# Impact of the equation of state on $f$\- and $p$\- mode oscillations of
neutron stars

Athul Kunjipurayil<EMAIL_ADDRESS>Department of Physics & Astronomy,
National Institute of Technology, Rourkela 769008, India  Tianqi Zhao
<EMAIL_ADDRESS>Department of Physics and Astronomy, Ohio University, Athens,
OH 45701, USA  Bharat Kumar  Correspondence<EMAIL_ADDRESS>Department
of Physics & Astronomy, National Institute of Technology, Rourkela 769008,
India  Bijay K. Agrawal<EMAIL_ADDRESS>Saha Institute of Nuclear
Physics, 1/AF Bidhannagar, Kolkata 700064, India  Homi Bhabha National
Institute, Anushakti Nagar, Mumbai 400094, India  Madappa Prakash
<EMAIL_ADDRESS>Department of Physics and Astronomy, Ohio University, Athens,
OH 45701, USA

###### Abstract

We investigate the impact of the neutron-star matter equation of state on the
$f$\- and $p_{1}$-mode oscillations of neutron stars obtained within the
Cowling approximation and linearized general relativity. The $f$\- and
$p_{1}$-mode oscillation frequencies, and their damping times are calculated
using representative sets of Skyrme Hartree-Fock and relativistic mean-field
models, all of which reproduce nuclear systematics and support $2M_{\odot}$
neutron stars. Our study shows strong correlations between the frequencies of
$f$\- and $p_{1}$-modes and their damping times with the pressure of
$\beta$-equilibrated matter at densities equal to or slightly higher than the
nuclear saturation density $\rho_{0}$. Such correlations are found to be
almost independent of the composition of the stars. The frequency of the
$p_{1}$-mode of $1.4M_{\odot}$ star is strongly correlated with the slope of
the symmetry energy $L_{0}$ and $\beta$-equilibrated pressure at density
$\rho_{0}$. Compared to GR calculations, the error in the Cowling
approximation for the $f$-mode is about 30% for neutron stars of low mass,
whereas it decreases with increasing mass. The accuracy of the $p_{1}$-mode is
better than 15% for neutron stars of maximum mass, and improves for lower
masses and higher number of radial nodes.

##  I Introduction

Oscillation modes are generated when a neutron star (NS) is perturbed by an
external or an internal disturbance. An oscillating NS can have different
quasinormal modes (QNMs), categorized according to the restoring force that
brings them back to equilibrium. QNMs can be fundamental mode ($f$-mode),
pressure mode ($p$-mode), gravity mode ($g$-mode), rotational mode ($r$-mode),
space-time mode ($w$-mode), etc Kokkotas and Schmidt (1999). The frequency of
these oscillations depends upon the interior structure, and in some cases, the
composition of the stars Zhao  _et al._ (2022). Moreover, these modes mainly
contribute to a NS’s gravitational wave emission. In order to study these
oscillation frequencies, we need to solve the fluid perturbation equations
along with the Tolman-Oppenheimer-Volkoff (TOV) equations in general
relativity (GR) Zhao  _et al._ (2022); Zhao and Lattimer (2022).

Among the various QNMs studied theoretically, the $f$-mode is the most
promising one to be observed first. The proto-neutron star (PNS) formed in a
core-collapse supernova has long been considered as a potential gravitational-
wave source for LIGO and Virgo detectors Ferrari  _et al._ (2003). Numerical
simulation shows that about 10% of the gravitational radiation is associated
with the $\ell=2,m=1,2$ $f$-mode oscillation Shibagaki  _et al._ (2020). The
total gravitational radiation energy could reach $10^{44}-10^{47}$ ergs
depending on the mass and rotation rate of the progenitor Radice  _et al._
(2019). Only galactic sources with distance $D<20$ kpc are likely to be
observed in advanced LIGO observations, at a rate of a few per century Li  _et
al._ (2011). The $f$-mode frequency of a proto-neutron star with high
temperature and high lepton fraction is lower than that for the NS which are
considered here. More likely to be observed are signals from the remnants of
NS mergers, but they will likely be complicated. Numerical simulation of an
equal-mass NS merger shows that the dominant fluid oscillation of a
supermassive NS remnant coincides with the $m=2$ $f$-mode Stergioulas  _et
al._ (2011), and has a strong correlation with the (zero temperature) isolated
NS $f$-mode frequency Lioutas  _et al._ (2021). The peak frequency in
supermassive NSs is almost equal to that of the non-rotating $f$-mode
frequency of isolated NSs with the same mass as each of the merging components
Ng  _et al._ (2020). With third generation gravitational wave telescopes such
as the Cosmic Explorer Reitze  _et al._ (2019) and Einstein telescope Punturo
_et al._ (2010), the predicted event rate improves further to 0.06 yr-1 to 4
yr-1, which is promising Abbott  _et al._ (2021); Zhao and Lattimer (2022).

Besides directly observing gravitational radiation from the fluid oscillations
in a proto-neutron star and supramassive NS remnant, there’s another direct
method to measure QNMs of NSs by analysing NS merger waveform through
dynamical tidal coupling Hinderer  _et al._ (2016); Steinhoff  _et al._
(2016); Schmidt and Hinderer (2019). During the inspiral of binary NSs, QNM
oscillation can be excited by the dynamical tide when the orbital frequency
approaches QNM frequency, which leads to loss of orbital angular momentum. The
significance of the signal is determined by the QNM frequency as well as its
coupling to dynamical tidal field. Because the $g$-mode has a low frequency,
tidal interactions could excite $g$-mode oscillations well before resonances
with the $f$-mode is reached during the last part of the inspiral. However,
the fluid perturbation of the $g$-mode peaks at the stellar center and thus
has a weaker coupling to the tidal field. In contrast, the fluid perturbation
of the $f$\- and $p$-modes peak at the stellar surface with a stronger
coupling to the tidal field. Indeed, the $f$\- and $p$-mode frequencies of
non-rotating NSs are too high to be excited in NS mergers. QNMs of rotating NS
can be promoted by the NS spin $\omega_{s}$, because the resonant frequency
could be reduced to $\omega_{f}-2\omega_{s}$, where $\omega_{s}$ is the spin
frequency of the NS. For a millisecond pulsar, $f$-mode resonances could cause
phase advances up to hundreds of cycles Wynn and Dong (1999); Steinhoff  _et
al._ (2021). A lower bound to the $f$-mode frequency has been estimated from
the non-detection of a significant phase shift due to $f$-mode tidal resonance
in GW170817, while an upper bound can be estimated from the
$\Omega_{f}-\Lambda$ universal relation. The resulting 90% credible interval
of $f$-mode frequency for GW170817 is 1.43 kHz $<\nu_{f}<2.90$ kHz (1.48 kHz
$<\nu_{f}<3.18$ kHz) for the more (less) massive NS in the binary.

The $f$-mode of neutron stars correlates with many NS properties including
compactness Andersson and Kokkotas (1998a), moment of inertia Lau  _et al._
(2010) and static tidal polarizability Chan  _et al._ (2014); Sotani and Kumar
(2021). Such universal relations hold even for bare quark stars without crust
or hybrid NSs with first order transitions (Zhao and Lattimer, 2022). However,
$p$-modes have very weak correlation with other NS properties Andersson and
Kokkotas (1998a). Because $p$-mode oscillation dominates in the stellar
surface, it is sensitive to the equation of state (EoS) at much lower
densities.

One of the objectives in this work is calculate $f$-mode and lowest order
$p$-mode frequencies and damping times by solving general relativistic non-
radial oscillation equations Lindblom and Detweiler (1983); Detweiler and
Lindblom (1985) in the form an ordinary differential equation (ODE) eigenvalue
problem. These equations are for even-parity modes that include $f$-, $g$\-
$p$\- and $w$-modes. We ignore the rotation of NSs which could slightly
increase the $f$\- and $p$-mode frequencies Kojima (1993); Krüger and Kokkotas
(2020). The results so obtained will then be compared with those obtained
using the Cowling approximation to assess its accuracy. Many previous studies
have used the Cowling approximation McDermott  _et al._ (1983); Ranea-Sandoval
_et al._ (2018); Sotani  _et al._ (2011a); Flores and Lugones (2014), which
lacks dissipation due to gravitational waves. The Cowling approximation
introduces about a 20-30% error in the $f$-mode frequency Yoshida and Kojima
(1997); Sotani  _et al._ (2001); Chirenti  _et al._ (2015a); Zhao and Lattimer
(2022), which is significantly less accurate than that of the
$\Omega_{f}-\bar{I}-\Lambda$ universal relations. A 10% error in the
compositional $g$-mode frequency Zhao  _et al._ (2022) and 19% error in the
discontinuity $g$-mode frequency Sotani  _et al._ (2001); Zhao and Lattimer
(2022) from the Cowling approximation has also been found. However, the
accuracy of the Cowling approximation for the and $p$-mode is not well
quantified. In addition such comparisons have only been made for schematic
EoSs such as piece-wise polytropes. Hence, another of our objective is to use
physics-based EoSs that reproduce laboratory as well as observational data.

Two classes of EoSs are chosen for our study. The first one is based on non-
relativistic zero-range Skyrme interactions using which calculations of the
EoSs are performed in the Hartree-Fock approach; these models are labelled
SHF. Care is taken to render these EoSs causal, at least for densities within
the cores of the NSs. The second class of EoSs are relativistic mean-field
models (RMF) that are naturally causal. A total of 35 EoSs are employed all of
which reproduce nuclear systematics at near saturation density of
$\rho_{0}\simeq 0.16\pm 0.01~{}{\rm fm}^{-3}$ and are able to to support a NS
of $\geq 2M_{\odot}$. We also compare these EoSs with recent chiral effective
field theoretical (EFT) calculations Drischler  _et al._ (2021) for densities
up to $2\rho_{0}$, the limiting density up to which such calculations are
valid. Many of the SHF and RMF EoSs fall within the $\pm 2\sigma$ region of
the chiral EFT results.

