#  $f$-mode oscillations of compact stars with realistic equations of state in
dynamical spacetime

Swarnim Shashank  Fatemeh Hossein Nouri  Anshu Gupta  Inter-University Centre
for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
Center for Field Theory and Particle Physics and Department of Physics, Fudan
University, 200438 Shanghai, China  Center for Theoretical Physics, Polish
Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland

###### Abstract

In this study, we perform full three dimensional numerical relativity
simulations of non-rotating general relativistic stars. Extending the studies
for polytropic equation of state, we investigate the accuracy and robustness
of numerical scheme on measuring fundamental ($f$)-mode frequency for
realistic equations of state (EoS). We use various EoS with varying range of
stiffness and numerically evolve perturbed stellar models for several mass
configurations (in the range of $1.2-2.0$ $M_{\odot}$) for each of these EoS.
Using the gravitational waveform obtained from the simulations we extract the
$f$-modes of the stars. The obtained results are tested against the pre-
existing perturbation methods and find good agreement. Validity and deviation
of universal relations have been carried out and are compared with earlier
results under Cowling approximation as well as perturbative approaches. We
also show that even using perturbed single star simulations can provide good
agreement with $f$-modes extracted from inspiral phase of binary neutron star
simulations which are computationally more expensive.

###### keywords:

neutron stars , stellar oscillations , equation of state , simulations

##  1 Introduction

Asteroseismology that enables us to understand the interior structure of
neutron stars is coming to the forefront, with added tools of gravitational
wave astronomy applied in detecting gravitational wave signal from two
possible binary neuron star mergers (GW170817 [1] & GW190425 [2]) and
electromagnetic observations like NICER data. This assists in resolving
degenerate parameters, puts stricter bounds on the internal composition of the
star [3, 4, 5, 6, 7] and tests universal relations (UR).

Some of the parameters of neutron stars which could be inferred and estimated
through multi-messenger astronomy are the mass, radius, spin, tidal
deformability based on the mode frequencies extracted from the signals. While
the joint mass, radius estimates using NICER data has put limits on the
possible equations of state (EoS) which describes the internal composition [3,
4, 5], incorporating tidal deformability information based on the GW170817
constraints further, the bounds on the permissible EoS [6, 8].

Some of the earlier works on gravitational asteroseismology includes Ref. [9,
10, 11, 12] (see Ref. [13] for the pulsations of relativistic stars and
references therein). Through multiple studies it has been shown that the
fundamental oscillation mode ($f$-mode) of the star has a strong resonance
with the orbital frequencies closer to merger of a coalescing binary neutron
star. Resonant tidal excitation of oscillation modes in merging binary neutron
stars has been carried out in Ref. [14, 15, 16]. Other oscillation modes of
the star, like $p$ and $g$-modes, may also become relevant throughout the
inspiral, due to nonlinear coupling to the tide, as discussed in Ref. [17, 18,
19, 20].

Phenomenological models connecting tidal deformability with frequency have
been discussed in Ref. [21, 22]. For GW170817 data, $f$-mode frequency have
been estimated in Ref. [23] by using an $f$-mode tidal model (fmtidal) in the
frequency domain described in Ref. [24]. Contribution due to spin during the
binary neutron star merger on $f$-modes and dynamical tides have been recently
carried out in Ref. [25, 26]. Gravitational-wave asteroseismology with
$f$-modes from neutron star binaries at the merger phase has been also carried
out recently [27], where they use NR BNS simulations carried out in Ref. [28,
29, 30] and find less than one percent difference between BNS merger frequency
(based on the merger peak amplitude) and $f$-modes computed for isolated
neutron stars. Frequency deviations in the universal relations of isolated
neutron stars and postmerger remnants have been discussed in Ref. [31],
whereas UR for damping time have been carried out in Ref. [32]. Universal
relations among compactness ($\mathcal{C}=M/R$), moment of inertia ($I$) and
tidal deformability, and their validity and deviations have been studied in
Ref. [33, 34, 35, 36] for the various choices of equations of state (see Ref.
[37] for review).

Series of studies have been performed in computing oscillation modes of
neutron star using linear perturbation theory [38, 39, 35]. These studies are
based on the standard approach laid out in Thorne and Campolattaro (1967) [40]
to solve the stellar perturbation equations in curved spacetime by applying
the appropriate boundary conditions. The $f$-mode oscillations have also been
studied using numerical simulations, employing the cowling approximation i.e.
by evolving hydrodynamic equations in the fixed background of general
relativistic spacetime for a single perturbed neutron star [41, 42, 43, 44,
35, 45], also in full GR to study bar mode instability [46], as well as under
conformally flatness condition (CFC) [47, 48, 49], where the 3-metric is
assumed to be conformally flat and the spacetime dynamics is coupled with
fluid dynamics.

Recently, Rosofsky et al. [50] carried out studies in the fully dynamical
spacetime and extracted the fundamental modes for the polytropic equations of
state. There have been studies using piece-wise poly-tropic EoS to impose
constraints on the neutron star structure [51, 8] or using the parametrised
EoS with continuous sound speed [52] as well.

