# Systematic analysis of the impacts of symmetry energy parameters on neutron
star properties

N. K. Patra1<EMAIL_ADDRESS>Anagh Venneti 2 Sk Md Adil Imam3,4
Arunava Mukherjee3,4<EMAIL_ADDRESS>B. K. Agrawal3,4
<EMAIL_ADDRESS>1Department of Physics, BITS-Pilani, K. K. Birla Goa
Campus, Goa 403726, India  2Department of Physics, BITS-Pilani, Hyderabad
Campus, Hyderabad - 500078, India  3Saha Institute of Nuclear Physics, 1/AF
Bidhannagar, Kolkata 700064, India  4Homi Bhabha National Institute, Anushakti
Nagar, Mumbai 400094, India.

###### Abstract

The impacts of various symmetry energy parameters on the properties of neutron
stars (NSs) have been recently investigated, and the outcomes are at variance,
as summarized in Table III of Phys. Rev. D 106, 063005 (2022). We have
systematically analyzed the correlations of slope and curvature parameters of
symmetry energy at the saturation density ($\rho_{0}=0.16\text{fm}^{-3}$) with
the tidal deformability and stellar radius of non-spinning neutron stars in
the mass range of $1.2-1.6M_{\odot}$ using a large set of minimally
constrained equations of state (EoSs). The EoSs at low densities correspond to
the nucleonic matter and are constrained by empirical ranges of a few low-
order nuclear matter parameters from the finite nuclei data and the pure
neutron matter EoS from chiral effective field theory. The EoSs at high
densities ($\rho>1.5-2\rho_{0}$) are obtained by a parametric form for the
speed of sound that satisfies the causality condition. Several factors
affecting the correlations between the NS properties and the individual
symmetry energy parameters usually considered in the literature are explored.
These correlations are quite sensitive to the choice of the distributions of
symmetry energy parameters and their interdependence. But, variations of NS
properties with the pressure of $\beta-$ equilibrated matter at twice the
saturation density remain quite robust which maybe due to the fact that the
pressure depends on the combination of multiple nuclear matter parameters that
describe the symmetric nuclear matter as well as the density dependence of the
symmetry energy. Our results are practically insensitive to the behavior of
EoS at high densities.

##  I Introduction

The equations of state (EoSs) of $\beta$-equilibrated charge neutral matter
and their connections to the properties of neutron stars (NSs) have been
studied for the last several decades Oppenheimer and Volkoff (1939); Tolman
(1939); Glendenning (1992). The precise knowledge of the properties of NS and
the data on Heavy ion collisions may constrain the behavior of EoSs at supra-
saturation densities Huth  _et al._ (2022). The radius and tidal deformability
from a population of neutron stars over a wide range of mass ($1-3M_{\odot}$)
would probe the EoS at densities upto a few times ($\sim 2-8$) of the
saturation density $\rho_{0}\approx 0.16$fm-3 encountered at the center of
finite nuclei. The tidal deformability parameter of NS, which encodes the
information about the EoS has been inferred for the first time from a
gravitational wave event GW170817 observed by the advanced-LIGO Aasi  _et al._
(2015) and advanced-Virgo detectors Acernese  _et al._ (2015) from a binary
neutron star (BNS) merger with a total mass of the system,
$2.74_{-0.01}^{+0.04}M_{\odot}$ Abbott et al. (2019a, b). Another subsequent
event GW190425, likely originating from the coalescence of BNSs was observed
Abbott et al. (2020). The BNS signals emitted from coalescing neutron stars
are likely to be observed more frequently in the upcoming runs of LIGO-Virgo-
KAGRA and the future detectors, e.g., Einstein Telescope Punturo  _et al._
(2010) and Cosmic Explorer Reitze  _et al._ (2019). The unprecedented
constraints on the EoS promised by gravitational wave astronomy through the
detailed analysis of gravitational wave parameter estimation have triggered
many theoretical investigations of the NS properties Abbott et al. (2017);
Malik  _et al._ (2018); De  _et al._ (2018); Fattoyev  _et al._ (2018); Forbes
_et al._ (2019); Landry and Essick (2019); Piekarewicz and Fattoyev (2019);
Biswas  _et al._ (2021); Abbott et al. (2020); Thi  _et al._ (2021). Recently,
two different groups of Neutron star Interior Composition Explorer (NICER)
X-ray telescopes provided neutron star’s mass and radius simultaneously for
PSR J0030+0451 with $R=13.02^{+1.24}_{-1.06}$km for mass
$1.44^{+0.15}_{-0.14}M_{\odot}$ Miller et al. (2019) and
$R=12.71^{+1.14}_{-1.19}$km for mass $1.34^{+0.15}_{-0.16}M_{\odot}$Riley et
al. (2019), which are the complementary constraints on the EoS. For heavier
pulsar PSR J0740+6620, $R=13.7^{+2.6}_{-1.5}$km with mass $2.08\pm
0.07M_{\odot}$ Miller et al. (2021) and $R=12.39^{+1.30}_{-0.98}$km with mass
$2.072^{+0.067}_{-0.066}M_{\odot}$ Riley et al. (2021) were reported. Current
observational lower bound on the maximum NS mass is $M_{\rm max}=2.35\pm
0.17M_{\odot}$ for the black-widow pulsar PSR J0952-0607 Romani  _et al._
(2022) that exceeds any previous measurements, including $M_{\rm
max}=2.27^{+0.17}_{-0.15}M_{\odot}$ for PSR J2215-5135Linares  _et al._
(2018). If the observational bounds are reliable, stiffer EoSs are required to
support the NS with a mass higher than $2M_{\odot}$.

The present lower bound on $M_{\rm max}$ suggests that the density at the
center of the star with canonical mass $1.4M_{\odot}$ may lie in the range
$\sim 2-3\rho_{0}$Hebeler  _et al._ (2013); Li and Steiner (2006). The
behavior of EoSs around $\rho_{0}$ may be important in determining the
properties of such NSs. It has been shown that the radius of a neutron star
with its mass in the range of $1-1.4M_{\odot}$ is strongly correlated with the
pressure for $\beta$-equilibrated matter at the densities $1-2\rho_{0}$
Lattimer and Prakash (2001). Similar analyses have been extended to the tidal
deformability, which is also found to be strongly correlated with pressure at
$2\rho_{0}$ Tsang  _et al._ (2019, 2020); Patra  _et al._ (2022). The EoS for
$\beta$-equilibrated matter can be decomposed into that for symmetric nuclear
matter and density-dependent symmetry energy. It may be important to constrain
them individually. Recently, it is shown in Ref. Imam  _et al._ (2022) that
the accurate knowledge of the equation of state of $\beta$-equilibrated matter
may not be resolved appropriately into its two main components, symmetric
nuclear matter, and density-dependent symmetry energy. Similar conclusions are
also drawn in references de Tovar  _et al._ (2021); Mondal and Gulminelli
(2022).

