# Rotating neutron stars with quark cores

Ishfaq A. Rather1<EMAIL_ADDRESS>Usuf Rahaman1 M. Imran1 H. C.
Das2,3 A. A. Usmani1 S. K. Patra2,3 1Department of Physics, Aligarh Muslim
University, Aligarh 202002, India  2Institute of Physics, Bhubaneswar 751005,
India  3 Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India

###### Abstract

The rotating neutron-star properties are studied to investigate a phase
transition to quark matter. The density-dependent relativistic mean-field
model (DD-RMF) is employed to study the hadron matter, while the vector-
enhanced bag (vBag) model is used to study the quark matter. The star matter
properties such as mass, radius, the moment of inertia, rotational frequency,
Kerr parameter, and other important quantities are studied to see their effect
on quark matter. The maximum mass of a rotating neutron star with the DD-LZ1
and DD-MEX parameter sets is found to be around 3$M_{\odot}$ for pure hadronic
phase and decreases to around 2.6$M_{\odot}$ upon phase transitioning to quark
matter, which satisfies the recent GW190814 possible maximum mass constraint,
implying that the secondary component of GW190814 could be a fast-rotating
hybrid star. For DDV, DDVT, and DDVTD parameter sets, the maximum mass
decreases to satisfy 2$M_{\odot}$. The moment of inertia calculated for
various DD-RMF parameter sets decreases with the increasing mass satisfying
constraints from various measurements. Other important quantities calculated
also vary with the bag constant and hence show that the presence of quarks
inside neutron stars can also allow us to constraint these quantities to
determine a proper equation of state. Also, the theoretical study along with
the accurate measurement of uniformly rotating neutron-star properties may
offer some valuable information concerning the high-density part of the
equation of state.

##  I Introduction

Compact objects such as neutron star (NSs) in the known universe are ideal
sources to study the properties and composition of highly dense matter. The
measurement of mass and the radius for spherically symmetric and static stars
impose constraints on the properties of matter at high density. The study of
rotating NS (RNS) properties may lead to significant new constraints. From the
past decade, the successful discoveries of various gravitational waves by LIGO
and Virgo collaborations (LVC) have allowed us to study the dense matter
properties with more constraints imposed on the NS equation of state (EoS).
The measurement of tidal deformability for static NSs ruled out many EoSs with
either too large or too small maximum mass.

The binary NS (BNS) merger event GW170817 Abbott  _et al._ (2017, 2018)
constrained the maximum mass and the tidal deformability of NSs and hence the
EoS. The total mass of the GW170817 event was around 2.7$M_{\odot}$ with the
heavier component mass 1.16$M_{\odot}$-1.60$M_{\odot}$ for low spin priors.
The maximum mass approached 1.9$M_{\odot}$ for high spin priors Abbott  _et
al._ (2019). The tidal deformability dependence on the NS radius
$\Lambda\propto R^{5}$ provided a more strong constraint on the high dense
nuclear EoS. A new gravitational wave event (GW190814) was observed recently
by LVC with a black hole merger of mass 22.2$M_{\odot}$-24.3$M_{\odot}$ and a
massive secondary component of mass 2.50$M_{\odot}$-2.67$M_{\odot}$ Abbott
_et al._ (2020a). The secondary component of GW190814 gained a lot of
attention about its nature whether it is a black hole, a NS, or some other
exotic object Dexheimer  _et al._ (2020); Tan  _et al._ (2020); Fishbach  _et
al._ (2020); Rather  _et al._ (2020a); Godzieba  _et al._ (2020); Most  _et
al._ (2020); Tan  _et al._ (2020); Zhang and Li (2020); Tsokaros  _et al._
(2020); Fattoyev  _et al._ (2020); Lim  _et al._ (2020); Tews  _et al._
(2020).

A proper knowledge of a NS maximum mass is assumed to be the most important
parameter determining the possible outcome of a BNS merger Shibata and
Taniguchi (2006); Sekiguchi  _et al._ (2011); Hotokezaka  _et al._ (2013);
Bauswein  _et al._ (2013); Palenzuela  _et al._ (2015); Bernuzzi  _et al._
(2016); Lehner  _et al._ (2016); Radice  _et al._ (2018); Köppel  _et al._
(2019). The constraints on the EoS at high density are imposed with accurate
information of a NS maximum mass and radius Hebeler  _et al._ (2010, 2013);
Lattimer (2012); Miller  _et al._ (2019); Annala  _et al._ (2020). The precise
measurement of masses of millisecond pulsars such as PSR J1614-2230
(1.928$\pm$0.017)$M_{\odot}$ Demorest  _et al._ (2010),PSR
J0348+0432(2.01$\pm$0.04)$M_{\odot}$ Antoniadis and Freire et al. (2013), and
PSR J0740+6620 (2.04${}^{+0.10}_{-0.09}$)$M_{\odot}$ Cromartie and Fonseca et
al. (2019) show that the theoretical maximum mass of a NS should be around
2$M_{\odot}$. Combining the GW observations of BNS with quasi-universal
relations, a maximum mass of $M_{max}\lesssim 2.17_{-0.15}^{+0.17}$
$M_{\odot}$ is attained for nonrotating NSs Rezzolla  _et al._ (2018). By
combining the total binary mass of GW170817 inferred from GW signal with
electromagnetic (EM) observations, an upper limit of $M_{max}\lesssim$ 2.17
$M_{\odot}$ is predicted Margalit and Metzger (2017). Further analysis
employing both energy and momentum conservations along with the numerical-
relativity simulations show that the maximum mass of cold NSs is weakly
constrained as $M_{max}\lesssim$ 2.3 $M_{\odot}$ Shibata  _et al._ (2019).
However, with the discovery of the recent secondary component of GW190814
predicting a maximum mass around 2.5$M_{\odot}$-2.67$M_{\odot}$, the maximum
mass limit for a NS seems to be weekly constrained.

The effect of the EoS on the properties of a RNS has been studied since the
late 90s by various groups Cook  _et al._ (1994); Margalit and Metzger (2017);
Stergioulas (1998); Paschalidis and Stergioulas (2017). To investigate the NS
structure and its properties, the choice of the EoS becomes the starting
point. There proper choice of EoS for NS matter invites theoretical
discussions. Every single EoS produces a NS with different properties. Despite
predicting several NS properties, the composition at several times the normal
nuclear density is still not known properly. The core of a NS is considered to
be a nuclear matter in $\beta$-equilibrium and charge-neutral conditions.
Neutron, proton, electron, and muon are the basic components of the core of a
NS. The NS structure with several exotic degrees of freedom like quarks,
kaons, and hyperons is also studied Rather  _et al._ (2020b, c, a); Ofengeim
_et al._ (2019); Sulaksono (2015). The presence of such exotic phases
significantly affects the NS properties.

NS matter containing only hadrons are studied by employing different model
parameters at high densities. Density functional theories (DFTs) have been
widely used to determine the saturation properties of high dense nuclear
matter (NM) Vautherin and Brink (1972); Shen  _et al._ (1998); Shen (2002);
Douchin, F. and Haensel, P. (2001); Bao and Shen (2014); Bao  _et al._ (2014).
At saturation density, the NM EoS is well constrained and its corresponding
properties are determined with less uncertainty. These EoSs at several times
the normal nuclear density describe the NS properties. The relativistic mean-
field (RMF) model has been very successful in describing both finite and
infinite NM Walecka (1974). The basic mechanism involves the interaction of
nucleons via mesons. Different mesons like $\rho$, $\sigma$, $\omega$, and
$\delta$ have reduced the large uncertainties present in the NM properties and
constrained the properties to well within the limits Horowitz and Piekarewicz
(2001); Sugahara and Toki (1994); Boguta and Bodmer (1977); Serot (1979);
Singh  _et al._ (2014); Kumar  _et al._ (2017a, 2018). The RMF EoS like NL3
Lalazissis  _et al._ (1997) and BigApple Fattoyev  _et al._ (2020); Das  _et
al._ (2020) determine NS with a maximum mass around 2.7$M_{\odot}$. The
density-dependent RMF (DD-RMF) model contains the density-dependent coupling
constants replacing the self- and cross-coupling of various mesons in the
basic RMF model Brockmann and Toki (1992). DD-RMF parameters like DD-ME1
Nikšić  _et al._ (2002), DD-ME2 Lalazissis  _et al._ (2005) generate very
massive NSs with a 2.3$M_{\odot}$-2.5$M_{\odot}$ maximum mass. Several new DD-
RMF parameter sets were proposed recently such as DD-LZ1 Wei  _et al._ (2020),
DD-MEX Taninah  _et al._ (2020), DDV, DDVT, and DDVTD Typel and Alvear Terrero
(2020). These recently proposed parameter sets are divided into two
categories. The DD-LZ1 and DD-MEX parameter sets produce very stiff EoSs and
hence a large NS maximum mass and belong to the stiff EoS group. Parameter
sets such as DDV, DDVT, and DDVTD produce soft EoSs and hence lie in the
softer EoS group. Both the stiff and soft EoS groups are used in the current
study to determine the NS properties for the static and rotating case.

