ns short=NS, long=neutron star, nsm short=NSM, long=neutron star matter, pnm
short=PNM, long=pure neutron matter, bns short=BNS, long=binary neutron star,
eos short=EOS, long=equation of state, gws short=GWs, long=gravitational
waves, qcd short=QCD, long=quantum chromodynamics, lqcd short=lQCD,
long=lattice quantum chromodynamics, pqcd short=pQCD, long=perturbative
quantum chromodynamics, qnm short=QNM, long=quasi-normal mode, ur short=UR,
long=universal relation, bps short=BPS, long=Bethe-Pethick-Sutherland, ddb
short=DDB, long=a nucleonic $\beta-$ equilibritated EOS based on a
relativistic description of hadrons through their density-dependent couplings
constrained by the existing observational, theoretical and experimental data
through a Bayesian analysis, ddbhyb short=DDB-Hyb, long=a hybrid set of EOSs
which consists of the DDB EOS at low density ($\leq 2\rho_{0}$) and the
deconfined quark matter at very high densities ($\geq 40\rho_{0}$) while the
region ($2\rho_{0}$-$40\rho_{0}$) is interpolated by piecewise polytropes, cft
short=chEFT, long=chiral effective field theory, rmf short=RMF,
long=relativistic mean field, tov short=TOV, long=Tolman-Oppenheimer-Volkoff,
gr short=GR, long=general relativistic,

# Robust universal relations in neutron star asteroseismology

Deepak Kumar  Theory Division, Physical Research Laboratory, Navarangpura,
Ahmedabad 380 009, India  Indian Institute of Technology Gandhinagar,
Gandhinagar 382 355, Gujarat, India<EMAIL_ADDRESS>Tuhin Malik  CFisUC,
Department of Physics, University of Coimbra, PT 3004-516 Coimbra, Portugal
<EMAIL_ADDRESS>Hiranmaya Mishra  Theory Division, Physical Research
Laboratory, Navarangpura, Ahmedabad 380 009, India<EMAIL_ADDRESS>School of
Physical Sciences, National Institute of Science Education and Research,
Jatni, 752050 India<EMAIL_ADDRESS>Constança Providência  CFisUC,
Department of Physics, University of Coimbra, PT 3004-516 Coimbra, Portugal
<EMAIL_ADDRESS>
###### Abstract

The non-radial oscillations of the neutron stars (NSs) have been suggested as
an useful tool to probe the composition of neutron star matter (NSM). With
this scope in mind, we consider a large number of equations of states (EOSs)
that are consistent with nuclear matter properties and pure neutron matter EOS
based on a chiral effective field theory (chEFT) calculation for the low
densities and perturbative QCD EOS at very high densities. This ensemble of
EOSs is also consistent with astronomical observations, gravitational waves in
GW170817, mass and radius measurements from Neutron star Interior Composition
ExploreR (NICER). We analyze the robustness of known universal relations (URs)
among the quadrupolar $f$ mode frequencies, masses and radii with such a large
number of EOSs and we find a new UR that results from a strong correlation
between the $f$ mode frequencies and the radii of NSs. Such a correlation is
very useful in accurately determining the radius from a measurement of $f$
mode frequencies in the near future. We also show that the quadrupolar $f$
mode frequencies of NS of masses 2.0 M⊙ and above lie in the range $\sim$ 2-3
kHz in this ensemble of physically realistic EOSs. A NS of mass 2M⊙ with a low
$f$ mode frequency may indicate the existence of non-nucleonic degrees of
freedom.

neutron stars, equation of state, non-radial oscillation

Introduction. The ns observations in the multi-messenger astronomy have piqued
a lot of interest in the field of nuclear astrophysics and strong interaction
physics. The recent radio, x-rays and gws observations in the context of nss
have provided interesting insights into the properties of matter at high
density. The core of such compact objects is believed to contain matter at few
times nuclear saturation density, $\rho_{0}$ ($\rho_{0}\approx$ 0.16 fm-3) [1,
2, 3, 4] and provides an unique window to get an insight into the behavior of
matter at these extreme densities. On the theoretical side, no controlled
reliable calculations are there that can be applicable to matter densities
relevant for the ns cores. The lqcd simulations are challenging at these
densities due to sign problem in Monte-Carlo simulations. On the other hand,
the analytical calculations like cft is valid only at low densities while pqcd
is reliable at extremely high densities. In recent approaches, the eoss
between these two limits have been explored by connecting these limiting cases
using a piecewise polytropic interpolation, speed of sound interpolation or
spectral interpolation [5, 6, 7, 8, 9, 10, 11].

The ns properties such as mass, radius and quadrupole deformation of merging
nss can constrain the uncertainty in eos. The discovery of massive ns with
masses of the order of $2M_{\odot}$ requires the eos to be stiff. However, the
fact that non-nucleonic degrees of freedom soften the eos at high density,
puts a constraint on the eos at the intermediate densities. The observations
of gws from bns inspiral by Advanced LIGO and Advanced Virgo gws observatories
have opened a new window in the field of multi-messenger astronomy and nuclear
physics. The inspiral phase of ns-ns merger leads to tidal deformation
($\Lambda$), which is strongly sensitive to the compactness. Since $\Lambda$
is related to the eos of the nsm, this measurement acts as another constraint
on the eos. On the other hand, recovering the nuclear matter properties from
the eos of $\beta$-equilibriated matter is rather non trivial. This further
requires the knowledge of the composition (e.g. proton fraction) of matter at
high densities [12, 13, 14, 15].

