# Thermal Relaxation of Dark Matter Admixed Neutron Star

Ankit Kumar1,2 , H. C. Das1,2 S. K. Patra1,2,
1 Institute of Physics, Sachivalaya Marg, Bhubaneswar-751005, India
2 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India<EMAIL_ADDRESS>
###### Abstract

Motivated by the various theoretical studies regarding the efficient capturing
of dark matter by neutron stars, we explore the possible indirect effects of
captured dark matter on the cooling mechanism of a neutron star. The equation
of states for different configurations of dark matter admixed star at finite
temperature is obtained using the relativistic mean-field formalism with the
IOPB-I parameter set. We show that the variation in the dark matter momentum
vastly modifies the neutrino emissivity through specific neutrino generating
processes of the star. The specific heat and the thermal conductivity of a
dark matter admixed star have also been investigated to explore the
propagation of cooling waves in the interior of the star. The dependence of
theoretical surface temperature cooling curves on the equation of state and
chemical composition of the stellar matter has also been discussed along with
the observational data of thermal radiation from various sources. We observed
that the dark matter admixed canonical stars with $k_{f}^{\rm DM}>0.04$ comply
with the fast cooling scenario. Further, the metric for internal thermal
relaxation epoch has also been calculated with different dark matter momentum
and we deduced that increment of dark matter segment amplify the cooling and
internal relaxation rates of the star.

###### keywords:

equation of state – stars: neutron – dark matter

##  1 Introduction

Neutron stars, undoubtedly one of the most astounding objects in the universe,
hold infinitude of possibilities in terms of density profile, structure,
symmetry, extreme surface temperature, magnetic field and composition, which
have been barely investigated using our limited acquirable knowledge. The
existence of parallel correspondence between its rearmost stellar and nuclear
properties makes it even more enigmatic and interesting. The physical aspects
of a neutron star can be explained in a complete and comprehensive manner only
if one can connect the drafts of various unassociated subdivisions of physics
like nuclear physics, neutrino physics, general relativity, quantum
chromodynamics, plasma physics and superfluid dynamics in a synergical manner.
The interpretation of pulsars as rotating stars during the numerous
observational events established the neutron stars as effectual objects to
explore the physics of dense matter systems and laid the foundation for
nuclear astrophysics as a distinct field of interest (Gold, 1968).

The well-established fact about the mysterious and uncanny neutron stars is
that the matter in the core of it can reach upto $5-10$ times of the nuclear
saturation density. The core is responsible for the major percentage of the
neutron star’s mass and holds extreme physical phenomena and conditions like
neutronization of matter, superconductivity and superfluidity at extremely
high temperatures, high magnetic field etc., which are far away from existence
in the subastral environment (Page et al., 2011). Since, the theories or the
model proposed to perceive the physics of matter at such high densities can
not be tested in terrestrial laboratories, therefore, the selection of an
accurate equation of state (EOS) to describe the structure and composition of
neutron star cores is a huge challenge for the physicists in the last few
decades. A number of EOSs obtained using different theoretical approaches have
been used in the past few years to unearth the physics of neutron stars and to
satisfy the constraints put by several observational events on various
properties of the neutron star (Epelbaum et al., 2009; Ruffert & Janka, 2010;
Lattimer, 2012; Dutra et al., 2012; Entem et al., 2015; Zhang & Chen, 2016;
Greif et al., 2020). An elaborated review with detailed discussions on various
methods such as meson-exchange and potential models; chiral effective field
theory and ab initio methods; Skyrme and Gogny interactions etc. to obtain the
equation of state for supernovae and compact stars is done by Oertel et. al.
(Oertel et al., 2017). One of the most successful and prominent theoretical
perspectives to obtain the EOS for the investigation of various properties of
a dense star is relativistic mean-field (RMF) formalism. The entire spectrum
of the EOS from the stiffest to the softest region can be obtained with the
help of existing RMF parameter sets in the literature, which corresponds to a
different order of coupling factors between mesons and nucleons and has its
own merits in terms of satisfying the various astrophysical observational
constraints (Walecka, 1974; Lalazissis et al., 1997; Shen, 2002; Dutra et al.,
2016; Kumar et al., 2017; Carbone & Schwenk, 2019).

The composition of the neutron star cores has also been a secret and topic of
debate in the last few years. Although, we can infer a broad idea about the
neutron star interiors using the astrophysical observations, but it is far
from reach to find out the exact creation of the dense star on the basis of
observational data. There are a lot of hypothesis available in the literature
regarding the availability of free quarks, hyperons, delta isobars, or the
phenomenon of kaon condensation in the interior shell of a neutron star
(Glendenning, 1985; Kaplan & Nelson, 1988; Collins & Perry, 1975; Waldhauser
et al., 1987; Li et al., 1997). The presence of exotic particles inside the
neutron star happens to soften the EOS which in turn will affect the
observables, e.g., reduce the mass and tidal deformability of the star
(Glendenning & Moszkowski, 1991; Weissenborn et al., 2012; Das et al., 2021b).
Apart from the standard exotic particles, theoretical studies also suggested
the presence of dark matter particles inside the neutron star. The ultra-dense
profile of the star and asymmetry between particles and anti-particles of dark
matter branch is supposed to be responsible for the gravitational trapping of
dark matter without self-annihilation in the core of a compact star (Goldman &
Nussinov, 1989; Kouvaris, 2008; Fan et al., 2012). The accumulated dark matter
inside a dense star with either being bosonic or fermionic in nature and with
different hypothesized dark matter candidates from the theoretical and
experimental point of view such as neutralinos, axions, technibaryons have
been explored in various theoretical studies (Jungman et al., 1996; Bertone et
al., 2005; Duffy & Bibber, 2009; Kouvaris & Tinyakov, 2010; Fan et al., 2012;
Xiang et al., 2014). Although, the nature, direct interaction or coupling of
dark matter with the other standard model particles is quite weak and
uncertain, but theoretical efforts have been made to explore the effects of
dark matter interaction on the properties of astral bodies (Li et al., 2012a;
Li et al., 2012b; Mukhopadhyay & Schaffner-Bielich, 2016). The peak in the
power spectral density of the gravitational wave observations of the GW170817
event which involves a neutron star-neutron star merger and coalescence event
involving a black hole and compact object (GW190814), also offers a new
approach for the search and effects of DM on the components of universe
(Abbott et al., 2017; Ellis et al., 2018; Abbott et al., 2020).

Another important aspect that can help a lot in understanding the physical
processes taking place inside a neutron star is the elaborated study of its
cooling dynamics. The first theoretical attempt to study the cooling mechanism
of a neutron star was done in the 1960s in order to establish them as the
newly discovered X-ray source detected by rocket and balloon experiment
(Giacconi et al., 1962; Chiu & Salpeter, 1964; Bowyer et al., 1964). Later,
the theoretical cooling simulations extended in a more comprehensive manner
within the context of the theory of general relativity for a hydrostatic
spherically symmetric neutron star, which ultimately accelerated the
importance of physical factors of neutron stars in X-ray astronomy (Glen &
Sutherland, 1980; Nomoto & Tsuruta, 1986; van Riper & Lamb, 1981). The study
of the thermal evolution of a neutron star has different dimensions engaged
with mass accretion, composition, magnetic field, superfluidity,
superconductivity etc. Theoretical studies suggest that the temperature of a
newly born proto-neutron star is around $10^{10}\,K$ and the initial rate of
heat loss is so intense that the temperature drips down to $10^{9}-10^{8}\,K$
within a few seconds. The excessive neutrino flux is much more dominant than
surface photon radiation and mainly responsible for the initial stage rapid
cooling of a neutron star (Burrows & Lattimer, 1986; Pons et al., 1999). As a
matter of fact, the neutrino flux is so enormous that even the Pauli effects
on the fermionic character of the neutrinos control the the dynamics of a
supernova explosion considerably (Bludman & Ruderman, 1975). The superfluid
effects can also influence neutrino emissivity and the phenomenon of heat
transport within the shell of a neutron star. Superfluidity leads to the
Cooper pairing of nucleons which can enhance the neutrino emissivity by
generating more neutrinos and hence can influence the thermal evolution of the
star (Yakovlev et al., 1999). The composition of a dense star controls the
production of neutrinos via different processes inside the star, which
indirectly influence its cooling mechanism (Pethick & Ravenhall, 1992; Page et
al., 2006). The high magnetic field of the neutron star is responsible for the
anisotropy in the thermal conductivity, which ultimately evolves into the
surface temperature gradient. Neutron stars remains apparent up to million
years via surface thermal emission and the thermal imaging of steady X-ray
spectra from the supernova remnants using the imaging proportional counter of
orbital X-ray observatories (such as Einstein Observatory, Chandra X-ray
Observatory, XMM-Newton, MAXI, NuSTAR, Astrosat, Max Valier Sat, NICER,
eROSITA, Spektr-RG etc.) can provide a lot of useful information about the
effective surface temperature of the stars (Giacconi et al., 1979; Boguta,
1981). The temperature-susceptive properties such as electron thermal
conductivity, specific heat, crust solidification etc., can also be explored
by deciphering the thermal evolution of the star. The outer layers of the
neutron star, generally known as the crust, control the surface photon
emission and yield a lot of information about the mechanism of thermal
relaxation inside the envelope of the star. The simple assumption about the
ingredient of the envelope of a neutron star i.e. whether the envelope is made
up of light elements or heavy elements, can subvene the surface temperature
versus relaxation time trajectory of the star quite considerably (Beznogov et
al., 2021).

Theoretical studies suggest that the accretion of weakly interacting massive
particles (WIMPs) onto the neutron star can induce heating in the aged stars,
which can be explored to put several constraints on the cross-section of the
dark matter inaccessible to the experiment (Goldman & Nussinov, 1989). The
accretion of interstellar gas and massive particles could possibly trigger the
reheating phenomenon inside the aged ($10^{6}$-$10^{7}$ years old) neutron
stars and ultimately lead its fate into a collapse (Kouvaris, 2008; Raj et
al., 2018). Also, the annihilation of WIMP in the core of a neutron star can
act as the energy source required for the phase transition to quark matter and
considerably affect the dynamics of supernova explosion (Hong et al., 2001; de
Lavallaz & Fairbairn, 2010; Kouvaris, 2008). However, in the present work, we
did not consider the heating effect due to WIMPs or the energy bump due to the
dark matter annihilation which corresponds to inappreciable cross-section, as
done in the above-cited references. In this work, we revisit the dynamics of
thermal relaxation for a neutron star in the context of nucleon interaction
with dark matter postulator indirectly. We explore the effects of indirect
interaction of dark matter on neutrino emissivity, specific heat, thermal
conductivity and surface temperature of the neutron star. The paper is
commenced with the formalism used to obtain the EOS of neutron stars along
with the effect of dark matter interaction. We derive the EOS of a dark matter
admixed neutron star using the temperature-dependent RMF formalism with
different variations of dark matter momentum. In the later sections, the
obtained EOS is used to study the mechanism of thermal relaxation of a neutron
star.

