feature from exerting undue influence on the PCA process due to its greater
magnitudes.

  3. 3.

The Covariance Matrix: Calculate the covariance matrix for the standardized
dataset. The covariance matrix is a mathematical representation that captures
both the variances and connections among different features. The computational
complexity of the covariance matrix is $O(ND\times min(N,D))$, which results
by multiplying two matrices of size $D\times N$ and $N\times D$, respectively.
Here, $N$ is the number of samples, and $D$ is the dimensionality or simply
the number of features.

  4. 4.

Eigenvalue decomposition: Determine the eigenvectors and eigenvalues of the
covariance matrix. The eigenvectors represent the PCs, while the eigenvalues
indicate the amount of variance explained by each PC. Arrange the eigenvalues
in descending order to assign higher priority to the principal components that
account for the greatest amount of variance.

  5. 5.

Selection of PCs: Determine the number of main components that will be kept.
The selection of this option depends upon our target to reduce dimensionality.
We can either select a certain number of top PCs or opt to keep a given
proportion of the total variance (e.g., 95%).

  6. 6.

Amplitude: An analysis is conducted to evaluate the square of the amplitude
values of the feature on each PC. Features that have larger absolute loadings
on a specific PC are regarded as having a greater contribution to that
component.

  7. 7.

Percentage Contribution: The final step is to find the percentage
contributions of all features to the target variable. The total contributions
of a given feature are obtained by summing their contributions from each of
the PCs weighted by the corresponding normalized eigenvalue.

The overall complexity of the PCA analysis is $O(ND\times min(N,D))$. By
following the above steps, One can identify the importance of PCs by reducing
the dimensionality of the data and can extract the key parameters from the
dataset for a specific target.

##  Chapter 3 Bayesian Analysis of Dense Matter Equation of State

In this chapter, we obtained the EOSs for $\beta$-equilibrated matter (BEM)
using Taylor and $\frac{n}{3}$ expansions as discussed in the previous Chapter
2.2 in Eqs. (2.38) and (2.43). The coefficients of the Taylor expansion are
the individual nuclear matter parameters, whereas they correspond to linear
combinations of nuclear matter parameters for the $\frac{n}{3}$ expansion. We
have constructed marginalized posterior distributions for the nuclear matter
parameters by applying a Bayesian approach to both the expansions considered.
The nuclear matter parameters or the corresponding EOSs are consistent with a
set of minimal constraints that includes basic properties of saturated nuclear
matter and low-density ($\rho=0.08-0.16$ fm-3) EOS for the pure neutron matter
from (N3LO) calculation in chiral effective field theory Hebeler et al.
[2013]. This large number of EOSs is employed to evaluate the properties of
neutron stars, such as tidal deformability, radius, and maximum mass. The
correlations of neutron star properties with the pressure of
$\beta$-equilibrated matter at a given density are studied. Most of these
correlations are sensitive to the neutron star mass and EOS choice at a given
density. Our results for the correlations of tidal deformability with pressure
for $\beta$-equilibrated matter are analogous to those obtained using a
diverse set of nonrelativistic and relativistic mean-field models (MFMs) that
re-emphasize their model independence. Such model-independent trends inspire
us to parametrize the pressure for $\beta$-equillibrated matter around
2$\rho_{0}$ in terms of neutron star mass and the corresponding tidal
deformability Patra et al. [2022a].

###  3.1 Priors, Likelihood, and Filters

We apply the Bayesian approach to obtain two large sets of EOSs corresponding
to the Taylor and $\frac{n}{3}$ expansions. The posterior distributions for
the NMPs are obtained by subjecting the EOSs to a set of minimal constraints,
which include some basic properties of nuclear matter evaluated at the
saturation density $\rho_{0}$ and EOS for the pure neutron matter at low
density. The constraints on the nuclear matter parameters are incorporated
through the priors and those from the EOS for the pure neutron matter through
the likelihood function. Not all the nuclear matter parameters are well-
constrained. Only a very few low-order nuclear matter parameters constrained
within narrow bounds are the binding energy per nucleon
$\varepsilon_{0}=-16.0\pm 0.3$ MeV , nuclear matter incompressibility
coefficients $K_{0}=240\pm 50$ MeV for the symmetric nuclear matter and
symmetry energy coefficient $J_{0}=32.0\pm 5$ MeV. The values of
$\varepsilon_{0}$ and $J_{0}$ are very well constrained by 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]. The value of $K_{0}$ is constrained from the experimental data on the
centroid energy of isoscalar giant monopole resonance in a few heavy nuclei
Garg & Colò [2018], Agrawal et al. [2005]. The values of $L_{0}$ have been
extracted from experimental data on variety of phenomena in the finite nuclei
as well as from neutron star observations. The model-independent estimates of
$L_{0}$ are expected to be derived from the measurement of neutron-skin
thickness in asymmetric nuclei. Recent measurement of neutron-skin thickness
in 208Pb nucleus yields $L_{0}=106\pm 37$ MeV Reed et al. [2021]. However,
this value of $L_{0}$ has only marginal overlap at the lower side with those
determined using experimental data on iso-vector giant dipole resonances in
several nuclei Roca-Maza et al. [2015] and recent neutron star observations
Essick et al. [2021b]. The remaining nuclear matter parameters, $Q_{0}$,
$Z_{0}$, $K_{\rm sym,0},Q_{\rm sym,0}$ and $Z_{\rm sym,0}$ are constrained
only poorly Tsang et al. [2020], Ferreira & Providência [2021], Dutra et al.
[2012, 2014], Mondal et al. [2017]. The priors for the nuclear matter
parameters employed in the present work are listed in Table 3.1. The prior
distributions of $\varepsilon_{0},K_{0}$ and $J_{0}$ are assumed to be
Gaussian with a smaller width, whereas the other higher-order nuclear matter
parameters correspond to Gaussian distribution with a very large width. We
have also repeated our calculations with uniform priors for the higher order
nuclear matter parameters, and the result for the median values are found to
be practically unaltered, and uncertainties are modified marginally, up to
10%(not shown). In what follows, we present only those results that are
obtained with priors as listed in Table 3.1.

Table 3.1:  The prior distributions of the nuclear matter parameters. The nuclear matter parameters considered are the binding energy per nucleon ( $\varepsilon_{0}$), incompressibility coefficient ($K_{0}$), symmetry energy coefficient ($J_{0}$), its slope parameter ($L_{0}$), symmetry energy curvature parameter ($K_{\rm sym,0}$) and $Q_{0}[Q_{\rm sym,0}]$ and $Z_{0}[Z_{\rm sym,0}]$ are related to third and fourth-order density derivatives of $E(\rho,0)$[ $E_{\rm sym}(\rho)$], respectively. All the nuclear matter parameters are evaluated at saturation density $\rho_{0}$ = 0.16 fm-3. The parameters of Gaussian distribution (G) are the mean ($\mu$) and standard deviation ($\sigma$). NMPs | Pr-Dist | $\mu$ | $\sigma$ | NMPs | Pr-Dist | $\mu$ | $\sigma$
---|---|---|---|---|---|---|---
(in MeV) |  |  |  | (in MeV) |  |  |
$\varepsilon_{0}$ | G | -16 | 0.3 | $J_{0}$ | G | 32 | 5
|  |  |  | $L_{0}$ | G | 50 | 50
$K_{0}$ | G | 240 | 50 | $K_{\rm sym,0}$ | G | -100 | 200
$Q_{0}$ | G | -400 | 400 | $Q_{\rm sym,0}$ | G | 550 | 400
$Z_{0}$ | G | 1500 | 1500 | $Z_{\rm sym,0}$ | G | -2000 | 2000

We know that the direct application of the lattice QCD simulations are
challenging to hadronic physics at finite density due to sign problems in
Monte Carlo simulations. However, analytical calculations in terms of the
effective degrees of freedom at low density ($\rho<\rho_{0}$), like chiral
effective theory, are valid with negligible uncertainty. The precise next-to-
next-to-next-to-leading-order (N3LO) calculation is usually fitted to the
nucleon–deuteron scattering cross section or few-body observables, and even
saturation properties of heavier nuclei Drischler et al. [2021b]. The low-
density EOS for the pure neutron matter obtained from a (N3LO) calculation in
chiral effective field theory Hebeler et al. [2013] is employed as pseudodata
to obtain a simple likelihood function as given by Eq. (2.71). The $d$s and
the $\sigma$s in Eq. (2.71) are the pseudodata for the energy per neutron and
the corresponding uncertainties taken from Ref. Hebeler et al. [2013]. This
has been employed in the past in many of the analyses as their pseudodata
Ekström et al. [2015], Lim & Holt [2018, 2019], Malik et al. [2022], Ghosh et
al. [2022b]. We have considered the values of energy per neutron over the
density range $\rho=0.08$ \- $0.16$ fm-3. At densities lower than $0.08$ fm-3,
the neutron star matter is expected to be clusterized.

We have filtered the nuclear matter parameters by demanding that (i) pressure
for the $\beta$-equilibrated matter should increase monotonically with density
(thermodynamic stability), (ii) The speed of sound must not exceed the speed
of light (causality) and (iii) maximum mass of neutron star must exceeds
$2M_{\odot}$ (observational constraint). The causality breaks down at higher
density mostly for the Taylor EOS. In such cases, we use the stiffest EOS,
$P(\epsilon)=P_{m}+(\epsilon-\epsilon_{m})$, where, $P_{m}$ and $\epsilon_{m}$
are the pressure and corresponding energy density at which the causality
breaks Glendenning [1992b].

###  3.2  Posterior distribution of nuclear matter parameters

To undertake the correlation systematics as proposed, we need a large number
of EOSs with diverse behaviour and corresponding neutron star properties. As
discussed above, the posterior distributions for the nuclear matter parameters
for the Taylor and $\frac{n}{3}$ expansions are obtained by subjecting the EOS
to a set of minimal constraints. The joint posterior distribution of the NMPs
for a given model depends on the product of the likelihood and the prior
distribution of nuclear matter parameters (Eq. (2.65)). The posterior
distribution of each individual parameter is obtained by marginalizing the
joint posterior distribution with the remaining model parameters. If the
marginalized posterior distribution of a nuclear matter parameter is localized
more than the corresponding prior distribution, then the nuclear matter
parameter is said to be well constrained by the data used for model fitting.