For Asteroseismology studies in curved spacetime, it is important to utilize
the full general relativistic hydrodynamical simulations to study neutron
stars’ EoS. The current standard nuclear EoS used in numerical simulations
(tables in https://stellarcollapse.org for instance) are in the tabulated
form. This means that the status of the fluid is identified by three
quantities $(\rho,T,Y_{e})$, where $\rho$ is the baryonic density, $T$ is the
temperature and $Y_{e}$ is the electron fraction. Obviously, the accuracy of
the numerical simulations with this type of EoS is affected by the resolution
of the three dimensional $(\rho,T,Y_{e})$ table. Moreover, the non-smoothness
features of various physical quantities in such EoS can be another source of
numerical errors for long-term evolutions. Therefore, it is crucial to perform
numerical simulations with tabulated EoS to examine the accuracy of the
current general relativistic hydrodynamical codes for measuring the
oscillation mode frequencies, and test their results against the analytical
results. In the current work, we present a few numerical tests to investigate
the accuracy challenges mentioned above, and then we extract the $f$-mode
frequencies by evolving nonrotating neutron star in the dynamical spacetime
while considering different tabulated nuclear EoS.

We also probe the possibility of using these single star simulations to study
the tidal effects expected during a binary inspiral. This is done by
perturbing the initial density field of the star with the dominant quadrupolar
term in the spherical harmonics expansion of the tidal field $Y_{22}$. Single
star simulations are computationally less expensive to carry out, providing
the possibility to reach higher resolutions in the future which may be
important for studying higher order harmonics. We, further, study the validity
of our results using the URs associated with tidal deformability and stellar
parameters.

This is a step towards extending our study incorporating spin effects which we
shall report in the follow up work. Our work provides leeway for future
studies of tidal effects of rotating stars in an inspiral and also post-merger
remnants with differential rotation.

In Sec. 2 we briefly outline mathematical and numerical framework for general
relativistic hydro-dynamical system, the initial setup and matter
configuration as described by a set of equations of state under consideration.
In Sec. 3 we discuss the accuracy challenges of mode frequency measurements
for tabulated EoS, and we present the results of the several numerical tests
to demonstrate the accuracy of our measurement. We analyse our simulation data
and describe the result findings in Sec. 4. We compute fundamental mode
($f$-mode frequency), its relations as a function of compactness and mass,
then compare our fits with some of the recently carried out works. Our
findings match well with the universal relations described with a deviation
less than of a few percent. Sec. 5 discusses our results and conclusions.

##  2 Basic Numerical Framework

We solve the Einstein equations using Numerical Relativity methods of 3+1
decomposition of the spacetime.

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi T_{\mu\nu}$ (1)
$ds^{2}=(-\alpha^{2}+\beta_{i}\beta^{i})dt^{2}+2\beta_{i}dx^{i}dt+\gamma_{ij}dx^{i}dx^{j}$
(2)

[53, 54, 55, 56] here, $\alpha$ is the lapse function, $\beta^{i}$ is the
shift vector and $\gamma_{ij}$ is the spatial metric. The units used are
$G=c=1$ unless mentioned otherwise.

###  2.1 Dynamical evolution

The sapcetime evolution is achieved using the Baumgarte-Shapiro-Shibata-
Nakamura-Oohara-Kojima (BSSNOK) formalism [57, 58, 59, 60] which is a
conformal formulation of the ADM equations [61, 62]. The (1+log) and Gamma-
driver gauge conditions are adopted for the evolution of lapse and shift [53,
63]. We use the MacLachlan code [64] for evolution of spacetime variables,
which is a publicly available code in the EinsteinToolkit [65, 66, 67, 68]
suite. A Kreiss-Oliger dissipation [54, 53, 56] is added to spacetime
variables for removing high frequency noise.
To model Neutron stars, a relativistic perfect fluid is assumed. The
conservation equations for the energy-momentum tensor $T_{\mu\nu}$ and the
matter current density $J_{\mu}$, are solved numerically after being recast
into a flux-conservative formulation [54, 69, 63]

$\nabla_{\mu}J^{\mu}=0,\nabla_{\mu}T^{\mu\nu}=0.$ (3)

The system of equations is complete with an Equation of State of the type
$p=p(\rho,Y_{e},T)$, which for our models is described in the Sec. 2.3. (For
more details readers can refer to Ref. [54, 69, 63, 56].) We carry out the
hydrodynamics using the publicly available WhiskyTHC code [70, 71, 72] which
works within the EinsteinToolkit framework and uses high-resolution shock
capturing methods. For the time integration, method of lines is used with
fourth-order Runge-Kutta methods. The fifth-order MP5 flux-reconstruction
method is used along with Harten-Lax-van Leer-Einfeldt (HLLE) Riemann solver.

###  2.2 Initial data

We use a perturbed Tolman–Oppenheimer–Volkoff (TOV) star for the initial data.
The initial data is generated using the PizzaTOV thorn for both polytropic and
tabulated equations of state. A perturbation is added for the density as:

$\delta\rho=\mathcal{A}\rho(r/R)Y_{22},$ (4)

where $\delta\rho$ is the perturbed density, $\rho$ is the density,
$\mathcal{A}$ is the perturbation amplitude, which we set to $0.01$ to
introduce a small perturbation, $r$ is the radial distance from centre of the
star and $R$ is the radius of the star. We use the $(2,2)$ eigenfunction which
is expected to be similar to tidal interactions of binary neutron stars [50,
23]. For all the models we choose the artificial background atmosphere density
as $\rho_{atm}=10^{-14}~{}M^{-2}\approx 6.17\times 10^{3}g~{}cm^{-3}$.
The simulations are performed at minimum grid spacing of $0.105M\approx 155m$.
For the tabulated EoS $1.4M_{\odot}$ cases we also perform simulations at
$0.07M$ and for polytrope at $0.06M$. The convergence test for the DD2
equation of state for three different resolutions is given in Sec. 3.2.