There have been several attempts to study the correlations of radius and tidal
deformability of a neutron star with individual nuclear matter parameters
which determine the density-dependence of symmetry energy Alam  _et al._
(2016); Carson  _et al._ (2019); Malik  _et al._ (2018); Tsang  _et al._
(2019); Güven  _et al._ (2020); Malik  _et al._ (2020); Tsang  _et al._
(2020); Malik and Agrawal (2021); Reed  _et al._ (2021); Pradhan  _et al._
(2022, 2023); Ghosh  _et al._ (2022a, b); Beznogov and Raduta (2022). The
nuclear matter parameters, often drawn randomly from uncorrelated uniform or
Gaussian distributions, are found to be weakly correlated with the NS
properties. The distributions of nuclear matter parameters obtained by fitting
the experimental data on finite nuclei properties within the mean-field models
are somewhat better correlated with the NS properties. Lately, the impact on
the NS properties due to constraints on the low-density EoS imposed by the
finite nuclei data have been investigated using more than four hundred mean-
field models derived from the non-relativistic Skyrme interactions and
relativistic lagrangians which describe the interactions of nucleons through
$\sigma$, $\omega$, and $\rho$ mesons Carlson  _et al._ (2023). The group of
models that describe very well the properties of symmetric and asymmetric
finite nuclei yields a stronger correlation between the tidal deformability
for the NS with canonical $1.4M_{\odot}$ mass star and the slope of the
symmetry energy at the saturation density ($\rho_{0}$). Several factors that
affect the correlations of NS properties with the nuclear matter parameters
are also summarized in Table III of Ref. Kunjipurayil  _et al._ (2022).

We have considered a large set of minimally constrained EoSs for the NS matter
in the present work to examine in detail the correlations between the
properties of a neutron star in the mass range $1.2-1.6M_{\odot}$ and the
parameters that govern the density dependence of symmetry energy. The EoSs at
low densities corresponds to the nucleonic matter in $\beta$-equilibrium and
are described by the nuclear matter parameters evaluated at $\rho_{0}$. These
EoSs are constrained by empirical values of the low-order nuclear matter
parameters determined by the experimental data on the bulk properties of
finite nuclei together with the pure neutron matter (PNM) EoS from a precise
next-to-next-to-next-to-leading-order (N3LO) calculation in chiral effective
field theory. The composition of NS matter at high density ($\rho>2\rho_{0}$)
is not very well known due to the possibility of the appearance of various new
degrees of freedom such as hyperons, kaons, and quarks Chatziioannou  _et al._
(2015); Chatterjee and Vidaña (2016); Stone  _et al._ (2021). Beyond a
transition density ($\rho_{\rm tr}$), taken to be $1.5-2\rho_{0}$, the EoSs
are constructed simply by imposing the causality condition on the speed of
sound and are independent of compositions of NS matter. The posterior
distributions of nuclear matter parameters that describe the low-density EoSs
are obtained within a Bayesian approach with minimal constraints. These
constraints introduce correlations among the nuclear matter parameters. The
joint posterior or correlated distribution of the nuclear matter parameters is
employed to study the sensitivity of NS properties to the parameters that
govern the density dependence of the symmetry energy. The calculations are
also performed for uncorrelated uniform and Gaussian distributions of the
nuclear matter parameters obtained by their marginalized posterior
distribution. The influence of the various correlations considered due to a
few other factors usually encountered in the literature are investigated.

The paper is organized as follows. We briefly outline our methodology in
Sec.II. The results for the correlations of NS properties with various
symmetry energy parameters at $\rho_{0}$ and the pressure for
$\beta$-equilibrated matter at $2\rho_{0}$ are discussed in detail in Sec.
III. The summary and outlook are presented in Sec. IV.

##  II Methodology

We discuss in brief the construction of the equation of state (EoS) at low and
high densities. The low-density EoS below 2$\rho_{0}$ is obtained using Taylor
expansion, and the high-density EoS is constructed keeping a check on the
speed of sound so that the causality is not violated.

###  II.1 EoS at Low density

We express energy per nucleon at a given density $\rho$ and asymmetry $\delta$
using parabolic approximation as,

$\displaystyle E(\rho,\delta)$ $\displaystyle=$ $\displaystyle
E(\rho,0)+E_{\rm sym}(\rho)\delta^{2}+...,$ (1)

where $\delta=\frac{\rho_{n}-\rho_{p}}{\rho}$ is determined using the
$\beta$-equilibrium and the charge neutrality conditions. The energy per
nucleon for the symmetric nuclear matter, $E(\rho,0)$ and the density-
dependent symmetry energy, $E_{\rm sym}(\rho)$ are expanded around $\rho_{0}$
using individual nuclear matter parameters as Chen  _et al._ (2005, 2009);
Newton  _et al._ (2014); Margueron  _et al._ (2018); Margueron and Gulminelli
(2019),

$\displaystyle E(\rho,0)$ $\displaystyle=$ $\displaystyle
e_{0}+\frac{1}{2}K_{0}\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)^{2}+\frac{1}{6}Q_{0}\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)^{3},$
(2) $\displaystyle E_{\rm sym}(\rho)$ $\displaystyle=$ $\displaystyle
J_{0}+L_{0}\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)+\frac{1}{2}K_{\rm
sym,0}\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)^{2}$ (3) $\displaystyle+$
$\displaystyle\frac{1}{6}Q_{\rm
sym,0}\left(\frac{\rho-\rho_{0}}{3\rho_{0}}\right)^{3}.$

In Eqs. (2) and (3), $e_{0}$ is the binding energy per nucleon, $K_{0}$ is the
incompressibility coefficient, $J_{0}$ is the symmetry energy coefficient, its
slope parameter $L_{0}$, $K_{\rm sym,0}$ the symmetry energy curvature
parameter, $Q_{0}[Q_{\rm sym,0}]$ is related to third-order density
derivatives of $E(\rho,0)$ $[E_{\rm sym}(\rho)]$.

###  II.2 EoS at High density

We impose the causality condition on the speed of sound to construct the EoS
beyond the transition density ($\rho_{\rm tr}$), which is taken to be
$1.5-2\rho_{0}$. The high-density part of the EoS ($\rho>\rho_{\rm tr}$) joins
smoothly to the one at the low density such that the velocity of the sound
never exceeds the velocity of light and asymptotically approaches the
conformal limit ($c_{s}^{2}$ = $\frac{1}{3}c^{2}$). The velocity of sound for
$\rho>\rho_{\rm tr}$ is given asTews  _et al._ (2018),

$\displaystyle\frac{c_{s}^{2}}{c^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{3}-c_{1}exp\left[{-\frac{(\rho-
c_{2})^{2}}{n_{b}^{2}}}\right]+h_{p}exp\left[-\frac{(\rho-
n_{p})^{2}}{w_{p}^{2}}\right]$ (4) $\displaystyle\left[1+erf(s_{p}\frac{\rho-
n_{p}}{w_{p}})\right].$

where the peak height $h_{p}$ determines the maximum speed of sound, the
position $n_{p}$ determines the density around which it happens, the width of
the curve controls by $w_{p}$ and $n_{b}$, and the shape or skewness parameter
$s_{p}$. For a given value of $n_{b}$, the parameters $c_{1}$ and $c_{2}$ are
determined by the continuity of the speed of sound and its derivative at the
transition density $\rho_{\rm tr}$. The values of $n_{b}$, $h_{p}$, $w_{p}$,
and $n_{p}$ are drawn from the uniform distribution with ranges in between
0.01-3.0 fm-3, 0.0-0.9, 0.1-5.0 fm-3, and ($\rho_{\rm tr}$ \+ 0.08) - 5.0
fm-3, respectively Tews  _et al._ (2018). We have taken $s_{p}$ equal to zero
throughout our calculations as it does not have much effect on the stiffness
of EoS.

