# Role of local anisotropy in hybrid stars

Luiz L. Lopes<EMAIL_ADDRESS>Centro Federal de Educação Tecnológica de
Minas Gerais Campus VIII, Varginha/MG, CEP 37.022-560, Brazil  H. C. Das
<EMAIL_ADDRESS>INFN Sezione di Catania, Dipartimento di Fisica,
Via S. Sofia 64, 95123 Catania, Italy

###### Abstract

Using the Bower-Liang model, we discuss how pressure anisotropies affect the
microscopic and macroscopic properties of hybrid stars. We find that
anisotropies affect the maximum mass, central density, and radius of the
canonical stars. Anisotropies also affect the minimum neutron star mass that
presents quarks in their core, as well as the total amount of quarks for the
maximally massive stars. We also confront our results with standard
constraints, such as the radius and the tidal parameter of the canonical star,
as well as the mass and radius of the PSR J0740+6620 pulsar. We observe that
moderate values for anisotropies could fulfill these constraints
simultaneously. On the other hand, within more extreme degrees of
anisotropies, more speculative constraints such as black widow pulsars PSR
J0952-0607 and the mass-gap object in the GW190814 event can be explained as
hybrid stars. We also investigate the role of anisotropies in the neutron
stars’ moment of inertia.

††preprint:

##  I Introduction

Neutron stars are compact and degenerate objects whose density at their core
can reach several times the density of the atomic nucleus. The theory of the
birth of neutron stars is well-established in the literature as supernova
remnants. However, other aspects are far from being determined. One of them is
the neutron stars’ internal composition. The standard model considers neutron
stars composed of nucleons and leptons with zero total charge net and in
chemical equilibrium. Still, the presence of exotic, non-Nucleonic degrees of
freedom is possible to exist in neutron stars’ core. One of the vastly studied
possibilities is the presence of hyperons [1, 2, 3, 4, 5]. Indeed, some
studies suggest that hyperons are inevitable [6, 7].

Another possibility is that at high densities, the quark matter becomes
energetically favorable, creating a hadron-quark phase transition in the
neutron star’s core. In this case, we have a neutron star constituted by
deconfined quark matter in the core, surrounded by hadronic matter in the
outer layers. Such a configuration is called a hybrid star [8, 9, 10]. The
study of hybrid stars is essential, once a recent work pointed out that the
presence of deconfined quarks in massive neutron stars is not only possible
but probable [11].

Another unknown feature of neutron stars is related to their isotropic
pressure. Most works (see ref. [12]) assume that neutron stars are isotropic
objects. However, several exotic phenomena, such as a strong magnetic field,
superfluity, phase transitions, and others (see ref. [13] for an extensive
discussion) can be present in neutron stars’ interiors, inducing a local
anisotropy. Several models have been proposed to explore the effects of
pressure anisotropy inside the star [14]. One of them is the so-called Bower-
Liang (BL) model [15]. In this simple model, the anisotropy is local,
gravitationally induced, and non-linear in pressure. Moreover, the BL model
has the advantage that the presence of local anisotropy does not break the
spherical symmetry of the star, as discussed in ref. [16]

In this work, we study the effects of local anisotropies in different
configurations of hybrid stars. In the hadronic phase, we construct hybrid
stars with and without hyperons, while in the quark phase, we use one soft and
one stiff equation of state (EOS). To accomplish this task, we use the quantum
hadrodynamic (QHD) for the hadron phase [17], and the thermodynamic consistent
vector MIT bag model for the quark phase [18, 19]. The presence of the
anisotropy is taken into account via BL model [15].

All our results are confronted with experimental and observational
constraints. From the microscopic nuclear physics point of view, there are
four well-constrained parameters at the saturation density [20]: the
saturation density itself $(n_{0})$, the binding energy per nucleon ($E/A)$,
the compressibility ($K$) and the effective nucleon mass ($M^{*}/M$). Two
additional parameters, the symmetry energy $(S_{0})$ and its slope $(L)$, are
still a matter of debate. Nevertheless, they are strongly constrained in a
recent study [21].