Exotic degrees of freedom like quarks have been studied over the past decades.
The presence of quarks in the core of NSs at very high densities has been
proposed Annala  _et al._ (2020). Thus the phase transition to quark matter
(QM) inside NSs is possible at very high density Witten (1984); Farhi and
Jaffe (1984). A NS with hadrons in the core followed by a phase transition to
the QM at several times the normal nuclear density is termed the hybrid star
(HS) Glendenning (1992); Özel  _et al._ (2010); Klähn  _et al._ (2013);
Douchin, F. and Haensel, P. (2001); Bombaci  _et al._ (2016).

The MIT bag model Farhi and Jaffe (1984); Chodos  _et al._ (1974); Freedman
and McLerran (1978) was first proposed to study strange and hybrid stars. The
Nambu-Jona-Lasinio (NJL) model Kubis and Kutschera (1997); Nambu and Jona-
Lasinio (1961a, b); Klevansky (1992); Buballa (2005) was later introduced and
explained the QM more precisely than the bag model. The modified NJL models
have been very successful in explaining the stable HSs and also satisfying the
recent GW170817 constraints Li  _et al._ (2017, 2018). The modified bag model,
termed the vector-enhanced bag model (vBag) Klähn and Fischer (2015) was
introduced as an effective model to study the astrophysical processes. The
vBag model is favored over the simple bag model and NJL model because it
accounts for the repulsive vector interactions along with the dynamic chiral
symmetry breaking (D$\chi$SB). The repulsive vector interaction and the
deconfinement for the construction of a mixed-phase allowed it to describe the
strange or hybrid stars which attain the 2$M_{\odot}$ limit. Recent work by
Roupas show the secondary component of GW190814 to be a strange star in the
color-flavor locked (CFL) phase Roupas  _et al._ (2020).

In the present work, we study the properties of a RNS by considering a phase
transition from hadron matter (HM) to QM. The star matter properties such as
mass, radius, moment of inertia, and Kerr parameter are studied along with
some other important properties. The dependence of these quantities on the NS
mass is discussed. Several properties of a static star such as mass, radius,
and tidal deformability are also discussed.

This article is organized as follows: the DD-RMF model for the HM and vBag
model for QM and the phase-transition properties are discussed in Sec. (II.
The static and rotating NS structure and various properties associated with
the star matter are discussed in Sec. (III). In Sec. (IV), the parameter sets
for the NM and the saturation density properties are defined. The EoS for the
hadronic and hybrid star configurations are explained. The static and RNS
properties like mass, radius, and moment of inertia are discussed in Sec.
(IV.3). Finally, the summary and concluding remarks are given in Sec. (V).

##  II Theory and Formalism

The RMF Lagrangian involves the interaction between the nucleons through
various mesons defined as Dirac particles. The most basic and simplest RMF
Lagrangian involves the scalar-isoscalar sigma $\sigma$ and vector-isoscalar
$\omega$ mesons without any interaction among themselves Horowitz and Serot
(1981), which results in large NM incompressibility $K_{0}$ Walecka (1974).
Boguta and Bodmer included a nonlinear self-coupling of the $\sigma$ field
which lowered the value of NM incompressibility to reasonable values Boguta
and Bodmer (1977). Apart from $\sigma$, $\omega$, and $\rho$ mesons, the
addition of the scalar-isoscalar $\delta$ meson is included to study the
isovector effect on the scalar potential of the nucleon. Both NM and NS matter
properties are obtained which lie well within the limits Kubis and Kutschera
(1997); Singh  _et al._ (2014). The effective field theory motivated RMF
(E-RMF) is the extended RMF model which includes all possible self- and cross-
couplings between the mesons Furnstahl  _et al._ (1996, 1997); Kumar  _et al._
(2017a). The RMF model gained a lot of success in investigating both finite
and infinite NM properties. The various nonlinear meson coupling terms can be
replaced by the density-dependent nucleon-meson coupling constants in the
density-dependent relativistic Hartree-Fock (DD-RHF) Bouyssy  _et al._ (1987);
Brockmann (1978); Long  _et al._ (2006) and DD-RMF Brockmann and Toki (1992).
The density-dependent models take into account the nuclear medium effect
caused by the relativistic Brueckner-Hartree-Fock mode Brockmann and Toki
(1992). Unlike the RMF model, the coupling constants in the DD-RMF are
density-dependent i.e., they vary with density. The DD-RMF coupling constants
depend either on the scalar density $\rho_{s}$ or the vector density
$\rho_{B}$, but the vector density parametrizations are usually considered
which does not influence the total energy of the system.

The DD-RMF Lagrangian density is

$\begin{split}\mathcal{L}=\sum_{\alpha=n,p}\bar{\psi}_{\alpha}\Biggl{\\{}\gamma^{\mu}\Bigg{(}i\partial_{\mu}-g_{\omega}(\rho_{B})\omega_{\mu}-\frac{1}{2}g_{\rho}(\rho_{B})\gamma^{\mu}\rho_{\mu}\tau\Bigg{)}-\Bigg{(}M-g_{\sigma}(\rho_{B})\sigma-
g_{\delta}(\rho_{B})\delta\tau\Bigg{)}\Biggr{\\}}\psi_{\alpha}\\\
+\frac{1}{2}\Bigg{(}\partial^{\mu}\sigma\partial_{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2}\Bigg{)}+\frac{1}{2}\Bigg{(}\partial^{\mu}\delta\partial_{\mu}\delta-
m_{\delta}^{2}\delta^{2}\Bigg{)}-\frac{1}{4}W^{\mu\nu}W_{\mu\nu}\\\
+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}R^{\mu\nu}R_{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\rho_{\mu}\rho^{\mu},\end{split}$
(1)

where $\psi_{\alpha},(\alpha=n,p)$ denotes the neutron and proton wave-
function. $g_{\sigma},g_{\omega},g_{\rho}$, and $g_{\delta}$ are the meson
coupling constants which are density-dependent, and
$m_{\sigma},m_{\omega},m_{\rho}$ and $m_{\delta}$ are the masses for
$\sigma,\omega,\rho$ and $\delta$ mesons respectively. The tensor fields
$W^{\mu\nu}$ and $R^{\mu\nu}$ are defined as

$\begin{gathered}W^{\mu\nu}=\partial^{\mu}W^{\nu}-\partial^{\nu}W^{\mu},\\\
R^{\mu\nu}=\partial^{\mu}R^{\nu}-\partial^{\nu}R^{\mu}\\\ \end{gathered}$ (2)

The coupling constants of $\sigma$ and $\omega$ mesons for the DD-MEX, DDV,
DDVT, and DDVTD parameter sets are expressed as a fraction of the vector
density. The density-dependent coupling constants for various parametrizations
are given as

$g_{i}(\rho_{B})=g_{i}(\rho_{0})f_{i}(x),$ (3)

where the function $f_{i}(x)$ is given by

$f_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}},i=\sigma,\omega$
(4)

as a function of $x=\rho_{B}/\rho_{0}$, where $\rho_{0}$ is the nuclear matter
saturation density.

For the function $f_{i}(x)$, the number of constraint conditions defined as
$f_{i}(1)=1$,$f^{{}^{\prime\prime}}_{\sigma}(1)=f^{{}^{\prime\prime}}_{\omega}(1)$,
$f^{{}^{\prime\prime}}_{i}(0)=0$ reduce the number of free parameters from
eight to three in Eq. (4). Out of them, the first two constraints are

$a_{i}=\frac{1+c_{i}(1+d_{i})^{2}}{1+b_{i}(1+d_{i})^{2}},3c_{i}d_{i}^{2}=1$
(5)

For the isovector $\rho$ and $\delta$ mesons, the coupling constants are given
by an exponential dependence as

$g_{i}(\rho_{B})=g_{i}(\rho_{0})exp[-a_{i}(x-1)]$ (6)

For the DD-LZ1 parameter set, the coefficient $g_{i}$ is fixed at $\rho_{B}$=0
for $i=\sigma,\omega$;

$g_{i}(\rho_{B})=g_{i}(0)f_{i}(x).$ (7)

There are only four constraint conditions for $\sigma$ and $\omega$ in the DD-
LZ1 parameter set. The constraint
$f^{{}^{\prime\prime}}_{\sigma}(1)=f^{{}^{\prime\prime}}_{\omega}(1)$ is
removed, which changes the coupling constant of $\rho$ meson as

$g_{\rho}(\rho_{B})=g_{\rho}(0)exp(-a_{i}x).$ (8)

Following the Euler-Lagrange equation, we obtain the equations of motion for
nucleons and mesons.

