit.>, considering four possible types of initial graviton states and estimating the decoherence time as a function of the gravitational energy scale (either a cutoff Λ_ g∼ξ^-1 <cit.> or the temperature β^-1 of a thermal state of gravitons)[The analysis done in Ref. <cit.> considered that the particle interacting with the gravitons was described by additional dynamical internal degrees of freedom that also coupled with the gravitational field. This coupling turned out to increase even more the decoherence rate. Although this is a feature to be taken advantage of, the internal degrees of freedom noise kernel also introduces a non-trivial modification to the equation of motion.]. Decoherence induced by gravitons was also analyzed in Ref. <cit.>, where it was established that decoherence occurs for particles with mass much greater than the Planck mass M_ pl∼10^-8 kg. For such particles, this allows us to give meaning to the concept of trajectory for the quantum system since the interference terms may be neglected due to decoherence. Within this scenario we derive a quantum fluctuation theorem for our system. 






§ FLUCTUATION THEOREM



In order to discuss the validity of the fluctuation theorem, we proceed as in Ref. <cit.>. By fluctuation theorem we mean Jarzynski-like relation, which is the integral form of Crook's fluctuation theorem <cit.>. For closed quantum systems, the derivation of the theorem relies on the hypothesis of initial thermal state of the entire system. However, our analysis of a quantum particle interacting with the gravitons was built under the assumption of initial product state, which is not a thermal state. 

To circumvent this issue, we take the initial time to be t_0=-∞. At this time instant, we assume that the total state of the system is described by a tensor product, ρ(-∞)=ρ_ sys(-∞)⊗ρ_ grav(-∞), where ρ_ grav is a thermal state of gravitons. We then allow the system to evolve according to the total action with f^i(t)=f^i(0) for t<0, so that we obtain a total thermal state at t=0. At t=0, the driving force starts changing according to some arbitrary protocol until a given time instant t=τ. Assuming the decoherence conditions discussed in the previous section hold, the work performed on the system in the interval [0,τ] is defined as

    W≡-∫_0^τ t ḟ_i(t)ξ^i(t),

where ξ^i(t) is the solution to the Langevin-like equation given in Eq. (<ref>). Physically, this is the work that must be performed on the system in order to move it through the bath of gravitons. By using the explicit forms of the solutions we can write W = W_f + W_n, with

    W_f=-1/m∫_0^τ t  t' ḟ^i(t)g(t-t')f_i(t'),

denoting the work associated with the driven protocol, while 

    W_n = -2/m∫_0^τ t t' ḟ^i(t)g(t-t')𝒩_ij(t')ξ^j_0(t'),

contains the quantum fluctuations coming from the noise. In these equations, g(t-t')=(t-t')θ(t-t'), with θ(x) being the step function. 

While W_f is entirely determined by the driven protocol, which is arbitrary, W_n encompasses both the driven contribution, as it includes f, and the noise term resulting from the interaction with the gravitons. Since gravity cannot be shielded and its quantum nature inherently involves fluctuations, this latter contribution, unlike the one arising from the driven protocol, is unavoidable.

We note that the work performed on the system is linear in 𝒩_ij(t), which is a Gaussian random process. Consequently, W itself must also be a Gaussian random variable. Its statistics are characterized by the first two moments W and σ_W^2=W^2-W^2. Therefore, we can write

    P(W)=1/√(2πσ_W^2)e^-(W-W)^2/2σ_W^2.

The brackets denote stochastic average with a Gaussian probability density.

Before we proceed, it is important to emphasize that the average with P(W) is equivalent to the stochastic average with P(𝒩) due to the linear dependence of W on 𝒩_ij(t). In conventional formulations of the classical fluctuation theorem, such stochastic forces are not typically considered, and the probabilistic nature of the work done on the system arises from the assumption of an initial thermal state. Each sampling from the initial state produces a trajectory, and the average is taken over an ensemble of such realizations. For the quantum fluctuation theorem, the inherent quantum uncertainty of the quantum state introduces an additional probabilistic aspect. However, we have opted to use an equivalent initial state preparation method based on a product initial state for the total system. This choice replaces the system's dependence on the initial state with the properties of noise statistics, thereby leaving only one probabilistic element instead of two. For further discussion on this topic, see Ref. <cit.>.

The moments of the work distribution can be directly computed as follows. First, since 𝒩^ij(t)=0, we have

    W=-1/m∫_0^τ dtdt' ḟ^i(t)g(t-t')f_i(t') = W_f.

Additionally, by using 𝒩^ij(t)𝒩^kl(t')=N^ijkl(t,t'), which is the noise kernel of gravitons, we also find

    σ_W^2=∫_0^τ dtdt' ḟ^i(t)σ_ij(t,t')ḟ^j(t'),

where

    σ_ij(t,t')   =   4/m∫_0^τ dt_1dt_2 g(t-t_1)g(t'-t_2) 
       ×    N_ijkl(t_1,t_2)ξ^k_0(t_1)ξ^l_0(t_2).

Note that, while W is independent of the noise kernel of gravitons, the second moment σ_W^2 is not. The noise kernel for an initial thermal distribution of gravitons was computed explicitly in Refs. <cit.>.

With the expressions for the first and second moments of work, the probability distribution is completely specified. One can now compute

    e^-β W=∫ dW P(W)e^-β W,

where β=T^-1 is the inverse temperature associated with the gravitons. Since Jarzynski equality holds for the entire system, we must have e^-β W=e^-βΔ F, where Δ F=F(τ)-F(0) is the total free energy difference. F(t) is defined with respect to the Hamiltonian of the system at time t, under the assumption that the temperature of the gravitons does not change. 

A direct calculation leads to

    e^-β W=e^-β(W_f-βσ_W^2/2),

allowing us to identify the free energy difference of the total system as

    Δ F=W_f-β/2σ_W^2,

which is a sum of a term that depends exclusively on the force protocol f [see Eq. (<ref>)] and another that also contains the contribution of gravitons via the noise kernel [see Eqs. (<ref>) and (<ref>)].

From this we can directly determine the entropy production Σ = β( W-Δ F) = β( W_f-Δ F), resulting in

    Σ = β^2/2σ_W^2.

This result intuitive says that spacetime fluctuations determine the entropy production in the system. There are two interesting limits to discuss here. First, when the gravitational field is turned off, σ_W^2 = 0, thus implying that there is no entropy production in the system. This is expected since we are dealing with a quantum system that is driven to follow a trajectory in spacetime. Thus, the absence of gravitational waves must imply that there will be no entropy being produced in the system. Secondly, by turning off the driven force, f=0, we also have Σ=0. This can be understood because, without the driving force, the system is in equilibrium with the fluctuating environment, and no net work or entropy production should be observed. 






§ DISCUSSIONS



Here, we analyze a driven quantum system interacting with a (weak) quantized gravitational field and derive an expression for the entropy production due to gravity by using an integral fluctuation theorem within the framework of decoherent histories. In the regime of low energy scales, perturbations of spacetime are treated quantum mechanically as a quantum field theory, leading to quantum fluctuations of spacetime. These fluctuations are unavoidable, as no system can be shield from gravity —there is no equivalent of a Faraday cage for gravitational effects. Thus, every system in nature experiences entropy production not only due to processes that potentially drive the system out of equilibrium but also because of the fundamental fluctuations of the spacetime in which the system live. The main result of this work is to characterize these fluctuations.

It is important to note that the key aspect of decoherent histories is the need for strong decoherence to eliminate interference among histories. Although crucial for the definition of work, this requirement is not a severe limitation in our case, as decoherence induced by quantum fluctuations of the gravitational field is indeed strong <cit.>. Furthermore, quantum particles typically possess additional internal degrees of freedom, which further increases the decoherence rate <cit.>.

An interesting question for future investigation is the impact of considering a curved spacetime instead of flat spacetime, as was done here. While this introduces many complexities, it should be possible to uncover some fundamental features of entropy production <cit.>.

The result that entropy is produced in quantum systems due to their interaction with gravitational waves could imply decoherence of spatial superpositions of massive particles. This raises the question of how long such superpositions persist. As shown in Ref. <cit.>, these effects are significant and should be considered when discussing the gravitational field generated by such states <cit.>.

Finally, we observed that no entropy is produced when the system is in free fall, in the absence of an external agent. This observation aligns with the equivalence principle. However, this does not imply that quantum spacetime fluctuations are undetected in such a frame, and further investigation is necessary. For example, instead of focusing on the work done by an external agent, one could analyze the work done by the stochastic force resulting from gravitational fluctuations. This distribution of work is not expected to be Gaussian, making the analysis more complex.






§ ACKNOWLEDGEMENTS

This work was supported by the National Institute for the Science and Technology of Quantum Information (INCT-IQ), Grant No. 465469/2014-0, by the National Council for Scientific and Technological Development (CNPq), Grants No 308065/2022-0, and by Coordination of Superior Level Staff Improvement (CAPES).





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 Paper Title: Comparing a gauge-invariant formulation and a "conventional complete gauge-fixing approach" for $l=0,1$ mode perturbations on the Schwarzschild background spacetime.









equationsection



















  Gravitational-Wave Science Project,
  National Astronomical Observatory of Japan,

  2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan











§ ABSTRACT 
  Comparison of the gauge-invariant formulation for l=0,1-mode
  perturbations on the Schwarzschild background spacetime proposed in
  [K. Nakamura, Class. Quantum Grav. 38 (2021), 145010.] and a
  “conventional complete gauge-fixing approach” in which we use the
  spherical harmonic functions Y_lm as the scalar harmonics from
  the starting point is discussed.
  Although it is often said that “gauge-invariant formulations in
  general-relativistic perturbations are equivalent to complete
  gauge-fixing approaches,” as the result of this comparison, we conclude
  that the derived solutions through the proposed gauge-invariant
  formulation and those through a “conventional complete gauge-fixing
  approach” are different.
  It is pointed out that there is a case where the boundary conditions
  and initial conditions are restricted in a conventional complete
  gauge-fixing approach.



  Comparing a gauge-invariant formulation and a “conventional
  complete gauge-fixing approach” for l=0,1 mode perturbations on the
  Schwarzschild background spacetime

    
  Kouji Nakamura
  [E-mail address: dr.kouji.nakamura@gmail.com]

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



















§ INTRODUCTION











Through the ground-based gravitational-wave detectors <cit.>,
many events of gravitational waves, mainly from black hole-black hole
coalescences, have now been detected.
We are now at the stage where there is no doubt about the existence of
gravitational waves due to their direct observations.
One of the future directions of gravitational-wave astronomy will be a
precise science through the statistics of many events.
Toward further development of gravitational-wave science, the projects
of future ground-based gravitational-wave
detectors <cit.> are also
progressing to achieve more sensitive detectors and some projects of
space gravitational-wave antenna are also
progressing <cit.>.
Although there are many targets of these detectors, the
Extreme-Mass-Ratio-Inspiral (EMRI), which is a source of gravitational
waves from the motion of a stellar mass object around a supermassive
black hole, is a promising target of the Laser Interferometer Space
Antenna <cit.>.
Since the mass ratio of this EMRI is very small, we can describe the
gravitational waves from EMRIs through black hole
perturbations <cit.>.
Furthermore, the sophistication of higher-order black hole
perturbation theories is required to support gravitational-wave
physics as a precise science.
The motivation of our series of papers,
Refs. <cit.>
and this paper, is in this theoretical sophistication of black hole
perturbation theories toward higher-order perturbations.





Although realistic black holes have their angular momentum and we have
to consider the perturbation theory of a Kerr black hole for direct
applications to the EMRI, we may say that further sophistications are
possible even in perturbation theories on the Schwarzschild background
spacetime.
From the pioneering works by Regge and
Wheeler <cit.> and
Zerilli <cit.>,
there have been many studies on the perturbations in the Schwarzschild
background
spacetime <cit.>.
In these works, perturbations are decomposed through the spherical
harmonics Y_lm because of the spherical symmetry of the background
spacetime, and l=0,1 modes should be separately treated.
These modes correspond to the monopole and dipole perturbations.
Due to these separate treatments, “gauge-invariant” treatments
for l=0 and l=1 modes were unclear.





Owing to this situation, in the previous
papers <cit.>,
we proposed the strategy of the gauge-invariant treatments of these
l=0,1 mode perturbations, which is declared as
Proposal <ref>
in Sec. <ref>
of this paper below.
One of the important premises of our gauge-invariant perturbations is
the distinction between the first-kind gauge and the second-kind
gauge.
The first-kind gauge is essentially the choice of the coordinate
system on the single manifold, and we often use this first-kind gauge
when we predict or interpret the measurement results of experiments
and observations.
On the other hand, the second-kind gauge is the choice of the
point identifications between the points on the physical spacetime
_ϵ and the background spacetime .
This second-kind gauge has nothing to do with our physical spacetime
_ϵ.
Although this difference is extensively explained in the Part I
paper <cit.>, we also emphasize this difference
in Sec. <ref> of this paper.
The proposal in the Part I paper <cit.> is a
part of our developments of the general formulation of a higher-order
gauge-invariant perturbation theory on a generic background spacetime
toward unambiguous sophisticated nonlinear general-relativistic
perturbation theories <cit.>.
Although we have been applied this general framework to cosmological
perturbations <cit.>,
we applied it to black hole perturbations in the series of papers,
i.e.,
Refs. <cit.>
and this paper.
Even in cosmological perturbation theories, the same problem as the
above l=0,1-mode problem exists as gauge-invariant treatments of
homogeneous modes of perturbations.
In this sense, we can expect that the proposal in the previous
paper <cit.> will be a clue to the same problem
in gauge-invariant perturbation theory on the generic background
spacetime.





In the Part I paper <cit.>, we also derived the
linearized Einstein equations in a gauge-invariant manner following
Proposal <ref>.
Perturbations on the spherically symmetric background spacetime are
classified into even- and odd-mode perturbations.
In the same paper <cit.>, we also gave the strategy
to solve the odd-mode perturbations including l=0,1 modes.
Furthermore, we also derived the formal solutions for the l=0,1
odd-mode perturbations to the linearized Einstein equations following
Proposal <ref>.
In the Part II paper <cit.>, we gave the
strategy to solve the even-mode perturbations including l=0,1 modes,
and we also derive the formal solutions for the l=0,1 even-mode
perturbations following
Proposal <ref>.
In the Part III paper <cit.>, we find the fact
that the derived solutions in the Part II
paper <cit.> realize the linearized version of
two exact solutions: one is the Lemaître-Tolman-Bondi (LTB)
solution <cit.> and the other is the
non-rotating
C-metric <cit.>.
Due to this fact, we conclude that the solutions for even-mode
perturbations derived in the Part II
paper <cit.> are physically reasonable.
This series of papers is the full paper version of our short
paper <cit.>.
Furthermore, brief discussions on the extension to the higher-order
perturbations are given in the short
paper <cit.>.





On the other hand, it is well-known the fact that we cannot construct
gauge-invariant variables for l=0,1 modes in a similar manner
to l≥ 2 modes if we decompose the metric perturbations through
the spherical harmonics as the scalar harmonics from the starting
point.
For this reason, we usually use gauge-fixing approaches.
Furthermore, it is often said that “gauge-invariant formulations in
general-relativistic perturbations are equivalent to complete
gauge-fixing approaches.”
In this paper, we check this statement through the comparison of our
proposed gauge-invariant formulation, in which we introduce singular
harmonics at once and regularize them after the derivation of the
mode-by-mode Einstein equation, and a “conventional complete
gauge-fixing approach”, in which we use the spherical harmonic
functions Y_lm as the scalar harmonics S_δ from the
starting point.
As the result of this comparison, we conclude that our gauge-invariant
formulation and the above “conventional complete gauge-fixing
approach” are different, though these two formulations lead similar
solutions for l=0,1-mode perturbations.
More specifically, there is a case where the boundary conditions and
initial conditions are restricted in a “conventional complete
gauge-fixing approach” where we use the decomposition of the metric
perturbation by the spherical harmonics Y_lm from the starting point.





The organization of this paper is as follows.
In Sec. <ref>, we briefly review the
premise of our series of
papers <cit.>,
which are necessary for the ingredients of this paper.
We also emphasize the difference between the concepts of the first-
and the second-kind gauges and summarize the linearized Einstein
equations for l=0,1 modes and their formal solutions in
Sec. <ref>.
In Sec. <ref>, we specify the rule of our
comparison between our gauge-invariant formulation and a conventional
gauge-fixing approach and summarize the gauge transformation rules for
the metric perturbations of l=0,1 modes, because the above statement
 “gauge-invariant formulations in
general-relativistic perturbations are equivalent to complete
gauge-fixing approaches” includes some ambiguous.
In Sec. <ref>, we discuss the
linearized Einstein equations for l=1 odd-modes and their solutions
in the conventional gauge-fixing approach.
In Sec. <ref>, we discuss the
linearized Einstein equations for l=1 even-modes and their solutions
in the conventional gauge-fixing approach.
In Sec. <ref>, we derive the solution
to the linearized Einstein equations for l=0 modes through the
complete gauge-fixing and discuss the comparison with linearized LTB
solution.
Sec. <ref> is devoted to the summary of
this paper and discussions based on our results.





We use the notation used in the previous
papers <cit.>
and the unit G=c=1, where G is Newton's constant of gravitation
and c is the velocity of light.











§ BRIEF REVIEW OF A GAUGE-INVARIANT TREATMENT OF L=0,1 MODES











In this section, we briefly review the premises of our series of
papers <cit.>
which are necessary for the ingredients of this paper.
In Sec. <ref>, we review
our framework of the gauge-invariant perturbation
theory <cit.>.
This is an important premise of the series of our
papers <cit.>
and this paper.
In Sec. <ref>, we review the
gauge-invariant perturbation theory on spherically symmetric
spacetimes which includes our proposal in
Refs. <cit.>.
In Sec. <ref>, we summarize
the l=1 odd-mode linearized Einstein equations and their “formal”
solutions.
In Sec. <ref>, we summarize
the l=0,1 even-mode linearized Einstein equations and their
“formal” solutions.
The equations and their solutions in
Sec. <ref> and
Sec. <ref> are derived
based on our proposal in Refs. <cit.>.
These are necessary for the arguments within this paper.










 §.§ General framework of gauge-invariant perturbation theory










In any perturbation theory, we always treat two spacetime manifolds.
One is the physical spacetime (_ ph,g̅_ab),
which is identified with our nature itself, and we want to describe
this spacetime (_ ph,g̅_ab) by perturbations.
The other is the background spacetime (,g_ab),
which is prepared as a reference by hand.
Note that these two spacetimes are distinct.
Furthermore, in any perturbation theory, we always write equations
for the perturbation of the variable Q as follows:

    Q(“p) = Q_0(p) + δ Q(p).

Equation (<ref>) gives a
relation between variables on different manifolds.
Actually, Q(“p) in
Eq. (<ref>) is a variable on
the physical spacetime _ϵ=_ ph, whereas
Q_0(p) and δ Q(p) are variables on the background spacetime
.
Because we regard Eq. (<ref>) as
a field equation, Eq. (<ref>)
includes an implicit assumption of the existence of a point
identification map _ϵ :
→_ϵ :
p∈↦“p∈_ϵ.
This identification map is a gauge choice in
general-relativistic perturbation theories.
This is the notion of the second-kind gauge pointed out by
Sachs <cit.>.
Note that this second-kind gauge is a different notion from the
degree of freedom of the coordinate transformation on a single
manifold, which is called the first-kind
  gauge <cit.>.
This distinction between the first- and the second-kind of gauges
extensively explained in the Part I paper <cit.> and
is also important to understand the ingredients of this paper.





To compare the variable Q on _ϵ with its
background value Q_0 on , we use the pull-back
_ϵ^* of the identification map
_ϵ : →_ϵ and
we evaluate the pulled-back variable _ϵ^*Q on the
background spacetime .
Furthermore, in perturbation theories, we expand the pull-back
operation _ϵ^* to the variable Q with respect
to the infinitesimal parameter ϵ for the perturbation as

    _ϵ^*Q
      =
      Q_0
      + ϵ^(1)_ Q
      + O(ϵ^2)
      .

Equation (<ref>) are evaluated on
the background spacetime .
When we have two different gauge choices _ϵ and
_ϵ, we can consider the gauge transformation,
which is the change of the point-identification
_ϵ→_ϵ.
This gauge transformation is given by the diffeomorphism
Φ_ϵ :=
(_ϵ)^-1∘_ϵ
:  → .
Actually, the diffeomorphism Φ_ϵ induces a pull-back from
the representation _ϵ^*Q_ϵ to the
representation _ϵ^*Q_ϵ as
_ϵ^*Q_ϵ(q)=Φ_ϵ^*_ϵ^*Q_ϵ(q)
at any point q∈.
From general arguments of the Taylor
expansion <cit.>,
the pull-back Φ_ϵ^* is expanded as

    _ϵ^*Q_ϵ(q)
         =   _ϵ^*Q_ϵ(q)
      + ϵ_ξ_(1)_ϵ^*Q_ϵ(q)
      + O(ϵ^2),

where ξ_(1)^a is the generator of Φ_ϵ.
From Eqs. (<ref>) and
(<ref>), the gauge transformation for the first-order
perturbation ^(1)Q is given by

    ^(1)_ Q(q) - ^(1)_ Q(q)    =   _ξ_(1)Q_0(q)

at any point q∈.
We also employ the order by order gauge invariance as a
concept of gauge invariance <cit.>.
We call the kth-order perturbation ^(k)_Q as
gauge invariant if and only if

    ^(k)_ Q(q) = ^(k)_ Q(q)

for any gauge choice _ϵ and _ϵ at any
point of q∈.





Based on the above setup, we proposed a formulation to construct
gauge-invariant variables of higher-order
perturbations <cit.>.
First, we expand the metric on the physical spacetime
_ϵ, which was pulled back to the background
spacetime  through a gauge choice _ϵ as

    ^*_ϵg̅_ab   =   
      g_ab + ϵ_h_ab
      + O(ϵ^2).

Although the expression (<ref>) depends
entirely on the gauge choice _ϵ, henceforth,
we do not explicitly express the index of the gauge choice
_ϵ in the expression if there is no
possibility of confusion.
The important premise of our formulation of higher-order
gauge-invariant perturbation theory was the following
conjecture <cit.> for the linear
metric perturbation h_ab:

  
  If the gauge transformation rule for a perturbative pulled-back
  tensor field h_ab from the physical spacetime _ϵ
  to the background spacetime  is given by _h_ab
  - _h_ab = _ξ_(1)g_ab with the
  background metric g_ab, there then exist a tensor field
  _ab and a vector field Y^a such that h_ab is
  decomposed as h_ab =: _ab + _Yg_ab,
  where _ab and Y^a are transformed as
  __ab - __ab = 0 and
  _Y^a - _Y^a = ξ^a_(1)
  under the gauge transformation, respectively.

We call _ab and Y^a as the
gauge-invariant and gauge-dependent parts
of h_ab, respectively.





The proof of Conjecture <ref> is
highly
nontrivial <cit.>,
and it was found that the gauge-invariant variables are essentially
non-local.
Despite this non-triviality, once we accept
Conjecture <ref>,
we can decompose the linear perturbation of an arbitrary tensor field
_^(1)Q, whose gauge transformation is given by
Eq. (<ref>), through the gauge-dependent part Y_a of
the metric perturbation in
Conjecture <ref> as

    _^(1)Q = ^(1) + __YQ_0,

where ^(1) is the gauge-invariant part of the perturbation
_^(1)Q.
As examples, the linearized Einstein tensor
_^(1)G_a^  b and the linear perturbation of the
energy-momentum tensor _^(1)T_a^  b are also
decomposed as

    _^(1)G_a^  b
      =
      ^(1)_a^  b[] + __YG_a^  b
      ,
        _^(1)T_a^  b
      =
      ^(1)_a^  b + __YT_a^  b
      ,

where G_ab and T_ab are the background values of the Einstein
tensor and the energy-momentum tensor,
respectively <cit.>.
Using the background Einstein equation
G_a^  b=8π T_a^  b, the linearized Einstein equation
_^(1)G_ab=8π_^(1)T_ab is
automatically given in the gauge-invariant form

    ^(1)_a^  b[]
      =
      8 π^(1)_a^  b[,ϕ]

even if the background Einstein equation is nontrivial.
Here, “ϕ” in Eq. (<ref>)
symbolically represents the matter degree of freedom.





Finally, we comment on the coordinate
transformation induced by the gauge transformation
Φ_ϵ of the
second-kind <cit.>.
To see this, we introduce the coordinate system
{O_α,ψ_α} on the background spacetime ,
where O_α are open sets on the background spacetime and
ψ_α are diffeomorphisms from O_α to ^4
(4=).
The coordinate system {O_α,ψ_α} is the set of
collections of the pair of open sets O_α and diffeomorphism
ψ_α:O_α↦^4.
If we employ a gauge choice _ϵ of the second kind, we
have the correspondence of the physical spacetime
_ϵ and the background spacetime .
Together with the coordinate system ψ_α on ,
this correspondence _ϵ between _ϵ and
 induces the coordinate system on _ϵ.
Actually, _ϵ(O_α) for each α is an
open set of _ϵ.
Then, ψ_α∘_ϵ^-1 becomes a
diffeomorphism from an open set _ϵ(O_α)⊂_ϵ to ^4.
This diffeomorphism ψ_α∘_ϵ^-1
induces a coordinate system of an open set on _ϵ.
When we have two different gauge choices _ϵ and
_ϵ of the second kind,
ψ_α∘_ϵ^-1 :
_ϵ ↦ ^4 ({x^μ}) and
ψ_α∘_ϵ^-1 : _ϵ
↦ ^4 ({y^μ}) become different coordinate systems
on _ϵ.
We can also consider the coordinate transformation from the coordinate
system ψ_α∘_ϵ^-1 to another
coordinate system ψ_α∘_ϵ^-1.
Because the gauge transformation
_ϵ→_ϵ is induced by the
diffeomorphism Φ_ϵ :=
(_ϵ)^-1∘_ϵ, this
diffeomorphism Φ_ϵ induces the coordinate transformation
as

    y^μ(q) := x^μ(p) = ((Φ_ϵ^-1)^*x^μ)(q)

in the passive point of
view <cit.>,
where p, q ∈  are identified to the same point
“p ∈ _ϵ by the gauge choices
_ϵ and _ϵ, respectively.
If we represent this coordinate transformation in terms of the Taylor
expansion (<ref>), we have the coordinate transformation

    y^μ(q) = x^μ(q) - ϵξ^μ_(1)(q) + O(ϵ^2).

