∎ e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>11institutetext: Institute of Physics, Sachivalaya Marg,
Bhubaneswar-751005, India 22institutetext: Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400085, India
33institutetext: Department of Astronomy, Xiamen University, Xiamen 361005, P.
R. China 44institutetext: Department of Physics $\&$ Astronomy, National
Institute of Technology, Rourkela, India 55institutetext: Center for
Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan

# Warm dense matter and cooling of supernovae remnants

Ankit Kumare1,addr1,addr2  H. C. Dase2,addr1,addr2  S. K. Biswal addr3  Bharat
Kumar addr4,addr5  S. K. Patra e3,addr1,addr2

(Received: xxxx / Accepted: xxxx)

###### Abstract

We study the thermal effects on the nuclear matter (NM) properties such as
binding energy, incompressibility, free symmetry energy and its coefficients
using NL3, G3 and IU-FSU parameter sets of relativistic mean-field models.
These models being consistent with the properties of cold NM, have also been
used to study the effect of temperature by incorporating the Fermi function.
The critical temperature for the liquid-gas phase transition in the symmetric
NM is found to be 14.60, 15.37 and 14.50 MeV for NL3, G3 and IU-FSU parameter
sets respectively, which is in excellent agreement with previous theoretical
and experimental studies. We inspect that the properties related to second
differential coefficient of the binding energy and free symmetry energy at
saturation density ( i.e. $K_{0}(n,T)$ and $Q_{sym,0}$ ) exhibit the contrary
effects for NL3 and G3 parameters as the temperature increases. We find that
the prediction of saturated curvature parameter ( $K_{sym,0}$ ) for G3
equation of state at finite temperature favour the combined analysis of
$K_{sym,0}$ for the existence of massive pulsars, gravitational waves from
GW170817 and NICER observations of PSR J0030+0451. Further, we investigate the
cooling mechanism of newly born stars through neutrino emissivity controlled
by direct Urca process and instate some interesting remarks about neutrino
emissivity. We also deliberate the effect of temperature on the M-R profile of
Proto-Neutron star.

###### Keywords:

nuclear matter equation of state neutron star neutrinos

††journal: Eur. Phys. J. C

##  1 Introduction

One of the most prominent energetic events of the universe is manifested by
the core-collapse supernovae (CCSN) explosion of the giant stars having mass
in the range of 8 - 40 times that of the mass of sun Gilmore1915 . The concept
that the gravitational instability during the evolution of a massive star
results in a rapid compression and then thermonuclear explosion, was first
suggested by Burbidge, Burbidge, Fowler, and Hoyle (designated as "$B^{2}FH$")
Colgate . The energy emerge during this explosion is carried off by the
photons and the neutrinos, which are billion trillion trillion in numbers and
hauled the most of the energy released. The observation of Einstein
Observatory (HEAO-2) first supported the fact that high neutrino emissivity is
mainly responsible for the rapid cooling of newly born dense star Boguta .
Studies suggest that the neutrinos also play a significant role in the
dynamics of supernovae explosion and control many important aspects of the
collapsed core, generally known as newly born neutron star Burrows ; Barbara .
In this exploring era of every tiny and massive object through modern science,
neutron stars still hold the mystery in itself and are poorly known objects.
These are the ideal play-field in the observable universe to inspect the
theories of dense matter physics and unfold new opportunities. A large part of
our perception about the dynamics of supernovae explosion and the composition
of neutron stars is based on the equation of state (EoS), which makes the EoS
a key ingredient of our study. Since it is believed that the matter evolved
during the core-collapse phenomenon is unable to attain $\beta-$equilibrium
condition for few seconds Stone , due to the promptness of the CCSN explosion,
so, usually the EoS for a dense matter with either same amount of protons and
neutrons or a proton fraction of $\sim 0.2$ is taken into account to cipher
the dynamics of collapse event Mezzacappa . Many other important aspects of
core-collapse events like how many protons are converted into neutrons,
quenching rate, the mass-radius profile of newly born neutron star and its
composition are determined by the EoS Schneider .

Another important dimension which makes the EoS of nuclear matter (NM) at
finite temperature more interesting is the study of the dynamics of heavy ion
collision reaction and structure of exotic nuclei. Under extreme conditions of
density, isospin and temperature the EoS can be explored through many
experimental facilities such as GANIL -SPIRAL2 France  facility in France, CSR
in China ZHIYU  and the FRIB technique in the United States US . Nuclear
collisions can produce hot and dense hadronic NM in terrestrial laboratories
momentarily, which can procure a lot of information about the thermodynamical
properties of NM like incompressibility, symmetry energy and its derivatives
Danielewicz . In heavy-ion collision experiments, the symmetry energy plays an
important role in understanding the isoscaling behaviour of multi-
fragmentation phenomena. The observations of the experiments at Texas A$\&$M
University and National Superconducting Cyclotron Laboratory suggest a
connection between the nuclear symmetry energy and primary fragment yield
distribution of statistical multi-fragmentation model AkiraOno ; Souliotis ;
Iglio . On the other hand, symmetry energy at finite temperature also plays an
important role in the cooling mechanism of newly born hot astrophysical
objects Prakash ; Takatsuka ; Bao-AnLi  and has effective impact on the
cooling rate through direct Urca and modified Urca processes 717Y ; refId0 .

In the recent years, the density dependence of the symmetry energy and its
derivatives ($L_{sym}$, $K_{sym}$, $Q_{sym}$) have been used to constrain the
EoS near saturation density, which make them more vital to decipher properly.
It is well known that the slope parameter ($L_{sym}$) and the size of the
neutron skin in super-heavy nuclei are connected by a strong linear
correlation. The slope ($L_{sym}$) and curvature parameter ($K_{sym}$) also
control the location of the neutron drip line, core-crust transition density
and gravitational binding energy of neutron star Tsang2 . The isovector
skewness parameter ($Q_{sym}$) is the most ambiguous quantity, due to the
large fluctuations in its value obtained from different models. Recent studies
show that $Q_{sym}$ is related to the incompressibility Typel  of the system
and also suggest an important role of $Q_{sym}$ in the cooling of newly born
proto-neutron star Schneider . We study in detail the effects of temperature
on all these nuclear parameters and the correlations among them using the most
familiar NL3, IU-FSU and recently developed G3 parameter sets. With the
thermal effects being the main focus of our study, we also explore the
dependence of thermal index ($\Gamma_{th}$) on the density for all the
parameter sets. In the extant work, we also investigate the dependence of
cooling mechanism of a hot dense matter on the EoS and the variation in the
mass-radius profile of a proto-neutron star with temperature. The temperature
in the interior of a newly born dense star just after the supernova explosion
can vary from 10 to 100 MeV Oechslin . The analyzation of the x-rays emitted
during the stellar evolution of the young star ensure the fact that it losses
most of its energy through extremely rapid neutrino emission enhanced by the
direct Urca process (named after a casino in Rio) Brown ; Kaminker ; Yakovlev
. However, the direct Urca process dominates only during the initial stage of
the cooling and can no longer operate once the proton fraction inside the core
reaches a threshold value. The proton fraction in a newly born proto-neutron
star depends on how the nuclear symmetry energy scales with density – which is
an important aspect of this work Brown . In the mean-field approximation, the
symmetry energy and the magnitude of the neutrino emissivity $(Q)$ seem to be
controlled by the EoS of different parameter sets.

The paper is organized as: In section 2, we explain the formalism for
temperature dependent quantum hadron- dynamics model. Section 3 is divided
into two parts, where 3.1 is devoted to the theoretical structure of the
relativistic mean-field formalism at finite temperature and acquaintance with
the NM parameters. In sub-section 3.2 we presented the results for various NM
parameters of symmetric NM using NL3, G3 and IU-FSU parameter sets. The
detailed framework and the results of neutrino emissivity through direct Urca
process are extended in section 4. The mass-radius profile of the proto-
neutron star is discussed in section 5. Finally, the discussion and the
concluding remarks are outlined in the section 6.