##  2 Equation of State

The RMF theory, introduced in the context of infinite nuclear matter, later
proved to be one of the most successful formalism to study the properties of
finite nuclei and compact astrophysical objects (Serot & Walecka, 1992;
Gambhir et al., 1990; Ring et al., 1997; Lalazissis et al., 1997). The
Lagrangian density introduced initially evolved with the addition of new
interactions and multi-mesonic couplings, to satisfy several constraints
imposed by the experimental and empirical data. Also, a number of RMF
parameter sets are developed by different group of researchers, each one of
which has its own importance and demerits in view of the flow data experiment
and observational constraints (Lalazissis et al., 1997; Furnstahl et al.,
1997; Horowitz & Piekarewicz, 2002; Todd-Rutel & Piekarewicz, 2005; Kumar et
al., 2006; Dhiman et al., 2007; Fattoyev et al., 2010; Dutra et al., 2014). We
adopted the newly developed RMF parameter set by Kumar et al. named as
"IOPB-I" (Kumar et al., 2018) and for which the corresponding Lagrangian
density representing the hadronic matter for a neutron star subsequent to
mean-field approximation can be written as (Liu et al., 2005; Ohnishi et al.,
2017; Kumar et al., 2021)

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

where, $M$ and $\psi$ are the mass and wave function for the nucleons (protons
and neutrons) and the last term of the Lagrangian corresponds to the leptons
(electrons and muons). The considered masses of the intermediating $\sigma$,
$\omega$ and $\vec{\rho}$ mesons for IOPB-I parameter set are $500.512$,
$782.5$, and $763$ MeV and the coupling constants of the corresponding mesons
i.e. $g_{\sigma}$, $g_{\omega}$ and $g_{\rho}$ are $-10.3954$, $13.3466$ and
$11.1242$ respectively. The self and cross-coupling coefficients i.e.
$\kappa_{3}$, $\kappa_{4}$, $\zeta_{0}$ and $\Lambda_{\omega}$ for the IOPB-I
parameter set are $1.4957$, $-2.9324$, $3.1039$ and $0.0243$ respectively. The
EOS for a neutron star acquired by using the IOPB-I parameter set is
consistent in view of the nuclear matter saturation properties and satisfy
almost all the bars imposed by the gravitational observational phenomenon
(GW170817), which can be verified in the references (Kumar et al., 2018; Das
et al., 2021a, b). Since, we want to explore the thermal mechanism in this
work, so, we infer the EOS representing the neutron star at finite
temperature. The total nucleonic ($n$) and scalar ($n_{s}$) density within RMF
formalism for a finite temperature neutron star will be represented by,

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

$M^{\ast}=M-g_{\sigma}\sigma$, is the effective mass of the nucleons in
mesonic field and $f_{j}(\mu_{j}^{\ast},T)$ and
$\bar{f}_{j}(\mu_{j}^{\ast},T)$ represents the Fermi distribution function for
the particle and anti-particle at effective chemical potential
$\mu_{j}^{\ast}$ and temperature $T$. The temperature dependent mean-field
mechanism to study the macroscopical properties of a neutron star at finite
temperature is explicitly discussed and analyzed in our recent work (Kumar et
al., 2020, 2021).

Now, in the present work, along with the temperature we incorporate the
effects of trapped fermionic dark matter also on the EOS of the neutron star.
The functional Lagrangian density to obtain the EOS for a finite temperature
dark matter admixed neutron star can be written as, (Panotopoulos & Lopes,
2017; Das et al., 2019; Das et al., 2020),

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{NM}+\bar{\chi}\left[i\gamma^{\mu}\partial_{\mu}-M_{\chi}+yh\right]\chi+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h$
(4) $\displaystyle-\frac{1}{2}M_{h}^{2}h^{2}+\frac{f\,M}{v}\bar{\psi}h\psi,$

where $\chi$ represents the lightest neutralino wave function, which for the
present case is assumed as fermionic dark matter candidate (Martin, 1998), $h$
stands for the Higgs field and, the last term is the nucleon-Higgs coupling.
The detailed expressions for the energy density and pressure using the above
defined Lagrangian density are derived in the references (Das et al., 2020;
Lourenço et al., 2021). The Yukawa coupling parameter for the nucleon-Higgs
interaction ($f\,M/v$) is set to $0.001336$ and since the theoretically
defined range for the neutralino-Higgs coupling ($y$) is $0.001-0.1$, so we
fix its value to be 0.07 (Cline et al., 2013; Alvarez-Ruso et al., 2013; Baer
et al., 2015). The effective mass of the nucleons in this scenario will be
modified as $M^{\ast}=M-g_{\sigma}\sigma-\frac{f\,M}{v}h$. The dark matter
density is considered around thousand times smaller than the nuclear
saturation density and the composition of dark matter inside the neutron star
is controlled through its momentum ($k_{f}^{\rm DM}$), which is varied upto
$0.06$ GeV and $k_{f}^{\rm DM}=0.00$ GeV reveals the equation of state of
$\beta$ equilibrated matter without any dark matter component (Panotopoulos &
Lopes, 2017; Das et al., 2019). We stabilize the star’s central temperature at
$10^{9}\,K$ and also preserve the beta equilibrium and charge neutrality to
get the EOS and fractional abundance of particles for a dark matter admixed
star. Also, the crustal segment of the star plays an important role in the
simulation dynamics of thermal evolution, so we use the stellar crust EOS data
derived by Chamel and Haensel (Chamel & Haensel, 2008). The combined EOS for
the crust and core of the dark matter admixed neutron star is depicted in Fig.
1.

Figure 1: Equation of state for a dark matter admixed neutron star at
$T=10^{9}\,K$. The dark matter momentum ($k_{f}^{\rm DM}$) is varied in-
between $0.02-0.06$ GeV and the black line curve corresponds to the neutron
star EOS without any dark matter component.

We publicized that the increase in the dark matter momentum at zero
temperature shifts the EOS towards the softer region which indirectly results
in the decreased mass of the star. The interaction between the nucleons and
the dark matter particle via Higgs and the inclusion of more degrees of
freedom is accountable to make the EOS softer. The system would like to
minimize its energy through low energy states of the dark matter, so, the
nucleons are replaced by the dark matter candidates depending on the system’s
density due to which a fraction of the gravitational mass is converted to
kinetic energy (Das et al., 2020; Das et al., 2021a). A similar pattern for
the EOS is observed at the finite temperature (Fig. 1). The maximum mass and
the corresponding radius of the neutron star for the EOS at $T=10^{9}\,K$
using IOPB-I parameter set with different dark matter momentum is calculated
using Tolman-Oppenheimer-Volkoff equations (Tolman, 1939; Oppenheimer &
Volkoff, 1939). The obtained canonical star radius ($R_{1.4}$) along with the
maximum mass and corresponding radius ($M,R$) for the neutron star with
$k_{f}^{\rm DM}=0.00-0.06$ GeV are stated in table 1.

Table 1: The maximum mass ($M$), radius for maximum mass star ($R_{M}$) and canonical star radius ($R_{1.4}$) for different values of dark matter Fermi momentum ($k_{f}^{\rm DM}$) using IOPB-I parameter set at $10^{9}\,K$. $k_{f}^{\rm DM}$ | Max. Mass ($M$) | $R_{M}$ | $R_{1.4}$
---|---|---|---
(GeV) | ($M_{\odot}$) | (km) | (km)
$0.00$ | $2.130$ | $11.987$ | $13.497$
$0.02$ | $2.099$ | $11.774$ | $13.190$
$0.03$ | $2.031$ | $11.296$ | $12.504$
$0.04$ | $1.918$ | $10.494$ | $11.380$
$0.05$ | $1.770$ | $9.655$ | $10.414$
$0.06$ | $1.604$ | $8.688$ | $9.250$

The thermal evolution mechanism in the further sections is explored for the
canonical star with the whole assumed momentum range and the above-listed
maximum mass star at the corresponding dark matter momentum.

This model of dark matter admixed cold neutron star has been adopted
previously in the listed literature references to explore its effects on the
mass-radius profile, curvature, in-spiral properties and $f$-mode oscillations
of the neutron star for various relativistic parameter sets (Das et al., 2019;
Das et al., 2020; Das et al., 2021c, b, a; Lourenço et al., 2021). However, in
this work, we dig the effects of the fermionic dark matter, with neutralino as
the suitable dark matter candidate, on the thermal evolution profile of a hot
beta equilibrated neutron star.

##  3 Thermal Evolution

The energy balance and flux dynamics in terms of relativistic framework to
depict the thermal mechanism of a hydrostatic spherically symmetric neutron
star were initiated by Thorne. The heat diffusion in the neutron stars within
the relativistic framework can be explored using these energy balance and
transport equations (Thorne, 1977),

$\displaystyle\frac{\partial}{\partial\,r}(Le^{2\phi(r)})=-\,\frac{4\pi
r^{2}e^{\phi(r)}}{\sqrt{1-\frac{2GM}{c^{2}r}}}\Big{(}C_{V}\frac{\partial\,T}{\partial\,t}-e^{\phi(r)}(Q_{\nu}+Q_{h})\Big{)},$
(5)
$\displaystyle\frac{\partial}{\partial\,r}(Te^{\phi(r)})=-\frac{L}{\kappa\,4\pi
r^{2}}\,\frac{e^{\phi(r)}}{\sqrt{1-\frac{2GM}{c^{2}r}}},$ (6)

where $t$ and $\phi$ are the time coordinate and Schwarzschild gravitational
potential respectively. $C_{V}$ is the specific heat, $Q_{\nu}$ represents
neutrino emissivity, $Q_{h}$ is associated to the heat production per unit
volume due to several factors (for instance, dissipation of rotational energy,
magnetic dissipation etc.) (Urpin et al., 1997; Page et al., 2000) and ignored
in the present calculations, $\kappa$ is the thermal conductivity and $L$
stands for the total luminosity with neutrino and radiation part combined. The
solution of these two coupled differential equations endue us with the surface
temperature ( $T_{s}(t)$ ) of the star and the distribution of the internal
temperature of the star ($T$) as a function of time. The star’s surface
temperature in a locally flat reference frame ($T_{s}$) is related to the
apparent surface temperature for a distant observer ($T_{s}^{\infty}$) as
$T_{s}^{\infty}=T_{s}\sqrt{1-2GM/(c^{2}R)}$. The numerical simulation to get
the solution of these equations would require some initial and boundary
conditions, which are adopted as: $L_{r}=0$ (no initial heat flux) ;
$L(r=R,t)=4\pi\sigma_{S}R^{2}T^{4}_{s}(t)$ (boundary) and
$Te^{\phi}(t=0)=T_{0}$ (initial redshiftted temperature), where $\phi$
determines the redshift and depends on the model used.