We construct the high-density equation of state starting from transition
density ($\rho_{\rm tr}$), where the energy density ($\epsilon(\rho_{\rm
tr})$), the pressure (P($\rho_{\rm tr}$)) and the derivative of energy density
($\epsilon^{\prime}(\rho_{\rm tr})$) are known. The successive values of
$\epsilon$ and P are obtained by assuming a step size $\Delta\rho=0.001$ fm-3
as follows,

$\displaystyle\rho_{i+1}$ $\displaystyle=$ $\displaystyle\rho_{i}+\Delta\rho,$
(5) $\displaystyle\epsilon_{i+1}$ $\displaystyle=$
$\displaystyle\epsilon_{i}+\Delta\epsilon$ (6) $\displaystyle=$
$\displaystyle\epsilon_{i}+\Delta\rho\frac{\epsilon_{i}+P_{i}}{\rho_{i}},$
$\displaystyle P_{i+1}$ $\displaystyle=$ $\displaystyle
P_{i}+c_{s}^{2}(\rho_{i})\Delta\epsilon.$ (7)

where, the index i = 0 refers to the transition density $\rho_{\rm tr}$ .
Note, in the Eq.(6) $\Delta\epsilon$ has been evaluated using the
thermodynamic relation $P=\rho\partial\epsilon/{\partial\rho}-\epsilon$ valid
at zero temperature. Once the EoS has been generated, the NS properties, such
as the radius and tidal deformability as a function of mass, are obtained by
solving the Tolman-Oppenheimer-Volkoff (TOV) equations by varying the central
density of the star.

###  II.3 Bayesian estimation of nuclear matter parameters

The detailed statistical analysis of the parameters of a model for a given set
of data can be carried out in the Bayesian approach. It gives the joint
posterior distributions of model parameters by which one can study the
distributions of given parameters and correlations among the parameters. Based
on the Bayes theorem, the joint posterior distribution of the parameters
$P(\bm{\theta}|D)$ is given as Gelman et al. (2013),

$P(\bm{\theta}|D)=\frac{{\mathcal{L}}(D|\bm{\theta})P(\bm{\theta})}{\mathcal{Z}},$
(8)

where D and $\bm{\theta}$ are the data and set of model parameters,
respectively. Here $P(\bm{\theta})$ is the prior for model parameters,
$\mathcal{L}(D|\bm{\theta})$ is the likelihood function, and $\mathcal{Z}$ is
the evidence. The marginalized posterior distribution for a parameter
$\theta_{i}$ can be obtained as

$P(\theta_{i}|D)=\int P(\bm{\theta}|D)\prod_{k\not=i}d\theta_{k}.$ (9)

We use the Gaussian likelihood function, defined as

$\displaystyle{\mathcal{L}}(D|\bm{\theta})$ $\displaystyle=$
$\displaystyle\prod_{j}\frac{1}{\sqrt{2\pi\sigma_{j}^{2}}}e^{-\frac{1}{2}\left(\frac{d_{j}-m_{j}(\bm{\theta)}}{\sigma_{j}}\right)^{2}}.$
(10)

where the index $j$ runs over all the data, $d_{j}$ and $m_{j}$ are the data
and corresponding model values, respectively. The $\sigma_{j}$s correspond to
the adopted uncertainties. The posterior distribution of Eq. (8) can be
evaluated by implementing a nested sampling algorithm. We have used the
Pymultinest nested sampling Buchner  _et al._ (2014) in the Bayesian Inference
Library Ashton et al. (2019).

##  III Results and Discussion

The EoSs for the asymmetric nuclear matter at low densities are expressed in
terms of nuclear matter parameters which are evaluated at the saturation
density as outlined in Sec. II.1 (Eqs.( 2) & ( 3)). The joint posterior
distribution of nuclear matter parameters is obtained within a Bayesian
approach using minimal constraints. These constraints introduce correlations
among the nuclear matter parameters. The EoSs at high density ($\rho>\rho_{\rm
tr}$) are constructed by simply imposing the causality condition on the speed
of sound as outlined in Sec. II.2. A large set of EoSs is constructed by
appropriately combining the low and high-density parts. These EoSs are
employed to evaluate various NS properties, like radius, tidal deformability,
and maximum mass. We assess the dependence of radius and tidal deformability
on the slope and curvature parameters of the symmetry energy evaluated at
$\rho_{0}$ as well as on the pressure $P(2\rho_{0})$ for the
$\beta$-equilibrated matter. The sensitivity of these dependencies to the
various factors as follows are analyzed,

  * (i)

the behavior of the high-density part of the EoS,

  * (ii)

the choice of distributions of nuclear matter parameters, their
interdependence, and uncertainties,

  * (iii)

the lower bound on the maximum mass of the stable neutron stars,

  * (iv)

the value of the transition density,

  * (v)

upper bound on the value of tidal deformability.

###  III.1 Priors and Posterior distributions of NMPs

Table 1: Uniform prior distributions are assumed for all the NMPs except for $e_{0}$, which is kept fixed to -16.0 MeV. The minimum (min.) and maximum (max.) values of the NMPs are listed in the units of MeV. | $K_{0}$ | $Q_{0}$ | $J_{0}$ | $L_{0}$ | $K_{\rm sym,0}$ | $Q_{\rm sym,0}$
---|---|---|---|---|---|---
min. | 190 | -1200 | 30 | 0 | -500 | -250
max. | 290 | 400 | 35 | 100 | 300 | 1350
Figure 1: (Color Online)Corner plot for the nuclear matter parameters (in
MeV). The one-dimensional marginalized posterior distributions (salmon) and
the prior distributions (green) lines are displayed along the diagonal plots.
The vertical dashed lines indicate 68$\%$ (1$\sigma$) confidence interval. The
confidence ellipses for two-dimensional posterior distributions are plotted
with 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence intervals along the off-
diagonal plots.

The posterior distributions of the NMPs are obtained by subjecting the EoSs to
a set of minimal constraints, which includes the basic nuclear matter
properties at the saturation density and the EoS for the pure neutron matter
at low densities from N3LO calculation in the chiral effective field theory
Hebeler  _et al._ (2013). Only a few low-order NMPs are well constrained, such
as the binding energy per nucleon in symmetric nuclear matter $e_{0}\sim-16.0$
MeV and symmetry energy coefficient $J_{0}=32.5\pm 2.5$ MeV from the binding
energy of finite nuclei over a wide range of nuclear masses Chabanat  _et al._
(1998, 1997); Malik  _et al._ (2019); Mondal  _et al._ (2015, 2016); Sulaksono
_et al._ (2009); Essick  _et al._ (2021a). The nuclear matter
incompressibility coefficient, $K_{0}=240\pm 50$ MeV for the symmetric nuclear
matter is constrained from the experimental data on the centroid energy of
isoscalar giant monopole resonance in a few heavy nuclei Garg and Colò (2018);
Agrawal  _et al._ (2005). The value of symmetry energy slope parameter has
been deduced by the neutron-skin thickness of 48Ca nucleus by CREX Adhikari
_et al._ (2022) and that in 208Pb by PREX-2 collaboration Adhikari  _et al._
(2021). The PREX-2 data analysis performed by Reed et al. in Ref. Reed  _et
al._ (2021); Reed and Horowitz (2020) places $L_{0}=106\pm 37$ MeV. Other
studies combining astronomical observations and PREX-2 data is
$L_{0}=53^{+14}_{-15}$ MeV Essick  _et al._ (2021b). A smaller value of
$L_{0}=54\pm 8$ MeV has also been inferred from PREX-2 data Reinhard  _et al._
(2021). The CREX data predicts $L_{0}=0-51$ MeV Tagami  _et al._ (2022). The
remaining nuclear matter parameters, $Q_{0}$, $K_{\rm sym,0}$ and $Q_{\rm
sym,0}$ appearing in Eqs.( 2) & ( 3) are only weakly constrained Tsang  _et
al._ (2020); Ferreira and Providência (2021); Dutra  _et al._ (2012, 2014);
Mondal  _et al._ (2017); Patra  _et al._ (2022). The prior for the binding
energy per nucleon is kept fixed to $e_{0}=-16.0$ MeV throughout. The prior
distributions of $J_{0}$ and $K_{0}$ are assumed to be uniform with a rather
small range, whereas the other higher-order nuclear matter parameters
correspond to uniform distributions with large ranges. We have listed the
assumed prior distributions for each of the nuclear matter parameters in Table
1. The values of $d$ and $\sigma$ in Eq.(10) are taken from Ref. Lattimer
(2021); Hebeler  _et al._ (2013) for the energy per neutron, and we consider a
$6\times$N3LO uncertainty band for our calculations. In addition to likelihood
and priors, we have imposed a few filters on the nuclear matter parameters:
(i) pressure for the $\beta$-equilibrated matter should increase monotonically
with density (thermodynamic stability),(ii) symmetry energy is positive semi-
definite and (iii) maximum mass of neutron star must exceed $2M_{\odot}$
(observational constraint).