In the context of gws, the non-radial oscillations of ns are particularly
interesting as they can carry information of the internal composition of the
stellar matter. These oscillations in the presence of perturbations
(electromagnetic or gravitational) can emit gws at the characteristic
frequencies of its qnm. The frequencies of qnm depend on the internal
structure of ns and it may be another probe to get an insight regarding the
composition of nsm also known as asteroseismology. Different qnms are
distinguished by the restoring forces that act on the fluid element when it
gets displaced from its equilibrium position. The important fluid modes
related to gws emission include fundamental $(f)$ modes, pressure $(p)$ modes
and gravity $(g)$ modes driven by the pressure and buoyancy respectively. The
frequency of $p$ modes is higher than that of $g$ modes while the frequency of
$f$ modes lies in between. The focus of the present investigation is on the
quadrupolar $f$ modes that are correlated with the tidal deformability during
the inspiral phase of ns merger [16] and have the strongest tidal coupling
among all the oscillation modes. More importantly, these modes lie within the
sensitivity range of the current as well as upcoming generation of the gws
detector networks [17]. In this context, qnms have been studied with various
eos models and some universal/quasi-universal behaviors for the frequency and
damping time which are insensitive to the eos models [18, 19, 20, 21, 22, 23,
24, 25]. This needs to be explored further regarding the robustness of these
relations for a large number of eoss consistent with recent observational
constraints.

In this letter we propose two major points of interest. Firstly we estimate,
within the Cowling approximation [26, 27], the $f$ mode oscillation
frequencies for nss using a large number of eoss and demonstrate that
observation of $f$ mode frequencies, apart from causality $c_{s}^{2}\leq 1$
and maximum mass constraints, further restrict the eoss. Secondly we verify
the robustness of few ur among the quadrupolar $f$ mode frequencies, masses
and radii studied earlier with limited eoss. It has been earlier found that
these urs between ns properties are strongly violated by hybrid eoss [28, 29,
30] and certain exotic phases [31]. We consider here a large number of eoss
and confirm that a known UR is almost insensitive to the eoss, while a second
one depends slightly on the composition of the eoss, i.e. the presence or not
of non-nucleonic degree of freedom, and, finally, we propose a new UR.

Setup -  The two ensembles of eoss that we consider here are constructed by
stitching together eoss valid for different segments in baryon densities. For
the outer crust the bps eos is chosen [32]. Outer crust and the core are
joined using a polytropic form
$p(\varepsilon)=a_{1}+a_{2}\varepsilon^{\gamma}$ in order to construct the
inner crust, where the parameters $a_{1}$ and $a_{2}$ are determined in such a
way that the eos for the inner crust matches with the outer crust at one end
($\rho=10^{-4}$ fm-3) and with the core at the other end ($\rho=0.04$ fm-3).
The polytropic index $\gamma$ is taken to be $4/3$ [33]. It is important to
note that the differences in nss radii between this treatment of the inner
crust eos and the unified inner crust description including the pasta phases
have been found to be less than 0.5 km, as discussed in [34]. The core eoss
are considered within two different approaches: (i) ddb, obtained in [34],
which satisfies pnm constraints at low densities obtained from next-to-next-
to-next-to leading order (N3LO) calculations in the cft [35, 36]. (ii) ddbhyb.
For the deconfined quark matter, we employ NNLO pqcd results of Refs. [6, 37]
which can be cast in a simple fitting function for the pressure as a function
of chemical potential ($\mu$) given as

$\displaystyle
P_{pQCD}(\mu)=\frac{\mu^{4}}{108\pi^{2}}\left(c_{1}-\frac{d_{1}X^{-\nu_{1}}}{(\mu/GeV)-d_{2}X^{-\nu_{2}}}\right)$
(1)

where the parameters are $c_{1}=0.9008$, $d_{1}=0.5034$, $d_{2}=1.452$,
$\nu_{1}=0.3553$ and $\nu_{2}=0.9101$ [38]. Here $X$ is a dimensionless
renormalization scale parameter, $X=3\bar{\Lambda}/\mu$ which is allowed to
vary $X\in\left[1,4\right]$. We use this pqcd eos for densities beyond
$\rho\simeq 40\rho_{0}$ which corresponds to $\mu_{\rm pQCD}=2.6$ GeV [38].
Between the region of the validity of pqcd and ddb i.e. $\mu_{\rm
DDB}\leq\mu\leq\mu_{\rm pQCD}$, where $\mu_{\rm DDB}$ is the chemical
potential of ddb eos at $\rho=2\rho_{0}$, we divide the interval into two
segments, ($\mu_{\rm DDB}$-$\mu_{c}$) and ($\mu_{c}$-$\mu_{\rm pQCD}$), and
assume eos has a polytropic form in each segment i.e.
$P_{i}(\rho_{i})=\kappa_{i}\rho_{i}^{\gamma_{i}}$ for the i-th segment [37].
The segments can be connected to each other by requiring that the pressure and
the energy density are continuous at $\mu_{c}$ as well as the pressure should
be an increasing function of the energy density and the eos must be
subluminal. We also ensure that there is no jump in the baryon number density.
This corresponds to assuming no first order phase transition between hadronic
matter and quark matter. If one wishes to include a first order phase
transition, an extra term to the number density at $\mu_{c}$ can be added
[37].