##  2 Temperature Dependent QHD Model

Temperature dependent QHD model is based on relativistic covariant field
theory, proposed to obtain the expediential in-medium nuclear properties
around the saturation density. The nucleons (neutrons and protons) are
represented by the Dirac spinor $\psi$ in the Lagrangian density. We take in
account the $\sigma$, $\omega$, $\rho$ and $\delta$ mesons in our Lagrangian,
represented by the $\sigma$, $\omega^{\mu}$, $\vec{\rho}^{\,\mu}$ and
$\vec{\delta}$ fields respectively. The mesons act as the mediators and
represent the effective nuclear interaction of the nucleons through the meson
fields $\sigma$, $\omega^{\mu}$, $\vec{\rho}^{\,\mu}$ and $\vec{\delta}$. The
basic Lagrangian density used in the present work, which includes the
interaction of nucleon fields $\psi$ with a scalar field ($\sigma$), a vector
field ($\omega$) and isovector fields ($\vec{\rho}^{\,\mu}$ and
$\vec{\delta}$) and the cross-coupled interactions of these meson fields up to
fourth order S.K.Singh , is given by

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\bar{\psi}_{\alpha}\Bigg{\\{}\gamma_{\mu}(i\partial^{\mu}-g_{\omega}\omega^{\mu}-\frac{1}{2}g_{\rho}\vec{\tau}_{\alpha}\\!\cdot\\!\vec{\rho}^{\,\mu})-(M-g_{\sigma}\sigma$
(1) $\displaystyle-
g_{\delta}\vec{\tau}_{\alpha}\\!\cdot\\!\vec{\delta})\Bigg{\\}}\psi_{\alpha}+\frac{1}{2}\partial^{\mu}\sigma\,\partial_{\mu}\sigma-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{\zeta_{0}}{4!}g_{\omega}^{2}(\omega^{\mu}\omega_{\mu})^{2}$
$\displaystyle\hbox{}-g_{\sigma}\frac{m_{\sigma}^{2}}{M}\Bigg{(}\frac{\kappa_{3}}{3!}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma}}{M}\sigma\Bigg{)}\sigma^{3}+\frac{1}{2}m_{\omega}^{2}\omega^{\mu}\omega_{\mu}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$
$\displaystyle\hbox{}+\frac{1}{2}\frac{g_{\sigma}\sigma}{M}\Bigg{(}\eta_{1}+\frac{\eta_{2}}{2}\frac{g_{\sigma}\sigma}{M}\Bigg{)}m_{\omega}^{2}\omega^{\mu}\omega_{\mu}+\frac{1}{2}\eta_{\rho}\frac{m_{\rho}^{2}}{M}g_{\sigma}\sigma(\vec{\rho}^{\,\mu}\\!\cdot\\!\vec{\rho}_{\mu})$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\rho^{\mu}\\!\cdot\\!\rho_{\mu}-\frac{1}{4}\vec{R}^{\mu\nu}\\!\cdot\\!\vec{R}_{\mu\nu}-\Lambda_{\omega}g_{\omega}^{2}g_{\rho}^{2}(\omega^{\mu}\omega_{\mu})(\vec{\rho}^{\,\mu}\\!\cdot\\!\vec{\rho}_{\mu})$
$\displaystyle+\frac{1}{2}\partial^{\mu}\vec{\delta}\,\partial_{\mu}\vec{\delta}-\frac{1}{2}m_{\delta}^{2}\vec{\delta}^{\,2},$

where M is the nucleon mass; $m_{\sigma}$, $m_{\omega}$, $m_{\rho}$ and
$m_{\delta}$ are the masses of mesons and $g_{\sigma}$, $g_{\omega}$,
$g_{\rho}$ and $g_{\delta}$ are the coupling constants for the $\sigma$,
$\omega$, $\rho$ and $\delta$ mesons respectively; $\kappa_{3}$ (or
$\kappa_{4}$) and $\zeta_{0}$ are the self-interacting coupling constants of
the $\sigma$ and $\omega$ mesons respectively; $\eta_{1}$, $\eta_{2}$,
$\eta_{\rho}$ and $\Lambda_{\omega}$ are the coupling constants of non-linear
cross-coupled terms. The quantities $F^{\mu\nu}$ and $\vec{R}^{\mu\nu}$ being
the field strength tensors for the $\omega$ and $\rho$ mesons respectively,
defined as $F^{\mu\nu}$ =
$\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu}$ and $\vec{R}^{\mu\nu}$
= $\partial^{\mu}\vec{\rho}^{\,\nu}-\partial^{\nu}\vec{\rho}^{\,\mu}$. The
$\vec{\tau}$ are the Pauli matrices and behave as the isospin operator, which
carry the isospin component of the nucleons. In the further calculations of
the present work, we will consider relativistic mean-field approximation and
isotropy in isospin space. Within the assumed approximation, the meson fields
are replaced by their expectation values and for rotationally invariant
systems only the time component $(\mu=0)$ of the isovector field survives
Aguirre ; Liu_Greco . The third component of the isospin operator ($\tau_{3}$)
when operates on neutron and proton gives, $\tau_{3}\ket{p}=(+1)\ket{p}$ and
$\tau_{3}\ket{n}=(-1)\ket{n}$. Applying the relativistic mean-field
approximation, the Lagrangian density yields the following Dirac equation for
the nucleon field

$\displaystyle\Bigg{\\{}i\gamma_{\mu}\partial^{\mu}-g_{\omega}\gamma_{0}\omega-\frac{1}{2}g_{\rho}\gamma_{0}\tau_{3\alpha}\rho-M^{\ast}_{\alpha}\Bigg{\\}}\psi_{\alpha}=0,$
(2)

where $M^{\ast}_{\alpha}$ is the effective mass of the nucleon. Taking into
account the third component of the isospin for neutron and proton as defined
above, the effective masses of the neutron and proton are given by

$\displaystyle M_{p}^{\ast}$ $\displaystyle=$ $\displaystyle M-g_{s}\sigma-
g_{\delta}\delta,$ (3) $\displaystyle M_{n}^{\ast}$ $\displaystyle=$
$\displaystyle M-g_{s}\sigma+g_{\delta}\delta,$ (4)

The Lagrangian density can be used to derive Euler-Lagrange equations of
motion for the meson fields with the help of relativistic mean-field
approximation as Bharat

$\displaystyle m_{\sigma}^{2}\sigma$ $\displaystyle=$ $\displaystyle
g_{\sigma}\sum_{\alpha=p,n}\langle\bar{\psi}_{\alpha}\gamma_{0}\psi_{\alpha}\rangle-\frac{m_{\sigma}^{2}g_{\sigma}}{M}\sigma^{2}\Bigg{(}\frac{\kappa_{3}}{2}+\frac{\kappa_{4}}{3!}\frac{g_{\sigma}\sigma}{M}\Bigg{)}$
(5)
$\displaystyle+\frac{g_{\sigma}}{2M}\Bigg{(}\eta_{1}+\eta_{2}\frac{g_{\sigma}\sigma}{M}\Bigg{)}{m_{\omega}^{2}\omega^{2}}+\frac{\eta_{\rho}}{2M}{g_{\sigma}}{m_{\rho}^{2}}{\rho^{2}},$
$\displaystyle m_{\omega}^{2}\omega$ $\displaystyle=$ $\displaystyle
g_{\omega}\sum_{\alpha=p,n}\langle\bar{\psi}_{\alpha}\psi_{\alpha}\rangle-\Bigg{(}\eta_{1}+\frac{\eta_{2}}{2}\frac{g_{\sigma}\sigma}{M}\Bigg{)}\frac{g_{\sigma}\sigma}{M}m_{\omega}^{2}\omega$
(6)
$\displaystyle-\frac{1}{3!}\zeta_{0}g_{\omega}^{2}\omega^{3}-2\;\Lambda_{\omega}g_{\omega}^{2}g_{\rho}^{2}\rho^{2}\omega,$
$\displaystyle m_{\rho}^{2}\rho$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{\rho}\sum_{\alpha=p,n}\langle\bar{\psi}_{\alpha}\tau_{3}\psi_{\alpha}\rangle-\eta_{\rho}\frac{g_{\sigma}\sigma}{M}m_{\rho}^{2}\rho$
(7)
$\displaystyle-2\;\Lambda_{\omega}g_{\omega}^{2}g_{\rho}^{2}\omega^{2}\rho,$
$\displaystyle m_{\delta}^{2}\delta$ $\displaystyle=$ $\displaystyle
g_{\delta}\sum_{\alpha=p,n}\langle\bar{\psi}_{\alpha}\tau_{3}\gamma_{0}\psi_{\alpha}\rangle.$
(8)