We should emphasize that the coordinate transformation
(<ref>) is not the starting
point of the gauge transformation but a result of the above framework.





On the other hand, we may consider the point replacement by the
one-parameter diffeomorphism s=Ψ_λ(r), where
s,r∈_ϵ and λ is an infinitesimal parameter
satisfying Ψ_λ=0(r)=r.
The pull-back Ψ_λ^* of any tensor field Q on
_ϵ is given by

    Q(s)
         =   
          Q(Ψ_λ(r)) = (Ψ_λ^*)Q(r)
          
       =   
          Q(r) + λ._ζQ|_λ=0(r)
          + O(λ^2)
          ,

where ζ^a is the generator of the pull-back
Ψ_λ^*.
Equation (<ref>) is just the formula of
the Taylor expansion on the manifold _ϵ without using
any coordinate system.
At the same time, Equation (<ref>) is the
definitions of the Lie derivative _ζ and its generator
ζ^a.
In many literatures, this formula is derived from the coordinate
transformation

    y^μ(s) := x^μ(r) + λζ^μ(r) + O(λ^2)

as explained in the Part I paper <cit.>.
The formula (<ref>) of the Taylor
expansion defined without any coordinate system and the second
term in the right-hand side of Eq. (<ref>)
represents the actual difference of the tensor fields Q(s) and
Q(r) in the different point s≠ r and its physical meaning.





If we consider a formal metric decomposition
g̅_ab=^(0)g_ab+λ h_ab+O(λ^2) within
_ϵ as an example of the tensor field Q in
Eq. (<ref>), the formula
(<ref>) is given by

    ^(0)g_ab(s) + λ h_ab(s)
      =
      ^(0)g_ab(r) + λ(
      h_ab(r) + ._ζ^(0)g_ab|_r)
      + O(λ^2).

Since the formula of the Taylor expansion
(<ref>) is derived from the infinitesimal
coordinate transformation rule
(<ref>) in many
literatures, the term ._ζ^(0)g_ab|_r
in the right-hand side of
Eq. (<ref>) is often
called “the degree of freedom of the coordinate transformation” and
“unphysical degree of freedom”.
However, since the formula
(<ref>) is just the
re-expression of the Taylor expansion
(<ref>) defined without any coordinate
system and the term ._ζ^(0)g_ab|_r
in Eq. (<ref>) should have
its physical meaning.
If we insist that the term
._ζ^(0)g_ab|_r which appears due
to the point replacement on the single manifold is “unphysical”,
this directly leads the statement that “the famous arguments of the
Killing vector and the symmetry of the spacetime are physically
meaningless.”
For this reason, we have to emphasize that we cannot regard the
second term of the right-hand side of
Eq. (<ref>) as an “unphysical degree of
freedom.”





In our series of
papers <cit.>
and in this paper, we classify the term
._ζ^(0)g_ab|_r in
Eq. (<ref>) as one of
gauge-degree of freedom of the first kind, since this term can
be eliminate the infinitesimal coordinate transformation
(<ref>) which is the
coordinate transformation within the single manifold _ϵ.
Furthermore, Equation
(<ref>) does not mean
h_ab(s) = h_ab(r) +
._ζ^(0)g_ab|_r but it just means
^(0)g_ab(s) = ^(0)g_ab(r) +
λ._ζ^(0)g_ab|_r
+ O(λ^2).
Moreover, we also have to emphasize that the infinitesimal
coordinate transformation
(<ref>) is
essentially different from the infinitesimal coordinate transformation
(<ref>).
Actually, the infinitesimal coordinate transformation
(<ref>) is the
replacement of the point within the single manifold
_ϵ.
On the other hand, the coordinate transformation
(<ref>) is just the change
of the coordinate label at the same point in the background spacetime
.










 §.§ Linear perturbations on spherically symmetric background spacetime










Here, we consider the 2+2 formulation of the perturbation of a
spherically symmetric background spacetime, which originally proposed
by Gerlach and Sengupta <cit.>.
Spherically symmetric spacetimes are characterized by the direct
product =_1× S^2 and their metric is

    g_ab   =   
      y_ab + r^2γ_ab
      , 
    
      y_ab   =    y_AB (dx^A)_a(dx^B)_b,   γ_ab = γ_pq (dx^p)_a (dx^q)_b,

where x^A = (t,r), x^p=(θ,ϕ), and γ_pq is the
metric on the unit sphere.
In the Schwarzschild spacetime, the metric
(<ref>) is given by

    y_ab   =   
          - f (dt)_a(dt)_b
          +
          f^-1 (dr)_a(dr)_b
          ,
            
          f = 1 - 2M/r
      ,
    γ_ab   =   
          (dθ)_a(dθ)_b
          +
          sin^2θ(dϕ)_a(dϕ)_b
          =
          θ_aθ_b + ϕ_aϕ_b
          ,
      
    θ_a   =   
          (dθ)_a,   ϕ_a
          =
          sinθ (dϕ)_a
          .






On this background spacetime (,g_ab), the components of
the metric perturbation is given by

    h_ab
      =
      h_AB (dx^A)_a(dx^B)_b
      +
      2 h_Ap (dx^A)_(a(dx^p)_b)
      +
      h_pq (dx^p)_a(dx^q)_b
      .

Here, we note that the components h_AB, h_Ap, and
h_pq are regarded as components of scalar, vector, and
tensor on S^2, respectively.
In the Part I paper <cit.>, we showed the
linear independence of the set of harmonic functions

    {
      S_δ,
      D̂_pS_δ,
      ϵ_pqD̂^qS_δ,
      1/2γ_pqS_δ,
      (D̂_pD̂_q-1/2γ_pq)S_δ,
      2ϵ_r(pD̂_q)D̂^rS_δ}
      ,

where D̂_p is the covariant derivative associated with
the metric γ_pq on S^2,
D̂^p=γ^pqD̂_q,
ϵ_pq=ϵ_[pq]=2θ_[pϕ_q] is the totally
antisymmetric tensor on S^2.
In the set of harmonic function (<ref>), the
scalar harmonic function S_δ is given by

    S_δ
      =
      {[      Y_lm                                   l≥ 2;; k_(Δ̂+2)m                                    l=1;;    k_(Δ̂)                                    l=0. ].

Here, functions k_(Δ̂) and k_(Δ̂+2)m are
the kernel modes of the derivative operator Δ̂ and
[Δ̂+2], respectively, and we employ the explicit form of
these functions as

    k_(Δ̂)   =   
          1 + δln(1-cosθ/1+cosθ)^1/2,
            δ∈
          ,
          
    
      k_(Δ̂+2,m=0)   =   cosθ
          +
          δ(1/2cosθln1+cosθ/1-cosθ -1)
          ,
           δ∈
         ,
         
    
      k_(Δ̂+2,m=±1)   =   [
          sinθ
          +
          δ(
          + 1/2sinθln1+cosθ/1-cosθ
          + θ)
          ]
          e^± iϕ
          .






Then, we consider the mode decomposition of the components
{h_AB,h_Ap,h_pq} as follows:

    h_AB   =   ∑_l,mh̃_AB S_δ
          ,
      
    
      h_Ap   =   
          r ∑_l,m[
          h̃_(e1)AD̂_pS_δ
          +
          h̃_(o1)Aϵ_pqD̂^qS_δ]
          ,
      
    
      h_pq   =   
          r^2∑_l,m[
          1/2γ_pqh̃_(e0) S_δ
          +
          h̃_(e2)(
          D̂_pD̂_q - 1/2γ_pqD̂^rD̂_r) S_δ
          +
          2 h̃_(o2)ϵ_r(pD̂_q)D̂^r S_δ]
          .

Since the linear independence of each element of the set of harmonic
function (<ref>) is guaranteed, the
one-to-one correspondence between the components {h_AB, h_Ap,
h_pq} and the mode coefficients {h̃_AB,
h̃_(e1)A, h̃_(o1)A, h̃_(e0),
h̃_(e2), h̃_(o2)} with the decomposition
formulae (<ref>)-(<ref>) is guaranteed
including l=0,1 mode if δ≠ 0.
Then, the mode-by-mode analysis including l=0,1 is possible when
δ≠ 0.
However, the mode functions
(<ref>)–(<ref>)
are singular if δ≠ 0.
When δ=0, we have k_(Δ̂)∝ Y_00 and
k_(Δ̂+2)m∝ Y_1m, but the linear dependence of
the set of harmonics (<ref>) is lost in this
case.
Because of this situation, we proposed the following strategy:

  
  We decompose the metric perturbation h_ab on the background
  spacetime with the metric
  (<ref>)–(<ref>)
  through Eqs. (<ref>)–(<ref>) with the
  harmonic function S_δ given by
  Eq. (<ref>).
  Then, Eqs. (<ref>)–(<ref>) become
  invertible including l=0,1 modes.
  After deriving the mode-by-mode field equations such as linearized
  Einstein equations by using the harmonic functions S_δ, we
  choose δ=0 as the regular boundary condition for solutions
  when we solve these field equations.






As shown in the Part I paper <cit.>, once we accept
Proposal <ref>,
the Conjecture <ref> becomes the
following statement:

  
  If the gauge-transformation rule for a perturbative pulled-back
  tensor field h_ab to the background spacetime  is
  given by _h_ab - _h_ab =
  _ξ_(1)g_ab with the background metric g_ab
  with spherically symmetry, there then exist a tensor field
  _ab and a vector  field Y^a such that h_ab is
  decomposed as h_ab =: _ab +
  _Yg_ab, where _ab and Y^a are
  transformed into __ab -
  __ab = 0 and _Y^a
  - _Y^a = ξ^a_(1) under the gauge
  transformation, respectively.

Actually, the gauge-dependent variable Y_a is given by

    Y_a   :=   ∑_l,mỸ_A S_δ (dx^A)_a
           +
           ∑_l,m(
           Ỹ_(e1)D̂_pS_δ
           +
           Ỹ_(o1)ϵ_pqD̂^qS_δ)
           (dx^p)_a
           ,

where

    Ỹ_A   :=   
           r h̃_(e1)A
           - r^2/2D̅_Ah̃_(e2)
           ,
      
    Ỹ_(e1)   :=   r^2/2h̃_(e2)
           ,
      
    Ỹ_(o1)   :=   
           - r^2h̃_(o2)
           .

Furthermore, including l=0,1 modes, the components of the
gauge-invariant part _ab of the metric perturbation
h_ab is given by

    _AB   =   ∑_l,mF̃_AB S_δ
          ,
      
    _Ap   =   
          r ∑_l,mF̃_Aϵ_pqD̂^qS_δ,   D̂^p_Ap = 0
          ,
      
    _pq   =   1/2γ_pq r^2∑_l,mF̃ S_δ
          ,

where F̃_AB, F̃_A, and F̃ are given by

    F̃_AB   :=   h̃_AB
           - 2 D̅_(AỸ_B)
           ,
      
    F̃_A   :=   h̃_(o1)A
           + r D̅_Ah̃_(o2)
           ,
      
    F̃   :=   h̃_(e0)
           - 4/rỸ_AD̅^Ar
           + h̃_(e2) l(l+1)
      .

Thus, we have constructed gauge-invariant metric perturbations on
the Schwarzschild background spacetime including l=0,1 modes.





Furthermore, from
Eqs. (<ref>)–(<ref>)
for the gauge-invariant variables _AB, _Ap, and
_pq and gauge-dependent variables Y_a defined by
Eqs. (<ref>)–(<ref>), the original
components {h_AB,h_Ap,h_pq} of the metric perturbation
(<ref>)–(<ref>) are given by

    h_ab   =   _ab + _Yg_ab .

This is the assertion of
Theorem <ref>.





The aim of our proposal is to add a greater degree of freedom of the
metric perturbation so that the decomposition
(<ref>) is guaranteed even in the case of
l=0,1 modes.
Therefore, it is impossible to reach the final expression
(<ref>) if we treat the metric perturbation
by using S_δ=0=Y_lm in the decomposition formulae
(<ref>)–(<ref>) from the starting point.





To see this more specifically, we show an explicit example of the
l=1 odd mode perturbation.
If we apply our proposal to the decomposition formulae
(<ref>)–(<ref>), l=1 odd-mode
perturbation is given by

    h_ab
      =
      2 r h̃_(o1)A^(l=1)ϵ_pqD̂^qS_δ
      (dx^A)_(a (dx^p)_b)
      +
      2 r^2h̃_(o2)^(l=1)ϵ_r(pD̂_q)D̂^r S_δ
      (dx^p)_(a (dx^q)_b)
      .

If we use  S_δ=0=Y_lm from the starting point,
2ϵ_r(pD̂_q)D̂^r S_δ=0.
So, the mode coefficient h̃_(o2)^(l=1) does not appear.
When δ≠ 0, the gauge-transformation rule of the variables
h̃_(o1)A^(l=1) and h̃_(o2)^(l=1) are given
by

    _h̃_(o1)A^(l=1)
      -
      _h̃_(o1)A^(l=1)
      =
      r D̅_A(1/rζ_(o))
      ,
        _h̃_(o2)^(l=1)
      -
      _h̃_(o2)^(l=1)
      =
      - 1/rζ_(o)
      ,

where the generator of the gauge transformation is given by
ξ_A=0 and ξ_p=r ζ_(o)ϵ_pqD̂^qS_δ.
If we use  S_δ=0=Y_lm from the starting point, the
gauge-transformation rule for the variable h̃_(o2)^(l=1)
does not appear, either.
However, in our case, this gauge transformation appears due to the fact
ϵ_r(pD̂_q)D̂^r S_δ≠ 0.
Inspecting these gauge-transformation rules, we rewrite
Eq. (<ref>) as

    h_ab   =   
          2 r [
          (
          h̃_(o1)A^(l=1)
          + r D̅_Ah̃_(o2))
          - r D̅_Ah̃_(o2)] ϵ_pqD̂^qS_δ
          (dx^A)_(a (dx^p)_b)
          
          +
          2 r^2h̃_(o2)^(l=1)ϵ_r(pD̂_q)D̂^r S_δ
          (dx^p)_(a (dx^q)_b)
       =   
          2 r [
          F̃_A
          - r D̅_A(1/r^2Ỹ_(o1))
          ] ϵ_pqD̂^qS_δ
          (dx^A)_(a (dx^p)_b)
          
          +
          2 r^2(1/r^2Ỹ_(o1)) ϵ_r(pD̂_q)D̂^r S_δ
          (dx^p)_(a (dx^q)_b)
          .

In Eq. (<ref>), F̃_A
is the gauge-invariant variable for the metric perturbation defined by
Eq. (<ref>) and Ỹ_(o1) is
the gauge-dependent part of the metric perturbation defined by
Eq. (<ref>).
The terms of Ỹ_(o1) in
Eq. (<ref>) can be written in the
form of the Lie derivative of the background metric as in
Eq. (<ref>).
When δ=0, Eq. (<ref>) is
given by

    h_ab   =   
          2 r [
          F̃_A
          - r D̅_A(1/r^2Ỹ_(o1))
          ] ϵ_pqD̂^qS_δ
          (dx^A)_(a (dx^p)_b)
          .

This is also given in the form (<ref>).





When we construct the gauge-invariant variable F̃_A in
Eqs. (<ref>) and
(<ref>), the existence of the
coefficient h̃_(o2)^(l=1)=Ỹ_(o1)/r^2 is
essential which does not appear in the treatment using the spherical
harmonics Y_lm from the starting point.
Actually, we cannot define the gauge-invariant variable for l=1
odd-mode perturbation in the same manner as l≥ 2-mode case.
Our
proposal <ref>,
make the degree of freedom of the metric perturbations increase.
We have to emphasize that we can reach the
decomposition (<ref>) owing to this
additional degree of freedom.





To discuss the linearized Einstein equation
(<ref>) and the linear perturbation of
the continuity equation

    ∇_a^(1)_b^  a = 0

of the gauge-invariant energy-momentum tensor
^(1)_b^  a:=g^ac^(1)_bc on a
vacuum background spacetime, we consider the mode-decomposition of the
gauge-invariant part ^(1)_bc of the linear perturbation
of the energy-momentum tensor through the set
(<ref>) of the harmonics as follows:

    ^(1)_ab   =   ∑_l,mT̃_AB
          S_δ
          (dx^A)_a(dx^B)_b
          +
          r
          ∑_l,m{T̃_(e1)AD̂_pS_δ
          +
          T̃_(o1)Aϵ_prD̂^rS_δ}
          2 (dx^A)_(a(dx^p)_b)
          
         +
         ∑_l,m{T̃_(e0)1/2γ_pq S_δ
         +
         T̃_(e2)(
         D̂_pD̂_qS_δ
         -
         1/2γ_pqD̂_rD̂^rS_δ)
         .
         
                .
         +
         T̃_(o2)
         2 ϵ_r(pD̂_q)D̂^rS_δ}
         (dx^p)_a(dx^q)_b
         .






In the Part I paper <cit.>, we derived the
linearized Einstein equations, discussed the odd-mode perturbation
F̃_Ap in
Eq. (<ref>), and derived the
l=1 odd-mode solutions to these equations.
The Einstein equation for even mode F̃_AB and F̃ in
Eqs. (<ref>) and
(<ref>) also derived in the
Part I paper <cit.>, and l=0,1
even-mode solutions are derived in the Part II
paper <cit.>.
Since these solutions include the Kerr parameter perturbation and the
Schwarzschild mass parameter perturbation of the linear order in the
vacuum case, these are physically reasonable.
Then, we conclude that our proposal is also physically reasonable.
Furthermore, we also checked that our derived solutions include the
linearized LTB solution and non-rotating C-metric with the
Schwarzschild background in the Part III
paper <cit.>.





The purpose of this paper is to compare our proposed gauge-invariant
treatments for l=0,1-mode perturbations on the Schwarzschild
background spacetime with conventional “complete gauge-fixing
treatments” in which we use the spherical harmonics Y_lm from the
starting point.
For this purpose, the linearized Einstein equations for the l=0,1-modes
and their solutions based on our proposal are necessary.
Therefore, we review them below.










 §.§ l=1 odd-mode linearized Einstein equations and solutions










As derived in the Part I paper <cit.>, the
l=1 odd-mode part in the linearized Einstein equations are
simplified as the constraint equation

    D̅_D(rF̃^D)
      =
      0
      ,

and the evolution equation

    - [ D̅^DD̅_D - 2/r^2] (rF̃_A)
         - 2/r^2 (D̅^Dr) (D̅_Ar) (rF̃_D)
         + 2/r (D̅^Dr) D̅_A(rF̃_D)
         
       =   
      16 π r T̃_(o1)A
      ,

where we choose T̃_(o2)=0 by hand.
Furthermore, we have the continuity equation

    D̅^CT̃_(o1)C
      + 3/r (D̅^Dr) T̃_(o1)D
      =
      0

for the l=1 odd-mode matter perturbation which is derived from the
divergence of the first-order perturbation of the energy-momentum
tensor.





We also note that the constraint
(<ref>) comes from the
traceless part of the (q,p)-component of the Einstein equation
(<ref>), which is the coefficient of
the mode function 2ϵ_r(pD̂_q)D̂^pS_δ.
If δ=0, i.e., the scalar harmonics S_δ is the
spherical harmonics Y_lm,
2ϵ_r(pD̂_q)D̂^pS_δ=0 for l=1.
Therefore, the constraint
(<ref>) does not appear when
we use the spherical harmonics Y_lm from the starting point.





The explicit strategy to solve these odd-mode perturbations and
l=0,1 mode solutions was discussed in the Part I
paper <cit.>.
As the result, we obtained the l=1 odd-mode “formal” solution as
follows:

    2 _Ap (dx^A)_(a(dx^p)_b)
         =
         6 M r^2[∫ dr a_1(t,r)/r^4]
         sin^2θ
         (dt)_(a (dϕ)_b)
         +
         _Vg_ab
         ,
      
          
         V_a
         =
         (β_1t + β_0 + W_(o)(t,r)) r^2sin^2θ (dϕ)_a
         ,

where

    a_1(t,r)
         =   
          - 16 π/3M r^3 f ∫ dt T̃_(o1)r + a_10
       =   
          - 16 π/3M∫ dr r^31/fT̃_(o1)t + a_10
          .

The constant a_10 in Eq. (<ref>)
corresponds to the Kerr parameter.
Furthermore, the function W_(o)(t,r) in
Eq. (<ref>) is related
to the solution to the l=1-mode Regge-Wheeler equation

    ∂_t^2Z_(o)
      -  f ∂_r( f ∂_rZ_(o))
      + 1/r^2 f [ l(l+1) - 3 (1-f) ] Z_(o)
      =
      16 π f^2T̃_(o1)r
      ,

where Z_(o) = r f ∂_rW_(o).
We have to solve Eq. (<ref>) to
obtain the function W_(o)(t,r).
In this sense, the solution
(<ref>)  should be
regarded as the “formal” one.










 §.§ l=0,1 even-mode linearized Einstein equations and solutions










The l=0,1 even-mode part of the linearized Einstein equation
(<ref>) is summarized as follows:

    F̃_D^   D   =   
          0
          ,
      
    D̅^D_AD
      - 1/2D̅_AF̃   =   
          16 π r T̃_(e1)A
          ,

where the variable _AB is the traceless part of the
variable F̃_AB defined by

    _AB := F̃_AB - 1/2 y_ABF̃_C^  C

and we choose the component of the energy-momentum tensor so that
T̃_(e2)=0 by hand.
Owing to the same choice T̃_(e2)=0, we also have the
evolution equations

    (
         D̅_DD̅^D
         + 2/r (D̅^Dr) D̅_D
         -  (l-1)(l+2)/r^2) F̃
         - 4/r^2 (D̅_Cr) (D̅_Dr) ^CD=
         16 π S_(F)
         ,
         
          
      S_(F)
      :=
           T̃_C^   C
           + 4 (D̅_Dr) T̃_(e1)^D
         ,
         
          [
         -  D̅_DD̅^D
         -  2/r (D̅_Dr) D̅^D
         + 4/r (D̅^DD̅_Dr)
         + l(l+1)/r^2]
         _AB
          
         + 4/r (D̅^Dr) D̅_(A_B)D
         -  2/r (D̅_(Ar) D̅_B)F̃
           =
          16 π S_()AB
          ,


    S_()AB   :=   T̃_AB - 1/2 y_ABT̃_C^   C
          - 2 ( D̅_(A(r T̃_(e1)B)) - 1/2 y_ABD̅^D(r T̃_(e1)D) )
          
          
          + 2 y_AB (D̅^Cr) T̃_(e1)C
         ,

for the variable F̃ and the traceless variable
_AB with l=0,1.
We also have to take into account the even-mode part of the
continuity equation as follows:

    D̅^CT̃_C^  B
         + 2/r (D̅^Dr)T̃_D^   B
         -  1/r l(l+1) T̃_(e1)^B
         -  1/r (D̅^Br) T̃_(e0)
         =
         0
         ,
         
          D̅^CT̃_(e1)C
         + 3/r (D̅^Cr) T̃_(e1)C
         + 1/2rT̃_(e0)
         =
         0
         ,

for l=0,1.





We also note that the constraint
(<ref>) comes from the
traceless part of the (q,p)-component of the Einstein tensor
(<ref>), which is the coefficient of
the mode function
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S_δ.
If δ =0, i.e., the scalar harmonics S_δ is the
spherical harmonics Y_lm,
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S_δ=0
for l=1.
Therefore, the constraint
(<ref>) does not appear when
we use the spherical harmonics Y_lm from the starting point.









  §.§.§ l=1 even-mode solution









In the Part II paper <cit.>, we derived the
l=1 solution to
Eqs. (<ref>),
(<ref>),
(<ref>),
(<ref>),
(<ref>), and
(<ref>) with l=1.
For m=0 mode, in the Part II paper <cit.>, we
derived the following “formal” solution to the linearized Einstein equation

    _ab   =   _Vg_ab
          - 16 π r^2/3(1-f)[
          f^2{1+f/2T̃_rr
          + r f ∂_rT̃_rr
          -  T̃_(e0)
          -  4 T̃_(e1)r}  (dt)_a(dt)_b.
          
                            .
          + 2r/f{∂_tT̃_tt
          - 3f(1-f)/2rT̃_tr} (dt)_(a(dr)_b).
          
                            .
          + r/f{∂_rT̃_tt
          - 3(1-3f)/2rfT̃_tt} (dr)_a(dr)_b.
          
                            .
          + r^2T̃_ttγ_ab] cosθ
          ,

where the vector field V_a is given by

    V_a   :=   
           -  r ∂_tΦ_(e)cosθ (dt)_a
           + ( Φ_(e) - r ∂_rΦ_(e)) cosθ (dr)_a
           -  r Φ_(e)sinθ (dθ)_a
           .

Here, the variable Φ_(e) is a solution to the l=1
Zerilli equation

    -  ∂_t^2Φ_(e)
         + f ∂_r[ f ∂_rΦ_(e)]
         -
         f(1-f)/r^2Φ_(e)
       =   4π r/3(1-f)[
         3(1-3f) T̃_tt
         + (1+f)f^2T̃_rr
         -  2 r f ∂_rT̃_tt.
          
                    .
         + 2 r f^3∂_rT̃_rr
         -  2 f^2T̃_(e0)
         -  8 f^2T̃_(e1)r]
         .

If we obtain the solution to the l=1 Zerilli equation
(<ref>), we can write the explicit
form of the solution (<ref>)
through the generator V_a defined by
Eq. (<ref>).
In this sense, we have to regard the solution
(<ref>) is a “formal” one.