To obtain the eos of the core, we proceed as follows. For the outer core,
which extends approximately until $\rho=2\rho_{0}$, we use a soft (stiff) ddb
eos as obtained in Ref. [34] within 90% CI. The corresponding value of
chemical potential at $\rho=2\rho_{0}$ is $\mu_{\rm DDB}=1.036\ (1.097)$ GeV
for a soft (stiff) ddb eos. We interpolate the region from $\mu=\mu_{\rm DDB}$
to $\mu=\mu_{c}$ and from $\mu=\mu_{c}$ to $\mu=\mu_{\rm pQCD}$ with two
piecewise polytropes. We select all those eoss which (i) match with pqcd at
$\mu=\mu_{\rm pQCD}$ (i.e. $X\in[1,4]$) (ii) have pressure as an increasing
function of energy density, and (iii) are subluminal. We refer this eos as
ddbhyb. The chemical potential $\mu_{c}$ is here chosen in such a way that the
EOS matches pqcd at $\mu=\mu_{\rm pQCD}$. We take $\mu_{c}\in[1.04,2.2]$ GeV
and the corresponding pressure $P_{c}\in[20,1260]$ MeV.fm-3. For an ensemble
of ddbhyb eoss we choose $\mu_{c},P_{c}$ randomly in the prescribed domain by
Latin-Hypercube-Sampling method [39] for an uniform distribution. For a given
$\mu_{c},P_{c}$ and $P_{\rm DDB}$, the parameters of the first polytrope,
$(\kappa_{1},\gamma_{1})$ get determined. Similarly for a given
$\mu_{c},P_{c}$ and $P_{2}$ (where $P_{2}$ is the pqcd pressure for a given
value of $X$ at $\mu=\mu_{\rm pQCD}$), the parameters of the second polytrope
($\kappa_{2},\gamma_{2}$) get determined. The domains for pressure ($P_{c}$)
and chemical potential ($\mu_{c}$) become $P_{c}\in[45,1255]$ MeV$\cdot$fm-3
and $\mu_{c}\in[1.07,2.09]$ GeV after constrained by pqcd. These domains
further squeeze to $P_{c}\in[53,680]$ MeV.fm-3 and $\mu_{c}\in[1.15,1.88]$ GeV
after putting the constraint of $M_{\rm max}\geq 2M_{\odot}$ and so we find
0.38 million eoss out of 54 million sampled eoss satisfying these constraints.
It may be mentioned here that, although we use two polytropes for the
interpolation between ($\mu_{\rm DDB}$-$\mu_{\rm pQCD}$), there have been
different interpolation functions like spectral decomposition [40, 41] and
speed of sound method [9, 42, 11].

Pulsating equations - To estimate the specific oscillation frequency of nss,
let us discuss the non-radial oscillation of a spherically symmetric ns
characterized by the background space-time metric where the line element is
given by

$\displaystyle
ds^{2}=-e^{2\Phi}dt^{2}+e^{2\Lambda}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (2)

We shall consider the pulsating equations within the Cowling approximation so
that our study is limited to the modes related to fluid perturbations and
neglect the metric perturbations. The Lagrangian fluid displacement vector is
given by

$\displaystyle\xi^{i}=\left(e^{-\Lambda}W,~{}-V\partial_{\theta},~{}-V\sin^{-2}\theta\partial_{\phi}\right)r^{-2}Y_{lm}$
(3)

Where $W(r,t)$ and $V(r,t)$ are the perturbation functions and $Y_{lm}$ are
the spherical harmonic function. The perturbation equations that describe
oscillations can be obtained by the perturbed Einstein field equations $\delta
G_{\alpha\beta}=8\pi\delta T_{\alpha\beta}$ with
$G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R$ being the
Einstein tensor. Linearising these equations in the perturbation, while
choosing a harmonic time dependence for the perturbation i.e. $W(r,t)\propto
W(r)e^{i\omega t}$ and $V(r,t)\propto V(r)e^{i\omega t}$ with frequency
$\omega$, the differential equations further fluid perturbation functions can
be obtained as [26, 24, 43]

$\displaystyle W^{\prime}$
$\displaystyle=\frac{d\epsilon}{dP}\Big{(}\omega^{2}r^{2}e^{\Lambda-2\Phi}V+W\Phi^{\prime}\Big{)}-l(l+1)e^{\Lambda}V,$
(4) $\displaystyle V^{\prime}$
$\displaystyle=2V\Phi^{\prime}-\frac{1}{r^{2}}We^{\Lambda},$ (5)

here, the ‘prime’ denotes the total derivative with respect to $r$. These
equations are solved with appropriate boundary conditions at the stellar
center $r=0$ and at the surface $r=R$. The $W$ and $V$ in the vicinity of the
stellar center are taken as $W(r)\sim Cr^{l+1}$ and $V(r)\sim-Cr^{l}/l$, where
$C$ is an arbitrary constant. The other boundary condition that needs to be
full-filled is that the Lagrangian perturbation to the pressure must vanish at
the stellar surface. This leads to [24, 43, 26]

$\displaystyle\omega^{2}r^{2}e^{\Lambda-2\Phi}V+W\Phi^{\prime}\big{|}_{r=R}=0$
(6)

This apart in the case of density discontinuity these equations have to be
supplemented by an extra junction condition at the surface of discontinuity.
We shall not consider here a density discontinuity. With these boundary
conditions, the problem becomes an eigenvalue problem for the parameter
$\omega$ which can be estimated numerically. We shall confine ourselves to
$l=2$ quadrupolar modes.