The scalar density $\rho_{s}$, baryon density $\rho_{B}$, isovector densities
$\rho_{s3}$, and $\rho_{3}$ are defined as

$\rho_{s}=\sum_{\alpha=n,p}\bar{\psi}\psi=\rho_{sp}+\rho_{sn}=\sum_{\alpha}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{\alpha}}d^{3}k\frac{M_{\alpha}^{*}}{E_{\alpha}^{*}},$
(9)
$\rho_{B}=\sum_{\alpha=n,p}\psi^{\dagger}\psi=\rho_{p}+\rho_{n}=\sum_{\alpha}\frac{2}{(2\pi)^{3}}\int_{0}^{k_{\alpha}}d^{3}k,$
(10) $\rho_{s3}=\sum_{\alpha}\bar{\psi}\tau_{3}\psi=\rho_{sp}-\rho_{sn},$ (11)
$\rho_{3}=\sum_{\alpha}\psi^{\dagger}\tau_{3}\psi=\rho_{p}-\rho_{n}.$ (12)

The effective masses of nucleons are given as

$M_{p}^{*}=M-g_{\sigma}(\rho_{B})\sigma-g_{\delta}(\rho_{B})\delta,$ (13)

, and

$M_{n}^{*}=M-g_{\sigma}(\rho_{B})\sigma+g_{\delta}(\rho_{B})\delta$ (14)

Also,

$E_{\alpha}^{*}=\sqrt{k_{\alpha}^{2}+M_{\alpha}^{*2}},$ (15)

is the effective energy of nucleon with nucleon momentum $k_{\alpha}$. The
energy-momentum tensor determines the total energy density and the pressure
for the NM as

$\begin{gathered}\mathcal{E}_{DD}=\mathcal{E}_{H}+\mathcal{E}_{kin},\\\
P_{DD}=P_{H}+P_{kin}\\\ \end{gathered}$ (16)

where $\mathcal{E}_{H}$ and $P_{H}$ are the energy density and the pressure of
hadronic matter, which are given as

$\begin{split}\mathcal{E}_{H}=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{2}m_{\omega}^{2}\omega^{2}-\frac{1}{2}m_{\rho}^{2}\rho^{2}+\frac{1}{2}m_{\delta}^{2}\delta^{2}+g_{\omega}(\rho_{B})\omega\rho_{B}+\frac{g_{\rho}(\rho_{B})}{2}\rho\rho_{3},\\\
P_{H}=-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{2}m_{\omega}^{2}\omega^{2}+\frac{1}{2}m_{\rho}^{2}\rho^{2}-\frac{1}{2}m_{\delta}^{2}\delta^{2}-\rho_{B}\sum_{R}(\rho_{B}),\\\
\end{split}$ (17)

and $\mathcal{E}_{kin}$ and $P_{kin}$ are the energy density and pressure from
the kinetic part,

$\begin{split}\mathcal{E}_{kin}=\frac{1}{8\pi^{2}}\Big{[}k_{\alpha}E_{\alpha}^{*}\big{(}2k_{\alpha}^{2}+M_{\alpha}^{2}\big{)}+M_{\alpha}^{4}ln\frac{M_{\alpha}}{k_{\alpha}+E_{\alpha}^{*}}\Big{]},\\\
P_{kin}=\frac{1}{24\pi^{2}}\Big{[}k_{\alpha}E_{\alpha}^{*}\big{(}2k_{\alpha}^{2}-3M_{\alpha}^{2}\big{)}+3M_{\alpha}^{4}ln\frac{k_{\alpha}+E_{\alpha}^{*}}{M_{\alpha}^{*}}\Big{]}.\end{split}$
(18)

For NS matter, the $\beta$-equilibrium condition is

$\mu_{e}=\mu_{\mu}=\mu_{n}-\mu_{p}.$ (19)

where

$\begin{gathered}\mu_{\alpha=n,p}=\sqrt{k_{\alpha}^{2}+M_{\alpha}^{*2}}\\\
+\Big{[}g_{\omega}(\rho_{B})\omega+\frac{g_{\rho}(\rho_{B})}{2}\rho\tau_{3}+\sum_{R}(\rho_{B})\Big{]},\\\
\mu_{l=\mu,e}=\sqrt{k_{l}^{2}+m_{l}^{2}}.\end{gathered}$ (20)

The charge neutrality condition implies

$q_{total}=\sum_{i=n,p}q_{i}k_{i}^{3}/(3\pi^{2})+\sum_{l}q_{l}k_{l}^{3}/(3\pi^{2})=0.$
(21)

To study the phase transition from HM to QM, the vBag Klähn and Fischer (2015)
is employed which is an extension of the simple bag model Chodos  _et al._
(1974); Freedman and McLerran (1978); Farhi and Jaffe (1984). The vBag model
accounts for D$\chi$SB and also the additional repulsive vector interactions
which allow the strange stars to achieve the 2$M_{\odot}$ limit on the maximum
mass and hence satisfy the constraints from recently measured masses of
pulsars such as PSR J1614-2230 Demorest  _et al._ (2010), PSR 0348+0432
Antoniadis and Freire et al. (2013), and MSP J0740+6620 Cromartie and Fonseca
et al. (2019).

The energy density and pressure in the vBag model follow as Cierniak (2018)

$\mathcal{E}_{Q}=\sum_{f=u,d,s}\mathcal{E}_{vBag,f}-B_{dc},$ (22)
$P_{Q}=\sum_{f=u,d,s}P_{vBag,f}+B_{dc},$ (23)

where $B_{dc}$ represents the deconfined bag constant introduced which lowers
the energy per particle, thus favoring stable strange matter. The energy
density and pressure of a single quark flavor are defined as

$\mathcal{E}_{vBag,f}(\mu_{f})=\mathcal{E}_{FG,f}(\mu_{f}^{*})+\frac{1}{2}K_{\nu}n_{FG,f}^{2}(\mu_{f}^{*})+B_{\chi,f},$
(24)
$P_{vBag,f}(\mu_{f})=P_{FG,f}(\mu_{f}^{*})+\frac{1}{2}K_{\nu}n_{FG,f}^{2}(\mu_{f}^{*})-B_{\chi,f},$
(25)

where FG denotes the zero-temperature Fermi gas. The coupling constant
parameter $K_{\nu}$ results from the vector interactions and controls the
stiffness of the star matter curve Wei  _et al._ (2019). The bag constant for
a single quark flavor is denoted $B_{\chi,f}$ . The chemical potential
$\mu_{f}^{*}$ of the system is

$\mu_{f}=\mu_{f}^{*}+K_{\nu}n_{FG,f}(\mu_{F}^{*}).$ (26)

An effective bag constant is defined in the vBag model so that the phase
transition to QM occurs at the same chemical potential

$B_{eff}=\sum_{f=u,d,s}B_{\chi,f}-B_{dc}.$ (27)

The effective bag constant $B_{eff}$ is an extension to the deconfined bag
constant to allow the phase transition to occur at the same chemical
potential. This also illustrates how $B_{eff}$ can be used in two and three
flavor QM.

The charge neutrality and $\beta$-equilibrium conditions for the QM are

$\frac{2}{3}\rho_{u}-\frac{1}{2}(\rho_{d}+\rho_{s})-\rho_{e}-\rho_{u}=0,$ (28)
$\mu_{s}=\mu_{d}=\mu_{u}+\mu_{e};\mu_{u}=\mu_{e}.$ (29)

The density range over which a phase transition exists between HM and QM is
determined by beta-equilibrium and charge-neutral conditions Glendenning
(1992); Schertler  _et al._ (1999); Sharma  _et al._ (2007); Burgio  _et al._
(2002); Orsaria  _et al._ (2014). The phase transition can be either by a
local charge condition (Maxwell construction) Logoteta and Bombaci (2013) or
global charge neutrality condition (Gibbs construction) Glendenning (1992).
The global charge neutrality condition allows the HM and QM to be separately
charged, unlike the local charge-neutrality condition. In this study, we used
the Gibbs method to construct the hadron-quark phase transition. The global
charge neutrality condition follows as

$\chi\rho_{Q}+(1-\chi)\rho_{H}+\rho_{l}=0,$ (30)

where the quark volume fraction in the mixed-phase is given by
$\chi=V_{Q}/(V_{T})$ which varies from $\chi=0$ to $\chi=1$ in the pure hadron
and pure quark phases respectively. The charge densities of quarks, hadrons,
and leptons are represented by $\rho_{Q}$, $\rho_{H}$, and $\rho_{l}$
respectively.

The equations governing the mixed-phase chemical potential, pressure, energy,
and baryon density are defined as:

$\mu_{B,H}=\mu_{B,Q};\mu_{l,H}=\mu_{l,Q},$ (31)

and

$P_{H}(\mu_{B},\mu_{l})=P_{Q}(\mu_{B},\mu_{l})=P_{MP}.$ (32)
$\varepsilon_{MP}=\chi\varepsilon_{Q}+(1-\chi)\varepsilon_{H}+\varepsilon_{l},$
(33)

and

$\rho_{MP}=\chi\rho_{Q}+(1-\chi)\rho_{H}.$ (34)

The above equations determine the properties of the mixed-phase and combined
with the hadron equations generate the overall properties of the star.

##  III Neutron star structure and properties

###  III.1 Static neutron star

For a spherically symmetric, static NS (SNS), the metric element has the
Schwarzschild form ($G$ = $c$ = 1)

$ds^{2}=-e^{2\phi(r)}dt^{2}+e^{2\Lambda(r)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (35)

where the metric functions $e^{-2\phi(r)}$ and $e^{2\Lambda(r)}$ are defined
as

$e^{-2\phi(r)}=(1-\gamma(r))^{-1},$ (36) $e^{2\Lambda(r)}=(1-\gamma(r)),$ (37)

with

$\gamma(r)=2M(r)/r$ (38)

The energy-momentum tensor reduces the Einstein field equations to well-known
Tolman-Oppenheimer-Volkoff coupled differential equations given byTolman
(1939); Oppenheimer and Volkoff (1939)

$\frac{dP(r)}{dr}=-\frac{[\mathcal{E}(r)+P(r)][M(r)+4\pi
r^{3}P(r)]}{r^{2}(1-2M(r)/r)}$ (39)

and

$\frac{dM(r)}{dr}=4\pi r^{2}\mathcal{E}(r)$ (40)

where $M(r)$ represents the gravitational mass at radius $r$ with fixed
central density. The boundary conditions $P(0)=P_{c}$, $M(0)=0$ allows one to
solve the above differential equations and determine the properties of a NS.