###  2.3 Equations of State

We use several finite-temperature, composition dependent nuclear-theory based
equations of state (Fig. 1). Three of them are based on relativistic mean
field (RMF) models. These equations of state are publicly available in
tabulated form at https://stellarcollapse.org. The resolution of tables used
are around 250-300 points.

  1. 1.

LS220 [73] (Lattimer & Swesty EoS with the incompressibility K = 220 MeV):
contains neutrons, protons, alpha particles and heavy nuclei. It is based on
the single nucleus approximation for heavy nuclei. LS220 has been widely used
in many supernova simulations.

  2. 2.

DD2 [74]: contains neutrons, protons, light nuclei such as deuterons, helions,
tritons and alpha particles and heavy nuclei. DD2 is an RMF with a density-
dependent nucleon-meson coupling for treating high density nuclear matter.

  3. 3.

SFHo [75]: Another RMF, and similar to DD2 it contains neutrons, protons,
light nuclei such as deuterons, helions, tritons and alpha particles and heavy
nuclei. However, the RMF parameters are tuned to fit the NS mass-radius
observation.

  4. 4.

BHB [76]: Another RMF similar to DD2 and SFHo with the same particle
composition, but BHB EoS additionally includes $\Lambda$ hyperons and hyperon-
hyperon interactions allowed by $\phi$ mesons.

  5. 5.

SLy [77]: contains only protons, neutrons and electrons. It is developed out
of a refined Skyrme-like effective potential, originating from the shell-model
description of the nuclei.

  6. 6.

APR4: is the complete version of a four-realistic-EoS series developed by
Akmal, Pandharipande and Ravenhall [78], named APR1 through 4, obtained from
potentials resulting from fits to nucleon-nucleon scattering. APR4 includes
the relativistic corrections and the three nucleon interaction potential
additionally, which makes it more complete in comparison with the other APRs.
This EoS is commonly used in neutron star simulations, as it occurs to be
compatible with astronomical observations.

  7. 7.

SRO-APR [79]: is based on APR potential model. However, similar to Lattimer &
Swesty EoS, it assumes compressible liquid droplet model of nuclei. It
contains neutrons, protons, a single type of heavy nucleus plus alpha
particles representing light nuclei.

For testing our methods, we also use a polytropic equation of state model

$p=\kappa\rho^{\Gamma}.$ (5)

We use $\Gamma=2$ and choosing accordingly the values of $\kappa$ and central
density, we create models for stars of masses $1.35M_{\odot}$ and
$1.4M_{\odot}$. The black dashed line in Fig. 1 represents the M-R relation of
a polytrope with $\kappa=94.29$. This parameter is applied to create the model
of the star with $1.35M_{\odot}$. The brown dotted line in Fig. 1 represents
M-R relation for a polytropic star with $\kappa=100$ for a star with the mass
of $1.4M_{\odot}$. These configurations are considered apart from the other
realistic EoSs to compare with the literature and the earlier studies which
considered polytropes. We evolve the polytropic models as well as the SLy and
APR4 models using a hybrid equation of state approach [80, 81] viz. a gamma-
law correction for finite temperature with $\Gamma_{th}=2$. The evolution
equation of state becomes:

$P(\rho,\epsilon)=P_{\mathrm{cold}}(\rho)+\rho[\epsilon-\epsilon_{\mathrm{cold}}(\rho)](\Gamma_{\mathrm{th}}-1)$
(6)

We find that the results obtained from the hybrid equation of state method for
our $1.4M_{\odot}$ polytrope to be consistent with the results from Ref. [50].

Figure 1: A comparison of all the equation of states used. The dotted line
represents our polytropic model with $\kappa=100$ and the dashed line
represents the polytropic model $\kappa=94.29$ (see Eq. 5). The shaded region
is the range of observed mass of pulsar $J0740+6620$ [82]. Purple-Blue region
in the middle is the parametrised EoS $M-R$ relation obtained from $GW170817$
[83].

###  2.4 Extracting the $f$-mode frequencies

In order to compute the $f$-modes, we use the Fourier transform of the time
series data for the gravitational waveform generated during the evolution of
each simulation (except for Cowling case where the frequency of $f$-mode is
extracted from Fourier transform of central density). For our analysis we
consider the waveform that is extracted at the radius $r=10M$ for the $l=2$,
$m=2$ mode of the $\Psi_{4}$ data, which is calculated using the Newman-
Penrose formalism [84, 85]. We choose this radius since it has the least
amount of noise among all the considered extraction radii while the computed
fundamental mode stays the same as could be seen in Fig. 3 (bottom right).

##  3 Numerical accuracy tests

Measuring the frequency of the oscillation modes from a fully general
relativistic hydrodynamic evolution can encounter several numerical
challenges. For an accurate frequency measurement, the space-time and hydro
evolutions are supposed to be done accurately with a reasonably-high grid
resolution within the convergence regime. Particularly, for tabulated
equations of state, the accuracy might be highly affected by the resolution of
the hydro variables from the original EOS table (the number of the divisions
over the density range for instance in our case). This accuracy limit appears
specifically in the first derivatives’ computation during each time step of
the hydro evolution [86]. The reconstruction algorithm can be another possible
source of errors for these measurements; Despite the fact that all commonly-
used reconstruction methods such as WENO, PPM, etc. formally converge at
second order in a standard finite volume/difference scheme, though the choice
of the numerical method to reconstruct data at cell faces is found to be
critical to correctly capture the stellar oscillations, and some of these
methods often have significantly smaller errors compared to other ones [50,
71].