Figure 3.1:  Corner plots for the nuclear matter parameters (in MeV) obtained
for Taylor expansions for the EOS of asymmetric nuclear matter. The one-
dimensional marginalized posterior distributions (salmon) and the prior
distributions (green lines) are displayed along the diagonal plots. The
vertical lines represent the nuclear matter parameters’ 68% confidence
interval. Along the off-diagonal plots, the confidence ellipses for two-
dimensional posterior distributions are plotted with confidence intervals of
1$\sigma$, 2$\sigma$, and 3$\sigma$. The distributions of nuclear matter
parameters are obtained by subjecting them to minimal constraints (see text
for details).  Figure 3.2:  The same as Fig. 3.1, but for $\frac{n}{3}$
expansions for the EOS of asymmetric nuclear matter.

The corner plots for the marginalized posterior distributions for the nuclear
matter parameters in one and two dimensions obtained for Taylor and
$\frac{n}{3}$ expansions are displayed in Figs.3.1 and 3.2, respectively. The
differences between the one-dimensional posterior distributions for the
nuclear matter parameters and corresponding prior distributions reflect the
role of low-density EOS for pure neutron matter in constraining the nuclear
matter parameters. The EOS for the pure neutron matter mainly constraints the
values of $J_{0}$, $L_{0}$ and $K_{\rm sym,0}$ and to some extent $Q_{\rm
sym,0}$ and $Z_{\rm sym,0}$. The shapes and orientations of the confidence
ellipses suggest that the correlations among most of the NMPs are weak. Most
Strong correlations exist only between $Q_{0}-Z_{0}$, $L_{0}-J_{0}$ and
$L_{0}-K_{\rm sym,0}$ for both the expansions with correlation coefficient r
$-$$\sim$ 0.8. The $K_{0}-Q_{0}$ correlation is slightly better in case of
$\frac{n}{3}$ expansion (r$\sim$ -0.6) as compared to Taylor (r$\sim$ -0.18).
The median values of the nuclear matter parameters and the corresponding
$68\%(90\%)$ confidence intervals obtained from the marginalized posterior
distributions are listed in Table 3.3(see Appendix ). We also provide the
values for the nuclear matter parameters obtained without the PNM constraints.
The low-density pure neutron matter mainly constrains those nuclear matter
parameters associated with the density dependence of symmetry energy. The
median values of $L_{0}$ and $K_{\rm sym,0}$, which determined the linear and
quadratic density dependence of the symmetry energy, become smaller,
suggesting softer symmetry energy ”at high-density” with the inclusion of pure
neutron matter constraints. Furthermore, the uncertainties on $L_{0}$ were
reduced by more than 50%. The median value of $Q_{\rm sym,0}$ remains more or
less unaltered. From the recent measurement of the neutron-skin thickness for
208Pb nucleus (PREX-II) Adhikari et al. [2021], Reed et al. [2021], $\Delta
R_{\rm skin}=0.283\pm 0.071$ fm, the value of $L_{0}$ has been determined to
be $106\pm 37$ MeV Reed et al. [2021] . This value of $L_{0}$ agrees with the
ones obtained in the present work with PNM constraints only within 90%
confidence interval.

###  3.3 Properties of Neutron Stars

Once the EOS for the core and crust are known, the values of NS mass, radius,
and tidal deformability corresponding to given central pressure can be
obtained by solving Tolman-Oppenheimer-Volkoff equations Oppenheimer & Volkoff
[1939], Tolman [1939]. The EOSs for the core region of a neutron star
correspond to the $\beta$-equilibrated matter over the density range
$0.5-8\rho_{0}$, are obtained from the posterior distributions of nuclear
matter parameters for the Taylor and $\frac{n}{3}$ expansions. The core EOSs
are matched to the crust EOSs to obtain the NS properties. The EOS for the
outer crust is taken to be the one given by Baym, Pethick, and Sutherland Baym
et al. [1971]. The inner crust that joins the inner edge of the outer crust
and the outer edge of the core is assumed to be polytropic Carriere et al.
[2003], $p(\varepsilon)=c_{1}+c_{2}\varepsilon^{\gamma}$. Here, the parameters
$c_{1}$ and $c_{2}$ 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 ($0.5\rho_{0}$ ). The polytropic index $\gamma$ is
taken to be equal to 4/3. The radii of neutron stars with mass $\sim$
1$M_{\odot}$ are more sensitive to the treatment of crust EOS Fortin et al.
[2016]. It is demonstrated that the treatment of crust EOS employed in the
present work may introduce the uncertainties of about 50-100 m in radii of NSs
having mass 1.4$M_{\odot}$. It is shown in Ref. Piekarewicz & Fattoyev [2019]
that the choice of EOS for the inner crust does not significantly impact the
values of tidal deformability, which depends on the Love number k2 as well as
the compactness parameter.

Figure 3.3:  The equation of state for pure neutron matter is shown. The
colored bands correspond to $\chi$EFT N3LO (light green), 90% confidence
interval for the Taylor EOSs (light blue) and for $\frac{n}{3}$ EOSs (light
magneta) obtained in our calculation (see text for details).

The 90% confidence interval for pure neutron matter Taylor EOSs (light blue)
and $\frac{n}{3}$ EOSs (light magenta) for the density range (
$0.5-1.0\rho_{0}$) is shown in Fig. 3.3. For comparison, colored bands
corresponding to N3LO (light green) from chiral effective field theory are
displayed. Our EOSs lie almost in the middle of the 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.

Figure 3.4: Corner plots representing the marginalized posterior distributions
(salmon) of the tidal deformability $\Lambda_{1.4}$, radii $R_{1.4}$ (km) and
$R_{2.07}$ (km) and the maximum mass $M_{\rm max}$ ($M_{\odot}$) for Taylor
(left) and $\frac{n}{3}$ (right) expansions. The green lines represent
effective priors obtained using the priors for nuclear matter parameters (see
also Table 3.1).

We have obtained the distributions of $\Lambda_{1.4}$, $R_{1.4}$, $R_{2.07}$
and $M_{\rm max}$ using the posterior distributions for the nuclear matter
parameters corresponding to the Taylor and $\frac{n}{3}$ expansions. The
corner plots for these NS properties are displayed in Fig. 3.4. The effective
priors for the NS properties, as shown by green lines, are obtained using the
priors for the nuclear matter parameters. The posterior distributions of NS
properties are narrower than the corresponding effective priors, indicating
the significance of the low-density EOS for pure neutron matter. The posterior
distributions of $\Lambda_{1.4}$ and $R_{1.4}$ for both the expansions are
quite close to each other. The differences begin to appear for the case of
$R_{2.07}$, which becomes even larger for the maximum mass. This is due to the
fact that the Taylor EOSs are much more stiffer than those for $\frac{n}{3}$.
The dichotomy in the high-density the behavior of the Taylor and $\frac{n}{3}$
expansions would help us to understand the extent to which the correlations of
the EOSs with the properties of NS, for masses in the range $1$ \-
$2M_{\odot}$, are model dependent. It is clear from off-diagonal plots that
$\Lambda_{1.4}$ is strongly correlated with $R_{1.4}$, the correlation
coefficient is r$\sim$ 0.9. The $\Lambda_{1.4}$ and $R_{1.4}$ also display
stronger correlations with $R_{2.07}$ (r $\sim$ 0.8) for the case of Taylor
and somewhat moderate correlations (r $\sim$ 0.7) for the $\frac{n}{3}$
expansion. As the EOSs are generated with the same models for each case and
there is no phase transition appears, So we found a strong correlation between
them. Otherwise, it will be weaker. The maximum mass of a neutron star is
almost uncorrelated with the other NS properties considered.

Figure 3.5:  The plot represents the joint probability distribution $P(M,R)$
as a function of mass and radius of neutron star obtained for $\frac{n}{3}$
expansion. The 90% confidence interval is represented by the red dashed line.
The 90% (solid) and 50% (dashed) confidence intervals of the LIGO-Virgo
analysis for the BNS component from the GW170817 event are shown by the outer
and inner gray shaded regions Abbott et al. [2018, 2019b, 2021]. The
constraints from the millisecond pulsar PSR J0030+0451 (purple & black) NICER
x-ray data Riley et al. [2019], Miller et al. [2019] and PSR J0740+6620
(green) Riley et al. [2021] are shown in the rectangular regions enclosed by
dotted lines.

We have summarized in Table 3.4 (see Appendix ) shows the median values of NS
properties along with 68% (90%) confidence intervals. Like in the case of
nuclear matter parameters, the NS properties get significantly constrained by
the EOS of pure neutron matter at low density. For instance, the median values
of $\Lambda_{1.4}$ become smaller by about 15% and the associated
uncertainties by about 40% with the pure neutron matter constraints. The
median values of $R_{1.4}$ and the corresponding uncertainties also become
noticeably smaller. The $R_{2.07}$ and $M_{\rm max}$ do not show significant
changes with the inclusion of low-density pure neutron matter constraints.
With the PNM constraints, the 90% confidence interval of the neutron stars
properties such as tidal deformability, radius, and mass overlap with the
currently available bounds, $\Lambda_{1.4}\in[70,580]$ Abbott et al. [2017],
$R_{1.4}\in[11.41,13.61]$ km Miller et al. [2021], $R_{2.07}\in[11.8,13.1]$ km
Riley et al. [2021] and $M_{\rm max}\geq 2.09M_{\odot}$ Romani et al. [2021].
The $M_{\rm max}=2.48_{-0.07}^{+0.06}M_{\odot}$ obtained for the Taylor EOSs
are on the slightly higher side in comparison to the ones derived by combining
the GW170817 observations of merging of binary neutron stars and
quasiuniversal relation Rezzolla et al. [2018]. The observed electromagnetic
emissions in the form of kilonova and the detection of a gamma-ray burst have
been linked to the formation of a black hole and, thus, have been utilized to
infer the maximum mass of a stable neutron star. However, such inference of
the maximum mass is subjected to uncertainties originating from model
dependence of postmerger dynamics. Recent observation of the GW190814 event, a
neutron star black hole/binary neutron star merger, has triggered an
assessment of the maximum mass of a stable neutron star Abbott et al. [2020b].
While there are different opinions available in the literature, the nature of
a compact object in the range of 2.5 - 2.67 $M_{\odot}$ being a neutron star
or black hole seems to be an unsettled issue to date Tsokaros et al. [2020],
Lim et al. [2021], Drischler et al. [2021a], Li et al. [2021], Rezzolla et al.
[2018], Abbott et al. [2020b]. So the maximum mass ($M_{max}$) we got for the
Taylor model supporting the static NS of mass greater than 2.5 M⊙ may not be
ruled out at present.