The joint posterior distribution of the NMPs for a given model is the product
of the likelihood and the prior distribution of NMPs (Eq. (8)). The posterior
distribution of individual parameters can be obtained by marginalizing the
joint posterior distribution with the remaining model parameters. If the
marginalized posterior distribution of the parameter is narrowed down as
compared to the corresponding prior distribution (uniform distribution in this
case), then the parameter is said to be constrained by the given data or the
likelihood functions. Therefore the narrower distributions of parameters
compared to their prior distributions indicate the importance of the
likelihood function. The likelihood function imposes additional constraints on
the multi-variate nuclear matter parameters of our model driven by the data.
The corner plots for the nuclear matter parameters, which yield the EoSs
consistent with the minimal constraints, are shown in Fig. 1. The median
values of the nuclear matter parameters and the $68\%$ confidence intervals
are given in the diagonal plots of the figure. The 68% confidence intervals
for $L_{0}$, $Q_{0}$, and $K_{\rm sym,0}$ are significantly smaller than their
prior ranges implying these parameters are well constrained by the low-density
EoS for the pure neutron matter. The values of $J_{0}$, and $Q_{\rm sym,0}$are
also constrained to some extent. Except for $L_{0}-K_{\rm sym,0}$ (r = 0.8)
and $L_{0}-J_{0}$ (r = 0.65), all other pairs of nuclear matter parameters do
not show any visible correlations.

###  III.2 Neutron star properties

The EoSs for $\beta$-equilibrated charge neutral matter in the density ranges
$0.5\rho_{0}$ to the transition density $\rho_{\rm tr}$ are obtained using
Taylor expansion with NMPs corresponding to the posterior distribution as
displayed in the Fig. 1. The calculations are performed assuming different
values for the $\rho_{\rm tr}$ in the range of $1.5-2\rho_{0}$. Each of the
EoSs beyond $\rho_{\rm tr}$ is smoothly joined by a diverse set of EoSs, which
are obtained simply by imposing the causality condition on the speed of sound
by following the Eqs.(4-7). The EoS for the density ranges $\rho<0.5\rho_{0}$
comprises outer and inner crusts. We have. for used the EoS for the outer
crust by Baym-Pethick-Sutherland Baym  _et al._ (1971) in the density range
$3.9\times 10^{-11}\rho_{0}<\rho<0.0016\rho_{0}$. We have assumed a polytropic
form of the EoS for the inner crust as followsCarriere  _et al._ (2003),

$p(\varepsilon)=\alpha+\beta\varepsilon^{\frac{4}{3}}.$ (11)

Here the parameters $\alpha$ and $\beta$ are determined in such a way that the
EoS for the inner crust matches with the outer crust at one end and with the
outer core at the other end. There is a greater sensitivity to the treatment
of crust EoS for neutron stars with mass $\sim 1M_{\odot}$ Fortin  _et al._
(2016). The treatment of crust EoS employed in the present work may introduce
the uncertainties of about 50-100 meters in radii of NSs having a mass
1.4$M_{\odot}$. In Ref. Piekarewicz and Fattoyev (2019), it is shown that the
choice of EoS for the inner crust affects both the Love number $k_{2}$ and
compactness parameter in such a way that the values of the tidal deformability
parameter remain practically unaltered. Once the EoSs for the core and crust
are determined, the values of neutron star mass, radius, and tidal
deformability corresponding to a given central pressure can be obtained by
solving Tolman-Oppenheimer-Volkoff equations Oppenheimer and Volkoff (1939);
Tolman (1939).

Figure 2: (Color Online) The energy per neutron ($E_{n}(\rho)=E(\rho,1)$) for
pure neutron matter as a function of neutron density. The colored bands
correspond to 6 $\times$ N3LO (light green), 2 $\times$ N3LO (light red), and
90% confidence interval for the EoSs (light blue) obtained in our calculation
(see text for details). Figure 3: (Color Online) The pressure for
$\beta$-equilibrated charge neutral matter as a function of nucleon density.
The colored bands correspond to 90% confidence intervals for the EoSs with
different ranges of the square of the speed of sound at the center of NS with
maximum mass ( $c_{s,\rm max}^{2}$)(see text for details).

The 90% confidence interval for pure neutron matter EoSs (light blue) for the
density range ( $0.5-2.0\rho_{0}$) is shown in Fig. 2. For comparison, colored
bands correspond to 6 $\times$ N3LO (light green), 2 $\times$ N3LO (light red)
from chiral effective field theory are displayed. Our EoSs lie almost in the
middle of the 6 $\times$ N3LO band and significantly satisfy the 2 $\times$
N3LO band. This indicates that our EoSs are well fitted with the pure neutron
matter EoS from a precise next-to-next-to-leading-order calculation in chiral
effective field theory. The 90$\%$ confidence interval for the pressure of the
$\beta$-equilibrated matter is plotted as a function of density in Fig. 3. The
results are divided into three groups depending upon the square of sound speed
($c_{s,\rm max}^{2}$) at the center of NS of its maximum stable mass
configuration. The three groups of EoSs correspond to $c_{s,\rm
max}^{2}=0.5-0.65c^{2}$, $0.65-0.8c^{2}$, and $0.8-1.0c^{2}$ are depicted by
different colors. All the EoSs in each group are plotted upto the same density
7$\rho_{0}$. The overall stiffness of the EoS increases with the $c_{s,\rm
max}^{2}$.

Figure 4: (Color Online) The mass-radius relationship (left panel) obtained
for the EoSs as shown in Fig. 3. The outer and inner gray shaded regions
indicate the 90% (solid) and 50% (dashed) confidence intervals from the LIGO-
Virgo analysis for BNS components of GW170817 event Abbott et al. (2018);
Abbott et al. (2019a); Abbott et al. (2021). The rectangular regions enclosed
by blue & black dashed lines indicate the constraints from the millisecond
pulsar PSR J0030+0451 from NICER X-ray observation Riley et al. (2019); Miller
et al. (2019), and PSR J0740+6620 orange dashed line Riley et al. (2021). The
right panel displays tidal deformability versus NS mass. The orange shaded
region is the observation for $\Lambda$ with 90% posterior interval from the
LIGO/Virgo Collaboration (GW170817 event) Abbott et al. (2018). The black line
correspond to observational bounds on $\Lambda_{1.4}=190^{+390}_{-120}$Abbott
et al. (2019a). For the comparison, we have shown violet and gold lines
corresponding to $\Lambda_{1.8}=70-270$, and $\Lambda_{2.0}=30-150$ obtained
from a few well-known theoretical models Piekarewicz and Fattoyev (2019); Xie
_et al._ (2022)

.