To address all these accuracy concerns, we designed several numerical tests
including convergence and Cowling tests to investigate the accuracy of our
simulations’ setup and numerical methods used for frequency measurements. The
results of these tests prove that our numerical setup and methods provide a
good accuracy and robustness for $f$-mode frequency studies.

###  3.1 Testing $f$-mode frequency in Cowling approximation

As a part of the accuracy investigations, we test the hydrodynamic evolution
of our numerical setup verses the analytical results from the linear
perturbation theory. We run a test simulation in the Cowling approximation
i.e. with spacetime evolution turned-off. In the perturbation theory, the
Cowling approximation is taken into account by neglecting the perturbation of
the metric components. The mode frequencies are derived by solving a set of
coupled ODEs for hydro perturbation equations by applying appropriate boundary
conditions (see Ref. [87, 20] for more details). We test the accuracy of our
code at our standard resolution for LS220 EoS case with mass of
$1.4M_{\odot}$. Applying the fast Fourier transform on the density
oscillations at the neutron star’s centre to extract modes frequencies, we
find our numerical results close enough to the $f$-mode and the first harmonic
$H1$ derived from the perturbation equations. For $f$-mode we get $4.059$ kHz
from simulation and $4.065$ kHz from the perturbation equations and for $H1$
$6.959$ kHz from simulation and $7.043$ kHz from perturbation equations. Fig.
2 illustrates the frequency spectrum from our numerical simulation compared
with the results from the analytical solution of the perturbation equations.

Figure 2: Test simulation in Cowling approximation to check against analytical
values (red lines) of $f$-mode frequencies. We find that the simulation
retrieves accurate frequencies for $f$-mode and the first harmonic.

###  3.2 $f$-mode frequency at different resolutions

Figure 3: The central lapse for three different resolutions (top left);
$\Psi_{4}$ plotted at three different resolution. The change in resolution
does not change the $\Psi_{4}$ data (top right). The FFT of $\Psi_{4}$ at
three different resolution shows the $f$-mode value does not change. The black
dashed line represents the value of $f$-mode frequency (bottom left). The FFT
of $\Psi_{4}$ at 5 different radii with a refinement layer between each
(bottom right).

The performance of WhiskyTHC code has been tested before [70]. Here we only
test the accuracy of our results within the selected resolution for our
simulations. In order to claim that the resolution is in the convergence
regime, we perform a multiple resolutions test for a single case, i.e. DD2 EoS
with $M=1.4M_{\odot}$. We choose our standard resolution with grid spacing
equals to $0.105M$ as the intermediate level, and $0.07M$ and $0.14M$ as the
higher and lower resolutions respectively. The $\Psi_{4}$ data extracted at
$10M$ for this test.

The results of these tests are presented in Fig. 3. The top-left panel shows
the lapse function varying with time in the simulation. One can see that
increasing the resolution result in the convergence toward one solution. On
the top-right panel, we present $\Psi_{4}$ for different resolutions. This
plot shows that increasing the resolution does not change the gravitational
wave significantly, but it reduces the noise as expected. The bottom-left
panel shows the FFT of $\Psi_{4}$. This figure shows that the changes in
frequency are negligible by varying the resolution, which confirms the
accuracy of our $f$-mode frequency measurements. These results indicate that
the observed quantities converge to one solution as we move from low to high
resolutions, and the value of $f$-mode frequencies do not change significantly
across different resolutions. This convergence study confirms that the
intermediate resolution used in our numerical simulations, to report the
$f$-mode frequencies, is in the convergence regime.

###  3.3 $f$-mode frequency at different $\Psi_{4}$ extraction radii

As the last numerical test, we investigate the accuracy of the $\Psi_{4}$
functions extracted from different radii. For this test we choose our
polytropic case with $\Gamma=2$, $\kappa=100$ and $M=1.4M_{\odot}$, with
minimum grid spacing equals to $0.06M$ for the resolution. The $\Psi_{4}$
outputs are extracted at $r=10,40,70,100,130M$ for the Fourier transform.

In Fig. 3 (bottom right), we show that the $f$-mode frequency does not change
with different radii of extraction of $\Psi_{4}$ for Polytropic equation of
state. We also notice that $f$-mode is most prominent and has less noise for
$r=10M$, and hence we chose it for comparison in with other equations of
state.

##  4 Analysis and Results

We evolve a set of non-rotating configurations in the mass range of
$1.2-2.0M_{\odot}$ for each of the EoS, described in Sec. 2.3 and test our
obtained results against the established results from perturbative approach.
The mass, radius, central density for these have been listed in the first four
columns of Tab. 2.

We also notice that change in resolution does not affect the extracted
$f$-mode frequencies (Fig. 3). We run the simulations for $2100M\approx
10.35ms$.

###  4.1 fundamental ($f$)-modes

Figure 4:  $f$-mode frequency for different EoS is shown in different panels
with respect to mass $M$ (top left); compactness $\mathcal{C}=M/R$, $R$ being
the radius of the star (top right); effective compactness
$\eta=\sqrt{M^{3}/I}$, where $I$ is the moment of inertia (bottom left) and
tidal deformability $\lambda_{2}$ as in Eq. 9 (bottom right). The softer EoS
has larger $f$-mode frequencies as compared to the intermediate (LS220) and
stiffer (DD2 and BHB) EoS.

