New articles on General Relativity and Quantum Cosmology

[1] 2306.03923

Glitch systematics on the observation of massive black-hole binaries with LISA

Detecting and coherently characterizing thousands of gravitational-wave signals is a core data-analysis challenge for the Laser Interferometer Space Antenna (LISA). Transient artifacts, or "glitches", with disparate morphologies are expected to be present in the data, potentially affecting the scientific return of the mission. We present the first joint reconstruction of short-lived astrophysical signals and noise artifacts. Our analysis is inspired by glitches observed by the LISA Pathfinder mission, including both acceleration and fast displacement transients. We perform full Bayesian inference using LISA time-delay interferometric data and gravitational waveforms describing mergers of massive black holes. We focus on a representative binary with a detector-frame total mass of $6 \times 10^7 M_\odot$ at redshift $7$, yielding a signal lasting $\sim 30~\mathrm{h}$ in the LISA sensitivity band. We explore two glitch models of different flexibility, namely a fixed parametric family and a shapelet decomposition. In the most challenging scenario, we report a complete loss of the gravitational-wave signal if the glitch is ignored; more modest glitches induce biases on the black-hole parameters. On the other hand, a joint inference approach fully sanitizes the reconstruction of both the astrophysical and the glitch signal. We also inject a variety of glitch morphologies in isolation, without a superimposed gravitational signal, and show we can identify the correct transient model. Our analysis is an important stepping stone toward a realistic treatment of LISA data in the context of the highly sought-after "global fit".

[2] 2306.03999

Exploring the Observational Signatures of $f(R,G,\mathcal{T})$ Gravity in FRW Universe

This research article presents a new cosmological model formulated within the $f(R,G,\mathcal{T})$ framework, focusing on the observational signatures and parameter constraints of the model. The Markov Chain Monte Carlo (MCMC) technique is employed to effectively explore the parameter space using data from 36 Cosmic Chronometers and 1701 Pantheon Plus data points. A comparative analysis is conducted between the proposed $f(R,G,\mathcal{T})$ model and the widely accepted $\Lambda$CDM model, considering various cosmological parameters, such as Deceleration, Snap, and Jerk. By evaluating these parameters, valuable insights into the dynamics and evolution of the universe within the context of the new model are obtained. Diagnostic tests including Statefinder and Om Diagnostic are performed to further investigate the behavior and consistency of the $f(R,G,\mathcal{T})$ model. These tests provide deeper insights into the properties of the model and its compatibility with observational data. The model is subjected to statistical analysis using Information Criteria to rigorously assess its goodness of fit to the data. This analysis helps determine the level of agreement between the $f(R,G,\mathcal{T})$ model and the observational data, establishing the viability and reliability of the proposed cosmological framework. The results highlight the potential of the $f(R,G,\mathcal{T})$ framework in understanding the fundamental aspects of the universe's evolution and dynamics. The comparative analysis with the $\Lambda$CDM model, along with the comprehensive diagnostic tests performed, demonstrates the efficacy and validity of the $f(R,G,\mathcal{T})$ model in explaining observed cosmological phenomena. These findings contribute to the ongoing pursuit of accurate and comprehensive models that provide a deeper understanding of the nature of our universe.

[3] 2306.04068

Effect of Ignoring Eccentricity in Testing General Relativity with Gravitational Waves

Detections of gravitational waves emitted from binary black hole coalescences allow us to probe the strong-field dynamics of general relativity (GR). One can compare the observed gravitational-wave signals with theoretical waveform models to constrain possible deviations from GR. Any physics that is not included in these waveform models might show up as apparent GR deviations. The waveform models used in current tests of GR describe binaries on quasicircular orbits, since most of the binaries detected by ground-based gravitational-wave detectors are expected to have negligible eccentricities. Thus, a signal from an eccentric binary in GR is likely to show up as a deviation from GR in the current implementation of these tests. We study the response of four standard tests of GR to eccentric binary black hole signals with the forecast O4 sensitivity of the LIGO-Virgo network. Specifically, we consider two parameterized tests (TIGER and FTI), the modified dispersion relation test, and the inspiral-merger-ringdown consistency test. To model eccentric signals, we use non-spinning numerical relativity simulations from the SXS catalog with three mass ratios $(1,2,3)$, which we scale to a redshifted total mass of $80M_\odot$ and luminosity distance of $400$ Mpc. For each of these mass ratios, we consider signals with eccentricities of $\sim0.05$ and $\sim 0.1$ at $17$ Hz. We find that signals with larger eccentricity lead to very significant false GR deviations in most tests while signals having smaller eccentricity lead to significant deviations in some tests. For the larger eccentricity cases, one would even get a deviation from GR with TIGER at $\sim 90\%$ credibility at a distance of $\gtrsim 1.5$ Gpc. Thus, it will be necessary to exclude the possibility of an eccentric binary in order to make any claim about detecting a deviation from GR.