An important further objective of our work is to explore possible universal
relations connecting the $f$\- and $p$\- mode frequencies with both nuclear
and observational properties, such as the NSs mass, tidal deformability, etc.,
as well as EOS characteristics such as the slope of symmetry energy at
$\rho_{0}$ and pressure at $\rho_{0}$ and $2\rho_{0}$. In all of these cases,
we find interesting correlations and proffer the underlying cause for such
behaviors.

This paper is organized as follows. Section II contains a description of non-
radial oscillations in GR and the Cowling approximation to full GR. The EoSs
used in this work are described in Sec. III. Our results and discussion are
contained in Sec. IV. A critical analysis of correlations including the
dependence on the sample sizes of EoSs and findings in earlier works is
provided in Sec. V. Finally, Sec. VI presents a summary along with our
conclusions.

##  II Non-radial oscillations in general relativity

NS oscillations that couple to gravitational radiation were first studied by
Thorne et al. Thorne and Campolattaro (1967). Such oscillations involve linear
scalar variations of pressure and density. Only even-parity perturbations of
the Regge-Wheeler metric are relevant. The appropriate line element is thus

$\begin{split}ds^{2}=-e^{\nu(r)}[1+r^{l}H_{0}(r)e^{i\omega
t}Y_{lm}(\phi,\theta)]c^{2}dt^{2}\\\ +e^{\lambda(r)}[1-r^{l}H_{0}(r)e^{i\omega
t}Y_{lm}(\phi,\theta)]dr^{2}\\\ +[1-r^{l}K(r)e^{i\omega
t}Y_{lm}(\phi,\theta)]r^{2}d\Omega^{2}\\\ -2i\omega r^{l+1}H_{1}(r)e^{i\omega
t}Y_{lm}(\phi,\theta)dt~{}dr\,,\end{split}$ (1)

where

$e^{\lambda(r)}=\frac{1}{1-2b(r)}$ (2)

and

$\displaystyle r\frac{d\nu(r)}{dr}$ $\displaystyle=$ $\displaystyle
2e^{\lambda(r)}\textrm{Q}(r)$ (3) $\displaystyle\textrm{Q}(r)$
$\displaystyle=$ $\displaystyle b(r)+\frac{4\pi Gr^{2}p(r)}{c^{4}}$ (4)

with the boundary condition $\nu(R)=-\lambda(R)$. Above,
$b(r)=2Gm(r)/(c^{2}r)$, $m(r)$ and $p(r)$ are the enclosed mass and pressure
at radius $r$, respectively. The angular part is characterized by the angular
quantum number $l$ and azimuthal quantum number $m$ in the spherical harmonics
$Y_{lm}$; $m$ is degenerate for the nonrotating NSs we calculate here.
Perturbations of the metric are described by the functions $H_{0}$, $H_{1}$,
and $K$. The real part of the complex eigenvalue $\omega$ is the oscillation
frequency, whereas the imaginary component is the inverse of the damping time
(positive) or instability time (negative).

The Lagrangian displacement vector

$\displaystyle\xi^{r}$ $\displaystyle=$ $\displaystyle
r^{l-1}e^{-\frac{\lambda}{2}}WY^{l}_{m}e^{i\omega t}$ (5)
$\displaystyle\xi^{\theta}$ $\displaystyle=$
$\displaystyle-r^{l-2}V\partial_{\theta}Y_{m}^{l}e^{i\omega t}$ (6)
$\displaystyle\xi^{\phi}$ $\displaystyle=$
$\displaystyle-\frac{r^{l-2}}{\sin^{2}\theta}V\partial_{\phi}Y_{m}^{l}e^{i\omega
t}\,,$ (7)

describes fluid perturbations inside the star with amplitudes $W$ and $V$ both
of which have the dimension $[R]^{2-l}$, where $R$ is the radius of NS. In
addition, an auxiliary function $X$, related to Lagrangian pressure
variations, is defined as

$\Delta p=-r^{l}e^{-\frac{\nu}{2}}XY_{m}^{l}e^{i\omega t}\,.$ (8)

In some range of frequency, the ODEs governing NS oscillation can exhibit a
singularity. In order to avoid such a singularity, Lindblom et al. Lindblom
and Detweiler (1983a); Detweiler and Lindblom (1985) choose the four degrees
of freedom to be $H_{1}$, $K$, $W$, and $X$. Evaluating the two remaining
functions $H_{0}$ and $V$ in terms of them yields

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle\left\\{8\pi
r^{2}e^{-\nu/2}X-\left[(n_{l}+1)\textrm{Q}-\omega^{2}r^{2}e^{-(\nu+\lambda)}\right]H_{1}\right.$
$\displaystyle+$
$\displaystyle\left.\left[n_{l}-\omega^{2}r^{2}e^{-\nu}-\textrm{Q}(e^{\lambda}\textrm{Q}-1)\right]K\right\\}(2b+n_{l}+\textrm{Q})^{-1},$
$\displaystyle V$ $\displaystyle=$
$\displaystyle\left[\frac{X}{\varepsilon+p}-\frac{\textrm{Q}}{r^{2}}e^{(\nu+\lambda)/2}W-e^{\nu/2}\frac{H_{0}}{2}\right]\frac{e^{\nu/2}}{\omega^{2}}\,,$
(10)

where $n_{l}=(l-1)(l+2)/2$ and $\varepsilon$ is the local energy density. By
expanding Einstein’s equation to first-order, the homogeneous linear
differential equations for $H_{1}$, $K$, $W$ and $X$ are Detweiler and
Lindblom (1985),

$\displaystyle r\frac{dH_{1}}{dr}$ $\displaystyle=$
$\displaystyle-[l+1+2be^{\lambda}+4\pi r^{2}e^{\lambda}(p-\varepsilon)]H_{1}$
(11) $\displaystyle+e^{\lambda}[H_{0}+K-16\pi(\varepsilon+p)V]\,,$
$\displaystyle r\frac{dK}{dr}$ $\displaystyle=$ $\displaystyle
H_{0}+(n_{l}+1)H_{1}$ (12)
$\displaystyle+[e^{\lambda}\textrm{Q}-l-1]K-8\pi(\varepsilon+p)e^{\lambda/2}W\,,$
$\displaystyle r\frac{dW}{dr}$ $\displaystyle=$
$\displaystyle-(l+1)[W+le^{\frac{\lambda}{2}}V]$ (13)
$\displaystyle+r^{2}e^{\lambda/2}\left[\frac{e^{-\nu/2}X}{(\varepsilon+p)c_{\rm
ad}^{2}}+\frac{H_{0}}{2}+K\right.,$ $\displaystyle r\frac{dX}{dr}$
$\displaystyle=$
$\displaystyle\left.-lX+\frac{(\varepsilon+p)e^{\nu/2}}{2}\right.$ (14)
$\displaystyle\left\\{\left.(1-e^{\lambda}\textrm{Q})H_{0}+(r^{2}\omega^{2}e^{-\nu}+n_{l}+1)H_{1}\right.\right.$
$\displaystyle+(3e^{\lambda}\textrm{Q}-1)K-\frac{4(n_{l}+1)e^{\lambda}\textrm{Q}}{r^{2}}V-2\left[\omega^{2}e^{\lambda/2-\nu}\right.$
$\displaystyle+4\pi(\varepsilon+p)e^{\lambda/2}\left.\left.-r^{2}\frac{d}{dr}\left(\frac{e^{{\lambda/2}}\textrm{Q}}{r^{3}}\right)\right]W\right\\}\,,$

where $c_{\rm ad}^{2}$ is the adiabatic sound speed of NS matter under
oscillations. In this work, we approximate this speed of sound with the
equilibrium sound speed $c_{\rm eq}^{2}=dp/d\varepsilon$. Generally, $c_{\rm
ad}^{2}$ is usually slightly larger than the equilibrium sound speed $c_{\rm
eq}^{2}$ due to lag in weak equilibrium Wei  _et al._ (2020); Jaikumar  _et
al._ (2021), and temperature equilibrium Reisenegger and Goldreich (1992);
Kuan  _et al._ (2022).

Perturbations at the center of the star $r=0$ are subject to the boundary
conditions

$\displaystyle W(0)$ $\displaystyle=$ $\displaystyle 1$ (15) $\displaystyle
X(0)$ $\displaystyle=$ $\displaystyle(\varepsilon_{0}+p_{0})e^{\nu_{0}/2}$
$\displaystyle\left\\{\left[\frac{4\pi}{3}(\varepsilon_{0}+3p_{0})-\frac{\omega^{2}}{l}e^{-\nu_{0}}\right]W(0)+\frac{K(0)}{2}\right\\}$
$\displaystyle H_{1}(0)$ $\displaystyle=$
$\displaystyle\frac{lK(0)+8\pi(\varepsilon_{0}+p_{0})W(0)}{n_{l}+1}$ (17)
$\displaystyle X(R)$ $\displaystyle=$ $\displaystyle 0\,.$ (18)

The last boundary condition above is obtained by first solving the two trial
solutions with $K(0)=\pm(\varepsilon_{0}+p_{0})$ and then constructing a
linear combination to obtain the correct solution that satisfies the boundary
condition $X(r=R)=0$. The latter boundary condition corresponds to no pressure
variations at the surface. By construction, $H_{0}(0)=K(0)$.