We present our models and results of our simulations in Tab. 2. In Fig. 4 (top
panels) we plot the $f$-mode frequency value in terms of mass and compactness
for the considered set of equations of state.

We notice that the $f$-mode frequency is much smaller for the stiff EoS such
as DD2 and BHB. It increases for the higher masses and softer EoS. Based on
the frequency band of the observed gravitational signal and the inferred mass,
it is possible to put bounds on the softness/stiffness of the EoS that
characterises the interior of the neutron star. For example, if the detected
$f$-mode frequency is below $1.8$ kHz for a star having mass above
$1.4M_{\odot}$, then all the considered soft EoS would be ruled out. On the
other hand, frequency above $1.8$ kHz would permit only some stiff EoS if the
object is more massive i.e. $\approx 1.8M_{\odot}$ or more and compactness
above $0.20$. Similar trends we see in terms of effective compactness $\eta$
and tidal deformability $\lambda_{2}$ (see bottom panels of Fig. 4). We also
observe that while few of the considered EoS show linear trend, some of the
other, such as BHB, LS220, SLy and SFHo deviate and show a faster rise in
$f$-mode values for larger parameter values (as could be seen in all the four
panels of Fig. 4). This doesn’t seem to be dependent only on the softness or
stiffness of the matter, but could be due to the finer micro-physics involved.
To understand this better, in our follow-up studies, we plan to consider a
larger set of EoS, including the ones having hyperons, strange quark matter,
etc.

In Fig. 5 we present the fits of our models with the relation as given in Ref.
[35] :

$f=a+b~{}\sqrt{\frac{M}{R^{3}}}$ (7)

We find a good linear fit for our data across the EoS used for our study and
list the values of $a$ and $b$ in Tab. 1. LS220 EoS shows the steepest change
(the red line), followed by BHB (the Beige color line) in Fig. 5.

EoS | a | b
---|---|---
DD2 | 0.657 | 32.061
BHB | 0.429 | 39.801
LS220 | 0.286 | 44.653
SFHo | 0.663 | 35.101
SRO-APR | 0.881 | 29.958
SLy | 0.557 | 36.544
APR4 | 0.795 | 30.791
Table 1: Data fitted for the various compact star models with Eq. 7. We find
that our results are in agreement with the Ref. [35].

We compare our fit obtained in Tab. 1 to the Table II in Ref. [35] for the
LS220 and APR4 equation of states, and find our results to be in agreement
(within $10\%$).

Figure 5: Fit from Eq. 7. The $a$ and $b$ values are listed on the figure
above and also Tab. 1.

###  4.2 Universal Relations

In order to study, whether the equations of state specific trends that we
notice above, contribute to the deviation from the established Universal
relations or not, we verify some of the URs. First, we compute the relation
between $f$-mode frequency and the tidal deformability as given in Ref. [23,
34]

$M\omega=\sum_{i}a_{i}~{}(\xi)^{i},$ (8)

where $a_{i}$ are the numerical coefficients presented in Tab. 3. $M$ is the
mass of the star and $\omega$ is the angular $f$-mode frequency.
$\xi=log(\lambda_{2})$, where $\lambda_{2}$ is the dimensionless electric
tidal deformability calculated from the tidal Love number $k_{2}$ as [34]:

$\lambda_{2}=\frac{2}{3}\frac{k_{2}}{(M/R)^{5}}$ (9)

The tidal Love number $k_{2}$ is calculated by solving the metric perturbation
equation as defined in Ref. [88]. For this computation we integrate Eq. (15)
from [88] for the metric perturbation function $H$, from center to surface
using fourth order Runge-Kutta method. We use the Runge-Kutta ODE solver with
adaptive step size routine from Numerical recipe [89]. Finally, the tidal Love
number $k_{2}$ is computed from Eq.(23) from [88]. Figure 4 (bottom-right
panel) shows the relation of tidal deformability $\lambda_{2}$ with the
$f$-mode.

The comparison and deviation for $f$-Love UR, Eq. 8 is carried out in two
ways: first, using these equations we compute the coefficients for our data.
Second, using the same values of coefficients as given in Ref. [34] but, with
the $f$, $\lambda_{2}$ and $\eta$ that we calculate for each of our
simulation. The last three columns of Tab. 2 show comparison and percent
deviations with $f$-Love UR’s. Table 3 lists the coefficients for Ref. [34]
and the ones computed for our data. In Fig. 6 we compare the our results with
that of Ref. [34].

Figure 6: The universality observed between tidal deformability and $f$-modes.
Here, we compare our fit with that of Ref. [34].

It has been observed that universal behaviour also exist between the $f$-mode
and the effective compactness $\eta$ [90, 35]. We also test these URs
described by the model

$M\omega=c_{1}+c_{2}~{}\eta+c_{3}~{}\eta^{2},$ (10)

where $\eta=\sqrt{M^{3}/I}$ and $I$ is the moment of inertia. We obtain the
fit as $c_{1}=-0.00747$, $c_{2}=0.1471$ and $c_{3}=0.55328$. We show this
universality in the Fig. 7. Our results agree well with the results of the
earlier works [90, 35]. The fundamental mode $f$-mode and effective
compactness $\eta$ that we compute for our configurations are also plotted in
Fig 4 (bottom left panel).

Figure 7: The universality observed between $\eta$ and $f$-mode. The black
dashed line is the fit that we obtained for Eq. 10. The brown dashed line is
the one presented in the Ref. [35].