###  3.1 Neutrino Emissivity

The exact metric for the total emissivity of a neutron star is a complicated
business due to the number of neutrino reactions in the neutron star crust as
well as in its core, which can be derived using the electroweak interaction
Glashow-Weinberg-Salam theory (Bilenky & Hošek, 1982). These neutrino
reactions are the most powerful source and are mainly responsible for the
initial cooling stage of the young stars. As we know, the neutron star’s crust
and core consist of different complex states of matter, so, the neutrino
emission in the crust and core is also controlled or dominated by an entirely
different mechanism. Since, it is irresolvable to include each neutrino
reaction taking place in the stellar objects and also the main aim of the work
is to observe the dark matter effects on the resultant emissivity, so, we
incorporated a few of them in our present work, most of which are mainly
responsible for the major contribution separately in the crust and the core
segment of the star.
(i) Plasmon decay [$\gamma\longrightarrow\nu\bar{\nu}$]. It occur due to the
electron-medium interaction and controlled by the electron plasma frequency,
which further depends on the number density and the chemical potential of the
electron in the defined beta equilibrium matter. This process is mainly
effective at lower densities and the adopted mathematical expression for the
same can be found in the listed references (Braaten, 1991; Itoh et al., 1992;
Kantor & Gusakov, 2007).
(ii) Pair annihilation [$e^{-}e^{+}\longrightarrow\nu\bar{\nu}$]. The neutrino
emissivity due to electron-proton pair annihilation is not much significant
due to the small fraction of positrons in the neutron stars (Chiu & Morrison,
1960; Yakovlev et al., 2001; Kantor & Gusakov, 2007).
(iii) $e^{-}$-nucleus Bremsstrahlung [ $e\,\\{A,Z\\}\longrightarrow
e\,\\{A,Z\\}\,\nu\bar{\nu}$ ], $\\{A,Z\\}$ represents the nuclei. The neutrino
emissivity due to $e^{-}$-nucleus Bremsstrahlung processes is affected by the
shape of the nuclei in the crust of the star and is considered as spherical
for the present calculations. For detailed analytical discussion and adopted
mathematical expression, the readers are advised to go through these
references (Flowers, 1973; Haensel et al., 1996; Kaminker et al., 1999;
Yakovlev et al., 2001).
(iv) Neutron-nucleus Bremsstrahlung [ $n\,\\{A,Z\\}\longrightarrow
n\,\\{A,Z\\}\,\nu\bar{\nu}$ ]. This process is operative in the crustal
envelope of the neutron star and enhance the neutrino emissivity magnitude of
crust in a considerate manner. The neutrino emissivity expression for the non-
superfluid stellar matter due to Neutron-nucleus Bremsstrahlung processes is
borrowed from (Flowers & Sutherland, 1977).
(v) Neutron-neutron Bremsstrahlung [$nn\longrightarrow nn\,\nu\bar{\nu}$];
proton-proton Bremsstrahlung [$pp\longrightarrow pp\,\nu\bar{\nu}$]; neutron-
proton Bremsstrahlung [$np\longrightarrow np\,\nu\bar{\nu}$];

$\displaystyle Q^{ij}=1.5\times
10^{20}\,\left(\frac{M_{i}^{\ast}}{M_{i}}\right)^{2}\left(\frac{M_{j}^{\ast}}{M_{j}}\right)^{2}\left(\frac{n_{j}}{n_{0}}\right)^{\frac{1}{3}}\,\alpha_{ij}\,\beta_{ij}\mathcal{N}_{\nu}\left(\frac{T}{10^{9}}\right)^{8},$
(7)

where, $\mathcal{N_{\nu}}\xrightarrow{}$ neutrino flavour number ($3$). The
pair $(ij)$ is defined as the combination of neutron-neutron (nn), proton-
proton (pp) and neutron-proton (np), for which the corresponding values of the
parameters $\alpha_{ij}$ and $\beta_{ij}$ are: $\alpha_{nn}=2.95$,
$\beta_{nn}=0.56$; $\alpha_{pp}=0.55$, $\beta_{pp}=0.7$; $\alpha_{np}=1.06$,
$\beta_{np}=0.66$ (Friman & Maxwell, 1979). These processes are the moderate
sources for the neutrino emissions in the stellar cores and also affected
considerably in presence of neutron superfluidity .
(vi) Coulomb Bremsstrahlung [ $ep\longrightarrow ep\,\nu\bar{\nu}$, $\mu
p\longrightarrow\mu p\,\nu\bar{\nu}$, $ee\longrightarrow ee\,\nu\bar{\nu}$,
$\mu\mu\longrightarrow\mu\mu\,\nu\bar{\nu}$, $e\mu\longrightarrow
e\mu\,\nu\bar{\nu}$ ]. These processes are similar to nucleon Bremsstrahlung
but they originate due to Coulomb forces instead of strong collisions, as in
case of baryon Bremsstrahlung and the mathematical expression for neutrino
emissivities due to the Coulomb Bremsstrahlung are borrowed from (Kaminker et
al., 1999).

Figure 2: Neutrino emissivity of a canonical star as a function of mass
density at $T=10^{9}\,K$ for different configuration of dark matter momentum.
The black colour curve ($k_{f}^{DM}=0.00$ GeV) represents the profile of a
canonical star obtained using IOPB-I parameter set without any dark matter
admixture.

(vii) Direct Urca processes [ $n\longrightarrow pe^{-}\bar{\nu}$,
$pe^{-}\longrightarrow n\nu$, $n\longrightarrow p\mu\bar{\nu}$,
$p\mu\longrightarrow n\nu$ ];

$\displaystyle
Q^{DU}=\frac{457\pi}{10080}\,G_{F}^{2}\,C^{2}\left(1+3g_{A}^{2}\right)\frac{M_{n}^{\ast}\,M_{p}^{\ast}\,m_{e}^{\ast}}{\hbar^{10}\,c^{3}}\,(k_{B}\,T)^{6}\,\Theta_{npe},$
(8)

where $G_{F}$ and $g_{A}$ are the Fermi weak and axial coupling constants
respectively, $C=0.973$ is known as Cabibbo factor and $\Theta_{npe}$ is the
step function which describes the onset of direct Urca process inside the core
of neutron star. The value of $\Theta_{npe}$ is unity only if the triangle
inequality between the Fermi momentum of neutrons, protons and electrons is
satisfied i.e. $(k_{F_{n}}-k_{F_{p}}-k_{F_{e}})\leq 0$, otherwise it is zero.
For direct Urca reactions involving muons, the expression for the neutrino
emissivity remains the same except the step function, which in case of muons
will be read as $\Theta_{np\mu}$. The threshold density for the muon Urca
processes is higher than that of electron involving events. Once the threshold
boundary density for the direct Urca to operate is reached, it dominates all
other neutrino processes significantly and responsible for the very fast
cooling of the dense cores. (Gamow & Schoenberg, 1941; Lattimer et al., 1991).
(viii) Modified Urca processes
(a) ( Neutron branch ) [ $nn\longrightarrow npe^{-}\bar{\nu}$,
$nn\longrightarrow np\mu\bar{\nu}$, $npe^{-}\longrightarrow nn\nu$,
$np\mu\longrightarrow nn\nu$ ]

$\displaystyle Q^{Mne}=6.224\times
10^{21}\left(\frac{M_{n}^{\ast}}{M_{n}}\right)^{3}\left(\frac{M_{p}^{\ast}}{M_{p}}\right)\left(\frac{n_{p}}{n_{0}}\right)^{1/3}\left(\frac{T}{10^{9}}\right)^{8}.$
(9)

Figure 3: Neutrino emissivity is shown for the maximum mass profile of the
star acquired with the corresponding dark matter momentum. The obtained
maximum mass of the star for $k_{f}^{\rm DM}=0.03$ and $0.06$ GeV are $2.031$
and $1.604\,M_{\odot}$ respectively.

(b) ( Proton branch ) [ $pn\longrightarrow ppe^{-}\bar{\nu}$,
$pn\longrightarrow pp\mu\bar{\nu}$, $ppe^{-}\longrightarrow pn\nu$,
$pp\mu\longrightarrow pn\nu$ ]

$\displaystyle
Q^{Mpe}=\left(\frac{M_{p}^{\ast}}{M_{n}^{\ast}}\right)^{2}\frac{\left(k_{F_{e}}+3k_{F_{p}}-k_{F_{n}}\right)^{2}}{8k_{F_{e}}k_{F_{p}}}Q^{Mne}\Theta_{Mpe}.$
(10)

The proton branch of the modified Urca process require a threshold density to
operate, which is controlled by the theta function ($\Theta_{Mpe}$) and is
non-zero for $k_{F_{n}}\,<\,3k_{F{p}}+k_{F_{e}}$. In the case of muon
involving cycles which depends on the appearance of muons inside the stars,
both the emissivity expressions should be multiplied by
$(n_{\mu}/n_{e})^{1/3}$ and the threshold for the proton branch is determined
by $k_{F_{n}}\,<\,3k_{F{p}}+k_{F_{\mu}}$. The reaction mechanism and neutrino
production rate of this process is different from that of the direct Urca
processes. It requires an additional nucleon for the conservation of momentum,
has a slower reaction rate and, happens to be inoperative in the presence of
direct Urca reactions (Chiu & Salpeter, 1964; Friman & Maxwell, 1979; Yakovlev
& Levenfish, 1995; Yakovlev et al., 2001).