Figure 1: We show pressure and energy density regions in MeV.fm-3 of our
sampled EOSs (ddb and ddbhyb). We consider nucleonic $\beta$-equilibrated eos
of the 90% CIs for ddb (light blue) as a full range and (dark blue) up to
$2\rho_{0}$ [34] and at very high density $\sim 40\rho_{0}$ the NNLO pQCD
(dark red) [6]. In the intermediate region, eos is evolved in
thermodynamically consistent way with two polytropic segments (see text for
details). Also included are the limits of the domain of eoss obtained in Ref.
[10] (red solid curve) and the dense PDF ($\geq 0.08$) calculated in Ref. [11]
(red dashed lines).

Results - We now proceed to analyze the ensembles of eoss that are consistent
with nuclear matter properties or pnm eos based on theoretically robust cft at
low densities and pqcd at very high densities. As mentioned earlier, we start
with $54$ million eoss. We discard those eos which do not match the two end
points or are superluminal (square of speed of sound $c_{s}^{2}>1$) as well as
the condition of positive speed of sound. This leaves us with an ensemble of
0.38 million ddbhyb eoss. This ensemble of eoss is represented in Fig. 1 by
the orange band. We next enforce the $M_{\rm max}\geq 2.0M_{\odot}$ constraint
resulting from solving the tov equations with this ensemble. This constraint
further reduces the number of eoss to 55,000 which are displayed in Fig. 1 as
the gray band, named here after ddbhyb set. The polytrope indices $\gamma_{1}$
and $\gamma_{2}$ are seen to vary over an intervals
$\gamma_{1}\in[1.67,13.76]$ and $\gamma_{2}\in[1.0,1.51]$. The tight
constraint on $\gamma_{2}$ has its origin on the matching to the pqcd
pressure. In Fig. 1, the light blue band is the $\beta$-equilibrated nuclear
matter $\approx 10$k eoss (ddb 90% CI) while the dark red band corresponds to
pqcd eos. For comparison, we also plot the domain of eoss obtained in Ref.
[10] (red solid curve) compatible with recent NICER and gws observations. The
red dashed lines refers to the dense PDF ($\geq 0.08$) obtained in Ref. [11]
with continuous sound speed and consistent not only with nuclear theory and
pqcd, but also with astronomical observations. It is to be noted that both of
ddb and ddbhyb sets are compatible with them.

Figure 2: ns mass ($M$)-radii ($R$) and $f$ mode frequency-mass ($M$) region
obtained from the 90% CI for the conditional probabilities $P(R|M)$ (left) and
$P(f|M)$ (right) for ddbhyb (black dotted) and ddb (dark red). The blue
horizontal bar on the left panel indicates the 90% CI radius for a
2.08$M_{\odot}$ star determined in [44] combining observational data from
GW170817 and NICER as well as nuclear saturation properties. The top and
bottom gray regions indicate, respectively, the 90% (solid) and 50% (dashed)
CI of the LIGO/Virgo analysis for each binary component from the GW170817
event [45]. The $1\sigma$ (68%) credible zone of the 2-D posterior
distribution in mass-radii domain from millisecond pulsar PSR J0030+0451 (cyan
and yellow) [46, 47] as well as PSR J0740 + 6620 (violet) [48, 44] are shown
for the NICER x-rays data. The horizontal (radius) and vertical (mass) error
bars reflect the $1\sigma$ credible interval derived for the same NICER data’s
1-D marginalized posterior distribution.

In Fig.2, we plot the ns mass-radii and $f$ mode frequency-mass regions
obtained at 90% CI for the conditional probabilities $P(R|M)$ (left) and
$P(f|M)$ (right) from the mass-radius clouds arising from the ensembles of
eoss of ddbhyb (black dotted) and ddb (dark red). The blue horizontal bar on
the left panel indicates the 90% CI radius for a 2.08$M_{\odot}$ star
determined in Ref. [44] combining observational data from GW170817 and NICER
as well as nuclear data. The top and bottom gray regions indicate, the 90%
(solid) and 50% (dashed) CI of the LIGO/Virgo analysis for each binary
component from the GW170817 event [45] respectively. The $1\sigma$ (68%)
credible zone of the 2-D posterior distribution in mass-radii domain from
millisecond pulsar PSR J0030+0451 (cyan and yellow) [46, 47] as well as PSR
J0740 + 6620 (violet) [48, 44] are shown for the NICER x-rays data. The
horizontal (radius) and vertical (mass) error bars reflect the $1\sigma$
credible interval derived for the same NICER data’s 1-D marginalized posterior
distribution. The mass-radius domain for the ddbhyb set sweeps a wider range
than the ddb set, restricted to nucleonic degrees of freedom. The ddbhyb set
constrained by pqcd at high density leads to larger radii for high mass ns. We
conclude that the present observational constraints either obtained from
GW170817 or NICER cannot rule out the existence of exotic degrees of freedom.
In the right panel, we see that the 90% CI for $P(f|M)$ $f$ mode frequency
$f\in\left[1.95,2.7\right]$ kHz for both the ddb and ddbhyb sets. The range is
smaller for low ns mass and as the mass increases the 90% CI for $f$ mode
frequency increases. The $f$ mode frequency of a ns above 2M⊙ mass is in the
range (2.1-2.7) kHz and (2.3-2.65) kHz for the ddbhyb and ddb sets,
respectively. As mentioned in the earlier sections, the solutions for $f$ mode
obtained in this work are within the Cowling approximation (neglecting
perturbations of the background metric). It was shown that the Cowling
approximation can overestimate the quadrupolar $f$ mode frequency of nss by up
to 30 to 10 % for ns masses in the range (1.0-2.5) M⊙ compared to the
frequency obtained in the linearised gr formalism [21, 49, 50]. The accurate
measurement of $f$ modes may further constrain eos to a narrower range.
Besides, a star of 2$M_{\odot}$ with a low $f$ mode frequency may indicate an
existence of non-nucleonic degrees of freedom.