Universal relation between the compactness and tidal Love numbers have been
discussed in A.

EoS |  Mass ($M_{\odot}$) |  Radius ($km$) |  Central Density ($g/cm^{3}$) |  $f$-mode ($kHz$) | k2 |  (%)-difference Chan et al. [34] v. This work
---|---|---|---|---|---|---
Polytrope | 1.35 | 13.80 | $8.22\times 10^{14}$ | 1.6237 | 0.0805 | 0.22
| 1.4 | 14.15 | $7.91\times 10^{14}$ | 1.5730 | 0.0793 | 0.26
| 1.2 | 13.20 | $5.25\times 10^{14}$ | 1.5522 | 0.1061 | 0.25
| 1.4 | 13.27 | $5.73\times 10^{14}$ | 1.5998 | 0.0953 | 0.33
DD2 | 1.6 | 13.31 | $6.27\times 10^{14}$ | 1.6715 | 0.0833 | 0.53
| 1.8 | 13.30 | $6.9\times 10^{14}$ | 1.7312 | 0.0709 | 0.05
| 2.0 | 13.24 | $7.67\times 10^{14}$ | 1.8028 | 0.0586 | 0.86
| 1.2 | 13.20 | $5.25\times 10^{14}$ | 1.5521 | 0.1058 | 0.24
| 1.4 | 13.27 | $5.84\times 10^{14}$ | 1.5998 | 0.0947 | 0.34
BHB | 1.6 | 13.26 | $6.73\times 10^{14}$ | 1.6834 | 0.0821 | 0.53
| 1.8 | 13.13 | $8.0\times 10^{14}$ | 1.7909 | 0.0675 | 0.08
| 2.0 | 12.71 | $1.02\times 10^{15}$ | 1.9461 | 0.0508 | 1.29
| 1.2 | 12.84 | $6.16\times 10^{14}$ | 1.5760 | 0.1056 | 0.14
| 1.4 | 12.75 | $7.19\times 10^{14}$ | 1.6954 | 0.0898 | 0.46
LS220 | 1.6 | 12.56 | $8.47\times 10^{14}$ | 1.8267 | 0.0734 | 0.35
| 1.8 | 12.23 | $1.03\times 10^{15}$ | 1.9936 | 0.0568 | 0.74
| 1.9 | 11.96 | $1.16\times 10^{15}$ | 2.0893 | 0.0482 | 1.41
| 2.0 | 11.53 | $1.37\times 10^{15}$ | 2.2446 | 0.0383 | 1.80
| 1.2 | 11.99 | $7.41\times 10^{14}$ | 1.7909 | 0.0922 | 0.21
| 1.4 | 11.92 | $8.4\times 10^{14}$ | 1.8864 | 0.0793 | 0.54
SFHo | 1.6 | 11.79 | $9.6\times 10^{14}$ | 1.9938 | 0.0653 | 0.06
| 1.8 | 11.56 | $1.12\times 10^{15}$ | 2.1252 | 0.0511 | 1.23
| 1.9 | 11.36 | $1.24\times 10^{15}$ | 2.1968 | 0.0436 | 1.71
| 2.0 | 11.05 | $1.43\times 10^{15}$ | 2.3042 | 0.0355 | 1.72
| 1.2 | 11.68 | $8.03\times 10^{14}$ | 1.8744 | 0.0862 | 0.32
| 1.4 | 11.61 | $8.9\times 10^{14}$ | 1.9699 | 0.0745 | 0.50
SRO-APR | 1.6 | 11.51 | $9.87\times 10^{14}$ | 2.0655 | 0.0620 | 0.26
| 1.8 | 11.35 | $1.11\times 10^{15}$ | 2.1610 | 0.0495 | 1.37
| 2.0 | 11.09 | $1.27\times 10^{15}$ | 2.2684 | 0.0368 | 1.75
| 1.2 | 11.47 | $7.79\times 10^{14}$ | 1.8028 | 0.1016 | 0.28
| 1.35 | 11.46 | $8.58\times 10^{14}$ | 1.8983 | 0.0885 | 0.55
| 1.4 | 11.46 | $8.86\times 10^{14}$ | 1.9103 | 0.0840 | 0.52
SLy | 1.6 | 11.37 | $1.02\times 10^{15}$ | 2.0296 | 0.0670 | 0.24
| 1.8 | 11.16 | $1.2\times 10^{15}$ | 2.1610 | 0.0506 | 1.43
| 2.0 | 10.62 | $1.55\times 10^{15}$ | 2.3639 | 0.0328 | 1.41
| 1.2 | 11.04 | $8.24\times 10^{14}$ | 1.9103 | 0.0984 | 0.40
| 1.4 | 11.09 | $9.12\times 10^{14}$ | 1.9938 | 0.0818 | 0.44
APR4 | 1.6 | 11.09 | $1.01\times 10^{15}$ | 2.0774 | 0.0661 | 0.41
| 1.8 | 11.01 | $1.14\times 10^{15}$ | 2.1729 | 0.0512 | 1.49
| 2.0 | 10.81 | $1.32\times 10^{15}$ | 2.2804 | 0.0371 | 1.66
Table 2: $f$-modes for the different EoS for different masses. A clear trend is visible. Frequency increases with mass and softness of the EoS. The last column lists the difference between the UR provided in Ref. [34] and our work. Fit | $a_{0}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$
---|---|---|---|---|---
Chan et al. [34] | $1.820\times 10^{-1}$ | $-6.836\times 10^{-3}$ | $-4.196\times 10^{-3}$ | $5.215\times 10^{-4}$ | $-1.857\times 10^{-5}$
This work | $1.442\times 10^{-1}$ | $3.005\times 10^{-2}$ | $-1.607\times 10^{-2}$ | $2.092\times 10^{-3}$ | $-9.247\times 10^{-5}$
Table 3: Fitting parameters $a_{i}$ from Eq. 8 for the $f$-Love universal
relations, comparing this work and Ref. [34]. The polytropic models are
excluded in $a_{i}$ values calculations.