The tidal deformability $\lambda$ is defined as the ratio of the induced
quadrupole mass $Q_{ij}$ to the external tidal field $\mathcal{E}_{ij}$ as
Hinderer  _et al._ (2010); Kumar  _et al._ (2017b)

$\lambda=-\frac{Q_{ij}}{\mathcal{E}_{ij}}=\frac{2}{3}k_{2}R^{5}$ (41)

The dimensionless tidal deformability $\Lambda$ is defined as

$\Lambda=\frac{\lambda}{M^{5}}=\frac{2k_{2}}{3C^{5}}$ (42)

where $C=M/R$ is the compactness parameter and $k_{2}$ is the second Love
number. The expression for the Love number is written as Hinderer  _et al._
(2010)

$\begin{split}k_{2}=\frac{8}{5}(1-2C)^{2}[2C(y-1)]\Bigl{\\{}2C(4(y+1)C^{4}+(6y-4)C^{3}\\\
+(26-22y)C^{2}+3(5y-8)C-3y+6)\\\
-3(1-2C)^{2}(2C(y-1)-y+2)log\Big{(}\frac{1}{1-2C}\Big{)}\Bigr{\\}}^{-1}.\end{split}$
(43)

The function $y=y(R)$ can be computed by solving the differential equation
Kumar  _et al._ (2017b); Hinderer (2008)

$r\frac{dy(r)}{dr}+y(r)^{2}+y(r)F(r)+r^{2}Q(r)=0,$ (44)

where

$F(r)=\frac{r-4\pi r^{3}[\mathcal{E}(r)-P(r)]}{r-2M(r)},$ (45)
$\begin{split}Q(r)=\frac{4\pi
r\Big{(}5\mathcal{E}(r)+9P(r)+\frac{\mathcal{E}(r)+P(r)}{\partial
P(r)/\partial\mathcal{E}(r)}-\frac{6}{4\pi r^{2}}\Big{)}}{r-2M(r)}\\\
-4\Bigg{[}\frac{M(r)+4\pi
r^{3}P(r)}{r^{2}(1-2M(r)/r)}\Bigg{]}^{2}.\end{split}$ (46)

The above equations are solved for spherically symmetric and static NS to
determine the properties like mass, radii, and tidal deformability.

###  III.2 Rotating neutron star

For a rapidly rotating NSs with a nonaxisymmetric configuration, they would
emit gravitational waves until they achieve axisymmetric configuration. The
rotation deforms the NS. Here we study the rapidly rotating NS assuming a
stationary, axisymmetric space-time. The energy-momentum tensor for such a
perfect fluid describing the matter is given by

$T^{\mu\nu}=(\mathcal{E}+P)u^{\mu}u^{\nu}+Pg^{\mu\nu},$ (47)

where the first term represents the contribution from matter. $u^{\mu}$
denotes the fluid-four-velocity, $\mathcal{E}$ is the energy density, and $P$
is the pressure. For RNS, the metric tensor is given by Butterworth and Ipser
(1976); Friedman  _et al._ (1986, 1989)

$\begin{gathered}ds^{2}=-e^{2\nu(r,\theta)}dt^{2}+e^{2\psi(r,\theta)}(d\phi-\omega(r)dt)^{2}\\\
+e^{2\mu(r,\theta)}d\theta^{2}+e^{2\lambda(r,\theta)}dr^{2},\\\
\end{gathered}$ (48)

where the gravitational potentials $\nu$, $\mu$, $\psi$, and $\lambda$ are the
functions of $r$ and $\theta$ only. The Einstein’s field equations are solved
for the given potential to determine the physical properties that govern the
structure of the RNS. Global properties like gravitational mass, equitorial
radius, moment of inertia, angular momentum and quadrupole moment are
calculated.

For a RNS, the angular momentum $J$ is easy to calculate. By defining the
angular velocity of the fluid relative to a local inertial frame,
$\bar{\omega}(r)=\Omega-\omega(r)$, $\bar{\omega}$ satisfies the following
differential equation

$\frac{1}{r^{4}}\frac{d}{dr}\Bigg{(}r^{4}j\frac{d\bar{\omega}}{dr}\Bigg{)}+\frac{4}{r}\frac{dj}{r}\bar{\omega}=0,$
(49)

where $j=j(r)=e^{-(\nu+\lambda)/2}$.

The angular momentum of the star is then given by the relation

$J=\frac{1}{6}R^{4}\Bigg{(}\frac{d\bar{\omega}}{dr}\Bigg{)}_{r=R},$ (50)

which relates the angular velocity as

$\Omega=\bar{\omega}(R)+\frac{2J}{R^{3}}.$ (51)

The moment of inertia defined by $I=J/\Omega$, is given by Lattimer and
Prakash (2000); Worley  _et al._ (2008)

$I\approx\frac{8\pi}{3}\int_{0}^{R}(\mathcal{E}+P)e^{-\phi(r)}\Big{[}1-\frac{2m(r)}{r}\Big{]}^{-1}\frac{\bar{\omega}}{\Omega}r^{4}dr,$
(52)

The properties of a RNS are calculated by using the RNS code Stergioulas
(2003); Stergioulas and Friedman (1995); Stergioulas (1996); Douchin, F. and
Haensel, P. (2001).

##  IV Results and Discussions

###  IV.1 Parameter sets

To determine the properties of SNSs and RNSs, we used several recent DD-RMF
parametrizations such as DD-MEX Taninah  _et al._ (2020), DD-LZ1 Wei  _et al._
(2020), and DDV, DDVT, DDVTD Typel and Alvear Terrero (2020). Apart from the
basic DD-MEX and DD-LZ1 parameter sets, the DDV, DDVT, and DDVTD sets include
the necessary tensor couplings of the vector mesons to nucleons.

Table 1 shows the necessary nucleon masses, meson masses, and the coupling
constants of the parameter sets used. The meson coupling constants in the DD-
LZ1 parameter set are the values at zero density while for the other parameter
sets, these coupling constants are obtained at the nuclear matter saturation
density $\rho_{0}$.

Table 1: Nucleon and meson masses and different coupling constants for various DD-RMF parameter sets.  | DD-LZ1 | DD-MEX | DDV | DDVT | DDVTD
---|---|---|---|---|---
$m_{n}$ | 938.9000 | 939.0000 | 939.5654 | 939.5654 | 939.5654
$m_{p}$ | 938.9000 | 939.0000 | 938.2721 | 938.2721 | 938.2721
$m_{\sigma}$ | 538.6192 | 547.3327 | 537.6001 | 502.5986 | 502.6198
$m_{\omega}$ | 783.0000 | 783.0000 | 783.0000 | 783.0000 | 783.0000
$m_{\rho}$ | 769.0000 | 763.0000 | 763.0000 | 763.0000 | 763.0000
$g_{\sigma}(\rho_{0})$ | 12.0014 | 10.7067 | 10.1369 | 8.3829 | 8.3793
$g_{\omega}(\rho_{0})$ | 14.2925 | 13.3388 | 12.7704 | 10.9871 | 10.9804
$g_{\rho}(\rho_{0})$ | 15.1509 | 7.2380 | 7.8483 | 7.6971 | 8.0604
$a_{\sigma}$ | 1.0627 | 1.3970 | 1.2099 | 1.2040 | 1.1964
$b_{\sigma}$ | 1.7636 | 1.3350 | 0.2129 | 0.1921 | 0.1917
$c_{\sigma}$ | 2.3089 | 2.0671 | 0.3080 | 0.2777 | 0.2738
$d_{\sigma}$ | 0.3799 | 0.4016 | 1.0403 | 1.0955 | 1.1034
$a_{\omega}$ | 1.0592 | 1.3926 | 1.2375 | 1.1608 | 1.1693
$b_{\omega}$ | 0.4183 | 1.0191 | 0.0391 | 0.04460 | 0.0264
$c_{\omega}$ | 0.5386 | 1.6060 | 0.0724 | 0.0672 | 0.0423
$d_{\omega}$ | 0.7866 | 0.4556 | 2.1457 | 2.2269 | 2.8062
$a_{\rho}$ | 0.7761 | 0.6202 | 0.3326 | 0.5487 | 0.5579

Table 2 displays the NM properties such as symmetry energy, incompressibility,
and slope parameter at saturation density for various DD-RMF parameter sets.
The $E/A$ for all the parameter sets lies well around -16 MeV. The value of
$J$ lies in the range $\approx$ 31-34 MeV which is compatible with the
measurement from various astrophysical observations, $J=31.6\pm 2.66$ MeV Li
and Han (2013). The $L$ value lies outside the constraints $L=59.57\pm
10.06$MeV for the DD-LZ1 parameter set, while others satisfy this constraint
properly Zhang  _et al._ (2020); Danielewicz and Lee (2014). The proton and
neutron effective masses are very large for the DDVT and DDVTD parameter sets
as compared with the DDV set.