Figure 3: urs obtained with our sets of eoss, namely ddbhyb and ddb. ur1
(left): The frequency of the $f$ mode is plotted as a function of the square
root of the average density, and corresponding GR scaled data according to
[51] (bottom band) as explained in the text. Also included are results from
Doneva et al. [49] obtained in the Cowling approximation, Andersson & Kokkotas
[19] and Benhar et al. [20] calculated with full GR; ur2a (center): The
universality among $\omega M$ and $M/R$ obtained with both datasets; and ur3
(right) the universal linear relations among $f$ mode frequency and radii of
ns with masses ranging from 1.6 to 2.4 M⊙ in a step of 0.2M⊙. The band
corresponds to 90% CI

In Fig. 3, we have studied two known urs involving the $f$ mode frequency with
global properties of ns, often studied in literature with a limited set of
eoss. In particular, we name ur1 the UR between the $f$ mode frequency and the
square root of the average star density $\sqrt{M/R^{3}}$, and ur2a the UR
involving the $\omega M$ versus the compactness $M/R$, where $\omega=2\pi f$.
We have analysed their robustness with our eos sets, ddbhyb and ddb. We have
also found a new and direct relation between the $f$ modes frequency, $f$, and
radius, $R$, with the help of the existing strong correlation between them. In
the left panel of the figure we show ur1:

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

It has been shown in Refs. [19, 52] that the average density can be well
parameterized via the $f$ mode frequency. The following values of $a$ and $b$
have been obtained: $a=22.27\pm 0.023~{}(26.76\pm 0.01)~{}{\rm kHz.km}$,
$b=1.520\pm 0.001~{}(1.348\pm 0.001)$ kHz for ddbhyb (ddb). The maximum
relative percentage error obtained for ur1 within 90% CI is 6.0%(4.5%) for
ddbhyb (ddb). We verify that the ur1 depends slightly on the eos, reflected in
a relative dispersion of $\sim 5\%$ at 90%CI. Also, the slope of the medians
depend on the dataset, with the nucleonic data set DDB presenting a 15% larger
slope, and similar to the one obtained in [49] which was calculated with
realistic nucleonic EOS, and is at the upper limit of our 90% CI. It is
important to take note that this particular work has been executed utilizing
the Cowling approximations [24, 43, 26] as previously referenced. In the
scholarly publication by Yoshida et al. [51], a comparative analysis was
performed between the outcomes obtained from complete linearized General
Relativity (GR) and those acquired through the Cowling approximations. The
findings of their study reveal that, for $l=2$, the $f$ mode is overestimated
by 30% and 15% when the compactness values of $M/R$ are 0.05 and 0.2,
respectively. Using this as a linear relation, we have scaled the solutions
obtained in the Cowling approximation, for both ddb and ddbhyb, see the bottom
band in Fig. 3 left panel designated by GR solutions. It is interesting to
notice that the scaled frequencies are compatible with the full GR solutions
obtained in the literature. Notice that the dispersion is smaller, but still
corresponds to a 5% relative uncertainty. In Andersson & Kokkotas (Benhar et
al) the authors have obtained the following parameters $a=35.9(33.0){\rm
kHz.km}$ and $b=0.78~{}(0.79)$ kHz [19, 52, 21], the difference between both
works being the eos considered in the study. In those studies the linearised
gr equations were solved, and, as expected, lower frequencies have been
determined. In Ref. [49], the oscillations of non-rotating and fast rotating
nss have been explored with a different set of eoss based on microscopic
theories within the Cowling approximation. The values of the coefficients of
the ur1 obtained were $a=25.32~{}{\rm kHz.km}$ and $b=1.562$ kHz, which are at
the 90% CI upper limit of the relations we have obtained.

In center panel of the Fig. 3 we display ur2a:

$\displaystyle\omega M=a\left(\frac{M}{R}\right)+b$ (8)