  §.§.§ l=0 even-mode solution









On the other hand, for the l=0-mode, we may choose
T̃_(e1)A=0 in
Eqs. (<ref>),
(<ref>),
(<ref>)–(<ref>)
with l=0.
Then, we derived the l=0 mode solution

    _ab   =   2/r(M_1+4π∫ dr r^2/f T_tt)
          ((dt)_a(dt)_a+ 1/f^2 (dr)_a(dr)_a)
          
          
          + 2 [4 π r ∫ dt (1/fT̃_tt + f T̃_rr)] (dt)_(a(dr)_b)
          + _Vg_ab
          ,

where

    V_a
      =
      (
      f/4Υ
      + rf/4∂_rΥ
      -  r Ξ(r)/(1-3f)
      + f ∫ dr 2 Ξ(r)/f(1-3f)^2) (dt)_a
      +
      1/4f r ∂_tΥ (dr)_a
      .

Here, the variable F̃=:∂_tΥ must satisfy the
equation

    - 1/f∂_t^2Υ
         + ∂_r(f ∂_rΥ)
         + 3(1-f)/r^2Υ
         - 8/r^3∫ dt m_1(t,r)
         - 4/1-3f∂_rΞ(r)
         
       =   
          16 π∫ dt ( - 1/fT̃_tt + f T̃_rr)
          ,

where m_1(t,r) is given by

    m_1(t,r) = 4 π∫ dr r^2/fT̃_tt + M_1
      = 4 π∫ dt r^2 f T̃_tr + M_1,

M_1 is the constant corresponding to the Schwarzschild mass
perturbation, and Ξ(r) is an arbitrary function of r.










§ RULE OF COMPARISON AND GAUGE-TRANSFORMATION RULES IN A
  CONVENTIONAL GAUGE-FIXING










The main purpose of this paper is to check the following statement.
“Gauge invariant formulations of perturbations are equivalent to
  complete gauge fixing approaches.”
However, this statement is too ambiguous to check if we have the
background knowledge explained in
Sec. <ref>.
For example, there is no explanation of the terminology “gauge” in this statement.
Therefore, we have to clarify this statement as follows:







  Our rule for comparison 
  First of all, we assume the terminology “gauge” in the
  statement “Gauge invariant formulations of perturbations are
    equivalent to complete gauge fixing approaches” is the second-kind
  gauge which is explained in
  Sec. <ref>.
  Therefore, the gauge-transformation rule for this statement is given
  by (<ref>).
  Furthermore, the degree of freedom that changes under this
  gauge-transformation rule is regarded as “unphysical.”
  The “gauge fixing” in the above statement is a specification
  of some perturbative variables through the degree of freedom of the
  generator ξ_(1)^a.
  Furthermore, “complete gauge fixing” in the above statement is
  a specification of some perturbative variables through the “entire” degree of freedom of the generator ξ_(1)^a.






These are all rules of our game to compare our gauge-invariant
formulation and a “conventional complete gauge-fixing approach” in
which we use the spherical harmonics Y_lm from the starting point.









 §.§ Metric perturbations









Based on the above conceptual premise, we consider the metric
perturbation h_ab on the background Schwarzschild spacetime
(,g_ab) whose metric g_ab is given by Eqs. (<ref>)–(<ref>).
We consider the components of the metric perturbation h_ab as
Eq. (<ref>) and the decomposition of
these components {h_AB, h_Ap, h_pq} as
Eqs. (<ref>)–(<ref>) but we concentrate
on the case S_δ=Y_lm=:S, where Y_lm is the conventional
spherical harmonics.





Since we choose S=Y_lm, we have D̂_pS =
ϵ_pqD̂^qS = 0,
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
= 2 ϵ_r(pD̂_q)D̂^rS = 0 for l=0
modes.
For l=1 modes,
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
= 2 ϵ_r(pD̂_q)D̂^rS = 0.
Due to these facts, we cannot construct gauge-invariant variables for
l=0,1 modes in a similar manner to the derivation of
Eqs. (<ref>)–(<ref>)
for l≥ 2 modes.
Then, we cannot use the gauge-invariant formulation reviewed in
Sec. <ref>.
Instead, we have to fix the second-kind gauge degree of freedom to
exclude the “unphysical” degree of freedom.









 §.§ Conventional gauge-transformation rules for l=0,1 metric perturbations









Here, we consider the gauge-transformation rule

    _h_ab
      -
      _h_ab
      =
      _ξg_ab
      =
      2 ∇_(aξ_b)
      .

From this gauge-transformation rule, we can derive the
gauge-transformation rule for the components
{h_AB,h_Ap,h_qp} defined by
Eq. (<ref>) is given by

    _h_AB
      -
      _h_AB   =   _ξg_AB
          =
          ∇_Aξ_B
          +
          ∇_Bξ_A
          =
          D̅_Aξ_B
          +
          D̅_Bξ_A
          ,
      
    _h_Ap
      -
      _h_Ap   =   _ξg_Ap
          =
          ∇_Aξ_p
          +
          ∇_pξ_A
          =
          D̅_Aξ_p
          +
          D̂_pξ_A
          -
          2/r (D̅_Ar) ξ_p
          ,
      
    _h_pq
      -
      _h_pq   =   _ξg_pq
          =
          ∇_pξ_q
          +
          ∇_qξ_p
          =
          D̂_pξ_q
          +
          D̂_qξ_p
          +
          2 r (D̅^Ar) γ_pqξ_A

from the formulae summarized in Appendix B of the Part I
paper <cit.>.
Here, we consider the Fourier transformation of
ξ_a=:ξ_A(dx^A)_a+ξ_p(dx^p)_a as

    ξ_A = ∑_l,mζ_A S,   ξ_p = r ∑_l,m(
      ζ_(e)D̂_pS
      +
      ζ_(o)ϵ_pqD̂^pS
      )
      .

From these Fourier transformation rules,
Eqs. (<ref>)–(<ref>)
are given by

    _h_AB
      -
      _h_AB   =   D̅_Aξ_B
          +
          D̅_Bξ_A
          =
          ∑_l,m 2 D̅_(Aζ_B) S
          ,
      
    _h_Ap
      -
      _h_Ap   =   D̅_Aξ_p
          +
          D̂_pξ_A
          -
          2/r (D̅_Ar) ξ_p
       =   
          r
          ∑_l,m[
          (
          1/rζ_A
          + D̅_Aζ_(e)
          - 1/r (D̅_Ar) ζ_(e)) D̂_pS
          .
          
                  .
         +
         (
         D̅_Aζ_(o)
         - 1/r (D̅_Ar) ζ_(o)) ϵ_pqD̂^pS
         ]
         ,
      
    _h_pq
      -
      _h_pq   =   D̂_pξ_q
          +
          D̂_qξ_p
          +
          2 r (D̅^Ar) γ_pqξ_A
       =   
          r^2∑_l,m[
          4/r(
          -  l (l+1)/2ζ_(e)
          + (D̅^Ar) ζ_A)
          1/2γ_pq S
          .
          
                  .
         +
         2/rζ_(e)(
         D̂_pD̂_qS
         -
         1/2γ_pqD̂^rD̂_rS
         )
          .
          
                  .
         -
         1/rζ_(o) 2ϵ_r(qD̂_p)D̂^rS
         ]
         .

Here, we used the property of the mode function S is an eigen
function of the Laplacian
D̂_rD̂^r:=γ^rsD̂_rD̂_s as 

    D̂_rD̂^rS = - l(l+1) S,    S=Y_lm.









  §.§.§ l≥ 2 perturbations








For l≥ 2 modes, the set of mode functions

    {
      S,
      D̂_pS,
      ϵ_pqD̂^qS,
      1/2γ_pq S,
      (
      D̂_pD̂_q - 1/2γ_pqD̂_rD̂^r) S,
      2 ϵ_r(pD̂^rD̂_q) S
      }

is a linear-independent set of the tensor field of the rank 0, 1, and
2, even if S=Y_lm.
Then, we may compare
Eqs. (<ref>)–(<ref>) and
Eqs. (<ref>)–(<ref>).
As the result of this comparison, we obtain the gauge-transformation
rules for the mode functions
{h̃_AB,h̃_(e1)A,h̃_(o1)A,h̃_(e0),h̃_(e2),h̃_(o2)}
as

    _h̃_AB
      -
      _h̃_AB   =   
          2 D̅_(Aζ_B)
          ,
      
    _h̃_(e1)A
      -
      _h̃_(e1)A   =   1/rζ_A
          + D̅_Aζ_(e)
          - 1/r (D̅_Ar) ζ_(e)
          ,
      
    _h̃_(o1)A
      -
      _h̃_(o1)A   =   D̅_Aζ_(o)
          - 1/r (D̅_Ar) ζ_(o)
          ,
      
    _h̃_(e0)
      -
      _h̃_(e0)   =   4/r(
          -  l (l+1)/2ζ_(e)
          + (D̅^Ar) ζ_A)
          ,
      
    _h̃_(e2)
      -
      _h̃_(e2)   =   2/rζ_(e)
          ,
      
    _h̃_(o2)
      -
      _h̃_(o2)   =   
         -
         1/rζ_(o)
          .

From these gauge-transformation rules, it is well-known that we can
construct gauge-invariant variables and the gauge-dependent variables
and Conjecture <ref> is valid for
l≥ 2 modes.








  §.§.§ l=1 perturbations







As shown in the Appendix A in Ref. <cit.>, for
l=1 modes,

    D̂_pY_1m≠ 0≠ϵ_pqD̂^qY_1m

but

    (
      D̂_pD̂_q - 1/2D̂_rD̂^r) Y_1m
      =
      0
      =
      2 ϵ_r(pD̂_q)D̂^rY_1m
      .

In this case, the Fourier transformation
(<ref>)–(<ref>)
of the metric perturbation with S_δ=S=Y_lm is given by

    h_AB   =   ∑_mh̃_AB S
          ,
      
    
      h_Ap   =   ∑_m r h̃_(e1)AD̂_pS
          +
          ∑_m r h̃_(o1)Aϵ_pqD̂^qS
          ,
      
    
      h_pq   =   ∑_mr^2/2γ_pqh̃_(e0) S
          .

The gauge-transformation rules
(<ref>)–(<ref>)
for each m mode are given by

    _h̃_AB
      -
      _h̃_AB   =   
          2 D̅_(Aζ_B)
          ,
      
    _h̃_(e1)A
      -
      _h̃_(e1)A   =   1/rζ_A
          + D̅_Aζ_(e)
          - 1/r (D̅_Ar) ζ_(e)
          ,
      
    _h̃_(o1)A
      -
      _h̃_(o1)A   =   D̅_Aζ_(o)
          - 1/r (D̅_Ar) ζ_(o)
          ,
      
    _h̃_(e0)
      -
      _h̃_(e0)   =   
          -  4/rζ_(e)
          + 4/r (D̅^Ar) ζ_A
          .

Comparing the gauge-transformation rules
(<ref>)–(<ref>)
for l≥ 2 with the gauge-transformation rules
(<ref>)–(<ref>),
it is easy to find the gauge-transformation rule
(<ref>) and
(<ref>) for l≥ 2 do not appear in
the l=1 mode gauge transformation.
This is due to the fact
Eqs. (<ref>), and the reason
why we cannot construct gauge-invariant variables for l=1 mode
perturbations in a similar method to the l≥ 2 case.








  §.§.§ l=0 perturbations








The spherical harmonic function S=Y_lm is constant when l=0.
In this case, the Fourier transformation
(<ref>)–(<ref>)
of the metric perturbation with S_δ=S=Y_lm is given by

    h_AB   =   h̃_AB S
          ,
      
    
      h_Ap   =   
          0
          ,
      
    
      h_pq   =   r^2/2γ_pqh̃_(e0) S
          .

The gauge-transformation rules
(<ref>)–(<ref>)
are given by

    _h̃_AB
      -
      _h̃_AB   =   
          2 D̅_(Aζ_B)
          =
          D̅_Aζ_B
          +
          D̅_Bζ_A
          ,
      
    _h̃_(e0)
      -
      _h̃_(e0)   =   4/r (D̅^Ar) ζ_A
          .

It is easy to see that many gauge-transformation rules in
(<ref>)–(<ref>)
are missing.
This is due to the fact

    D̂_pS = ϵ_pqD̂^qS = 0,   (D̂_qD̂_p- 1/2γ_pqD̂_rD̂^r)S
      =
      2 ϵ_r(pD̂_q)D̂^r S = 0

This is the reason why we cannot construct gauge-invariant variables
for l=0 mode perturbations in a similar method to the l≥ 2 case.










§ L=1 ODD-MODE PERTURBATION IN THE CONVENTIONAL APPROACH










As discussed in
Sec. <ref>, l=1 mode
perturbations are given by
Eqs. (<ref>)–(<ref>).
In particular, among these expressions of the l=1 mode perturbation, 
odd-mode perturbation is given by

    h_AB   =   
          0
          ,   
      h_Ap
      =
          r h̃_(o1)Aϵ_pqD̂^qS
          ,   
      h_pq
      =
          0
          .

As the property of the l=1 spherical harmonics, we obtain the
condition (<ref>).
For odd-mode perturbation we have

    2 ϵ_r(pD̂_q)D̂^rY_1m
      =
      0
      .






Here, we apply the notation which is introduced by
Eqs. (<ref>)–(<ref>) and we obtain

    h̅_AB = 0,   h̅_A^  D = 0,   h̅^pD := γ^pq y^DE r h̃_(o1)Eϵ_qrD̂^rS
    ,   h̅_p^  q = 0,   h̅^pq = 0,

for l=1 odd mode perturbations.
Through this notation of the metric perturbation, the linear
perturbations
(<ref>)–(<ref>)
of the Einstein tensor are given by

    ^(1)G_A^  B   =   
          0
          ,


    ^(1)G_A^  q   =   1/2r[
          D̅_AD̅^Ch̃_(o1)C
          -  D̅_CD̅^Ch̃_(o1)A
          -  2/r (D̅^Cr) D̅_Ch̃_(o1)A.
          
            .
          + 3/r (D̅^Cr) D̅_Ah̃_(o1)C
          -  1/r (D̅_Ar) D̅^Ch̃_(o1)C
          + 1/2r^2 (D̅_Cr) (D̅^Cr) h̃_(o1)A.
          
            .
          -  2/r^2 (D̅_Ar)  (D̅^Cr) h̃_(o1)C
          + 3/2r^2h̃_(o1)A]
          ϵ^qrD̂_rS
          
         ,


    ^(1)G_p^  q   =   
          - 1/2r^2D̅^C( r h̃_(o1)C)
          γ^qr
          2
          ϵ_s(pD̂_r)D̂^sS
      =
          0
          .

The final equality in
Eq. (<ref>) is due to
the property of the spherical harmonics Y_lm
(<ref>).
Furthermore, the expression
(<ref>) is gauge
invariant under the gauge transformation rule
(<ref>).
On the other hand, the D̅^C( r h̃_(o1)C)
in Eq. (<ref>) is not
gauge invariant as shown in below.
However, the gauge-invariance of the linear-order Einstein tensor
^(1)G_a^  b is guaranteed by the identity
2ϵ_s(pD̂_r)D̂^sS=0 for the l=1 odd-mode
perturbations.





For the l=1 odd-mode perturbations, the components of the linearized
energy-momentum tensor are summarized as

    ^(1) T_A^   B
      =
          0
          ,
    
        ^(1) T_A^  q
      =
          1/rT̃_(o1)Aϵ^qrD̂_rS
          ,
    
        ^(1) T_p^  q
      =
          0
         .

Then, the linearized Einstein equations for the odd-mode perturbations
are given by

    D̅_AD̅^Ch̃_(o1)C
         -  D̅_CD̅^Ch̃_(o1)A
           
          -  2/r (D̅^Cr) D̅_Ch̃_(o1)A
          + 3/r (D̅^Cr) D̅_Ah̃_(o1)C
          -  1/r (D̅_Ar) D̅^Ch̃_(o1)C
           
         + 1/2r^2 (D̅_Cr) (D̅^Cr) h̃_(o1)A
         -  2/r^2 (D̅_Ar)  (D̅^Cr) h̃_(o1)C
         + 3/2r^2h̃_(o1)A
       =   
          16πT̃_(o1)A
          ,






Although the equation

    D̅^C( r h̃_(o1)C) = 0

does not appear from
Eq. (<ref>) due to the
fact that 2ϵ_s(pD̂_r)D̂^sY_1m=0,
we may use
Eq. (<ref>) as a
gauge condition.
As noted in
Sec. <ref>, the
gauge-transformation rule for the variable h̃_(o1)C is
given by Eq. (<ref>).
From the gauge transformation rule
(<ref>), we consider the
gauge-transformation rule for the left-hand side of
Eq. (<ref>) as

    D̅^A(r _h̃_(o1)A)
      -
      D̅^A(r _h̃_(o1)A)
         =   
          r D̅^AD̅_Aζ_(o)
          - (D̅^AD̅_Ar) ζ_(o)
          .

Using the background Einstein equation (Eq. (B67) of Appendix B in
Ref. <cit.>), we consider the equation

    r D̅^CD̅_Cζ_(o)
      - 1/r(
      1 - (D̅^Cr)(D̅_Cr)
      ) ζ_(o)
      =
      - D̅^C(r _h̃_(o1)C)

More explicitly, Eq. (<ref>)

    -    ∂_t^2ζ_(o)
      + f ∂_r(f∂_rζ_(o))
      - f/r^2(
      1 - f
      ) ζ_(o)
      =
      - f/rD̅^C(r _h̃_(o1)C)

If we choose ζ_(o) as a special solution to
Eq. (<ref>) or equivalently
Eq. (<ref>),
Eq. (<ref>) is
regarded as a gauge condition in the -gauge, i.e.,

    D̅^C(r _h̃_(o1)C)
         =   
          0
          .






We have to emphasize that this is not a complete gauge fixing, since
there is a room of the choice of ζ_(o) which satisfy the
homogeneous equation of
Eq. (<ref>):

    r D̅^CD̅_Cζ_(o)
      - 1/r(
      1 - (D̅^Cr)(D̅_Cr)
      ) ζ_(o)
      =
      0
      ,

i.e.,

    -    ∂_t^2ζ_(o)
      + f ∂_r(f∂_rζ_(o))
      - f/r^2(
      1 - f
      ) ζ_(o)
      =
      0
      .






Under the gauge choice (<ref>),
Eq. (<ref>) is given by

    [
      D̅_CD̅^C
      - 2/r^2] (r h̃_(o1)A)
      -  2/r^2 (D̅_Ar) (D̅^Cr) (r h̃_(o1)C)
      + 2/r (D̅^Cr) D̅_A(r h̃_(o1)C)
       
       =   
      16π r T̃_(o1)A
      .

Although the variable rh̃_(o1)C is a gauge-dependent
variable which is different from the gauge-invariant variable
rF̃_A, we can obtain the equation
(<ref>) if we
replace the variable rF̃_A in
Eq. (<ref>) with the
variable rh̃_(o1)C.





We also note that the gauge condition
(<ref>) coincides with
Eq. (<ref>), though we have
to replace the gauge-invariant variable r F̃^D with the
gauge-dependent variable r h̃_(o1)C in
Eq. (<ref>) to confirm this
coincidence.
Furthermore, we also take into account the continuity equation
(<ref>) of the l=1 odd-mode
perturbation of the energy-momentum tensor.
Thus, the equations to be solved for the l=1 odd-mode perturbation
are the gauge condition
(<ref>), the
evolution equation
(<ref>), and the
continuity equation (<ref>).
These equations coincide with
Eqs. (<ref>)–(<ref>)
except for the fact that the variable to be obtained is not the
gauge-invariant rF̃^D variable but the gauge-dependent
variable r h̃_(o1)C.
Then, through the same logic in Ref. <cit.>, we
obtain the solution to
the gauge condition (<ref>),
the evolution equation
(<ref>), and the
continuity equation (<ref>) as follows:

    2 r h̃_(o1)Csin^2θ (dx^C)_(a (dϕ)_b)
         =
         6 M r^2[∫ dra_1(t,r)/r^4]
         sin^2θ (dt)_(a(dϕ)_b)
                                      
         + _Vg_ab,
         
         , 
          
         V_a = (β_1t + β_0 + W_(o)(t,r)) r^2sin^2θ (dϕ)_a,

Here, we concentrate only on the m=0 solution, β_1 and
β_0 are constant, and Z_(o)=rf∂_rW_(o)(t,r) is
an arbitrary function which satisfies the equation

    ∂_t^2Z_(o)
      -
      f∂_r(f∂_rZ_(o))
      +
      1/r^2 f [
      2 - 3 (1-f)
      ]
      Z_(o)
      =
      16π f^2T̃_(o1)r
      .

a_1(t,r) in Eq. (<ref>) is
given by Eq. (6.44) in Ref. <cit.>, i.e.,

    a_1(t,r)
         =   
          - 16π/3M r^3 f ∫ dt T̃_(o1)r + a_10
       =   
          - 16π/3M∫ dr r^31/fT̃_(o1)t + a_10
          ,

where a_10 is the perturbative Kerr parameter.





Note that the formal solution described by
Eqs. (<ref>)–(<ref>)
has the same form as the formal solution described by
Eqs. (<ref>)–(<ref>).
However, we have to emphasize that the variable _Ap in
Eq. (<ref>) is gauge invariant
as noted by Eq. (<ref>) and the
vector field V_a in
Eq. (<ref>) is also gauge
invariant.
On the other hand, the variable h̃_(o1)C is still
gauge-dependent.
Actually, there are remaining gauge degrees of freedom of the
generator ζ_(o) which satisfy
Eq. (<ref>) as noted
above.
This is clear from the fact that the gauge transformation rule
(<ref>) with
Eq. (<ref>) gives a still
non-trivial transformation rule.
This remaining gauge degree of freedom is so-called “residual gauge”.
For this reason, there is a possibility that V_a in
Eq. (<ref>) includes this
residual gauge.





To clarify whether V_a in
Eq. (<ref>) includes the
“residual gauge”, we have to confirm
Eq. (<ref>).
Within our rules to compare our gauge-invariant formulation with a
conventional gauge-fixed approach, we regard the terms in V_a in
Eq. (<ref>) which
satisfies Eq. (<ref>) is
the second-kind gauge degree of freedom and we regard these degrees of
freedom as “unphysical degree of freedom.”
To clarify this “unphysical degree of freedom,” we introduce the
indicator function _(o)[*] as

    _(o)[ζ_(o)]
      :=
      -    ∂_t^2ζ_(o)
      + f ∂_r(f∂_rζ_(o))
      - f/r^2(
      1 - f
      ) ζ_(o)
      .

If _(o)[ζ_(o)]=0, we should regard ζ_(o) is
the second-kind gauge and we regard ζ_(o) as “unphysical
degree of freedom.”





We can easily confirm that

    _(o)[(β_1t+β_0)r] = 0
      .

Then, according to our above rule of comparison, we have to conclude
that the rigid rotating term (β_1t+β_0)r in V_a in
Eq. (<ref>) should be regarded
as the second-kind gauge degree of freedom which is “unphysical degree
of freedom.”





Next, we consider the term W_(o)(t,r)r in
Eq. (<ref>).
In this case, the direct calculation of the indicator
_(o)[W_(o)(t,r)r] is useless.
However, it is useful to consider the r-derivative of
_(o)[W_(o)(t,r)r] and we can show that

    r f ∂_r(
      1/r_(o)[W_(o)(t,r)r]
      )
         =   
          -  ∂_t^2Z_(o)
          + f ∂_r(f ∂_rZ_(o))
          -  f/r^2( 2 - 3 (1-f) ) Z_(o)
          ,

where we used Z_(o)=rf∂_rW_(o)(t,r).
Since the left-hand side of
Eq. (<ref>) coincides with
the right-hand side of Eq. (<ref>).
Through Eq. (<ref>), we obtain

    r f ∂_r(
      1/r_(o)[W_(o)(t,r)r]
      )
         =   
          - 16 π f^2T̃_(o1)r
          .

or, equivalently,

    _(o)[W_(o)(t,r)r]
         =   
          - 16 π r ∫ dr f/rT̃_(o1)r
         .

Then, if we consider the case where T̃_(o1)r≠ 0, we
have nonvanishing _(o)[W_(o)(t,r)r].
This means that the W_(o)(t,r) term does not belong to the
second-kind gauge, and we have to regard the degree of freedom
W_(o)(t,r) in Eq. (<ref>)
is a “physical one”.





On the other hand, even in the case where T̃_(o1)r=0,
Eq. (<ref>) is valid.
If W_(o)(t,r) belongs to the second-kind gauge, i.e.,
_(o)[W_(o)(t,r)r]=0,
Eq. (<ref>) yields

    -  ∂_t^2Z_(o)
      + f ∂_r(f ∂_rZ_(o))
      -  f/r^2( 2 - 3 (1-f) ) Z_(o)
      =
      0
      .

However, even if Z_(o) is a solution to
Eq. (<ref>), we cannot directly yield
_(o)[W_(o)(t,r)r]=0.
Therefore, considering the following two sets of function
W_(o)(t,r)r as

    _(o)   :=   {.
           r W_(o)(t,r)
           |
           _(o)[W_(o)(t,r)r] = 0
           }
           ,
      
    _(o)   :=   {.
           r W_(o)(t,r)
           |
           Z_(o) = rf ∂_rW_(o)(t,r),
           .
           
                      .
           - ∂_t^2Z_(o)
           + f ∂_r(f ∂_rZ_(o))
           -
           f/r^2( 2 - 3(1-f) ) Z_(o)
           = 0
           .
           }
           ,

we obtain the relation

    _(o)⊂_(o).

This indicates that a part of solutions to
Eq. (<ref>) should be regarded
as a gauge degree of freedom of the second kind which is the
“unphysical degree of freedom.”





Furthermore, to obtain the explicit solution W_(o)(t,r), we have
to solve Eq. (<ref>) with
appropriate boundary conditions.
Equation (<ref>) is an inhomogeneous
second-order linear differential equation for Z_(o) and its boundary conditions
are adjusted by the homogeneous solutions to
Eq. (<ref>), i.e., the element of the
set of function _(o).
However, a part of this homogeneous solution to
Eq. (<ref>) should be regarded as
an unphysical degree of freedom in the “complete gauge fixing approach”
as mentioned above.
Therefore, in the conventional “complete gauge fixing approach”, the
boundary conditions for Eq. (<ref>) is
restricted.
In this sense, a conventional “complete gauge-fixing approach”
includes a stronger restriction than our proposed gauge-invariant
formulation.