From the macroscopic nuclear astrophysics point of view, there are also some
constraints. Maybe the more important is the PSR J0740+6620 with a mass of
$2.08\pm 0.07\ M_{\odot}$ and a radius of $12.35\pm 0.75$ km [22]. Concerning
the canonical $1.4\ M_{\odot}$ star, two NICER results point that their radius
must be in the range $11.52<R_{1.4}<13.85$ km [23], and $11.96<R_{1.4}<14.26$
km [24]. Later, these results were refined to $11.80<R_{1.4}<13.10$ km [22].
Besides these standard constraints, there are some unorthodox and speculative
ones presented in the literature. The first is the HESS J1731-347 object [25]
whose mass and radius are $M=0.77_{-0.17}^{+0.20}~{}M_{\odot}$ and
$R=10.4_{-0.78}^{+0.86}$ km respectively. The true nature of the HESS object
is one of the hot topics in nuclear astrophysics. In ref. [25], it was shown
that some chiral theory can explain this object as an ordinary hadronic
neutron star. The hadronic nature was also explored in ref. [26]. On the other
hand, the HESS object as a strange star was studied in refs. [27, 28, 29].
Finally, in ref. [30] the authors show that the HESS object can be a hadronic
neutron star with a soft EOS or a hybrid star with an early deconfinement
phase transition. A second one is the speculative mass of the black widow
pulsars PSR J0952-0607, $M=2.35\pm 0.17M_{\odot}$ [31]. Finally, there is the
mass-gap object in the GW190814 event [32], whose mass was estimated to be
$2.50-2.67\ M_{\odot}$. Although we are not sure about its true nature [33,
34]. We check if some degree of anisotropy can explain some or even all of
these exotic objects as hybrid stars.

In addition, we also use the constraints from the gravitational wave
observations by LIGO/VIRGO/KAGRA. The GW170817 event [35] put the constraints
on the dimensionless tidal parameter of the canonical star $\Lambda_{1.4}<800$
[35]. This result was then refined in ref. [36], to $70<\Lambda_{1.4}<580$.
Moreover, assuming that the mass-gap object in the GW190814 event was not a
black hole implies that the dimensionless tidal parameter for the canonical
star must be in the range of $458<\Lambda_{1.4}<889$ [32].

Ultimately, we discuss the moment of inertia (MOI) of the hybrid stars. Till
now, we don’t have any observational data for any NS. The authors of ref. [37]
have obtained the MOI for several pulsars using the universal relations
between the mass and the tidal deformability. Here, we calculate the MOI of
the anisotropic hybrid stars with different hadronic and quark
parametrizations by varying the degrees of anisotropicity. One can constrain
the value of MOI from the future observational data for the anisotropic
compact star. There are other ways to constrain the magnitude of the MOI for
different systems, such as Millisecond Pulsars (MSP), Double NS (DNS), and Low
Mass X-ray Binary (LMXB), as done in Refs. [37, 38]. The MOI of these pulsars
is expected to be measured soon; for example, the PSR J0737-3039(A) is the
only known DNS up to date. Since the NS EOS is believed to be universal, the
tidal deformability constraints from GW170817 have implications for all NSs,
including PSR J0737-3039(A), the MOI has been obtained to be
$1.15_{-0.24}^{+0.38}\times 10^{45}$ g cm2 [37]. One can also estimate the MOI
for other MSPs and LMXBs.

##  II Microscopic Formalism

###  II.1 Quantum Hadrodynamics

To simulate the interaction between baryons in dense cold matter, we use an
extended version of the QHD model, whose Lagrangian density reads [17, 39]:

$\displaystyle\mathcal{L}_{\rm QHD}$
$\displaystyle=\bar{\psi}_{B}[\gamma^{\mu}(i\partial_{\mu}-g_{B\omega}\omega_{\mu}-g_{B\rho}\frac{1}{2}\vec{\tau}\cdot\vec{\rho}_{\mu})$
$\displaystyle-(M_{B}-g_{B\sigma}\sigma)]\psi_{B}-U(\sigma)$
$\displaystyle+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{s}^{2}\sigma^{2})-\frac{1}{4}\Omega^{\mu\nu}\Omega_{\mu\nu}+\frac{1}{2}m_{v}^{2}\omega_{\mu}\omega^{\mu}$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\vec{\rho}_{\mu}\cdot\vec{\rho}^{\;\mu}-\frac{1}{4}\bf{P}^{\mu\nu}\cdot\bf{P}_{\mu\nu}+\mathcal{L}_{\omega\rho}+\mathcal{L}_{\phi},$
(1)

in natural units. $\psi_{B}$ is the baryonic Dirac field, where $B$ can stand
either for nucleons only ($N$) or can run over $N$ and hyperons ($H$). The
$\sigma$, $\omega_{\mu}$ and $\vec{\rho}_{\mu}$ are the mesonic fields, while
$\vec{\tau}$ are the Pauli matrices. The $g^{\prime}s$ are the Yukawa coupling
constants that simulate the strong interaction, $M_{B}$ is the baryon mass,
$m_{s}$, $m_{v}$, and $m_{\rho}$ are the masses of the $\sigma$, $\omega$ and
$\rho$ mesons respectively. The $U(\sigma)$ is the self-interaction term
introduced in Boguta and Bodmer [40]:

$\displaystyle
U(\sigma)=\frac{1}{3}\lambda\sigma^{3}+\frac{1}{4}\kappa\sigma^{4},$

and $\mathcal{L}_{\omega\rho}$ is a non-linear $\omega$-$\rho$ coupling
interaction as in ref. [39]:

$\displaystyle\mathcal{L}_{\omega\rho}=\Lambda_{\omega\rho}(g_{N\rho}^{2}\vec{\rho^{\mu}}\cdot\vec{\rho_{\mu}})(g_{N\omega}^{2}\omega^{\mu}\omega_{\mu}),$
(2)

which is necessary to correct the slope of the symmetry energy and has a
strong influence on the radii and tidal deformation of the neutron stars [41,
42, 43]; $\mathcal{L}_{\phi}$ is related to the strangeness of hidden $\phi$
vector meson, which couples only with the hyperons, not affecting the
properties of symmetric nuclear matter:

$\mathcal{L}_{\phi}=g_{H\phi}\bar{\psi}_{H}(\gamma^{\mu}\phi_{\mu})\psi_{H}+\frac{1}{2}m_{\phi}^{2}\phi_{\mu}\phi^{\mu}-\frac{1}{4}\Phi^{\mu\nu}\Phi_{\mu\nu},$
(3)

as pointed out in refs. [44, 4], this vector channel is crucial to obtain
massive Hyperonic neutron stars.

As neutron stars are stable macroscopic objects, we need to describe a
neutral, chemically stable matter, and hence, leptons are added as free Fermi
gases, whose Lagrangian density is the usual one.

$\displaystyle\mathcal{L}_{l}=\bar{\psi}_{l}[\gamma^{\mu}(i\partial_{\mu}-m_{l})]\psi_{l}.$

To solve the equations of motion, we use the mean field approximation (MFA),
where the meson fields are replaced by their expectation values. Applying the
Euler-Lagrange formalism and using the quantization rules ($E=\partial^{0}$,
$k=i\partial^{j}$), we easily obtain the eigenvalue for the energy:

$E_{B}=\sqrt{k^{2}+M_{B}^{*2}}+g_{B\omega}\omega_{0}+g_{B\phi}\phi_{0}+\frac{\tau_{3B}}{2}g_{B\rho}\rho_{0},$
(4)

where $M^{*}_{B}~{}\equiv~{}M_{B}-g_{B\sigma}\sigma_{0}$ is the effective
baryon mass and $\tau_{3B}$ assumes the value of +1 for p, $\Sigma^{+}$,
$\Sigma^{0}$ and $\Xi^{0}$; and -1 for n, $\Lambda^{0}$ $\Sigma^{-}$ and
$\Xi^{-}$. Within this approach, we can add the mixed term
$g_{\Sigma\Lambda\rho}$ as discussed in ref. [45]. This term is necessary to
recover the relations of completeness and closure. For the leptons, we have:

$E_{l}=\sqrt{k^{2}+m_{l}^{2}},$ (5)

and the mesonic fields in MFA are given by:

$\displaystyle
m_{\sigma}^{2}\sigma_{0}+\lambda\sigma_{0}^{2}+\kappa\sigma_{0}^{3}=\sum_{B}g_{B\sigma}n_{B}^{s},$
$\displaystyle(m_{\omega}^{2}+2\Lambda_{v}\rho_{0}^{2})\omega=\sum_{B}g_{B\omega}n_{B},$
$\displaystyle m_{\phi}^{2}\phi_{0}=\sum_{B}g_{B\phi}n_{B},$
$\displaystyle(m_{\rho}^{2}+2\Lambda_{v}\omega_{0}^{2})\rho_{0}=\sum
g_{B\rho}\frac{\tau_{3B}}{2}n_{B},$ (6)

where
$\Lambda_{v}~{}\equiv~{}\Lambda_{\omega\rho}g_{N\omega}^{2}g_{N\rho}^{2}$, and
$n_{B}^{s}$ and $n_{B}$ are, respectively, the scalar density and the number
density of the baryon $B$. Finally, applying Fermi-Dirac statistics to baryons
and leptons and with the help of Eq. (6), we can write the total energy
density as [46]:

$\displaystyle\epsilon$
$\displaystyle=\sum_{B}\frac{1}{\pi^{2}}\int_{0}^{k_{Bf}}dkk^{2}\sqrt{k^{2}+M_{B}^{*2}}+U(\sigma_{0})$
$\displaystyle+\frac{1}{2}m_{\sigma}^{2}\sigma_{0}^{2}+\frac{1}{2}m_{\omega}^{2}\omega_{0}^{2}+\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}+\frac{1}{2}m_{\rho}^{2}\rho_{0}^{2}$
$\displaystyle+3\Lambda_{v}\omega_{0}^{2}\rho_{0}^{2}+\sum_{l}\frac{1}{\pi^{2}}\int_{0}^{k_{lf}}dkk^{2}\sqrt{k^{2}+m_{l}^{2}},$
(7)

and the pressure is easily obtained by thermodynamic relations:
$p=\sum_{f}\mu_{f}n_{f}-\epsilon$, where the sum runs over all the fermions,
and $\mu_{f}$ is the corresponding chemical potential. Now, to determine each
particle population, we impose that the matter is $\beta$ stable and has a
total electric net charge equals to zero.

###  II.2 Parametrization

In this work, we use the enhanced L3$\omega\rho$ parametrization
(eL3$\omega\rho$), as originally proposed in ref. [5], such parametrization
virtually fulfills all constraints at the saturation density. The parameters,
constraints, and predictions of this model are presented in Tab. 1.

Enhanced L3$\omega\rho$
---
$(g_{N\sigma}/m_{s})^{2}$ | 12.108 fm2
$(g_{N\omega}/m_{v})^{2}$ | 7.132 fm2
$(g_{N\rho}/m_{\rho})^{2}$ | 5.85 fm2
$\kappa$ | 0.04138
$\lambda$ | $-0.0390$
$\Lambda_{\omega\rho}$ | 0.0283

Quantity | Constraint | This model
---|---|---
$n_{0}$ ($fm^{-3}$) | 0.148–0.170 | 0.156
$M^{*}/M$ | 0.60–0.80 | 0.69
$K$ (MeV) | 220–260 | 256
$S_{0}$ (MeV) | 31.2–35.0 | 32.1
$L$ (MeV) | 38–67 | 66
$B/A$ (MeV) | 15.8–16.5 | 16.2

Table 1: eL3$\omega\rho$ [5] parameterization (top) and its predictions for
nuclear matter quantities (bottom). The phenomenological constraints are taken
from Ref. [20, 21].

If hyperons are present, we must fix their coupling constants with the mesonic
fields as well. The coupling constants of the vector mesons can be fixed
within the flavor SU(3) group symmetry. Assuming an ideal angle mixing,
$\theta_{v}=35.264$, and a rating $z=1\sqrt{6}$ (see refs. [47, 48, 46, 7, 5,
45] and the references therein for a more complete discussion about group
theory and its relation with the meson-hyperon coupling constants) the only
free parameter left is the $\alpha_{v}$. As we wish to produce neutron stars
with $M\geq 2.0\ M_{\odot}$, we use $\alpha_{v}$ = 0.50.

For the coupling with the scalar meson, the hyperon potential depths are taken
as $U_{\Lambda}=-28$ MeV, $U_{\Sigma}=+30$ MeV [49], and $U_{\Xi}=-4$ MeV [50]
respectively. The complete set of coupling constants used in this work are
given as follows:

$\displaystyle
g_{\Lambda\omega}/g_{N\omega}=0.714,\,g_{\Sigma\omega}/g_{N\omega}=1.0,\,g_{\Xi\omega}/g_{N\omega}=0.571,$
$\displaystyle g_{\Lambda\rho}=0.0,\quad g_{\Sigma\rho}=1.0,\quad
g_{\Xi\rho}/g_{N\rho}=0.0,$ $\displaystyle
g_{\Lambda\sigma}/g_{N\sigma}=0.646,\,g_{\Sigma\sigma}/g_{N\sigma}=0.663,\,g_{\Xi\sigma}/g_{N\sigma}=0.453,$
$\displaystyle
g_{\Lambda\phi}/g_{N\omega}=-0.808,\,g_{\Sigma\phi}/g_{N\omega}=-0.404,$
$\displaystyle g_{\Xi\phi}/g_{N\omega}=-1.01,\,g_{\Sigma\Lambda\rho}=0.577,$
(8)

where we explicitly add the mixed Yukawa coupling in order to restore the
relations of completeness and closure of the SU(3) Clebsch-Gordan coefficients
[45].