All the above-introduced processes circumstantially depend on the superfluid
effects, high magnetic field, presence of hyperons, condensation of pions or
kaons and a high dense quark core of the neutron stars. The inspection of
neutrino emission rate from the neutron stars along with the inclusion of all
these factors is beyond the scope of this work and may be adopted later in
different context for the forthcoming projects. The keen readers are advised
to go through the corresponding listed references, mainly (Yakovlev et al.,
2001) & (Potekhin et al., 2015a), for more detailed derivations and explicit
discussions of each neutrino emission process along with the superfluid
effects. The effects of the dark matter component on the neutrino emission
processes in a canonical and maximum mass star are depicted in figs. 2 and 3.
The thermal contour of the neutron star in the early stage of the star is
controlled by the outermost crust of the star, which in turn is mostly
influenced by the neutrino emissivity due to plasmon decay (Nomoto & Tsuruta,
1987; Levin & Ushomirsky, 2001). The increment of the dark matter factor in a
coequal mass neutron star amplifies the plasma frequency by means of electron
Fermi momentum and develops the neutrino sources in the outer layers of the
star, which results in the increased neutrino emissivity (fig. 2). Since, the
initial age cooling of a neutron star is independent of what is happening in
the core and the crustal part is the prime source of neutrino emissions, it
turns out that a dark matter admixed neutron star cools more expeditiously in
formative years. The compositional profile of a pure canonical neutron star
(i.e. without any dark matter admixture) obtained using the IOPB-I parameter
set does not support the direct Urca process and the neutrino emissivity
through modified Urca processes is responsible for the fastest cooling
scenario of such stars. The activation of the proton branch for electron and
muon case takes place separately depending upon the fulfillment of triangle
inequality by Fermi momenta of the particles and the activation point as
pointed in figs. 2 and 3 in the form of bumps within the modified Urca
dominance region. The proton branch enhances the emissivity of the modified
Urca process by a factor of 10 and the activation density is higher for a
neutron star with greater dark matter composition. However, we observe that if
the theoretically considered availability of dark matter segment in such stars
is such that $k_{f}^{\rm DM}>0.04$ GeV, then the compositional abundance of
the star enables the direct Urca reactions in the core (fig. 2). The dark
matter admixture increases the central density of the star, as shown in fig.
1, which is accompanied by an increase of proton and lepton fraction. With the
high fraction of protons and leptons in such dark matter admixed canonical
stars ($k_{f}^{\rm DM}>0.04$ GeV), the inevitable triangle inequality for
direct Urca process is satisfied and an additional highly efficient neutrino
emission channel is operative in such stars. The metric for the neutrino
emissivity of the direct Urca process is almost $10^{5}$ times higher than the
other neutrino emission processes, so navigating the threshold density for
direct Urca reaction results in a very rapid cooling of the canonical star.
The turning on density for direct Urca processes of DM admixed canonical
neutron star in case of $k_{f}^{\rm DM}=0.05$ and $0.06$ GeV are $1.052\times
10^{15}$ and $1.195\times 10^{15}$ g cm-3 respectively. The direct Urca
neutrino emissions are present in both cases of the maximum mass star i.e.
with or without dark matter component and the triangle inequality for muons is
also satisfied in the case of maximum mass star (fig. 3), which opens an extra
channel for neutrino production and further enhance the neutrino emissivity.
The electron and muon channel threshold density of direct Urca process in the
maximum mass scenario of a pure neutron star ($k_{f}^{\rm DM}=0.00$ GeV) with
IOPB-I parameter set are $8.585$ and $14.974$ ($10^{14}$ g cm-3) respectively.
The threshold value of density for DM admixed maximum mass neutron star with
$k_{f}^{\rm DM}=0.06$ GeV is same as that of a canonical star.

###  3.2 Heat Capacity and Thermal Conductivity

The energy required to increase the temperature of the star is theoretically
studied in terms of the specific heat capacity, measured by summing the
individual contribution of heat from all the considered particles in the
neutron star model. The total heat capacity of a neutron star in the crustal
part is dominated by the relativistic electron degenerate gas and the lattice
of atomic nuclei present in different phases of the neutron star’s crust. The
contribution in the core part of the star received from the neutrons, protons,
electrons and muons for the present case and other exotic particles like
hyperons, quarks etc., which are not considered in the present model. The
theoretical mechanism for the specific heat capacity generated in the crust
part of the star due to the coulomb ion vibrations in the lattice depends on
anharmonicity, phase transition, Debye frequency, order parameter ($\Gamma$)
etc. (Carr, 1961; Slattery et al., 1982). The final adopted mathematical
expressions can be found in the reference (van Riper, 1991), and the
interested readers for detailed procedure of derivations are advised to go
through the listed references (Baiko et al., 2001a; Potekhin & Chabrier,
2010).

An updated and generally used expressions for the specific heat of the ions in
star’s crust are derived by Haensel et. al. and advised to the readers for
relevant simulations in future projects (Potekhin & Chabrier, 2018). Apart
from the ions, the second component that majorly contributes to the specific
heat of the crust is the relativistic degenerate electron gas.

Figure 4: Specific heat of a dark matter admixed canonical star as the
function of density at $10^{9}\,K$ temperature. Dark matter momentum is varied
from $0.00$ GeV to $0.06$ GeV. The stellar core portion is depicted in the
inset. The region of the appearance of muons is incited in rectangular box.

For temperature of the interest in the present case $T<<\mu_{i}$ and in the
absence of superfluidity, the specific heat of each constituent particle can
be stated as,

$\displaystyle C_{V}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{M_{i}^{\ast}\,n_{i}}{k_{F_{i}}^{2}}\,\pi^{2}\,k_{B}^{2}\,T,$
(11)

where $i$ sums over neutrons, protons, electrons and muons (Page et al., 2004;
Potekhin et al., 2015b). The effective mass of leptons (electron and muon) is
defined as $\mu_{e,\mu}/c^{2}$, while for nucleons (proton and neutron) it is
the in-medium effective mass acquired for the IOPB-I parameter set in the
presence of dark matter as defined in section 2. The cumulative specific heat
of a dark matter admixed canonical star and maximum mass star profiles for
IOPB-I parameter set is depicted in figs. 4 and 5 respectively at $T=10^{9}K$.
The influence of dark matter on the heat capacity of the star is embodied only
through the interference of dark matter particles on the effective mass and
compositional distribution of the constituent particles. As for the crustal
segment, the major contribution to the specific heat is through the ionic
oscillations and degenerate electron gas which is not much influenced by the
dark matter contribution and depends on the shape, which is considered
spherical in the present case, structure and nature of the envelope considered
(heavy or light element), so, there is no influential change is observed in
the specific heat with dark matter variation at low density. With the increase
in density, neutron drips out of the nuclei and sharp enhancement in the heat
capacity is recorded at the boundary. In the high dense region (core of the
star), the effective mass and abundance of the nucleons and leptons, mainly
neutrons, is influenced considerably by the involvement of dark matter
contribution which is reflected on the specific heat of the star too. The
increase in the dark matter momentum hints for the suppression of nucleon’s
effective mass and number density product in the core which results in the
reduction of heat capacity for a larger value of $k_{f}^{\rm DM}$ at a certain
density. The appearance of the muons which is controlled by the beta
equilibrium conditions of the star results in the further increment of the
heat capacity inside the core. It should be noted that, however this algorithm
scheme predicts the consistent pattern about specific heat capacity of the
stellar core but is largely uncertain due to the absence of cooper pairing of
neutrons, magnetic field, quark degrees of freedom and superfluidity.

Another important ingredient required to unfold the thermal dynamics of young
non-relaxed neutron stars is thermal conductivity. Thermal conduction is also
extremely responsible for the internal thermal relaxation of a neutron star.

Figure 5: Same as Fig. 4, but for the profile of maximum mass star obtained
for different values of $k_{f}^{\rm DM}$ with IOPB-I parameter set as apprised
in section 2.

The radiative luminosity of the stars is determined by the mechanism of heat
transfer in the outermost layers and the ocean, which is carried out mostly by
electrons. In the absence of magnetic field the heat transfer through the
phonons (ionic thermal conductivity) is not of much importance. So, we mainly
consider the electron heat conduction in outer crust of the neutron star. The
expression for the electron thermal conductivity coefficient in the outer
envelopes with multiphonon phenomenon and long-range nucleus correlations in
Coulomb solid and liquid of nuclei had been derived explicitly by Potekhin and
Yakovlev, and is given by (Potekhin & Yakovlev, 2001; Potekhin et al., 2015a)

$\displaystyle\kappa_{e}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}\,k_{B}^{2}\,n_{e}\,T}{3\,m_{e}^{\ast}\,\nu_{e}},$
(12)

where, $m_{e}^{\ast}$ is the relativistic kinematic electron mass and
$\nu_{e}$ is the effective collision rate due to electron-ion ($\nu_{ei}$) and
electron-electron ($\nu_{ee}$) scattering. The kinematic electron mass is
defined as $m_{e}^{\ast}=\sqrt{1+x_{r}^{2}}$ with
$x_{r}=\Big{(}\frac{1.027\,\rho\,Z}{10^{6}\,A}\Big{)}^{1/3}$. The effective
collision frequency in the outermost crust due to the electron-electron and
electron-ion components can be simply add up in accordance with the
Matthiessen rule and depends on the phase state and composition of the nuclei
and ions adopted in the crustal part.

Figure 6: Thermal conductivity for a pure canonical star ($k_{f}^{\rm
DM}=0.00$ GeV) with IOPB-I parameter set as a function of temperature at
different values of density. Mainly outer and inner crust is shown here.

The electron-ion collision frequency in case of degenerate and non-degenerate
electron gas for Coulomb solid and ion liquid regime was first proposed by
Yakovlev and Urpin (Yakovlev & Urpin, 1980), which later had been unified in a
single regime by Potekhin et al. and adopted in the present work for
simulations. Accordingly, the electron-electron scattering in the stellar
crust like environment, first entertained by Flowers and Itoh at small
temperatures (Flowers & Itoh, 1976), and then undergo several modifications
with the inclusion of charge-current interactions, Landau damping, magnetic
field etc. by various authors and the final admittable expression of
$\nu_{ee}$ for the present simulations is stated in the same work by Potekhin
et al. as mentioned above (Potekhin et al., 1999). For the intermediate
density (inner crust), the thermal conductivity due to neutrons also comes
into play. Also, the electron conductivity will be affected by the shape and
size of the nuclei in the crust and one should include the form factor
modifications in the collision frequency calculations (Gnedin et al., 2001).
The thermal conduction coefficient of neutrons in that region can be estimated
using the same equation as that for electron (eq. 12) by replacing the
electron number density ($n_{e}$) and kinematic mass ($m_{e}^{\ast}$) with the
corresponding neutron number density ($n_{n}$) and the neutron in-medium
effective mass ($M_{n}^{\ast}$) as defined in section 2. The effective neutron
collision frequency ($\nu_{n}$) for the said region can also be acquired in
the same way by adding the neutron-ion ($\nu_{ni}$) and neutron-neutron
collision ($\nu_{nn}$) counterparts. For the case of degenerate neutrons and
in the negligence of ion correlated collisions, the neutron-ion collision rate
will be represented by,

$\displaystyle\nu_{ni}$ $\displaystyle=$
$\displaystyle\frac{\pi\,n_{i}\,R_{n}^{2}\,k_{F_{n}}\,M_{n}^{\ast}}{\sqrt{M_{n}^{\ast\,2}c^{2}+k_{F_{n}}^{2}}},$
(13)

where $n_{i}$ is the ion number density and $R_{n}$ is the neutron radius of
atomic nuclei in the crust (Bisnovatyi-Kogan & Romanova, 1982; Potekhin et
al., 2013). The neutron-neutron collision frequency as a function of
temperature and effective self scattering cross-section ($S_{nn}$) in the
stellar medium evaluated by Shternin et al. as (Shternin et al., 2013),

$\displaystyle\nu_{nn}$ $\displaystyle=$
$\displaystyle\frac{12.8\,k_{B}^{2}\,T^{2}\,}{\hbar^{3}}\,\left(\frac{M_{n}^{\ast\,3}}{M_{n}^{2}}\right)\,S_{nn}.$
(14)

The mechanism for the in-medium cross-section was later revisited by Baiko et
al. using the Bonn potential and provided in the form of simple analytical
expression (Baiko et al., 2001b).