The EoSs displayed in Fig. 3 are employed to obtain the mass-radius
relationship for static neutron stars as presented in Fig. 4 (see left panel).
For the comparison, we also display the constraints obtained from the GW170817
event and NICER x-ray observation. The maximum mass of a neutron star lies in
the range of $2.1-2.7M_{\odot}$, and the radius for a neutron star with mass
1.4$M_{\odot}$ lies in the range 11.8-14 km. Our mass-radius relationships
exclude smaller values of radius for a given mass, as predicted by the
GW170817 event. This is due to the choice of priors for the low-order nuclear
matter parameters constrained by the experimental data on bulk properties of
the finite nuclei. In the right panel of Fig. 4, we plot the variations of
tidal deformability as a function of mass. We display the constraints obtained
from the GW170817 event for comparison. The value of tidal deformability
$\Lambda_{1.4}$ is highlightedAbbott et al. (2019a). Further, we depict the
constraints on $\Lambda$ for NS of mass 1.8$M_{\odot}$, and 2.0$M_{\odot}$
within 90% CI obtained from ten realistic models that can accurately describe
the finite nuclei properties and support the $2M_{\odot}$ neutron star
massesPiekarewicz and Fattoyev (2019); Xie  _et al._ (2022). The values of
tidal deformability obtained with our minimally constrained EoSs have a
reasonable overlap with the ones inferred from the GW170817 event.

Figure 5: (Color Online) Corner plot for the central density ($\rho_{c}$ in
$\rho_{0}$) and corresponding pressure ($P_{c}$ in MeV.fm-3) for the neutron
star with canonical and maximum mass, radius ($R_{1.4}$ in km), tidal
deformability ($\Lambda_{1.4}$) and maximum mass ($M_{\rm max}$ in
$M_{\odot}$) of NS. The confidence ellipses for two-dimensional posterior
distributions are plotted with $1\sigma$(solid line) and $2\sigma$ (dashed
line) confidence intervals along the off-diagonal plots. The vertical dashed
lines indicate 68% confidence intervals.

In Fig. 5, we display the corner plot describing various quantities associated
with neutron stars, such as the central density $\rho_{c}$ and corresponding
pressure $P_{c}$ for the neutron star with canonical ($1.4M_{\odot}$) and
maximum mass, radius $R_{1.4}$, tidal deformability $\Lambda_{1.4}$ and
maximum mass $M_{\rm max}$. To understand the impact of the high-density EoS,
the distributions of all the quantities are segregated into three different
groups of stiffness for the EoSs according to the range of $c_{s,\rm max}^{2}$
at the higher density part as indicated with different colors. It can be seen
from the diagonal plots that the distributions for $\rho_{c,1.4}$,
$P_{c,1.4}$, $\Lambda_{1.4}$ and $R_{1.4}$ are more-or-less independent of
$c_{s,\rm max}^{2}$. The median values of $\rho_{c,\rm max}$ decrease with
$c_{s,\rm max}^{2}$, but, $M_{\rm max}$ and $P_{c,\rm max}$ increase. The
$\rho_{c,1.4}$, $P_{c,1.4}$, $R_{1.4}$ and $\Lambda_{1.4}$ are strongly
correlated with each other, the absolute values of Pearson’s correlation
coefficients being r$\sim 0.86-0.98$. The $\rho_{c,\rm max}$ and $P_{c,\rm
max}$ are moderately correlated with the properties of NS with canonical mass
($\mid r\mid\sim 0.70-0.85$), except for $\rho_{c,1.4}$. The $M_{\rm max}$
show strong correlations with $\rho_{c,\rm max}$ ($\mid r\mid\sim 0.94$), but,
relatively weakly correlated with $P_{c,\rm max}(\mid r\mid\sim 0.78)$. The
maximum mass of a stable neutron star seems to be weakly correlated with the
properties of NS with canonical mass.

###  III.3  Neutron star properties and symmetry energy parameters

Figure 6:  (Color Online) The tidal deformability ($\Lambda_{1.4}$) as a
function of slope parameter ($L_{0}$) , curvature parameter ($K_{\rm sym,0}$)
and the pressure of $\beta$-equilibrated matter (P(2$\rho_{0}$)). Figure 7:
(Color Online) Variations of tidal deformability ($\Lambda_{1.4}$) with slope
parameter ($L_{0}$) for three different distributions of the nuclear matter
parameters which are with uniform uncorrelated (U-Unc), Gaussian uncorrelated
(G-Unc) and Gaussian correlated posterior distributions (G-Cor) as discussed
in the text in details. The results are obtained by considering the EoSs
associated with maximum NS mass $M_{\rm max}$ $\geqslant$ 2.1$M_{\odot}$. The
circle (orange) symbols represent the results obtained by varying all
parameters, whereas the (blue) star symbols represent those obtained by fixing
$K_{0}=240$MeV and $J_{0}=32$ MeV.

The correlations of NS properties with various symmetry energy parameters and
the pressure of $\beta-$equilibrated matter have been extensively investigated
earlier using non-parametric Essick  _et al._ (2021b), parametric Tsang  _et
al._ (2020); Biswas (2021), and physics-based models Malik  _et al._ (2020);
Reed  _et al._ (2021); Pradhan  _et al._ (2022); Carson  _et al._ (2019);
Essick  _et al._ (2021b) and found to yield the results which are sometimes at
variance, as summarized in Table III of Ref. Kunjipurayil  _et al._ (2022).
Some of these studies include constraints imposed by bulk nuclear properties,
while others have also imposed those through the constraints on nuclear matter
parameters assuming them to be independent of each other. We now study the
correlations of various NS properties with the $L_{0}$, $K_{\rm sym,0}$ and
$P(2\rho_{0})$ using our minimally constrained EoSs and assess how they are
affected by the several factors as listed at the beginning of this section.

We have seen in the previous subsections that our EoSs yield the various
properties of neutron stars within reasonable observational as well as
theoretical bounds. As mentioned earlier, the EoSs for $\rho\leqslant\rho_{\rm
tr}$ are obtained with the nuclear matter parameters, which are interdependent
due to the minimal constraints. We employ these EoSs to study the variations
of radius and tidal deformability with the slope ($L_{0}$), curvature ($K_{\rm
sym,0}$) and $P(2\rho_{0})$ for NS with mass $1.2-1.6M_{\odot}$. The
parameters $L_{0}$ and $K_{\rm sym,0}$ determine the density dependence of
symmetry energy. We first consider in detail our results for the tidal
deformability corresponding to the NS with canonical mass assuming $\rho_{\rm
tr}=2\rho_{0}$. We show the dependence of tidal deformability $\Lambda_{1.4}$
on the $L_{0}$, $K_{\rm sym,0}$ and $P(2\rho_{0})$ in Fig. 6. The results for
all three groups of EoSs corresponding to different values of $c_{s,\rm
max}^{2}$ overlap with each other, indicating that the values of
$\Lambda_{1.4}$ seem to be more-or-less insensitive to the behavior of EoS at
high densities $(\rho>\rho_{\rm tr}$). The $\Lambda_{1.4}$ tend to increase
with $L_{0}$, $K_{\rm sym,0}$ and $P(2\rho_{0})$. The values of correlation
coefficients, as indicated in Fig. 6, are practically independent of the
choice of $c_{s,\rm max}^{2}$. Hereafter, we show the results obtained by
combining all three groups of the EoSs.