Exterior solutions

Outside the NS, fluid perturbations $W$, $V$ and $X$ vanish and the metric
perturbations follow Zerilli’s differential equation

$\displaystyle{d^{2}Z}/{dr^{*2}}$ $\displaystyle=$
$\displaystyle(V_{Z}(r)-\omega^{2})Z\,.$ (19)

The function $Z$ is defined through Fackerell (1971)

$\displaystyle\begin{pmatrix}K(r)\\\ H_{1}(r)\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}g(r)&1\\\
h(r)&k(r)\end{pmatrix}\begin{pmatrix}{Z(r^{*})/r}\\\
{dZ(r^{*})/dr^{*}}\end{pmatrix},$ $\displaystyle g(r)$ $\displaystyle=$
$\displaystyle\frac{n(n+1)+3n{b}+6{b}^{2}}{(n+3{b})},$ (20) $\displaystyle
h(r)$ $\displaystyle=$
$\displaystyle\frac{(n-3n{b}-3{b}^{2})}{(1-2{b})(n+3{b})},$ $\displaystyle
k(r)$ $\displaystyle\equiv$ $\displaystyle\frac{dr^{*}}{dr}=\frac{1}{1-2{b}},$

and an effective potential

$\displaystyle
V_{Z}(r)=(1-2{b})\frac{2n^{2}(n+1)+6n^{2}{b}+18n{b}^{2}+18{b}^{3}}{r^{2}(n+3{b})^{2}}\,,$
(21)

where $b={GM}/{(c^{2}r)}$ since $m(r>R)=M$. We first solve Eqs. (11-14) from
the stellar center to the surface. Then, we evaluate $Z(R)$ by the matching
condition Eq. (20) and use it as the initial boundary condition to solve Eq.
(19) from the stellar surface to $r\gtrsim 100$ km. In the far-field limit,
the solution $Z$ can be decomposed into incoming ($Z_{+}$) and outgoing
($Z_{-}$) spherical gravitational radiation:

$\displaystyle\begin{pmatrix}Z(\omega)\\\ {dZ}/{dr^{*}}\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}Z_{-}(\omega)\quad
Z_{+}(\omega)\\\
{dZ_{-}}/{dr^{*}}\quad{dZ_{+}}/{dr^{*}}\end{pmatrix}\begin{pmatrix}A_{-}(\omega)\\\
A_{+}(\omega)\end{pmatrix},$ $\displaystyle Z_{-}$ $\displaystyle=$
$\displaystyle e^{-i\omega
r^{*}}\left[\alpha_{0}+\frac{\alpha_{1}}{r}+\frac{\alpha_{2}}{r^{2}}+\mathcal{O}(r^{-3})\right],$
$\displaystyle\frac{dZ_{-}}{dr^{*}}$ $\displaystyle=$ $\displaystyle-i\omega
e^{-i\omega r^{*}}[\alpha_{0}+\frac{\alpha_{1}}{r}$
$\displaystyle+\frac{\alpha_{2}+{i\alpha_{1}}(1-2b)/\omega}{r^{2}}+\mathcal{O}(r^{-3})],$
$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\frac{-i(n+1)\alpha_{0}}{\omega},$ $\displaystyle\alpha_{2}$
$\displaystyle=$
$\displaystyle\frac{[-n(n+1)+iM\omega({3}/{2}+{3}/{n})]\alpha_{0}}{2\omega^{2}}\,,$

where $A_{+}(\omega)$ and $A_{-}(\omega)$ are the corresponding amplitudes,
and $Z_{+}$ is the complex conjugate of $Z_{-}$.

Numerical notes

To evaluate $A_{+}(\omega)$, we used the complex-valued ODE solver _zvode_
Brown  _et al._ (1989). The solver _zvode_ uses implicit Adams method for non-
stiff problems and a method based on backward differentiation formulas for
stiff problems. The integration steps are automatically adjusted to yield a
specified relative error tolerance. In order to take care of the imaginary
part of $\omega$, which is over 1000 times smaller than the real part, we need
to keep the relative error of our ODE solver to $10^{-6}$ for $H_{1}$, $K$,
$W$, $X$ and $Z$. The quantity $A_{-}(\omega)$ is evaluated by Eq. (II) at
$r=25\omega^{-1}$ and $r=50\omega^{-1}$ to guarantee convergence within
$10^{-6}$.

Up to this point, we have discussed the ODEs and boundary conditions for the
initial value problem of all perturbation functions with a given oscillation
frequency $\omega$. The eigenvalue problem is defined by the pure outgoing
gravitational wave boundary condition at far-field, $A_{+}(\omega)=0$. In
order to solve the eigenvalue problem, we use a complex root finding algorithm
which takes about 8 Newton–Raphson iterations to converge.

By solving the eigenvalue problem we can calculate even parity quasinormal
modes of any order ($n=0,1,$ etc.), and angular number ($\ell-2,3,$ etc.).
Radial perturbation functions can have different number of turnovers (nodes)
$n$ for different quasinormal modes. The $n=0$ modes that have no nodes are
referred to as the $f$-modes, whereas those with $n$=1,2, etc., have the
corresponding number of nodes and are termed as the $p$-modes. While we
calculate $f$-and $p$-modes of arbitrary order $n$, in this work we compute
the $\ell=2$ quasinormal $f$-mode ($n=0$ of $W$ and $V$) and the lowest order
$p$-mode ($n=1$ and $\ell=2$ of $W$ and $V$) as frequencies of higher order
$p$-modes are too large compared with the detection limits. Henceforth, we
will refer to the $p$-modes with $n=1$ as $p_{1}$ modes.

The non-radial modes calculated by our codes are in agreement to four
significant digits with the results given in the tables of the pioneering
works of Lindblom and Detweiler (1983); Miniutti  _et al._ (2003); Finn (1987)
for $f$\- and $p$-modes as well as the discontinuity $g$-mode Zhao and
Lattimer (2022) and the compositional $g$-mode Zhao  _et al._ (2022). Profiles
of the $f$\- and $p$-modes will be presented in the section on results.

### Relativistic Cowling approximation

In Newtonian theory of stellar pulsations, the perturbation of the gravity
field is neglected for fluid modes. This simplification is known as the
Cowling approximation Cowling (1941). In the relativistic theory, the
perturbation of the GR metric is often neglected as well, which leads to the
relativistic Cowling approximation McDermott  _et al._ (1983). The
relativistic Cowling equations are obtained by setting $H_{0}=H_{1}=K=0$ in
Eqs. (10, 13, 14) and furthermore, dropping the term
$-4\pi(\varepsilon+p)^{2}e^{(\nu+\lambda)/2}W$ in Eq. (14). These
simplifications lead to Zhao  _et al._ (2022),

$\displaystyle\frac{dW}{d\ln r}$ $\displaystyle=$
$\displaystyle-(l+1)\left[W-le^{\nu+\lambda/2}U\right]$ (23)
$\displaystyle-\frac{e^{\lambda/2}(\omega
r)^{2}}{c_{ad}^{2}}\left[U-\frac{e^{\lambda/2}\textrm{Q}}{(\omega
r)^{2}}W\right]\,,$ $\displaystyle\frac{dU}{d\ln r}$
$\displaystyle=e^{\lambda/2-\nu}\left[W-le^{\nu-\lambda/2}U\right]\,,$ (24)

where $W=e^{\lambda/2}r^{1-l}\xi^{r}$ and
$U=-e^{-\nu}V=r^{-l}\omega^{-2}\delta p/(\varepsilon+p)$, $\xi^{r}$ are radial
Lagrangian displacements defined in Eq. (5) and $\delta P$ is the Eulerian
perturbation of pressure, which is related to the the Lagrangian perturbation
by $\Delta P=\delta P-(\varepsilon+p)\frac{d\Phi}{dr}\xi^{r}$. The boundary
conditions can be written explicitly as,

$\displaystyle\left.\frac{W}{U}\right|_{r=0}$ $\displaystyle=$ $\displaystyle
le^{\nu|_{r=0}}$ (25) $\displaystyle\left.\frac{W}{U}\right|_{p=0}$
$\displaystyle=$
$\displaystyle\frac{\omega^{2}R^{3}}{GM}\sqrt{1-\frac{2GM}{c^{2}R}}\,.$ (26)

These equations determine the eigenmode frequency of the oscillation in the
relativistic Cowling approximation.

##  III Equations of state

We have considered a diverse set of EoSs for charge neutral and
$\beta$-equilibrated neutron star matter that are obtained using relativistic
and non-relativistic mean-field models. Two different variants of the
relativistic mean-field (RMF) models: (1) models with nonlinear self and/or
mixed interaction terms with constant coupling strengths, and (2) models with
only linear interaction terms, but density-dependent coupling strengths have
been employed. The RMF models with density-independent coupling considered are
BSR2, BSR6 Dhiman  _et al._ (2007); Chen and Piekarewicz (2014a), GM1
Glendenning and Moszkowski (1991), NL3 Lalazissis  _et al._ (1997),
NL3$\omega\rho 03$ Carriere  _et al._ (2003), IOPB-I Kumar  _et al._ (2018);
Parmar  _et al._ (2022), G3 Kumar  _et al._ (2017); Parmar  _et al._ (2022),
TM1 Sugahara and Toki (1994), and FSUG Chen and Piekarewicz (2014b). The RMF
models with density-dependent couplings are DD2 Typel  _et al._
(2010),DDH$\delta$ Gaitanos  _et al._ (2004), and DDME2 Lalazissis  _et al._
(2005). The non-relativistic mean-field models considered employ the Skyrme
Hartree-Fock (SHF) approach and they are SKa, SKb Köhler (1976), SkI2, SkI3,
SkI4, SkI5 Reinhard and Flocard (1995), SkI6 Nazarewicz  _et al._ (1996),
SLY2, SLY9 Chabanat , SLY230a Chabanat  _et al._ (1997) SLY4 Chabanat  _et
al._ (1998), SkMP Bennour  _et al._ (1989), SKOp Reinhard (1999), KDE0V1
Agrawal  _et al._ (2005), SK255, SK272 Agrawal  _et al._ (2003), Rs Friedrich
and Reinhard (1986), BSk20, BSk21 Goriely  _et al._ (2010), BSk22, BSk23,
BSk24, and BSk25 Goriely  _et al._ (2013).