We obtain joint probability distribution $P(M,R)$ for a given mass and radius
for both the Taylor and $\frac{n}{3}$ expansions. They display qualitatively
very similar trends. In Fig. 3.5, we plot the $P(M,R)$ obtained for the
$\frac{n}{3}$ expansion. The red dashed line represents the 90% confidence
interval. The color gradient from orange to dark purple represents the lowest
to highest probability. The most probable values for $R_{1.4}$ and $R_{2.07}$
are approximately 13.3 and 12.3 km, respectively. The $P(M,R)$ is maximum for
$M\sim 1.4-2.0M_{\odot}$, $R\sim 12.4-13.4$ km. The $90\%$ confidence interval
has significant overlap with LIGO-Virgo and NICER estimations. It may be
however pointed out that the main objective of the present work is to
construct large sets of EOSs with diverse behavior to assess various
correlation systematics as follows.

###  3.4 Correlations of Neutron Star properties with EOS

We randomly select 1000 EOSs and corresponding NS properties from marginalized
posterior distributions obtained for the Taylor as well as $\frac{n}{3}$
expansions. They are used to study the correlations of various NS properties
with key quantities determining the behavior of the EOS. The correlations of
$\Lambda_{1.4}$, $R_{1.4}$, $R_{2.07}$ and $M_{\rm max}$ with the pressure of
$\beta$-equilibrated matter over a wide range of densities are evaluated. The
values of correlation coefficients are plotted as a function of density in
Fig.3.6. We also display the values of correlation coefficients for NS
properties with the pressure of $\beta$-equilibrated matter calculated using
unified EOSs for a diverse set of 41 non-relativistic and relativistic
microscopic mean-field models (MFMs) Fortin et al. [2016]. The various NS
properties considered show strong correlations with $P_{\scriptscriptstyle{\rm
BEM}}(\rho)$ around a particular density. The density at which the correlation
is at its maximum increases with the NS mass. The values of $\Lambda_{1.4}$
and $R_{1.4}$ are strongly correlated with $P_{\scriptscriptstyle{\rm
BEM}}(\rho)$ at density $\sim$ $1.5-2.5\rho_{0}$. The $R_{2.07}$ is strongly
correlated with $P_{\scriptscriptstyle{\rm BEM}}(\rho)$ around 3$\rho_{0}$.
The $M_{\rm max}$ is strongly correlated with $P_{\scriptscriptstyle{\rm
BEM}}(\rho)$ around 4.5$\rho_{0}$. Our results for the Taylor and
$\frac{n}{3}$ expansions for the region of maximum correlations are in line
with those obtained using a diverse set of mean-field models, except for the
case of $R_{2.07}$. Thus, it seems possible that the EOS over a range of
densities beyond $\rho_{0}$ can be constrained in a nearly model-independent
manner with the help of various NS observables.

Figure 3.6:  The correlation coefficients r(x,$P_{\scriptscriptstyle{\rm
BEM}}(\rho)$), as approximated by both Taylor and $\frac{n}{3}$ expansion
along with the mean-field theory calculations, is shown in this figure. Here,
x represents either of the tidal deformability $\Lambda_{1.4}$, radii
$R_{1.4}$, $R_{2.07}$, or maximum mass $M_{\rm max}$ of the neutron star,
whereas, $P_{\scriptscriptstyle{\rm BEM}}(\rho)$ represents the pressure for
$\beta$-equilibrated matter at a density $\rho$. The calculations are
performed with neutron star properties obtained using marginalized posterior
distributions of nuclear matter parameters in Taylor and $\frac{n}{3}$
expansions. For the comparison, the results are also displayed for a diverse
set of non-relativistic and relativistic microscopic mean-field models (MFMs).
Figure 3.7: The variations of pressure for $\beta$-equilibrated matter
[$P_{\scriptscriptstyle{\rm BEM}}(\rho)$] at selected densities versus tidal
deformability $\Lambda_{1.4}$, radii $R_{1.4}$ and $R_{2.07}$ and maximum mass
$M_{\rm max}$ of neutron star. The red dashed lines are obtained by linear
regression [see Eq. (3.4) in Sec.3.4]. Figure 3.8:  Dependence of correlation
coefficients between tidal deformability ($\Lambda_{M}$) and the pressure of
$\beta$-equilibrated matter ($P_{\scriptscriptstyle{\rm BEM}}(\rho)$) on
neutron star mass (M) and density ($\rho$) is depicted in this plot. Here
$\rho_{0}$=0.16 fm-3 is used only for scaling purposes.

In Table 3.5(see the Appendix ), we list the values of correlation
coefficients obtained between the NS properties and the EOS at some selected
densities. The correlation coefficients are obtained using 100 and 1000 EOSs,
corresponding to Taylor and $\frac{n}{3}$ expansions, randomly selected from
the posterior distributions. We also present the results which are obtained by
combining 1000 EOSs corresponding to each of the expansions. The values of
correlation coefficients for the combined set of EOSs are close to those
obtained separately. The values of the correlation coefficients are close to
those obtained for mean-field models, which are listed in the second column.
We plot in Fig. 3.7, the variations of $P_{\scriptscriptstyle{\rm
BEM}}(\rho)$, at selected densities, with $\Lambda_{1.4}$, $R_{1.4}$,
$R_{2.07}$ and $M_{\rm max}$ for which the correlations are stronger. We
compare our results with those obtained from a diverse set of mean-field
models. The correlation lines obtained by combining results of the Taylor and
$\frac{n}{3}$ expansions are also plotted to estimate the values of
$P_{\scriptscriptstyle{\rm BEM}}(\rho)$ at selected densities with the help of
NS properties. The equations for the correlation lines are obtained using
linear regression as,

$\displaystyle\frac{P_{\scriptscriptstyle{\rm BEM}}(2\rho_{0})}{\rm MeV\rm
fm^{-3}}$ $\displaystyle=$ $\displaystyle(0.96\pm 0.10)+(0.0473\pm
0.0002)\Lambda_{1.4},$ $\displaystyle\frac{P_{\scriptscriptstyle{\rm
BEM}}(2\rho_{0})}{\rm MeV\rm fm^{-3}}$ $\displaystyle=$
$\displaystyle(-85.63\pm 0.89)+(8.01\pm 0.06)\frac{R_{1.4}}{\rm km},$
$\displaystyle\frac{P_{\scriptscriptstyle{\rm BEM}}(3\rho_{0})}{\rm MeV\rm
fm^{-3}}$ $\displaystyle=$ $\displaystyle(-233.16\pm 2.85)+(25.86\pm
0.23)\frac{R_{2.07}}{\rm km},$ $\displaystyle\frac{P_{\scriptscriptstyle{\rm
BEM}}(4.5\rho_{0})}{\rm MeV\rm fm^{-3}}$ $\displaystyle=$
$\displaystyle(-895.85\pm 4.00)+(524.75\pm 1.70)\frac{M_{\rm
max}}{M_{\odot}}.$ (3.1)

Table 3.2:  The median values and associated 68%(90%) uncertainties for the parameters, appearing in Eq. (3.2), obtained from their marginalized posterior distributions. The values of parameters $b_{0}$, $b_{1}$, and $b_{2}$ as listed are scaled up by a factor of 10. Pressure | $a_{0}$ | $a_{1}$ | $a_{2}$ | $b_{0}$ | $b_{1}$ | $b_{2}$
---|---|---|---|---|---|---
(in MeV fm-3) |  |  |  |  |  |
$P_{\scriptscriptstyle{\rm BEM}}(1.5\rho_{0})$ | $0.544^{+0.031(0.050)}_{-0.029(0.060)}$ | $1.869^{+0.158(0.260)}_{-0.161(0.309)}$ | $7.451^{+0.237(0.390)}_{-0.234(0.450)}$ | $0.176^{+0.001(0.001)}_{-0.001(0.002)}$ | $0.740^{+0.004(0.007)}_{-0.004(0.008)}$ | $1.152^{+0.013(0.021)}_{-0.012(0.025)}$
$P_{\scriptscriptstyle{\rm BEM}}(2\rho_{0})$ | $0.146^{+0.030(0.050)}_{-0.030(0.059)}$ | $-0.598^{+0.163(0.269)}_{-0.159(0.320)}$ | $27.909^{+0.233(0.397)}_{-0.240(0.469)}$ | $0.493^{+0.001(0.001)}_{-0.001(0.002)}$ | $2.234^{+0.004(0.007)}_{-0.004(0.008)}$ | $3.728^{+0.012(0.021)}_{-0.012(0.025)}$
$P_{\scriptscriptstyle{\rm BEM}}(2.5\rho_{0})$ | $7.345^{+0.030(0.050)}_{-0.030(0.060)}$ | $-15.102^{+0.161(0.272)}_{-0.167(0.321)}$ | $68.411^{+0.239(0.396)}_{-0.238(0.475)}$ | $0.906^{+0.001(0.001)}_{-0.001(0.002)}$ | $4.518^{+0.004(0.007)}_{-0.004(0.008)}$ | $8.115^{+0.012(0.021)}_{-0.012(0.025)}$
Figure 3.9:  The median values of pressure for $\beta$-equilibrated matter is
shown here as a function of neutron star mass and its tidal deformability at
densities 1.5$\rho_{0}$ (top), 2.0$\rho_{0}$ (middle) and 2.5$\rho_{0}$
(bottom).