§ L=1 EVEN-MODE PERTURBATION IN THE CONVENTIONAL APPROACH










As discussed in
Sec. <ref>, l=1 mode
perturbations are given by
Eqs. (<ref>)–(<ref>).
In particular, among these expressions of the l=1 mode
perturbations, even-mode perturbation is given by

    h_AB
      =
          h̃_AB S
          ,
        
      h_Ap
      =
          r h̃_(e1)AD̂_pS
          ,
        
      h_pq
      =
          r^2/2γ_pqh̃_(e0) S
          .

As the property of the l=1 spherical harmonics, we obtain the
condition (<ref>).
For even-mode perturbation, we have

    (
      D̂_pD̂_q - 1/2γ_pqD̂_rD̂^r) Y_1m
      =
      0
      .






The gauge-transformation rules for the l=1 even-mode perturbations
are given by Eqs. (<ref>),
(<ref>), and
(<ref>).
Inspecting the gauge-transformation rules
(<ref>) and
(<ref>), we define the
variable H̃ by

    H̃   :=   h̃_(e0) - 4 (D̅^Ar) h̃_(e1)A
          .

The gauge-transformation rule of H̃ is given by

    _H̃
      -
      _H̃   =   
          - 4 (D̅^Ar) D̅_Aζ_(e)
          -  4/r( 1 - (D̅^Ar) (D̅_Ar) ) ζ_(e)
          .

Furthermore, inspecting gauge-transformation rules
(<ref>) and
(<ref>) we also define
the variable H̃_AB by

    H̃_AB   :=   h̃_AB
           -
           D̅_A( r h̃_(e1)B)
           -
           D̅_B( r h̃_(e1)A)
           .

The gauge-transformation of H̃_AB is given by

    _H̃_AB
      -
      _H̃_AB   =   
          - 2 r D̅_AD̅_Bζ_(e)
          + 2 (D̅_AD̅_Br) ζ_(e)
          .






The definitions (<ref>)
and (<ref>) of the
variables H̃ and H̃_AB, respectively, are
analogous to the gauge-invariant variable F̃ and
F̃_AB define by Eqs. (<ref>)
and (<ref>) for l≥ 2 modes,
respectively.
However, the variables H̃ and H̃_AB are not gauge
invariant.
The employment of the variables H̃ and H̃_AB
corresponds to the gauge fixing of the gauge degree of freedom of the
generator ζ_A and the specification of the component
h̃_(e1)A of the metric perturbation.
On the other hand, the gauge-transformation rules
(<ref>),
(<ref>), and
(<ref>) includes the gauge
degree of freedom of the generator ζ_(e).
This gauge degree of freedom appears in the gauge-transformation rules
(<ref>) and
(<ref>).





The linearized Einstein tensor for l=1 even modes in terms of
H̃_AB and H̃ are given as

    y_CB^(1)G_A^  C/S
         =   [
          -  1/2D̅_CD̅^C
          + 3/r^2
          -  1/r (D̅^Cr) D̅_C] H̃_AB
          + D̅_(AD̅^CH̃_B)C
          -  1/2D̅_AD̅_BH̃_C^  C
          
         +  2/r (D̅^Cr) [
         D̅_(AH̃_B)C
         -  1/r (D̅_Cr) H̃_AB]
         -  1/2D̅_AD̅_BH̃
         -  1/r (D̅_(Ar) D̅_B)H̃
          
         + 1/2 y_AB[
         [
         D̅_CD̅^C
         + 3/r (D̅^Cr) D̅_C] H̃
         +
         {D̅_DD̅^D
         -  5/r^2}H̃_C^  C.
         
                  .
         + 1/r (D̅^Cr) {
         2 D̅_CH̃_D^  D
         + 3/r (D̅_Cr) H̃_D^  D}
         -  D̅^CD̅^DH̃_CD.
         
                  .
         - 2/r (D̅^Dr) {
         2 D̅^CH̃_CD
         + 1/r (D̅^Cr) H̃_CD}]
         ,


    ^(1)G_A^  q   =   [
          1/2r^2D̅^CH̃_AC
          -  1/2r^2D̅_AH̃_C^  C
          + 1/2r^3 (D̅_Ar) H̃_C^  C
         -  1/4r^2D̅_AH̃] D̂^qS
         ,


    γ_qr^(1)G_p^  r   =   [
          + 1/4D̅_CD̅^CH̃
          + 1/2r (D̅^Cr) D̅_CH̃
         + 1/2{
         + D̅_CD̅^C
         + 1/r (D̅^Cr) D̅_C
         -  1/r^2}H̃_D^  D.
         
            .
         - 1/2{D̅_C
         + 2/r (D̅_Cr)
         }D̅_DH̃^CD] γ_pq S
          
          
         -  1/2r^2H̃_C^  C(
         D̂_pD̂_qS
         - 1/2γ_pqD̂^rD̂_r S
         )
         .

Since there is the identity
(<ref>) for l=1 mode
perturbations, the last term of
Eq. (<ref>)
does not appear for l=1 even modes.
Furthermore, the expressions
(<ref>) and
(<ref>) are
gauge invariant under the gauge-transformation rules
(<ref>) and
(<ref>).
Even in
Eq. (<ref>)
the component γ_qr^(1)G_p^  r is gauge invariant
except for the last term in
Eq. (<ref>).
Actually, the variable H̃_C^  C is gauge dependent as
shown below.
However, the gauge-invariance γ_qr^(1)G_p^  r is
guaranteed by the identity
(<ref>). 





On the other hand, the l=1 even components of them are given by

    ^(1) T_A^   B   =   
          S
          T̃_A^   B
          ,


    γ_qr^(1) T_A^  r   =   1/rT̃_(e1)AD̂_qS
          ,


    γ_qr^(1) T_p^  r   =   T̃_(e0)1/2γ_pq S
         +
         T̃_(e2)(
         D̂_pD̂_qS
         -
         1/2γ_pqD̂_rD̂^rS
         )

Here again, we note that
Eq. (<ref>) satisfies as
a mathematical identity for l=1 mode perturbations.
Therefore, the last term of
Eq. (<ref>) does not appear
for l=1 even modes.





If equation (<ref>) is
not satisfied, we have the equation

    [
         -  1/2r^2H̃_C^  C]
         (
         D̂_pD̂_qS
         - 1/2γ_pqD̂^rD̂_r S
         )
      =
        8πT̃_(e2)(
         D̂_pD̂_qS
         -
         1/2γ_pqD̂_rD̂^rS
         )

as one of the components of the linearized Einstein equation.
However, Eq. (<ref>)
implies that this equation is identically satisfied.
Therefore, this equation does not restrict H̃_C^  C nor
T̃_(e2) and there is no other restriction of them.
Here, we note that the tensor field H̃_AB is not
gauge-invariant as shown in
Eq. (<ref>).
Since we may freely choose H̃_C^  C and
T̃_(e2), we choose T̃_(e2)=0 by hand and choose

    H_C^  C = 0.

as a gauge condition.





Actually, from the gauge-transformation rule
(<ref>), we obtain

    y^AB_H̃_AB
      -
      y^AB_H̃_AB   =   
          - 2 r [
          -  1/f∂_t^2ζ_(e)
          + ∂_r( f ∂_rζ_(e))
          - 1-f/r^2ζ_(e)]
          .

Then, if we choose ζ_(e) as a special solution to the equation

    1/2r y^AB_H̃_AB   =   
          -  1/f∂_t^2ζ_(e)
          + ∂_r( f ∂_rζ_(e))
          - 1-f/r^2ζ_(e)
          ,

we may regard that _H̃_C^C=0 in the
-gauge.
We have to note that this gauge fixing
(<ref>) is not complete
gauge fixing.
There is a remaining degree of freedom in the choice of ζ_(e)
as the homogeneous solution ζ_(e)h to the wave equation

    -  1/f∂_t^2ζ_(e)h
      + ∂_r( f ∂_rζ_(e)h)
      - 1-f/r^2ζ_(e)h
      =
      0
      .






Due to the gauge condition (<ref>),
the components
(<ref>)–(<ref>)
of the linearized Einstein tensor and components
(<ref>)–(<ref>)
of the linear-order energy-momentum tensor yield the linearized
Einstein equations.
The (A,p)-components (equivalently, (p,B)-components) of the
linearized Einstein equations are given by

    D̅^CH̃_AC
      -  1/2D̅_AH̃
      =
      16 π r
      T̃_(e1)A
      .

Since the tensor H̃_AC is traceless due to the gauge
condition (<ref>), equation
(<ref>)
coincides with Eq. (<ref>).





From the trace part of the (p,q)-component of the linearized
Einstein equation and
Eq. (<ref>)
yields the component (<ref>) of the
continuity equation of the linearized energy-momentum tensor.





The trace part of (A,B)-component of the linearized Einstein
equation with
Eq. (<ref>)
gives

    D̅_CD̅^CH̃
         + 2/r (D̅_Cr) D̅^CH̃
         -  4/r^2 (D̅^Cr) (D̅^Dr) H̃_CD
      =
          16 π(
          T̃_C^  C
          +  4 (D̅^Dr) T̃_(e1)D)
          .

Since H̃_CD is a traceless tensor,
Eq. (<ref>)
coincides with the l=1 version of
Eq. (<ref>) with the source term
(<ref>).
However, we have to emphasize that H̃ nor H̃_CD
are not gauge invariant as shown above.





Finally, the traceless part of (A,B)-component of the linearized
Einstein equation with
Eqs. (<ref>) and
(<ref>)
yields

    [
          -  D̅_CD̅^C
          -  2/r (D̅^Cr) D̅_C
          + 4/r (D̅^CD̅_Cr)
          + 2/r^2] H̃_AB
         + 4/r (D̅^Cr) D̅_(AH̃_B)C
          
         -  2/r (D̅_(Ar) D̅_B)H̃
       =   
          16 π[
          T̃_AB
          -
          1/2 y_ABT̃_C^  C
         -  2 (
          D̅_(A(r T̃_(e1)B))
          - 1/2 y_ABD̅^C(r T̃_(e1)C)
          )
          .
          
                .
         + 2 y_AB (D̅^Dr) T̃_(e1)D]
          ,

Since the variable H_AB is traceless due to the gauge condition
(<ref>),
Eq. (<ref>)
coincides with the l=1 version of
Eq. (<ref>) with the
source term (<ref>).





In addition to these linearized Einstein equations, the following
linearized perturbations of the continuity equation should be
satisfied

    D̅^CT̃_C^  B
         + 2/r (D̅^Dr)T̃_D^   B
         -  2/rT̃_(e1)^B
         -  1/r (D̅^Br) T̃_(e0)
         =
         0
         ,
         
          D̅^CT̃_(e1)C
         + 3/r (D̅^Cr) T̃_(e1)C
         + 1/2rT̃_(e0)
         =
         0
         .






Together with the gauge condition
(<ref>),
Eqs. (<ref>)–(<ref>)
coincide with
Eqs. (<ref>)–(<ref>)
with l=1 in our gauge-invariant formulation developed in Refs. <cit.>.
In Ref. <cit.>, we derived the formal solution to
Eqs. (<ref>)–(<ref>)
with l=1 as Eq. (<ref>) with
Eqs. (<ref>),
(<ref>), and m=0.
Due to the coincidence of the set of equations, this formal solution
should be the m=0 formal solution to
Eqs. (<ref>), and Eqs. (<ref>)–(<ref>).
Then, as the m=0 solution, we obtain

    _ab   :=   
           H_ABcosθ (dx^A)_(a(dx^B)_b) + r^2/2 H cosθγ_ab
       =   _Vg_ab
          - 16π r^2/3(1-f)[
          f^2{1+f/2T̃_rr + r f ∂_rT̃_rr -
          T̃_(e0) - 4 T̃_(e1)r} (dt)_a (dt)_b.
          
                            .
          + 2r/f{∂_tT̃_tt - 3f(1-f)/2rT̃_tr} (dt)_(a(dr)_b).
          
                            .
          + r/f{∂_rT̃_tt - 3(1-3f)/2rfT̃_tt} (dr)_a (dr)_b)
          .
          
                            .
          + r^2T̃_ttγ_ab] cosθ
         ,

where γ_ab = γ_pq(dx^p)_a(dx^q)_b,
V_a=g_abV^b is the vector field defined by

    V_a := - r ∂_tΦ_(e)cosθ (dt)_a +
      (Φ_(e) - r ∂_rΦ_(e)) cos (dr)_a - r Φ_(e)sinθ (dθ)_a

and Φ_(e) is a solution to the equation

    - 1/f∂_t^2Φ_(e) + ∂_r[f ∂_rΦ_(e)]
      - 1-f/r^2Φ_(e)
      =
      16π r/3(1-f) S_(Φ_(e)),
      
          
         S_(Φ_(e)) = 1/2 r ∂_t T_tr - 1/2
         r ∂_rT̃_tt + 1-4f/2fT̃_tt
         - f/2T̃_rr - f T̃_(e1)r
         .






As in the case of the l=1 odd-mode perturbation, the tensor field
_ab in Eq. (<ref>) is
gauge invariant.
Then, V_a in Eq. (<ref>),
which defined by
Eq. (<ref>) is also
gauge invariant.
Therefore, we regarded V_a in
Eq. (<ref>) as the first kind gauge.
On the other hand, the tensor field _ab in
Eq. (<ref>) is gauge-dependent in
the sense of the second kind due to the gauge-transformation rules
(<ref>) and
(<ref>) for the components
H̃ and H̃_AB, respectively.
Actually, there is a remaining gauge degree of freedom of the generator
ζ_(e) which satisfies
Eq. (<ref>) as noted
above.
This remaining gauge degree of freedom is so-called “residual
gauge”.
For this reason, there is a possibility that V_a in
Eq. (<ref>) includes this “residual
gauge”, while V_a in
Eq. (<ref>) is gauge invariant.





Since we already fixed the gauge-degree of freedom ζ_(e) so that
Eq. (<ref>) is satisfied, the
remaining gauge degree of freedom ζ_(e)h must satisfy the
equation (<ref>).
Therefore, to clarify whether V_a in
Eq. (<ref>) includes the “residual
gauge,” we have to confirm
Eq. (<ref>).
Within our rules to compare our gauge-invariant formulation with a
conventional gauge-fixed approach, we regard the term in V_a in
Eq. (<ref>) satisfies
Eq. (<ref>) is the
second-kind gauge degree of freedom and we regard this degree of
freedom as an “unphysical degree of freedom.”
As in the case of the l=1 odd-mode perturbations, we introduce the
indicator function _(e)[*] as

    _(e)[ζ_(e)h]
      :=
      -  1/f∂_t^2ζ_(e)h
      + ∂_r( f ∂_rζ_(e)h)
      - 1-f/r^2ζ_(e)h
      .

If _(e)[ζ_(e)]=0, we should regard ζ_(e) is
the second-kind gauge degree of freedom and we regard ζ_(e) as
an “unphysical degree of freedom.”





To clarify the second-kind gauge degree of freedom, we consider the
gauge-transformation rule of the tensor field _ab through the
gauge-transformation rules
(<ref>) and
(<ref>) as

    __ab
      -
      __ab
       :=   (
           _H_AB
           -
           _H_AB) cosθ (dx^A)_(a(dx^B)_b)
           + r^2/2(
           _H
           -
           _H
           ) cosθγ_ab


    =   
          - 2 r [
          ∂_t^2ζ_(e)h
          -  f(1-f)/2r∂_rζ_(e)h
          +  f(1-f)/2r^2ζ_(e)h] cosθ
          (dt)_a(dt)_b
          
          - 4 r [
          ∂_t∂_rζ_(e)h
          -  1-f/2rf∂_tζ_(e)h] cosθ
         (dt)_(a(dr)_b)
          
          - 2 r [
          ∂_r^2ζ_(e)h
          + 1-f/2rf∂_rζ_(e)h
          -  1-f/2r^2fζ_(e)h] cosθ
         (dr)_a(dr)_b
          
         - 2 r^2(
         f ∂_rζ_(e)h
         + 1/r (1-f) ζ_(e)h) cosθγ_ab
       =   _Wg_ab
          ,

where

    W_a   =   
          - r ∂_tζ_(e)hcosθ (dt)_a
          + ( ζ_(e)h - r ∂_rζ_(e)h) cosθ(dr)_a
          - r ζ_(e)hsinθ (dθ)_a
          .

Comparing Eq. (<ref>) with the
generator (<ref>) and
Eq. (<ref>) with the generator
(<ref>), there is the
possibility that the Lie derivative term _Vg_ab in the
solution (<ref>) is “residual gauge
degree of freedom” with the identification

    Φ_(e) = ζ_(e)h.

For this reason, we check the indicator
(<ref>).
From Eq. (<ref>), we obtain

    _(e)[Φ_(e)]
      :=
      16π r/3(1-f) S_(Φ_(e))
      .

Therefore, the variable Φ_(e) should be regarded as the
second-kind gauge degree of freedom and is the “unphysical degree of
freedom” if S_(Φ_(e))=0, while in the non-vacuum case
S_Φ_(e)≠ 0, the indicator
(<ref>) yields
_(e)[Φ_(e)]≠ 0.
This means that Φ_(e) is not the second-kind gauge degree of
freedom but is the “physical degree of freedom” in the non-vacuum
case S_Φ_(e)≠ 0.
Due to this existence of the source term, the identification
(<ref>) is impossible in the case of the
non-vacuum situation.
Rather, this term is gauge-invariant in the sense of the second kind,
and we regard this term as a gauge-degree of freedom of the first
kind in the non-vacuum case.





Although we have _(e)[Φ_(e)]= 0 in the case where
S_Φ_(e)=0 and we should regard Φ_(e) is a “gauge
degree of freedom of the second kind”,
Eq. (<ref>) indicates that

    _(e)[Φ_(e)]
      =
      -  1/f∂_t^2Φ_(e)
      + ∂_r( f ∂_rΦ_(e))
      - 1-f/r^2Φ_(e)
      =
      0
      .

This coincides with the left-hand side of
Eq. (<ref>).
Therefore, as in the case of the l=1 odd-mode perturbations, we
consider the following two sets of function Φ_(e) as

    _(e)   :=   {.
           Φ_(e)|
           _(o)[Φ_(e)] = 0
           }
           ,
      
    _(e)   :=   {.
           Φ_(e)|
           -  1/f∂_t^2Φ_(e)
           + ∂_r( f ∂_rΦ_(e))
           - 1-f/r^2Φ_(e)
           =
           0
           }
           .

From Eq. (<ref>), we
obtain the relation

    _(e) = _(e).

This indicates that any homogeneous solution (without source term) to
Eq. (<ref>) should be
regarded as a gauge degree of freedom of the second kind which is
the “unphysical degree of freedom.”





On the other hand, to obtain the explicit solution Φ_(e) in the
case S_Φ_(e)≠ 0, which is regarded as a “physical degree
of freedom,” we have to solve
Eq. (<ref>) with appropriate
boundary conditions.
Equation (<ref>) is an inhomogeneous
second-order linear differential equation for Φ_(e) and its boundary conditions
are adjusted by the homogeneous solutions to
Eq. (<ref>), i.e., the
element of the set of function _(e).
However, any homogeneous solution to
Eq. (<ref>) should be
regarded as an “unphysical degree of freedom” in the “complete
gauge fixing approach” as mentioned above.
Therefore, in the conventional “complete gauge fixing approach”, we
have to impose the boundary conditions for
Eq. (<ref>) using a homogeneous
solution, which is regarded as an “unphysical degree of freedom,” to
obtain a “physical solution” to
Eq. (<ref>) with nonvanishing source
term S_Φ_(e)≠ 0.
This situation is a dilemma.
In this sense, as in the case of l=1 odd-mode perturbations, a
conventional “complete gauge-fixing approach” includes a stronger
restriction than our proposed gauge-invariant formulation.










§ L=0 MODE PERTURBATION IN THE CONVENTIONAL APPROACH














 §.§ Einstein equations for l=0 mode perturbations









Here, we consider the l=0 mode perturbations which are described by
Eqs. (<ref>)–(<ref>).
Substitution of
Eqs. (<ref>)–(<ref>)
into
Eqs. (<ref>)–(<ref>),
we obtain the l=0 mode perturbations of the linearized Einstein
tensor.
We use the background Einstein equations (Eq. (B67) and (B68) of
Appendix B in the Part I paper <cit.>).
Then, we obtain the non-trivial components of the l=0 mode Einstein
equation ^(1)G_a^  b=8π^(1)T_a^  b are  given by

    -  1/2D̅_CD̅^Ch̃_A^  B
          + 1/2D̅^BD̅^Ch̃_AC
          + 1/2D̅_AD̅_Ch̃^BC
          -  1/2D̅_AD̅^Bh̃_C^  C
          
         -  1/r (D̅^Cr) D̅_Ch̃_A^  B
         + 1/r (D̅^Cr) D̅^Bh̃_AC
         +  1/r (D̅_Cr) D̅_Ah̃^BC
         -  2/r^2 (D̅^Cr) (D̅_Cr) h̃_A^  B
          
         -  1/2r (D̅^Br) D̅_Ah̃_(e0)
         -  1/2r (D̅_Ar) D̅^Bh̃_(e0)
         -  1/2D̅_AD̅^Bh̃_(e0)
         + 2/r^2h̃_A^  B
          
         + y_A^  B(
          -  1/2D̅_CD̅_Dh̃^CD
         + 1/2D̅_CD̅^Ch̃_D^  D
         -  2/r (D̅_Dr) D̅_Ch̃^CD
         + 1/r (D̅^Cr) D̅_Ch̃_D^  D.
         
                  .
         -  1/r^2 (D̅_Cr) (D̅_Dr) h̃^CD
         + 3/2r^2 (D̅^Dr) (D̅_Dr) h̃_C^  C
         -  3/2r^2h̃_C^  C.
         
                  .
         + 1/2r^2h̃_(e0)
         + 3/2r (D̅^Cr) D̅_Ch̃_(e0)
         + 1/2D̅_CD̅^Ch̃_(e0))
         
       =   
             8 πT̃_A^   B
             ,
         
          D̅_CD̅^Ch̃_D^  D
         -  D̅_CD̅_Dh̃^CD
         -  2/r (D̅_Cr) D̅_Dh̃^CD
         + 1/r (D̅^Cr) D̅_Ch̃_D^  D
          
         + 1/2D̅_CD̅^Ch̃_(e0)
         + 1/r (D̅^Cr) D̅_Ch̃_(e0)
       =   
         8 πT̃_(e0)
         .






Here, we decomposition of the component h̃_AB as

    h̃_AB =: _AB + 1/2 y_ABh̃_C^  C.

In terms of the variables defined by
Eq. (<ref>), the linearization of
the Einstein equations for l=0 mode are summarized as follows.
The trace of ^(1)G_A^  B=8π^(1)T_A^  B
for l=0 mode is given by

    -  1/r^2h̃_C^  C
         - 2/r (D̅_Cr) (
         + D̅_D^DC
         + 1/r (D̅_Dr)  ^CD)
         
          
         + 1/2D̅_CD̅^Ch̃_(e0)
         + 2/r (D̅_Cr) D̅^Ch̃_(e0)
         + 1/r^2h̃_(e0)
         =
          8 πT̃_C^   C
          .

The traceless part of ^(1)G_A^  B=8π^(1)T_A^  B for l=0 mode is given by

    - 1/2[
         D̅_CD̅^C
         + 2/r (D̅^Cr) D̅_C
         + 4/r^2 (D̅^Cr) (D̅_Cr)
         - 4/r^2] _AB
          
         + D̅_(AD̅^C_B)C
         -  1/2 y_ABD̅_CD̅_D^CD
         + 2/r (D̅^Cr) (
         D̅_(A_B)C
         -  1/2 y_ABD̅^D_DC)
         
          
         + 1/r(
         (D̅_(Ar) D̅_B)h̃_E^  E
         -  1/2 y_AB (D̅^Cr) D̅_Ch̃_D^  D)
         - 1/2[
         D̅_AD̅_Bh̃_(e0)
         - 1/2 y_ABD̅_CD̅^Ch̃_(e0)]
         
          
         - 1/r[
         (D̅_(Ar) D̅_B)h̃_(e0)
         -  1/2 y_AB (D̅^Cr) D̅_Ch̃_(e0)]
         =
          8 π[
          T̃_AB - 1/2 y_ABT̃_C^   C]
          .

Furthermore, ^(1)G_p^  q=8π^(1)T_p^  q
for l=0 mode is given by

    1/2D̅_CD̅^Ch̃_(e0)
          + 1/r (D̅^Cr) D̅_Ch̃_(e0)
         - D̅_CD̅_D^CD
         -  2/r (D̅_Cr) D̅_D^CD
         + 1/2D̅_CD̅^Ch̃_D^  D
       =   
          8 πT̃_(e0)
          .










 §.§ Gauge-fixing for l=0 mode perturbations









We consider the gauge-transformation rules
(<ref>) and
(<ref>) of the mode
coefficient h̃_AB and h̃_(e0).
If these gauge-transformation rules are those of the second kind, we
should exclude these gauge degrees of freedom through some
gauge-fixing procedure, because the degree of freedom of the
second-kind gauge is the unphysical degree of freedom.





In the static chart in which the metric y_AB is given by
Eq. (<ref>).
Through the static chart
(<ref>), the
gauge-transformation rule
(<ref>) is given by

    _h̃_(e0)
      -
      _h̃_(e0)
      =
      4/r f  ζ_r
      .

Then we may choose ζ_r so that

    _h̃_(e0)   =   _h̃_(e0)
          + 4/r f  ζ_r
          =
          0
          ,

i.e.,

    ζ_r
      =
      - r/4f_h̃_(e0)
      .

Through this gauge-fixing, we may regard h̃_(e0)=0 in the
-gauge.