The EoS for the outer crust region is taken to be the one obtained by Baym-
Pethick-Sutherland Baym  _et al._ (1971). The EoSs for the inner crust in the
case of RMF models are determined within a Thomas-Fermi approach up to the
crust-core transition density. In the case of SHF models, the EoSs for the
inner crust are constructed with the compressible liquid-drop model Gulminelli
and Raduta (2015); Fortin  _et al._ (2016). These unified EoSs models closely
reproduce the properties of finite nuclei, nuclear matter and neutron stars
(Kumar  _et al._ , 2017, 2018; Fortin  _et al._ , 2016). Moreover, these EoSs
are causal Malik  _et al._ (2018) and support a $2.0\,M_{\odot}$ star
(Antoniadis  _et al._ , 2013). All of the employed unified EoSs are for
$npe\mu$ matter, and apply from the neutron star crust to its core.

The SHF and RMF EoSs used are displayed in terms of their pressure-energy
density relations in Fig. 1. The inset in this figure contrasts the SHF and
RMF EOSs with the chiral EFT results of Ref. Drischler  _et al._ (2021) up to
$2\rho_{0}$. Note that many SHF EOSs lie within the $2\sigma$ error estimates
of the chiral EFT results, several RMF EoSs fall outside these limits. Such
EoSs are nevertheless included in our analyses for the sake of comparison and
completeness.

The behaviour of EoS is often understood in terms of various nuclear matter
parameters for a given EoS. The lower order nuclear matter parameters, govern
the behaviour of EoS at lower densities, are strongly correlated with several
bulk properties of finite nuclei. The higher order nuclear matter parameters
are expected to be correlated with the bulk properties of NS.These parameters
are evaluated as follows. For densities up to $2\rho_{0}$ \- $3\rho_{0}$ and
for neutron-proton asymmetry
$\delta=\frac{\rho_{n}-\rho_{p}}{\rho},(\rho=\rho_{n}+\rho_{p}))$, where
$\rho_{n}$ and $\rho_{p}$ are the neutron and proton densities, respectively,
the energy per nucleon as a function of density $\rho$ and asymmetry $\delta$
can be expressed as,

$\displaystyle e(\rho,\delta)$ $\displaystyle=$ $\displaystyle e_{\rm
snm}(\rho)+e_{\rm sym}(\rho)\delta^{2}+\mathcal{O}(\delta^{4})\,,$ (27)
$\displaystyle e_{\rm snm}(\rho)$ $\displaystyle=$ $\displaystyle
e_{0}+\frac{K_{0}}{2}x^{2}+\frac{Q_{0}}{6}x^{3}+\mathcal{O}(x^{4})$ (28)
$\displaystyle e_{\rm sym}(\rho)$ $\displaystyle=$ $\displaystyle
J_{0}+L_{0}x+\frac{K_{\rm sym,0}}{2}x^{2}+\mathcal{O}(x^{3}),\,$ (29)

where $x=\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)$. The binding energy
$e_{0}$, incompressibility $K_{0}$, and the skewness coefficient $Q_{0}$ can
be determined from the energy per nucleon for the symmetric nuclear matter
$e_{\rm snm}(\rho)$. Likewise, the symmetry energy coefficient $J_{0}$, its
slope $L_{0}$ and the curvature $K_{\rm sym,0}$ of the symmetry energy can be
determined from $e_{\rm sym}(\rho)$. All the parameters appearing in the right
hand side of Eqs. (28-29) are evaluated at the saturation density $\rho_{0}$.
Once a functional form for $e(\rho,\delta)$ is known, the EoS for the
$\beta$-equilibrated matter for near-nuclear densities can be obtained by
adding the ideal gas contributions from electrons and muons so that the
conditions of chemical equilibrium and charge neutrality are satisfied.

The parametrization in Eqs. (27)-(29) is valid only for densities such that
$x<<1$. These equations are employed here only to define the nuclear matter
parameters. In constructing EoSs relevant for NS structure, the full equations
from the various SHF and RMF models considered here have been used.

##  IV Results and Discussion

The values of the EoS parameters at saturation density show a diverse
behaviour across the models Fortin  _et al._ (2016). In the present study, we
shall investigate the correlations between the NS observables and saturation
properties of matter expressed via parameters such as $K_{0}$, $Q_{0}$,
$L_{0}$ and $K_{\rm sym,0}$. We shall also consider the correlations of NS
properties with the presure of $\beta$-equilibrated matter at densities in the
range $1.0$ \- $2.5\rho_{0}$.

Figure 1:  Pressure vs energy density for a representative set of EoSs
corrosponding to the SHF ( red solid ) and RMF (blue dashed ) models
considered in the present work. The orange region in the inset shows results
for the Chiral EFT $\pm 1\sigma$ and $\pm 2\sigma$ bounds up to twice
saturation density Drischler  _et al._ (2021). Figure 2:  The NS mass-radius
relationships obtained for the EoSs for SHF and RMF models as shown in Fig. 1.
The RMF models give higher masses than SHF models. The lower boxes at slightly
different central masses show the constraints on mass and radius from NICER
(2019) data of PSR J0030+0451Miller  _et al._ (2019); Riley  _et al._ (2019).
The upper horizontal bands at the same central mass (bands shifted for
clarity) show the NICER data with X-ray Multi-Mirror (NICER XMM-Newton 2021)
observations of PSR J0740+6620Fonseca  _et al._ (2021); Miller  _et al._
(2021). The Orange region shows results for the Chiral EFT $\pm 1\sigma$ and
$\pm 2\sigma$ bounds up to twice saturation density Drischler  _et al._
(2021).

###  IV.1 Masses and radii of neutron stars

Figure 2 shows mass-radius curves obtained by solving the TOV equations in
general relativity Tolman (1939); Oppenheimer and Volkoff (1939) with the
various EoSs from SHF and RMF models used as input data. Results for the SHF
models are shown by red lines, whereas, those from the RMF models are in blue
dashed lines. The mass of a NS increases with increasing central density up to
the maximum mass. The ‘x’s marked on the curves indicate the maximum masses of
the various EoSs shown in Fig. 1. The orange region shows results of the
chiral EFT calculations up to the maximum mass such EoSs can support. Note
that without any extrapolation of the chiral EFT EoS beyond $2\rho_{0}$, the
maximum mass falls well below even the canonical value of $1.4M_{\odot}$.

In Fig. 2, we have also shown the observational data and confirmed that all
the SHF and RMF EoSs considered in this work could predict stars $\geq
2M_{\odot}$. The lower boxes show the constraint on the mass and radius from
the Neutron star Interior Composition Explorer (NICER) data of PSR J0030+0451.
From the data analysis of this NS, Miller et al. reported that the mass-radius
of the pulsar therein as $M=1.44^{+0.15}_{-0.14}M_{\odot}$ and
$R=13.02^{+1.24}_{-1.06}$ km Miller  _et al._ (2019), whereas Riley et al.
gave it as $M=1.34^{+0.15}_{-0.16}M_{\odot}$ and $R=12.71^{+1.14}_{-1.19}$ km
Riley  _et al._ (2019). Cromartie et al. Cromartie  _et al._ (2020) and
Antoniadis et al. Antoniadis  _et al._ (2013) used radio observations from PSR
J0740+6620 to determine the NS mass as $M=2.14^{+0.1}_{-0.09}$ $M_{\odot}$ and
$M=2.01\pm 0.04$ $M_{\odot}$, respectively. The model-averaged estimate of the
mass of this pulsar by Fonseca et al. Fonseca  _et al._ (2021) gives the lower
limit for the maximum mass of the NS as $2.08^{+0.07}_{-0.07}M_{\odot}$. With
NICER and X-ray Multi-Mirror (XMM-Newton) observations of the same pulsar,
Miller et al. and Riley et al. infer the radius as $R=13.7^{+2.6}_{-1.5}$ km
Miller  _et al._ (2021) and $R=12.39^{+1.30}_{-0.98}$ km Riley  _et al._
(2021), respectively.

###  IV.2 $f$-mode and tidal deformabilities of a neutron star

Figure 3: The Cowling $f$-mode frequency vs the tidal deformability parameter
for the NSs obtained by varying the central density. The crosses on the curves
indicate maximum mass configurations with the unstable branches extending
toward the left.

The historical event of the binary neutron star merger (BNS), GW170817,
started a new era in multi-messenger astronomy. This event gave important
information about the tidal deformability of a NS. The later stage of the
inspiring phase of the NS-NS merger creates a strong gravitational field,
which deforms a NS’s multipolar structure. This is quantified by the
dimensionless tidal deformability parameter,

$\Lambda=\frac{2}{3}k_{2}C^{-5}\,,$ (30)

where, $k_{2}$, is the tidal Love number Hinderer (2009); Hinderer  _et al._
(2010). As $k_{2}\propto R/M$, $\Lambda$ varies as the sixth power of $R$
Malik  _et al._ (2018); Zhao and Lattimer (2018). The tidal deformability of
canonical neutron star $\Lambda_{1.4}=190^{+390}_{-120}$ is extracted from the
combined tidal deformability of binary neutron stars Abbott  _et al._ (2020).
It is interesting to know how the tidal deformability helps constrain the
$f$-mode oscillations, thereby inferring the properties and underlying EoS
through an inverse approach. Figure 3 shows the dimensionless tidal
deformability $\Lambda$ as a function of $f$-mode frequency obtained by
varying the central density of the star. The later quantity is obtained within
the Cowling approximation. Our results demonstrate that the tidal
deformability has a strong dependence on $f$-mode frequency. Both the
quantities $f$-mode and $\Lambda$ are anti-correlated to each other.