obtained for both ddbhyb and ddb sets, with $a=0.6474\pm 4.6\times 10^{-5}$
($a=0.6549\pm 2.6\times 10^{-5}$) and $b=-0.0085\pm 1.05\times 10^{-5}$
($b=-0.0103\pm 6.18\times 10^{-6}$) for ddbhyb (ddb) set. Both the
coefficients are dimensionless. The maximum relative percentage error obtained
for ur2a within 90% CI is 3.78% (2.20%) for ddbhyb (ddb) set. The values of
the slope and intercept for ur2a are also compatible with the ones obtained in
Ref. [53] within the Cowling approximation with a few nucleonic and hyperonic
eoss as $a=0.65765$ and $b=0.0127866$, respectively. We have also obtained a
relation as ur2b for $\omega R$ as $\omega
R=a\left(\frac{M}{R}\right)^{2}+b\left(\frac{M}{R}\right)+c$. The coefficients
are found to be $a=-3.0369\pm 0.0013(-3.1844\pm 0.0020)$, $b=1.5829\pm
0.0005(1.6288\pm 0.0008)$ and $c=0.4095\pm 5\times 10^{-5}(0.4087\pm 7\times
10^{-5})$ for ddbhyb (ddb) set, all the coefficients are dimensionless. In
this case the maximum relative percentage error is 2.6% (1.6%) in the set
ddbhyb (ddb). Compared with ur1, the relative maximum uncertainty is smaller
for ur2a and ur2b for both ddbhyb and ddb sets. Using these relations we
predict $f$ mode frequencies for the PSR J0740+6620. For this pulsar, the mass
and radius are determined as $2.08\pm 0.7$ M⊙ and $12.35\pm 0.75$ km in [44]
combining observational data from GW170817 and NICER as well as nuclear data.
The corresponding mean values of $f$ mode frequency is calculated as 2.35 kHz
and 2.36 kHz for ur2a and ur2b, respectively, with a $\sim 1-4\%$ intrinsic
error in the urs and additional $\sim 10-12\%$ error due to uncertainty
present in mass and radius. A comment regarding the Cowling approximation may
be in order.

We have identified a strong linear correlation between the $f$ mode frequency
and ns radius $R$ and we are naming it as ur3. The values $r\in[0.98,0.99]$ of
the Pearson correlation coefficient were obtained between $f$ and $R$ for ns
with a mass $M\in[1.6,2.4]$ with our two sets of eoss. These results can also
be traced back from ur1 by keeping fixed ns mass while noting that the
correlation is stronger only for the ns of large mass. In the right panel of
Fig. 3, we plot the linear relations between $f$ and $R$. The values of slope
$m\in[-0.23,-0.21]$ and intercept are $c\in[5.21,5.0]$ for ns mass
$M\in[1.6~{}{\rm to}~{}2.0,2.0~{}{\rm to}~{}2.4]~{}M_{\odot}$. We also plot a
marginalized ur3 obtained with ns masses in the range of 1.6 to 2.4 M⊙ with a
slope, ($m=-0.22$) and an intercept, ($c=5.1$). This gives $\approx 1.5\%$
relative residual within 90% CI. We expect that the correlation is also
present if the full gr solutions are considered. Taking this correction factor
into account, the new relation (ur3) will be very useful for the upcoming
future detection in order to constrain ns radius of massive ns precisely. For
example, in order to measure a radius of a ns with $\sim 0.2$ km uncertainty,
the $f$ mode frequency needs to be measured within $\sim 2\%$ uncertainty.

Summary and conclusion -  The qnms are related with the viscous properties of
matter. In the future, precise measurements of them can put constraints on the
eos of dense matter. We have studied the $f$ mode frequency among the qnms,
which is in the sensitivity band of the future gravitational waves networks
[17]. We have calculated the $f$ mode frequency within the Cowling
approximation with a nucleonic set of 14,000 eoss (ddb set), obtained in Ref.
[34] based on the rmf theory, constrained by existing observational,
theoretical and experimental data through Bayesian analysis. We have also
generated an ensemble of eoss using ddb below twice saturation density
($\rho\leq 2\rho_{0}$) and pqcd at high densities ($\rho\geq\rho_{0}$) as in
Ref.[9]. Two piecewise polytropes have been used to interpolate the region
from $2\rho_{0}$ to $40\rho_{0}$. Implementing the constraints of causality
and maximum mass $M_{\rm max}\geq 2.0M_{\odot}$ a set of 55000 ddbhyb typed
eoss has obtained. The mass-radius cloud that we obtain from the ensembles of
these eoss is consistent with the $\rm{GW}170817$ joint probability
distribution as well as the recent NICER observations of mass and radius. We
have analyzed the robustness of a few previously known universal relations,
UR1 and UR2, and confirmed the robustness of UR2. UR1 shows a dispersion of 5%
relative uncertainty at 90% CI, and a 15% smaller slope for the ddbhyb
compared with the ddb set. We also found a novel strong correlation between
the $f$ mode frequency, $f$, and the radius, $R$, for a ns of mass in the
range (1.6-2.4) M⊙. These new direct relations between $f$ and $R$ will allow
an accurate determination of the radius of ns using future $f$ mode detection.

We show that the quadrupolar $f$ mode frequencies obtained in Cowling
approximation of ns of masses 2.0M⊙ and above lie in the range (2.1-2.7) kHz
and (2.3-2.65) kHz for ddbhyb and ddb sets, respectively. We use this urs to
predict the $f$ mode frequencies of the NICER observations and obtain
$\sim$2.35 (2.0) kHz in Cowling approximation (linearized GR) for the PSR
J0740+6620 which interestingly lies within the sensitivity band of the future
gravitational wave detector networks [17] for the detection of gravitational
waves. It was shown that a two solar mass star with a low $f$ mode frequency
may indicate the existence of non-nucleonic degrees of freedom. In the future,
a detailed investigation of how this frequency is correlated with the
individual component of the eos or different particle compositions in ns core
will be carried out.