We extend our analysis for the correlations of the pressure for the
$\beta$-equilibrated matter with tidal deformability over a wide range of
neutron stars’ mass. In Fig. 3.8, we display color-coded graph for the
correlations of tidal deformability of neutron star for the mass
$1.2-2.0M_{\odot}$ with the pressure for $\beta$-equilibrated matter at
densities $0.5-5\rho_{0}$ (r($\Lambda_{M},P_{\scriptscriptstyle{\rm
BEM}}(\rho)$)). One can easily obtain the value of the correlation coefficient
as a function of density at a given NS mass. The $P_{\scriptscriptstyle{\rm
BEM}}(\rho)$ at $\rho\sim 1.5-2.5\rho_{0}$ are strongly correlated ($r\sim
0.8-1$ ) with tidal deformability for NS masses in the range
$1.2-2.0M_{\odot}$. Hence, $P_{\scriptscriptstyle{\rm BEM}}(\rho)$ can be
parametrized at a given $\rho$ as,

$\displaystyle\frac{P_{\scriptscriptstyle{\rm BEM}}(\rho)}{\rm MeV\rm
fm^{-3}}$ $\displaystyle=$ $\displaystyle a(M)+b(M)\Lambda_{M},$ (3.2)

with mass-dependent coefficients $a(M)$ and $b(M)$ expanded as

$\displaystyle a(M)$ $\displaystyle=$
$\displaystyle(a_{0}+a_{1}(M-M_{0})+a_{2}(M-M_{0})^{2}),$ (3.3) $\displaystyle
b(M)$ $\displaystyle=$
$\displaystyle(b_{0}+b_{1}(M-M_{0})+b_{2}(M-M_{0})^{2}),$ (3.4)

respectively, where $M_{0}$ is taken to be 1.4$M_{\odot}$ and the values of
$a_{i}$ and $b_{i}$ are estimated using a Bayesian approach with the help of
$P_{\scriptscriptstyle{\rm BEM}}(\rho)$ and tidal deformability obtained for
Taylor and $\frac{n}{3}$ expansions. For a given $\rho$, the Eq. (3.2) is
fitted using the tidal deformability corresponding to NS mass
$1.2-2.0M_{\odot}$. The priors for $a_{i}$ and $b_{i}$ are taken to be uniform
in the range of -100 to 100. The calculations are performed for $\rho$= 1.5,
2.0, and 2.5 $\rho_{0}$. All the $a_{i}$s are strongly correlated with
corresponding $b_{i}$s. The median values of parameters $a_{i}$ and $b_{i}$
and associated uncertainties are summarized in Table 3.2. It may be noticed
that the values of $a_{0}$ and $b_{0}$ for the case of
$P_{\scriptscriptstyle{\rm BEM}}(2\rho_{0})$ are not the same as those in Eq.
(3.4). This may be partly due to the strong correlations between $a_{0}$ and
$b_{0}$ of Eq. (3.2). Moreover, Eq. (3.4) is fitted to the values of tidal
deformability at a fixed NS mass 1.4$M_{\odot}$. To validate our parametrized
form for $P_{\scriptscriptstyle{\rm BEM}}(\rho)$, we have calculated the
values of $P_{\scriptscriptstyle{\rm BEM}}(2\rho_{0})$ using Eq. (3.2) with
the help of tidal deformability for 1.4$M_{\odot}$ obtained for a large number
of mean-field models, which includes the once considered in Fig. 3.6 along
with those taken from Tsang et al. [2019], Malik et al. [2022], Ferreira et
al. [2020]. The average deviation of $P_{\scriptscriptstyle{\rm
BEM}}(2\rho_{0})$, obtained using Eq. (3.2), from the actual values, is about
$10\%$. We find marginal improvement when the terms corresponding to quadratic
in tidal deformability are included in Eq. (3.2).

In Fig.3.9, we display the variations of tidal deformability as a function of
mass and pressure for $\beta$-equilibrated matter at $\rho$=1.5, 2.0 and 2.5
$\rho_{0}$. These results are obtained using the parametrized form for
$P_{\scriptscriptstyle{\rm BEM}}(\rho)$ as given by Eq. (3.2). One can easily
estimate the values of $P_{\scriptscriptstyle{\rm BEM}}(\rho)$ for $\rho\sim
2\rho_{0}$ once the values of tidal deformability known in NS mass ranges
$1.2-2.0M_{\odot}$.

###  3.5 Conclusion

We have used Taylor and $\frac{n}{3}$ expansions of equations of state to
construct marginalized posterior distributions of the nuclear matter
parameters, which are consistent with the minimal constraints. Only a few low-
order nuclear matter parameters, such as the energy per nucleon,
incompressibility coefficient for the symmetric nuclear matter, and symmetry
energy coefficients at the saturation density ($\rho_{0}$), are constrained in
narrow windows along with the low-density pure neutron matter EOS obtained
from a precise next-to-next-to-next-to-leading-order (N3LO) calculation in
chiral effective field theory. The tidal deformability, radius, and maximum
mass are evaluated using large sets of minimally constrained EOSs.

The correlations of neutron star properties over a wide range of mass with
various key quantities characterizing the EOS are investigated. We find that
the values of tidal deformability and radius for the neutron star with
1.4$M_{\odot}$ are strongly correlated with the pressure for the
$\beta$-equilibrated matter at density $\sim 2\rho_{0}$. The radius for
$2.07M_{\odot}$ neutron star is strongly correlated with the pressure for
$\beta$-equilibrated matter at density $\sim 3\rho_{0}$. The maximum mass of a
neutron star is correlated with the pressure for the $\beta$-equilibrated
matter at density $\sim 4.5\rho_{0}$. These correlation systematics align with
those obtained for unified EOSs for the $\beta$-equilibrated matter available
for a diverse set of non-relativistic and relativistic mean-field models. We
exploit the model independence of correlations to parametrize the pressure for
$\beta$-equilibrated matter in the density range $1.5-2.5\rho_{0}$, in terms
of the mass and corresponding tidal deformability of neutron star. Such
parametric form may facilitate back-of-the-envelope estimation of the pressure
at densities around $2\rho_{0}$ for a given value of tidal deformability of
neutron stars with mass in the range of $1.2-2.0M_{\odot}$.

Appendix A : Supplemental for Chapter 3

We present our results in tabular form, which are obtained with minimal
constraints. The values of the nuclear matter, properties of neutron stars,
and their correlations with various key quantities associated with EOS are
listed in Tables 3.3–3.5. These results are depicted in Figs. 3.1 \- 3.6.

Table 3.3:  The median values and associated 68%(90%) uncertainties for the nuclear matter parameters from their marginalized posterior distributions. The results are obtained for Taylor and $\frac{n}{3}$ expansions with and without pure neutron matter (PNM) constraints. NMPs | without PNM | with PNM
---|---|---
(in MeV) | Taylor | $\frac{n}{3}$ | Taylor | $\frac{n}{3}$
$\varepsilon_{0}$ | $-16.02^{+0.23(0.41)}_{-0.28(0.56)}$ | $-15.99^{+0.27(0.43)}_{-0.27(0.51)}$ | $-16.00^{+0.27(0.42)}_{-0.30(0.54)}$ | $-16.00^{+0.27(0.44)}_{-0.28(0.56)}$
$K_{0}$ | $236.42^{+42.78(74.34)}_{-42.58(79.62)}$ | $233.38^{+48.94(76.14)}_{-42.73(83.95)}$ | $237.43^{+44.24(72.25)}_{-45.75(83.22)}$ | $231.96^{+44.80(72.94)}_{-41.33(76.63)}$
$Q_{0}$ | $-436.23^{+273.36(419.17)}_{-306.50(603.76)}$ | $-411.84^{+207.53(301.56)}_{-210.88(409.00)}$ | $-419.81^{+262.95(437.69)}_{-272.47(531.58)}$ | $-418.89^{+187.43(300.76)}_{-179.25(377.42)}$
$Z_{0}$ | $1441.51^{+792.45(1298.64)}_{-696.39(1381.30)}$ | $1600.07^{+1067.33(1883.00)}_{-1362.28(2615.10)}$ | $1403.84^{+704.56(1133.85)}_{-690.82(1386.25)}$ | $1638.14^{+1241.83(1906.75)}_{-1277.48(2244.23)}$
$J_{0}$ | $32.37^{+4.08(6.79)}_{-4.26(8.83)}$ | $32.37^{+4.69(7.22)}_{-4.71(10.23)}$ | $31.88^{+0.87(1.43)}_{-0.92(-1.85)}$ | $31.87^{+0.93(1.49)}_{-0.82(1.68)}$
$L_{0}$ | $59.88^{+41.14(65.90)}_{-39.84(78.17)}$ | $55.60^{+37.59(63.89)}_{-43.88(84.62)}$ | $51.25^{+13.32(21.60)}_{-13.91(25.54)}$ | $52.25^{+13.55(22.73)}_{-12.76(23.04)}$
$K_{\rm sym,0}$ | $-85.86^{+192.67(327.83)}_{-151.57(266.76)}$ | $-40.03^{+161.60(271.89)}_{-135.08(234.67)}$ | $-96.65^{+141.41(225.69)}_{-127.49(216.74)}$ | $-67.44^{+127.18(206.09)}_{-114.80(200.38)}$
$Q_{\rm sym,0}$ | $731.13^{+308.54(543.01)}_{-347.82(669.47)}$ | $705.36^{+311.23(511.39)}_{-352.72(727.86)}$ | $699.56^{+324.38(521.95)}_{-323.52(639.30)}$ | $726.49^{+300.40(510.33)}_{-358.51(631.86)}$
$Z_{\rm sym,0}$ | $-2.07^{+1190.67(2153.84)}_{-820.92(1473.09)}$ | $-1390.39^{+1518.69(2526.53)}_{-1856.18(3623.74)}$ | $55.34^{+1205.62(2255.28)}_{-782.52(1415.84)}$ | $-1622.35^{+1606.61(2788.70)}_{-1911.81(3468.40)}$
Table 3.4:  Similar to Table 3.3, but, for the neutron star properties, namely the tidal deformability ($\Lambda_{1.4}$), radii ($R_{1.4}$ and $R_{2.07}$) and maximum mass ($M_{\rm max}$) . NS properties | without PNM | with PNM
---|---|---
Taylor | $\frac{n}{3}$ | Taylor | $\frac{n}{3}$
$\Lambda_{1.4}$ | $527.72^{+250.72(477.68)}_{-186.11(292.57)}$ | $455.85^{+223.65(465.72)}_{-163.05(243.23)}$ | $426.20^{+139.93(224.58)}_{-130.32(205.18)}$ | $386.52^{+132.76(213.24)}_{-102.84(199.09)}$
$R_{1.4}$(km) | $14.69^{+1.78(3.43)}_{-1.63(2.74)}$ | $14.15^{+1.87(3.34)}_{-1.69(2.58)}$ | $13.37^{+0.67(1.03)}_{-0.75(1.60)}$ | $13.22^{+0.64(0.99)}_{-0.67(1.59)}$
$R_{2.07}$(km) | $13.24^{+0.82(1.49)}_{-0.82(1.42)}$ | $12.27^{+0.88(1.52)}_{-0.80(1.52)}$ | $12.72^{+0.55(0.85)}_{-0.59(1.07)}$ | $12.02^{+0.54(0.88)}_{-0.58(1.23)}$
$M_{\rm max}$($M_{\odot}$) | $2.45^{+0.07(0.11)}_{-0.06(0.13)}$ | $2.19^{+0.10(0.19)}_{-0.09(0.09)}$ | $2.48^{+0.06(0.10)}_{-0.07(0.14)}$ | $2.20^{+0.10(0.16)}_{-0.09(0.11)}$
Table 3.5:  The comparison of values for Pearson’s correlation coefficient (r) obtained from randomly selected 100 and 1000 EOSs using both Taylor and $\frac{n}{3}$ expansions. The values of correlation coefficients are also obtained by combining 1000 EOSs from each of the expansions. For comparison, the values of $r$ obtained for a diverse set of mean-field models are also presented in 2nd column.  Name of Pairs | MFMs | Taylor | $\frac{n}{3}$ | combined
---|---|---|---|---
41 | 100 | 1000 | 100 | 1000 | 2000
$\Lambda_{1.4}$-$P_{\scriptscriptstyle{\rm BEM}}(2\rho_{0})$ | 0.90 | 0.98 | 0.98 | 0.99 | 0.98 | 0.98
$R_{1.4}$-$P_{\scriptscriptstyle{\rm BEM}}(2\rho_{0})$ | 0.83 | 0.93 | 0.93 | 0.94 | 0.93 | 0.93
$R_{2.07}$-$P_{\scriptscriptstyle{\rm BEM}}(3\rho_{0})$ | 0.81 | 0.93 | 0.91 | 0.92 | 0.92 | 0.93
$M_{\rm max}$-$P_{\scriptscriptstyle{\rm BEM}}(4.5\rho_{0})$ | 0.99 | 0.97 | 0.98 | 0.95 | 0.96 | 0.99