As the gauge-fixing for h̃_AB, we fix the gauge so that
h̃_AB is traceless.
This gauge-fixing so that h̃_AB is traceless makes easy to
compare with our gauge-invariant expression in Refs. <cit.>.
The trace of the gauge transformation rue
Eq. (<ref>), we obtain

    y^AB_h̃_AB
      -
      y^AB_h̃_AB   =   _h̃_C^  C
          -
          _h̃_C^  C
          =
          2
          D̅^Cζ_C
          .

In the static chart
(<ref>) of the
Schwarzschild spacetime, the gauge transformation rule
(<ref>) is given by

    _h̃_C^  C
      -
      _h̃_C^  C   =   
          - 2 f^-1∂_tζ_t
          + 2 ∂_r(fζ_r)
          .

From this gauge-transformation rule and the gauge fixing
(<ref>), we may fix the gauge
degree of freedom ζ_t so that

    0
         =   _h̃_C^  C
       =   _h̃_C^  C
          - 2 f^-1∂_tζ_t
          + 2 ∂_r(
          - r/4_h̃_(e0))
          .

Through this gauge-fixing, we may regard h̃_C^  C=0.
However, we have to emphasize that the degree of freedom of the choice
of the generator ζ_t remains the degree of freedom of an
arbitrary function of r.





Thus, through the gauge-fixing
(<ref>) and
(<ref>), we
may regard

    h̃_(e0) = h̃_C^  C = 0

Under the gauge-fixing condition
(<ref>) linearization of the Einstein equations
for l=0 mode are summarized as

    - (D̅_Cr) D̅_D^DC
      -  1/r (D̅_Cr) (D̅_Dr)  ^CD
      =
      4 π r T̃_C^   C
      ,


    - 1/2[
         D̅_CD̅^C
         + 2/r (D̅^Cr) D̅_C
         + 4/r^2 (D̅^Cr) (D̅_Cr)
         - 4/r^2] _AB
         + D̅_(AD̅^C_B)C
          
         -  1/2 y_ABD̅_CD̅_D^CD
         + 2/r (D̅^Cr) (
         + D̅_(A_B)C
         -  1/2 y_ABD̅^D_DC)
         
       =   
          8 π[
          T̃_AB - 1/2 y_ABT̃_C^   C]
          ,

and

    - D̅_CD̅_D^CD
      - 2/r (D̅_Cr) D̅_D^CD
      =
      8 πT̃_(e0)
      .










 §.§ Component expression of the linearized l=0 gauge-fixed
  field equations









Here, we consider the component representations of
Eqs. (<ref>)–(<ref>).
To do this, we consider the components of the traceless tensor
_AB as

    _AB
      =:
      X_(e)[(dt)_A (dt)_B + f^-2 (dr)_A(dr)_B]
      +
      2 Y_(e) (dt)_(A (dr)_B)
      .

Here, we use the background Einstein equation (B65) of Appendix B in
the Part I paper <cit.>, i.e.,

    ∂_rf = 1-f/r.

Then, we obtain

    ∂_r^2f
      =
      ∂_r(1-f/r)
      =
      - 2(1-f)/r^2






Through Eqs. (<ref>) and
(<ref>),
Eq. (<ref>) is given by

    r f ∂_tY_(e)
      - r f ∂_rX_(e)
      -  f X_(e)
      =
      - 4 π r^2( T̃_tt - f^2T̃_rr)
      .

The (A,B)=(t,t) and (A,B)=(r,r) components of
Eq. (<ref>) yield the
same equation as

    r f ∂_tY_(e)
      =
      4 π r^2( T̃_tt + f^2T̃_rr)
      .

The (A,B)=(t,r) component of
Eq. (<ref>) is given by

    ∂_tX_(e)
      =
      8 π r f T̃_tr
      .

In terms of the components (<ref>),
Eq. (<ref>) is
given by

    -  ∂_t^2X_(e)
      -  f^2∂_r^2X_(e)
      -  2/r f^2∂_rX_(e)
      + 2 f^2∂_t∂_rY_(e)
      + 1/r f(1+f) ∂_tY_(e)
      =
      8 π f^2T̃_(e0)
      .






On the other hand, the even-mode perturbation of the divergence of the
energy-momentum tensor is given by Eqs. (<ref>)
and (<ref>).
However, Eq. (<ref>) does not appear due to
D̂_pS=ϵ_pqD̂^qS=0.
Then, the non-trivial l=0 components of the divergence of the
energy-momentum tensor are given by

    D̅^CT̃_C^  B
      + 2/r (D̅^Dr)T̃_D^   B
      -  1/r (D̅^Br) T̃_(e0)
      =
      0
      .

The B=t component of
Eq. (<ref>) is given by

    ∂_tT̃_tt
      -  f^2∂_rT̃_rt
      - 1/r f (1+f) T̃_rt
      =
      0
      .

and B=r component of
Eq. (<ref>) is given by

    -  f ∂_tT̃_tr
      + f^3∂_rT̃_rr
      + 1/2r (1-f) T̃_tt
      + 1/2r f^2 (3+f) T̃_rr
      -  1/r f^2T̃_(e0)
      =
      0
      .






Substituting
Eq. (<ref>)
into
Eq. (<ref>),
we obtain

    ∂_r(r X_(e))
      =
      r ∂_rX_(e)
      + X_(e)
      =
      8 πr^2/fT̃_tt
      .

Substituting
Eqs. (<ref>),
(<ref>), and (<ref>)
into
Eq. (<ref>),
we obtain

    0
         =   
          -  f ∂_tT̃_tr
          + f^3∂_rT̃_rr
          + 1/2r (1-f) T̃_tt
          + 1/2r f^2 (3+f) T̃_rr
          -  1/r f^2T̃_(e0)
          .

This coincides with
Eq. (<ref>)
which indicates that
Eq. (<ref>)
does not give any new information other than
Eq. (<ref>).





Here, we consider the integrability of
Eq. (<ref>)
and
(<ref>) as follows:

    ∂_r(∂_t(rX_(e)))
      -
      ∂_t(∂_r(rX_(e)))
         =   ∂_r(
          8 π r^2 f T̃_tr)
          -
          ∂_t(
          8 πr^2/fT̃_tt)
          
       =   
          - 8 π r^2[
          + ∂_tT̃_tt
          - f^2∂_rT̃_tr
          - 1/r f (1+f) T̃_tr]
          
       =   
          0.

Here, we used
Eq. (<ref>) in
the last equality.
This means that the t-component
(<ref>) of the
continuity equation guarantees the integrability of
Eqs. (<ref>)
and
(<ref>).
Then, from
Eq. (<ref>),
we may write the solution to
Eqs. (<ref>)
and
(<ref>)
under the integrability condition
(<ref>) as

    X_(e)
      =
      1/r[2M_1 + 8π∫ dr r^2/fT̃_tt]
      ,

where M_1 is the constant of integration.
This M_1 corresponds to the perturbation of the Schwarzschild mass
parameter.





On the other hand, in the linearized Einstein equation, there is no
equation for ∂_rY_(e) which guarantees the integrability
condition for
Eq. (<ref>).
However, we may write the solution to the equation
(<ref>) as

    Y_(e)
      =
      4π r/f∫ dt ( T̃_tt + f^2T̃_rr)
      +
      Y_(e)0(r)
      .

where Y_(e)0(r) is an arbitrary function of r.
There is no equation that determines the arbitrary function
Y_(e)0(r) of r within the linearized Einstein equations.





From the definition (<ref>) of the metric
perturbation _AB and the solutions
(<ref>) and
(<ref>), we
obtain

    _AB   =   2/r(M_1 + 4π∫ dr r^2/fT̃_tt)
          ((dt)_A (dt)_B + f^-2 (dr)_A(dr)_B)
          
          
          2 [
          4π r ∫ dt ( 1/fT̃_tt + f T̃_rr)
          +
          Y_(e)0(r)
          ] (dt)_(A (dr)_B)
          .

Here, the components of the energy-momentum tensor satisfy the
continuity equations
(<ref>) and
(<ref>).





To interpret the term of Y_(e)0(r) in
Eq. (<ref>), we consider the term
_Vg_ab with the generator V_a whose components are
given by

    V_a = V_t(r) (dt)_a.

Then, the nonvanishing components of _Vg_ab are given
by

    _Vg_tr   =   
          f ∂_r( 1/f V_t)
          .

Choosing V_t so that

    f ∂_r( 1/f V_t)
      =
      Y_(e)0(r)
      ,   
      V_t
      =
      f ∫ dr 1/f Y_(e)0(r)
      ,

we obtain

    _Vg_tr   =   
          Y_(e)0(r)
          .

Then, the solution (<ref>) is given by

    _ab   =   2/r(M_1 + 4π∫ dr r^2/fT̃_tt)
          ((dt)_a (dt)_b + f^-2 (dr)_a(dr)_b)
          
          
         +
         8π r ∫ dt ( 1/fT̃_tt + f T̃_rr)
         (dt)_(a (dr)_b)
         +
         _Vg_ab
         ,

where

    V_a
      =
      (
      f ∫ dr 1/f Y_(e)0(r)
      ) (dt)_a
      .

As noted just after
Eq. (<ref>),
there is still the remaining degree of freedom of gauge whose
generator is given by

    ζ_a = ζ_t(r) dt_a,

where ζ_t(r) is an arbitrary function of r.
Therefore, we may regard the degree of freedom of V_a given in
Eq. (<ref>) can be
eliminated as the second-kind gauge degree of freedom
(<ref>).
According to our rule of the comparison between our gauge-invariant
formulation and a “conventional complete gauge-fixing approach,” we
have to regard that the degree of freedom of V_a given in
Eq. (<ref>) is
“unphysical.”





Thus, as the “physical solution” for l=0-mode linearized Einstein
equation based of a “conventional complete gauge-fixing approach,” we
obtain

    h_ab   =   2/r(M_1 + 4π∫ dr r^2/fT̃_tt)
          ((dt)_a (dt)_b + f^-2 (dr)_a(dr)_b)
          
          
         +
         8π r ∫ dt ( 1/fT̃_tt + f T̃_rr)
         (dt)_(a (dr)_b)
         .

This coincides with the solution obtained in
Refs. <cit.> except for the
terms of the Lie derivative of the background metric g_ab.
Therefore, we conclude that the l=0 solution except for the
terms of the Lie derivative of the background metric g_ab
obtained in Refs. <cit.> can
be also obtained as
(<ref>)
through a complete gauge-fixing approach.





The difference between our solution in
Refs. <cit.> is in the term
of the Lie derivative of the background metric.
As shown above, according to our rule of comparison between our
gauge-invariant formulation and a “conventional complete gauge-fixing
approach,” all terms of the Lie derivative of the background metric
should be regarded as the second-kind gauge degree of freedom,
and these are “unphysical degree of freedom” in the above solution
(<ref>).
On the other hand, in our gauge-invariant formulation developed in
Refs. <cit.>,
l=0,1-mode metric perturbations are given in a gauge-invariant form
and the terms of the Lie derivative of the background metric is
included in these gauge-invariant variables.
Since the second-kind gauge degree of freedom is completely excluded
in our gauge-invariant formulation, we cannot regard these terms of
the Lie derivative of the background metric as an “unphysical degree
of freedom.”
Therefore, we regard the terms of the Lie derivative of the
background metric in these gauge-invariant variables as
first-kind gauges that have some physical meaning.
This point is the essential difference of the solutions in
Refs. <cit.>
and the above solutions based on a “conventional complete
gauge-fixing approach.”
This difference leads to the confusion of the interpretation of the
perturbative Tolman Bondi solution, as shown in the next subsection.









 §.§ Comparing with Lemaître-olman-Bondi solution












  §.§.§ Perturbative expression of the LTB solution on
  Schwarzschild background spacetime








Here, we consider the Lemaître-Tolman-Bondi (LTB)
solution <cit.> which is an exact solution to
the Einstein equation with the matter field

    T_ab = ρ u_au_b,    u_a = - (dτ)_a,

and the metric

    g_ab   =   
          - (dτ)_a (dτ)_b
          + (∂_Rr)^2/1+f(R) (dR)_a(dR)_b
          + r^2γ_ab
          ,
            
          r = r(τ,R)
         .

This solution is a spherically symmetric solution to the Einstein equation.
The function r=r(τ,R) satisfies the differential equation

    (∂_τr)^2
      =
      F(R)/r
      +
      f(R)
      .

Here, we note that F(R) is an arbitrary function of R representing
the dust matter's initial distribution.
f(R) is also an arbitrary function of R that represents the
initial distribution of the energy of the dust field in the Newtonian
sense.
The solution to Eq. (<ref>) is given in the
three cases


  (i) f(R)>0 : 
  
    r = F(R)/2f(R)(coshη - 1),   τ_0(R) - τ  = F(R)/2f(R)^3/2 (sinhη -η)
        ,


  (ii) f(R)<0 : 
  
    r = F(R)/-2f(R)(1-cosη),   τ_0(R) - τ = F(R)/2(-f(R))^3/2 (η - sinη)
        ,


  (iii) f(R)=0 : 
  
    r = (9F(R)/4)^1/3[τ_0(R)-τ]^2/3
        .


The energy density ρ is given by

    8πρ = ∂_RF/(∂_Rr)r^2.

The LTB solution includes the three arbitrary functions f(R),
F(R), and τ_0(R).





Here, we consider the vacuum case ρ=0.
In this case, from Eq. (<ref>), we have

    ∂_RF = 0,    F = 2M
      ,

where M is the Schwarzschild metric with the mass parameter M.
Furthermore, we consider the case f(R)=0.
Here, we chose τ_0=R, i.e., ∂_Rτ_0=1.
In this case, Eq. (<ref>) yields

    (dR)_a   =   
          (dτ)_a
          +
          (2M/r)^-1/2
          (dr)_a
          .

Then, the metric (<ref>) is given by

    g_ab   =   
          - (dτ)_a (dτ)_b
          + (∂_Rr)^2 (dR)_a(dR)_b
          + r^2γ_ab
       =   
          -
          f
          (dt)_a(dt)_b
         +
         f^-1
         (dr)_a
         (dr)_b
         + r^2γ_ab
         ,
            
          f = 1 - 2M/r
          ,

where (dt)_a is defined by

    (dt)_a
      :=
      (dτ)_a
      -
      f^-1(1-f)^1/2
      (dr)_a
      .

We also note that the degree of freedom of the choice of the
coordinate R is completely fixed through the choice τ_0=R,
though there remains a degree of freedom of the choice of
R=R(R̃) in the exact solution (<ref>).





Now, we consider the perturbation of the Schwarzschild spacetime which
is derived by the exact LTB solution (<ref>) so that

    F(R)
         =   
          2 [
          M
          +
          ϵ
          m_1(R)
          ]
          + O(ϵ^2)
          ,
          
    
      f(R)
         =   
          0
          +
          ϵ
          f_1(R)
          + O(ϵ^2)
          ,
          
    τ_0(R)
         =   
          R
          +
          ϵτ_1(R)
          + O(ϵ^2)
          .

Through these perturbations
(<ref>)–(<ref>), we
consider the perturbative expansion of the function r which is
determined by Eq. (<ref>):

    r(τ,R) = r_s(τ,R) + ϵ r_1(τ,R)
      + O(ϵ^2)
      .

Here, the function r_s(τ,R) is given by
Eq. (<ref>), i.e.,

    r_s(τ,R)
      =
      r(τ,R)
      =
      (9M/2)^1/3[R-τ]^2/3
      .

In Eqs. (<ref>) and
(<ref>), we chose the background
value of the function τ_0(R) to be R.





Through this perturbative expansion, we evaluate
O(ϵ^1) perturbation of Eq. (<ref>)
through Eq. (<ref>) as

    (1-f)^1/2
         (∂_τr_1)
         +
         m_1(R)/r
         -
         M/r^2
         r_1
         +
         1/2
         f_1(R)
         =
         0
         ,

where we used Eq. (<ref>)
and the replacement r_s→ r.
The solution to Eq. (<ref>) is
given by

    r_1   =   (M/6)^1/3m_1(R)/M[R-τ]^2/3
          -
          3/20(6/M)^1/3
          f_1(R)
          [R-τ]^+4/3
          
          +
          B(R)
          [R-τ]^-1/3
          .

From the comparison with Eq. (<ref>),
B(R) is the perturbation of the τ_1(R) as
τ_0(R)=R+τ_1(R) in the exact solution
(<ref>)–(<ref>).
Furthermore, the solution
(<ref>) can be also derived
from the exact solution
(<ref>)–(<ref>).
From Eq. (<ref>), the perturbative dust energy
density given by

    8πρ = 2∂_Rm_1(R)/(∂_Rr)r^2.






Through the perturbative solution
(<ref>), the metric
(<ref>) is given by

    g_ab   =   
          - (dτ)_a (dτ)_b
          +
          (∂_Rr)^2
          (dR)_a(dR)_b
          +
          r^2γ_ab
          
         +
         ϵ[
         (
         2(∂_Rr_1)
         -
         f_1
         (∂_Rr)
         )
         (∂_Rr)
         (dR)_a(dR)_b
         +
         2
         r
         r_1γ_ab]
         + O(ϵ^2)
         
       =:   
           g_ab^(0)
           +
           ϵ_h_ab
           + O(ϵ^2)
           .

As shown in Eq. (<ref>),
the background metric g_ab^(0) is given by the Schwarzschild
metric in the static chart.
On the other hand, the linear order perturbation _h_ab
(in the gauge _ϵ) is given by

    _h_ab
      :=
      (
      2
      (∂_Rr_1)
      -
      f_1(R)
      (∂_Rr)
      )
      (∂_Rr)
      (dR)_a(dR)_b
      +
      2
      r
      r_1γ_ab
      .

Here, we fixed the second-kind gauge so that

    _ϵ :
      (τ,R,θ,ϕ)∈_ph↦
      (τ,R,θ,ϕ)∈.

Through this second-kind gauge choice and the choice of the background
radial coordinate R in
Eq. (<ref>), the degree of
freedom of the choice of the radial coordinate R=R(R̃) is
also completely fixed even in the linearized version of the exact
solution (<ref>), which
though there remains a degree of freedom of the choice of
R=R(R̃) in the exact solution (<ref>).





Of course, if we employ the different gauge choice
_ϵ from the above gauge-choice
_ϵ, we obtain the different expression of the
metric perturbation _h_ab.
Actually, we may choose _ϵ as the identification of

    _ϵ :
      (τ+ϵξ^τ(τ,R),R+ϵξ^R(τ,R),θ,ϕ)∈_ph↦
      (τ,R,θ,ϕ)∈.

In this identification, the metric on _ph pulled back to
 is given by

    __ϵg_ab   =   __ϵg_ab
          +
          ϵ_ξg_ab^(0)
          +
          O(ϵ^2)
          
       =   
          g_ab^(0)
          +
          ϵ(
          _h_ab
          +
          _ξg_ab^(0))
          +
          O(ϵ^2)
           ,

where
ξ^a=ξ^τ(∂_τ)^a+ξ^R(∂_R)^a is
the generator of second-kind gauge transformation
_ϵ→_ϵ.








  §.§.§ Expression of the perturbative LTB solution in static chart








Here, we consider the expression of the linear perturbation
_h_ab given by
Eq. (<ref>).
From Eqs. (<ref>) and
(<ref>) with F=2M, we obtain

    (dR)_a   =   
          (dt)_a
          +
          f^-1
          (1-f)^-1/2
          (dr)_a
          ,
            
          f = 1 - 2M/r
          ,
          
    
      (dτ)_a   =   
          (dt)_a
          +
          f^-1
          (1-f)^1/2
          (dr)_a
          .






First, we consider the perturbation of the energy-momentum tensor of
the matter field.
In the case of the LTB solution, the matter field is characterized by
the dust field whose energy-momentum tensor
(<ref>) is given by

    T_ab = ρ u_a u_b,    u_a = - (dτ)_a,
         u^a = (∂_τ)^a.

In our case, the linearized Einstein equation gives
Eq. (<ref>), i.e.,

    8πρ
      =
      2∂_Rm_1(R)/(∂_Rr)r^2
      =
      ∂_Rm_1(R)/4π r^2 (1-f)^-1/2
      .

On the other hand, substituting
Eq. (<ref>) into
Eq. (<ref>), we obtain

    T_ab   =   ρ (dτ)_a (dτ)_b
       =   ρ(
          (dt)_a
          +
          f^-1
          (1-f)^1/2
          (dr)_a)
          (
          (dt)_b
          +
          f^-1
          (1-f)^1/2
          (dr)_b)
          
       =   ρ
          (dt)_a
          (dt)_b
          +
          ρ(1-f)^1/2/f
          2
          (dt)_(a (dr)_b)
          +
          ρ1-f/f^2
          (dr)_a
          (dr)_b
          .

Then, we obtain the components of the energy-momentum tensor for the
static coordinate (t,r) as

    T̃_tt
      =
      ρ
      ,
        T̃_tr
      =
      (1-f)^1/2/fρ
      ,
        T̃_rr
      =
      1-f/f^2ρ
      ,
        T̃_(e0)
      =
      0
      .

From this component, we can confirm the continuity equations (<ref>)
and (<ref>).





As derived in Ref. <cit.>, the linearized
Tolman-Bondi solution (<ref>)
with the Schwarzschild background is given by

    _h_ab   =   2m_1(R)/r[
          (dt)_a (dt)_b + 1/f^2 (dr)_a (dr)_b]
          +
          2-f/f (1-f)^1/2m_1(R)/r
          2 (dt)_(a (dr)_b)
          
         + _V_(LTB)g_ab
         ,

where V_(LTB)a are given by

    V_(LTB)a
      :=
      [(1-f)^1/2 r_1 + 1/2 f ∫ dt f_1(R)] (dt)_a
      +
      r_1/f (dr)_a
      .

As shown in Ref. <cit.>, the l=0 solution
(<ref>) with
the components (<ref>) of the linearized
energy-momentum tensor realize the first line of
Eq. (<ref>).
In this sense, the solutions
(<ref>) of
the l=0 mode perturbations are justified by the LTB solutions.





However, we emphasize that the linearized LTB solution
(<ref>) does have the
term _V_LTBg_ab.
As noted by Eqs. (<ref>) and
(<ref>), we can always
eliminate the terms of the Lie derivative of the background metric as
the second-kind gauge-degree of freedom.
Therefore, according to our rule of the comparison, the term
_V_(LTB)g_ab should be regarded as the second-kind
gauge-degree of freedom in the “conventional gauge-fixing approach”
discussed in this paper.
In this case, we have to regard that the perturbation f_1(R) of
the initial distribution f(R) of the energy of dust field in
Eq. (<ref>) is an “unphysical degree of
freedom” in spite that the behavior of the LTB solution crucially
depends on the signature of the function f(R) as shown in
Eqs. (<ref>)–(<ref>).










§ SUMMARY AND DISCUSSIONS










In this paper, we have discussed comparison of our gauge-invariant
formulation for l=0,1 perturbations on the Schwarzschild background
spacetime proposed in the series of papers <cit.>
and a “conventional complete gauge-fixing approach.”
It is well-known that we cannot construct gauge-invariant variables
for l=0,1 mode perturbations through a similar manner to the
construction of the gauge-invariant variables for l≥ 2 mode
perturbations if we use the decomposition formula
(<ref>)–(<ref>) with the spherical
harmonic functions Y_lm as the scalar harmonics S_δ.
In our gauge-invariant formulation for l=0,1 perturbations on the
Schwarzschild background spacetime, we proposed the introduction of
the singular harmonic function at once.
Due to this, we can construct gauge-invariant variables for
l=0,1-mode perturbations in a similar manner to the l≥ 2
modes of perturbations.
After deriving the mode-by-mode perturbative Einstein equations in
terms of the gauge-invariant variables, we impose the regularity on
the introduced singular harmonics when we solve the derived Einstein
equations.
Based on this proposal, we derived formal solutions to the
l=0,1-mode linearized Einstein equations without the specification
of the components of the linear perturbation of the energy-momentum tensor <cit.>.
Our proposal enables us to develop higher-order perturbations of the
Schwarzschild spacetime <cit.>.
Furthermore, we check that our derived solutions realized the
linearized version of the LTB solution and non-rotating
C-metric <cit.>.
In this sense, we conclude that our proposal is physically
reasonable.
On the other hand, it is often said that “gauge-invariant
formulations in general-relativistic perturbations are equivalent to
complete gauge-fixing approaches.”
For this reason, we check this statement through the comparison of our
gauge-invariant formulation and a “conventional complete gauge-fixing
approach” in which we use the spherical harmonic functions Y_lm as
the scalar harmonics S_δ from the starting point.





After reviewing the concept of “gauges” in general relativistic
perturbation theories and our proposed gauge-invariant formulation for
the l=0,1-mode perturbations and our derived l=0,1-mode solutions,
we considered l=1 odd-mode perturbations, l=1 even-mode
perturbations, and l=0 even-mode perturbations, separately.
As a result, it is shown that we can actually derive similar solutions
even in the treatment of the “conventional complete gauge-fixing
approach.”
However, we have to emphasize that the derived solutions are slightly
different from those derived based on our gauge-invariant formulation
from a conceptual point of view.