Table 2: NM properties binding energy ($E/A$), incompressibility ($K_{0}$), symmetry energy ($J$), slope parameter ($L$) at saturation density for various DD-RMF parameter sets.  Parameter | DD-LZ1 | DD-MEX | DDV | DDVT | DDVTD
---|---|---|---|---|---
$\rho_{0}(fm^{-3})$ | 0.158 | 0.152 | 0.151 | 0.154 | 0.154
$E/A$(MeV) | -16.126 | -16.140 | -16.097 | -16.924 | -16.915
$K_{0}$(MeV) | 231.237 | 267.059 | 239.499 | 239.999 | 239.914
$J$(MeV) | 32.016 | 32.269 | 33.589 | 31.558 | 31.817
$L$(MeV) | 42.467 | 49.692 | 69.646 | 42.348 | 42.583
$M_{n}^{*}/M$ | 0.558 | 0.556 | 0.586 | 0.667 | 0.667
$M_{p}^{*}/M$ | 0.5582 | 0.556 | 0.585 | 0.666 | 0.666

###  IV.2 Equation of State

Figure 1: (color online) EoS profile for DD-LZ1, DD-MEX, DDV, DDVT, and DDVTD
parameter sets. The recent combined constraints from GW170817 Abbott  _et al._
(2018) and GW190814 Abbott  _et al._ (2020a) are also shown Abbott  _et al._
(2020b).

Figure 1 displays the various EoS for various DD-RMF parameter sets for a NS
in beta-equilibrium and charge-neutrality conditions. The DDVTD parameter set
produces the stiffest EoS at low densities and softest EoS at high density as
compared with other parameter sets. DDV and DDVT sets produce soft EoS at high
densities which represent a NS with small maximum mass. The DD-LZ1 and DD-MEX
parameter sets produce stiff EoSs at high densities and hence larger NS
maximum masses. The recently combined constraints from the gravitational wave
data GW170817 and GW190814 in the shaded region are adopted from Ref. Abbott
_et al._ (2020a). This joint constraint was introduced by considering the
GW190814 event as neutron star-black hole (NSBH) merger, with its secondary
component assumed to be a NS. For this scenario, the maximum mass was assumed
to be not less than secondary component of GW190814, which constraints the
distribution of EoSs compatible with astrophysical data. For a unified EoS,
the Baym-Pethick-Sutherland (BPS) EoS Baym  _et al._ (1971) is used for the
outer crust part which lies in the density region 104-1011 g/cm3. Since the
outer crust EoS does not effect the NS maximum and the radius, therefore the
it is chosen for the outer crust part of the NS for all parameter sets. The
inner crust EoS has a high impact on the NS radius, $R_{1.4M_{\odot}}$ at the
canonical mass, while a small change is seen in the maximum mass and radius
Rather  _et al._ (2021). For the parameter sets used in this work, the inner
crust EoS is not available. Thus, we have employed the DD-ME2 inner crust EoS
Lalazissis  _et al._ (2005) for all the parameter sets but with matching
symmetry energy and slope parameter Pais and Providência (2016); Grill  _et
al._ (2014).

For the mixed-phase HM and QM, the Gibbs construction method, which
corresponds to the global charge neutrality between two different phases, has
been employed. The effective bag model with an effective bag constant
$B^{1/4}$ is used to study the QM. The coupling constant parameter $K_{\nu}$
is fixed at 6 GeV-2 for the three flavor configuration. Three different values
of effective bag constant are used $B_{eff}^{1/4}$=130, 145, and 160 MeV.

Figure 2: (color online) Equation of state for the hadron-quark phase
transition for DDV, DDVT, DDVTD, DD-LZ1, and DD-MEX hadronic parameter sets
and vBag QM at different effective bag constants. The solid lines represents
the hybrid EoS at $B_{eff}^{1/4}$=130 MeV while dashed and dot-dashed lines
represent the hybrid EoS at $B_{eff}^{1/4}$=145 & 160 MeV, respectively.

Figure 2 shows the hadron-quark phase transition with DD-RMF parameter sets
for hadronic matter and vBag model for QM using the Gibbs method for
constructing mixed-phase which ensures a smooth transition between the two
different phases. With the increasing effective bag constant $B_{eff}^{1/4}$,
the phase transition density increases, and the mixed-phase region also
expands. For bag constant $B_{eff}^{1/4}$=130 MeV, the mixed-phase region
starts from $\rho=2.47\rho_{0}$ and extends up to 4.03 $\rho_{0}$. For
$B_{eff}^{1/4}$=145 and 160 MeV, the mixed-phase region lies in the density
range (3.03-4.82) $\rho_{0}$ and (3.69-5.31)$\rho_{0}$, respectively. DD-LZ1
and DD-MEX parameter sets produce stiff EoS and thus the mixed-phase region
lies in a higher pressure region than the DDV, DDVT, and DDVTD parameter sets.
The mixed-phase region in the DD-LZ1 parameter sets lies in the density range
(2.56-4.23)$\rho_{0}$ for 130 MeV, (2.73-4.95)$\rho_{0}$ for 145 MeV ,and
(3.04-5.43)$\rho_{0}$ for 160 bag constants. Thus, DD-LZ1 and DD-MEX sets
predict a large mixed-phase region as compared with the other parameter sets.

###  IV.3 Neutron star properties

Figure 3: (color online) Mass vs Radius profiles for pure DD-LZ1, DD-MEX, DDV,
DDVT, and DDVTD parameters for a static NS. The recent constraints on mass
from the various gravitational wave data and the pulsars (shaded region)
Abbott  _et al._ (2020a); Demorest  _et al._ (2010); Antoniadis and Freire et
al. (2013); Cromartie and Fonseca et al. (2019) and radii Miller  _et al._
(2019); Riley  _et al._ (2019) are also shown.

Figure 3 displays the hadronic mass vs radius curves for DD-LZ1, DD-MEX, DDV,
DDVT, and DDVTD parameter sets. The DD-LZ1 set produces a NS with a maximum
mass of 2.55$M_{\odot}$ and with a radius of 12.30 km. DD-MEX set produces a
2.57$M_{\odot}$ NS with a 12.46 km radius. Both these parameter sets satisfy
the constraints from recent gravitational wave data GW190814 and recently
measured mass and radius of PSR J0030+0451, $M=1.34_{-0.16}^{+0.15}M_{\odot}$
and $R=12.71_{-1.19}^{+1.14}$ km by NICER Miller  _et al._ (2019); Riley  _et
al._ (2019). The DDV, DDVT, and DDVTD predict a maximum mass of
1.95$M_{\odot}$, 1.93$M_{\odot}$, and 1.85$M_{\odot}$ for a static NS with
12.11, 11.40, and 11.33 km radius at canonical mass, $R_{1.4}$, respectively.
DDV and DDVT satisfy the mass constraint from PSR J1614-2230 and radius
constraint from PSR J0030+0451. The DDVTD parameter set produces a NS with a
slightly lower maximum mass than PSR J1614-2230. The shaded regions display
the constraints on the maximum mass of a NS from PSR J1614-2230
(1.928$\pm$0.017$M_{\odot}$)Demorest  _et al._ (2010), PSR J0348+0432
(2.01$\pm$0.04$M_{\odot}$) Antoniadis and Freire et al. (2013), MSP J0740+6620
(2.14${}^{+0.10}_{-0.09}$$M_{\odot}$) Cromartie and Fonseca et al. (2019), and
GW190814 (2.50-2.67 $M_{\odot}$) Abbott  _et al._ (2020a).

Figure 4: (color online) Mass-Radius profile for pure hadronic and hybrid
rotating NSs for (a) DD-LZ1 and (b) DD-MEX parameter sets at bag values
$B_{eff}^{1/4}$=130, 145 & 160 MeV. The shaded regions represent recent
constraints on the mass from various measured astronomical observables.

The RNS mass-radius profile for DD-LZ1 and DD-MEX parameter sets are shown in
Fig. 4. The solid lines represent the pure hadronic star while the dashed
lines represent the HS at different bag constants. The effective bag constant
$B_{eff}^{1/4}$ is written as $B^{1/4}$ for convenience. The DD-LZ1 EoS
produces a pure hadronic RNS with a maximum mass of 3.11$M_{\odot}$ with a
radius of 18.23 km. With the phase transition from HM to QM, the maximum mass
and the corresponding radius decrease with the increase in the bag constant.
For the DD-LZ1 set, the maximum mass decreases from 3.11$M_{\odot}$ to
2.98$M_{\odot}$ for $B^{1/4}$=130 MeV, and to 2.75$M_{\odot}$ and
2.64$M_{\odot}$ for $B^{1/4}$=145 and 160 MeV, respectively. The radius
$R_{1.4}$ decreases from 18.32 km for pure HM to 16.64 km for hybrid star
matter at 160 MeV bag value. Similarly, for the DD-MEX parameter set, the
maximum mass for pure hadronic matter is 3.15$M_{\odot}$ at radius 16.53 km
which reduces to 2.69$M_{\odot}$ at 16.63 km for bag constant of 160 MeV.
Thus, while the pure hadronic RNS predict a large maximum mass, the phase
transition to QM lowers the maximum mass and the radius thereby satisfying the
maximum mass constraint from GW190814. These results imply that the secondary
component of GW190814 could be a possible fast-rotating hybrid star.