Acknowledgments- The authors acknowledge the Laboratory for Advanced Computing
at the University of Coimbra for providing HPC resources for this research
results reported within this paper, URL: https://www.uc.pt/lca. T.M and C.P
would like to thank national funds from FCT (Fundação para a Ciência e a
Tecnologia, I.P, Portugal) under Projects No. UID/FIS/04564/2019, No.
UIDP/04564/2020, No. UIDB/04564/2020, and No. POCI-01-0145-FEDER-029912 with
financial support from Science, Technology, and Innovation, in its FEDER
component, and by the FCT/MCTES budget through national funds (OE).

###### Acknowledgements.

## References

  * Glendenning [1996] N. K. Glendenning,  _Compact Stars_ (Springer New York, NY, 1996).
  * Haensel  _et al._ [2007] P. Haensel, A. Y. Potekhin, and D. G. Yakovlev,  _Neutron Stars 1 : Equation of State and Structure_, Vol. 326 (Springer New York, NY, 2007).
  * Rezzolla  _et al._ [2018] L. Rezzolla, P. Pizzochero, D. I. Jones, N. Rea, and I. Vidaña, eds.,  _The Physics and Astrophysics of Neutron Stars_, Vol. 457 (Springer, 2018).
  * Schaffner-Bielich [2020] J. Schaffner-Bielich,  _Compact Star Physics_ (Cambridge University Press, 2020).
  * Lindblom and Indik [2012] L. Lindblom and N. M. Indik, Phys. Rev. D 86, 084003 (2012), arXiv:1207.3744 [astro-ph.HE] .
  * Kurkela  _et al._ [2014] A. Kurkela, E. S. Fraga, J. Schaffner-Bielich, and A. Vuorinen, Astrophys. J. 789, 127 (2014), arXiv:1402.6618 [astro-ph.HE] .
  * Most  _et al._ [2018] E. R. Most, L. R. Weih, L. Rezzolla, and J. Schaffner-Bielich, Phys. Rev. Lett. 120, 261103 (2018), arXiv:1803.00549 [gr-qc] .
  * Lope Oter  _et al._ [2019] E. Lope Oter, A. Windisch, F. J. Llanes-Estrada, and M. Alford, J. Phys. G 46, 084001 (2019), arXiv:1901.05271 [gr-qc] .
  * Annala  _et al._ [2020a] E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, Nature Phys. 16, 907 (2020a), arXiv:1903.09121 [astro-ph.HE] .
  * Annala  _et al._ [2022] E. Annala, T. Gorda, E. Katerini, A. Kurkela, J. Nättilä, V. Paschalidis, and A. Vuorinen, Phys. Rev. X 12, 011058 (2022), arXiv:2105.05132 [astro-ph.HE] .
  * Altiparmak  _et al._ [2022] S. Altiparmak, C. Ecker, and L. Rezzolla, Astrophys. J. Lett. 939, L34 (2022), arXiv:2203.14974 [astro-ph.HE] .
  * de Tovar  _et al._ [2021] P. B. de Tovar, M. Ferreira, and C. Providência, Phys. Rev. D 104, 123036 (2021), arXiv:2112.05551 [nucl-th] .
  * Imam  _et al._ [2022] S. M. A. Imam, N. K. Patra, C. Mondal, T. Malik, and B. K. Agrawal, Phys. Rev. C 105, 015806 (2022), arXiv:2110.15776 [nucl-th] .
  * Mondal and Gulminelli [2022] C. Mondal and F. Gulminelli, Phys. Rev. D 105, 083016 (2022), arXiv:2111.04520 [nucl-th] .
  * Essick  _et al._ [2021] R. Essick, I. Tews, P. Landry, and A. Schwenk, Phys. Rev. Lett. 127, 192701 (2021).
  * Chirenti  _et al._ [2012] C. Chirenti, P. R. Silveira, and O. D. Aguiar, Int. J. Mod. Phys. Conf. Ser. 18, 48 (2012), arXiv:1205.2001 [gr-qc] .
  * Pratten  _et al._ [2020] G. Pratten, P. Schmidt, and T. Hinderer, Nature Commun. 11, 2553 (2020), arXiv:1905.00817 [gr-qc] .
  * Andersson and Kokkotas [1996] N. Andersson and K. D. Kokkotas, Phys. Rev. Lett. 77, 4134 (1996), arXiv:gr-qc/9610035 .
  * Andersson and Kokkotas [1998] N. Andersson and K. D. Kokkotas, Mon. Not. Roy. Astron. Soc. 299, 1059 (1998), arXiv:gr-qc/9711088 .
  * Benhar  _et al._ [1999] O. Benhar, E. Berti, and V. Ferrari, Mon. Not. Roy. Astron. Soc. 310, 797 (1999), arXiv:gr-qc/9901037 .
  * Benhar  _et al._ [2004] O. Benhar, V. Ferrari, and L. Gualtieri, Phys. Rev. D 70, 124015 (2004), arXiv:astro-ph/0407529 .
  * Tsui and Leung [2005] L. K. Tsui and P. T. Leung, Mon. Not. Roy. Astron. Soc. 357, 1029 (2005), arXiv:gr-qc/0412024 .
  * Chan  _et al._ [2014] T. K. Chan, Y. H. Sham, P. T. Leung, and L. M. Lin, Phys. Rev. D 90, 124023 (2014), arXiv:1408.3789 [gr-qc] .