Assuming NM as a uniform system, where the fields of $\sigma$, $\omega$ and
$\rho$ mesons are independent of position and taking the thermodynamic
argument of free gas into consideration, the baryon, scalar and isovector
densities for the appropriate Fermi momentum at finite temperature can be
evaluated as Wang ; Yang ; Bharat

$\displaystyle n$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\langle\bar{\psi}_{\alpha}\psi_{\alpha}\rangle=n_{p}+n_{n}$
(9)
$\displaystyle=\sum_{\alpha=p,n}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{\alpha}}d^{3}k\,[f_{\alpha}(\mu_{\alpha}^{\ast},T)-\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T)],$
$\displaystyle n_{s}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\langle\bar{\psi}_{\alpha}\gamma_{0}\psi_{\alpha}\rangle=n_{sp}+n_{sn}$
(10)
$\displaystyle=\sum_{\alpha=p,n}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{\alpha}}d^{3}k\,\frac{M_{\alpha}^{\ast}}{(k^{2}_{\alpha}+M_{\alpha}^{\ast
2})^{\frac{1}{2}}}\Big{[}f_{\alpha}(\mu_{\alpha}^{\ast},T)$
$\displaystyle+\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T)\Big{]},$ $\displaystyle
n_{3}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\langle\bar{\psi}_{\alpha}\tau_{3}\psi_{\alpha}\rangle=n_{p}-n_{n},$
(11) $\displaystyle n_{s3}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\langle\bar{\psi}_{\alpha}\tau_{3}\gamma_{0}\psi_{\alpha}\rangle=n_{sp}-n_{sn},$
(12)

where $n_{p}$ and $n_{n}$ are the proton and neutron densities respectively,
$k_{\alpha}$ is the nucleon Fermi momentum, T is the temperature,
$f_{\alpha}(\mu_{\alpha}^{\ast},T)$ and
$\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T)$ are thermal Fermi distribution
function of the nucleon and the anti-nucleon, and $\mu_{\alpha}^{\ast}$ is the
effective chemical potential of the nucleon. The familiar Fermi distribution
function for the particle and the anti-particle can be written as Wang

$\displaystyle f_{\alpha}(\mu_{\alpha}^{\ast},T)$ $\displaystyle=$
$\displaystyle\frac{1}{e^{({\cal
E}_{\alpha}^{\ast}-\mu_{\alpha}^{\ast})/k_{B}T}+1},$ (13)
$\displaystyle\bar{f_{\alpha}}(\mu_{\alpha}^{\ast},T)$ $\displaystyle=$
$\displaystyle\frac{1}{e^{({\cal
E}_{\alpha}^{\ast}+\mu_{\alpha}^{\ast})/k_{B}T}+1},$ (14)

where $k_{B}$ is the Boltzmann constant and ${\cal E_{\alpha}^{\ast}}$ is the
effective energy of the nucleon which can be written as

$\displaystyle{\cal E_{\alpha}^{\ast}}=\sqrt{k_{\alpha}^{2}+M_{\alpha}^{\ast
2}}.$ (15)

The effective chemical potential of the proton and the neutron can be derived
as Wang

$\displaystyle\mu_{p}^{\ast}$ $\displaystyle=$
$\displaystyle\mu_{p}-g_{\omega}\omega-\frac{1}{2}g_{\rho}\rho,$ (16)
$\displaystyle\mu_{n}^{\ast}$ $\displaystyle=$
$\displaystyle\mu_{n}-g_{\omega}\omega+\frac{1}{2}g_{\rho}\rho,$ (17)

where $\mu_{p}$ and $\mu_{n}$ are the usual the chemical potential of proton
and neutron relative to free nucleon mass. Another important thing we would
like to mention for the formalism is the contribution of loop correction for
the renormalized theory. Recently, M. Prakash et. al. Xilin  reported that the
two-loop approximation improves the energy density functional by adding
density-dependent contributions to the Hartree terms of mean-field theory from
the exchange of isoscalar, isovector and pseudoscalar mesons. It is found that
the thermal properties of proto-neutron stars differ significantly if we
include the two-loop correction in the mean-field theory. However, it has also
been reported in the references ( Frun1989 , Serot1995  and Hu_2007 ) that the
loop contribution in the mean-field theory can be mocked up by refitting the
parameters using experimentally available data. We calculate the thermal
properties with the mean-field approach without loop correction because in our
work the coupling constants of the Lagrangian are obtained by fitting the
experimental and observational data.

##  3 Symmetric Nuclear Matter

###  3.1 Theoretical Formalism

In the following section, the formalism used for the calculations of symmetric
and asymmetric NM properties has been discussed. To describe the asymmetric
NM, we introduce the asymmetric parameter $t$, which is defined as,

$\displaystyle t=\frac{n_{n}-n_{p}}{n_{n}+n_{p}}.$ (18)

We can assign the desired asymmetry in the NM by varying the value of $t$. For
symmetric NM (SNM), $t=0$ and $t=1$ for pure neutron matter. In the present
work, we extend our calculations for SNM ($t=0$). The energy density and the
pressure for the NM can be calculated from the Lagrangian using the expression
for energy-momentum tensor Walecka , which is

$\displaystyle T^{\mu\nu}=\sum_{j}\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi_{j})}\partial^{\nu}\phi_{j}-\eta^{\mu\nu}{\cal
L},$ (19)

where $\phi_{j}$ includes all the fields present in the Lagrangian. Using this
expression of the energy density and pressure for a warm nuclear system can be
naively derived as S.K.Singh

$\displaystyle E$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{\alpha}}d^{3}k\,{\cal
E}_{\alpha}^{\ast}(k)\Big{[}f_{\alpha}(\mu_{\alpha}^{\ast},T)+\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T)\Big{]}$
(20)
$\displaystyle+n\,g_{\omega}\,\omega+m_{\sigma}^{2}{\sigma}^{2}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma}\sigma}{M}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma}^{2}\sigma^{2}}{M^{2}}\Bigg{)}-\frac{1}{4!}\zeta_{0}\,{g_{\omega}^{2}}\,\omega^{4}$
$\displaystyle-\frac{1}{2}m_{\omega}^{2}\,\omega^{2}\Bigg{(}1+\eta_{1}\frac{g_{\sigma}\sigma}{M}+\frac{\eta_{2}}{2}\frac{g_{\sigma}^{2}\sigma^{2}}{M^{2}}\Bigg{)}+\frac{1}{2}n_{3}\,g_{\rho}\,\rho$
$\displaystyle-\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}g_{\sigma}\sigma}{M}\Bigg{)}m_{\rho}^{2}\,\rho^{2}-\Lambda_{\omega}\,g_{\rho}^{2}\,g_{\omega}^{2}\,\rho^{2}\,\omega^{2}+\frac{1}{2}m_{\delta}^{2}\,\delta^{2},$
$\displaystyle and$ $\displaystyle P$ $\displaystyle=$
$\displaystyle\sum_{\alpha=p,n}\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{\alpha}}d^{3}k\,\frac{k^{2}}{{\cal
E}_{\alpha}^{\ast}(k)}\Big{[}f_{\alpha}(\mu_{\alpha}^{\ast},T)+\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T)\Big{]}$
(21) $\displaystyle-
m_{\sigma}^{2}{\sigma}^{2}\Bigg{(}\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{g_{\sigma}\sigma}{M}+\frac{\kappa_{4}}{4!}\frac{g_{\sigma}^{2}\sigma^{2}}{M^{2}}\Bigg{)}+\frac{1}{4!}\zeta_{0}\,{g_{\omega}^{2}}\,\omega^{4}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\,\omega^{2}\Bigg{(}1+\eta_{1}\frac{g_{\sigma}\sigma}{M}+\frac{\eta_{2}}{2}\frac{g_{\sigma}^{2}\sigma^{2}}{M^{2}}\Bigg{)}+\Lambda_{\omega}\,g_{\rho}^{2}\,g_{\omega}^{2}\,\rho^{2}\,\omega^{2}$
$\displaystyle+\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}g_{\sigma}\sigma}{M}\Bigg{)}m_{\rho}^{2}\,\rho^{2}-\frac{1}{2}m_{\delta}^{2}\,\delta^{2}.$

Since we deal with the temperature dependent NM, we also did some study
regarding phase coexistence, which unfolds the liquid-gas phase transition in
thermal NM. The NM system can remain only in one phase (gaseous) once it
reaches the critical point Wang . For symmetric NM, the inflection point of
the pressure curve with respect to the total nucleon density determines the
critical point Yang , that is