[4] 2306.04069

Superluminal propagation along the brane in space with extra dimensions

We demonstrate that a model with extra dimensions formulated in Phys. Rev. D, 62, 045015 , which fatefully reproduces Friedmann-Robertson-Walker (FRW) equations on the brane, allows for an apparent superluminal propagation of massless signals. Namely, a massive brane curves the spacetime and affects the trajectory of a signal in a way that allows a signal sent from the brane through the bulk to arrive (upon returning) to a distant point on the brane faster than the light can propagate along the brane. In particular, the signal sent along the brane suffers a greater gravitational time delay than the bulk signal due to the presence of matter on the brane. While the bulk signal never moves with the speed greater than the speed of light in its own locality, this effect still enables one to send signals faster than light from the brane observer's perspective. For example, this effect might be used to resolve the cosmological horizon problem. In addition, one of the striking observational signatures would be arrival of the same gravitational wave signal at two different times, where the first signals arrives before its electromagnetic counterpart. We used GW170104 gravitational wave event to impose a strong limit on the model with extra dimensions in question.

[5] 2306.04145

Generalized Gibbons-Werner method for stationary spacetimes

The Gibbons-Werner (GW) method is a powerful approach in studying the gravitational deflection of particles moving in curved spacetimes. The application of the Gauss-Bonnet theorem (GBT) to integral regions constructed in a two-dimensional manifold, enables the deflection angle to be expressed and calculated from the perspective of geometry. However, different techniques are required for different scenarios in the practical implementation. This fact leads to different GW methods. For the GW method for stationary axially symmetric (SAS) spacetimes, we identify two problems: (a) the integral region constructed by people is infinite, which is ill-defined for some asymptotically nonflat spacetimes whose metric possesses singular behavior, and (b) the calculation is too complicated, especially for highly accurate results and complex spacetimes. To address these issues, a generalized GW method is proposed in which the infinite region is replaced by a flexible region to avoid the singularity, and a new calculation formula which greatly simplifies the computation is derived with such a method. Our method provides a comprehensive framework for describing the GW method for various scenarios. Additionally, the generalized GW method and simplified calculation formula are applied to three different kinds of spacetimes (Kerr spacetime, Kerr-like black hole in bumblebee gravity, and rotating solution in conformal Weyl gravity), effectively verifying the validity and superiority of our method.

[6] 2306.04172

From entropy to gravitational entropy

The concept of entropy forms the backbone of the principles of thermodynamics. R.C. Tolman initiated a correlation between gravity and thermodynamics. The development of black hole thermodynamics and the generalized second law of thermodynamics led to Penrose's conjecture that the Weyl tensor should serve as a measure of the entropy of the free gravitational field. This entropy reflects the degrees of freedom associated with the free gravitational field. The proposition of gravitational entropy justifies the initial entropy of the universe. This entropy function had to be associated with the dynamics of the free gravitational field from the time of the big bang, so that a gravity-dominated evolution of the universe preserves the second law of thermodynamics. Moreover, the concept of black hole entropy emerges as a particular case of the entropy of the free gravitational field. However, a self-consistent notion of gravitational entropy in the context of cosmological structure formation has eluded us till today. Various proposals have been put forward, initially based on Penrose's Weyl Curvature Hypothesis, and subsequently modified to fit the needs of specific geometries and matter distributions. Such proposals were basically geometric in nature. A few years back a new definition of gravitational entropy was proposed from the considerations of the relativistic Gibb's equation and based on the square root of the Bel-Robinson tensor, the simplest divergence-free tensor derived from the Weyl tensor. Even this proposal is valid only for a restricted class of spacetimes. A complete self-consistent description of gravitational entropy encompassing black hole physics and cosmological dynamics is yet to emerge. In this article, we gather an overview of the concept of gravitational entropy, following it up with the development of the various proposals of gravitational entropy.