In the case of l=1 odd-mode perturbations, we obtained the
formal solution to the linearized Einstein equation through our proposed
gauge-invariant formulation.
This formal solution includes the term of the Lie derivative of the
background metric.
In our gauge-invariant formulation, we describe the solutions
only through gauge-invariant variables.
Therefore, we should regard the term of the Lie derivative of the
background metric is gauge invariant.
On the other hand, in the conventional gauge-fixing approach where we
use the spherical harmonics Y_lm from the starting point, we
cannot construct gauge-invariant variable for l=1 odd-mode
perturbations in a similar manner to those for l≥ 2 mode
perturbations.
For this reason, we have to treat gauge-dependent variables for
perturbations.
However, the linearized Einstein equations and the continuity
equations of the linearized energy-momentum tensor for l=1 odd-mode
perturbations in terms of these gauge-dependent variables have the
completely same form that is derived through our gauge-invariant
formulation.
Therefore, the same formal solutions derived through our
gauge-invariant formulation should be the formal solutions to these
linearized Einstein equations and the continuity equation of the
linearized energy-momentum tensor in terms of gauge-dependent
variables.
As noted above, we treat gauge-dependent variables in the
conventional gauge-fixing approach.
Therefore, the above Lie derivative terms of the background metric may
include the gauge degree of freedom of the second kind, which should be
regarded as an “unphysical degree of freedom.”
Checking the residual gauge degree of freedom, we conclude that the above
Lie derivative terms of the background metric include the second-kind
gauge degree of freedom.
However, in our formal solution, there is a variable that should be
obtained by solving the l=1 Regge-Wheeler equation.
We conclude that the solution to this l=1 Regge-Wheeler equation is
not the gauge degree of freedom of the second kind but a physical
degree of freedom.
Furthermore, we have to impose appropriate boundary conditions to
solve this l=1 Regge-Wheeler equation.
Since the l=1 Regge-Wheeler equation is an inhomogeneous linear
second-order partial differential equation, the boundary conditions
for an inhomogeneous linear second-order partial differential equation
are adjusted by the homogeneous solutions to this linear second-order
partial differential equation.
According to the check of the residual gauge degree of freedom, we
have to conclude that a part of homogeneous solutions to this equation
is the gauge degree of freedom of the second kind.
We have to eliminate this part from our consideration because this
gauge degree of freedom is “unphysical.”
This is the restriction of the boundary conditions of the linearized
Einstein equations.





In the case of l=1 even-mode perturbations, the situation is
worse than the l=1 odd-mode case.
As in the l=1 odd-mode case, we obtained the
formal solution to the linearized Einstein equation through our proposed
gauge-invariant formulation.
This formal solution includes the term of the Lie derivative of the
background metric, which is gauge-invariant within our proposed
gauge-invariant formulation.
On the other hand, in the conventional gauge-fixing approach where we
use the spherical harmonics Y_lm from the starting point, we
cannot construct gauge-invariant variable for l=1 even-mode
perturbations in a similar manner to those for l≥ 2 mode
perturbations.
For this reason, we have to treat gauge-dependent variables for
perturbations.
As in the l=1 odd-mode case, the linearized Einstein equations and
the continuity equations of the linearized energy-momentum tensor for
l=1 even-mode perturbations in terms of these gauge-dependent
variables have the completely same form that is derived through our
gauge-invariant formulation.
Therefore, the same formal solutions derived through our
gauge-invariant formulation should be the formal solutions to these
linearized Einstein equations and the continuity equation of the
linearized energy-momentum tensor in terms of gauge-dependent
variables.
As noted above, we treat gauge-dependent variables in the
conventional gauge-fixing approach as in the l=1 odd-mode case.
Therefore, the above Lie derivative terms of the background metric may
include the second-kind gauge degree of freedom, which should be
regarded as an “unphysical degree of freedom.”
However, in our formal solution, there is a variable that should be
obtained by solving the l=1 Zerilli equation.
We conclude that the solution to this l=1 Zerilli equation is
not the gauge degree of freedom of the second kind but the physical
degree of freedom in the non-vacuum case.
Furthermore, we have to impose appropriate boundary conditions to
solve this l=1 Zerilli equation in the non-vacuum case.
Since the l=1 Zerilli equation is an inhomogeneous linear
second-order partial differential equation.
The boundary conditions for an inhomogeneous linear second-order
partial differential equation are adjusted by the homogeneous
solutions to this linear second-order partial differential equation.
According to the check of the residual gauge degree of freedom, we
have to conclude that all homogeneous solutions to this equation
are the gauge degree of freedom of the second kind, i.e., all
homogeneous solution to the l=1 Zerilli equation is “unphysical.”
We have to eliminate these homogeneous solutions from our
consideration because these are regarded as “unphysical.”
Due to this situation, we have to say that we have to impose the
boundary conditions using these “unphysical degrees of freedom” to
obtain the “physical solution” through l=1 Zerilli equation.
This is a dilemma.





In the case of l=0 even-mode perturbations, we obtain the complete
gauge fixed solution.
This solution does not include any term of the Lie derivative of the
background metric.
All terms of the Lie derivative of the background metric are regarded
as the gauge degree of freedom of the second kind.
On the other hand, the solution that is derived through our proposed
gauge-invariant formulation includes the terms of the Lie derivatives
of the background metric which is regarded as the first-kind gauge in
our gauge-invariant formulation, i.e., these are “physical.”
This difference leads a problem when we compare the derived solution
with the linearized LTB solution.
In the linearized LTB solution, the initial energy distribution
f_1(R) of the dust field in the Newtonian sense is included in the
terms of the Lie derivative of the background metric.
When we realize this linearized exact solution by the solution
obtained by our gauge-invariant formulation, this initial energy
distribution is regarded as a physical degree of freedom.
On the other hand, when we realize this linearized exact solution
through the solution obtained by a conventional complete gauge-fixing
approach, we have to regard this initial energy distribution of the dust
field as the gauge degree of freedom of the second kind and have to
regard it “unphysical.”
Since the behavior of the exact LTB solution crucially depends on this
initial energy distribution of the dust field, it is not natural to
regard that this initial degree of freedom is unphysical.





In summary, we have to conclude that there is a case where the
boundary conditions and initial conditions are restricted in the
conventional complete gauge-fixing approach where we use the
decomposition of the metric perturbation by the spherical harmonics
Y_lm from the starting point.
On the other hand, such a situation does not occur in our proposed
gauge-invariant formulation.
As a theory of physics, this point should be regarded as the
incompleteness of the conventional complete gauge-fixing approach where
we use the decomposition of the metric perturbation by the spherical
harmonics Y_lm from the starting point.
This is the main result of this paper.





Let us discuss why such a difference occurs between our proposed
gauge-invariant formulation and a conventional complete gauge-fixed
approach where we use the decomposition of the metric perturbation by
the spherical harmonics Y_lm from the starting point.
The aim of the introduction of the singular harmonic functions in our
proposed gauge-invariant formulation is to increase the degree of
freedom to clarify the distinction between the gauge degree of freedom
of the second kind and the physical degree of freedom.
As emphasized in Sec. <ref>, if we choose
S=Y_lm, we have D̂_pS = ϵ_pqD̂^qS =
0,
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
= 2 ϵ_r(pD̂_q)D̂^rS = 0 for l=0
modes and
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
= 2 ϵ_r(pD̂_q)D̂^rS = 0 for l=1 modes.
This is the essential reason for the fact that we cannot construct
gauge-invariant variables for l=0,1 mode metric perturbation in a
similar manner to the derivation of gauge-invariant variables for
l≥ 2 modes.
Due to these vanishing vector- or tensor-harmonics, the associated
mode coefficients do not appear, and we cannot construct
gauge-invariant variables for l=0,1-modes in a similar manner to the
case of l≥ 2 modes.
In our series of
papers <cit.>,
we regarded that this is due to the lack of the degree of freedom.
For this reason, in our proposed gauge-invariant formulation, we
introduced singular harmonic functions k_Δ̂ and
k_Δ̂+̂2̂m for l=0 and l=1 modes, respectively.
Owing to the introduction of these singular harmonic functions, we have
D̂_pS ≠ 0 ≠ ϵ_pqD̂^qS and
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
≠ 0 ≠ 2 ϵ_r(pD̂_q)D̂^rS for l=0
modes and (D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
≠ 0 ≠ 2 ϵ_r(pD̂_q)D̂^rS for l=1
modes.
Thanks to these non-vanishing vector- or tensor-harmonics, the
associated mode coefficients do appear, and we can construct
gauge-invariant variables for l=0,1-modes in a similar manner
to the case of l≥ 2 modes.
This leads to the development of the gauge-invariant perturbation
theory for all
modes <cit.>
and the development of the higher-order gauge-invariant
perturbations <cit.>.





As noted in the Part I paper <cit.>, the
decomposition using the spherical harmonics Y_lm from the starting
point corresponds to the imposition of the regular boundary condition
on S^2 on the metric perturbations from the starting point.
In this sense, our introduction of the singular harmonic functions
corresponds to the change of the boundary conditions on S^2.
As shown in the Part I paper <cit.>, if we
introduce this change of boundary conditions on S^2, we can
clearly distinguish the gauge degree of freedom of the second kind and
the physical degree of freedom, i.e., we can easily construct
gauge-invariant variables for l=0,1-mode perturbations.
This indicates that the imposition of the boundary conditions on
S^2 and the construction of the gauge-invariant variables does not
commute as the procedure of calculations.
This is the appearance of the non-locality of l=0,1-mode
perturbations as pointed out in Ref. <cit.>.
Due to these reasons, we reached the conceptual difference between
the l=0,1-mode solutions discussed in this paper.
Namely, in the conventional complete gauge-fixing approach where we use
the decomposition by the harmonic function Y_lm from the starting
point, the degree of freedom of the metric perturbations is not
sufficient so that we can distinguish the gauge degree of freedom of
the second kind, i.e., “unphysical modes” and “physical modes.”
In spite of this situation of the lack of degree of freedom, if we
dare to carry out the “complete gauge-fixing” as the “elimination
of unphysical modes,” this “complete gauge-fixing” leads to the
restriction of the boundary conditions and initial conditions as
shown in this paper.





On the other hand, in our proposed gauge-invariant formulation, there
are no conceptual difficulties, such as the restriction of the boundary
conditions and initial conditions pointed out in this paper, due to the
sufficient degree of freedom of the metric perturbations.
Incidentally, due to this sufficient degree of freedom of the metric
perturbations through the introduction of the singular harmonic
functions, our proposed gauge-invariant variables are equivalent to
variables of the complete gauge-fixing within our proposed formulation
in which the degree of freedom of the metric perturbations is
sufficiently extended.
Furthermore, we can develop higher-order perturbation theory without
gauge-ambiguities if we apply our proposal and there are wide
applications of the higher-order gauge-invariant perturbations on
the Schwarzschild background spacetime as briefly discussed in
Ref. <cit.>.
We leave these further developments of the application of our
formulation to concrete problems of perturbations on the Schwarzschild
background spacetime as future works.




















§ LINEARIZED EINSTEIN TENSOR










In this Appendix, we derive components of the linearized Einstein
tensor.
Although similar formulae are derived in Appendix C in the Paper
I <cit.> in the gauge-invariant form.
However, in this paper, we have to derive the components of the
Einstein tensor with the spherically background spacetime without any
gauge fixing.
In this case, the formulation of the “gauge-ready formulation”
proposed in Ref. <cit.> in the context of cosmological
perturbation theories is an appropriate formulation.
Therefore, in this Appendix, we derive the components of the
linear-order Einstein tensor without any gauge fixing following the
philosophy of the “gauge-ready formulation” 'in
Ref. <cit.>.
The derived formulae in the philosophy of “gauge ready formulation”
is useful in the ingredient of this manuscript.





Here, we consider the metric perturbation as

    _ Xg̅_ab
      =
      g_ab + ϵ_ Xh_ab + O(ϵ^2).

The connection C^c_  ab between the covariant derivative
∇̅_a associated with the metric g̅_ab and the
covariant derivative ∇_a associated with the metric g_ab
is given by

    C^c_  ab = 1/2g̅^cd(
      ∇_ag̅_db + ∇_bg̅_da - ∇_dg̅_ab)
      ,

where g̅^ab is the inverse of g̅_ab.
Here, we expand the connection C^c_  ab with respect to
ϵ as

    C^c_  ab =
      ϵ^(1)C^c_  ab
      +
      O(ϵ^2)
      .

Then, we have

    ^(1)C^c_  ab
      =
      1/2 g^cd(
      ∇_ah_db + ∇_bh_da - ∇_dh_ab)
      .

The relation between the Riemann curvature
R̅_abc^      d associated with the metric
g̅_ab and the curvature R_abc^      d associated
with the background metric g_ab is given by

    R̅_abc^      d
      =
      R_abc^      d
      - 2 ∇_[a^C^d_  b]c
      + 2 C^e_  c[aC^d_  b]e,

Then, we have

    R̅_abc^      d
      =
      R_abc^      d
      - 2 ϵ∇_[a^^(1)C^d_  b]c
      +
      O(ϵ^2)
      .

The perturbative expansion of the Ricci tensor R̅_ac is given
by

    R̅_ac
      =
      R_ac
      - 2 ϵ∇_[a^^(1)C^b_  b]c
      +
      O(ϵ^2)
      .

The perturbative expansion of the curvature R̅_a^  c is
given by

    R̅_a^  d   =   g̅^cdR̅_ac
       =   g̅^cdR̅_ac
          + ϵ(
          - 2 g^cd∇_[a^^(1)C^b_  b]c
          -  h^cd R_ac)
          +
          O(ϵ^2)
          .

The perturbation of the scalar curvature is given by

    R̅   =   g̅^acR̅_ac
       =   
          R
          + ϵ(
          - h^ac R_ac
          - 2 g^ac∇_[a^^(1)C^b_  b]c)
          +
          O(ϵ^2)
          .

Then, the perturbative expansion of the Einstein tensor
G̅_a^  d is given by

    G̅_a^  d   =   R̅_a^  d - 1/2δ_a^  dR̅
       =   
          G_a^  d
          + ϵ(
          - 2 g^cd∇_[a^^(1)C^b_  b]c
          +  δ_a^  d  g^ec∇_[e^^(1)C^b_  b]c
          -  h^cd R_ac
          +  1/2δ_a^  d h^ec R_ec)
          +
          O(ϵ^2)
          
       =:   
          G_a^  d
          + ϵ^(1)G_a^  d
          +
          O(ϵ^2)
          .

Namely, the 1st-order perturbation of the Einstein tensor is given by

    ^(1)G_a^  d   =   
          -  2 g^cd∇_[a^^(1)C^b_  b]c
          + δ_a^  d  g^ec∇_[e^^(1)C^b_  b]c
          -  h^cd R_ac
          + 1/2δ_a^  d h^ec R_ec
          .

Substituting Eq. (<ref>) into
Eq. (<ref>), we obtain

    ^(1)G_a^  d   =   
          -  1/2∇_b∇^bh_a^  d
          + R_abc^      d h^bc
          
          + 1/2(
          g_ac∇^d_
          - 2 δ_[a^  d∇_c]^) ∇_bh^cb
          - 1/2(
          ∇_a∇^d - δ_a^  d∇_c∇^c) h_b^  b
          
          -  1/2 R_ac h^dc
          + 1/2 R^df h_af
          + 1/2δ_a^  d h^ec R_ec
          .

Since we consider the vacuum background spacetime R_ab=0, we
obtain

    ^(1)G_a^  d   =   
          -  1/2∇_b∇^bh_a^  d
          + R_abc^      d h^bc
          
          + 1/2(
          g_ac∇^d_
          - 2 δ_[a^  d∇_c]^) ∇_bh^cb
          - 1/2(
          ∇_a∇^d - δ_a^  d∇_c∇^c) h_b^  b
          .






To derive the component of ^(1)G_a^  d, we denote the
components of the metric perturbation and the derivative operator as

    h̅_AB := h_AB,   h̅_A^  D := y^DE h_AE,   h̅^pD := γ^pq y^DE h_qE,   
          h̅_p^  q := γ^qr h_pr,   h̅^pq := γ^prγ^qs h_rs,

and

    D̅^C = y^CED̅_E,   D̂^p = γ^pqD̂_q.






Now, the components of ^(1)G_a^  d in terms of the
variable defined by Eq. (<ref>) as follows:

    ^(1)G_A^  B   =   
          -  1/2D̅_CD̅^Ch̅_A^  B
         -  1/2r^2D̂_pD̂^ph̅_A^  B
         -  2/r^2 (D̅^Cr) (D̅_Cr) h̅_A^  B
         + 2/r^2h̅_A^  B
          
          + 1/2D̅^BD̅^Ch̅_AC
          + 1/2D̅_AD̅_Ch̅^BC
         -  1/r (D̅^Cr) D̅_Ch̅_A^  B
          -  1/2D̅_AD̅^Bh̅_C^  C
          
         + 1/r (D̅^Cr) D̅^Bh̅_AC
         +  1/r (D̅_Cr) D̅_Ah̅^BC
          
         + 1/2r^2D̅^BD̂^ph̅_Ap
         + 1/2r^2D̅_AD̂_ph̅^Bp
          
         -  1/2r^2D̅_AD̅^Bh̅_r^  r
         + 1/2r^3 (D̅_Ar) D̅^Bh̅_r^  r
         + 1/2r^3 (D̅^Br) D̅_Ah̅_r^  r
         -  1/r^4 (D̅_Ar) (D̅^Br) h̅_r^  r
          
         + y_A^  B(
          -  1/2D̅_CD̅_Dh̅^CD
         + 1/2D̅_CD̅^Ch̅_D^  D
         + 1/2r^2D̂_pD̂^ph̅_C^  C
         -  2/r (D̅_Dr) D̅_Ch̅^CD.
         
                .
         + 1/r (D̅^Cr) D̅_Ch̅_D^  D
         -  1/r^2 (D̅_Cr) (D̅_Dr) h̅^CD
         + 3/2r^2 (D̅^Dr) (D̅_Dr) h̅_C^  C.
         
                .
         -  1/r^2D̅_CD̂_ph̅^pC
         -  1/r^3 (D̅_Cr) D̂_ph̅^Cp.
         
                .
         + 1/2r^2D̅_CD̅^Ch̅_r^  r
         + 1/2r^4D̂_pD̂^ph̅_r^  r
         -  1/2r^3 (D̅^Cr) D̅_Ch̅_r^  r
         -  1/2r^4D̂_pD̂_sh̅^ps.
         
                .
         + 1/2r^4 (D̅_Cr) (D̅^Cr) h̅_r^  r
         -  3/2r^2h̅_C^  C)
         .


    ^(1)G_A^  q   =   1/2r^2D̂^qD̅^Ch̅_AC
          -  1/2r^2D̅_AD̂^qh̅_C^  C
          + 1/2r^3 (D̅_Ar) D̂^qh̅_C^  C
          
          -  1/2r^2D̅_CD̅^Ch̅_A^  q
          -  1/2r^4D̂_rD̂^rh̅_A^  q
          + 1/2r^4h̅_A^  q
          + 1/2r^4D̂^qD̂^ph̅_Ap
          
          + 1/2r^2D̅_AD̅_Dh̅^qD
          -  1/r^3 (D̅_Ar) D̅_Dh̅^qD
          -  1/r^4 (D̅_Ar) (D̅_Dr) h̅^qD
          + 1/r^3 (D̅_Dr) D̅_Ah̅^qD
          
          + 1/2r^4D̅_AD̂_sh̅^qs
          -  1/r^5 (D̅_Ar) D̂_sh̅^qs
          -  1/2r^4D̅_AD̂^qh̅_r^  r
          + 1/r^5 (D̅_Ar) D̂^qh̅_r^  r
          ,


    ^(1)G_p^  q   =   
          -  1/2r^2D̅_CD̅^Ch̅_p^  q
          -  1/2r^4D̂_sD̂^sh̅_p^  q
          + 1/r^3 (D̅_Cr) D̅^Ch̅_p^  q
          -  2/r^4 (D̅_Cr) (D̅^Cr) h̅_p^  q
          + 2/r^4h̅_p^  q
          
         -  1/2r^2D̂_pD̂^qh̅_C^  C
          + 1/2r^2D̂_pD̅_Ch̅^qC
          + 1/2r^2D̂^qD̅^Ch̅_pC
          
         + 1/2r^4D̂_pD̂_rh̅^qr
         + 1/2r^4D̂^qD̂^rh̅_pr
         -  1/2r^4D̂_pD̂^qh̅_r^  r
          
          + γ_p^  q(
          -  1/2D̅_CD̅_Dh̅^CD
          -  1/r (D̅_Cr) D̅_Dh̅^CD
          + 1/2D̅_CD̅^Ch̅_D^  D
          + 1/2r (D̅^Cr) D̅_Ch̅_D^  D.
          
                .
          + 1/2r^2D̂_sD̂^sh̅_C^  C
          -  1/r^2D̅_CD̂_rh̅^Cr.
          
                .
          -  1/2r^4D̂_rD̂_sh̅^sr
          -  3/2r^4h̅_r^  r
          + 1/2r^2D̅_CD̅^Ch̅_r^  r
          -  1/r^3 (D̅^Cr) D̅_Ch̅_r^  r.
          
                .
          + 2/r^4 (D̅_Cr) (D̅^Cr) h̅_r^  r
         + 1/2r^4D̂_sD̂^sh̅_r^  r)
          .













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 Paper Title: Separating deterministic and stochastic gravitational wave signals in realistic pulsar timing array datasets.

 

    
   




    Dipartimento di Fisica “G. Occhialini", Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy

    i.ferranti@campus.unimib.it
    INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
    INAF - Osservatorio Astronomico di Cagliari, via della Scienza 5, 09047 Selargius (CA), Italy
    INAF - Osservatorio Astronomico di Brera, via Brera 20, I-20121 Milano, Italy
    ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
    
Not anymore
Ferranti et al





    











  Recent observations by pulsar timing arrays (PTAs) suggest the presence of gravitational wave (GW) signals, potentially originating from supermassive black hole binaries (SMBHBs). These binaries can generate two kinds of signals: a stochastic gravitational wave background (GWB) or deterministic continuous gravitational waves (CGW). Being able to correctly recognize and separate them is crucial for accurate signal recovery and astrophysical interpretation. This paper aims at investigating the interaction between stochastic GWB and deterministic CGW signals within current analysis pipelines. We focus on understanding potential misinterpretations and biases in the parameter estimation when these signals are analysed separately or together. To this end, we performed several realistic simulations based on the European PTA 24.8yr dataset. We first injected either a GWB or a CGW into five datasets (three GWB realisations and two CGW realisations) with identical noise and analysed each signal type independently, and then analysed data sets containing both a stochastic GWB and a single resolvable CGW. We compared parameter estimation using different search models, including Earth term (ET) only or combined Earth and pulsar term (ET+PT) CGW templates, and correlated or uncorrelated power law GWB templates. We show that, when searched independently, the GWB and CGW signals can be misinterpreted as each other, and only a combined search is able to recover the true signal present. For datasets containing both a GWB and a CGW, failure to account for the latter biases the recovery of the GWB, but when performing a combined search, both GWB and CGW parameters can be recovered without any strong bias. Care must be taken with the method used to perform combined searches on these multi-component datasets, as the CGW PT can be misinterpreted as a common uncorrelated red noise. However, this can be avoided by direct searches for a correlated GWB plus a CGW (ET+PT). Our study underscores the importance of combined searches to ensure unbiased recovery of GWB parameters in the presence of strong CGWs. This is crucial to interpret the signal recently found in PTA data, and is a first step towards a robust framework for disentangling stochastic and deterministic GW components in future, more sensitive datasets.
    
    
 


  

  

  

  

  





   Separating deterministic and stochastic gravitational wave signals in realistic pulsar timing array datasets
    Irene Ferranti1,2, Golam Shaifullah1,2,3, Aurelien Chalumeau1,2,5 & Alberto Sesana1,2,4
    August 5, 2024
============================================================================================================







§ INTRODUCTION


Pulsar Timing Array (PTA) collaborations search for gravitational waves (GWs) with frequencies in the nanoHertz (nHz) band (10^-9-10^-7Hz). The primary target of PTA experiments is a stochastic gravitational wave background (GWB) generated by the incoherent superposition of the GWs emitted by all the super-massive black hole binaries (SMBHBs) that populate our universe <cit.>. The signal manifests itself in the data as a red noise process common to all the pulsars in the array, with a specific angular correlation, first derived by <cit.>. Recently, the PTA community found evidence for a signal with both these characteristics: the collaborations reporting this finding were the European PTA collaboration along with the Indian PTA collaboration <cit.>, the North American Nanohertz Observatory for Gravitational Waves collaboration <cit.>, the Parkes PTA collaboration <cit.> and the Chinese PTA collaboration <cit.>. 

While the Hellings and Downs (HD hereinafter) correlation is distinctive of a GW origin of the signal, separating it from possible correlated noise of different nature <cit.>, its source cannot yet be confidently established. In fact, besides SMBHBs, a nHz cosmic GWB can also be produced by a number of physical processes occurring in the early universe, such as cosmic strings networks <cit.>, primordial curvature perturbations <cit.>, or QCD phase transitions in the early universe <cit.>, all of which can provide a viable explanation of the observed signal <cit.>.
Moreover, the HD correlation is not unique of stochastic GWBs; as shown in <cit.> and in <cit.>, the signal generated by a single SMBHB and observed by an array of pulsars results in an angular correlation with the HD expectation value, albeit with a somewhat different variance. Note that the quality of the current data does not allow a description of the signal variance and  therefore, the possibility that a particularly loud source – a massive and nearby SMBHB – can be mistaken by a GWB is real. By loud source, we mean a source that can be singled out from the noise and, if present, from the GWB by means of a deterministic, template based search. Since the SMBHBs producing such signals are far from mergers and persistent in the data, their signal is generally referred to as continuous gravitational wave <cit.>.

CGWs searches have been performed on the latest data release of EPTA <cit.> and on the NANOGrav 15yr dataset <cit.>. They both found evidence in favour of the presence of a CGW with a 2.5-3σ significance, compared to a model where only pulsar noise is present in the data. This value is comparable to the evidence in favour of the presence of a GWB, which has a 3-to-4σ significance). Some simulations carried out in <cit.> already showed how a CGW can be misinterpreted as a GWB and, vice versa, a GWB can be misinterpreted as a CGW. In both the EPTA and NANOGrav analyses, when a GWB is also searched for, evidence for a CGW significantly decreases, indicating that its presence is at the very least not necessary to explain the observations. However, the degeneracy between the pulsar response to the two signals does not allow to conclusively rule out the presence of a CGW. 

The presence of these resolvable sources in the PTA data on the one hand poses a number of interesting challenges, on the other hand opens the door to invaluable opportunities. In fact, while the analysis pipelines currently used to search for a GWB rely on the assumption that the background is a Gaussian, isotropic, and stationary process characterised by a power-law Fourier spectrum <cit.>, the presence of a CGW can result in an excess of power at specific frequencies, which produces some level of anisotropy and non Gaussianity <cit.>. Thus, the standard analysis may depend on inapplicable assumptions and yield biases on the estimated GWB parameters <cit.>. This is a recognised issue within the PTA community and analysis techniques are being developed to go beyond the power law, isotropic Gaussian assumption <cit.> and to pull out of that data GWBs and CGWs simultaneously <cit.>.