##  Chapter 4 Systematic Analysis of impacts of Symmetry energy parameters

In this chapter, we have considered a large set of minimally constrained EOSs
for the NS matter 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 Patra et al. [2023b]. The
EOSs at low densities correspond to the nucleonic matter in
$\beta$-equilibrium and are described by the nuclear matter parameters
evaluated at $\rho_{0}$ (see Chapter 2.2.1). 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 & Vidaña
[2016], Stone et al. [2021]. Beyond the density ($\rho_{c_{s}}$), 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 (see Chapter 2.3). 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 nuclear matter parameters’ joint
posterior or correlated distribution 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. These factors are as follows,

  * (i)

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

  * (ii)

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

  * (iv)

the value of the density ($\rho_{c_{s}}$),

  * (v)

upper bound on the value of tidal deformability.

###  4.1 Priors and Posterior distributions of NMPs

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], Lattimer [2021]. 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], 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 & 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
& 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.( 2.36) & ( 2.37) are
only weakly constrained Tsang et al. [2020], Ferreira & Providência [2021],
Dutra et al. [2012, 2014], Mondal et al. [2017], Patra et al. [2022a]. 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 nuclear matter parameter
in Table 4.1. The values of $d$s and $\sigma$s in Eq.(2.71) 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).

Table 4.1: 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 4.1: 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. Along
the off-diagonal plots, the confidence ellipses for two-dimensional posterior
distributions are plotted with confidence intervals of 1$\sigma$, 2$\sigma$,
and 3$\sigma$.

The joint posterior distribution of the NMPs for a given model is the product
of the likelihood and the prior distribution of NMPs (Eq. (2.65)). 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. 4.1. 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 somewhat constrained. 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.

###  4.2 Neutron Star properties

The EOSs for $\beta$-equilibrated charge neutral matter in the density ranges
$0.5\rho_{0}$ to the density $\rho_{c_{s}}$ are obtained using Taylor
expansion with NMPs corresponding to the posterior distribution as displayed
in the Fig. 4.1. The calculations are performed assuming different values for
the $\rho_{c_{s}}$ in the range of $1.5-2\rho_{0}$. Each of the EOSs beyond
$\rho_{c_{s}}$ 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.(2.48-2.51) in chapter 2.3. 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 follows Carriere
et al. [2003],

$\displaystyle p(\varepsilon)=\alpha+\beta\varepsilon^{\frac{4}{3}}.$ (4.1)

Here, the parameters $\alpha$ and $\beta$ are determined so that the EOS for
the inner crust matches with the outer crust at one end and the outer core at
the other. 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 & 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 & Volkoff [1939], Tolman [1939].

Figure 4.2:  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).

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. 4.2. 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. 4.3. The results are divided into three groups depending
upon the square of sound speed ($c_{s,\rm max}^{2}$) at the centre 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 4.3:  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).

Figure 4.4:  The mass-radius relationship (left panel) obtained for the EOSs
as shown in Fig. 4.3. The 90% (solid) and 50% (dashed) confidence intervals of
the LIGO-Virgo analysis for the BNS component from the GW170817 event are
shown by the outer and inner gray shaded regions Abbott et al. [2018, 2019b,
2021]. The constraints from the millisecond pulsar PSR J0030+0451 (blue &
black) NICER x-ray data Riley et al. [2019], Miller et al. [2019] and PSR
J0740+6620 (green) Riley et al. [2021] are shown in the rectangular regions
enclosed by dotted lines. 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. [2019b]. 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 & Fattoyev [2019], Xie et al. [2022]

.

The EOSs displayed in Fig. 4.3 are employed to obtain the mass-radius
relationship for static neutron stars as presented in Fig. 4.4 (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. 4.4, 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 highlighted Abbott et al. [2019b].
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 masses Piekarewicz & 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 4.5: 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. 4.5, 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.

###  4.3  Neutron Star properties and symmetry energy parameters

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.

Figure 4.6:  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 4.7:  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.

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
$\rho\leqslant\rho_{c_{s}}$ 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 $\rho_{c_{s}}=2\rho_{0}$. 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. 4.6. 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 $(\rho>\rho_{c_{s}}$).
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.
4.6, 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. 4.6, we have generated two different distributions of
nuclear matter parameters with the help of their posterior distributions, as
shown in Fig. 4.1. 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. 4.1. 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. 4.7, 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
labelled 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. 4.1) 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 4.8: 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 that 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 4.9: 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 that 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. 4.8. 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 4.2:  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

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. 4.9. 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
desirable Imam 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 4.7 \- 4.9 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 4.2, 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
$\Lambda_{1.4}-K_{\rm sym,0}$.

Table 4.3:  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 $\rho_{c_{s}}$ and for the lower bound on maximum neutron star mass $M_{\rm max}$ $\geqslant$ 2.1$M_{\odot}$. $\frac{\rho_{c_{s}}}{\rho_{0}}$ | $\frac{M}{M_{\odot}}$ |  | $\Lambda_{M}$ |  |  | $R_{M}$ |
---|---|---|---|---|---|---|---
$L_{0}$ | $K_{\rm sym,0}$ | $P(2\rho_{0})$ | $L_{0}$ | $K_{\rm sym,0}$ | $P(2\rho_{0})$
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

So far, we have considered only the case of tidal deformability for a neutron
star with the canonical mass obtained for the EOSs assuming
$\rho_{c_{s}}=2.0\rho_{0}$, 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 4.3, 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 $\rho_{c_{s}}=1.5,1.75$ 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. 4.1), which determine the EOS
at $\rho\leqslant\rho_{c_{s}}$. The correlations increase a little with the
$\rho_{c_{s}}=1.5-2.0\rho_{0}$. 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 the correlation coefficient
presented in Table 4.3 may not be significantly affected by the inclusion of
any exotic degrees of freedom beyond 2$\rho_{0}$, since the EOSs beyond the
$\rho_{c_{s}}$ 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. 4.1. Similar to Table 4.3, 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 4.4. The values of correlation
coefficients show similar trends as listed in Table 4.3, 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.

Table 4.4:  Same as Table 4.3 but, the results are obtained using the uncertainty on the PNM EOS reduced to 3$\times$ N3L0. $\frac{\rho_{c_{s}}}{\rho_{0}}$ | $\frac{M}{M_{\odot}}$ |  | $\Lambda_{M}$ |  |  | $R_{M}$ |
---|---|---|---|---|---|---|---
$L_{0}$ | $K_{\rm sym,0}$ | $P(2\rho_{0})$ | $L_{0}$ | $K_{\rm sym,0}$ | $P(2\rho_{0})$
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

We have assessed 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
$\rho_{c_{s}}=2\rho_{0}$ by using different crust EOSs such as
NL3$\omega\rho$-L55 Grill et al. [2014], Pais & Providência [2016] and TM1e
Grill et al. [2014], Boukari et al. [2021] available in the CompOSE 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 %.

###  4.4 Conclusion

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
($\rho\leq\rho_{c_{s}}$) 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
$\rho_{c_{s}}=1.5-2\rho_{0}$ 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 $\rho_{c_{s}}$ 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 $\rho_{c_{s}}$. The diverse behavior of EOSs at high-density
($\rho\geqslant\rho_{c_{s}}$), 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 robust and 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.

##  Chapter 5 Multivariate Analysis of Neutron Star Properties

In this Chapter, we perform a Principal Component Analysis (PCA) ( see Chapter
2.7) to investigate the connection between multiple nuclear matter parameters
and the tidal deformability as well as the radius of neutron stars across a
wide range of masses Patra et al. [2023a]. The analysis is carried out using
two different sets of EOSs. The EOSs at low densities (using Taylor expansion
2.2.1) have been derived using the uncorrelated uniform and joint posterior
distributions of nuclear matter parameters. The joint posterior distributions
of nuclear matter parameters are obtained by imposing the constraints on 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 Hebeler et al. [2013], Lattimer [2021] within
the Bayesian Inference (see the chapter 2.6). The EOSs at high density
($\rho>2\rho_{0}$) are constructed by imposing the causality condition on the
speed of sound and are independent of compositions of NS matter (see the
chapter 2.3). Our analysis reveals that more than one principal component is
necessary to appropriately describe the NS properties, such as tidal
deformability and radius. The role of iso-scalar nuclear matter parameters
becomes increasingly important with neutron star mass.