To study the impact of the interdependence of nuclear matter parameters on the
results shown in Fig. 6, we have generated two different distributions of
nuclear matter parameters with the help of their posterior distributions, as
shown in Fig. 1. These distributions of the nuclear matter parameters are (a)
uncorrelated Uniform (U-Unc) and (b) uncorrelated Gaussian (G-Unc). The
parameters of U-Unc and G-Unc distributions are obtained from the 95%
confidence interval of the marginalized distributions of Fig. 1. Often, U-Unc
and G-Unc distributions have been employed to study the correlations of
$\Lambda_{1.4}$ with various symmetry energy parameters Malik  _et al._
(2020); Pradhan  _et al._ (2022); Carson  _et al._ (2019); Tsang  _et al._
(2020).

In Fig. 7, we plot the variations of dimensionless tidal deformability
$\Lambda_{1.4}$ with $L_{0}$ for the three different distributions of nuclear
matter parameters as indicated (orange circles). The extreme right panels are
labeled as G-Cor, corresponding to the ones obtained using correlated or joint
posterior distribution of nuclear matter parameters. The results shown are
obtained by considering the EoSs associated with the maximum NS mass $M_{\rm
max}$ $\geqslant$ 2.1$M_{\odot}$. For the comparison, the results obtained by
fixing the lower order parameters $K_{0}=240$MeV and $J_{0}=32$MeV are also
displayed (blue stars). In the figure, Pearson’s correlation coefficients are
given for all three distributions of NMPs. The correlations of $\Lambda_{1.4}$
are very sensitive to the choice of the distributions of nuclear matter
parameters. The $\Lambda_{1.4}$ is very weakly correlated with $L_{0}$ for the
case of U-Unc with a correlation coefficient $r\sim 0.39$.. for The situation
somewhat improves for the case of nuclear matter parameters corresponding to
uncorrelated Gaussian distribution as indicated by G-Unc ($r\sim 0.48$). The
posterior distribution (G-Cor) of nuclear matter parameters obtained from
minimal constraints (See Fig. 1) yields relatively stronger correlations of
$\Lambda_{1.4}$ with $L_{0}$ ($r\sim 0.72$). The correlations also increase
marginally when the values of low-order nuclear matter parameters such as
$K_{0}$ and $J_{0}$ are kept fixed (blue stars). Evidently, the correlations
of $\Lambda_{1.4}$ with $L_{0}$ depend on the various factors, such as the
constraints imposed on the nuclear matter parameters that govern the low-
density behavior of the EoSs.

Figure 8: (Color Online)The variation of the tidal deformability
($\Lambda_{1.4}$) with curvature parameter ($K_{\rm sym,0}$) for three
different nuclear matter parameter distributions with uniform uncorrelated
(U-Unc), Gaussian uncorrelated (G-Unc) and posterior distributions (G-Cor) are
discussed in details in the text. The results are shown for those EoSs which
are associated with a maximum mass of NS $\geqslant$ 2.1$M_{\odot}$. The
symbols in circles (orange) represent the results obtained by varying all
parameters, whereas stars (blue) symbols represent the results obtained when
$K_{0}=240$MeV and $J_{0}=32$MeV are fixed. Figure 9: A variation in tidal
deformability ($\Lambda_{1.4}$) with the pressure of $\beta$-equilibrated
matter at 2$\rho_{0}$ based on three different nuclear matter parameter
distributions, namely uniform uncorrelated (U-Unc), Gaussian uncorrelated
(G-Unc) and posterior distributions (G-Cor), are discussed in detail in the
text. The results are shown for those EoSs which are associated with a maximum
mass of NS $\geqslant$ 2.1$M_{\odot}$. As shown by the circles (orange)
symbols, results obtained with all parameters varied, while those obtained
with $K_{0}=240$MeV and $J_{0}=32$MeV fixed are represented by the stars
(blue) symbols.

The variations of $\Lambda_{1.4}$ with $K_{\rm sym,0}$ for different
distributions of nuclear matter parameters are plotted in Fig. 8. The
correlations for $\Lambda_{1.4}$ with $K_{\rm sym,0}$ for different cases are
stronger in comparison to those obtained with $L_{0}$. It appears that the
correlations of $\Lambda_{1.4}$ with various symmetry energy parameters are
quite sensitive to the distributions of nuclear matter parameters employed.
Our results for the correlations for the U-Unc case are qualitatively similar
to those obtained in Refs. Pradhan  _et al._ (2022); Carson  _et al._ (2019)
with a similar strategy for nuclear matter parameters but with different
models. Similar qualitative trends are observed for the G-Unc Tsang  _et al._
(2020). The correlations significantly improve even with the inclusion of
minimal constraints, as indicated by G-Cor. Our results for the case of G-Cor
are in harmony with those obtained very recently using about 400 non-
relativistic and relativistic mean field models Carlson  _et al._ (2023),
which demonstrates the impact of low-density EoSs on the properties of NS of
canonical mass. It was found that tighter constraints on the bulk properties
of finite nuclei, such as binding energy, charge radii, and isoscalar giant
monopole resonance energy, yield stronger correlations of $\Lambda_{1.4}$ with
$L_{0}$ and $K_{\rm sym,0}$. The correlations of $\Lambda_{1.4}$ with various
symmetry energy parameters are stronger only when the nuclear matter
parameters evaluated at saturation densities that govern the low-density
behavior of EoSs, are appropriately constrained.