$\displaystyle\frac{\partial P}{\partial
n}\Bigg{|}_{T=T_{C}}=\frac{\partial^{2}P}{\partial
n^{2}}\Bigg{|}_{T=T_{C}}=0,$ (22)

$T_{C}$ being the critical temperature. One of the basic and fundamental
quantity of the NM is incompressibility, also known as the isoscalar
incompressibility or compression modulus $(K)$ Schneider . $K$ directly
influences the curvature of the equation of state and gives adequate
information about the nature of the equation of state. Higher the value of K,
more stiff the equation of state will be NgoHaiTan . The incompressibility is
related to the equation of state through Walecka

$\displaystyle K(n,T)$ $\displaystyle=$ $\displaystyle
9\,n^{2}\,\frac{\partial^{2}(E/n)}{\partial n^{2}}.$ (23)

Another important quantity which controls the equation of state of NM is
symmetry energy. Symmetry energy has a significant contribution to the
pressure of the astrophysical objects which is responsible for gravitational
attraction Goudarzi . In case of the hot NM, we can define two forms of
symmetry energy, one we call as the nuclear symmetry energy $(E_{sym})$ and
the other as free symmetry energy $(F_{sym})$. To calculate the free symmetry
energy of the hot NM, we need entropy density $S$ and free energy density $F$
of that system. So the free energy density is given by

$\displaystyle F$ $\displaystyle=$ $\displaystyle E-TS,$ (24)

and the entropy density for a hot NM is calculated by Walecka

$\displaystyle S$ $\displaystyle=$
$\displaystyle-\sum_{\alpha=p,n}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{\alpha}}d^{3}k\,\Bigg{[}f_{\alpha}(\mu_{\alpha}^{\ast},T)\ln{f_{\alpha}(\mu_{\alpha}^{\ast},T)}$
(25)
$\displaystyle+(1-f_{\alpha}(\mu_{\alpha}^{\ast},T))\ln{(1-f_{\alpha}(\mu_{\alpha}^{\ast},T))}+\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T)$
$\displaystyle\ln{\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T)}+(1-\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T))\ln{(1-\bar{f}_{\alpha}(\mu_{\alpha}^{\ast},T))}\Bigg{]}.$

Various theoretical studies have shown that we can calculate the temperature
and density dependence of nuclear symmetry energy in hot symmetric NM using
empirical parabolic approximation, which can be estimated as the difference of
the energy per nucleon of pure neutron matter and symmetric NM NgoHaiTan . In
a similar way, we can find the free symmetric energy for a hot NM, which is of
utmost importance for astrophysical phenomena. The free symmetric energy of a
symmetric NM can be defined as NgoHaiTan ; Bao-AnLi

$\displaystyle F_{sym}(n,T)=\frac{F(n,T,t=1)}{n}-\frac{F(n,T,t=0)}{n},$ (26)

which can be interpreted as the remnant of free energy per nucleon for pure
neutron matter and symmetric NM. The free symmetry energy can be expanded in a
Taylor series expansion around $\chi$. It is a dimensionless variable which
describes the density deviation from saturation density and for mathematical
convention and simplification of the expressions of higher order derivatives
defined as, $\chi=({n-n_{0}})/{3n_{0}}$, $n_{0}$ being the saturation density.
So the expansion can be written as,

$\displaystyle F_{sym}(n,T)$ $\displaystyle=$ $\displaystyle
F_{sym}(n_{0})+L_{sym}\,\chi+\frac{K_{sym}}{2!}\,\chi^{2}$ (27)
$\displaystyle+\frac{Q_{sym}}{3!}\,\chi^{3}+O(\chi^{4}),$

where $L_{sym}$, $K_{sym}$ and $Q_{sym}$ are the slope parameter, curvature
parameter and skewness parameter respectively. Recently it is convinced that
these parameters are of utmost importance in nuclear and astrophysics by
showing the correlation of these parameters with different nuclear and
astrophysical properties Schneider . The expressions for these parameters can
be extracted as Chen

$\displaystyle L_{sym}(n,T)$ $\displaystyle=$ $\displaystyle
3n\,\frac{\partial F_{sym}(n,T)}{\partial n},$ (28) $\displaystyle
K_{sym}(n,T)$ $\displaystyle=$ $\displaystyle
9n^{2}\,\frac{\partial^{2}F_{sym}(n,T)}{\partial n^{2}},$ (29) $\displaystyle
Q_{sym}(n,T)$ $\displaystyle=$ $\displaystyle
27n^{3}\,\frac{\partial^{3}F_{sym}(n,T)}{\partial n^{3}}.$ (30)

Although we extract the results from this expansion at high densities, we
should not exclude the fact that at very high nuclear densities the error bar
in higher-order coefficients extracted through series expansion is quite high
Schneider . Thermal index commits an important aspect on core-collapse
supernovae simulations Yasin . The thermal index can be calculated with the
help of simple formula discussed in Carbone

$\displaystyle\Gamma_{th}$ $\displaystyle=$ $\displaystyle
1+\frac{E_{th}}{P_{th}}$ (31)

where $E_{th}$ and $P_{th}$ are the thermal energy density and pressure
respectively. $E_{th}$ and $P_{th}$ are given by $E_{th}=E(T)-E(0)$ and
$P_{th}=P(T)-P(0)$, where $E(T)$ and $P(T)$ are the energy density and
pressure at temperature T.

###  3.2 Results

Throughout our calculations in this paper, we use NL3, G3 and IU-FSU parameter
sets, where NL3 endue us the stiff EoS and the other two (G3 and IU-FSU)
facilitate us to examine the softer region. The obtained results for the
variation in the properties of the NM with temperature and density are
discussed in the present section. The coupling constants and the empirical
values of nuclear properties at saturation of cold EoS for the assumed
parameter sets (NL3, G3 and IU-FSU) are given in Table 1 Lalazissis ;
KUMAR2017197 .

Table 1: The coupling constants and the NM properties at saturation for cold EoS of NL3 Lalazissis , G3 KUMAR2017197  and IU-FSU Carbone  parameter sets. The nucleon mass ($M$) is 939.0 MeV. All of the coupling parameters are dimensionless and the NM parameters are in MeV, except $k_{3}$ and $n_{0}$ which are in fm-1 and fm-3 respectively. The NM parameters are given at saturation point and T = 0 K for NL3, G3 and IU-FSU parameter sets in the lower panel. The references are $[a]$,$[b]$, $[c]$ $\&$ $[d]$ Zyla_2020 , $[e]$&$[f]$ Bethe_1971 , $[g]$ Garg_2018 , $[h]$&$[i]$ Danielewicz_2014 , and $[j]$ zimmerman . Parameter | NL3 | G3 | IU-FSU | Empirical/Expt. Value
---|---|---|---|---
$m_{\sigma}/M$ | 0.541 | 0.559 | 0.523 | 0.426 – 0.745 $[a]$
$m_{\omega}/M$ | 0.833 | 0.832 | 0.833 | 0.833 – 0.834 $[b]$
$m_{\rho}/M$ | 0.812 | 0.820 | 0.812 | 0.825 – 0.826 $[c]$
$m_{\delta}/M$ | 0.0 | 1.043 | 0.0 | 1.022 – 1.064 $[d]$
$g_{\sigma}/4\pi$ | 0.813 | 0.782 | 0.793 |
$g_{\omega}/4\pi$ | 1.024 | 0.923 | 1.037 |
$g_{\rho}/4\pi$ | 0.712 | 0.962 | 1.081 |
$g_{\delta}/4\pi$ | 0.0 | 0.160 | 0.0 |
$k_{3}$ | 1.465 | 2.606 | 1.1593 |
$k_{4}$ | -5.688 | 1.694 | 0.0966 |
$\zeta_{0}$ | 0.0 | 1.010 | 0.03 |
$\eta_{1}$ | 0.0 | 0.424 | 0.0 |
$\eta_{2}$ | 0.0 | 0.114 | 0.0 |
$\eta_{\rho}$ | 0.0 | 0.645 | 0.0 |
$\Lambda_{\omega}$ | 0.0 | 0.038 | 0.046 |
$n_{0}$ | 0.148 | 0.148 | 0.154 | 0.148 – 0.185 $[e]$
$B.E.$ | -16.29 | -16.02 | -16.39 | -15.00 – 17.00 $[f]$
$K_{0}$ | 271.38 | 243.96 | 231.31 | 220 – 260 $[g]$
$F_{sym,0}$ | 37.43 | 31.84 | 32.71 | 30.20 – 33.70 $[h]$
$L_{sym,0}$ | 120.65 | 49.31 | 49.26 | 35.00 – 70.00 $[i]$
$K_{sym,0}$ | 101.34 | -106.07 | 23.28 | -174 – -31 $[j]$
$Q_{sym,0}$ | 177.90 | 915.47 | 536.46 | ———–