In Newtonian physics, the $f$-mode frequency is related to the stellar average
density $M/R^{3}$ Andersson and Kokkotas (1998b); CHIRENTI  _et al._ (2012).
In Fig. 4, normalized eigenvalues with average density
$\omega(R^{3}/M)^{1/2}$, as a function of NS mass for each SHF and RMF EoSs
are plotted. We see that results for most of the EoSs roughly preserve the
scaling relation $1.1\lesssim\omega(R^{3}/M)^{1/2}\lesssim 1.6$ for
$M>1M_{\odot}$, which is comparable to results for the Tolman VII (1.155) and
Buchdal (1.237) EOSs, and larger than the incompressible EOS (0.894) Zhao and
Lattimer (2022). Furthermore, it is interesting to notice that the normalized
eigenvalues with the average density of $f$-modes are independent of the
adopted EoSs. Therefore, it is not easy to distinguish the EoSs with the help
of observation of stellar properties Sotani  _et al._ (2011b).

Figure 4: Normalised Cowling $f$-mode frequency vs NS mass.

Table 1 summarizes results of the essential NS properties: canonical and
maximum masses (in $M_{\odot}$), their corresponding radii R (in km), the
dimensionless tidal deformability $\Lambda$, compactness $C$, $f$-mode
frequency (in kHz) and its damping time for QNMs.

Table 1: Values of neutron star properties: maximum mass ($M_{\odot}$), radius
at the maximum mass $R_{\rm{max}}$ (km) and radius at the canonical mass
$R_{1.4}$ (km), compactness parameter $C_{1.4}$ and the dimensionless tidal
deformability for canonical mass, $\Lambda_{1.4}$. The values of the $f$-mode
and $p_{1}$-mode frequencies (kHz) are obtained by using the Cowling
approximation and a full general relativistic treatment for the EoSs
considered in this work. The values of damping time $\tau_{1.4}$ (s) are also
listed.

|  |  |  |  |  | Cowling appr. | General Relativity
---|---|---|---|---|---|---|---
EoS | $M_{\rm{max}}$ | $R_{\rm{max}}$ | $R_{1.4}$ | $C_{1.4}$ | $\Lambda_{1.4}$ | $f_{\rm{max}}$ | $f_{1.4}$ | $\nu_{1.4}^{p_{1}}$ | $f_{max}$ | $f_{1.4}$ | $\tau_{1.4}^{f}$ | $\nu_{1.4}^{p_{1}}$ | $\tau_{1.4}^{p_{1}}$
SKa | 2.208 | 10.853 | 12.906 | 0.160 | 568.694 | 2.670 | 2.114 | 5.867 | 2.382 | 1.684 | 0.251 | 5.391 | 4.666
SKb | 2.188 | 10.621 | 12.198 | 0.169 | 474.683 | 2.702 | 2.212 | 6.551 | 2.415 | 1.752 | 0.232 | 6.006 | 5.626
SkI2 | 2.163 | 11.047 | 13.474 | 0.153 | 775.670 | 2.639 | 1.976 | 5.449 | 2.352 | 1.572 | 0.289 | 4.964 | 4.401
SkI3 | 2.240 | 11.241 | 13.545 | 0.153 | 789.865 | 2.586 | 1.979 | 5.613 | 2.302 | 1.565 | 0.291 | 5.140 | 5.354
SkI4 | 2.169 | 10.639 | 12.363 | 0.167 | 472.443 | 2.713 | 2.212 | 6.503 | 2.415 | 1.754 | 0.232 | 5.973 | 5.514
SkI5 | 2.240 | 11.499 | 14.074 | 0.147 | 1000.071 | 2.537 | 1.868 | 5.184 | 2.282 | 1.478 | 0.328 | 4.718 | 4.909
SkI6 | 2.190 | 10.717 | 12.481 | 0.166 | 490.162 | 2.694 | 2.192 | 6.443 | 2.397 | 1.737 | 0.236 | 5.923 | 5.657
SLY2 | 2.053 | 10.082 | 11.773 | 0.176 | 310.385 | 2.877 | 2.388 | 6.783 | 2.578 | 1.916 | 0.195 | 6.243 | 4.245
SLY230a | 2.099 | 10.209 | 11.822 | 0.175 | 332.692 | 2.831 | 2.371 | 6.960 | 2.513 | 1.892 | 0.200 | 6.418 | 5.187
SLY4 | 2.050 | 10.050 | 11.694 | 0.177 | 299.732 | 2.884 | 2.406 | 6.861 | 2.591 | 1.931 | 0.192 | 6.314 | 4.273
SLY9 | 2.156 | 10.630 | 12.457 | 0.166 | 453.543 | 2.728 | 2.223 | 6.351 | 2.436 | 1.770 | 0.227 | 5.852 | 4.984
SkMP | 2.107 | 10.522 | 12.487 | 0.166 | 482.124 | 2.758 | 2.183 | 6.051 | 2.467 | 1.747 | 0.234 | 5.536 | 4.068
SKOp | 1.973 | 10.152 | 12.116 | 0.171 | 363.930 | 2.884 | 2.294 | 6.163 | 2.587 | 1.855 | 0.208 | 5.633 | 3.160
KDE0v1 | 1.969 | 9.866 | 11.615 | 0.178 | 266.465 | 2.961 | 2.444 | 6.712 | 2.664 | 1.978 | 0.184 | 6.169 | 3.406
SK255 | 2.144 | 10.774 | 13.135 | 0.157 | 589.750 | 2.710 | 2.082 | 5.519 | 2.413 | 1.670 | 0.256 | 5.059 | 3.787
SK272 | 2.231 | 11.006 | 13.304 | 0.155 | 647.483 | 2.641 | 2.052 | 5.550 | 2.351 | 1.637 | 0.266 | 5.103 | 4.480
Rs | 2.116 | 10.796 | 12.921 | 0.160 | 594.582 | 2.698 | 2.084 | 5.697 | 2.424 | 1.667 | 0.257 | 5.195 | 3.942
BSk20 | 2.165 | 10.288 | 11.719 | 0.176 | 322.916 | 2.790 | 2.389 | 7.081 | 2.506 | 1.904 | 0.198 | 6.536 | 5.560
BSk21 | 2.278 | 11.080 | 12.539 | 0.165 | 520.829 | 2.599 | 2.189 | 6.787 | 2.314 | 1.719 | 0.241 | 6.282 | 8.148
BSk22 | 2.265 | 11.175 | 13.015 | 0.159 | 633.050 | 2.589 | 2.090 | 6.191 | 2.307 | 1.646 | 0.263 | 5.711 | 6.638
BSk23 | 2.271 | 11.136 | 12.799 | 0.162 | 576.473 | 2.593 | 2.135 | 6.459 | 2.307 | 1.678 | 0.253 | 5.968 | 7.295
BSk24 | 2.279 | 11.086 | 12.547 | 0.165 | 522.583 | 2.598 | 2.188 | 6.784 | 2.311 | 1.717 | 0.241 | 6.280 | 8.173
BSk25 | 2.224 | 11.072 | 12.344 | 0.167 | 485.592 | 2.609 | 2.228 | 7.135 | 2.314 | 1.744 | 0.234 | 6.613 | 9.451
BSR2 | 2.383 | 11.728 | 12.796 | 0.162 | 743.777 | 2.435 | 2.025 | 6.117 | 2.142 | 1.583 | 0.285 | 5.676 | 8.671
BSR6 | 2.430 | 11.946 | 13.229 | 0.156 | 825.331 | 2.401 | 1.966 | 5.425 | 2.117 | 1.551 | 0.297 | 5.027 | 6.546
GM1 | 2.361 | 11.884 | 13.680 | 0.151 | 909.926 | 2.435 | 1.931 | 5.618 | 2.159 | 1.516 | 0.311 | 5.158 | 6.727
NL3 | 2.774 | 13.245 | 14.601 | 0.142 | 1284.080 | 2.162 | 1.793 | 5.095 | 1.909 | 1.400 | 0.368 | 4.699 | 8.311
NL3$\omega\rho$03 | 2.753 | 12.994 | 13.722 | 0.151 | 948.459 | 2.193 | 1.953 | 6.167 | 1.935 | 1.504 | 0.316 | 5.833 | 16.933
TM1 | 2.179 | 12.384 | 14.249 | 0.145 | 1059.437 | 2.368 | 1.855 | 5.139 | 2.086 | 1.465 | 0.334 | 4.705 | 5.573
DD2 | 2.418 | 11.884 | 13.151 | 0.157 | 695.868 | 2.426 | 2.062 | 6.183 | 2.148 | 1.612 | 0.274 | 5.779 | 9.386
DDH$\delta$ | 2.138 | 11.113 | 12.591 | 0.164 | 589.032 | 2.606 | 2.135 | 6.714 | 2.299 | 1.671 | 0.255 | 6.165 | 7.855
DDME2 | 2.483 | 12.053 | 13.190 | 0.157 | 715.231 | 2.385 | 2.058 | 6.293 | 2.106 | 1.601 | 0.278 | 5.923 | 11.674
IOPB-I | 2.149 | 11.891 | 13.288 | 0.156 | 694.552 | 2.463 | 2.048 | 5.849 | 2.139 | 1.612 | 0.274 | 5.455 | 7.185
G3 | 1.997 | 10.986 | 12.586 | 0.164 | 465.918 | 2.679 | 2.208 | 6.178 | 2.369 | 1.760 | 0.230 | 5.724 | 4.817
FSUG | 2.066 | 11.830 | 13.155 | 0.157 | 634.019 | 2.485 | 2.097 | 6.147 | 2.175 | 1.645 | 0.263 | 5.787 | 8.172

###  IV.3 Pearson’s correlation coefficients

Linear correlations between any given pair of quantities is measured by
Pearson’s correlation coefficient defined as Benesty  _et al._ (2009)

$r(a,b)=\frac{\sigma_{ab}}{\sqrt{\sigma_{aa}\sigma_{bb}}}\,\\\ $ (31)

with the covariance, $\sigma_{ab}$, given by

$\sigma_{ab}=\frac{1}{N_{m}}\sum_{i}a_{i}b_{i}-\left(\frac{1}{N_{m}}\sum_{i}a_{i}\right)\left(\frac{1}{N_{m}}\sum_{i}b_{i}\right)\,,$
(32)

where the index $i$ runs over the number of models $N_{m}$Brandt . In what
follows, $a_{i}$ and $b_{i}$ correspond to the NS properties for a fixed mass
obtained for the different models. A correlation coefficient close to unity in
absolute value indicates a strong linear relation between the pair of
quantities that are considered.