[7] 2306.04211

Cosmological perturbations out of the box I

Using the tool of Hodge-Morrey decomposition of forms, we prove a new decomposition of symmetric rank-2 tensors on flat manifolds with boundary. Using this we reconstruct a new cosmological perturbation theory that allows for the scalar-vector-tensor type separation of the linearized Einstein equations with general boundary conditions. We discuss gauge transformations, gauge invariant quantities and as an example how the new decomposition works out in the single-field inflation scenario. For the scalar modes we get two copies of Mukhanov-Sasaki equation, one of them with a slight modification. Additionally we run a Weinberg-like argument for the existence adiabatic modes, and find some gauge-invariant solutions to the perturbations that exists whatever the constituents of the universe are.

[8] 2306.04253

A general method for solving light-like geodesics

A universal method to solve the differential equations of light-like geodesics is developed. The validity of this method depends on a new theorem, which is introduced for light-like geodesics in analogy to Beltrami's "geometrical" method for time-like geodesics. we apply the method to the Schwarzschild and Kerr spacetime as two examples. The general solutions of the light-like geodesic equations in the two spacetimes are derived straightforwadly. After setting $\theta=\pi/2$, the general light-like geodesics in Schwarzschild spacetime reduce to the same expression as that in literatures. The method developed and results obtained in this paper may be useful in modeling dynamical phenomena in strong gravitaional fields like black holes since the solutions are expressed in terms of elliptic integrals, which can be calculated effectively.

[9] 2306.04298

Regular black hole from regular initial data

Recently there has been an interest in exploring black holes that are regular in that the central curvature singularity is avoided. Here, we give a recipe to obtain a regular black hole spacetime from the unhindered gravitational collapse from regular initial data of a spherically symmetric perfect fluid. While the classic Oppenheimer-Snyder collapse model necessarily produces a black hole with a Schwarzschild singularity at the centre, we show here that there are classes of regular initial conditions when collapse gives rise to a regular black hole.

[10] 2306.04373

Deep multimessenger search for compact binary mergers in LIGO, Virgo and Fermi/GBM data from 2016-2017

GW170817 - GRB 170817A provided the first observation of gravitational waves from a neutron star merger with associated transient counterparts across the entire electromagnetic spectrum. This discovery demonstrated the long-hypothesized association between short gamma-ray bursts and neutron star mergers. More joint detections are needed to explore the relation between the parameters inferred from the gravitational wave and the properties of the gamma-ray burst signal. We developed a joint multi-messenger analysis of LIGO, Virgo, and Fermi/GBM data designed for detecting weak gravitational-wave transients associated with weak gamma-ray bursts. As such, it does not start from confident (GWTC-1) events only. Instead, we take the full list of existing compact binary coalescence triggers generated with the PyCBC pipeline from the second Gravitational-Wave Observing Run (O2), and reanalyze the entire set of public Fermi/GBM data covering this observing run to generate a corresponding set of gamma-ray burst candidate triggers. We then search for coincidences between the gravitational-wave and gamma-ray burst triggers without requiring a confident detection in any channel. The candidate coincidences are ranked according to a statistic combining each candidate's strength in gravitational-wave and gamma-ray data, their time proximity, and the overlap of their sky localization. The ranking is then converted to a false-alarm rate using time shifts between the gravitational-wave and gamma-ray burst triggers. We present the results using O2 triggers which allowed us to check the validity of our method against GW170817 - GRB 170817A. We also discuss the different configurations tested to maximize the significance of the joint detection.

[11] 2306.04397

A rapid method for preliminary identification of subthreshold strongly lensed counterparts to superthreshold gravitational-wave events

Gravitational waves (GWs) from stellar-mass compact binary coalescences (CBCs) are expected to be strongly lensed when encountering large agglomerations of matter, such as galaxies or clusters. Searches for strongly lensed GWs have been conducted using data from the first three observing runs of the LIGO-Virgo GW detector network. Although no confirmed detections have been reported, interesting candidate lensed pairs have been identified. In this work, we delineate a preliminary analysis that rapidly identifies pairs to be further analyzed by more sophisticated Bayesian parameter estimation (PE) methods. The analysis relies on the Gaussian/Fisher approximation to the likelihood and compares the corresponding approximate posteriors on the chirp masses of the candidate pair. It additionally cross-correlates the rapidly produced localization sky areas (constructed by Bayestar sky-localization software). The analysis was used to identify pairs involving counterparts from targeted sub-threshold searches to confidently detected super-threshold CBC events. The most significant candidate ``super-sub'' pair deemed by this analysis was subsequently found, by more sophisticated and detailed joint-PE analyses, to be among the more significant candidate pairs, but not sufficiently significant to suggest the observation of a lensed event [1].