Table 2:  The Pearson’s correlation coefficients for $\Lambda_{1.4}$ with the slope parameter ($L_{0}$), curvature parameter ($K_{\rm sym,0}$) of the symmetry energy and the pressure for $\beta$-equilibrated matter at a density $2\rho_{0}$ ($P(2\rho_{0})$) obtained using joint posterior distribution of the nuclear matter parameters. The results are obtained assuming different upper bounds on $\Lambda_{1.4}$ and those are associated with a maximum mass of NS $\geqslant$ 2.1$M_{\odot}$. Upper bound | $\Lambda_{1.4}-L_{0}$ | $\Lambda_{1.4}-K_{\rm sym,0}$ | $\Lambda_{1.4}-P(2\rho_{0})$
---|---|---|---
on $\Lambda_{1.4}$ |  |  |
400 | 0.29 | 0.42 | 0.88
600 | 0.54 | 0.71 | 0.97
800 | 0.67 | 0.81 | 0.98
1000 | 0.72 | 0.84 | 0.98
Table 3:  The Pearson’s correlation coefficients for $\Lambda_{M}$ and $R_{M}$, corresponding to different neutron star mass $M$ with the slope parameter ($L_{0}$), curvature parameter ($K_{\rm sym,0}$) of the symmetry energy and the pressure for $\beta$-equilibrated matter at a density $2\rho_{0}$ ($P(2\rho_{0})$). The correlation coefficients are obtained at different transition densities and for the lower bound on maximum neutron star mass $M_{\rm max}$ $\geqslant$ 2.1$M_{\odot}$. $\frac{\rho_{\rm tr}}{\rho_{0}}$ | $\frac{M}{M_{\odot}}$ |  | $\Lambda_{M}$ |  |  | $R_{M}$ |
---|---|---|---|---|---|---|---
$L_{0}$ | $K_{\rm sym,0}$ | $P(2\rho_{0})$ | $L_{0}$ | $K_{\rm sym,0}$ | $P(2\rho_{0})$
1.5 | 1.2 | 0.76 | 0.85 | 0.91 | 0.83 | 0.67 | 0.76
1.4 | 0.70 | 0.81 | 0.93 | 0.79 | 0.70 | 0.81
1.6 | 0.60 | 0.76 | 0.91 | 0.72 | 0.70 | 0.86
1.75 | 1.2 | 0.77 | 0.87 | 0.94 | 0.84 | 0.70 | 0.81
1.4 | 0.71 | 0.83 | 0.96 | 0.80 | 0.74 | 0.88
1.6 | 0.61 | 0.77 | 0.97 | 0.73 | 0.72 | 0.91
2.0 | 1.2 | 0.80 | 0.88 | 0.97 | 0.85 | 0.73 | 0.86
1.4 | 0.72 | 0.84 | 0.98 | 0.81 | 0.75 | 0.91
1.6 | 0.65 | 0.80 | 0.98 | 0.75 | 0.75 | 0.94
Table 4:  Same as Table 3 but, the results are obtained using the uncertainty on the PNM EoS reduced to 3$\times$ N3L0. $\frac{\rho_{\rm tr}}{\rho_{0}}$ | $\frac{M}{M_{\odot}}$ |  | $\Lambda_{M}$ |  |  | $R_{M}$ |
---|---|---|---|---|---|---|---
$L_{0}$ | $K_{\rm sym,0}$ | $P(2\rho_{0})$ | $L_{0}$ | $K_{\rm sym,0}$ | $P(2\rho_{0})$
1.5 | 1.2 | 0.47 | 0.73 | 0.92 | 0.68 | 0.65 | 0.74
1.4 | 0.36 | 0.64 | 0.93 | 0.57 | 0.66 | 0.83
1.6 | 0.26 | 0.58 | 0.88 | 0.45 | 0.65 | 0.84
1.75 | 1.2 | 0.50 | 0.76 | 0.95 | 0.70 | 0.67 | 0.81
1.4 | 0.39 | 0.72 | 0.96 | 0.59 | 0.68 | 0.88
1.6 | 0.31 | 0.67 | 0.96 | 0.49 | 0.66 | 0.91
2.0 | 1.2 | 0.51 | 0.77 | 0.97 | 0.70 | 0.70 | 0.84
1.4 | 0.40 | 0.73 | 0.98 | 0.59 | 0.70 | 0.90
1.6 | 0.31 | 0.69 | 0.98 | 0.49 | 0.68 | 0.94

The variations of $\Lambda_{1.4}$ with the pressure of $\beta$-equilibrated
charge neutral matter at twice the saturation densities $P(2\rho_{0})$ are
plotted in Fig. 9. The correlations of $\Lambda_{1.4}$ with $P(2\rho_{0})$ are
quite robust, independent of distributions of nuclear matter parameters. It
may be pointed out that the pressure is related to the density derivative of
the energy per particle and would depend on several nuclear matter parameters,
except for $\rho=\rho_{0}$. Its value at $\rho_{0}$ is mainly governed by the
$L_{0}$. The strong $\Lambda_{1.4}-P(2\rho_{0})$ correlations may not allow
one to reconstruct the EoS of symmetric nuclear matter and the density-
dependent symmetry energy very accurately, though they are highly
desirableImam  _et al._ (2022). The accurate determination of lower-order
nuclear matter parameters from the bulk properties of finite nuclei in
conjunction with tighter constraints on $P(2\rho_{0})$ may shed some light on
the value of high-order nuclear matter parameters.

The results presented in Figs 7 \- 9 may also be sensitive to the range of
values for the $\Lambda_{1.4}$ as well as to the bounds on the nuclear matter
parameters. In Table 2, we have listed the values of Pearson’s correlation
coefficient for $\Lambda_{1.4}-L_{0}$, $\Lambda_{1.4}-K_{\rm sym,0}$ and
$\Lambda_{1.4}-P(2\rho_{0})$ obtained for the joint posterior distribution of
nuclear matter parameters. The correlation coefficients increase with the
increase in the upper bound on $\Lambda_{1.4}$. In particular, these effects
are quite strong for the case of $\Lambda_{1.4}-L_{0}$ and
$\Lambda_{1.4}-K_{\rm sym,0}$.

So far, we have considered only the case of tidal deformability for a neutron
star with the canonical mass obtained for the EoSs assuming $\rho_{\rm
tr}=2.0\rho_{0}$, beyond which the high-density part of the EoS is switched
on. We now consider the tidal deformability and radius with neutron star
masses M$=1.2,1.4$, and $1.6M_{\odot}$ and study their correlations with
symmetry energy slope $L_{0}$, curvature parameters $K_{\rm sym,0}$ and
pressure $P(2.0\rho_{0})$. In Table 3, we present the values of Pearson’s
correlation coefficients for $\Lambda_{M}$ and $R_{M}$ with $L_{0}$, $K_{\rm
sym,0}$ and $P(2\rho_{0})$. These values of correlation coefficients are
calculated for $\rho_{\rm tr}=1.5,1.75$ and $2.0\rho_{0}$ and $M_{\rm
max}\geqslant 2.1M_{\odot}$. The results are obtained for the joint posterior
distribution of nuclear matter parameters (Fig. 1), which determine the EoS at
$\rho\leqslant\rho_{\rm tr}$. The correlations increase a little with the
$\rho_{\rm tr}=1.5-2.0\rho_{0}$. It is interesting to note that the $\Lambda-
L_{0}$ correlations are moderate, while the $\Lambda-K_{\rm sym,0}$
correlations become relatively stronger which further increase for the
$\Lambda-P(2\rho_{0})$ case. The correlations involving the NS radius display
little different trends; the $R-L_{0}$ correlations are stronger than those
for $R-K_{\rm sym,0}$. The correlations of $R-P(2\rho_{0})$ are weaker than
those for $\Lambda-P(2\rho_{0})$. The values of correlation coefficient
presented in Table 3 may not be significantly affected by the inclusion of any
exotic degrees of freedom beyond 2$\rho_{0}$, since the EoSs beyond the
$\rho_{\rm tr}$ have diverse behavior as they are obtained simply by imposing
the causality condition. We have also repeated the calculations to obtain the
joint posterior distribution of nuclear matter parameters by reducing the
uncertainty on the pure neutron matter EoS by a factor of two. The values of
the median and the 68% confidence interval for the nuclear matter parameters
with the reduced uncertainties are $K_{0}=239_{-32}^{+34}$,
$Q_{0}=-5_{-343}^{+258}$, $J_{0}=32.19_{-0.9}^{+0.9}$, $L_{0}=52_{-8}^{+8}$,
$K_{\rm sym,0}=-102_{-60}^{+63}$ and $Q_{\rm sym,0}=722_{-510}^{+379}$. The
values of symmetry energy coefficient $J_{0}$, slope parameter $L_{0}$, and
curvature parameter $K_{\rm sym,0}$ are now more constrained as compared to
those in Fig. 1. Similar to Table 3, we have listed the values of Pearson’s
correlation coefficients for $\Lambda_{M}$ and $R_{M}$ with $L_{0}$, $K_{\rm
sym,0}$ and $P(2\rho_{0})$ in Table 4. The values of correlation coefficients
show similar trends as listed in Table 3, but the values of correlation
coefficients for $\Lambda_{M}$ and $R_{M}$ with $L_{0}$, $K_{\rm sym,0}$ are
reduced. But, the correlations of the $\Lambda_{M}$ and $R_{M}$ with
$P(2\rho_{0})$ seem to be more-or-less independent of the ranges of the
nuclear matter parameters, which once again implies that the NS properties may
be more sensitive to the combination of several nuclear matter parameters
rather than the individual ones. We have also examined the influence of the
lower bound on the maximum mass of NS for 2.1 $M_{\odot}$ to 2.4 $M_{\odot}$
on our correlation systematics. The correlations do not change significantly;
for example, the correlation coefficient value for $\Lambda_{1.4}$ with
$L_{0}$ and $K_{\rm sym,0}$ changes from 0.72 to 0.78 and from 0.84 to 0.86,
respectively.