(a)Cowling approximation

(a)

(b) Full GR

(b)

Figure 5:  Heat map depicting the correlation of nuclear matter parameters to
neutron star properties. The $f$\- and $p_{1}$-mode frequencies are obtained
within (a) the Cowling approximation and (b) full GR.

We present the correlation matrices as heat maps in Figs. 5(a) and 5(b) for NS
properties, nuclear matter properties, and pressure for beta-equilibrated
matter. The NS properties considered are the $f$\- and $p_{1}$-mode
frequencies, radii, and tidal deformability at fixed masses. For comparison,
the $f$\- and $p_{1}$-mode frequencies obtained with the Cowling approximation
and full GR are also shown. The inferences for Cowling approximation as shown
in Fig. 5(a) are as follows:

  1. 1.

The $p-$ mode frequency for a NS with $1.4M_{\odot}$ is strongly correlated
with $L_{0}$. The radius, tidal deformality and $f-$ mode frequency for a NS
with $1.4M_{\odot}$ are moderately correlated with $L_{0}$ and $K_{\rm sym,0}$
(Malik  _et al._ , 2018, 2020). consideration.

  2. 2.

The beta-equilibrated matter pressure corresponding to two times the
saturation density has a substantial correlation with $f_{1.4}$, whereas the
correlation becomes marginally stronger at higher masses; and

  3. 3.

The $f$-mode frequency also strongly correlates with the corresponding tidal
deformability and radius Pradhan  _et al._ (2022a). However, the $p_{1}$-mode
frequency is only moderately correlated with tidal deformability, but displays
a strong correlation with the slope $L_{0}$ of symmetry energy curve.

A similar analysis is made for $f$\- and $p_{1}$-mode values obtained by the
full GR approach as shown in Fig. 5(b). The damping time that appears only in
full GR is correlated with various quantities associated with the EoS in the
following manner:

  1. 1.

The damping time $\tau_{1.4}^{f}$ is strongly correlated with beta-
equilibrated matter pressure at $2\rho_{0}$, $R_{1.4}$, and $\Lambda_{1.4}$;
and

  2. 2.

The damping time $\tau_{1.4}^{p_{1}}$ is moderately correlated with beta-
equilibrated matter pressure at $2.5\rho_{0}$, and $Q_{0}$.

###  IV.4 Correlations involving pressure

Figure 6: Pressure at twice the saturation density plotted against $f$\- mode
frequency in Cowling approximation, tidal deformability and radius calculated
for 1.4$M_{\odot}$ neutron star using SHF and RMF models. The thick solid grey
line gives the best fit. Values of Pearson’s coefficient $r$ are as indicated.

The analysis of GW170817 has provided pivotal information about the tidal
deformability and pressure as a function of density Abbott (2018). The
behaviour of the pressure at twice nuclear saturation density is measured at
$21.84^{+16.85}_{-10.61}$ MeV fm-3 at the $90\%$ confidence limit Abbott
(2018). In Fig. 6 we plot the $f_{1.4}$, $\Lambda_{1.4}$, and $R_{1.4}$ for NS
with $1.4M_{\odot}$ mass versus pressure at $2\rho_{0}$ for the sets of SHF
(red stars) and RMF (blue stars) models considered. The thick solid grey line
in the figure is obtained by linear regression and the correlation
coefficients are displayed for each case considered. The expressions obtained
by fitting the variation of pressure $P(2\rho_{0})$ with NS properties using
linear regression are:

$\frac{P(2\rho_{0})}{\text{MeV
fm}^{-3}}=83.24-27.35\,\,\frac{f_{1.4}}{\text{kHz}}\\\ $ (33)

$\frac{P(2\rho_{0})}{\text{MeV fm}^{-3}}=12.87+0.02\,\,\Lambda_{1.4}\\\ $ (34)

$\frac{P(2\rho_{0})}{\text{MeV
fm}^{-3}}=-52.27+6.02\,.\frac{R_{1.4}}{\text{km}}\\\ $ (35)

Figure 7:  Same as Fig 6, but , for the $f$\- mode frequency and damping time
within full GR. Values of Pearson’s coefficient $r$ are as indicated.

From the heat map Fig. 5(a), we noticed that the correlations between the
$f$-mode and pressure $P(\rho_{0})$ of beta-equilbrated matter decrease with
the increasing neutron star mass and are not significantly strong to make a
meaningful statement. However, the correlations of $f$-mode frequencies with
$P(2\rho_{0})$ for beta-equilibrated matter are quite strong (r = -0.86 to
-0.90). The correlations are marginally higher for larger NS mass.

Figure 7 shows the frequency and damping time of $1.4M_{\odot}$ star, which is
obtained with full GR calculations done for all EoSs considered in this work.
We note that the correlation between $P(2\rho_{0})$ and $f_{1.4}$ is almost
identical as in the case of the Cowling approximation. Thus, we look into the
correlations of the damping time (imaginary part of the eigenvalue) with the
pressure $P(2\rho_{0})$ of beta-equilibrated matter. The behaviour is
qualitatively similar to that for $\Lambda_{1.4}$ as shown in Fig. 6
exhibiting a strong correlations between $P(2\rho_{0})$ and $\tau_{1.4}$. We
have obtained following expressions from linear-regressions:

$\frac{P(2\rho_{0})}{\text{MeV
fm}^{-3}}=79.10-32.09\,\,\frac{f_{1.4}^{GR}}{\text{kHz}}\\\ $ (36)
$\frac{P(2\rho_{0})}{\text{MeV
fm}^{-3}}=-2.41+106.93\,\,\frac{\tau_{1.4}^{f}}{\text{sec}}\,.\\\ $ (37)

Figure 8: The $p_{1}$-mode frequency $\nu_{1.4}^{p1}$ vs symmetry energy slope
parameter $L_{0}$ and pressure at saturation density obtained for NS mass
$1.4M_{\odot}$ within full GR. Values of Pearson’s coefficient $r$ are as
indicated.

Finally, in Fig. 8, we plot the frequency of $p_{1}$-mode versus pressure at
saturation density and slope of the symmetry energy $L_{0}$ for our
representative sets of SHF and RMF models. It is interesting to note that the
$\nu_{1.4}^{p_{1}}-L_{0}$ correlation is stronger than the $f_{1.4}-L_{0}$
correlation. Similar trends are observed for the cases of
$\nu_{1.4}^{p_{1}}-P(\rho_{0})$ and $f_{1.4}-P(\rho_{0})$ correlations (see
Figs. 5(a) and 5(b)). These correlations indicate that the $p_{1}$-mode
frequency is more sensitive to the isovector EoS parameters $L_{0}$ and
pressure at saturation density. The correlations involving $p_{1}$-mode
frequency take the form

$\frac{\nu_{1.4}^{p_{1}}}{\text{kHz}}=7.12-0.02\,\,\frac{L_{0}}{\text{MeV}}\\\
$ (38)

$\frac{\nu_{1.4}^{p_{1}}}{\text{kHz}}=7.43-0.58\,\,\frac{P(\rho_{0})}{\text{MeV
fm}^{-3}}\,.\\\ $ (39)

In summary, we notice from the various correlations that the information
containing the $f$\- and $p_{1}$-mode frequencies are complementary. These
modes probe different density regions of the NS. The $f$-mode frequency is
more sensitive to the region at $2\rho_{0}$, and the $p_{1}$-mode frequency is
more sensitive to the region of the NS in the vicinity of the outer core. The
reason for this is that the $p_{1}$-mode oscillation peaks stronger at the
stellar surface than the $f$-mode, see Fig. 13.

##  V Critical analysis of correlations

Correlations of $R_{1.4}$ and $\Lambda_{1.4}$ with $L_{0}$ and $K_{\rm sym,0}$
have also been reported in several earlier investigations (Pradhan  _et al._ ,
2022a; Carson  _et al._ , 2019; Tsang  _et al._ , 2020; Reed  _et al._ , 2021;
Essick  _et al._ , 2021; Biswas, 2021). The EoSs employed in these studies
range from non-parametric Essick  _et al._ (2021), parametric Tsang  _et al._
(2020), and physics-based models Malik  _et al._ (2020); Reed  _et al._
(2021). While in some cases, constraints imposed by bulk nuclear properties
have been used Malik  _et al._ (2020); Tsang  _et al._ (2020); Reed  _et al._
(2021), other cases have also imposed those provided by the bulk properties of
nuclei through the constraints on nuclear matter parameters assuming them to
be independent of each other. The sample sizes used in these works also
differ. In what follows, we first explore the dependence on the sample sizes
for the EoSs used in our work, and thereafter recount the main findings in the
above mentioned studies.