  * Sotani [2021] H. Sotani, Phys. Rev. D 103, 123015 (2021), arXiv:2105.13212 [astro-ph.HE] .
  * Sotani and Kumar [2021] H. Sotani and B. Kumar, Phys. Rev. D 104, 123002 (2021), arXiv:2109.08145 [gr-qc] .
  * McDermott  _et al._ [1983] P. N. McDermott, H. M. van Horn, and J. F. Scholl, Astrophys. J. 268, 837 (1983).
  * Yoshida and Lee [2002] S. Yoshida and U. Lee, Astron. Astrophys. 395, 201 (2002), arXiv:astro-ph/0210591 .
  * Lau  _et al._ [2019] S. Y. Lau, P. T. Leung, and L. M. Lin, Phys. Rev. D 99, 023018 (2019), arXiv:1808.08107 [astro-ph.HE] .
  * Bandyopadhyay  _et al._ [2018] D. Bandyopadhyay, S. A. Bhat, P. Char, and D. Chatterjee, Eur. Phys. J. A 54, 26 (2018), arXiv:1712.01715 [astro-ph.HE] .
  * Han and Steiner [2019] S. Han and A. W. Steiner, Phys. Rev. D 99, 083014 (2019), arXiv:1810.10967 [nucl-th] .
  * von Doetinchem  _et al._ [2020] P. von Doetinchem _et al._ , JCAP 08, 035 (2020), arXiv:2002.04163 [astro-ph.HE] .
  * Baym  _et al._ [1971] G. Baym, C. Pethick, and P. Sutherland, Astrophys. J. 170, 299 (1971).
  * Carriere  _et al._ [2003] J. Carriere, C. J. Horowitz, and J. Piekarewicz, Astrophys. J. 593, 463 (2003), arXiv:nucl-th/0211015 .
  * Malik  _et al._ [2022] T. Malik, M. Ferreira, B. K. Agrawal, and C. Providência, Astrophys. J. 930, 17 (2022), arXiv:2201.12552 [nucl-th] .
  * Tews  _et al._ [2013] I. Tews, T. Krüger, K. Hebeler, and A. Schwenk, Phys. Rev. Lett. 110, 032504 (2013), arXiv:1206.0025 [nucl-th] .
  * Hebeler  _et al._ [2013] K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk, Astrophys. J. 773, 11 (2013), arXiv:1303.4662 [astro-ph.SR] .
  * Kurkela  _et al._ [2010] A. Kurkela, P. Romatschke, and A. Vuorinen, Phys. Rev. D 81, 105021 (2010), arXiv:0912.1856 [hep-ph] .
  * Fraga  _et al._ [2014] E. S. Fraga, A. Kurkela, and A. Vuorinen, Astrophys. J. Lett. 781, L25 (2014), arXiv:1311.5154 [nucl-th] .
  * Florian [1992] A. Florian, Probabilistic Engineering Mechanics 7, 123 (1992).
  * Lindblom [2010] L. Lindblom, Phys. Rev. D 82, 103011 (2010), arXiv:1009.0738 [astro-ph.HE] .
  * Lindblom [2018] L. Lindblom, Phys. Rev. D 97, 123019 (2018), arXiv:1804.04072 [astro-ph.HE] .
  * Annala  _et al._ [2020b] E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, Nature Phys. 16, 907 (2020b), arXiv:1903.09121 [astro-ph.HE] .
  * Kumar  _et al._ [2023] D. Kumar, H. Mishra, and T. Malik, JCAP 02, 015 (2023), arXiv:2110.00324 [hep-ph] .
  * Miller  _et al._ [2021] M. C. Miller _et al._ , Astrophys. J. Lett. 918, L28 (2021), arXiv:2105.06979 [astro-ph.HE] .
  * Abbott  _et al._ [2019] B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. X 9, 011001 (2019), arXiv:1805.11579 [gr-qc] .
  * Riley  _et al._ [2019] T. E. Riley _et al._ , Astrophys. J. Lett. 887, L21 (2019), arXiv:1912.05702 [astro-ph.HE] .
  * Miller  _et al._ [2019] M. C. Miller _et al._ , Astrophys. J. Lett. 887, L24 (2019), arXiv:1912.05705 [astro-ph.HE] .
  * Riley  _et al._ [2021] T. E. Riley _et al._ , Astrophys. J. Lett. 918, L27 (2021), arXiv:2105.06980 [astro-ph.HE] .
  * Doneva  _et al._ [2013] D. D. Doneva, E. Gaertig, K. D. Kokkotas, and C. Krüger, Phys. Rev. D 88, 044052 (2013), arXiv:1305.7197 [astro-ph.SR] .
  * Pradhan  _et al._ [2022] B. K. Pradhan, D. Chatterjee, M. Lanoye, and P. Jaikumar, Phys. Rev. C 106, 015805 (2022), arXiv:2203.03141 [astro-ph.HE] .
  * Yoshida and Kojima [1997] S. Yoshida and Y. Kojima, Mon. Not. Roy. Astron. Soc. 289, 117 (1997), arXiv:gr-qc/9705081 .
  * Kokkotas  _et al._ [2001] K. D. Kokkotas, T. A. Apostolatos, and N. Andersson, Mon. Not. Roy. Astron. Soc. 320, 307 (2001), arXiv:gr-qc/9901072 .
  * Pradhan and Chatterjee [2021] B. K. Pradhan and D. Chatterjee, Phys. Rev. C 103, 035810 (2021), arXiv:2011.02204 [astro-ph.HE] .