Within this context, it is of paramount importance to study the interplay of CGWs and GWB when both are present in the data and what challenges arise in the interpretation of the analysis results. From an exquisitely scientific perspective, conversely, the prospects of detecting CGWs is particularly appealing. In fact due to their deterministic nature, PTAs are able to estimate the source parameters, in particular the sky location via triangulation <cit.> and possibly the distance to the source <cit.>. The detection of one or more resolvable sources <cit.> would be very precious as it would allow to track the orbital evolution of the binary and thus obtain direct information on the Massive Black Holes (MBHs) coupling to the environment <cit.>. This information cannot be well constrained by a measure of the GWB power spectral density, because it is strongly degenerate with the MBH mass and eccentricity distribution. In addition, CGWs offer the exciting opportunity to search for electromagnetic counterparts of the GW signal, opening the prospects of low frequency GW multimessenger astronomy <cit.>. Although PTAs can realistically localise the CGW source only within several deg^2 <cit.> they must be originated by particularly massive and/or nearby SMBHBs, making the search of their galaxy host viable <cit.>. Identifying the host and detecting accretion activity onto the binary will shed light about the physics governing the formation and (thermo)dynamical evolution of SMBHBs and the accretion discs surrounding them. 

In this paper, we carry out the first detailed investigation of the interplay between stochastic GWBs and deterministic CGWs in realistic PTA datasets, assessing the performance of currently used analysis pipelines when both signals are present in the data. We do so by building state of the art mock datasets capturing the complexity of real data, and by injecting them with as realistic as possible GW signals, mimicking the one found in the recent EPTA analyses <cit.>. Realistic GWBs are injected as the incoherent sum of individual waves consistent with astrophysical observations <cit.>, and CGW are modelled according to the candidate identified in <cit.>. Datasets are then analysed with the standard models implemented in the  <cit.> analysis suite, which models the GWB as a common red process characterized by a single power law spectrum, while we also test CGW models including only the Earth term or the full Earth plus pulsar components. 

The paper is organized as follows. In Section <ref> we describe in detail the simulations: the properties of the adopted datasets, the injected signals and the model used for the recovery. In Section <ref> we present the main results of our analysis. Using the methods described in the previous section, we carried out several investigations. We first assess the importance of including the pulsar term in the CGW model (Section <ref>). We then study the degeneracy of signal recovery on datasets containing only a GWB or a CGW (Section <ref>), showing that datasets injected only with a CGW can produce evidence of a GWB (Section <ref>) and vice versa (Section <ref>), and that a joint search is able to pin down the correct nature of the signal (Section <ref>). The performance of a combined analysis of a GWB and a CGW on datasets containing both signals is then evaluated in Section <ref> and we discuss our main findings in Section <ref>.
 Throughout the paper we assume a Λ cold dark matter cosmology with parameters (H_0, Ω_m, Ω_Λ)=(70 km s^-1Mpc^-1, 0.3, 0.7). 
















  


§ METHODS

In this section, we describe both the simulations structure and the models and methods used in the analysis.



 §.§ Simulations




Our simulations are based on the second data release of EPTA <cit.>. In particular, we have reproduced the DR2full dataset: 25 pulsars with a maximum observation time of 24.8 yr. The main properties of the real time of arrivals (ToAs) of each pulsar are reproduced: the observation time, the number of ToAs, the timing model (TM), the multiband measures – the frequency coverage is identical to that of the EPTA DR2full pulsars – and the noise levels. The uneven sampling of the ToAs is also reproduced, but large gaps, which are common in the real data streams, are not present in the simulations (see Fig. <ref> for a comparison between real and simulated residuals).
All the properties of the arrival times were injected in the dataset using the  package <cit.>. The same package is used to inject the three components of the noise budget: white noise (WN), achromatic – independent of the observation radio frequency –  red noise (RN) and chromatic red noise (due to dispersion measure variations, and customarily referred to as DM). 

To model the white noise, the ToAs are sampled from a Gaussian distribution centred at the value predicted by the TM and with a width given by:

    σ_i = √(EFAC_i^2σ_ ToA^2 + EQUAD_i^2),

where σ_ ToA is the errorbar associated to each ToA according to the template-fitting errors obtained in EPTA DR2, the EFAC takes into account for the ToA measurement errors and the EQUAD accounts for any putative extra source of white noise. The two parameters EFAC and EQUAD are specific for each observing backend. In the simulations, each pulsar's backend, as well as the ToAs' uncertainties, are identical those of DR2full; therefore, each pulsar has ToAs taken from ∼12 different backends, which means 12 different couples of values for EFAC and EQUAD. The values used for EFAC and EQUAD are the maximum likelihood values obtained by the single pulsar noise analysis of EPTA DR2full <cit.>.

In addition to the white noise, achromatic and chromatic red noise contribute to the single-pulsar noise budget. These are time-correlated noise components modelled as a stationary Gaussian process with a power-law power spectral density (PSD) of the form

    P(f; A, γ) = A^2/12π^2(f/ yr^-1)^-γ yr^3.

For achromatic red noise, in each ToA, the delay induced in the residuals is proportional to the red noise PSD, 

    Δ_ RN∝√(P(f; A_ RN, γ_ RN)).


This noise component accounts for the long-term variability of the pulsar spin along with other sources, such as unmodelled objects orbiting the pulsar. For chromatic red noise, on the other hand, the induced delay also depends on the observing radio frequency because it is due to the interaction of the radio signal with the ionised interstellar medium (IISM), the interplanetary medium in the solar system, and the Earth's ionosphere. This effect is taken into account during the observations and inside the timing model which considers its value at a reference epoch together
with its first and second derivatives. However, the inhomogeneous and turbulent nature of the IISM also induces stochastic variations in the DM value, which are modelled as chromatic red noise. The residual DM delay is modelled as 

    Δ_ DM∝√(P(f; A_ DM, γ_ DM))×ν^-2

where ν is the observing radio frequency. The values of A_ RN, γ_ RN, A_ DM and γ_ DM used in the simulations are  reported in Table <ref>, together with the number of Fourier components used in the modelling of these processes. These values – amplitude, slope and number of Fourier components – correspond to maximum likelihood values obtained from the single pulsar noise analysis of EPTA DR2full. An example of simulated data compared to the real EPTADR2 dataset is shown in Fig. <ref>.






  §.§.§ Gravitational wave signal injection


Having specified the properties of the adopted PTA, we now turn to the description of the injected GW signals. The signal produced by a SMBHB is composed by two terms, customarily referred to as Earth term (ET) and pulsar term (PT), which are respectively associated to the GW strain hitting the Earth at the ToA of the considered pulse and the strain hitting the pulsar at the time of emission of the same pulse.

Throughout the paper we will always assume circular SMBHBs, evolving under the effect of GW backreaction only (i.e. we neglect any possible coupling with the local stellar or gaseous environment). Under these assumptions, the delays induced in the ToAs at an epoch t are fully defined by 8 parameters: the chirp mass ℳ[Note that in the quadrupolar approximation employed here for the injected waveform, the signal does not depend on the individual binary masses.], the observed frequency f_gw, the redshift z, the inclination angle , the sky location (θ, ϕ) – respectively polar and azimuthal angle – , the initial phase ϕ_0 and the polarization angle ψ. Following <cit.>, we model them as:


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

where 

    Δ s_A(t) = s_A(t_p) - s_A(t_e),

the functions F^A(Ω̂) – with A indexing the polarization and Ω̂ being the unit vector pointing from the GW source to the Solar System Barycenter ˆ
(SSB) – are the antenna pattern functions that depend on the relative position of the pulsar and the source <cit.>. s_A(t_p) and s_A(t_e) are the PT and ET respectively. They both take the form


    s_+(t) = ℳ^5/3/d_Lω(t)^1/3(-sin(2(Φ(t)-Φ_0))(1+cos^2)cos(2ψ) 
       - 2cos(2(Φ(t)-Φ_0))cos sin(2ψ))
       s_×(t) = ℳ^5/3/d_Lω(t)^1/3(-sin(2(Φ(t)-Φ_0))(1+cos^2)sin(2ψ) 
       +2cos(2(Φ(t)-Φ_0))cos cos(2ψ)),

where 

    Φ(t) = Φ_0 + 1/32ℳ^5/3(ω_0^-5/3-ω(t)^-5/3),

and 

    ω(t) = (ω_0^-8/3-256/5ℳ^5/3t)^-3/8.

The only difference between PT and ET is that while the latter is the delay induced at the time of observation on Earth t_e, the former has to be evaluated at the time of emission of the radio pulse at the pulsar t_p = t_e - L(1 + Ω̂·p̂), where L is the pulsar distance and p̂ is the pulsar position in the sky. We inject in our data two different GW components: a stochastic GWB and a resolvable CGW. Since the evolution of the signal over the observation time is small, the phase approximation is used both in the injection and in the recovery: ω_ Earth = ω_0 at each epoch t.

Stochastic GWB. The GWB is injected as a realistic superposition of GWs coming from a cosmic population of SMBHBs. The injection pipeline starts from a catalog of SMBHBs, computes the delays induced by each binary in each pulsar ToA according to Eq. (<ref>) and then sums all the delays in the time domain. The distribution of the sources' ℳ, f_gw, z and  determines the observed power spectrum generated by the SMBHBs population: the binned total characteristic strain, where the frequency bin width is Δ f = 1/T with T the time span of the pulsar array, can be evaluated as <cit.>

    h_c^2(f_i) = ∑_j∈Δ f_ih_j^2(f_r)f_r/Δ f_i,

where f_i is the central frequency of the bin Δ f_i, and the sum runs over all the binaries for which the observed frequency f=f_r/(1+z)∈Δ f_i, where f_r is the GW frequency in the binary rest frame. The strain of each GW signal is given by 

    h(f_r) = 2√(1/2(a()^2 + b()^2))(Gℳ)^5/3(π f_r)^2/3/c^4d(z),
 
where a() = 1+cos^2, b() = -2cos and d(z) is the comoving distance of the binary, which is computed as 

    d(z) = c/H_0∫_0^z1/√(Ω_m (1+z)^3 + Ω_Λ)dz.

For the population of circular GW driven SMBHBs, assumed here, the characteristic strain given by Eq. (<ref>) can be approximated as a power law. It is therefore customary to describe the signal as

    h_c(f) = A_ GW(f/ yr^-1)^α,

where α=(3-γ)/2=-2/3 being γ the PSD induced in the timing residual (cf. Eq. <ref>).

The power spectrum h_c^2(f_i) thus depends on the number of emitting sources per unit redshift, mass and frequency d^3N/(dzdℳdf_r). A list of ∼ 120k binaries is constructed by extracting ℳ, f_gw, z as Monte Carlo samples of a numerical distribution d^3N/(dzdℳdf_r), obtained from the observation-based models described in <cit.> and <cit.>. In those papers, the authors construct hundreds of thousands of SMBHB populations in agreement with current constraints on the galaxy merger rate, the relation between SMBHs and their hosts, the efficiency of SMBH coalescence and the accretion processes following galaxy mergers. From this set of models, we selected 100 populations producing a GWB amplitude A ∼ 2.4×10^-15, consistent with the signal observed by EPTA <cit.>. The other parameters of the binaries are then randomly sampled such that sources are isotropically distributed in the sky – cos θ∈Uniform(-1, 1) and ϕ∈Uniform(0, 2π) –, their orbital angular momenta are uniformly distributed on the unit sphere – cos ι∈Uniform(-1, 1) –, the phase and the polarization angle are random – ϕ_0 ∈Uniform(0, 2π), ψ∈Uniform(0, π). 


Resolvable CGW. In some of the dataset, an individually resolvable CGW is injected. The waveform is again injected in each pulsar in the time domain residual according to Eq. (<ref>). The only difference, compared to the SMBHBs contributing to the unresolved GWB is that the amplitude of the signal is large enough that the source can be identified above the pulsar noise and the unresolved GWB by a signal search employing a deterministic filter to match the signal (see next Section). We stress that we always inject both the ET and the PT for every SMBHB, regardless of whether it provides an unresolvable contribution to the GWB or it is a resolvable CGW.

Datasets. For the purpose of this investigation, seven different datasets were produced, with the exact same noise realization, but different gravitational signals:

    
  * dataset injGWB01, injGWB02, injGWB03. These three datasets contain a GWB-only. They are produced injecting three background selected from the 100 populations mentioned above. The three populations are chosen among those displaying a smooth behaviour in the lowest frequency bins, without particularly loud sources.

    
  * dataset injCGW5, injCGW20. These two datasets only contain a single loud CGW-only. The injection procedure is similar to the GWB datasets, but the population file used contains only one binary. The two injected sources have Earth term GW frequencies of 5nHz (injCGW5) and 20nHz (injCGW20). The parameters of the 5nHz source are chosen to approximately reproduce frequency and amplitude of the candidate CGW found in the EPTA DR2new, <cit.>, which in fact has a frequency of ∼5nHz and an amplitude h∼1.1× 10^-14. The source is then located in the Virgo cluster; this choice determines sky location and distance, and consequently the chirp mass. Inclination, initial phase and polarization angles are randomly extracted as in the GWB case. The source injected in injCGW20 has identical parameters except for the higher frequency (20nHz instead of 5nHz), thus resulting in a higher strain (h∼ 2.7× 10^-14). The delays induced in pulsar PSR J1022+1001 by these two CGWs are shown in Fig. <ref>.
    
    
  * dataset injGWB03+CGW5, injGWB03+CGW20. These two datasets are injected with a GWB and a CGW. The GWB is in both cases the one resulting from population 03, while the CGWs are either at 5nHz or 20nHz.










 §.§ Signal modelling and analysis methods

All the analyses are carried using the  <cit.> package. The starting point of the analysis is the likelihood function of the timing residuals δt

    P(δ𝐭|ξ, ζ) = exp[-1/2(δ𝐭 - 𝐌ϵ - 𝐬)^T𝐂^-1(δ𝐭 - 𝐌ϵ - 𝐬)]/√((2π)^ldet𝐂).

The term Mϵ is the timing model error expressed as a linear function of the offset from the nominal timing model parameters, ϵ. ξ are the timing model parameters and ζ are the stochastic components parameters. The likelihood function sampled to search for the gravitational signal is marginalized over the timing model parameters as described in <cit.>. 𝐂=𝐂(ζ) is the covariance matrix that contains all the contributions from stochastic processes, which are:

    
  * WN, modelled according to Eq. (<ref>). All the white noise parameters are fixed at the injected values in the GW searches;
    
  * RN, modelled as a Gaussian process with amplitude of the Fourier components given by Eq. (<ref>)  (<cit.>;
    
  * DM, modelled as a Gaussian process with amplitude of the Fourier components given by Eq. (<ref>) and amplitude depending on the observed radio frequency (see Eq. <ref>). As for RN, the DM is not present in all the pulsars and, if present, the number of Fourier bins used in the Gaussian process depends on the pulsar and it is equal to the number of components used in the injection. For a summary of the noise model characteristics, see Table <ref>.
    
  * common red noise (CRN), if searched for. A red noise process which is common to all the pulsars can be accounted for in the covariance matrix. If included in the analysis, it is also modelled as a Gaussian process. The amplitude of the components are modelled in two ways: as a power law function of the frequency, and therefore the parameters are the slope γ_ CRN and the amplitude A_ CRN; or as a binned spectrum, and thus the parameters are the amplitudes of the common process in each frequency bin f_i. In both cases, the number of components used in the Gaussian process is 30.
    

Although we inject the GWB as  a superposition of unresolved SMBHBs, we follow the standard PTA analysis practice and model it in our likelihood function as a CRN characterized by an amplitude and a spectral index, as in Eq. (<ref>), which can be directly translated in a characteristic strain of the form given by Eq. (<ref>). Note that, depending on the dataset, we use two different flavours of CRN. In most searches, the GWB is modelled as a common uncorrelated red noise (CURN). This means that inter-pulsar correlations induced by the GWB are neglected and the GWB posterior distribution is then reconstructed a posterirori with the reweighting technique <cit.>, when possible. The advantage of this model is that it is much faster since it keeps the correlation matrix in the likelihood function diagonal and it naturally provides the Bayes factor of the reconstructed model versus the  sampled model, besides the posterior distribution of both models. However, the reweighting technique is not always efficient and, as we will see later, using a CURN to model a GWB is not always appropriate; therefore, when needed, we use the HD-correlated CRN model which we referred to as GWB model. In this case, GWB induced inter-pulsar spatial correlation are also modelled, according to the HD curve <cit.>

    Γ_ab=3/21-cosθ_ab/2ln( 1-cosθ _ab/2)-1/41-cosθ_ab/2 +1/2

where a and b are indexes identifying two pulsars in the array and θ_ ab is their angular separation in the sky. Note that searching for an HD-correlated CRN is much more computationally expensive, since the correlation matrix is not diagonal anymore.

The last term in the likelihood function is s, which is the delay induced by an individually resolvable CGW, and is analytically described by Eqs.(<ref>)–(<ref>) which are implemented in the CGW model of . An approximation that can be adopted is to consider in the CGW model the ET only, neglecting the PT. This is computationally very convenient, since the former only depends on the 8 source parameters. Conversely, the latter also depends on the distance of each pulsar in the array. Since those  distances are generally known with large uncertainties (of the order of 20%), they must be considered as additional free parameters in the search. Therefore, the complete model has at least one more parameter per each pulsar with respect to the ET only model. Moreover, as prescribed in <cit.>, in order to ease the search of the best parameters, one extra parameter is added per each pulsar, which is an initial phase at the pulsar, Φ_p. We will assess the performance of an ET only search on realistic signals including ET and PT in Section <ref>.

We mainly adopt Bayesian techniques in our analysis, although we also exploit the frequentist optimal statistics, when useful. The priors used in the search are listed in table <ref> and are all uninformative, except for the pulsar distances' priors that exploit information obtained from independent electromagnetic observations. The prior distributions have a width which is taken as the measurement error listed in <cit.>; for the pulsars that are not included in this paper, the relative error is fixed at 20%. The mean of the prior is chosen such that the pulsar distance used for the injection is within 1σ of the prior distribution for each pulsar. Parameter estimation is then performed by sampling the posterior distribution with the Markov
Chain Monte Carlo (MCMC) sampler , <cit.>. For the 25 pulsar PTA we employ, the dimensionality of the posterior distribution is 62 (2 parameters – slope and amplitude – for each red noise and dispersion measure process included in the model) plus the GW model dimension (2 for the GWB power-law model, 8 or 58 for an ET only or a full ET+PT CGW model search respectively). Model selection is done by computing the Bayes factor, either with the reweighting technique, when it is efficient, or with the product space method implemented in  <cit.>.


































  


§ RESULTS


Using the methods described in the previous section, we carried out several investigations. In Section <ref>, we assess the importance of modelling a CGW including both the Earth and pulsar terms. In Section <ref>, we investigate the degeneracy between the analysis results obtained with a GWB-only dataset and with a CGW-only dataset. In Section <ref>, it is shown how CGW-only datasets can produce strong evidence in favour of an HD-correlated GWB. Viceversa, in Section <ref>, it is shown how a GWB-only dataset can be misinterpreted as a continuous wave. This degeneracy is solved in Section <ref>, by means of a combined analysis that searches simultaneously for a GWB and a CGW on both GWB-only and CGW-only datasets . Finally, in Section 
<ref>, we evaluate the performance of a combined analysis of a GWB and a CGW on datasets containing both signals. 






 §.§ On the importance of the pulsar term in CGW modelling





















As described in section <ref>, the signal imprinted by a CGW in the ToAs is the sum of two terms: the Earth term and the pulsar term. The ET is common to all pulsars and only depends on the 8 parameters describing the source. Conversely, the PT depends on the distance of each pulsar and the phase of the GW signal at the pulsar's location. 
This means that modelling of the full CGW ET+PT in a search performed on an array of 25 pulsars (like the one we are considering here) requires 50 extra free parameters, that must be added to the 8 source parameters and to the 62 parameters of the individual pulsars noise components. It is therefore interesting to compare searches of CGW modelled as ET only and ET+PT on realistic datasets, to see whether the computational efficiency of the ET only search bears any significant drawback.


The two templates have been tested on both CGW-only datasets (injCGW5 and injCGW20), and the main results are shown in Fig. <ref>. From the figure it is clear that the performance of the ET only template depends on the CGW frequency. In fact, when applied to the injCGW20 dataset (right panel of Fig. <ref>), the ET only model performs well on all parameters that it is sensitive to, i.e. all source parameters except the chirp mass and the luminosity distance. Conversely, the same model applied to the injCGW5 dataset results in some non-negligible biases in the recovery. As it can be seen from the left panel of Fig. <ref>, the injected values for the frequency and sky location are not compatible with the 90% C.I. of the recovered posterior distributions. 

The different performance of the ET only template in the two cases is due to the different degree of separability of the ET and PT terms for the two injected sources. For the injected system chirp mass, ℳ=1.16· 10^9M_⊙, the average frequency difference between ET and PT is Δ f∼ 
3.8×10^-6nHzyr^-1Δ t∼ 1.0× 10^-2 nHz∼ 0.02Δ f_bin for the 5nHz CGW and Δ f∼ 1.4× 10^-3nHzyr^-1Δ t∼ 4.0 nHz∼ 3.2Δ f_bin for the 20nHz CGW. Therefore, while in the 20nHz CGW, ET and PT falls at very distinct frequencies, they blend together in the same frequency bin for the 5nHz CGW. In the former case, the ET only template only picks the common power produced by the ET at its frequency, disregarding the PT term; while in the latter, the ET only template is trying to fit a superposition of ET and PT essentially overlapping at the same frequency. This concept is demonstrated in Fig. <ref>, where the reconstructed waveforms are compared to the injected ones. In the 20nHz case (injCGW20 dataset), the ET only template almost perfectly fits the ET of the full injected waveform, while in the 5nHz case (injCGW5), the ET only template absorbs power from both ET and PT and the reconstructed waveforms sits, for many pulsars, somewhat in between the ET and the full injected signal and is also way more uncertain as shown by the much larger credible region.





Conversely, the full ET+PT template is able to correctly reconstruct the signal and, because it is sensitive to the frequency evolution, it also allows to constrain both the source chirp mass and the luminosity distance. The performance is very different for the 5 nHz CGW and the 20 nHz CGW: for the former the chirp mass recovery is ℳ=0.52^+9.98_-0.48× 10^8M_⊙, while for the latter it is ℳ=1.01^+0.07_-0.06×10^9M_⊙, where we give the median and the 68% credible interval. The superior recovery of the chirp mass in the 20 nHz case is due to the much larger frequency evolution of the source between ET and PT, allowing to place a much tighter constrain on ḟ and thus on M.
Another consequence of the large frequency evolution of the 20 nHz CGW is that the model is sensitive to the pulsar distances. This is shown in Fig.<ref> where the reconstructed waveform (left) and the posterior distribution of the pulsar distance (right) are shown for PSR J1713+0747 for the injCGW20 (top row) and injCGW5 (bottom row) datasets. In the first case, the posterior distribution on the distance is narrower than the prior and centred at the injected value: this is allowed by a correct modelling of the frequency evolution, which is reflected in the excellent reconstruction of the waveform. Conversely, for the 5 nHz case, the posterior distribution of the pulsar distance is almost indistinguishable from the prior, but signal reconstruction still matches the injection, as the frequency evolution is too small for the search to be sensitive to it. We note that the GW signal reconstruction and distance determination is not always as good for the 20 nHz CGW. This is shown in the middle row of Fig.<ref> for pulsar PSR J1918-0642. In this case the posterior distribution resembles the prior, and while the reconstructed signal does not strongly disagree with the injection, the median fails to capture the signal shape accurately. In fact, only about 13 pulsars, whose evolution is accurately reconstructed, significantly contribute to the chirp mass recovery.



In addition to better and unbiased recovery of the CGW parameters, there is another rationale for preferring the complete CGW template over the ET-only template: in a joint CGW + GWB search, the contribution to the CGW signal due to the pulsar term, if not taken into account by the CGW model, is absorbed by the GWB model, since it is a common red noise process. It will therefore add extra spurious power to the GWB, biasing the recovery of its parameters. An example of this behaviour is shown in Fig. <ref>, where the GWB+CGW search is applied to the dataset injCGW5: when the CGW is searched with the full template, the posterior distribution of the GWB parameters is unconstrained, when the search is done with the ET-only template, the posterior distribution of the GWB parameters remains well constrained because it absorbs all the power coming from the unmodelled pulsar term.

We therefore conclude that while the ET-only template can be used to search for possible CGW candidates and provides a reasonable estimate of the source parameters <cit.>, the full ET+PT model must be preferred, especially in a joint search with a stochastic GWB. Despite the much lower computational efficiency, we will therefore always model our CGWs as ET+PT in the following.











 §.§ Degeneracy between GWB and CGW

In this section, searches for GWB, CGW and combined searches of the two signals together are applied to the 3 GWB realizations (injGWB01, injGWB02, injGWB03) and to the 2 CGW realizations (injCGW5, injCGW20). The aim is to study the degeneracy between the two signals and how it can be solved.





  §.§.§ CGW misinterpreted as a GWB


We start by performing a GWB analysis on the three injGWB and the two injCGW datasets. To this end we make use of both Bayesian and frequentist techniques following this procedure:

    
  * a power law-shaped and HD-correlated stochastic process is assumed for the GWB, then the slope γ_ GWB and the amplitude log_10A_ GWB of the powerlaw process are sampled, together with the noise parameters to obtain the posterior distribution;
    
  * the Bayes factor ℬ^ GWB_ CURN is evaluated to compare the evidence for an HD-correlated and an uncorrelated red noise process. The bayes factor is computed either with the reweigthing method or with the product space when reweighting is not efficient;
    
  * the averaged angular correlation between the pulsars is estimated with the optimal statistics method averaging over 10 angular bins, each one containing 30 pulsar pairs. A powerlaw-shaped background with variable γ is assumed;
    
  * the SNR of the HD cross-correlation marginalized over the noise parameters is obtained with the optimal statistics method, assuming a powerlaw-shaped background with variable γ.