Figure 7: Total thermal conductivity profile of a dark matter admixed
canonical star with IOPB-I parameter set as a function of neutron star’s
density. Here thermal conductivity is shown for a fixed temperature
($10^{9}\,K$).

Total thermal conductivity of a canonical star due to electron and neutron
contribution for the considered crustal EOS (Chamel & Haensel, 2008) is
depicted in fig. 6 within the temperature range $10^{5}-10^{9}\,K$. High
thermal conductivity results in more rapid cooling scenarios, so, once the
temperature around $10^{8}\,K$ is achieved by the stellar crust, the smaller
thermal conductivity indicates the slow cooling era of the neutron star. The
thermal conductivity of electrons in the outer envelopes of the star is
suppressed up to several orders of magnitude due to the finite size atomic
nuclei scatterers and the scattering due to charged impurities is neglected in
the present calculations. Also, at a constant temperature the thermal
conductivity of the crustal part increases with an increase in density and we
observe that this effect is reversed in the core with an increment of the dark
matter percentage, as shown in fig. 7.

The heat transport in the stellar core is mainly categorized into two parts:
baryon heat conduction and lepton heat conduction. The total thermal
conductivity magnitude is the result of the solution of the multi-component
system of transport equations and the analytical expression for each type of
particle ($i$) is analogous to equation (12),

$\displaystyle\kappa_{i}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}\,k_{B}^{2}\,n_{i}\,T\,\tau_{i}}{3\,m_{i}^{\ast}},$
(15)

where $i=n,p,e^{-}\,\&\,\mu$ and the other variables have their usual meaning
as defined above. It should be noted here that for core simulations the
effective mass of leptons is determined by their dynamical chemical potential
which is controlled by the beta equilibrium condition for the stellar core.
The effective mass of the particle and the corresponding number density is
acquired using mean-field formalism for IOPB-I parameter set with different
dark matter momenta, as discussed in section 2. The formal solution of
effective relaxation times for baryons within the high dense environment using
the differential collision probability and solving the set of algebraic
equations are obtained by Shternin et al. as (Shternin & Yakovlev, 2008;
Shternin et al., 2013),

$\displaystyle\sum_{i,j=\\{p,n\\}}\frac{64\,M_{i}^{\ast}M_{j}^{\ast\,2}\,T^{2}}{5\,M_{N}^{2}\,\hbar^{3}}\,S_{ij}\,\tau_{j}=1.$
(16)

The detailed calculations for the effective cross-section ($S_{ij}$) in beta
equilibrated matter are performed in (Baiko et al., 2001b) and borrowed from
the same for present calculations. Coulomb potential is the main mechanism for
scattering in the case of leptons, and the effective relaxation time for
electrons and muons will be given by (Gnedin & Yakovlev, 1995a),

$\displaystyle\tau_{e}=\frac{\nu_{\mu}-\nu_{e\mu}^{{}^{\prime}}}{\nu_{e}\nu_{\mu}-\nu_{e\mu}^{{}^{\prime}}\nu_{\mu
e}^{{}^{\prime}}};\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\tau_{\mu}=\frac{\nu_{e}-\nu_{\mu
e}^{{}^{\prime}}}{\nu_{e}\nu_{\mu}-\nu_{e\mu}^{{}^{\prime}}\nu_{\mu
e}^{{}^{\prime}}},$ (17)

with $\nu_{i}=\sum_{j}\nu_{ij}$; where $i$ denotes the leptons ($e\,\&\,\mu$)
and $j$ stands for all the considered charged particles in the star core
($p,e,\mu$). The $\nu_{\mu e}^{{}^{\prime}}$ and $\nu_{e\mu}^{{}^{\prime}}$
are crossed collision frequencies that coupled the heat transport of electrons
and muons. The total self and crossed collision frequencies for electrons and
muons in the ultra-dense cores are calculated by solving the multidimensional
integral over colliding particles with Coulomb screening momentum ($q_{0}$)
taken into account and are expressed in the form of simple analytical
expressions (Gnedin & Yakovlev, 1995b),

$\displaystyle\nu_{ii}$ $\displaystyle=$
$\displaystyle\frac{5}{2}\frac{\beta_{i}}{\mu_{i}^{2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,(i=e,\mu),$
(18) $\displaystyle\nu_{ip}$ $\displaystyle=$
$\displaystyle\beta_{i}\,\frac{M_{p}^{\ast\,2}}{k_{F_{i}}^{4}}\,\,\,\,\,\,\,(i=e,\mu),$
(19) $\displaystyle\nu_{ij}$ $\displaystyle=$
$\displaystyle\frac{\mu_{ip}}{M_{p}^{\ast\,2}}\frac{\nu_{j}^{2}}{c^{4}}\Bigg{[}1+\Big{(}\frac{k_{F_{i}}k_{F_{j}}c^{2}}{4\mu_{i}\mu_{j}}\Big{)}^{2}\Bigg{]}(i,j=e,\mu\,;\,i\neq
j),$ (20) $\displaystyle\nu^{{}^{\prime}}_{ij}$ $\displaystyle=$
$\displaystyle\frac{\beta_{i}}{k_{F_{i}}k_{F_{j}}c^{2}}\,\,\,\,\,\,\,\,\,\,\,\,(i,j=e,\mu\,;\,i\neq
j),$ (21) with $\displaystyle\beta_{i}$ $\displaystyle=$
$\displaystyle\frac{4\pi^{2}}{5\hbar}\Bigg{(}\frac{e^{2}}{\hbar
c}\Bigg{)}^{2}\Bigg{(}\frac{k_{F_{i}}}{q_{0}}\Bigg{)}^{3}\mu_{i}(k_{B}T)^{2}\,\,\,\,\,\,\,(i=e,\mu).$

We presented here all the possible components of collision frequency with
proton being the only charged nucleon, however, in presence of other exotic
charged particles (baryon octet or deconfined quarks) there may also be other
components, whose impressions on the thermal transport can be explored in
future. It should also be noted here that we neglected the possible effects of
superfluidity and magnetic field for the time being. After implementing all
the above-listed expressions with proper factors and extracted variables from
the RMF formalism together, the results for the thermal conductivity of a
canonical star with varied dark matter momentum are illustrated in fig. 7. The
thermal conductivity curve reflects the same nature as that of EOS for the
canonical star with varying dark matter proportions. The appearance of muons
in the stellar core further enhances the metric for conductivity but does not
seem to be much influential as in the case of neutrino emissivity and specific
heat. The magnitude of thermal conductivity, in general, increases as we move
from the surface to the core of the star. However, for the high dense portion,
the increment in the dark matter percentage at a fixed density reduces the
thermal conduction of the star. On the other hand, we also observe that the
core of a dark matter concentrated canonical star is highly dense and much
more conductive than the no dark matter admixed star. In our calculations, we
noticed that at a certain ultra dense core site of dark matter admixed star
the thermal transport is greater for higher temperature, which from the stated
mathematical expressions can be alluded as the suppression of high-temperature
collision effects by the higher momentum metric of nucleons for the core.

All the above three examined properties (neutrino emissivity, specific heat
and thermal conductivity) are required for the numerical simulations related
to cooling dynamics and thermal energy transport of young neutron stars, which
are performed in the next section.

###  3.3 Cooling and Internal Thermal Equilibrium

Figure 8: The theoretically calculated curves of surface temperatures for a
dark matter admixed canonical star as function of stellar age. The
observational data of several thermal emitting sources i.e. Cas A NS (Ho &
Heinke, 2009), PSR J1119-6127 (Safi-Harb & Kumar, 2008), RX J0822-4247 (Zavlin
et al., 1999), 1E 1207.4-5209 (Pavlov et al., 2002), PSR J1357–6429 (Zavlin,
2007b), RX J0002+6246 (Page et al., 2004), PSR B0833-45 (Pavlov et al., 2001),
PSR B1706-44 (McGowan et al., 2004), PSR J0538+2817 (Zavlin & Pavlov, 2004),
PSR B2334+61 (McGowan et al., 2006), PSR J0243+2740 (Zavlin, 2007a), XMMU
J1732-344 (Ruffert & Janka, 2010) had also been illustrated with upper and
lower bounds of surface temperature and age.

The computed values of $Q$, $C_{v}$ and $\kappa$ are employed in eqs. 5 and 6
to obtain the surface and internal temperature profiles of the canonical and
maximum mass stars. Some of the earlier works which provide the solution
algorithm to these equations are (Richardson et al., 1979, 1982; Fujimoto et
al., 1984; van Riper, 1991), later developed through finite-difference Euler
and Henyey methods by several researchers in various works related to cooling
dynamics of neutron star (Gnedin et al., 2001; Potekhin et al., 2019). The key
point to solve these equations is to divide the spatial points in
infinitesimal spherical layers of thickness $dr_{i}$
($\sum_{i=1}^{N}dr_{i}=R$) and similarly discretize time coordinate $dt_{n}$
($\sum_{n=1}^{N^{{}^{\prime}}}dt_{n}=t$). Intrinsic quantities such as
internal temperature and conductivity should be defined at the spatial zone
centres, and the extrinsic quantities such as luminosity at the spatial zone
boundaries. The time steps and the spatial zones should be very thin due to
the ultra-dense and highly sensitive conductive nature of neutron star layers.
To facilitate the numerical simulations further, one can solve the eqs. 5 and
6 separately by subdividing the star into an outer thin heat blanketing
envelope ($\sim$ $100\,m$), which is composed of either heavy or light
elements depending on the model used and acts as a thermal insulator for rest
of the star. In our numerical conventions, we adopted the backward Euler
method with the stated initial and boundary conditions using the surface-
boundary temperature relations ($T_{s}-T_{b}$) of the carbon-iron (C-Fe
mixture) composed envelope (Beznogov et al., 2016, 2021). The superfluid
effects are not considered in the present study. We followed the similar
mechanism for cooling simulations as presented by Gnedin et al., and the keen
readers are advised to track the listed references and the references therein
for more detailed technical particularity (van Riper, 1991; Potekhin et al.,
1997; Gnedin et al., 2001).

Figure 9: Redshifted surface temperature profile of the maximum mass star
versus age. The curves for the dark matter admixed maximum mass star profile
with $k_{f}^{\rm DM}=0.00,0.03\,\&\,0.06$ are shown. The maximum mass of the
star decreases with an increase in the dark matter fermi momentum, but the
calculated mass for all the cases is greater than $1.4M_{\odot}$ and so the
fast cooling scenario is followed in all the cases.