Figure 5: (color online) Same as Fig. 4, but for (a) DDV, (b) DDVT, and (c)
DDVTD EoSs.

Figure 5 displays the mass-radius relation for hadronic and hybrid rotating NS
with DDV, DDVT, and DDVTD EoSs. The maximum mass for a RNS with DDV EoS is
2.37$M_{\odot}$ with a 17.41 km radius at the canonical mass. Both the maximum
mass and the radius decrease to 2.23$M_{\odot}$, 2.13$M_{\odot}$,
2.01$M_{\odot}$ and 16.91, 16.68, 16.13 km for bag constants $B^{1/4}$=130,
145, and 160 MeV, respectively, thereby satisfying the 2$M_{\odot}$
constraint. For DDVT, the maximum mass reduces from 2.28$M_{\odot}$ to
1.99$M_{\odot}$. $R_{1.4}$ also decreases from 17.82 km to 16.01 km. Similarly
for the DDVTD EoS, the RNS maximum mass reduces to 1.93$M_{\odot}$ from
2.21$M_{\odot}$ at $B^{1/4}$=160 MeV. For all the parameter sets, the phase
transition to QM lowers the maximum mass which satisfies the 2$M_{\odot}$
limit.

The measurement of the NS moment of inertia is important because it follows a
universal relation with the tidal deformability and the compactness of a NS.
The moment of inertia as a function of gravitational mass for the RNS is
displayed in Fig. 6. The constraint on the moment of inertia obtained from the
joint PSR J0030+0451, GW170817, and the nuclear data analysis predicting
$I_{1.4}=1.43_{-0.13}^{+0.30}\times 10^{38}$ kg.m2 is given in Ref. Jiang  _et
al._ (2020). The predicted moment of inertia of pulsar PSR J0737-3093A,
$I_{1.338}=1.36_{-0.32}^{+0.15}\times 10^{45}$ g.cm2 is also given Landry  _et
al._ (2020). For pure hadronic matter, DD-LZ1 and DD-MEX EoSs predicts an NS
with a moment of inertia 2.22 and 2.35 $\times$ 1045 g.cm2, respectively. The
phase transition to the QM reduces the moment of inertia to a value 1.65 and
1.93$\times$ 1045 g.cm2 for DD-LZ1 and DD-MEX parameter sets at bag constant
$B^{1/4}$=160 MeV, which satisfies the constraint from Refs. Jiang  _et al._
(2020); Landry  _et al._ (2020); Lim  _et al._ (2019).

Figure 6: (color online) Moment of inertia variation with the gravitational
mass for (a) DD-LZ1 and (b) DD-MEX EoSs. The constraints on canonical moment
of inertia are also shown Lim  _et al._ (2019). The constraint from joint PSR
J0030+0451, GW170817, and the nuclear data analysis are shown by green bar
Jiang  _et al._ (2020). The predicted moment of inertia of pulsar J0737-3039A
using Bayesian analysis of nuclear EoS is shown by brown bar Landry  _et al._
(2020).  Figure 7: (color online) Same as Fig. 6 but for (a) DDV, (b) DDVT,
and (c) DDVTD parameter sets. The constraints on the moment of inertia of MSPs
obtained from universal relations with GW170817 are shown Kumar and Landry
(2019).

Figure 7 displays the moment of inertia variation with the gravitational mass
for DDV, DDVT, and DDVTD parameter sets. The solid lines represent the pure
hadronic matter, while the dashed lines represent the hadron-quark mixed phase
at bag constants $B^{1/4}$=130, 145, and 160 MeV. The constraints on the
moment of inertia obtained from millisecond pulsars (MSPs) with GW170817
universal relations are shown in Ref. Kumar and Landry (2019). For the DDV
EoS, the moment of inertia of a pure hadronic star is found to be 2.01$\times$
1045 g.cm2 while for the DDVT and DDVTD EoSs, the value is found to be 1.95
and 1.88 $\times$ 1045 g.cm2, respectively. For the hybrid EoS, the moment of
inertia is lowered to a value of 1.71$\times$ 1045 g.cm2 for the DDV set at
bag constant 160 MeV. For DDVT and DDVTD sets, this value reduces to 1.68 and
1.64$\times$ 1045 g.cm2 respectively for a 160 MeV bag constant. The phase
transition to QM produces a NS with the moment of inertia that satisfies the
constraints from various measurements.

For a static NS, the maximum mass is usually determined as the first maximum
of a $M$-$\varepsilon_{c}$ curve, i.e., $\partial
M/\partial\varepsilon_{c}$=0, where $\varepsilon_{c}$ is the central energy
density. For RNSs, the situation becomes complicated. To determine the
axisymmetric instability points, several methods have been used in the
literature. Friedman et al. Friedman  _et al._ (1988) described a method to
determine the points at which instability is reached in rotating NSs Sorkin
(1981, 1982).

$\Bigg{|}\frac{\partial
M(\varepsilon_{c},J)}{\partial\varepsilon_{c}}\Bigg{|}_{J=constant}=0,$ (53)

where $J$ is the angular momentum of the star, which is obtained self-
consistently in the solution of the Einstein’s equation for a rotating NS.
Once the secular instability is initiated, the star evolves until it reaches a
point of dynamical instability where the gravitational collapse starts
Stergioulas (2003). The maximum mass of the rotating star lies at the
termination point of uniformly rotating star.

The above equation defining an upper limit on the mass at a given angular
momentum is sufficient but not a necessary condition for the instability. The
limit on the dynamic instability is shown in Ref. Takami  _et al._ (2011).

Figure 8: (color online) Gravitational mass versus central density for (a) DD-
LZ1 and (b) DD-MEX EoSs. The solid lines represent pure hadronic rotating
stars while dashed lines represent the hybrid stars at different bag
constants. Figure 9: (color online) Same as Fig. 8 but for (a) DDV, (b) DDVT,
and (c) DDVTD EoSs.

Figure 8 shows the variation in the gravitational mass of a rotating NS with
the central density for DD-LZ1 and DD-MEX parameter sets. Figure 9 repreents
the same for DDV, DDVT, and DDVTD parameter sets. The maximum mass of
3.11$M_{\odot}$ for DD-LZ1 EoS is produced at a density of 1.40$\times$1015
g/cm3. The phase transition to QM at bag constant $B^{1/4}$=160 MeV reduces
the maximum mass to 2.64$M_{\odot}$ at 1.17$\times$1015 g/cm3 energy density.
For the DD-MEX parameter set, the maximum mass of 3.15$M_{\odot}$ occurs at
1.47$\times$1015 g/cm3 reduces to 2.69$M_{\odot}$ at 1.25$\times$1015 g/cm3.

A star rotating at a Keplerian rate becomes unstable due to the loss of mass
from its surface. The mass shedding limit angular velocity which is the
maximum angular velocity of a rotating star is the Keplerian angular velocity
evaluated at the equatorial radius $R_{e}$, i.e., $\Omega_{K}^{J\neq
0}=\Omega_{orb}(r=R_{e}).$

Figure 10: (color online) Variation of rotational frequency with the NS
gravitational mass at Keplerian velocity for (a) DD-LZ1 and (b) DD-MEX EoSs.
Solid lines represent a pure hadronic star while the dashed lines represent a
hybrid star at bag constants $B^{1/4}$=130, 145, and 160 MeV. The vertical
lines represent the observational limits imposed on the frequency from rapidly
rotating pulsars like PSR B1937+21 ($\nu_{k}$=633 Hz) Backer  _et al._ (1982),
PSR J1748-2446ad ($\nu_{k}$=716 Hz) Hessels  _et al._ (2006), and XTE
J1739-285($\nu_{k}$=1122 Hz) Kaaret  _et al._ (2007).

Figures 10a and 10b display the NS gravitational mass as a function of the
Kepler frequency $\nu_{k}$ for the DD-LZ1 and DD-MEX EoSs, respectively. The
limits imposed on the rotational frequency by various pulsars such as PSR
B1937+21 ($\nu_{k}$=633 Hz) Backer  _et al._ (1982), PSR J1748-2446ad
($\nu_{k}$=716 Hz) Hessels  _et al._ (2006), and XTE J1739-285 ($\nu_{k}$=1122
Hz) Kaaret  _et al._ (2007) are also shown. For the DD-LZ1 EoS, the pure
hadronic star rotates with a maximum frequency of 1525 Hz. For a HS at bag
value $B^{1/4}$=130 MeV, the star rotates with a frequency of 1405 Hz. For bag
values of 145 and 160, the frequency obtained is 1431 and 1497 Hz,
respectively. Similarly for DD-MEX EoS, the maximum rotational frequency for a
pure hadronic star is found to be 1503 Hz, which changes to 1361 Hz at a bag
constant of 130 MeV, 1408 and 1438 Hz for the HS at 145 and 160 MeV bag
values. Both pure hadronic and HSs rotate at a frequency greater than
$\nu_{k}$=1122 Hz. Also, the hybrid star $M$-$\nu_{k}$ curves coincide with
the pure hadronic curves upto $\nu_{k}\approx$ 400 Hz, which then show a
transition towards higher frequency depending upon the bag constant.