[12] 2306.04475

Palatini $f(R)$ gravity tests in the weak field limit: Solar System, seismology and galaxies

Palatini $f(R)$ gravity is probably the simplest extension of general relativity (GR) and the simplest realization of a metric-affine theory. It has the same number of degrees of freedom as GR and, in vacuum, it is straightforwardly mapped into GR with a cosmological constant. The mapping between GR and Palatini $f(R)$ inside matter is possible but at the expense of reinterpreting the meaning of the matter fields. The physical meaning and consequences of such mapping will depend on the physical context. Here we consider three such cases within the weak field limit: Solar System dynamics, planetary internal dynamics (seismology), and galaxies. After revising our previous results on the Solar System and Earth's seismology, we consider here the possibility of $f(R)$ Palatini as a dark matter candidate. For any $f(R)$ that admits a polynomial approximation in the weak field limit, we show here, using SPARC data and a recent method that we proposed, that the theory cannot be used to replace dark matter in galaxies. We also show that the same result applies to the Eddington-inspired Born-Infeld gravity. Differently from the metric $f(R)$ case, the rotation curve data are sufficient for this conclusion. This result does not exclude a combination of modified gravity and dark matter.

[13] 2306.04496

Electromagnetic high-frequency gravitational wave detection

Ultra-high frequency gravitational waves in the MHz to THz regime promise a unique possibility to probe the very early universe, particle physics at very high energies and exotic astrophysical objects - but achieving the sensitivity required for detection is an immense challenge. This is a brief summary of recent progress in electromagnetic high-frequency gravitational wave searches, which are based on classical electromagnetism in a space-time perturbed by gravitational waves. A particular focus is given to synergies with axion searches and atomic precision measurements. This article was prepared as proceedings for Moriond EW 2023.

[14] 2306.03915

Optimal Estimation of the Binned Mask-Free Power Spectrum, Bispectrum, and Trispectrum on the Full Sky: Tensor Edition

We derive optimal estimators for the binned two-, three-, and four-point correlators of statistically isotropic tensor fields defined on the sphere, in the presence of arbitrary beams, inpainting, and masking. This is a conceptually straightforward extension of the associated scalar field Philcox (2023), but upgraded to include spin-$2$ fields such as Cosmic Microwave Background polarization and galaxy shear, and parity-violating physics in all correlators. All estimators can be realized using spin-weighted spherical harmonic transforms and Monte Carlo summation and are are implemented in the public code PolyBin, with computation scaling, at most, with the total number of bins. We perform a suite of validation tests verifying that the estimators are unbiased and, in limiting regimes, minimum variance. These facilitate general binned analyses of higher-point functions, and allow constraints to be placed on various pheomena, such as non-separable inflationary physics (novelly including polarized trispectra), non-linear evolution in the late Universe, and cosmic parity-violation.

[15] 2306.03930

Pole-skipping in a non-black hole geometry

The pole-skipping has been discussed in black hole backgrounds, but we point out that the pole-skipping exists even in a non-black hole background, the AdS soliton. For black holes, the pole-skipping points are typically located at imaginary Matsubara frequencies $\omega=-(2\pi T)ni$ with an integer $n$. The AdS soliton is obtained by the double Wick rotation from a black hole. As a result, the pole-skipping points are located at $q_z=-(2\pi n)/l$, where $l$ is the $S^1$ periodicity and $q_z$ is the $S^1$ momentum. The ``chaotic" and the ``hydrodynamic" pole-skipping points lie in the physical region. We also propose a method to identify all pole-skipping points instead of the conventional method.