1.4 





The results of the powerlaw spectrum parameter estimation are listed in the last two columns of Table <ref> for all five datasets and two examples of the recovered GWB parameters' posterior distributions are shown in Fig. <ref> for datasets injGWB02 and injCGW5. In general, the GWB search returns well constrained posteriors regardless of the actual content of the dataset, which in practice means that a CGW can easily be detected as (and mistaken for) a GWB. However, the signal recovered from the two CGW-only datasets (injCGW5, injCGW20) is higher and flatter than the one obtained from the background datasets. This can be clearly seen both from the A and γ values of Table <ref> and from the 2D contours in Fig. <ref>. This is expected since the extra power due to the two single sources is at frequencies which are higher than the loudest GWB frequency bins that dominate the powerlaw spectrum recovery in the GWB-only datasets.






The Bayes factors reported in Table <ref> show that an HD-correlated common process (i.e. a GWB) is generally strongly favoured regardless of the content of the data. In fact, dataset injCGW5 produces a Bayes factor ℬ^ GWB_ CURN of ∼ 10 and HD S/N of 3.0^+0.5_-0.5 (fixing γ to 3), while injCGW20 
gives ℬ^ GWB_ CURN of ∼ 60 and S/N = 3.2^+0.6_-0.6 (fixing γ to 2). Interestingly, injGWB03 does not favour an HD correlated process, displaying ℬ^ GWB_ CURN of ∼ 0.15. This is consistent with <cit.>, who found that, due to the stochastic nature of the signal, individual realizations of the same SMBHB population can result in Bayes factor that differ by several orders of magnitude.

Although the Bayes factors obtained on the  injCGW5 and injCGW20 are compatible with a GWB detection, the resulting HD S/N is significant only in a small range of values of γ that are consistent with the posterior distributions obtained by the Bayesian search, as shown in Fig. <ref>. By fixing γ to the nominal value of 13/3, the HD S/N drops to 1.3^0.4_-0.4 and to 1.1^1.0_-0.9 for the injCGW5 and injCGW20 datasets respectively. Conversely, the 
injGWB01 and injGWB02 datasets (also shown in Fig. <ref>) result in distributions of the HD S/N that are consistently high for all values of γ≥ 4.


Moreover, at higher values of γ, the CGW-only dataset has an increasing evidence for a dipolar-correlated signal, whose S/N surpasses that of HD. This behaviour is not observed in the GWB-only datasets, for which the dipole S/N always stays lower than the HD S/N. Finally, two examples of HD reconstruction are shown in Fig. <ref> for datasets injGWB02 and injCGW5. While the quadrupolar pattern can be seen in both datasets, the scatter around the expected relation appears to be much larger when a CGW only is present in the data, consistent with <cit.> and <cit.>. Moreover, the HD correlation for the injCGW5 dataset is reconstructed assuming γ=3, while it tends to disappear for higher values of γ.














  §.§.§ GWB misinterpreted as a CGW


We now search for a CGW in the same 5 datasets analysed in the previous section. In the search, we estimate the posterior distributions of all the noise parameters – except for the WN, which is fixed – and the CGW parameters – 8 source parameters plus 50 additional parameters for the pulsars' distances and phases. We then use hypermodeling to evaluate the Bayes factor between a model containing a CGW versus a model featuring intrinsic pulsar noise only. 

We can see from Table <ref> that also in this case, regardless on whether the dataset contains a CGW or a GWB, a CGW is recovered with high confidence, with a Bayes factor ℬ^ CGW_ PSRN>1000 in all cases. Table <ref> also shows that the CGW frequency and amplitude are fairly well constrained. The performance of the CGW search on the injCGW datasets has already been assessed in Section <ref>. As shown in Fig. <ref> when a full ET+PT template is used for the CGW, the model can correctly recover the injected signals and estimate their parameters, although with differences depending on the frequency of the source. Here it is particulary interesting to check the CGW recovery on the three injGWB datasets, which is shown in Fig. <ref>: all source parameters are fairly well constrained, including the sky location, which is different in each dataset. These parameters, however, do not correspond to any specific source present in the datasets, as can be seen in the three panels of Fig. <ref>. The recovered CGW frequency always corresponds to the frequency bin that is the most constrained by the HD-correlated free spectrum search; moreover, the amplitude recovered is always compatible with the GWB amplitude in that frequency bin. Thus, it can be concluded that the CGW parameters are constrained because the CGW model absorbs part of the HD-correlated common red noise produced by the incoherent superposition of all the SMBHBs composing the GWB. 

Finally, we note that the CGW amplitudes recovered from the three injGWB datasets are respectively 1.2^+1.5_-0.7× 10^-14, 1.4^+0.6_-0.5× 10^-14  and 1.8^+0.5_-0.4× 10^-14 (median and 68% credible interval). These values are all well compatible with the amplitude of the DR2new CGW candidate, which is 1.0^+0.5_-2.6× 10^-14 (median and 90% credible interval) <cit.>.














  §.§.§ Breaking the degeneracy: combined search for a GWB and a CGW




We now perform a joint CGW + GWB search on the five datasets considered thus far. We perform two flavours of combined searches: (i) common uncorrelated red noise plus CGW (CURN+CGW), and (ii) HD-correlated common red noise plus CGW (GWB+CGW). The CGW is always modelled as the sum of ET and PT, and together with the signal parameters, all the RN and DM parameters are also searched over.

The results of these two searches are reported in Table <ref> for all datasets, while posterior distributions of selected parameters of the model components are shown in Fig. <ref> for datasets injGWB02 and injCGW5. In figure, orange and light purple distributions are obtained respectively from the GWB+CGW and CURN+CGW searches, and results of GWB-only and CGW-only searches are also shown for comparison. To see how searching for both type of signals simultaneously allows to properly characterize the actual content of the data, let us discuss the injGWB and injCGW separately. 

In the injGWB cases, when a common red noise component is added to the model alongside the CGW, the evidence for the latter strongly decreases. This can be seen both from the drop of the Bayes factors in the first three lines of Table <ref>, and from the posterior distributions of the CGW parameters in the left panel of  Fig. <ref>. In Section <ref>, when searching for a CGW on a dataset containing only a GWB, we found  ℬ^ CGW_ PSRN>1000. This is because the GWB produces a common noise that the model prefers to assign to the CGW signal (common to all pulsars) rather than to the individual pulsars' RN components. 
When a CURN is included in the model, ℬ^ CURN+CGW_ CURN drops significantly but is still slightly larger than unity, suggesting that the presence of CGW component is still preferred. This can also be seen in the posterior distribution of the CGW amplitude shown in purple in the left panel of  Fig. <ref> that, while featuring a long tail that extends to the lower end of the prior range, still shows a significant peak around logA≈-14.3. This behaviour is shared by all three injGWB datasets and is due to the fact that the GWB manifests itself primarily as a common red noise process, with a weaker, emerging 'off diagonal' HD-correlated component. The CURN models it as a common red noise process, but is unable to account for the emerging HD correlation, which the model then assigns to the CGW.
Conversely, when a GWB is added to the model, we get ℬ^ GWB+CGW_ GWB<1 (first three lines of Table <ref>), strongly suggesting that the GWB component is now able to describe the full content of the data. This is also clear by looking at the CGW posterior parameters in   the left panel of  Fig. <ref>. The orange distributions show that both the frequency and amplitude posteriors are almost perfectly flat, with the latter displaying a cut-off, which we find to be at a similar value for all three realizations. From this we can extract  a 95% upper limit of the CGW amplitude which is ≈4.0×10^-15, ≈3.8×10^-15 and ≈4.6×10^-15 for datasets injGWB01, injGWB02 and injGWB03 respectively. Note that these upper limits are lower than the GWB level, since the total GW content of the dataset is correctly accounted for by the GWB component. Finally, the recovered source is not localized in the sky, with the posterior distributions of the sky location almost uniform across the sphere, consistent with the prior. 







Differently from the CGW parameters, the GWB parameters' posterior distributions are not significantly affected by the addition of the CGW component to the model: the GWB posterior distributions obtained from the GWB+CGW search and with the GWB-only search are well consistent with each other (see left panel of Fig. <ref> for the injGWB02 dataset and table <ref> for all the datasets) confirming again that the GWB component of the model is correctly accounting for all the HD-correlated power. Only a small variation can be noticed in the CURN+CGW recovery of the background parameters: the posterior distributions are slightly wider with a longer low amplitude tail. This is because the CURN cannot absorb any HD correlated power, which is (spuriously) assigned to the CGW component.  


 
Conversely, when the same analysis is applied to the injCGW datasets, the posterior distributions of the CGW parameters are always well constrained, both with the CURN+CGW and with the GWB+CGW search, as shown in the right panel of Fig. <ref>. Searching for CGW-only, CURN+CGW or GWB+CGW, in this case, gives consistent results for the CGW parameter, with no significant changes in the posterior distributions' shape and width (see right panel of Fig. <ref>). Moreover, GWB+CGW is able to recover the sky location with a precision which is comparable to the one obtained with the CGW-only search and the model comparison strongly favours the GWB+CGW model over the GWB-only model (see table <ref>). As expected, the GWB posterior distribution is instead completely unconstrained when the model applied in the analysis contains both a common red noise and a CGW: the power that generates the GWB signal recovered with the GWB-only search is now correctly completely absorbed by the CGW component (see Fig. <ref>).





  §.§.§ Noise-only datasets

As a further check, we tested the analysis pipelines on a dataset containing only the noise and no GW signal, neither deterministic nor stochastic. The noise realization is the one contained in all the datasets analysed. In this case, neither a GWB nor a CGW is detected. As for the GWB, we find a value of the Bayes factor ℬ^ HD_ CURN∼1.5 and the posterior distributions of A and γ are poorly constrained, with a 95% upper limit on the GWB amplitude of ∼ 4.47× 10^-15. As for the CGW, the Bayes factor ℬ^ CGW_ PSRN is ∼ 0.5 and the CGW parameters are totally unconstrained, with the amplitude showing a 95% upper limit of ∼4.5×10^-15. Although many more noise realizations should be tested to draw robust statistical conclusions, these tests show on fully realistic datasets that current PTA detection pipelines are unlikely to return a spurious detection in absence of an actual GW signal in the data.





 §.§ Joint GWB+CGW analysis on combined CGW+CGW datasets


In this section we present the results of simultaneous searches for a GWB and a CGW, applied to datasets injGWB03+CGW5 and injGWB03+CGW20. We remind the reader that both datasets contain the same noise and GWB realisation from dataset injGWB03. 

In each dataset a loud CGW is superimposed to the GWB: in one case the CGW is at 5nHz (injGWB03CGW5), in the second one it is at 20nHz (injGWB03CGW20). This is done in order to see how the joint analysis performs in presence of a slowly evolving source (the 5nHz CGW) and on a fast evolving one (the 20nHz CGW), and how the two of them interact with the GWB.









  §.§.§ Bias induced in the GWB recovery by the presence of a resolvable source in the SMBHBs population

As already discussed by <cit.> and  <cit.>, the presence of a particularly loud source among the systems contributing to the GWB, if not accounted for in the analysis, can significantly bias the recovery of the overall GWB signal. As a preliminary step in our investigation, we therefore perform a GWB only search on the injGWB03CGW5 and injGWB03CGW20 datasets. As expected, the analysis produces a biased recovery of the GWB parameters: the results are shown in the right panels of Fig. <ref> and <ref>: in both cases, the recovered background is flatter and higher than the injected signal due to the extra power introduced by the loud source. In the two following sections, we demonstrate how this bias can be solved by performing a joint analysis of the GWB and the CGW components.





  §.§.§ Results for the injGWB03+CGW5 dataset


















The results presented here are obtained by sampling the posterior distributions of the CURN+CGW model and then reweighting them to obtain the posteriors of the GWB+CGW model.
The results of the combined search applied to the dataset injGWB03+CGW5 provides good overall results, with the majority of the CGW parameters recovered within 2σ of the injected values. 

The posterior distributions of the CGW parameters are shown in the left panel of Fig. <ref>, while the recovery of the background parameters is shown in the right panel. The recovery of the CGW parameters is successful: the frequency and the sky localization of the source are well constrained – the width of the posterior distributions is ≲10% of the prior width for all three parameters – and compatible within 68% with the injected values: injected frequency is f = 5.02nHz, recovered frequency is f = 5.04^+0.04_-0.04nHz; injected sky location is (12^ h19^ min50^ s, 12.38^∘), recovered sky location is (12^ h 33^+20  min_-19, 10^+5_-5^∘). However, the 3D source location cannot be properly constrained, since the luminosity distance posterior distribution is  extremely wide due to its strong covariance with the chirp mass. Both posterior distributions are roughly bimodal, with a narrow peak close to the injected value – in particular for the chirp mass – and a long, bumpy tail extending to low values of luminosity distance and chirp mass. Note that the posterior of the luminosity distance spans the interval (0.02, 24)Mpc within the 90% credible region; for such a nearby putative source, ancillary information coming from galaxy catalogues can effectively enable the identification of the source host <cit.>. The inclination angle, initial phase and polarization angle are well constrained and compatible with the injected values.

Focusing now on the GWB parameters, the posterior distribution in the right panel of Fig. <ref> shows how accounting for the resolvable source in the model solves the bias in the GWB parameters' recovery. In fact, the posterior distribution obtained with the combined search is well consistent with the one obtained with the GWB-only search applied to the GWB-only dataset injGWB03: they peak at similar values and have comparable widths, showing that the parameters of the stochastic and of the deterministic components are not significantly correlated and the interplay between the two signals does not affect the search.





  §.§.§ Results for the injGWB03+CGW20 dataset













 





The joint search applied to the injGWB03CGW20 dataset unveils some new interesting behaviour of the recovery pipeline. In fact, in this case, when a joint CURN+CGW (ET+PT) search is performed and the GWB is obtained with the reweighting technique, the estimate of the GWB parameters is completely off, as demonstrated by the filled purple distributions shown in the right panel of 
Fig. <ref>. The result is indeed quite similar to a GWB search only (shown in red), meaning that a significant fraction of the CGW power is absorbed in the common background component and the CURN+CGW search followed by reweighting fails in this case, producing strong biases in the results.



The reason of this failure is the following: as shown in Section <ref>, due to the significant difference in the 20nHz CGW between the Earth and the pulsar term GW frequency, in order to correctly account for both terms with the CGW model, the majority of the 50 PT parameters (i.e. 25 pulsar distances and 25 individual PT GW phases) have to be constrained. This is a difficult task for the sampler, which in the CURN+CGW case converges to an easier, alternative solution: the pulsar terms are all absorbed by the common uncorrelated red noise component, which is possible because they are indeed a common sinusoidal process that is uncorrelated, since the pulsars are all at different sky locations and distances from Earth. This justifies the much higher and flatter recovered background shown in the right panel of Fig. <ref>: the recovered amplitude is 1.83^+0.28_-0.26×10^-14, while the injected value is 2.4×10^-15. 

This interpretation is confirmed by the recovered CGW parameters (not shown). Most of them are biased and in particular the chirp mass, which determines the frequency evolution of the signal, is much lower than the injected value: the obtained value is 6. 9^+18.0_-5.0×10^6 M_⊙, which corresponds to a 95% upper limit on the frequency derivative of 4.2×10^-6nHz·yr^-1, ∼300 times smaller than the ḟ of the injected SMBHB. This is consistent with the fact that the values of the CGW parameters are such that the recovered time domain signal matches the injected Earth term and completely neglects the pulsar term contribution. This is confirmed by the shape of the reconstructed CGW shown in  Fig.<ref>. In the top panel we can see that in the CURN+CGW search the recovered CGW waveform for pulsar PSR J1713+0747 reproduces exactly the injected ET contribution. In fact, being HD-correlated, in presence of a CGW component in the model, the Earth term is not absorbed by the CURN component. Indeed, the recovered background is slightly lower than the one obtained by a GWB-only search on the same dataset (shown by the red distributions in the right panel of Fig. <ref>), because in that case both the Earth and pulsar terms are absorbed by the background component. 

We stress that this bias indeed corresponds to the maximum of the posterior distribution. This can be explained by the fact that the CURN model does not exactly match the injected common red noise, which is HD correlated, and therefore it prefers to absorb the pulsar term, since it is much cheaper to describe it as a two-parameter CURN signal, rather than to determine the 2×25 parameters describing each individual sinusoidal pulsar term.

In this dataset, the injected signal components are  properly recovered only by a direct GWB+CGW (ET+PT) search, which is more computationally expensive due to the off-diagonal HD terms that must be computed when evaluating the correlation matrix in Eq. (<ref>). In fact, in a GWB+CGW search the pulsar term is not absorbed by the HD-correlated common red noise component and both signals are properly reconstructed. In the right panel of Fig.<ref> the GWB recovery obtained by this search is shown in filled purple: it agrees very well with that obtained with the GWB-only template applied to the GWB-only dataset (empty purple), although it is slightly larger, which can be explained by the fact that the GWB+CGW model contains 58 more parameters describing the single source. A tiny correlation with this high number of parameters can easily explain the less precise recovery of the background parameters.

The correct recovery of the GWB parameters is a direct consequence of the correct recovery of the CGW parameters. In the left panel of Fig. <ref>, the posterior distributions of the 8 source parameters are shown and in Fig. <ref> the median and 90% credible region of the recovered CGW signal is shown for pulsar PSR J1713+4707. From the recovery, it can be seen how the model is able to account for the frequency evolution of the signal: the frequency recovered is 19.99^+0.21_0.18nHz (injected is 20nHz) and the chirp mass 1.14^+0.11_0.10×10^9M_⊙ (injected is 1.16× 10^9M_⊙). This allows the luminosity distance to be much better constrained than in the low frequency case, with a median and 68% credible interval of 32.7^+15.1_-9.3Mpc. Since the injected value is 21.6Mpc, the value obtained is a mild overestimate, which is probably due to the anti correlation with the inclination angle and the initial phase, which are both slightly underestimated. The 2D sky location recovery is also improved with respect to the 5nHz case, resulting in a declination angle of 12^ h38^+17 min_-16 and a right ascension of 11.5^+3.0_-3.4^∘.

  


§ DISCUSSION AND CONCLUSIONS


In this paper, we have carried out the first comprehensive study of the interplay between stochastic GWBs and deterministic CGWs in realistic PTA datasets, assessing the performance of currently used analysis pipelines. To this end we constructed mock datasets featuring all the complexity of EPTA DR2full <cit.>; we simulated up to 24.8 years of unevenly sampled, multi frequency ToAs for the 25 pulsars of DR2full and injected WN, RN and DM consistent with the maximum likelihood values found in the DR2full single pulsar analysis <cit.>. The injected GW signals were also as realistic as possible, tailored to mimic the findings of the recent EPTA analyses <cit.>. A GWB with nominal amplitude of A_ GW=2.4× 10^-15 was injected as the incoherent sum of individual waves coming from observationally based cosmic populations of SMBHBs <cit.>, while the deterministic CGW was modelled according to the candidate signal identified in the analysis of EPTA DR2new <cit.>. In all cases, SMBHBs were assumed to be circular and both the ET and PT of the GW induced time delays were injected. These GW signals were searched with the standard models implemented in the  analysis suite. The GWB was therefore modelled as a common red process characterized by a single power law spectrum, while the model CGW matched the one used for injecting the signal (i.e., circular, GW driven SMBHB), either including only ET or ET+PT. With this set-up we carried a number of investigations, whose results are summarized in the following.

We first compared the performance of ET-only and ET+PT templates on two datasets containing only a CGW either at 5nHz (injCGW5) or at 20nHz (injCGW20). In both cases, the ET+PT template was able to fully recover the signal, providing an unbiased estimate of all its parameters, including chirp mass, distance and sky location. Things were not so straightforward when the ET only template was employed. In this case, the 5nHz CGW dataset resulted in a biased recovery of the frequency and the sky location, whereas in the 20nHz case, the recovery was unbiased. In fact, in the latter case Earth and pulsar term can be easily disentangled and the ET-only template correctly matches the ET component of the signal. Conversely, in the 5 nHz case, ET and PT fall in the same frequency bin and the ET template tries to fit a mixture of the two, leading to a bias in the signal reconstruction. In both cases, the ET-only model is not sensitive to the chirp mass and therefore to the luminosity distance, preventing 3D source localization.

We also stress that in both cases, modelling ET only leaves behind some unmodelled residual (e.g. the full PT in the 20 nHz case), which will likely be absorbed by the GWB template in a joint search, leading to a biased recovery of the background parameters. These are the reasons why the complete CGW template (ET+PT) must be preferred to the ET-only, even if this means having 2 times the number of pulsars extra parameters (50 in our case) and therefore much larger computational costs for the sampling.

We therefore adopted the full ET+PT template for the CGW and turned to the investigation of datasets in which only one of the two types of signal was injected (either a GWB or a CGW) and the analyses were performed separately with one model at the time (eiter a GWB or a CGW). We analysed 5 datasets: 3 realizations of a realistic GWB (dataset injGWB01,injGWB02, injGWB03) and 2 realizations of single resolvable sources (datasets injCGW5, injCGW20). All datasets contained the same noise realization. These searches showed that a GWB can easily be misinterpreted as a CGW and vice versa. In fact, all sources returned evidence in favour of either GW signal and well constrained recovery of the model parameters. However, when a combined search is performed – meaning that the model used for the recovery contains both an HD-correlated GWB and a CGW – this degeneracy is broken: when only a GWB is present in the data, the posterior distributions of the CGW parameters obtained with the combined search are unconstrained and the CGW amplitude shows an upper limit; on the contrary, when a CGW is the only signal injected, the combined search gives unconstrained posterior distributions for the GWB parameters. No CGW evidence is absorbed by the HD-correlated background if the single source is present in the data. Therefore, the combined search is able to correctly recover the signal that is actually present in the data. Focusing on GWB-only datasets, we highlight that the CGW evidence is totally absorbed when an HD-correlated GWB is present in the model, while it does not completely disappear if only a CURN component is modelled. In this case, even if poorly constrained, the CGW parameters posterior distributions still peak at specific values. Therefore, modelling the common red noise as HD-correlated is necessary to totally absorb the spurious CGW. These results are similar to the ones obtained with the real EPTA DR2new data <cit.>: CURN + CGW searches peaking at a CGW amplitude of ∼10^-14, and peak disappearing once the background is modelled as HD-correlated and the Bayes factor ℬ^ GWB+CGW_ GWB being of order unity. Our simulations show that this behaviour arises when a CGW is indeed not present in the data, strengthening 
  the conclusion reached in 
the EPTA DR2 analysis.

We then analysed datasets containing both a stochastic GWB and a CGW, either at 5 nHz (injGWB03+CGW5) or at 20 nHz (injGWB03+CGW20). The low frequency CGW is an almost monochromatic source, while the high frequency one is drifting and the signal frequency significantly changes between the  PT and the ET. Performing a combined search on these types of datasets is particularly important because a single resolvable source, if not properly taken into account in the search, will induce a bias in the GWB parameters recovery <cit.>.

The results of the combined searches indicate that the bias in the GWB recovery induced by the presence of the single source is resolved when the CGW is correctly accounted for by the deterministic CGW model, so long as it contains both ET and PT. This result is obtained with both datasets, injGWB03CGW5 and injGWB03CGW20. However, in the injGWB03CGW5 case, the same unbiased result can be obtained with two methods: by performing a CURN+CGW search and then reweighing to obtain the HD+CGW posterior distribution, and by directly sampling the HD+CGW posterior distribution. The first method is computationally advantageous, as the CURN model is considerably faster to compute than the HD one. In the injGWB03CGW20 case, however, the first method (reweighting) is unable to recover the unbiased posterior distribution, as the CGW template solely accounts for the ET contribution, while the PT is more easily absorbed by the CURN. The posterior distribution on the CURN parameters is therefore strongly biased with respect to the GWB injection. Performing a direct sampling of the HD+CGW model, on the contrary, gives an unbiased recovery of the GWB parameters: the posterior distributions of the CURN and HD parameters are consequently disjoint, which makes the reweighing technique extremely inefficient. Therefore, while computationally much more expensive, the only method that is able to converge to unbiased results is the direct HD+CGW search. As for the GWB, the CGW recovery is successful, with most of the parameters constrained and compatible with the injection.

Given the recent PTA results indicating the presence of a gravitational signal in the data, it is crucial to be able to distinguish between the two types of signal that can be produced by SMBHBs. This work represents a first thorough study of this problem. Despite the limited parameter exploration, which is constrained by the high computational cost of the analyses, the investigation decisively points towards the importance of performing combined searches. Indeed, in the cases analysed, combined searches were able to both identify which signal was present in the data, without misinterpretation, and to guarantee an unbiased recovery of the GWB parameters in the presence of a loud individual source in the SMBHB population. Looking ahead, it will be important to extend this analysis to richer and more sensitive datasets, investigating the possibility of disentangling multiple searches, also in the presence of eccentricity or environmental coupling. In fact, the high amplitude of the currently observed signal implies that the future SKA might be able to resolve several tens of individual SMBHBs at S/N>5 <cit.>. Being able to successfully identify and disentangle those signals will have invaluable payouts for the understanding of SMBHB evolution and dynamics in the context of multimessenger astrophysics. Given the complexity and high dimensionality of these sources, it is of paramount importance to design more efficient search techniques <cit.>.


       I.F. thanks Stanislav Babak for his hospitality and mentoring at the APC, where this investigation was initiated.
       We thank the B-Massive group at Milano-Bicocca University for useful discussions and comments. A.C., A.S., G.M.S. acknowledge the financial support provided under the European Union’s H2020 ERC Consolidator Grant “Binary Massive Black Hole Astrophysics” (B Massive, Grant Agreement: 818691). A.C. acknowledges financial support provided under the European Union's Horizon Europe ERC Starting Grant "A Gamma-ray Infrastructure to Advance Gravitational Wave Astrophysics" (GIGA).

  

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