We start our discussion with the energy and pressure of hot NM and procurement
of critical temperature for liquid-gas phase transition. In Fig.1, we depicted
the calculated results of binding energy per nucleon and pressure density from
$T=0$ to $20$ MeV as a function of density. We observed an increase in the
binding energy and saturation density of the SNM with temperature for all the
three supposed parameter sets. The marked point on each curve of the upper
panel of the Fig.1 represents its minima. It has been already reported that NM
saturates around $0.148$ fm-3 with binding energy per nucleon around $-16$ MeV
for both the parameter sets (NL3 and G3) at zero temperature Bharat  while the
saturation density for IU-FSU parameter set is 0.154 fm-3. Here we found that
as we increase the temperature from $0$ to $20$ MeV, $n_{0}$ increases
linearly from $0.148$ to $0.196$ fm-3 and similarly the binding energy also
goes on increasing which is clear from the Fig.1. The increase in the binding
energy per nucleon with temperature indicates that the system becomes more
loosely bound at the higher temperature.

Figure 1: (colour online) Binding energy and pressure as a function of nucleon
density for symmetric NM at different temperatures. The left, middle and right
panels show the results for NL3, G3 and IU-FSU parameter sets respectively.

The lower panel of Fig.1 shows the variation of pressure with temperature as a
function of density. This adaptation of pressure is utterly important in
determining the critical parameters of the liquid-gas phase transition,
especially the critical temperature, $T_{C}$. Various theoretical and
experimental studies predict the value of $T_{C}$ for SNM in the range of
$10-20~{}$MeV SubrataPal ; KUPPER ; BoNan ; Moretto ; Tatsuyuki . We also
found the value of $T_{C}$ using equation 22 as $14.60$, $15.37$ and $14.50$
MeV for NL3, G3 and IU-FSU parameter sets respectively. Other important
characteristics quantities of the liquid-gas phase transition like
$P_{C},\rho_{C}$ can also be determined using $T_{C}$ with the help of
correlations derived in reference BoNan .

Figure 2: (colour online) Incompressibility $(K)$ as a function of nucleon
density for SNM ($t=0$). The left panel shows the results for NL3, middle
panel for G3 and right panel for IU-FSU parameter set.

The variation of the incompressibility modulus has also been depicted in Fig.
2 and 3. As the value of n increases, the chart of $K$ exhibits anomalous
behaviour. We observed that the magnitude of $K$ shows a maximum around $4-5$
times of the saturation density for all the three parameter sets. However, the
magnitude of $K$ is quite high for NL3 parameter set in comparison to G3 and
IU-FSU, which indicates a strong dependence of $K$ on the nature of the EoS.
We also observed an unfamiliar proneness in the K values at saturation density
(denoted as $K_{0}$) for different temperature which are shown in Fig. 3.For
NL3 parameter set $K_{0}$ shows an increment with increase in temperature
while for G3 and IU-FSU parameter sets the case is reversed. This
contemplation of $K_{0}$ confirms the sensitiveness of incompressibility on
the choice of parametrization and the cross-coupling of field variables
responsible for the softening of EoS. This fact has also been supported by the
study in reference Mekjian .

Figure 3: (colour online) Variation of incompressibility at saturation density
with temperature for SNM ($t=0$).

We also analysed that the G3 and IUFSU parameter sets indulge all the
constraints of the elliptic flow heavy-ion collision experiment Chenchen  on
the value of $K_{0}$, which is $K_{0}$ = $220\pm 40$ MeV. NL3, on the other
hand, being the stiff parameter set depicts the higher value of $K_{0}$ and
does not support the experimental results. The effect of the temperature on
free symmetry energy and its derivatives has also been studied from lower to
higher density of NM. Fig. 4 reflects the effect of temperature on $F_{sym}$
and its first derivative $L_{sym}$ (i.e slope parameter) of SNM. The rise in
temperature does not hint any significant change in the symmetry energy of the
system at high density. However, the magnitude of the symmetry energy at
saturation density ($F_{sym,0}$) increases as we increase the temperature.
Another important aspect that we observed here is the difference in the
magnitude of the $F_{sym}$ for NL3 and G3 parameter sets. The magnitude range
of $F_{sym}$ for NL3 is quite high in comparison to G3 and IU-FSU, which
indicates that the stiff EoS allocate the higher value of symmetry energy.

Figure 4: (colour online) Symmetry energy ($F_{sym}$) and $L_{sym}$ as a
function of nucleon density for SNM ($t=0$). The left panel shows the results
for NL3, middle panel for G3 and right panel for IU-FSU parameter set.

Again the determination of the higher value of $F_{sym,0}$ along with $K_{0}$
by NL3 forces suggested that we need a better parameterized EoS, so that we
can match the prediction of nuclear parameters made by various theoretical and
experimental studies. On the other hand, the spectrum of $F_{sym}$ at finite
temperature determined by our recently developed G3 parameter and the popular
IU-FSU, fall in the expected range which is verified by other studies also
Yingxun ; Agrawal_2014 ; Baldo ; Colonna ; Carbone . $L_{sym}$ changes
slightly as we change the temperature for NL3 parameter set, while for G3 and
IU-FSU it remains unaffected. The change in slope parameter at saturation
density ($L_{sym,0}$) with temperature is shown in Fig. 5. The effect of
temperature on the second and third derivatives of $F_{sym}$, which are known
as curvature and isovector skewness parameter respectively (denoted by
$K_{sym}$ and $Q_{sym}$) is shown in Fig. 6 and 7 for both the parameter sets.

Figure 5: (colour online) Variation of symmetry energy and slope parameter at
saturation density ($F_{sym,0}$ and $L_{sym,0}$) with temperature for SNM
($t=0$).

The variation of $K_{sym}$ and $Q_{sym}$ with density somehow reflects the
sight of the sinusoidal wave, with extremum of the curve near twice the
saturation density as can be seen in Fig. 6. $K_{sym}$, being the higher order
parameter, can be constrained with the help of neutron star observations
zimmerman . Recently, a study done by Josef Zimmerman et al. merged the data
reported by two most vital experiments (PSR J0030+0451 by NICER Collaboration
Riley_2019  and GW170817 by LIGO/Virgo PhysRevLett.119.161101 ; Abbott_2018 )
and derive a joint 1-$\sigma$ constraint on curvature parameter at saturation
density ($K_{sym,0}$). It is believed to be the most reliable bound on
$K_{sym,0}$ till date and reported within the 90 % confident bounds as
$K_{sym,0}=-102^{+71}_{-72}$ MeV zimmerman .

Figure 6: (colour online) $K_{sym}$ and $Q_{sym}$ as a function of nucleon
density for SNM ($t=0$). The left panel shows the results for NL3 and right
panel for G3 parameter set. Figure 7: (colour online) Variation of Curvature
parameter and Skewness parameter at saturation density ($K_{sym,0}$ and
$Q_{sym,0}$ respectively) with temperature for SNM.

In Fig. 7 we can see that the value of $K_{sym,0}$ estimated by G3 parameter
set for entire temperature spectrum falls into the range of 1-$\sigma$
constraint discussed above. NL3 and IU-FSU parameter sets, on the other hand,
not only inconsistent with the estimated constraint but surprisingly falls
entirely in the opposite magnitude range. This observation clearly rejects the
NL3 and IU-FSU phenomenological model in determining the NM parameters for
astrophysical observations and hints for G3 EoS, best suited for the study of
astrophysical phenomena. Studies show that there is a strong correlation
between $K_{sym,0}$, tidal deformability and $R_{1.4}$ (radius of NS with mass
$1.4$ $M_{\odot}$) Fortin ; Steiner ; Yagi . By obligating $K_{sym,0}$ in the
befitting range, we can measure tidal deformability theoretically, which is
quite a complicated quantity to measure independently, precisely up to a
certain level of accuracy. $Q_{sym}$ being the most ambiguous quantity also
shows entirely different behaviour for the NL3 and G3 parameter set. For G3 it
falls in the positive magnitude range while for NL3 and IU-FSU, it beholds the
negative magnitude as the temperature increases. Although there is no reported
constraint on $Q_{sym,0}$ till now, but we believe that since G3 parameter set
satisfy all the desired results for other NM parameters, so it is obvious to
assume it of the right kind. However, some predictions made on the basis of
skyrme interactions supports the negative magnitude of skewness parameter
Tews_2017 , so it is difficult to state anything about $Q_{sym}$ with
certainty.