We have asset the influence of crust EoS on our correlation systematics. We
have repeated our calculations for the correlations of $\Lambda_{1.4}$ and
$R_{1.4}$ with $L_{0}$, $K_{\rm sym,0}$ and $P(2\rho_{0})$ for $\rho_{\rm
tr}=2\rho_{0}$ by using different crust EoSs such as NL3$\omega\rho$-L55Grill
_et al._ (2014); Pais and Providência (2016) and TM1eGrill  _et al._ (2014);
Boukari  _et al._ (2021) available in the CompOSE Com ; Typel  _et al._
(2022). The results for the correlations involving $\Lambda_{1.4}$ change
almost by 1%. The correlation of $R_{1.4}$ with $L_{0}$ also remains
practically unaffected. However, the correlations of $R_{1.4}$ with $K_{\rm
sym,0}$ and $P(2\rho_{0})$ improve by 5-10 %.

##  IV Conclusions

We have constructed a large set of minimally constrained EoSs for the NS
matter and performed a detailed investigation of the correlations of NS
properties with several nuclear matter parameters that determine the density
dependence of symmetry energy. The joint posterior distribution of nuclear
matter parameters that determine the EoSs at low densities ($\rho\leq\rho_{\rm
tr}$) is obtained by employing our minimal constraints within a Bayesian
approach. These EoSs are consistent with (i) the pure neutron matter EoS from
a precise next-to-next-to-next-to-leading-order ( N3LO) calculation in chiral
effective field theory and (ii) empirical ranges of low-order nuclear matter
parameters determined by the experimental data on the bulk properties of
finite nuclei. The EoSs beyond $\rho_{\rm tr}=1.5-2\rho_{0}$ are constrained
only by imposing the causality condition on the speed of sound. The large set
of EoSs so obtained is employed to study the sensitivity of NS properties to
the symmetry energy slope parameter $L_{0}$ and curvature parameter $K_{\rm
sym,0}$ as well as to the pressure of $\beta-$equilibrated matter at
2$\rho_{0}$. The calculations are also performed with uncorrelated Uniform and
Gaussian distributions of NMPs that ignore the interdependence among them as
present in their joint posterior distribution.

The tidal deformability and radius of NS, as a function of mass, are evaluated
using our minimally constrained EoSs. The NS properties at canonical mass and
the maximum NS mass are found to be consistent with the observational
constraints. The correlations of tidal deformability and radius for different
NS masses ($1.2-1.6M_{\odot}$) with the slope and curvature parameter of
symmetry energy and the pressure for $\beta-$equilibrated charge neutral
matter at $2\rho_{0}$ are studied. We have examined the sensitivity of these
correlations to several factors such as, (i) the behavior of the high-density
part of the EoS,(ii) the choice of distributions of nuclear matter parameters,
their interdependence and uncertainties,(iii) the lower bound on the maximum
mass of the stable neutron stars, (iv) the value of transition density beyond
which the low-density EoSs are smoothly joined with a diverse set of EoSs
constrained only by the causality condition on the speed of sound, (v) the
upper bound on the value of tidal deformability.

The tidal deformability is reasonably correlated with the symmetry energy
slope parameter $L_{0}$ for the EoSs obtained from the joint posterior
distribution of nuclear matter parameters. The correlation of tidal
deformability with $K_{\rm sym,0}$ is slightly stronger. These correlations
become even stronger when the priors of lower-order nuclear matter parameters
corresponding to incompressibility and symmetry energy coefficients are kept
fixed. The correlations become noticeably weaker in the absence of
interdependence among nuclear matter parameters. For instance, the Pearson’s
correlations coefficients for $\Lambda_{1.4}-L_{0}$ ( $\Lambda_{1.4}-K_{\rm
sym,0}$) are $r\sim 0.4(0.6)$ for independent distribution of nuclear matter
parameters which become $r\sim 0.8(0.9)$ for the joint posterior distribution.
This implies that the correlations of NS properties with individual symmetry
energy parameters are masked in the absence of appropriate constraints on the
EoSs at low densities. It also partly explains why the outcome of the similar
correlations studied in the earlier publications Malik  _et al._ (2020);
Pradhan  _et al._ (2022); Carson  _et al._ (2019); Tsang  _et al._ (2020);
Kunjipurayil  _et al._ (2022) are at variance. These correlations improve a
little bit with the increase in the transition density. The diverse behavior
of EoSs at high-density ($\rho\geqslant\rho_{\rm tr}$), as modeled by wide
variations of the speed of sound within the causal limit, do not affect the
sensitivity of the NS properties to the symmetry energy parameters evaluated
at saturation density. The results for the correlation of tidal deformability
with the pressure at 2$\rho_{0}$ are found to be robust, practically
independent of all the factors considered. The correlations of NS radius with
the symmetry energy parameters are also sensitive to various factors
considered. The vulnerability of correlations of NS properties with individual
parameters of symmetry energy and, on the contrary, the robustness of their
correlations with $P(2\rho_{0})$ needs to be further investigated to pin down
the combination of the optimum number of nuclear matter parameters required to
describe the NS properties for the masses $\sim 1.4M_{\odot}$. With the
observations of heavier NS, the correlation between NS properties and symmetry
energy parameters presented in this paper may be improved.

##  V Acknowledgements

The authors are grateful to Sarmistha Banik for critical reading and the
suggestions to improve the presentation of the paper. The authors are thankful
to Philippe Landry for providing useful feedback to improve the clarity of the
paper. N.K.P. would like to acknowledge the Department of Science and
Technology, Ministry of Science and Technology, India, for the support of
DST/INSPIRE Fellowship/2019/IF190058. A. M. acknowledges support from the
Department of Science and Technology (DST)-Science and Engineering Research
Board (SERB) Start-up Research Grant No. SRG/2020/001290. The authors
sincerely acknowledge the usage of the analysis software BILBY Ashton et al.
(2019); Romero-Shaw et al. (2020) and open data from GWOSC Abbott et al.
(2021). This article has been assigned LIGO Document No. LIGO-P2200392. BKA
acknowledge partial support from the SERB, Department of Science and
Technology, Government of India with grant no. SIR/2022/000566 and
CRG/2021/000101, respectively. AV would like to acknowledge the CSIR for the
support through the CSIR-JRF 09/1026(16303)/2023-EMR-I.

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