###  5.1 Distributions of Nuclear Matter Parameters

Table 5.1:  The lower (Min.) and upper (Max.) bounds of the uniform distributions (D1) along with the means ($\mu$) and uncertainties ($2\sigma$) of marginalized distributions (D2) for each nuclear matter parameter except for $e_{0}$, which is kept fixed to -16.0 MeV are listed. NMPs | D1 | D2
---|---|---
(in MeV) | Min. | Max. | $\mu$ | 2$\sigma$
$K_{0}$ | 200 | 290 | 242 | 45
$Q_{0}$ | -500 | 450 | -25 | 466
$J_{0}$ | 30 | 35 | 32.2 | 2
$L_{0}$ | 30 | 80 | 54.2 | 24
$K_{\rm sym,0}$ | -300 | 100 | -89 | 180
$Q_{\rm sym,0}$ | 0 | 1000 | 772 | 700

We have generated two distinct sets of EOSs that correspond to different
distributions of nuclear matter parameters. One of these sets is based on
uncorrelated uniform distributions of nuclear matter parameters, and the other
one is obtained from their joint posterior distribution. The joint posterior
distribution of the nuclear matter parameters is taken from Ref. Patra et al.
[2023b], which was obtained by imposing minimal constraints that include some
selected 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], Lattimer [2021]. The
conditions of thermodynamic stability, causality speed of sound on the EOS,
and the resulting maximum mass of neutron star larger than 2$M_{\odot}$ were
also imposed. Some of the nuclear matter parameters are correlated due to
minimal constraints from the chiral effective field theory applied within the
Bayesian statistical method Gelman et al. [2013], Buchner et al. [2014],
Ashton et al. [2019]. The marginalized distributions of each nuclear matter
parameter are derived from the joint posterior distribution in order to put
the bounds on their uniform distributions. The bounds on uniform distributions
of nuclear matter parameters roughly the 90% confidence intervals of the
corresponding marginalized distributions. In Table 5.1, we have listed the
lower and upper bounds of the uniform distributions, as well as the means and
variances of the marginalized distributions for each nuclear matter parameter.
The iso-scalar nuclear matter parameters, the binding energy per nucleon
$e_{0}$, and the saturation density $\rho_{0}$ for symmetric nuclear matter
remain fixed at -16.0 MeV and 0.16 fm-3, respectively. References Patra et al.
[2023b], Kunjipurayil et al. [2022] have shown that the dependence of neutron
star properties on individual nuclear matter parameters is sensitive to the
choice of the distributions of nuclear matter parameters. We would like to
explore how multivariate analysis is sensitive to different distributions of
these nuclear matter parameters.

Figure 5.1:  The correlation among various nuclear matter parameters is shown.
The results for the left and right panels are for uncorrelated (D1) and
correlated (D2) distributions of nuclear matter parameters, respectively.
Color codes indicate the values of Pearson’s correlation coefficients among
the various NMPs.

We have generated 10,000 samples of nuclear matter parameters for each of the
uniform uncorrelated and joint posterior distributions, hereafter referred to
as D1 and D2, respectively. For each set of nuclear matter parameters, the
equations of the state beyond $2\rho_{0}$ are obtained for a random set of the
speed of sound parameters $n_{b},h_{p},w_{p},$ and $n_{p}$ as discussed in the
previous Chapter 2.3. Out of these, around 2,500 samples from both
distributions have been selected after applying the filters mentioned earlier.
In Fig. 5.1, we have illustrated the correlations among nuclear matter
parameters for both D1 and D2. The distribution D1 displays poor correlations
among the nuclear matter parameters, whereas D2 reveals stronger correlations
among some of the parameters due to the minimal constraints, such as the
notable correlation coefficient r($L_{0}$, $K_{\rm sym,0}$) = 0.80. The
significant correlations between $L_{0}$ and $K_{\rm sym,0}$ have been
previously documented as well Patra et al. [2022a, 2023b].

###  5.2 Fit of NS Properties to Nuclear Matter Parameters

The EOSs for $\beta$-equilibrated charge neutral matter in the density ranges
$0.5\rho_{0}$ to $2\rho_{0}$ are constructed using Taylor expansion with the
D1 and D2 distributions of nuclear matter parameters. Each of these low-
density EOS is smoothly joined by a diverse set of EOSs that satisfy causality
conditions as given by Eqs. (2.48-2.51). The EoS for outer and inner crusts
for density ranges $\rho<0.5\rho_{0}$ is used as follows. We employed Baym-
Pethick-Sutherland’s EOS in the density range $3.9\times
10^{-11}\rho_{0}<\rho<0.0016\rho_{0}$ for the outer crust. The inner crust EOS
is polytropic Carriere et al. [2003],

$\displaystyle p(\varepsilon)=\alpha+\beta\varepsilon^{\frac{4}{3}}.$ (5.1)

The inner crust EOS is matched with the outer crust at one end and the outer
core at the end by appropriately adjusting the values of the coefficients
$\alpha$ and $\beta$. The inner crust’s EOS affects the Love number $k_{2}$
and compactness parameter, but not the tidal deformability parameter
Piekarewicz & Fattoyev [2019]. After determining the core and crust EOSs, the
neutron star mass, radius, and tidal deformability for a given central
pressure can be computed using TOV equations. In order to demonstrate our
approach, we use a linear fit function of nuclear matter parameters as listed
in Table 5.1 to calculate the tidal deformability ($\Lambda_{M}$) and radius
($R_{M}$) for a given NS mass as,

$\displaystyle{\Lambda_{M}}$ $\displaystyle=$
$\displaystyle\sum_{i}W_{i}P_{i}+b=\sum_{i}\Lambda_{i}+b$ (5.2)
$\displaystyle{R_{M}}$ $\displaystyle=$
$\displaystyle\sum_{i}W^{\prime}_{i}P^{\prime}_{i}+b^{\prime}=\sum_{i}R_{i}+b^{\prime}$
(5.3)

where $W_{i}$ and $W^{\prime}_{i}$ are the weight factors of given nuclear
matter parameters. The $b$ and $b^{\prime}$ are the biases. The $P\in$ {
$K_{0}$, $Q_{0}$, $J_{0}$, $L_{0}$, $K_{\rm sym,0}$, $Q_{\rm sym,0}$} and M
stands for the neutron star masses. The $\Lambda_{i}$ and $R_{i}$ in the right
part of above Eqs.(5.2) and (5.3) correspond to the $W_{i}P_{i}$ and
$W^{\prime}_{i}P^{\prime}_{i}$, respectively. We assess the quality of fit for
our regression model using the $\mathcal{R}^{2}$ value as a measure Wackerly
et al. [2002]. The $\mathcal{R}^{2}$ value ranges from 0 to 1, where 0
signifies no variability in the dependent variable, and 1 represents complete
variance. The $\mathcal{R}^{2}$ close to unity indicates that the NS
properties from the solutions of TOV equations are almost equal to the ones
obtained by their linear fit to the nuclear matter parameters. In Fig. 5.2, we
present the $\mathcal{R}^{2}$ values obtained from fitting the tidal
deformability and radius of neutron stars within the mass range of
$M=1.2-2.0M_{\odot}$. As the neutron star mass increases, the associated
$\mathcal{R}^{2}$ value tends to decrease. Specifically, the $\mathcal{R}^{2}$
values for the fitted properties of neutron stars with a mass of 2$M_{\odot}$
fall below 0.9. Consequently, in what follows, we focus on the multivariate
analysis of neutron stars within the mass range of $1.2-1.8M_{\odot}$, guided
by the $\mathcal{R}^{2}$ value considerations.

Figure 5.2:  The $\mathcal{R}^{2}$ values of the neutron star properties, such
as tidal deformability and radius in the mass range of $1.2-2M_{\odot}$, are
shown. The different color symbols correspond to different NS properties and
the data. Figure 5.3:  The bar plot shows correlation coefficient values of NS
properties with nuclear matter parameters only for mass 1.2 and 1.8
$M_{\odot}$. For the comparison, the values of correlation coefficients from
the fitted model are shown with different color bars. The upper and lower
panels correspond to uncorrelated (D1) and correlated (D2) NMPs distributions.

In Fig.5.3, we display the correlation coefficient values obtained for exact
values of tidal deformability and radius with individual nuclear matter
parameters for the masses 1.2 and 1.8 $M_{\odot}$ by green and red bars,
respectively. We also juxtapose similar results obtained from the fitted
values of tidal deformability and radius through Eqs. (5.2) and (5.3) for the
masses 1.2 and 1.8 $M_{\odot}$ by purple and blue bars, respectively. The
results are presented for two different distributions of NMPs as indicated by
labels D1 (Upper) and D2 (Lower). This visual representation demonstrates that
all the correlation coefficient values remain consistent even after the
fitting process. The application of the fit does not alter these correlation
trends. The correlations of the tidal deformability with NMPs are
qualitatively similar for the D1 and D2 cases. In general, the D2
distributions of NMPs result in stronger correlations. The correlations of the
radius with the NMPs appear to be sensitive to the choice of their
distributions. In particular, the radius is very weakly correlated with
$K_{\rm sym,0}$ for the D1 case and shows reasonably strong correlations for
the D2 case. It is also interesting to note that the correlations of tidal
deformability and radius with $Q_{0}$ increase somewhat with an increase in NS
mass.

###  5.3 Principal Component Analysis of NS Properties

As discussed in Chapter 2.7, the methodology of PCA analysis comprises the
following steps: (1) Construction of Covariance matrix (2) Diagonalization of
the covariance matrix. (3) The principal components’ proportionate variance
captured by the eigenvectors and eigenvalues. First, we calculate the values
of weights $W_{i}$s and $W^{\prime}_{i}$s appear in Eqs. (5.2) and (5.3). The
PCA is performed using $\Lambda_{i}$s and $R_{i}$s as features corresponding
to the target variables $\Lambda$ and R, respectively. Then, $6\times 6$
covariance matrices for the tidal deformability and radius of NS are
constructed for a given mass. The eigenvalues and the corresponding
eigenvectors obtained by diagonalizing the covariance matrices are arranged in
descending order. The most important principal component, PC1 corresponds to
the eigenvector with the highest eigenvalue. The succeding eigenvalues and the
eigenvectors are labelled as PC2, PC3, etc. The nuclear matter parameter
having the largest contribution to the eigenvector for PC1 is the most
dominant one in determining the NS property in consideration. Likewise, the
eigenvectors associated with remaining PCs together with eigenvalues, can be
used to identify other important nuclear matter parameters for the NS
properties.

Table 5.2:  The normalized eigenvalues associated with the principal components are listed for the NS properties with masses range $1.2-1.8M_{\odot}$. D1 and D2 correspond to the uncorrelated and correlated distributions of the nuclear matter parameters.  NS | $\frac{M}{M_{\odot}}$ | D1 | D2
---|---|---|---
PC1 | PC2 | PC3 | PC4 | PC5 | PC6 | PC1 | PC2 | PC3 | PC4 | PC5 | PC6
$\Lambda_{M}$ | 1.2 | 1.00 | 0.62 | 0.32 | 0.17 | 0.13 | 0.01 | 1.00 | 0.35 | 0.23 | 0.13 | 0.04 | 0.00
1.4 | 1.00 | 0.40 | 0.35 | 0.20 | 0.14 | 0.00 | 1.00 | 0.52 | 0.27 | 0.16 | 0.04 | 0.00
1.6 | 1.00 | 0.53 | 0.25 | 0.22 | 0.15 | 0.00 | 1.00 | 0.67 | 0.33 | 0.20 | 0.03 | 0.00
1.8 | 1.00 | 0.70 | 0.31 | 0.16 | 0.13 | 0.00 | 1.00 | 0.71 | 0.36 | 0.21 | 0.01 | 0.00
RM | 1.2 | 1.00 | 0.12 | 0.10 | 0.07 | 0.01 | 0.00 | 1.00 | 0.18 | 0.09 | 0.02 | 0.00 | 0.00
1.4 | 1.00 | 0.19 | 0.12 | 0.03 | 0.00 | 0.00 | 1.00 | 0.29 | 0.13 | 0.01 | 0.00 | 0.00
1.6 | 1.00 | 0.31 | 0.18 | 0.01 | 0.00 | 0.00 | 1.00 | 0.46 | 0.20 | 0.01 | 0.00 | 0.00
1.8 | 1.00 | 0.54 | 0.29 | 0.03 | 0.02 | 0.00 | 1.00 | 0.70 | 0.32 | 0.04 | 0.00 | 0.00
Figure 5.4:  The variance (in %) of different principal components
corresponding to the NS properties is displayed. The results in the upper and
lower panels represent the uncorrelated (D1) and correlated (D2) nuclear
matter parameter distributions, respectively.