###  4.3 Calculation of damping times

We compute fundamental frequency from $\Psi_{4}$ data of our simulations (see
discussion in the beginning of Sec. 4) by evolving each initial configuration
for about $2100M$ ($\approx 10$ ms). This duration is insufficient, and it is
required to have a longer simulation and higher resolution to extract reliable
damping times. Thus, we choose the recently established relations, which use
compactness and effective compactness for calculating the damping times [32,
35]:

$\frac{M}{\tau_{1}}=0.112\left(\frac{M}{R}\right)^{4}-0.53\left(\frac{M}{R}\right)^{5}+0.628\left(\frac{M}{R}\right)^{6},$
(11)

$\frac{I^{2}}{M^{5}\tau_{2}}=0.0068-0.025~{}\eta^{2},$ (12)

where $\tau_{1}$ and $\tau_{2}$ are the damping times in terms of compactness
$\mathcal{C}=\frac{M}{R}$ and effective compactness $\eta$, respectively .
Figure 8 shows damping time vs frequency plot, where the damping time is taken
to be the average of $\tau_{1}$ and $\tau_{2}$. We find that for the polytrope
case one of the relations overestimates the damping time obtained from linear
perturbations methods [91, 50] while the other underestimates, taking an
average of $\tau_{1}$ and $\tau_{2}$ gives us a value close to the expected
value for the polytrope, hence we report the average values. In the subsequent
study, we intend to extract damping times by evolving our systems for a much
longer time and at a much higher resolution as one needs to be sure that the
damping obtained is not due to numerical errors.

Figure 8: $f$-mode vs the damping time for all the models. The damping time
here is calculated as $(\tau_{1}+\tau_{2})/2$ from Eq. 11 and Eq. 12.

###  4.4 Comparison with compact binary merger simulations

In the context of a binary system, the tidal force comes from a companion star
in circular motion. The spherical harmonic expansion of the full tidal
potential is given by Eq.(2.2) from Lai (1994) [14]:

$U=-GM\sum_{l,m}W_{lm}\frac{r^{l}}{A(t)^{l+1}}Y_{lm}(\theta,\phi)e^{-im\Omega
t},$ (13)

where $G$ is the gravitational constant, $M$ is the mass of the companion
star, the coefficients $W_{lm}$ depend on $(l,m)$ with equation (2.3) from
[14], $A(t)$ is the binary separation and $\Omega$ is the orbital frequency.

The initially seeded perturbation with $Y_{22}$ harmonics given by Eq.(4),
corresponds to the leading quadrupole term of the spherical harmonic expansion
of the tidal potential in Eq.(13). Therefore, the initial setup of our single
neutron star simulation helps us to artificially mimic the perturbation caused
by the tidal field of the companion star during inspiral phase up to an
acceptable order of accuracy.

As a comparison we simulate couple of configuration to compare the $f$-mode
frequency with Ref. [26], where they extract the $f$-mode frequency using
numerical relativity simulation of binary systems. We obtain an excellent
match with their results, comparing to their black hole - neutron star binary
simulation using $1.35M_{\odot}$ polytrope, we find a deviation of $0.6\%$ and
with their neutron star - neutron star binary simulation using $1.35_{\odot}M$
SLy EoS we see a deviation of only $0.03\%$. We further note that even by
interpolating the mass vs $f$-mode data based on our simulations to get the
$f$-mode frequency at $1.35M_{\odot}$, the value which we get differs only by
$0.9\%$ for the SLy EoS from the one reported in Ref. [26].

##  5 Conclusions

In this work, we evolve isolated non-spinning neutron star in fully dynamical
spacetime with nuclear EoS. We investigate the accuracy of our numerical
methods by conducting several tests including the convergence, and Cowling
evolution described in Sec. 3. These tests confirm that our selected grid
resolution is in the convergence regime, and their results match very well
with the linear perturbation theory’s solution for $f$-mode frequency
measurement in the Cowling approximation.

For the main simulations using full general relativistic hydrodynamic
evolution, we consider a set of realistic EoS as listed in Sec. 2.3 to
describe internal composition of the star. For each of the considered EoS, we
evolve 5-6 configurations having mass in the range of $1.2-2.0M_{\odot}$. We
compute $f$-mode frequency for each of the case and its dependence on the
equation of state. We also compute tidal deformability for each of the case
using perturbative approach.

Our analysis show evidence of approximate universal relations between $f$-mode
and other neutron star parameters such as tidal deformability, compactness
etc. However, there is still a possibility to distinguish and constrain the
possible stiff or soft equations of state based on the observed gravitational
wave signal, as $f$-mode frequency value differs almost by $0.5$ kHz for soft
EoS (APR4) and stiff EoS (DD2 or BHB) when the mass of the neutron star is
$1.4M_{\odot}$. The difference is higher for the more massive cases, in fact,
for $M\geq 1.8M_{\odot}$, frequency differs by $\approx 300$ Hz between DD2
(stiff EoS) and LS220 (having intermediate stiffness). Frequency computed are
also well within the bounds reported by Ref. [23] through Bayesian estimates.