Figure 11: (color online) Same as Fig. 10 but for (a) DDV, (b) DDVT, and (c)
DDVTD EoSs.

Figure 11 displays the same gravitational mass variation with the Kepler
frequency for DDV, DDVT, and DDVTD parameter sets. For the DDV set, the pure
hadronic star rotates with a rotational frequency of 1498 Hz. The HSs produced
with bag constants $B^{1/4}$=130, 145, and 160 MeV have a rotational frequency
of 1454, 1446, and 1520 Hz, respectively. Similarly for DDVT and DDVTD EoSs,
the pure hadronic star has a rotational frequency of 1473 and 1418 Hz
respectively which then changes to 1503 and 1456 Hz respectively for HS at 160
MeV bag constant. Thus it is seen that HSs with a hadron-quark phase
transition initially produce a low mass NS with a low rotating frequency than
the pure hadronic star at low bag constant ($B^{1/4}$=130 MeV). Thus the HSs
can withstand higher rotation as the star is denser and has low maximum mass
as compared with the pure hadronic star.

A useful parameter to characterize the rotation of a star is the ratio of
rotational kinetic energy $T$ to the gravitational potential energy $W$,
$\beta=T/W$. For a RNS, if $\beta>\beta_{d}$, where $\beta_{d}$ is the
critical value, the star will be dynamically unstable. The critical value
$\beta_{d}$ for a rigidly rotating star is found to be 0.27 McNall (1970);
Tohline  _et al._ (1985). However, for different angular-momentum
distributions, the value lies in the range 0.14 to 0.27 Pickett  _et al._
(1996); Imamura  _et al._ (1995); Centrella  _et al._ (2001).

The variation in the $T/W$ ratio of the pure hadron and HS with the
gravitational mass is shown in Fig. (12). The $T/W$ ratio for pure hadronic
stars is 0.147 and 0.145 for DD-LZ1 and DD-MEX parameter sets, respectively.
The HSs have large $T/W$ ratio and increase with bag constant. For DD-LZ1 set,
the ratio increases from 0.150 at $B^{1/4}$=130 MeV to 0.153 at $B^{1/4}$=160
MeV. For the DD-MEX set, the ratio increases to 0.149 and 0.151 for bag values
130 and 160 MeV, respectively. The large value of the $T/W$ ratio in HSs is
since the quark stars being bound by the strong interaction, unlike hadron
stars which are bound by gravity.

Figure 12: (color online) Variation in the ratio of rotational kinetic energy
to the gravitational potential energy $T/W$ with gravitational mass for (a)
DD-LZ1 and (b) DD-MEX EoSs. Solid lines represent pure hadronic stars while
the dashed lines represent hybrid stars at bag constants $B^{1/4}$=130, 145,
and 160 MeV. Figure 13: (color online) Same as Fig. 12 but for (a) DDV, (b)
DDVT, and (c) DDVTD EoSs.

Figure 13 depicts the $T/W$ variation with the gravitational mass for DDV,
DDVT, and DDVTD parameter sets. For the DDV EoS, the pure hadronic star
predicts a $T/W$ ratio of 0.127, which lies below the critical value
$\beta_{d}$. For hybrid stars, this ratio increases 0.142 for a bag constant
of 160 MeV thereby satisfying the critical $\beta_{d}$ limit and hence becomes
dynamically unstable and emits gravitational waves. Similarly, for DDVT and
DDVTD EoS, the pure hadron star produces a ratio of 0.115 and 0.108 while the
HS at $B^{1/4}$=160 MeV gives a value of 0.127 and 0.125, respectively.

The Einstein’s field equations provide Kerr space-time for so-called Kerr
black holes which can be fully described by the angular momentum $J$ and
gravitational mass $M$ of rotating black holes Lo and Lin (2011); Cipolletta
_et al._ (2015). The condition $J\geq GM^{2}/c$ must be satisfied to define a
stable Kerr black hole. The gravitational collapse of a massive RNS
constrained to angular-momentum conservation creates a black hole with mass
and angular momentum resembling that of a NS. Thus, it is an important
quantity used in the study of black holes as well as RNSs. The Kerr parameter
leads to the possible limits on the compactness of a NS and also can be an
important criterion for determining the final fate of the collapse of a
rotating compact star Koliogiannia and Moustakidis (2020); Lo and Lin (2011).
The Kerr parameter is described by the relation

$\kappa=\frac{cJ}{GM^{2}}$ (54)

where $J$ is the angular momentum and $M$ is the gravitational mass of the
rotating NS. The Kerr parameter for black holes is an important and
fundamental quantity with a maximum value of 1, but it is important for other
compact stars as well.

Figure 14: (color online) Kerr parameter $\kappa$ as a function of
gravitational mass for (a) DD-LZ1 and (b) DD-MEX EoSs. The plot shows both
pure hadronic stars (solid lines) and hybrid stars (dashed lines) at different
bag constants.

To constrain the Kerr parameter for NSs, we studied the dependence of the Kerr
parameter on the NS gravitational mass as displayed in Figs. 14 and 15 for the
given parameter sets. From Fig. 14, the Kerr parameter for pure hadronic DD-
LZ1 and DD-MEX parameter sets is found to be 0.64 and 0.67 respectively. This
parameter increases for the hybrid stars with a maximum value of 0.73 at
$B^{1/4}$=160 MeV for the DD-LZ1 set. For the DD-MEX set, the maximum value of
the Kerr parameter is 0.75 at 160 MeV bag constant. For the DD-LZ1 parameter
sets, the Kerr parameter remains almost unchanged once the star reaches a mass
of around 1.4$M_{\odot}$ for pure hadronic matter and around 1.2$M_{\odot}$
for hybrid configurations. For DDV, DDVT, and DDVTD parameter sets as shown in
Fig. 15, the Kerr parameter value for pure hadronic stars at the maximum mass
is 0.64, 0.62, and 0.61 respectively. For hybrid star configurations, the
value increases to 0.75 for all parameter sets at bag constant $B^{1/4}$=160
MeV. The Kerr parameter for HS configurations remains almost identical to the
hadron star up to almost 0.4$M_{\odot}$. Therefore, by definition, the
gravitational collapse of a RNS cannot form a Kerr black hole.

Figure 15: (color online) Same as Fig. 14 but for (a) DDV, (b) DDVT, and (c)
DDVTD EoSs.

Another important quantity related to the NSs is the redshift which has been
investigated deeply Xia and Yong-Jiu (2009); Cook  _et al._ (1994); Friedman
_et al._ (1986). The measurement of redshift can impose constraints on the
compactness, and in turn, on the NS EoS. For a RNS, if the detector is placed
in the direction of the polar plane of the star, the polar redshift, also
called gravitational redshift, can be measured. For a detector directed
tangentially, the forward and backward redshifts can be measured. The
expression for the polar redshift is given as

$Z_{P}(\Omega)=e^{-2\nu(\Omega)}-1$ (55)

where $\nu$ is the metric function. The variation of the polar redshift with
the gravitational mass is depicted in Fig. 16 for DD-LZ1 and DD-MEX EoSs. For
pure hadronic stars, the polar redshift is found to be around 1.1 for both
EoSs. With the QM present in the NSs, the polar redshift for DD-LZ1 decreases
to a value 0.89, 0.84, and 0.64 for bag constants $B^{1/4}$=130, 145, and 160
MeV, respectively. Similarly for the DD-MEX set, the redshift decreases up to
0.68 for the 160 MeV bag constant. The observational limits imposed on the
redshift from 1E 1207.4-5209 ($Z_{P}$=0.12-0.23) Sanwal  _et al._ (2002) , RX
J0720.4-3125 ($Z_{P}$=0.205${}_{-0.003}^{+0.006}$) Douchin, F. and Haensel, P.
(2001), and EXO 07482-676 ($Z_{P}$=0.35) Cottam  _et al._ (2002) are also
shown. The redshift prediction of $Z_{P}$=0.35 for EXO 07482-676 was based of
the narrow absorption lines in the x-ray bursts. However, it was later seen
that the spectral lines from EXO 07482-676 may be narrower than predicted
Bauböck  _et al._ (2013). Therefore the estimates of the redshift from EXO
07482-676 are uncertain.

Figure 16: (color online) Polar redshift vs gravitational mass for pure hadron
stars and hybrid star configurations for (a) DD-LZ1 and (b) DD-MEX EoSs. The
observational limits imposed on the polar redshift from 1E 1207.4-5209 (grey
band)Sanwal  _et al._ (2002), RX J0720.4-3125 (brown band)Douchin, F. and
Haensel, P. (2001), and EXO 07482-676 (orange horizontal line)Cottam  _et al._
(2002) are shown.  Figure 17: (color online) Same as Fig. 16 but for (a) DDV,
(b) DDVT, and (c) DDVTD EoSs.