The results of the numerical simulations for the effective surface temperature
($T_{s}^{\infty}$) as s function of star’s age and the redshifted interior
temperature ($\tilde{T}\equiv Te^{\phi}$) as a function of star’s density for
various stellar age profiles are exemplified in figs. 8 \- 11. The results are
shown for both canonical and maximum mass stars with the different dark matter
occupied segments as discussed in the former sections. We also confront the
theoretically obtained cooling curves with the thermal emissions of
observational pulsar data.

Figure 10: Redshifted internal temperature of a dark matter admixed canonical
star calculated with the IOPB-I parameter equation of states as a function of
density. The dark matter momentum $k_{f}^{\rm DM}$ is varied from $0.00$ GeV
to $0.06$ GeV. Each curve corresponds to a specific stellar age and the time
required for the thermal relaxation of each segment is mentioned at the
bottom. The stellar age key for the curves is presented at the right hand side
of the graph. Figure 11: Temperature profiles for the interior of a dark
matter admixed maximum mass star calculated with the IOPB-I parameter equation
of states. The red-shiftted temperature profile curves are illustrated for
three different dark matter momentum cases i.e. $k_{f}^{\rm
DM}=0.00,\,0.03\,\&\,0.06$ GeV. The stellar age key is provided at the bottom
and some of the initial time epochs are chosen similar to that of the case for
canonical star plot.

A detailed spectral analysis of several thermally emitting observed pulsars
along with the non-thermal originated neutron stars is done by Potekhin et al.
in (Potekhin et al., 2020). The authors in the same reference also discuss the
effects of the magnetic field, superfluidity and outer heat blanketing
envelope on the extracted values of age and thermal luminosities through
various techniques from the observationally recorded data by X-ray
laboratories. The cooling era of a neutron star can be recognized in three
distinct stages: initial thermal relaxation, neutrino and photon cooling
stages (Lattimer et al., 1994). Initially, the crust segment of the star
remains thermally decoupled from the core and the cooling curve reflects the
thermal emission dynamics of the crust in the initial years, which is
considerably affected by the chemical composition of the envelope (Page &
Reddy, 2006). Thus, the initial cooling stage of both the canonical and
maximum mass dark matter admixed star is independent of the EOS or core
composition and demonstrates similar cooling pattern for various considered
values of dark matter momenta. The neutrino cooling era lasts up to $10^{4}$
\- $10^{5}$ years and mainly influenced by the neutrino emissions from the
stellar core. The final cooling stage is, however, referred as the photon
emissions from the stellar surface and almost insensitive to the core segment
of the star, but during final epoch the temperature decreases rapidly due to
the overlap of various cooling scenarios and seems to be contingent on the
specific heat of the star (Page, 1998). The effects of dark matter admixture
on the photon cooling stage of a maximum mass star turn out to be minimal and
the cooling curves of the canonical star tend to approach the same region too
after $10^{5}$ years of age. This duration of the cooling era also seems to be
independent of the EOS of stellar matter obtained through mean-field formalism
for various dark matter admixed stars. We reviewed that the cooling rate of
the neutron star depends on the compositional distribution of the stellar core
and the EOS of the model used to describe the stellar structure rather than
the overall mass of the star. Mathematically, the overall luminosity and the
integrated heat capacity of the star depend on the mass of the star in the
same manner and therefore the effect of the total mass in the energy balance
equation cancels out and drives the effective surface temperature insensitive
to mass. It is quite straightforward from fig. 8 that even if we fix the mass
of the star at some constant value and vary only the internal configuration
and compositional structure, the cooling rate undergoes thorough the
transition from slow cooling to fast cooling scenario. The rapid decrease in
the surface temperature of canonical star with dark matter Fermi momentum
participation greater than $0.04$ GeV during the neutrino cooling epoch is due
to the excess neutrino production in the stellar core as a result of the
activation of direct Urca process channel for such stars. In the case of
maximum mass star, the direct Urca process is observed for all the
configurations (with or without dark matter component), and so the fast
cooling tendency is also followed in all such cases (fig. 9). The canonical
and maximum mass neutron star configuration with the larger dark matter
component turns out to be cool more rapidly as compared to smaller or no dark
matter admixed star. The small dip in the apparent surface temperature curve
for canonical star with $k_{f}^{\rm DM}<0.05$ GeV around hundred years of
stellar age is the consequence of enhanced neutrino emissivity due to the
modified Urca process and the activation of proton branch for the same. The
effect of the modified Urca process can also be noticed in the fast cooling
scenarios ($k_{f}^{\rm DM}>0.04$ GeV) around the same stellar age with further
scaling down of curves around $3\times 10^{5}\,K$ apparent surface
temperature. The slope change of the cooling curves around $10^{5}$ years
stellar age indicates the transition to the photon cooling stage.

The thermal relaxation of the dark matter admixed canonical and maximum mass
stars in terms of internal temperature profiles is illustrated in figs. 10 and
11 respectively. We observe that up to 5 years of stellar age, the inner
crust, outer crust, outer core and inner core of the neutron star form almost
decoupled type of thermal reservoirs for all the cases of dark matter
admixture, which again manifests the independency of initial cooling of a star
on the EOS and internal composition. The effect of fast thermal conduction for
the case of high dark matter momentum admixed neutron stars is also reflected
in these cooling curves. The canonical star with dark matter momentum $0.06$
GeV achieve internal thermal equilibrium in $30$ years whereas the star with
$k_{f}^{\rm DM}=0.02$ GeV took $350$ years for the outer crust and inner core
to be in the same thermal phase. Also, for the case of canonical star with
$k_{f}^{\rm DM}=0.05\,\&\,0.06$ GeV, the central core is much much colder than
the outer core and the crustal section of the star, which is the manifestation
of powerful direct Urca process in the central core of these stars. A similar
kind of scenario with much lower core’s temperature can also be noticed in all
cases of the maximum mass star profiles. The thermal conduction of a neutron
star is not strong enough to overcome the effects of Urca processes and unable
to establish a throughout thermal relaxation for several years. The
transmission of cooling waves in case of slow and fast cooling scenarios
impacts the relaxation mechanism of the crust in a drastic manner. As we
mentioned previously too (fig. 8) that a canonical star with $k_{f}^{\rm
DM}=0.05\,\&\,0.06$ GeV follow fast cooling scenario while the other admixed
stars cools down slowly. We here noticed that for the fast cooling scenario
the temperature gradient of the crustal part is almost eroded after one-tenth
of a year, while in the case of slow cooling scenario it took more than $10$
years for the crust temperature profile to dissipate gradient. A huge abrupt
drop of surface (crustal) temperature is also observed for all the fast
cooling scenarios before attaining internal equilibrium while in the case of
slow cooling structure the equilibrium is attained through a consistent
reduction. The effect of neutron dripping on the outer and inner crust
equilibrium almost vanishes after $\sim 1$ year in both the scenarios, which
means that the dominance of plasmon decay in the neutrino emissivity vanishes
and the next epoch will be controlled by the Bremsstrahlung processes.

Overall, despite the large temperature gradient between the crust and core of
high dark matter segment canonical and maximum mass stars, the internal
thermal relaxation is achieved quicker in these cases. In the case of pure
canonical star with IOPB-I parameter set it took $400$ years to attain the
thermal relaxation while for the dark matter admixed canonical star with
$k_{f}^{\rm DM}=0.06$ GeV, the same was achieved only in $30$ years of stellar
age. Similarly, for the case of maximum mass star the relaxation time of the
dark matter admixed star is half as compared to the one without any dark
matter admixture. We also realized that the thermal relaxation of a heavier
star is much quicker than the lighter one with the same chemical and
compositional structure. A pure canonical star with IOPB-I set is thermally
relaxed in 400 years while a pure maximum mass star with the same parameter
set ($2.130\,M_{\odot}$) instates the equal temperature for crust and core
only in 40 years. Similarly, there is a vast difference between the thermal
relaxation time period of a canonical and maximum mass dark matter admixed
star of $k_{f}^{\rm DM}=0.03$ GeV. We, therefore, deduce that, however, the
surface temperature of a star is almost independent its mass whereas the
internal thermal relaxation contradicts this fact and seems to be strongly
influenced by the gravitational mass of the star. The thermal relaxation time
of a star also depends considerably on the magnetic field and superfluid
effects through specific heat and thermal conductivity, which are not adopted
in the present simulations. An analytical approach to establish the
mathematical estimates between the relaxation time, crust thickness, specific
heat, thermal conductivity and radius of the star is also explored in a number
of papers (Lattimer et al., 1994; Gnedin et al., 2001) and it has been
emphasized that the crust and core have distinct relaxation times and
mechanism, which can also be verified through our present simulations.

##  4 Conclusions

We have incorporated the neutralino as a dark matter candidate to derive the
EOS for a neutron star using RMF formalism and investigated the indirect
effects of fermionic dark matter particles on various neutrino emission
processes inside a stellar matter. The threshold density for the activation of
the direct Urca process inside a canonical star with different compositional
structures has been explored. The stellar mass threshold for the direct Urca
process to operate is significantly lowered due to the presence of dark matter
admixture. The presence of an enhanced neutrino cooling mechanism sends
cooling wave from the stellar core to the crust which causes a sudden drop in
the surface temperature and mainly responsible for the fast cooling mechanism
of the star. We inferred through our theoretical simulations that these
powerful neutrino emitting reactions are greatly influenced from the dark
matter segment trapping dynamics and configuration. The two other basic
physical ingredients, specific heat and thermal conductivity, required for the
numerical modeling of cooling dynamics of a neutron star have also been
explored. The specific heat of the crustal segment of the star appears to be
independent of the dark matter inclusion while the heat capacity of the core
is interrupted and further enhanced by the presence of muons along with
electrons. The heat conduction in the crust is also less pronounced due to the
presence of dark matter candidate, however, it sight to be sensitive to the
shape and size of nuclei in the outer crust and the temperature dependency of
the thermal conductivity in the crustal segment has also been studied, which
increases up to $10^{5}$ orders of magnitude as the temperature is lowered.
The thermal relaxation of a canonical star with different admixtures of dark
matter momentum is also sensitive to the heat capacity and thermal conduction
of the stellar crust and core part, which is not explored separately in the
present calculations but the indirect effects on the fast cooling scenarios
are manifested in the similar way as depicted by Lattimer et al. (Lattimer et
al., 1994). Some of the observational limits for the surface temperature of
the canonical star also seem to be consistent with the current theoretical
simulations. The variation in the dark matter admixtures of the canonical star
vastly influences the surface temperature decrement with the stellar age,
which solidifies the fact that the cooling mechanism depends on the internal
composition and EOS too, rather than solely on the gravitational mass. We also
observed that the internal thermal relaxation time period of the star is very
sensitive to the percentage of dark matter segment and the theoretical
consideration of greater dark matter percentage enhances both the surface
temperature and the internal relaxation of the star drastically.