###  V.1 Dependence on the sample sizes

Figure 9: The distributions of correlation coefficients for $\Lambda_{1.4}$
with $L_{0}$ and $K_{\rm sym,0}$ obtained using 1500 samples. Each sample
consists of a set of 20 models selected randomly from the pool of 35 SHF and
RMF models considered in the previous section. The dashed red vertical line
indicates the mean value.

The correlation systematics, up to a large extent, is also driven by the
selection criteria of models. Even not too large a number of models but having
diverse behaviour, if randomly selected, may serve the purposeMargueron  _et
al._ (2018). The various models obtained by systematically varying one or few
of the base model introduces a strong bias and usually result in large
correlations under consideration.

Table 2: The mean and standard deviations for the distributions of correlation coefficients for $\Lambda_{1.4}$ and $R_{1.4}$ with $L_{0}$ and $K_{\rm sym,0}$ obtained with ensembles of 10, 20 and 30 models. Results for the correlations of $L_{0}$ and $K_{\rm sym,0}$ with $P(\frac{2}{3}\rho_{0})$ and $P(2\rho_{0})$ are also presented. |  |  |  Number of EoSs |
---|---|---|---|---
|  |  10 |  20 |  30
|  $L_{0}$ |  0.7$\pm$0.18 |  0.73$\pm$0.09 |  0.74$\pm$0.04
$\Lambda_{1.4}$ |  $K_{sym,0}$ |  0.73$\pm$0.15 |  0.74$\pm$0.07 |  0.74$\pm$0.03
|  $L_{0}$ |  0.76$\pm$0.15 |  0.78$\pm$0.07 |  0.78$\pm$0.03
$R_{1.4}$ |  $K_{\rm sym,0}$ |  0.71$\pm$0.17 |  0.73$\pm$0.08 |  0.74$\pm$0.04
|  $P(\frac{2}{3}\rho_{0})$ |  0.94$\pm$0.04 |  0.94$\pm$0.02 |  0.94$\pm$0.01
$L_{0}$ |  $P(2\rho_{0})$ |  0.61$\pm$0.22 |  0.62$\pm$0.12 |  0.62$\pm$0.06
|  $P(\frac{2}{3}\rho_{0})$ |  0.34$\pm$0.36 |  0.38$\pm$0.19 |  0.39$\pm$0.09
$K_{\rm sym,0}$ |  $P(2\rho_{0})$ |  0.74$\pm$0.15 |  0.73$\pm$0.08 |  0.73$\pm$0.04

We investigate how the selection of EoSs affects the correlations of
$\Lambda_{1.4}$ and $R_{1.4}$ with $L_{0}$ and $K_{\rm sym,0}$. For this
purpose, sets of 10, 20 and 30 EoS randomly selected from our pool of 35 EoSs
corresponding to the SHF and RMF models are constructed. The distribution of
correlation coefficients are studied using 1500 such samples. In Fig. 9,
results obtained for sets of 20 EoSs are displayed. The mean and standard
deviations for the distributions of correlation coefficients corresponding to
different number of EoS models are presented in Table 2. The standard
deviations are found to decrease with increase in number of EoSs in a sample,
but the mean values show only marginal changes.

The distributions of correlation coefficients for the correlations of $L_{0}$
and $K_{\rm sym,0}$ with the pressures $P(\frac{2}{3}\rho_{0})$ and
$P(2\rho_{0})$ are also subjected to a similar analysis. The trend for $K_{\rm
sym,0}-P$ correlations is opposite to that obtained for the $L_{0}-P$
correlations. The $K_{\rm sym,0}$ is better correlated with $P(2\rho_{0})$
than with $P(\frac{2}{3}\rho_{0})$ since as the density increases, the
contribution to the pressure from the $K_{\rm sym,0}$ increases faster
compared to the ones for $L_{0}$.

###  V.2 Correlation findings in earlier works

The correlations of various NS properties with nuclear matter parameters were
studied using large set of RMF models in Ref. Pradhan  _et al._ (2022a). The
coupling constants that appear in the effective Lagrangian were determined
from various nuclear matter parameters. Values of these nuclear matter
parameters were drawn from uniform distributions with their ranges guided by
available data on finite nuclei assuming them to be uncorrelated with each
other. The correlations of $R_{1.4}$ and $\Lambda_{1.4}$ with $L_{0}$ were
found to be very weak. A similar investigation performed in Ref. Carson  _et
al._ (2019) using metamodels Margueron  _et al._ (2018) also yielded weak
correlations of the combined tidal deformability $\tilde{\Lambda}$ with
$L_{0}$. However, the correlations of $\tilde{\Lambda}$ with $K_{\rm sym,0}$
was stronger than those with $L_{0}$. The values of $\tilde{\Lambda}$ are
expected to be strongly correlated with $\Lambda_{1.4}$ Malik  _et al._
(2018); De  _et al._ (2018). The $\Lambda_{1.4}$ in Ref. Tsang  _et al._
(2020) was found to be moderately sensitive to the $L_{0}$ and $K_{\rm sym,0}$
within a metamodel. The nuclear matter parameters in Ref. Tsang  _et al._
(2020) were considered to have Gaussian distributions with the mean and
standard deviations obtained by combining the results from the SHF and RMF
models fitted to some selected properties of finite nuclei. The parameters
were further constrained using current estimates for
$\Lambda_{1.4}=190^{+390}_{-120}$ that led to very weak correlations in the
values of $L_{0}$ and $K_{\rm sym,0}$. Reference Biswas (2021) presented an
interesting study on the correlations of $R_{1.4}$ with $L_{0}$ and $K_{\rm
sym,0}$ obtained by combining the impact of PREX-II and Radio/NICER/XMM-
Newton’s mass–radius measurement of PSR J0740+6620 on the dense matter EoS.
The EoSs employed in Ref. Biswas (2021) were obtained using Taylor expansion
up to $1.25\rho_{0}$ and piece wise polytrope (PP) were used at higher
densities. It was shown that the $R_{1.4}$ \- $L_{0}$ correlations somewhat
improves with the increase in the $L_{0}$ \- $K_{\rm sym,0}$ correlations
which is enforced by demanding higher accuracy in the neutron-skin measurement
of PREX-II for the 208Pb nucleus. The strong correlations of $\Lambda_{1.4}$
with $L_{0}$ reported in Refs. Malik  _et al._ (2020); Agrawal and Malik
(2021) for the SHF models stem from the model parameters informed by the bulk
properties of finite nuclei. Similar conclusions were drawn for the RMF models
fitted to data on finite nuclei Reed  _et al._ (2021).

Table 3:  Overview of the correlations of $R_{1.4}$ and $\Lambda_{1.4}$ with
the symmetry energy parameters $L_{0}$ and $K_{\rm sym,0}$ obtained using
different models. The model parameters are determined either in terms of the
nuclear matter parameters or by fitting directly some selected properties of
finite nuclei. The nuclear matter parameters are assumed to have uniform (U),
Gaussian (G) or Hybrid (H) distributions. Furthermore, these parameters may be
uncorrelated(Unc) or correlated (Cor); see the text for details. The ‘W’, ‘M’
and ‘S’ denote the weak ($r\leq 0.5$), moderate ($0.5<r\leq 0.75$) and strong
($r\geq 0.75$), correlations.
Model |  Dist. NMPs |  $L_{0}$ |  $K_{\rm sym,0}$ |  Ref.
---|---|---|---|---
RMF |  U-Unc |  W |  – |  Pradhan  _et al._ (2022a)
Metamodel |  U-Unc |  W |  S |  Carson  _et al._ (2019)
Metamodel |  G-Unc |  M |  M |  Tsang  _et al._ (2020)
Taylor+PP |  H-Unc |  W |  W |  Biswas (2021)
SHF |  G-Unc |  M |  M |  Malik  _et al._ (2020)
SHF |  G-Cor |  S |  S |  Malik  _et al._ (2020)
SHF |  Finite nuclei |  S |  S |  Agrawal and Malik (2021)
RMF |  Finite nuclei |  S |  S |  Reed  _et al._ (2021)

The outcomes described above are summarized in Table 3 for the sake of
convenience. It seems that the correlations of $R_{1.4}$ and $\Lambda_{1.4}$
with various symmetry energy parameters are sensitive to the joint
distribution of the nuclear matter parameters that enter into the
investigation. The nuclear matter parameters drawn from uncorrelated uniform
distributions tend to yield weaker correlation of $R_{1.4}$ and
$\Lambda_{1.4}$ with $L_{0}$ and $K_{\rm sym,0}$. On the other extreme, models
fitted to finite nuclei yield stronger correlations. The bulk properties of
finite nuclei not only constrain the values of few low order nuclear matter
parameters, but also introduce the correlations among them Malik  _et al._
(2020). For instance, the values of iso-scalar giant monopole resonance
energies impose stringent constraints on the slope of incompressibility
coefficient at crossing density ($\sim 0.7\rho_{0}$) that can be expressed in
terms of $K_{0}$ and $Q_{0}$ Khan  _et al._ (2012); Khan and Margueron (2013).
The correlations between $L_{0}$ and $K_{\rm sym,0}$ have also been
extensively studied using mean field models such as SHF and RMF Tews  _et al._
(2017); Mondal  _et al._ (2017); Li and Magno (2020) and these correlations
were found to be strong. The existence of universal correlations among various
symmetry energy parameters $J_{0},L_{0}$ and $K_{\rm sym,0}$ which are
independent of the details of the nuclear forces used in the calculation has
been demonstrated Holt and Lim (2018). Given the above conflicting findings in
the literature, it appears that the dependence of $R_{1.4}$ and
$\Lambda_{1.4}$ on $L_{0}$ and other higher order nuclear matter parameters
are not yet conclusive. Detailed analysis must be done in this regard for
arriving at some meaningful conclusions.