In Table 5.2, we provide a list of eigenvalues that were obtained by
diagonalizing the covariance matrix for a specific property and mass of a
neutron star. The highest eigenvalues are normalized to unity, representing
PC1. The principal component with a normalized eigenvalue less than 0.1 does
not contribute significantly. It is evident that a larger number of principal
components contribute to tidal deformability for most of the cases than those
for the corresponding radius. For instance, consider a neutron star with a
mass of $1.2M_{\odot}$; there are four significant principal components with
normalized eigenvalues higher than 0.1 for tidal deformability, while only
three for the radius.

In Fig. 5.4, we display the percentage of variation in neutron star properties
explained by the first four principal components. The upper and lower panels
illustrate the outcomes for uncorrelated (D1) and joint posterior (D2)
distributions of nuclear matter parameters. The bars in green correspond to
properties at a neutron star mass of $1.2M_{\odot}$, while the red bars
represent $1.8M_{\odot}$. This figure depicts how much each PC contributes to
capturing the variance. From PC1 to PC4, the portion of variance decreases.
The total number of PCs needed to account for over 90% of the variance is
smaller for the radius than for the tidal deformability. As an example,
consider $\Lambda_{1.2}$ in cases D1 and D2: to achieve over 94% and 97%
variance, respectively, at least four PCs are necessary. On the other hand,
for $R_{1.2}$, only three PCs are sufficient to achieve more than 94% and 98%
variance in the respective cases.

Figure 5.5:  The square of the amplitude values of various nuclear matter
parameters for each principal component. The results are presented for the
uncorrelated nuclear matter parameter distributions (D1). The different PCs
are indicated by different color bars. Figure 5.6:  Same as Fig.5.5 but for
correlated nuclear matter parameters distributions (D2).

The squared amplitude components of eigenvectors for a specific principal
component provide insights into the contributions from different nuclear
matter parameters. This is illustrated in Fig. 5.5 and 5.6, corresponding to
D1 and D2 distributions, respectively. The upper panels pertain to tidal
deformability, while the lower panels relate to radii. Various nuclear matter
parameters contribute differently to the principal components, and this
sensitivity depends on the chosen distribution of nuclear matter parameters.
In the case of D1 distribution (Fig. 5.5), PC1 is primarily composed of a
single parameter, $K_{\rm sym,0}$, for both $\Lambda_{1.2}$ and
$\Lambda_{1.8}$. The PC2 is dominated by $L_{0}$ for $\Lambda_{1.2}$ and by
$Q_{0}$ for $\Lambda_{1.8}$. The contribution to PC3 for $\Lambda_{1.2}$ is
mainly from $Q_{0}$, while for $\Lambda_{1.8}$, it is primarily $Q_{\rm
sym,0}$. In PC4, $Q_{\rm sym,0}$ has the most significant contribution for
$\Lambda_{1.2}$, while for $\Lambda_{1.8}$, it is from $K_{0}$. Combining the
outcomes of figures 5.4 \- 5.6, we can infer that the behavior of
$\Lambda_{1.2}$ can be primarily explained by the linear combination of
$K_{\rm sym,0}$, $L_{0}$, and $Q_{0}$ in the D1 distribution. However, in the
case of the D2 distribution, the influence of $K_{0}$ becomes evident as an
additional factor. Shifting the focus to $\Lambda_{1.8}$, the impact of
$L_{0}$ diminishes, while the contributions from iso-scalar parameters like
$K_{0}$ and $Q_{0}$, along with the iso-vector parameter $Q_{\rm sym,0}$,
increase. This suggests that $\Lambda_{1.8}$ is influenced by a simultaneous
interplay of various nuclear matter parameters. For radii ($R_{1.2}$ and
$R_{1.8}$), PC1 is dominated by $L_{0}$, while PC2 by $Q_{0}$. In PC3 for
$R_{1.2}$, contributions from $K_{\rm sym,0}$ and $Q_{\rm sym,0}$ are notable,
whereas for $R_{1.8}$, it’s mainly $Q_{\rm sym,0}$. In Fig. 5.6, similar
outcomes are depicted, but for D2 distributions, which are notably different
from those in Fig. 5.5. Multiple nuclear matter parameters contribute to most
PCs due to correlations in the D2 distribution. Strong correlations, such as
between $L_{0}$ and $K_{\rm sym,0}$, lead to their combined contributions. For
instance, in PC1, both $\Lambda_{1.2}$ and R1.2 are influenced by $L_{0}$ and
$K_{\rm sym,0}$, while $\Lambda_{1.8}$ is influenced by $Q_{0}$, $K_{\rm
sym,0}$, $Q_{\rm sym,0}$ and $R_{1.8}$ by $Q_{0}$, $L_{0}$, and $Q_{\rm
sym,0}$. The PC2’s contributions include $Q_{0}$ and $Q_{\rm sym,0}$ for
$\Lambda_{1.2}$ and $R_{1.2}$, $Q_{0}$, $K_{\rm sym,0}$, and $Q_{\rm sym,0}$
for $\Lambda_{1.8}$, and $Q_{0}$, $L_{0}$, and $Q_{\rm sym,0}$ for $R_{1.8}$.
The PC3 encompasses all NMPs except $J_{0}$ for $\Lambda_{1.2}$, $Q_{0}$ and
$Q_{\rm sym,0}$ for $\Lambda_{1.8}$, $Q_{0}$, $K_{\rm sym,0}$, and $Q_{\rm
sym,0}$ for $R_{1.2}$, and $Q_{0}$, $Q_{\rm sym,0}$ for $R_{1.8}$. Similarly,
PC4 involves $K_{0}$, $Q_{0}$, and $Q_{\rm sym,0}$ for $\Lambda_{1.2}$, while
$K_{0}$ dominates PC4 for $\Lambda_{1.8}$. The contributions of PC4 for radii
are negligible.

Figure 5.7:  The values of the percentage contributions of nuclear matter
parameters to the NS properties with masses range $1.2-1.8M_{\odot}$. The
results in the upper panels and lower panels correspond to the D1 and D2
distributions of nuclear matter parameters, respectively. The colors of the
bar are related to the nuclear matter parameters.

We computed the total contributions for each of the nuclear matter parameters
by summing their contributions to each of the PCs. The weights for this
calculation are derived from the product of the square of amplitude for each
nuclear matter parameter and the corresponding eigenvalues of the PCs listed
in Table 5.2. These weights are then normalized so that the total sum of
contributions from all nuclear matter parameters equals unity. The percentages
representing the contributions of individual nuclear matter parameters to
neutron star properties within the mass range of $1.2-1.8M_{\odot}$ are shown
in Fig. 5.7. The upper and lower panels show the results for D1 and D2
distributions, respectively. Different colors are used to depict the
percentage contributions of each nuclear matter parameter. It is important to
note that the choice of nuclear matter parameter distributions influences the
contributions of different nuclear matter parameters to neutron star
properties. For both D1 and D2 distributions, the contributions of specific
nuclear matter parameters to tidal deformability and radius can differ by up
to 20%. However, these differences diminish when categorized broadly into iso-
scalar and iso-vector parameters. Iso-scalar parameters, $K_{0}$ and $Q_{0}$,
contribute together, while the remaining parameters contribute to the iso-
vector category. The total contributions from iso-scalar parameters increase,
while they decrease for the iso-vector parameters with the increase in NS
mass. For instance, the contribution from iso-scalar parameters in the case of
D1 (D2) distributions increases from 15% to 40% (24% to 44%) for tidal
deformability with the NS mass increasing from 1.2$M_{\odot}$ to
1.8$M_{\odot}$. Concerning radius, iso-scalar parameter contributions increase
from approximately 10% to 30% as the neutron star mass increases from
$1.2M_{\odot}$ to $1.8M_{\odot}$.

Finally, it may be emphasized that the present work highlights the necessity
of multivariate analysis of neutron star properties through PCA as one of the
tools. A more comprehensive investigation addressing improved treatment of
crust EOSs and high-density EOSs is warranted for a more realistic assessment
Carreau et al. [2019].

###  5.4 Conclusion

We have addressed an unresolved issue of connecting the nuclear matter
parameters to the key neutron star properties, such as tidal deformability and
radius. The outcomes of the majority of the investigations exploring the
correlations between properties of neutron stars and individual nuclear matter
parameters, which describe the equations of state, are at variance. We have
exploited the efficacy of Principal Component Analysis, a sophisticated
analytical tool, in order to establish a comprehensive connection between
multiple nuclear matter parameters and the key properties of neutron stars,
with the ultimate aim of shedding some light on the existing issue. The EOSs
essential for describing neutron star matter within the core region up to a
density of $2\rho_{0}$ have been derived. This was accomplished by utilizing
both uncorrelated uniform and joint posterior distributions of nuclear matter
parameters. To ensure continuity and consistency, each of these distinct EOSs
is joined smoothly at $2\rho_{0}$ by a diverse set of the EOSs obtained by
parameterizing the speed of sound such that it remains causal and approaches
the conformal limit gradually.