Apart from the present scenario, there are a number of opportunities in which
the present study can be extended. To begin with, we did not consider the
reheating mechanism of old neutron stars due to the accretion of dark matter
and heat dissipation due to the very high rotational frequency of the stars.
The energy released due to the annihilation of dark matter can expedite the
reheating dynamics which can also be responsible for the core-collapse and
gravitational bursts. The interesting possibility of the presence of strange
hyperons, phase transition to quark matter and kaons had also not been
considered. The cooper pairing, superfluidity and high magnetic field of the
neutron star can also be included in the simulations for a more practical
picture of cooling dynamics. Detailed exploration of all these possibilities
along with the dark matter inclusion is beyond the scope of present work and
may be investigated in future projects.

## Data Availability

This manuscript has no associated data and data sharing is not applicable to
this article as no data sets were generated during the current study.

## References

  * Abbott et al. (2017) Abbott B. P., et al., 2017, Phys. Rev. Lett., 119, 161101
  * Abbott et al. (2020) Abbott R., et al., 2020, ApJ, 896, L44
  * Alvarez-Ruso et al. (2013) Alvarez-Ruso L., Ledwig T., Martin Camalich J., Vicente-Vacas M. J., 2013, Phys. Rev. D, 88, 054507
  * Baer et al. (2015) Baer H., Barger V., Huang P., Mickelson D., Padeffke-Kirkland M., Tata X., 2015, Phys. Rev. D, 91, 075005
  * Baiko et al. (2001a) Baiko D. A., Potekhin A. Y., Yakovlev D. G., 2001a, Phys. Rev. E, 64, 057402
  * Baiko et al. (2001b) Baiko D. A., Haensel P., Yakovlev D. G., 2001b, A&A, 374, 151
  * Bertone et al. (2005) Bertone G., Hooper D., Silk J., 2005, Physics Reports, 405, 279–390
  * Beznogov et al. (2016) Beznogov M. V., Potekhin A. Y., Yakovlev D. G., 2016, Monthly Notices of the Royal Astronomical Society, 459, 1569
  * Beznogov et al. (2021) Beznogov M., Potekhin A., Yakovlev D., 2021, Physics Reports, 919, 1
  * Bilenky & Hošek (1982) Bilenky S., Hošek J., 1982, Physics Reports, 90, 73
  * Bisnovatyi-Kogan & Romanova (1982) Bisnovatyi-Kogan G. S., Romanova M. M., 1982, Soviet Physics - JETP, 56, 243
  * Bludman & Ruderman (1975) Bludman S. A., Ruderman M. A., 1975, ApJ, 195, L19
  * Boguta (1981) Boguta J., 1981, Physics Letters B, 106, 255
  * Bowyer et al. (1964) Bowyer S., Byram E. T., Chubb T. A., Friedman H., 1964, Science, 146, 912
  * Braaten (1991) Braaten E., 1991, Phys. Rev. Lett., 66, 1655
  * Burrows & Lattimer (1986) Burrows A., Lattimer J. M., 1986, ApJ, 307, 178
  * Carbone & Schwenk (2019) Carbone A., Schwenk A., 2019, Phys. Rev. C, 100, 025805
  * Carr (1961) Carr W. J., 1961, Phys. Rev., 122, 1437
  * Chamel & Haensel (2008) Chamel N., Haensel P., 2008, Living Reviews in Relativity, 11, 10
  * Chiu & Morrison (1960) Chiu H.-Y., Morrison P., 1960, Phys. Rev. Lett., 5, 573
  * Chiu & Salpeter (1964) Chiu H.-Y., Salpeter E. E., 1964, Phys. Rev. Lett., 12, 413
  * Cline et al. (2013) Cline J. M., Scott P., Kainulainen K., Weniger C., 2013, Phys. Rev. D, 88, 055025
  * Collins & Perry (1975) Collins J. C., Perry M. J., 1975, Phys. Rev. Lett., 34, 1353
  * Das et al. (2019) Das A., Malik T., Nayak A. C., 2019, Phys. Rev. D, 99, 043016
  * Das et al. (2020) Das H. C., Kumar A., Kumar B., Biswal S. K., Nakatsukasa T., Li A., Patra S. K., 2020, Monthly Notices of the Royal Astronomical Society, 495, 4893
  * Das et al. (2021a) Das H. C., Kumar A., Biswal S. K., Patra S. K., 2021a, Phys. Rev. D, 104, 123006
  * Das et al. (2021b) Das H. C., Kumar A., Patra S. K., 2021b, Monthly Notices of the Royal Astronomical Society, 507, 4053
  * Das et al. (2021c) Das H., Kumar A., Kumar B., Biswal S., Patra S., 2021c, Journal of Cosmology and Astroparticle Physics, 2021, 007
  * Dhiman et al. (2007) Dhiman S. K., Kumar R., Agrawal B. K., 2007, Phys. Rev. C, 76, 045801
  * Duffy & Bibber (2009) Duffy L. D., Bibber K. v., 2009, New Journal of Physics, 11, 105008
  * Dutra et al. (2012) Dutra M., Lourenço O., Sá Martins J. S., Delfino A., Stone J. R., Stevenson P. D., 2012, Physical Review C, 85
  * Dutra et al. (2014) Dutra M., et al., 2014, Phys. Rev. C, 90, 055203
  * Dutra et al. (2016) Dutra M., Lourenço O., Menezes D. P., 2016, Phys. Rev. C, 93, 025806
  * Ellis et al. (2018) Ellis J., Hektor A., Hütsi G., Kannike K., Marzola L., Raidal M., Vaskonen V., 2018, Physics Letters B, 781, 607
  * Entem et al. (2015) Entem D. R., Kaiser N., Machleidt R., Nosyk Y., 2015, Phys. Rev. C, 91, 014002
  * Epelbaum et al. (2009) Epelbaum E., Hammer H.-W., Meißner U.-G., 2009, Rev. Mod. Phys., 81, 1773
  * Fan et al. (2012) Fan Y.-z., Yang R.-z., Chang J., 2012, arXiv e-prints, p. arXiv:1204.2564
  * Fattoyev et al. (2010) Fattoyev F. J., Horowitz C. J., Piekarewicz J., Shen G., 2010, Phys. Rev. C, 82, 055803
  * Flowers (1973) Flowers E., 1973, ApJ, 180, 911
  * Flowers & Itoh (1976) Flowers E., Itoh N., 1976, ApJ, 206, 218
  * Flowers & Sutherland (1977) Flowers E. G., Sutherland P. G., 1977, Ap&SS, 48, 159
  * Friman & Maxwell (1979) Friman B. L., Maxwell O. V., 1979, ApJ, 232, 541
  * Fujimoto et al. (1984) Fujimoto M. Y., Hanawa T., Iben I. J., Richardson M. B., 1984, ApJ, 278, 813
  * Furnstahl et al. (1997) Furnstahl R., Serot B. D., Tang H.-B., 1997, Nuclear Physics A, 615, 441
  * Gambhir et al. (1990) Gambhir Y., Ring P., Thimet A., 1990, Annals of Physics, 198, 132
  * Gamow & Schoenberg (1941) Gamow G., Schoenberg M., 1941, Phys. Rev., 59, 539
  * Giacconi et al. (1962) Giacconi R., Gursky H., Paolini F. R., Rossi B. B., 1962, Phys. Rev. Lett., 9, 439
  * Giacconi et al. (1979) Giacconi R., et al., 1979, ApJ, 234, L1
  * Glen & Sutherland (1980) Glen G., Sutherland P., 1980, ApJ, 239, 671
  * Glendenning (1985) Glendenning N. K., 1985, ApJ, 293, 470
  * Glendenning & Moszkowski (1991) Glendenning N. K., Moszkowski S. A., 1991, Phys. Rev. Lett., 67, 2414
  * Gnedin & Yakovlev (1995a) Gnedin O. Y., Yakovlev D. G., 1995a, Nuclear Phys. A, 582, 697
  * Gnedin & Yakovlev (1995b) Gnedin O., Yakovlev D., 1995b, Nuclear Physics A, 582, 697
  * Gnedin et al. (2001) Gnedin O. Y., Yakovlev D. G., Potekhin A. Y., 2001, Monthly Notices of the Royal Astronomical Society, 324, 725
  * Gold (1968) Gold T., 1968, Nature, 218, 731
  * Goldman & Nussinov (1989) Goldman I., Nussinov S., 1989, Phys. Rev. D, 40, 3221
  * Greif et al. (2020) Greif S. K., Hebeler K., Lattimer J. M., Pethick C. J., Schwenk A., 2020, The Astrophysical Journal, 901, 155
  * Haensel et al. (1996) Haensel P., Kaminker A. D., Yakovlev D. G., 1996, A&A, 314, 328
  * Ho & Heinke (2009) Ho W. C. G., Heinke C. O., 2009, Nature, 462, 71
  * Hong et al. (2001) Hong D. K., Hsu S. D., Sannino F., 2001, Physics Letters B, 516, 362–366
  * Horowitz & Piekarewicz (2002) Horowitz C. J., Piekarewicz J., 2002, Phys. Rev. C, 66, 055803
  * Itoh et al. (1992) Itoh N., Mutoh H., Hikita A., Kohyama Y., 1992, ApJ, 395, 622
  * Jungman et al. (1996) Jungman G., Kamionkowski M., Griest K., 1996, Physics Reports, 267, 195–373
  * Kaminker et al. (1999) Kaminker A. D., Pethick C. J., Potekhin A. Y., Thorsson V., Yakovlev D. G., 1999, A&A, 343, 1009
  * Kantor & Gusakov (2007) Kantor E. M., Gusakov M. E., 2007, Monthly Notices of the Royal Astronomical Society, 381, 1702
  * Kaplan & Nelson (1988) Kaplan D., Nelson A., 1988, Nuclear Physics A, 479, 273
  * Kouvaris (2008) Kouvaris C., 2008, Phys. Rev. D, 77, 023006
  * Kouvaris & Tinyakov (2010) Kouvaris C., Tinyakov P., 2010, Phys. Rev. D, 82, 063531
  * Kumar et al. (2006) Kumar R., Agrawal B. K., Dhiman S. K., 2006, Phys. Rev. C, 74, 034323
  * Kumar et al. (2017) Kumar B., Singh S., Agrawal B., Patra S., 2017, Nuclear Physics A, 966, 197