##  VI Damping times of QNMs in General Relativity

Figure 10: Variation of the damping time for $f$-mode oscillations vs neutron
star mass.

We have computed the damping time of the $f$\- modes constructed from the EoSs
of RMF and SHF models. The dependence of the damping time on the NS mass is
shown in Fig. 10. For a typical NS, the frequency of the $f$-mode is around
1-3 kHz, whereas their damping times $\tau_{f}$ can be a few tenths of a
seconds Kokkotas and Schmidt (1999). The damping time decreases as the neutron
star mass increases. The variation of the curves shows the nature of the
softer and stiffest EoSs, which can be understood from Table 1. For example,
the soft EoS, such as G3, has the shortest damping time, 0.230 s, and the
highest $f$-mode frequency, 1.760 kHz of $1.4M_{\odot}$. The nature of a short
damping time would be to enable sizeable emission of gravitational radiation
Lindblom and Detweiler (1983b); Burgio  _et al._ (2011).

Figure 11:  Variation of the imaginary part $\omega$ of $p_{1}$-mode
oscillations (inverse damping time) vs neutron star mass.

Fig. 11 shows the the imaginary part of the eigen-frequency (inverse damping
time) of the $p_{1}$-mode. We found the damping times of the $p_{1}$-mode
$\tau^{p_{1}}$ is significantly longer than that of the $f$-mode $\tau^{f}$.
The damping times are sensitive to the structure of neutron stars and are
smallest for stars slightly lower than the maximum mass configurations.
Results for $1.4M_{\odot}$ stars can also be seen in the rightmost column of
Table 1.

### Accuracy of the Cowling approximation

(a) $f$-mode

(a)

(b) $p_{1}$-mode

(b)

Figure 12:  Relative deviations of the $f$\- and $p_{1}$-mode frequencies
obtained using the Cowling approximation with those for full GR.

Figure 13: Fluid and metric perturbation (real part) amplitude profiles of
$f$-mode (top panels) and $p_{1}$-mode (bottom panels) oscillations for a NS
with 1$M_{\odot}$ (left panels) and $M_{max}$ (right panels). $W$ and $V$ are
dimensionless, $H_{0}$, $H_{1}$ and $K$ are in units of
$\varepsilon_{s}=152.26$ MeV fm-3, and $X$ is in units of
$\varepsilon_{s}^{2}$. While the SLY4 EoS is used as an example in this
figure, all EoSs have qualitatively similar behavior.

The frequencies calculated by the Cowling approximation and full GR show
significant differences although their qualitative trends are similar. For the
$\ell=2$ modes considered, We calculate the relative error in the $f$-mode
frequencies using

$\frac{\left|f_{Cow}-f_{GR}\right|}{f_{GR}}$ (40)

Similar calculation has done in case of $p_{1}$-mode as depicted in Fig.
12(b). The Cowling approximation has an error of around $15-30\%$ for $f$-mode
frequencies as shown in Fig. 12(a). The error exhibits a linear trend with
increasing mass with a more rapid variation occurring as the maximum is
reached. These features were also noted in Chirenti  _et al._ (2015b); Yoshida
and Eriguchi (1997).

The $f$-mode oscillations have fluid perturbations ($W$ and $V$) that peak
near the stellar surface, whereas the metric perturbations ($H_{0}$, $H_{1}$
and $K$) peak close to the stellar center; see top two panels in Fig. 13. The
Cowling approximation neglects metric perturbations which introduce a smaller
error in NSs with higher mass, as massive NSs have a stronger fluid
perturbation peak at the stellar surface and couples weakly to the metric
perturbation at the stellar center. Fluid perturbations of the $p_{1}$-mode
have a radial node within the NS; see the bottom two panels in Fig. 13. Also,
the fluid perturbation is much weaker at the stellar center which results in a
much higher peak at the stellar surface. As a result, the $p_{1}$-mode
oscillation frequency calculated by the Cowling approximation has an error of
around 10%, less than the error in the $f$-mode frequency. However, unlike for
the $f$-mode, fluid perturbations peak at the stellar surface the more massive
NSs exhibiting a higher peak than stars of lesser mass. Therefore, the error
in the Cowling approximation for $p_{1}$-modes increases with increasing mass.
Since the $p$-modes with more radial nodes generally have weak fluid
perturbations in the stellar center, the accuracy of the Cowling approximation
increases with the order of the mode Lindblom and Splinter (1990). Therefore,
the error of the Cowling approximation in $p$-modes of any order is less than
15%, which corresponds to $p_{1}$-mode for stable NS with maximum mass.

Since the $f-mode$ frequency can be estimated by the $1\%$-accurate universal
relations between the $f$-mode frequency vs tidal deformability and vs
momentum of inertia Lau  _et al._ (2010); Zhao and Lattimer (2022), the 30%
error of the Cowling approximation is unacceptable. Due to the lack of such a
tight universal relation for $p$-modes, a 15$\%$ error from the Cowling
approximation is regarded as tolerable for explorative purposes. For example,
the Cowling approximation can greatly simplify the equations especially in
non-linear full GR simulations Rosofsky  _et al._ (2019); Shashank  _et al._
(2021).

##  VII Summary and Conclusions

We have used a representative set of EoSs from non-relativistic and
relativistic mean-field models to explore the possibility of strong
correlations of $f$-mode, $p_{1}$-mode frequencies and damping time with the
properties of nuclear matter and neutron stars. Most of the EoSs are
consistent with that for the chiral EFT model upto $2\rho_{0}$. Our chosen
EoSs also satisfy experimental data on the bulk properties of finite nuclei
and are also compatible with the various properties of NS observables such as
maximum mass, radius, and tidal deformability. For our correlation
systematics, the values of nuclear matter incompressibility $K_{0}$, skewness
parameter $Q_{0}$ for symmetric nuclear matter, and the slope of the symmetry
energy coefficients $L_{0}$ as well as pressure for the $\beta$-equilibrated
matter in the range of $\rho_{0}$-$2.5\rho_{0}$ are employed. We also present
a critical analysis of the correlations of nuclear matter parameters, such as
aspects of the density-dependent symmetry energy, with the tidal deformability

The differences in the $f$\- and $p_{1}$-mode frequencies calculated using
full GR and the Cowling approximation are also studied.

Our principal findings are as follow. The $f$-mode frequencies corresponding
to NS masses in the range $1.2-1.8M_{\odot}$ are strongly correlated with the
radius and tidal deformability of a $1.4M_{\odot}$ neutron star. The pressure
at $2\rho_{0}$ for $\beta$-equilibrated matter is also strongly correlated
with the $f$-mode frequencies. This correlation grows marginally with increase
in the NS mass. The $L_{0}$ is found to be moderately correlated with $f$-mode
frequencies whereas it decreases with increasing NS mass. The trends of the
correlations for the $f$-mode frequencies obtained with the Cowling
approximation and full GR are quite similar.

The frequency of the $p_{1}$-mode for a NS with mass $1.4M_{\odot}$ is
strongly correlated with $L_{0}$ and pressure at $\rho_{0}$. However, the
correlation significantly decreases with pressure at higher densities. This
can be understood as being due to the $p$-mode oscillation peaking stronger at
the stellar surface than the $f$-mode, see Fig. 13. The correlation of the
$p_{1}$-mode frequency with the canonical radius and tidal deformability are
weaker than those for the $f$-mode frequency. These results imply that the
$f$\- and $p_{1}$-mode frequencies are sensitive to the behavior of EoS at
different densities.

The correlation of the damping time for $f$-mode exhibits the same behavior as
those for the $f$-mode frequencies. However, the damping time for the
$p_{1}$-mode displays opposite trends. For instance, the damping time for the
$p_{1}$-mode is only weakly correlated with $L_{0}$ and pressure for
$\beta$-equlibrated matter at saturation density $\rho_{0}$.

Our examination of the accuracy and its mass dependence of the Cowling
approximation reveals that the error in the $f$-mode frequency decreases from
about $30\%$ for a $1M_{\odot}$ NS to about $15\%$ for a NS with the highest
mass, which is consistent with previous calculations Pradhan  _et al._
(2022b); Zhao and Lattimer (2022). Recently, calculation of the Newtonian
Kelvin frequency has been shown to be a simpler and more accurate
approximation than the Cowling approximation Zhao and Lattimer (2022).
However, the Cowling approximation gives reasonable results for the
$p_{1}$-mode with errors $\lesssim 15\%$. Unlike non-linear simulation of the
$f$-mode oscillation which requires full GR Rosofsky  _et al._ (2019);
Shashank  _et al._ (2021), numerical simulation of the $p_{1}$-mode could be
simplified with the Cowling approximation, which is sufficiently accurate for
most purposes.

Our work here has concentrated on NS matter in which the strongly interacting
components are nucleons only. In work to follow, we will be considering the
possible presence of quarks treated via the Gibbs construction and the case in
which a nucleon-to-quark crossover transition occurs.

###### Acknowledgements.

T.Z. and M. P. are supported by the Department of Energy, Grant No. DE-
FG02-93ER40756. B.K. and B.K.A. acknowledge partial support from the
Department of Science and Technology, Government of India with grant no.
CRG/2021/000101.

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