Figure 8: (colour online) The thermal index at different temperatures as a
function of nucleon density for NL3 (solid line), G3 (dashed line) and IU-FSU
(dotted) parameter sets. $T_{C}$ is the critical temperature for the
corresponding parameter set

Fig. 8 enounce the variation of the thermal index for the avowed parameter
sets. The nature of the EoS of the astrophysical phenomenon is also dictated
in some manner by the density dependence of the thermal index. As we can see
in Fig. 8 that the thermal index in the low density region approaches the
value of 5/3 for all temperatures, which indicates that the system described
by our model for all the parameter sets almost behaves like a non-relativistic
ideal gas Carbone . We also observe that as the temperature goes on increasing
the maximum value of the thermal index decrease, which is consistent with the
fact that at high temperature, the system approaches relativistic behaviour.
The value of $\Gamma_{th}$ around 2 has been appraised as an important aspect
in the neutron-star merger simulations, which has also been reflected in our
calculations Bauswein . The thermal index curve for NL3 is comparatively
steeper than G3 and IU-FSU parameter sets and approaches towards the
relativistic gas behaviour more rapidly. The variations in the thermal index
with density for different EoS solidify the frequency of gravitational wave
oscillations and also decides the delay time of black hole collapse Bauswein ;
Carbone .

##  4 Cooling through neutrino emission

In this section, we provide an approach to study the thermal evolution of the
newly born neutron star through EOS. It is widely believed that the dense
neutron star formed just after the supernova explosion contains tantamount
nucleons, i.e. almost the equal number of protons and neutrons Lattimer .
After that, the process of cooling and neutronization Lattimer ; stgaard
happens slowly, and finally, it achieves the thermal stabilization and the
beta equilibrium, which we call cold neutron star. The direct Urca processes
$n\longrightarrow p+e^{-}+\bar{\nu}_{e}$ and $p+e^{-}\longrightarrow
n+\nu_{e}$, governs the cooling of newly born neutron stars Boguta  through
neutrino emission and requires a super threshold proton fraction inside the
super-dense star core to operate. To realize the effect of neutrino emissivity
on cooling mechanism and EOS, we consider a degenerate dense matter containing
nucleons (neutrons and protons) and the electrons at finite temperature, for
which the Lagrangian is redefined as

$\displaystyle{\cal L}_{total}$ $\displaystyle=$ $\displaystyle{\cal
L}+\bar{\phi}\,(i\gamma_{\mu}\partial^{\mu}-m_{e})\phi,$ (32)

where ${\cal L}$ is the Lagrangian defined in Eq. (1), $m_{e}$ being the mass
of the electron and the last term represents the electrons contribution in the
matter, which are considered to be non-interacting particles. Following the
same procedure as defined in section 2, we will get the total energy density
and pressure for this system

$\displaystyle E_{total}$ $\displaystyle=$ $\displaystyle
E+\frac{2}{(2\pi)^{3}}\int_{0}^{k_{e}}d^{3}k\,{\cal
E}_{e}(k)\,[f_{e}(T)+\bar{f}_{e}(T)],$ (33) $\displaystyle P_{total}$
$\displaystyle=$ $\displaystyle
P+\frac{2}{3(2\pi)^{3}}\int_{0}^{k_{e}}d^{3}k\,\frac{k^{2}}{{\cal
E}_{e}}\,\Big{[}f_{e}(T)+\bar{f}_{e}(T)\Big{]},$ (34)

where E and P is the energy density and pressure defined in equation 20,
$k_{e}$ is the Fermi momentum of the electron, $f_{e}$ is the Fermi
distribution of electron for finite temperature and ${\cal E}_{e}$ is the
energy of electron given by

$\displaystyle{\cal E}_{e}$ $\displaystyle=$
$\displaystyle\sqrt{k^{2}_{e}+m^{2}_{e}}.$ (35)

We maintained the same number density for proton and electron, i.e.
$n_{p}=n_{e}$, to achieve charge neutrality in the described system.

The expression for the neutrino emissivity ($Q$) in high dense (neutron star)
system was first estimated by Lattimer et al. in non-relativistic manner
LattimerUrca . Since the movement of nucleons in neutron star cores is
relativistic, so the relativistic expression for neutrino emissivity was later
calculated by L. B. Leinson and A. Perez. The non-relativistic emissivity is
quite small than what is predicted by the relativistic formalism. The detailed
explanations and the derivation for the formula of neutrino emissivity ($Q$)
in the relativistic framework and mean-field approximation, which is used here
can be found in the reference Leinson_2001 ; Leinson_2002 .

The formula for neutrino emissivity is given by Leinson_2002

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{457\pi}{10080}G_{F}^{2}\,C^{2}\,T^{6}\,\Theta(k_{e}+k_{p}-k_{n})\Bigg{\\{}\left(C_{A}^{2}-C_{V}^{2}\right)$
(36) $\displaystyle M_{p}^{\ast}\,M_{n}^{\ast}\,{\cal
E}_{e}+\frac{1}{2}\left(C_{V}^{2}+C_{A}^{2}\right)\bigg{[}4\,{\cal
E}_{n}\,{\cal E}_{p}\,{\cal E}_{e}-\left({\cal E}_{n}-{\cal E}_{p}\right)$
$\displaystyle\left(\left({\cal E}_{n}+{\cal
E}_{p}\right)^{2}-k_{e}^{2}\right)\bigg{]}+C_{V}\,C_{M}\,\frac{\sqrt{M_{p}^{\ast}\,M_{n}^{\ast}}}{M}$
$\displaystyle\left[2\left({\cal E}_{n}-{\cal
E}_{p}\right)k_{e}^{2}-\left(3\left({\cal E}_{n}-{\cal
E}_{p}\right)^{2}-k_{e}^{2}\right){\cal E}_{e}\right]$
$\displaystyle+C_{A}\left(C_{V}+2\frac{\sqrt{M_{p}^{\ast}M_{n}^{\ast}}}{M}C_{M}\right)\left({\cal
E}_{n}+{\cal E}_{p}\right)$ $\displaystyle\left(k_{e}^{2}-\left({\cal
E}_{n}+{\cal
E}_{p}\right)^{2}\right)+C_{M}^{2}\frac{1}{4M^{2}}\bigg{[}8M^{\ast
2}\left({\cal E}_{n}-{\cal E}_{p}\right)$
$\displaystyle\left(k_{e}^{2}-\left({\cal E}_{n}-{\cal E}_{p}\right){\cal
E}_{e}\right)+\left(k_{e}^{2}-\left({\cal E}_{n}-{\cal
E}_{p}\right)^{2}\right)$ $\displaystyle\left(2{\cal E}_{n}^{2}+2{\cal
E}_{p}^{2}-k_{e}^{2}\right){\cal E}_{e}-\left(k_{e}^{2}-\left({\cal
E}_{n}-{\cal E}_{p}\right)^{2}\right)$ $\displaystyle\left({\cal E}_{n}+{\cal
E}_{p}\right)\left(2{\cal E}_{n}-2{\cal E}_{p}-{\cal
E}_{e}\right)\bigg{]}\Bigg{\\}},$

where $G_{F}=1.166\times 10^{-11}$ MeV-2 is the Fermi weak coupling constant,
$C$ = 0.973 is the Cabibbo factor, $C_{V}$ = 1 and $C_{A}$ = 1.26 are the
vector and axial-vector constants respectively and the constant $C_{M}$ = 3.7
represents the weak magnetism effects. The condition inevitable for Urca
processes to go is represented by $\Theta(k_{e}+k_{p}-k_{n})$ = 1, if
$k_{e}+k_{p}-k_{n}\geq 0$ and zero otherwise, where $k_{e}$, $k_{p}$ and
$k_{n}$ are the Fermi momenta of electron, proton and neutron respectively.
Since our defined system of nucleons and leptons (n-p-e) clearly satisfy the
above necessary condition required for the initialisation of direct Urca
process, so, we calculated the neutrino emissivity for both NL3, G3 and IU-FSU
parameter sets which is displayed in Fig. 9. The dashed lines in Fig. 9
represent the results for G3, the solid lines stand for NL3 and the dotted
lines for IU-FSU parameter set. Some very interesting remarks can be concluded
based on these curves of $Q$ depicted at different temperatures. The most
vital observation of this work is that the neutrino emissivity is responsible
for the cooling of newly born neutron star only in its initial stage, i.e.
when the temperature is quite high, and the neutrons are enough thermally
excited to trigger the direct Urca process. As the star cools down, the
magnitude of Q decreases substantially and then cooling mainly takes place
through photon emission. This behaviour can be seen clearly in Fig. 9 for all
the assumed parameter sets. The magnitude of $Q$ for 40 MeV (represented by
the black line) temperature is negligible in comparison to 80 MeV curve (blue
line).