  * Kumar et al. (2018) Kumar B., Patra S. K., Agrawal B. K., 2018, Phys. Rev. C, 97, 045806
  * Kumar et al. (2020) Kumar A., Das H. C., Biswal S. K., Kumar B., Patra S. K., 2020, The European Physical Journal C, 80, 775
  * Kumar et al. (2021) Kumar A., Das H., Bhuyan M., Patra S., 2021, Nuclear Physics A, 1015, 122315
  * Lalazissis et al. (1997) Lalazissis G. A., König J., Ring P., 1997, Phys. Rev. C, 55, 540
  * Lattimer (2012) Lattimer J. M., 2012, Annual Review of Nuclear and Particle Science, 62, 485
  * Lattimer et al. (1991) Lattimer J. M., Pethick C. J., Prakash M., Haensel P., 1991, Phys. Rev. Lett., 66, 2701
  * Lattimer et al. (1994) Lattimer J. M., van Riper K. A., Prakash M., Prakash M., 1994, ApJ, 425, 802
  * Levin & Ushomirsky (2001) Levin Y., Ushomirsky G., 2001, Monthly Notices of the Royal Astronomical Society, 324, 917
  * Li et al. (1997) Li Z., Mao G., Zhuo Y., Greiner W., 1997, Phys. Rev. C, 56, 1570
  * Li et al. (2012a) Li A., Huang F., Xu R.-X., 2012a, Astroparticle Physics, 37, 70
  * Li et al. (2012b) Li X., Wang F., Cheng K., 2012b, Journal of Cosmology and Astroparticle Physics, 2012, 031–031
  * Liu et al. (2005) Liu B., Guo H., di Toro M., Greco V., 2005, European Physical Journal A, 25, 293
  * Lourenço et al. (2021) Lourenço O., Frederico T., Dutra M., 2021, Dark matter component in hadronic models with short-range correlations (arXiv:2112.07716)
  * Martin (1998) Martin S. P., 1998, Advanced Series on Directions in High Energy Physics (Perspectives On Supersymmetry), 18, 1–98
  * McGowan et al. (2004) McGowan K. E., Zane S., Cropper M., Kennea J. A., Cordova F. A., Ho C., Sasseen T., Vestrand W. T., 2004, The Astrophysical Journal, 600, 343
  * McGowan et al. (2006) McGowan K. E., Zane S., Cropper M., Vestrand W. T., Ho C., 2006, The Astrophysical Journal, 639, 377
  * Mukhopadhyay & Schaffner-Bielich (2016) Mukhopadhyay P., Schaffner-Bielich J., 2016, Phys. Rev. D, 93, 083009
  * Nomoto & Tsuruta (1986) Nomoto K., Tsuruta S., 1986, ApJ, 305, L19
  * Nomoto & Tsuruta (1987) Nomoto K., Tsuruta S., 1987, ApJ, 312, 711
  * Oertel et al. (2017) Oertel M., Hempel M., Klähn T., Typel S., 2017, Rev. Mod. Phys., 89, 015007
  * Ohnishi et al. (2017) Ohnishi A., Tsubakihara K., Harada T., 2017, in Kubono S., Kajino T., Nishimura S., Isobe T., Nagataki S., Shima T., Takeda Y., eds, 14th International Symposium on Nuclei in the Cosmos (NIC2016). p. 020811, doi:10.7566/JPSCP.14.020811
  * Oppenheimer & Volkoff (1939) Oppenheimer J. R., Volkoff G. M., 1939, Phys. Rev., 55, 374
  * Page (1998) Page D., 1998, in Buccheri R., van Paradijs J., Alpar A., eds, NATO Advanced Study Institute (ASI) Series C Vol. 515, The Many Faces of Neutron Stars.. p. 539 (arXiv:astro-ph/9706259)
  * Page & Reddy (2006) Page D., Reddy S., 2006, Annual Review of Nuclear and Particle Science, 56, 327
  * Page et al. (2000) Page D., Geppert U., Zannias T., 2000, A&A, 360, 1052
  * Page et al. (2004) Page D., Lattimer J. M., Prakash M., Steiner A. W., 2004, The Astrophysical Journal Supplement Series, 155, 623
  * Page et al. (2006) Page D., Geppert U., Weber F., 2006, Nuclear Physics A, 777, 497
  * Page et al. (2011) Page D., Prakash M., Lattimer J. M., Steiner A. W., 2011, Phys. Rev. Lett., 106, 081101
  * Panotopoulos & Lopes (2017) Panotopoulos G., Lopes I., 2017, Phys. Rev. D, 96, 083004
  * Pavlov et al. (2001) Pavlov G. G., Zavlin V. E., Sanwal D., Burwitz V., Garmire G. P., 2001, The Astrophysical Journal, 552, L129
  * Pavlov et al. (2002) Pavlov G. G., Zavlin V. E., Sanwal D., Trümper J., 2002, The Astrophysical Journal, 569, L95
  * Pethick & Ravenhall (1992) Pethick C. J., Ravenhall D. G., 1992, Philosophical Transactions: Physical Sciences and Engineering, 341, 17
  * Pons et al. (1999) Pons J. A., Reddy S., Prakash M., Lattimer J. M., Miralles J. A., 1999, ApJ, 513, 780
  * Potekhin & Chabrier (2010) Potekhin A. Y., Chabrier G., 2010, Contributions to Plasma Physics, 50, 82
  * Potekhin & Chabrier (2018) Potekhin A. Y., Chabrier G., 2018, A&A, 609, A74
  * Potekhin & Yakovlev (2001) Potekhin A. Y., Yakovlev D. G., 2001, A&A, 374, 213
  * Potekhin et al. (1997) Potekhin A. Y., Chabrier G., Yakovlev D. G., 1997, A&A, 323, 415
  * Potekhin et al. (1999) Potekhin A. Y., Baiko D. A., Haensel P., Yakovlev D. G., 1999, A&A, 346, 345
  * Potekhin et al. (2013) Potekhin A. Y., Fantina A. F., Chamel N., Pearson J. M., Goriely S., 2013, A&A, 560, A48
  * Potekhin et al. (2015a) Potekhin A. Y., Pons J. A., Page D., 2015a, Space Science Reviews, 191, 239
  * Potekhin et al. (2015b) Potekhin A. Y., Pons J. A., Page D., 2015b, Space Sci. Rev., 191, 239
  * Potekhin et al. (2019) Potekhin A. Y., Chugunov A. I., Chabrier G., 2019, A&A, 629, A88
  * Potekhin et al. (2020) Potekhin A. Y., Zyuzin D. A., Yakovlev D. G., Beznogov M. V., Shibanov Y. A., 2020, Monthly Notices of the Royal Astronomical Society, 496, 5052
  * Raj et al. (2018) Raj N., Tanedo P., Yu H.-B., 2018, Phys. Rev. D, 97, 043006
  * Richardson et al. (1979) Richardson M. B., Savedoff M. P., van Horn H. M., 1979, ApJS, 39, 29
  * Richardson et al. (1982) Richardson M. B., van Horn H. M., Ratcliff K. F., Malone R. C., 1982, ApJ, 255, 624
  * Ring et al. (1997) Ring P., Gambhir Y., Lalazissis G., 1997, Computer Physics Communications, 105, 77
  * Ruffert & Janka (2010) Ruffert M., Janka H. T., 2010, A&A, 514, A66
  * Safi-Harb & Kumar (2008) Safi-Harb S., Kumar H. S., 2008, The Astrophysical Journal, 684, 532
  * Serot & Walecka (1992) Serot B. D., Walecka J. D., 1992, Relativistic Nuclear Many-Body Theory. Springer US, Boston, MA, pp 49–92, doi:10.1007/978-1-4615-3466-2_5, https://doi.org/10.1007/978-1-4615-3466-2_5
  * Shen (2002) Shen H., 2002, Phys. Rev. C, 65, 035802
  * Shternin & Yakovlev (2008) Shternin P. S., Yakovlev D. G., 2008, Phys. Rev. D, 78, 063006
  * Shternin et al. (2013) Shternin P. S., Baldo M., Haensel P., 2013, Phys. Rev. C, 88, 065803
  * Slattery et al. (1982) Slattery W. L., Doolen G. D., DeWitt H. E., 1982, Phys. Rev. A, 26, 2255
  * Thorne (1977) Thorne K. S., 1977, ApJ, 212, 825
  * Todd-Rutel & Piekarewicz (2005) Todd-Rutel B. G., Piekarewicz J., 2005, Phys. Rev. Lett., 95, 122501
  * Tolman (1939) Tolman R. C., 1939, Phys. Rev., 55, 364
  * Urpin et al. (1997) Urpin V., Konenkov D., Urpin V., 1997, MNRAS, 292, 167
  * Waldhauser et al. (1987) Waldhauser B. M., Theis J., Maruhn J. A., Stöcker H., Greiner W., 1987, Phys. Rev. C, 36, 1019
  * Walecka (1974) Walecka J., 1974, Annals of Physics, 83, 491
  * Weissenborn et al. (2012) Weissenborn S., Chatterjee D., Schaffner-Bielich J., 2012, Phys. Rev. C, 85, 065802
  * Xiang et al. (2014) Xiang Q.-F., Jiang W.-Z., Zhang D.-R., Yang R.-Y., 2014, Phys. Rev. C, 89, 025803
  * Yakovlev & Levenfish (1995) Yakovlev D. G., Levenfish K. P., 1995, A&A, 297, 717
  * Yakovlev & Urpin (1980) Yakovlev D. G., Urpin V. A., 1980, Soviet Ast., 24, 303
  * Yakovlev et al. (1999) Yakovlev D. G., Levenfish K. P., Shibanov Y. A., 1999, Physics-Uspekhi, 42, 737–778
  * Yakovlev et al. (2001) Yakovlev D., Kaminker A., Gnedin O., Haensel P., 2001, Physics Reports, 354, 1
  * Zavlin (2007a) Zavlin V. E., 2007a, in Neutron Stars and Pulsars: About 40 Years After the Discovery: 363rd Heraeus Seminar. (arXiv:astro-ph/0702426)
  * Zavlin (2007b) Zavlin V. E., 2007b, The Astrophysical Journal, 665, L143
  * Zavlin & Pavlov (2004) Zavlin V. E., Pavlov G. G., 2004, Mem. Soc. Ast. It., 75, 458
  * Zavlin et al. (1999) Zavlin V. E., Trumper J., Pavlov G. G., 1999, The Astrophysical Journal, 525, 959
  * Zhang & Chen (2016) Zhang Z., Chen L.-W., 2016, Phys. Rev. C, 94, 064326
  * de Lavallaz & Fairbairn (2010) de Lavallaz A., Fairbairn M., 2010, Phys. Rev. D, 81, 123521
  * van Riper (1991) van Riper K. A., 1991, ApJS, 75, 449
  * van Riper & Lamb (1981) van Riper K. A., Lamb D. Q., 1981, ApJ, 244, L13