[16] 2306.03973

Three-Dimensional General-Relativistic Simulations of Neutrino-Driven Winds from Magnetized Proto-Neutron Stars

Formed in the aftermath of a core-collapse supernova or neutron star merger, a hot proto-neutron star (PNS) launches an outflow driven by neutrino heating lasting for up to tens of seconds. Though such winds are considered potential sites for the nucleosynthesis of heavy elements via the rapid neutron capture process ($r$-process), previous work has shown that unmagnetized PNS winds fail to achieve the necessary combination of high entropy and/or short dynamical timescale in the seed nucleus formation region. We present three-dimensional general-relativistic magnetohydrodynamical (GRMHD) simulations of PNS winds which include the effects of a dynamically strong ($B \gtrsim 10^{15}$ G) dipole magnetic field. After initializing the magnetic field, the wind quickly develops a helmet-streamer configuration, characterized by outflows along open polar magnetic field lines and a ``closed'' zone of trapped plasma at lower latitudes. Neutrino heating within the closed zone causes the thermal pressure of the trapped material to rise in time compared to the polar outflow regions, ultimately leading to the expulsion of this matter from the closed zone on a timescale of $\sim$60 ms, consistent with the predictions of \citet{Thompson03}. The high entropies of these transient ejecta are still growing at the end of our simulations and are sufficient to enable a successful 2nd-peak $r$-process in at least a modest $\gtrsim 1\%$ of the equatorial wind ejecta.

[17] 2306.04053

Detection of the Cosmological Time Dilation of High Redshift Quasars

A fundamental prediction of relativistic cosmologies is that, due to the expansion of space, observations of the distant cosmos should be time dilated and appear to run slower than events in the local universe. Whilst observations of cosmological supernovae unambiguously display the expected redshift-dependent time dilation, this has not been the case for other distant sources. Here we present the identification of cosmic time dilation in a sample of 190 quasars monitored for over two decades in multiple wavebands by assessing various hypotheses through Bayesian analysis. This detection counters previous claims that observed quasar variability lacked the expected redshift-dependent time dilation. Hence, as well as demonstrating the claim that the lack of the redshift dependence of quasar variability represents a significant challenge to the standard cosmological model, this analysis further indicates that the properties of quasars are consistent with them being truly cosmologically distant sources.

[18] 2306.04394

Dominant Spacetime in Three Dimensional de Sitter Gravity

In three dimensions, Kerr-de Sitter spacetime as a solution of Einstein gravity with positive cosmological constant has a single cosmological horizon. In this paper, we calculate the free energy of this spacetime and compare it with the free energy of the three-dimensional de Sitter spacetime. We investigate which one of these two spacetimes will dominate in the semi-classical approximation for estimating the partition function. It is shown that for the same temperature of cosmological horizon of two spacetimes this is the de Sitter spacetime which is always dominant.

[19] 2306.04427

Melvin Space-times in Supergravity

We consider Melvin-like cosmological and static solutions for the theories of ${\cal N}=2$, $D=4$ supergravity coupled to vector multiplets. We analyze the equations of motion and give some explicit solutions with one scalar and two gauge fields. Generalized Melvin solutions with four charges are also constructed for an embedding of a truncated ${\cal N}=8$ supergravity theory. Our results are then extended to supergravity theories with the scalar manifolds $SL(N, R)/SO(N, R)$. It is shown that solutions with $N$ charges only exist for $N=8$, $6$ and $5$ corresponding to theories with space-time dimensions $D=4$, $5$ and $7$.

[20] 2306.04545

Gravito-thermal transports, Onsager reciprocal relation and gravitational Wiedemann-Franz law

Using the near-detailed-balance distribution function obtained in our recent work, we present a set of covariant gravito-thermal transport equations for neutral relativistic gases in a generic stationary spacetime. All relevant tensorial transport coefficients are worked out and are presented using some particular integration functions in $(\alpha,\zeta)$, where $\alpha = -\beta\mu$ and $\zeta =\beta m$ is the relativistic coldness, with $\beta$ being the inverse temperature and $\mu$ being the chemical potential. It is shown that the Onsager reciprocal relation holds in the gravito-thermal transport phenomena, and that the heat conductivity and the gravito-conductivity tensors are proportional to each other, with the coefficient of proportionality given by the product of the so-called Lorenz number with the temperature, thus proving a gravitational variant of the Wiedemann-Franz law. It is remarkable that, for strongly degenerate Fermi gases, the Lorenz number takes a universal constant value $L=\pi^2/3$, which extends the Wiedemann-Franz law into the Wiedemann-Franz-Lorenz law.