Figure 9: (colour online) Neutrino emissivity at different temperatures as a
function of nucleon density for NL3 (solid line), G3 (dashed line) and IU-FSU
(dotted line) parameter sets.

Also, initially, the diffusion rate of neutrinos is so high that the matter
cools down within a fraction of seconds and it is believed that it is so
effective that it lowers the temperature to about 1 billion Kelvin during this
momentary span Dmitry ; Cumming . Another important dimension that we observe
in this curve is the difference in the magnitude of the $Q$ values for NL3, G3
and IU-FSU parameter sets. This tells us about the dependence of cooling
property of newly born NS on the EoS. As we know that NL3 provides the
stiffest EoS, which means the high mass of the proto-neutron star and G3 being
the softer EoS in comparison to NL3 predicts lower mass. Keeping this fact in
mind, we concluded that the cooling of proto-neutron star through direct Urca
process is slow in heavier star and fast in the lighter one.

##  5 Proto-Neutron Star

To understand the complete penchant of temperature on a dense matter system,
we extend our calculations to obtain the mass-radius (M-R) profile of proto-
neutron star (PNS) in this section. The newly born proto-neutron star is hot,
lepton-rich and has a core in the evolutionary stage which ultimately grow
either in a cold neutron star or crash into a black hole. As we discussed in
the previous section that a PNS losses a significant amount of energy through
neutrino emission and neutrino emissivity plays an important role in the
evolution of the newly born star. However, some of the neutrinos with small
mean free path unable to escape the core of the neutron star and got trapped
inside the star. These trapped neutrinos affect the early evolutionary
illation and properties of the PNS camelio2018 ; HuanyuandHong ; Pons_1999 .
We modify our Lagrangian mentioned in Eq. (32) and added the terms responsible
for neutrino effects to derive the EoS for the proto-neutron star system
envisaged in the present section. The modified Lagrangian is

$\displaystyle{\cal L}_{PNS}$ $\displaystyle=$ $\displaystyle{\cal
L}_{total}+\bar{\phi}_{\nu_{e}}\,(i\gamma_{\mu}\partial^{\mu})\phi_{\nu_{e}},$
(37)

where $\phi_{\nu_{e}}$ is the wave-function of electron neutrino. The relevant
energy density and pressure for the above mentioned Lagrangian can be derived
as HuanyuandHong

$\displaystyle E_{PNS}$ $\displaystyle=$ $\displaystyle
E_{total}+\sum_{\nu_{e}}\Bigg{(}\frac{7\pi^{2}}{120}+\frac{\mu^{2}_{\nu_{e}}T^{2}}{12}-\frac{\mu^{4}_{\nu_{e}}}{24\pi^{2}}\Bigg{)},$
(38) $\displaystyle P_{PNS}$ $\displaystyle=$ $\displaystyle
P_{total}+\sum_{\nu_{e}}\frac{1}{360}\Bigg{(}7\pi^{2}T^{4}+30\mu^{2}_{\nu_{e}}T^{2}+\frac{15\mu^{4}_{\nu_{e}}}{\pi^{2}}\Bigg{)}$
(39)

where the summation is over the total number of trapped neutrinos; $E_{total}$
and $P_{total}$ are the energy density and pressure defined in eq. 33 and
$\mu_{\nu_{e}}$ is the chemical potential of electron neutrino. We also
maintained the necessary $\beta-$ equilibrium and charge neutrality
conditions, i.e.

$\displaystyle\mu_{n}$ $\displaystyle=$
$\displaystyle\mu_{p}+(\mu_{e}-\mu_{\nu_{e}}),$ (40) $\displaystyle n_{p}$
$\displaystyle=$ $\displaystyle n_{e},$

where $\mu_{n},\mu_{p},\mu_{e}$ and $\mu_{\nu_{e}}$ are the chemical
potentials of the neutron, proton, electron and neutrino respectively; $n_{p}$
and $n_{e}$ are the number densities of proton and electron. We can also
define the lepton fraction as

$\displaystyle Y_{L}$ $\displaystyle=$
$\displaystyle\frac{n_{e}+n_{\nu_{e}}}{n}$ (41)

where $n_{e}$, $n_{\nu_{e}}$ and n are the number density of electrons,
neutrinos and baryons respectively. The effect of trapped neutrinos on the
early evolutionary stages of PNSs can be represented by fixing $Y_{L}$ in a
certain defined range, i.e. $Y_{L}\sim 0.1-0.4$. We fix $Y_{L}=0.4$ in our
calculations, which is the best precedence to explore the properties of newly
born PNS HuanyuandHong ; Prakash_1997 ; Pons_1999 . Also, the entropy per
baryon ($S$) of the star matter can be derived as Prakash_1997

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{E_{PNS}+P_{PNS}-\sum_{\alpha=p,n}n_{\alpha}\mu_{\alpha}}{nT}$
(42)

For a fixed entropy, the effect of the neutrino trapping is to keep the
electrons concentration high so that matter is more proton-rich in comparison
to the case in which the neutrinos are not trapped Prakash . We study the M-R
profile for the PNS using a different kind of EoS, first by placing the
temperature constant throughout the star (fixed temperature EoS) MESQUITA ;
Dexheimer_2008 ; Manka_2001  and another by fixing the certain defined values
of entropy per baryon ($S=1\&2$) for the star matter (fixed entropy EoS)
Burgio ; BIN2016340 ; Hong_2016 ; Strobel_2001 . Although we have used the
constant temperature EoS to describe the mass-radius profile of proto-neutron
star, but this assumption is valid only for low temperature spectrum. If the
matter’s temperature is significantly larger than the corresponding critical
Fermi temperature, the star becomes unstable. Also, the temperature of the
star varies significantly from core to the surface of the star. So, to debar
all these restrictions and for a better undisputed M-R outline of PNS, we
extended our calculations for fixed entropy EoS. The method of fixed entropy
EoS is more appropriate to study the properties of PNS at finite temperature.
By imposing the above stated conditions on the described Lagrangian, we can
easily calculate the mass and radius of the static isotropic proto-neutron
star for both kind of EoSs using the Tolman-Oppenheimer-Volkov (TOV) equations
tolman ; Oppenheimer . The TOV equations are

$\displaystyle\frac{dP_{PNS}(r)}{dr}$ $\displaystyle=$
$\displaystyle-\frac{[E_{PNS}(r)+P_{PNS}(r)][M(r)+{4\pi
r^{3}P_{PNS}(r)}]}{r^{2}\Big{(}1-\frac{2M(r)}{r}\Big{)}},$ (43)
$\displaystyle\frac{dM(r)}{dr}$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}{E_{PNS}(r)}.$ (44)

Where $E_{PNS}$ and $P_{PNS}$ are the energy density and pressure defined in
eqs. (38) and (39). These equations are integrated from $r=0$ to the stellar
surface $r=R$, where $P_{PNS}(R)=0$, for a particular choice of central
density $\rho_{c}=\rho(0)$ to determine the NS mass $M=M(R)$. The value of
$\rho_{c}$ that produces the maximum mass $M_{max}$ for a given EoS is
$\rho_{max}$. The crust part of the neutron star also plays an important role
in determining the properties of the proto-neutron star Strobel_1999 . We have
also used the crust EoS of the corresponding temperature for a complete
contact of Mass-Radius profile Togashi_2017 .

Figure 10: (colour online) M-R profile of proto-neutron star at different
temperatures for NL3 (left panel), G3 (middle) and IUFSU (right panel)
parameter sets respectively.