Further, the computed frequencies match very well with the studies performed
by other groups while studying the fundamental modes during the binary neutron
star merger [26], as well as the frequencies extracted through perturbative
studies [35]. These findings indicate that using single perturbed star
simulations could give us similar results to that of a binary simulation,
saving on computational time and resources for measuring $f$-mode frequency at
inspiral phase. We also observe, similar tidal behaviour for SRO-APR and APR4
EoS, where the former is one of the most updated EoS in the family of APR EoS,
hence it could be a good choice for binary merger simulations for future
studies.

Our $f$-mode study of a perturbed neutron star in this paper is limited in
many ways. Our numerical scheme for hydrodynamic evolution is limited to the
second-order convergent finite volume method. In future studies, more advance
numerical scheme such as discontinuous Galerkin method with higher accuracy
and better efficiency can be used to study the perturbed neutron stars’
evolution [45, 92]. We considered only non-rotating stars, and our EoS
selection includes only one EoS with exotic matter i.e. BHB with hyperons. In
the follow-up studies, it is important to perform dynamical spacetime
simulations of a perturbed rotating neutron star to investigate the
relativistic corrections on $f$-mode frequency shift due to spin effects [26].
Further, these studies can be enhanced by including a broad range of EoS with
hyperons [93], muons [94], and also considering the parametrized EoS with
piece-wise polytropic EoS [51, 8] or with continuous sound speed [52] to
impose constraints on EoS, as are being considered in some of the recent
studies. In addition, we intend to consider longer evolution with higher
resolution to have a better accuracy for the higher mode measurements, as well
as for the $f$-mode damping time measurements. Similar asteroseismological
studies can also be extended to theories of gravity beyond general relativity.
Many models for compact stars in different types of modified gravity theories
exist (for example, see Refs. [95, 96, 97, 98]). Large-scale cosmological and
gravitational wave observations may help probe these alternative gravity
theories [99].

## Acknowledgments

The authors thank Sukanta Bose for frequent helpful discussions, scientific
advice and comments over the entire course of this project. We thank D. Radice
for help and support with WhiskyTHC code. We are also grateful to Sukanta Bose
and P. Pnigouras for their useful inputs on the manuscript. S.S. acknowledges
L. Baiotti, I. Hawke and International Centre for Theoretical Sciences (ICTS)
for discussions during the program - Gravitational Wave Astrophysics (Code:
ICTS/Prog-gws2020/05). Part of this research is supported by the Navajbai
Ratan Tata Trust and LIGO-India funds at IUCAA, India. F.H. acknowledges grant
No. 2019/35/B/ST9/04000 from the Polish National Science Center, Poland. S.S.
also acknowledges support from the China Scholarship Council (CSC), Grant No.
2020GXZ016646. The numerical simulations for this work were performed on
Pegasus cluster, which is a part of the high performance computing (HPC)
facility at The Inter-University Centre for Astronomy and Astrophysics
(IUCAA), Pune, India.

##  Appendix A Compactness - Tidal deformability universal relations

Figure 9: The universality observed between tidal deformability and
compactness. Here, we compare our fit (black dashed line) with that of Ref.
[100] (brown dashed line).

For our compact star models, we also study another universal relation in
addition to the discussion in Sec. 4.2 between the compactness and tidal Love
number using the initial data that we use in our simulations. We also use this
as a check for our initial data against already established results [33, 100].
The universal relations are given as,

$\mathcal{C}=\sum_{i}b_{i}~{}(log(\lambda_{2}))^{i}$ (14)

The universal relation between compactness and tidal deformability provides us
the constraint on the radius of the star as a less compact star will be
deformed more by a tidal potential for a given mass [100], providing a
relation between radius and $\lambda_{2}$. We present our obtained fit for
$b_{i}$ in Tab. 4. In Fig. 9, we compare our results to a recent study by
[100]. Our results are consistent with the results of Ref. [33] as presented
in Tab. 4.

Fit | Maselli et al. [33] | This work | Godzieba et al. [100] | This work
---|---|---|---|---
$b_{0}$ | $3.71\times 10^{-1}$ | $3.770\times 10^{-1}$ | $3.388\times 10^{-1}$ | $7.951\times 10^{-1}$
$b_{1}$ | $-3.91\times 10^{-2}$ | $-4.137\times 10^{-2}$ | $-2.30\times 10^{-2}$ | $-5.839\times 10^{-1}$
$b_{2}$ | $1.056\times 10^{-3}$ | $1.149\times 10^{-3}$ | $-4.651\times 10^{-4}$ | $2.877\times 10^{-1}$
$b_{3}$ | - | - | $-2.636\times 10^{-4}$ | $-7.908\times 10^{-2}$
$b_{4}$ | - | - | $5.424\times 10^{-5}$ | $1.205\times 10^{-2}$
$b_{5}$ | - | - | $-3.188\times 10^{-6}$ | $-9.628\times 10^{-4}$
$b_{6}$ | - | - | $6.181\times 10^{-8}$ | $3.155\times 10^{-5}$
Table 4: Fitting parameters $b_{i}$ from Eq. 14 for the $\mathcal{C}$-Love
universal relations. Comparing this work with the coefficient values in Ref.
[33] and Ref. [100]. The polytrope cases are excluded from $b_{i}$’s
calculations.

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