[21] 2306.04577

Cosmological significance of the early bright galaxies observed with JWST

The recent discovery of objects with redshift $z>10$ with the help of James Webb Space Telescope (JWST) poses serious challenges to the $\Lambda$CDM cosmological model, which has been in vogue for some time now. The new data indicate that galaxy formation must have taken place much earlier than expected in this model. Another viable class of cosmological models is that of the so-called coasting models, in which the scale factor of the universe varies proportionately with time. In these models, the universe at redshift $z=12$ has ample time ($\sim 1070$ Myrs) for galaxy formation. The earliest such model is the one proposed by E.A. Milne, based on his `kinematic relativity', but it is considered unrealistic for not treating gravity as relevant at cosmological scales. A closed version of an eternal coasting FLRW model was proposed by the present authors even before SNe Ia data began to pour in. Subsequently we developed a more general model of the same class, which is valid for all the three possible geometries, with open, closed or flat spatial sections. In the nonrelativistic era, this model makes the falsifiable prediction that the ratio of matter density to dark energy density is 2. This avoids the cosmic coincidence problem. Moreover, this eternal coasting model allows room for creation of matter from dark energy, that may speed up galaxy and structure formation at the early epochs, as implied by the JWST data. The paper also attempts to review some similar coasting models, but emphasizes the eternal coasting cosmology as the most probable candidate model capable of explaining the presence of high redshift galaxies discovered by JWST.

[22] 2306.04589

Unifying fractons, gravitons and photons from a gauge theoretical approach

We revisit the first principles gauge theoretical construction of relativistic gapless fracton theory recently developed by A. Blasi and N. Maggiore. The difference is that, instead of considering a symmetric tensor field, we consider a vector field with a gauge group index, (i.e.) the usual Einstein-Cartan variable used in the first order formalism of gravity. After discussing the most general quadratic action for this field, we explore the physical sectors contained in the model. Particularly, we show that the model contains not only linear gravity and fractons, but also ordinary Maxwell equations, suggesting an apparent electrically charged phase of, for instance, spin liquids and glassy dynamical systems. Moreover, by a suitable change of field variables, we recover the Blasi-Maggiore gauge model of fractons and linear gravity.

[23] 2306.04626

Constraining nuclear parameters using f-modes from glitching pulsars

Gravitational waves (GW) emanating from unstable quasi-normal modes in Neutron Stars (NS) could be accessible with the improved sensitivity of the present gravitational wave (GW) detectors or with the next-generation GW detectors and therefore employed to study the NS interior. By taking into account potential GW candidates detectable by A+ and Einstein Telescope (ET) originating from f-modes excited by glitches in isolated pulsars, we demonstrate the inverse problem of NS asteroseismology in a Bayesian formalism to constrain the nuclear parameters within a relativistic mean field (RMF) description of NS interior. We find that for a single detected GW event from the Vela pulsar in A+ and ET, with the considered RMF model, the nucleon effective mass ($m^*$) can be restricted (within $90\%$ credible interval) within $10\%$ and $5\%$, respectively. With the considered RMF model, the incompressibility ($K$) and the slope of the symmetry energy ($L$) are only loosely constrained. With a single observed event in A+ and ET, the f-mode frequency of a $1.4M_{\odot}$ ($f_{1.4M_{\odot}}$) inside a 90\% symmetric credible interval (SCI) can be confined to 100 Hz and 50 Hz, respectively. Additionally, we consider multiple GW candidates in our analysis. For detecting multiple (ten) events with A+ and ET, $m^*$ can be constrained to $3\%$ and $2\%$, respectively. All the other nuclear saturation parameters get well constrained. In particular, $K$ and $L$ can be constrained within $10\%$ and $20\%$ (< $90\%$ SCI), respectively. Within the 90\% SCI, $f_{1.4M_{\odot}}$ can be estimated within 50 Hz and 20 Hz in A+ and ET, respectively. Uncertainty of other NS properties such as radius of a $1.4M_{\odot}$ ($R_{1.4M_{\odot}}$), f-mode damping time of a $1.4M_{\odot}$ ($\tau_{1.4M_{\odot}}$) and few equations of state (EOS) properties including squared speed of sound ($c_s^2$) are also estimated.