The mass-radius profile of the proto-neutron star for fixed temperature EoS
along with trapped neutrinos is plotted in Fig.10. The results from the
precisely measured neutron stars masses, such as PSR J1614-2230 with mass
$M=1.97\pm 0.04M_{\odot}$ Demorest_2010  and PSR J0740+6620 with
$M=2.15^{+0.10}_{-0.09}M_{\odot}$ Cromartie_2019  are shown in the horizontal
bars in pink color. These observations suggest that the maximum mass predicted
by any theoretical model should reach the limit of $\sim 2.0M_{\odot}$, and
this condition is satisfied by all of the EoSs taken into consideration.
However, G3 parameter set follows the observational constraint of PSR
J0740+6620 for the entire assumptive temperature range. We observe that the
inclusion of neutrino trapping flattens the M-R curve at the top for G3 and
IUFSU parameter sets, which affect the radius of the proto-neutron star
considerably. We also notice that the inclusion of temperature increases the
pressure at a given baryon density which yields the increase in the mass-
radius profile of the proto-neutron star. The proto-neutron star has a little-
bit large mass compared with that of the neutron star at zero temperature
because the EoS is stiffer in the former case. The mass-radius profile of
proto-neutron star using constant entropy EoS have also been studied and
depicted in Fig. 11.

Figure 11: (colour online) M-R profile of proto-neutron star at different
entropy for NL3 (left panel), G3 (middle) and IU-FSU (right panel) parameter
sets respectively.

By analysing Fig. 10 and 11, we concluded that the maximum mass determined by
the G3 parameter set strongly satisfy the constraints of the PSR J0740+6620
and PSR J1614-2230. We observe an increase in maximum mass and radius in
comparison to the cold NS for both kind of EoS (fixed T and S). The maximum
mass, radius and central temperature ($T_{Ce}$) of the PNS for all the three
parameter sets at different entropy is given in Table 2. The maximum mass of
the PNS for all the three parameter sets goes on increasing with increase in
entropy of the system. Also, we observe that the massive PNSs have larger
radius, so we concluded that more the entropy per baryon of the star, larger
the radius of the system. In other words, we can say that the evolutionary
process of the newly born star favours contraction in terms of size. It has
also been reported that the maximum mass of a proto-neutron star is
considerably affected by the earliest stage of its evolution, i.e. ratheripe
type proto-neutron star ($\sim$ 1 sec after core bounce) have larger mass than
the late-type proto-neutron starStrobel_2001 . So, the properties of the PNS
depends strongly on how immaculately we gravitate the entropy per baryon of
the system in its early stage.

Table 2: Maximum mass and radius of the PNS for NL3, G3 and IU-FSU parameter
sets calculated using constant entropy EoS for S = 0, 1 and 2. $T_{Ce}$
denotes the central temperature of the PNS.

Entropy ($S$) | NL3 | G3 | IU-FSU
---|---|---|---
| $M$
---
($M_{\odot}$)
| $R$
---
(km)
| $T_{Ce}$
---
| $M$
---
($M_{\odot}$)
| $R$
---
(km)
| $T_{Ce}$
---
| $M$
---
($M_{\odot}$)
| $R$
---
(km)
| $T_{Ce}$
---
0 | 2.850 | 13.32 | 0 | 2.004 | 10.95 | 0 | 1.930 | 11.12 | 0
1 | 2.905 | 14.10 | 44.21 | 2.016 | 11.07 | 40.38 | 1.950 | 11.37 | 39.91
2 | 2.908 | 14.34 | 82.24 | 2.034 | 11.49 | 79.87 | 1.960 | 11.63 | 78.85

Also, to reconcile the results for maximum mass and radius determined by the
two different approaches (i.e. constant entropy and constant temperature), we
calculated the mass and radius of the PNS using constant temperature approach
by fixing the temperature at $T=T_{Ce}$ for the corresponding parameter set.
The results for the same have been presented in Table 3. We observe that the
fixed temperature approach provide a larger mass and radius at $T=T_{Ce}$ of
the corresponding entropy value. G3 parameter set demonstrates $\sim 2.5\%$
and IU-FSU set shows $\sim 4.5\%$ difference in the radius of the PNS
calculated using constant entropy and temperature approach for the
corresponding $T=T_{Ce}$.

Table 3:  Maximum mass and radius of the PNS for NL3, G3 and IU-FSU parameter
sets calculated using constant temperature EoS. The central temperature at S =
0, 1 and 2 for the corresponding parameter set have been used as the constant
temperature.

Temp. ($T$) | NL3 | G3 | IU-FSU
---|---|---|---
| $M$
---
($M_{\odot}$)
| $R$
---
(km)
| $M$
---
($M_{\odot}$)
| $R$
---
(km)
| $M$
---
($M_{\odot}$)
| $R$
---
(km)
$T_{Ce}(S=0)$ | 2.850 | 13.32 | 2.004 | 10.95 | 1.930 | 11.12
$T_{Ce}(S=1)$ | 2.880 | 14.29 | 2.035 | 11.36 | 1.987 | 11.89
$T_{Ce}(S=2)$ | 2.909 | 14.60 | 2.053 | 11.79 | 2.021 | 12.15

At last, we concluded that the properties of a PNS can be deciphered more
appropriately by using the constant entropy perspective and as discussed in
the previous sections also, since G3 parameter set respects all the
experimental and observational constraints thoroughly, so it is one of the
most compatible RMF parameter set.

##  6 Summary and Conclusions

We have studied the consequences of finite temperature on the nuclear
properties of SNM and the crucial section required for the cooling of remnants
of supernovae explosion. We used well-known NL3 and IU-FSU and the recently
developed G3 parameter sets of the RMF model for a comparative study. We
concluded that NL3 with the possession of stiffest EoS does not provide the
empirical values for most of the nuclear properties. However, IU-FSU satisfies
most of the constraints on the NM properties but also neglect some of them.
The variations of the binding energy and pressure with baryon density for
different temperatures are qualitatively similar for all the three acquired
parameter sets. However, G3 predicts the higher value of critical temperature
for liquid-gas phase transition, which is more proximate to the reported
experimental values. We also observed a contrarious development in the value
of $K_{0}$ for the defined parameter sets with the increase of temperature.
Both $K$ and $K_{sym}$ being the second derivatives of the different forms of
energy, this behaviour of $K_{0}$ may be influenced by the magnitude of
$K_{sym}$ which is positive for NL3 and negative for the G3 parameter set.
$K_{sym}$ value of the G3 parameter set lies in the range reported by the
NICER and LIGO collaboration. This result shows that the G3 parameter set is
more suitable to reproduce the appropriate form of the EoS and can be used to
study the properties of neutron stars more accurately. We did not observe any
significant variation in the $Q_{sym}$ parameter at saturation density for the
G3 parameter set, while its value decreases for the NL3 and IU-FSU parameter
sets.
We emphasized on the cooling mechanism of highly dense matter and studied the
effects of all three parameter sets on the neutrino emissivity. We used the
relativistic approach to derive the detailed expression of neutrino
emissivity, which is more effective than the non-relativistic approach. We
concluded some important remarks about the cooling mechanism of the newly born
proto-neutron star on the basis of the outcomes of our detailed calculation.
We observed that the magnitude of neutrino emission is directly proportional
to the temperature of the neutron star. As the body cools down, the magnitude
of the neutrino emission decreases. Also, the neutrino emissivity has maximum
value around the saturation density, which indicates that the saturated matter
cools more rapidly through direct Urca process. Another important aspect that
we inspect is that the neutrino emissivity is higher for that parameter set
which provides softer EoS. As we can see in the previous section that the
stiff EoS measures higher mass NS, so we concluded that the lighter remnant of
the supernovae explosion cools down more expeditiously through neutrino
emissivity of the direct Urca process. Moreover, We look forward to see if any
post-merger signal is explored observationally in near future by LIGO /VIRGO
and NICER collaboration, which will help us to comprehend finite temperature
NM physics more appropriately and extensively. We also take notice that the
mass numerated by the G3 parameter set for the whole temperature range fits in
the envelope determined by the GW170817 experimental data. Finally, we adduced
that G3 parameter set is more appropriate to study the properties of stellar
objects through EoS.
Acknowledgement
This work is partially supported by National Natural Science Foundation of
China Grant No. 11873040.

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