We have found that the variability in the considered neutron star properties
requires the incorporation of more than one principal component. These
observations emphasize that the neutron star properties depend on multiple
nuclear matter parameters. In particular, tidal deformability demands the
inclusion of three or more principal components to account for over 90% of its
variations, while for the radius, two principal components suffice to explain
similar variations. The dominance of nuclear matter parameters contributing to
the principal components depends on the specific NS properties and their mass.
For instance, in the case of the tidal deformability of a neutron star with a
mass of 1.2$M_{\odot}$, the symmetry energy curvature parameter $K_{\rm
sym,0}$ emerges as the primary contributor to the first principal component.
The second and third principal components are significantly influenced by the
symmetry energy slope parameter $L_{0}$ and the skewness parameter $Q_{0}$ for
symmetric nuclear matter, respectively. The significance of iso-scalar nuclear
matter parameters, specifically the incompressibility coefficient K0 and the
skewness parameter $Q_{0}$ of symmetric nuclear matter, becomes more
pronounced with an increase in the mass of the neutron star. When the NS mass
reaches 1.8$M\odot$, the incompressibility coefficient $K_{0}$ surpasses the
importance of the symmetry energy slope parameter $L_{0}$. When considering
the radius of a neutron star at lower masses, the symmetry energy slope
parameter $L_{0}$ stands out as the primary driver behind the observed
variations. Overall, when analyzing the collective impact of iso-scalar
parameters ($K_{0}$ and $Q_{0}$), these contributions exhibit an approximately
25% increase with the increase in neutron star mass from 1.2$M_{\odot}$ to
1.8$M_{\odot}$.

##  Chapter 6 Summary and Conclusion

This thesis investigated the connection of nuclear physics inputs to the
neutron star properties. For this analysis, we employed several statistical
tools. The Bayesian analysis is used to construct the equation of the state of
neutron stars by using recently available astrophysical observational
constraints. The Principal Component Analysis is employed to connect the
various neutron star properties directly to nuclear physics inputs.

Bayesian analysis is a common statistical approach for estimating the joint
posterior distribution of model parameters. It is based on Bayes’s theorem,
which combines the prior, likelihood, and evidence to compute a posterior
probability distribution, representing updated beliefs based on observed data.
The nuclear part of EOS is expressed in terms of nuclear matter parameters. We
use the nuclear part of EOS to construct marginalized posterior distributions
of the nuclear matter parameters that are consistent with the minimal
constraints. A few low-order nuclear matter parameters, such as the binding
energy per nucleon, the incompressibility coefficient for the symmetric
nuclear matter, and symmetry energy coefficients at the saturation density
($\rho_{0}$), are constrained in narrow windows along with the low-density
pure neutron matter EOS obtained from a precise next-to-next-to-next-to-
leading-order (N3LO) calculation in chiral effective field theory. Principal
Component Analysis (PCA) is a statistical method used for dimensionality
reduction of a dataset. The process begins with standardizing the data,
ensuring variables are on a comparable scale. The diagonalized covariance
matrix is then computed, and its eigenvalues and eigenvectors are calculated.
Sorting the eigenvalues in descending order allows for the selection of the
top eigenvectors, known as principal components. The largest eigenvalue
associated with the first PC captures the largest variance, and so on. PCs are
the composition of linear combinations of original variables to capture the
shared variation patterns. By analyzing the correlation between key principal
components and the target variable, we can identify the key parameters
essential for the properties of neutron stars.

In Chapter-3, we used Taylor and $\frac{n}{3}$ expansions to obtain the
marginalized posterior distributions of the nuclear matter parameters by
applying a Bayesian approach to both expansions. The nuclear matter parameters
or the corresponding EOSs are consistent with a set of minimal constraints
that includes basic properties of saturated nuclear matter and low-density
($\rho=0.08-0.16$fm-3) EOS for the pure neutron matter from (N3LO) calculation
in chiral effective field theory Hebeler et al. [2013]. The NS properties,
such as tidal deformability, radius, and maximum mass, are evaluated using
large sets of minimally constrained EOSs. The correlations of the
$\beta-$equilibrium EOSs as a function of density with various neutron star
properties over a wide mass range are investigated. We found that the pressure
for the $\beta$-equilibrated matter at density $\sim 2\rho_{0}$ is strongly
correlated with NS properties such as tidal deformability and radius at
1.4$M_{\odot}$. The pressure for $\beta$-equilibrated matter at density $\sim
3\rho_{0}$ substantially correlates with the radius of $2.07M_{\odot}$ neutron
star. The maximum mass of a neutron star is linked to the pressure of
$\beta$-equilibrated matter at density $\sim 4.5\rho_{0}$. These correlation
systematics match those of unified EOSs for $\beta$-equilibrated matter in
several non-relativistic and relativistic mean-field models. We found model
independence in the correlations of pressure for $\beta$-equilibrated matter
with NS properties. The variations of NS properties with the pressure of
$\beta-$equilibrated matter at $2\rho_{0}$ remain quite robust, which may be
due to the pressure depending on the combination of multiple nuclear matter
parameters that describe the symmetric nuclear matter as well as the density
dependence of the symmetry energy.

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 & Agrawal [2021], Reed
et al. [2021], Pradhan et al. [2022, 2023], Ghosh et al. [2022b, a], Beznogov
& Raduta [2023]. Depending upon the distribution of nuclear matter parameters,
the correlations of NS properties with the nuclear matter parameters are also
summarized in Table III of Ref. Kunjipurayil et al. [2022]. There may be other
factors that affect these correlations 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 the density
($\rho_{c_{s}}$), (v) upper bound on the value of tidal deformability, are
studied in Chapter-4. The low-density ($\rho\leq\rho_{c_{s}}$) EOSs are
constructed by employing our minimal constraints within a Bayesian approach.
The EOSs beyond $\rho_{c_{s}}=1.5-2\rho_{0}$ are constrained only by imposing
the causality condition on the speed of sound. The correlation of tidal
deformability is found to be stronger with $K_{\rm sym,0}$ than with slope
parameter $L_{0}$. 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. The high-density part of the EOSs does not influence to the
correlation of NS properties with symmetry energy parameters. With an increase
in the upper bound of $\Lambda_{1.4}$, these correlations improved much. These
correlations improve a little bit with the increase in the density
$\rho_{c_{s}}$. The results for the correlation of tidal deformability with
the pressure at 2$\rho_{0}$ are robust and practically independent of all the
factors considered. This is because the pressure is not determined by single
parameters. It is composed of multivariate parameters.

A multivariate analysis is required to identify the key nuclear matter
parameters that have the greatest impact on the neutron star’s properties,
such as tidal deformability and radius at canonical mass 1.4 $M_{\odot}$. In
Chapter-5, we perform a Principal Component Analysis (PCA) to investigate the
connection between multiple nuclear matter parameters and the tidal
deformability as well as the radius of neutron stars within the mass range of
$1.2-1.8M_{\odot}$. The analysis is carried out using two different sets of
EOSs. The EOSs at low densities (using Taylor expansion 2.2.1) have been
derived using the uncorrelated uniform and joint posterior distributions of
nuclear matter parameters. The EOSs at high density ($\rho>2\rho_{0}$) are
constructed by imposing the causality condition on the speed of sound and are
independent of compositions of NS matter (see the chapter 2.3). We have found
that the variability in the considered neutron star properties requires the
incorporation of more than one principal component. These observations
emphasize that the neutron star properties depend on multiple nuclear matter
parameters. In particular, tidal deformability demands the inclusion of three
or more principal components to account for over 90% of its variations, while
for the radius, two principal components suffice to explain similar
variations. In the case of the tidal deformability of a neutron star, the
symmetry energy curvature parameter $K_{\rm sym,0}$ emerges as the primary
contributor to the first principal component. When considering the radius of a
neutron star, the symmetry energy slope parameter $L_{0}$ stands out as the
primary driver behind the observed variations. Overall, when analyzing the
collective impact of iso-scalar parameters ($K_{0}$ and $Q_{0}$), these
contributions exhibit an approximately 25% increase with the increase in
neutron star mass from 1.2$M_{\odot}$ to 1.8$M_{\odot}$.

##  Chapter 7 Future Prospective

In recent years, our understanding of the equation of state for dense matter
has made significant progress. However, there is still space for expansion of
the research presented in this thesis, particularly with regard to neutron
stars. We have already started work on potential expansions, taking into
consideration recent and upcoming developments in the available facilities
around the world. In addition, multiple theoretical, observational,
experimental, and empirical perspectives are being considered. Detailed below
are some of our ongoing initiatives:-

  1. 1.

Constraints the EOS in a model-independent way:-
Using different nuclear models, researchers have been looking into the
possibility of discovering links between the parameters of nuclear matter and
the properties of neutron stars. However, these association investigations
have shown that there is a strong model reliance. This dependence on the model
is caused by the fact that different models might result in distinct equations
of state, even though the nuclear matter properties of the models are
comparable. We plan to use a Bayesian analysis to study further the
correlations between nuclear matter parameters and various astrophysical
observables in order to overcome this issue and get a more model-independent
point of view. This will allow us to better understand how various
astrophysical observables relate to one another. We can evaluate these
associations using a method that is less dependent on particular nuclear
models as a result of this approach.

  2. 2.

Machine Learning approach to nuclear matter studies:-
The correlation analysis is mostly used to find factors that are linearly
dependent on each other. It is important to note, though, that the connection
between observables in astrophysics and properties of nuclear matter might be
complicated and not follow a straight line. Supervised machine learning
methods can be used to find these non-linear connections. In neutron star
physics, for example, a Deep Neural Network (DNN) has been used as a useful
way to connect a limited set of mass-radius observed data to the equation of
state plane. This methodology facilitates the investigation of intricate and
non-linear correlations that might not be apparent alone through conventional
correlation analysis.

  3. 3.

Neutron stars with high magnetic field:-
The magnetic fields of neutron stars are extraordinarily powerful, usually
between $10^{12}$ and $10^{15}$ Gauss. They are created over millions of years
from interstellar gases. Anomalous X-ray pulse stars are frequently linked to
magnetars, which are neutron stars with considerably stronger fields,
typically between $10^{12}$ and $10^{15}$ Gauss. Significant to the physics of
compact star mergers, these magnetic fields also affect the structure and
composition of neutron stars. In addition to the fact that magneto-rotational
instability can cause magnetic fields to grow extraordinarily enormous, which
can have an impact on the dense matter equations of state, gravitational waves
released during mergers can also shed light on these equations.

  4. 4.

Exploring Neutron Star Oscillations and Hybrid Star Phase Transitions:-
There is significant potential for future research on phase transitions in
hybrid stars and oscillations in neutron stars. It is still an exciting and
difficult study path to look at the equation of state governing these
oscillation modes and to look for phase transitions within hybrid stars.
Comprehending the basic characteristics of neutron stars, like their internal
makeup and behavior, will contribute to our understanding of extreme
astrophysical environments and clarify the complex physics of matter in
extreme circumstances. In the end, this investigation might provide
significant new information on the high-density matter domain and further our
knowledge of the cosmos.

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