For the softer EoS group, the polar redshift variation with the gravitational
mass is shown in Fig. 17 for both pure HM and HS configurations. For the DDV
set, the polar redshift is found to be 0.75 for the maximum mass of a pure
hadronic star and decreases to 0.50 for the hybrid star at a bag constant of
160 MeV. For DDVT and DDVTD EoSs, the redshift decreases from 0.72 and 0.70
for pure HM to 0.55 and 0.53 respectively for a hybrid star at $B^{1/4}$=160
MeV. The NS redshift provided by measuring the $\gamma$-ray burst annihilation
lines has been interpreted as gravitationally redshifted 511 keV electron-
positron pair annihilation from the NS surface Liang (1986). If this
interpretation is correct, then it will support a NS with redshift in the
range $0.2\leq Z_{P}\leq 0.5$ and thus will rule out almost every EoS studied
in this work.

For the static NS, the phase transition to the QM for DD-LZ1 and DD-MEX
parameter sets is studied in Ref. Wei  _et al._ (2020). For DDV, DDVT, and
DDVTD sets, the maximum mass obtained is around 2$M_{\odot}$ and hence the
phase transition to QM will decrease the maximum mass to a value not
satisfying any recent constraints on the mass and other NS properties.
However, to study the properties of a pure hadronic EoS, the mass-radius
profile for static stars is explained in Fig. 3. In addition to this, we study
the tidal deformability of the given parameter sets. The equations describing
the tidal deformation and its dependence on the star matter properties are
described above.

Figure 18: (color online) Dimensionless tidal deformability as a function of
NS mass for DD-LZ1, DD-MEX, DDV, DDVT, and DDVTD EoSs. The recent constraint
using Bayesian analysis Li  _et al._ (2021) and an upper limit on
dimensionless tidal deformability set by measurement from GW170817 Abbott  _et
al._ (2017) are shown.

The dimensionless tidal deformability $\Lambda$ as a function of NS mass for
the hadronic EoSs is shown in Fig. 18. The constraint on dimensionless tidal
deformability obtained using Bayesian analysis is shown,
$\Lambda_{1.4}=500_{-367}^{+186}$ Li  _et al._ (2021). The orange dotted curve
represents an uper limit set on the tidal deformability from the measurement
of GW170817 Abbott  _et al._ (2017). The tidal deformability depends upon the
NS mass and the radius. The value decreases with increasing mass and becomes
very small at the NS maximum mass. The dimensionless tidal deformability for
DD-LZ1, DD-MEX, DDV, DDVT, and DDVTD EoSs at the canonical mass is found to be
727.17, 791.60, 391.23, 337.51, and 281.05 respectively. All these values lie
well below the upper limit set by the GW170817 event. Using Bayesian analysis,
Lim et al. Lim and Holt (2018) showed a 90% and 65% credibility interval on
the dimensionless tidal deformability at 1.4$M_{\odot}$, 136 $<\Lambda<$ 519
and 256 $<\Lambda<$ 442, respectively. The DD-MEX set produces a little higher
value of the tidal deformability. The value of $\Lambda_{1.4}$ for softer
group EoS (DDV, DDVT, and DDVTD) is significantly lower than the stiffer group
(DD-LZ1 and DD-MEX) because of the small maximum mass and the corresponding
radius. However, the stiffer group EoSs cannot be neglected in comparison to
the softer group. The tidal deformability of softer group satisfies all the
constraints imposed. The precise measurement of the tidal deformability for
the BNS mergers with a maximum mass around 2$M_{\odot}$ by future
gravitational wave detectors will lower the uncertainties in these values
thereby constraining the EoSs.

##  V Summary and Conclusion

The properties of static and rotating NSs are studied with a hadron-quark
phase transition. The hadronic matter is studied by employing the DD-RMF
model. Recent parameter sets like DDV, DDVT, and DDVTD along with the DD-LZ1
and DD-MEX are used to study the hadronic EoS. The QM is studied using a
modified version of the bag model, the vBag model. The vBag model includes the
necessary repulsive vector interactions and D$\chi$SB. The vBag model coupling
parameter $K_{\nu}$ controlling the stiffness of the EoS curve is held
constant at 6 GeV-2. The effective bag constant $B_{eff}^{1/4}$ is varied by
taking the values 130, 145, and 160 MeV. The Gibbs technique is used to
construct the mixed-phase between hadrons and quarks which accounts for the
global charge neutrality of the system. The properties such as mass, radius,
and the tidal deformability of static NS are studied. For RNS, the variation
in the NS properties like maximum mass, radius, the moment of inertia,
rotational frequency, Kerr parameter, etc are studied in the presence of QM.

For static NSs, the maximum mass for DD-LZ1 and DD-MEX is found to be
2.55$M_{\odot}$ and 2.57$M_{\odot}$ respectively, forming a stiffer EoS group.
For DDV, DDVT, and DDVTD EoSs, the maximum is found to be around
1.9$M_{\odot}$, thus lying in the softer EoS group. The phase-transition
properties for SNS are not studied for the softer EoS group because it would
result in a very low maximum mass not satisfying any mass constraints.

For RNSs, the maximum mass is found to be 3.11$M_{\odot}$ for the DD-LZ1 set
which in presence of QM reduces to 2.64$M_{\odot}$ satisfying the recent
GW190814 possible maximum mass constraint. The DD-MEX set also predicts a
maximum mass of 3.15$M_{\odot}$ decreasing to 2.69$M_{\odot}$ for
$B_{eff}^{1/4}$=160 MeV bag constant. For the softer EoS group, the RNS mass
lies in the range 2.2$M_{\odot}$-2.3$M_{\odot}$ which then reduces with
increasing bag constant to satisfy the 2$M_{\odot}$ limit. The radius also
decreases with increasing bag constant. The moment of inertia for the stiffer
group lies in the range (2.2-2.3)$\times$1045g.cm2 for pure hadron EoSs. The
phase transition to QM reduces the value to 1.7$\times$1045g.cm2 satisfying
the recent constraints. For the softer group of EoSs, the moment of inertia is
lowered in the presence of QM to satisfy the constraints from GW170817 with
universal relations.

The variation in the rotational frequency of a NS with the gravitational mass
is also studied. The pure hadronic EoSs produce NSs with high rotational
frequencies. For DD-LZ1 and DD-MEX, the rotational frequency at the maximum
mass is 1525 and 1503 Hz, respectively. For DDV, DDVT, and DDVTD EoSs, the
frequency obtained is in the range 1400-1500 Hz. The quarks produce the hybrid
star configurations with larger rotational frequencies as the quark star are
more compact than hadron stars. Initially, for HS configuration at
$B^{1/4}$=130 MeV, the rotating with frequency smaller than a pure hadronic
star is formed. As the bag constant increases, the maximum mass decreases, and
the corresponding frequency increases. All the pure hadronic and hybrid star
configurations produce NSs with a frequency higher than the highest measured
frequency of $\nu$=1122 Hz.

The ratio of rotational kinetic energy to the gravitational potential energy
$\beta=T/W$ is studied to determine the dynamical stability of the RNS. For
$\beta>\beta_{d}$ (=0.14-0.27), the star is considered to be dynamically
unstable and hence emits gravitational radiation. The $T/W$ ratio for rotating
pure hadronic stars is found to be 0.147 and 0.145 for DD-LZ1 and DD-MEX EoSs.
The QM phase transition tends to increase the $T/W$ ratio with decreasing
mass. For a bag constant of 160 MeV, the ratio is found to be 0.153 and 0.151
for DD-LZ1 and DD-MEX EoSs, respectively. For a softer EoS group, this ratio
lies below the critical limit for pure hadronic stars, but increases to a
value well within the critical limit.

The Kerr parameter is calculated for the RNSs whose measurement allows us to
constrain the compactness of a star and hence the EoS. The precise value of
the Kerr parameter for a NS is not known yet, but a maximum value of 0.75 is
seen in most of the theoretical works. For the given parametrization sets, the
Kerr parameter value lies around 0.65 for the stiffer group and 0.6 for the
softer group. Following the inverse relationship with the gravitational mass,
the Kerr parameter increases in the presence of quarks. For both stiffer and
softer EoS groups, the value attains a maximum value of 0.75, which remains
almost unchanged as the mass increases beyond 1$M{\odot}$. The dependence of
polar redshift on the NS mass is also calculated. It is seen that the polar
redshift decreases in presence of quarks. The redshift parameter measured for
all hybrid star configurations lies well above the predicted value from EXO
07482-676, $Z_{P}$=0.35.

For static, spherically symmetric stars, we have also calculated the
dimensionless tidal deformability. It is seen that all the parameter sets
predict a value of tidal deformability satisfying the constraints from various
measurements.

Thus, it is clear that the presence of quarks inside the NS affects both
static and rotating NS properties. Eliminating the uncertainties present in
the values of these quantities will allow us to rule out very stiff and very
soft EoSs. The measurement of tidal deformability for RNS will help us to
constraint its properties and hence determine a proper EoS in the near future.
Additional gravitational-wave observations of binary NS mergers and more
accurate measurements of other NS properties like mass, radius, tidal
deformability will allow the universal relation-based bounds on canonical
deformability to be further refined. The theoretical study of a uniformly RNS,
along with the accurate measurements, may offer new information about the
equation of state in high density regime. Besides, NSs through their evolution
may provide us with a criterion to determine the final fate of a rotating
compact star.

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