###  II.3 Vector MIT bag model

The quark matter is studied within the vector MIT bag model, which is an
extension of the original MIT bag model [51] that incorporates some features
of the QHD. In its original form, the MIT bag model considers that each baryon
is composed of three non-interacting quarks inside a bag. The bag, in turn,
corresponds to an infinite potential that confines the quarks. As a
consequence, the quarks are free inside the bag and are forbidden to reach its
exterior. All the information about the strong force relies on the bag
pressure value, which mimics the vacuum pressure.

In the vector MIT bag model, the quarks are still confined inside the bag, but
now they interact with each other through a vector meson exchange. This vector
meson plays a role analog to the $\omega$ meson of the QHD [17]. Moreover, the
contribution of the Dirac sea can be taken into account through a self-
interaction of the vector meson [52]. The Lagrangian of the vector MIT bag
model, therefore, consists of the Lagrangian of the original MIT, plus the
Yukawa-type Lagrangian of the vector field exchange, plus the Dirac sea
contribution. We must also add the mesonic mass term to maintain the
thermodynamic consistency. It then reads [18, 19]:

$\mathcal{L}=\mathcal{L}_{\rm MIT}+\mathcal{L}_{\rm V}+\mathcal{L}_{\rm
DIRAC},$ (9)

where

$\mathcal{L}_{\rm
MIT}=\sum_{i}\\{\bar{\psi}_{i}[i\gamma^{\mu}\partial_{\mu}-m_{i}]\psi_{i}-B\\}\Theta(\bar{\psi_{i}}\psi_{i}),$
(10) $\mathcal{L}_{\rm
V}=\sum_{i}\\{\bar{\psi}_{i}g_{iV}(\gamma^{\mu}V_{\mu})\psi_{i}-\frac{1}{2}m_{V}^{2}V^{\mu}V_{\mu}\\}\Theta(\bar{\psi}_{i}\psi_{i}),$
(11) $\mathcal{L}_{\rm DIRAC}=b_{4}\frac{(g^{2}V_{\mu}V^{\mu})^{2}}{4},$ (12)

where $\psi_{i}$ is the Dirac quark field, $B$ is the constant vacuum
pressure, $m_{V}$ is the mass of the $V_{\mu}$ mesonic field, $g_{iV}$ is the
coupling constant of the quark $i$ with the meson $V_{\mu}$, and $g$ =
$g_{uV}$. The $\Theta(\bar{\psi}_{i}\psi_{i})$ is the Heaviside step function
included to assure that the quarks exist only confined to the bag.

Applying MFA and the Euler-Lagrange equations, we can obtain the energy
eigenvalue for the quark fields and the equation of motion for the mesonic
$V_{0}$ field:

$E_{i}=\sqrt{m_{i}^{2}+k_{i}^{2}}+g_{iV}V_{0},\,$ (13)
$gV_{0}+\bigg{(}\frac{g}{m_{v}}\bigg{)}^{2}\bigg{(}b_{4}(gV_{0})^{3}\bigg{)}=\bigg{(}\frac{g}{m_{v}}\bigg{)}\sum_{i}\bigg{(}\frac{g_{iV}}{m_{v}}\bigg{)}n_{i}.\\\
$ (14)

To construct an EOS in MFA, we now consider the Fermi-Dirac distribution of
the quarks, the Hamiltonian of the vector field, and the bag pressure value,
$\mathcal{H}=-\langle\mathcal{L}\rangle$. We obtain

$\varepsilon_{i}=\frac{\gamma_{i}}{2\pi^{2}}\int_{0}^{k_{f}}E_{i}~{}k^{2}dk,$
(15)
$\varepsilon=\sum_{i}\varepsilon_{i}+B-\frac{1}{2}m_{V}^{2}V_{0}^{2}-b_{4}\frac{(g^{2}V_{0}^{2})^{2}}{4},$
(16)

with $\gamma_{i}=6~{}(3~{}{\rm colors}\times 2~{}{\rm spins})$ is the
degeneracy factor.

To construct a hybrid star, the quark matter must lie outside the so-called
stability window [53, 54], i.e., the energy per baryon must be higher than 930
MeV. Otherwise, the entire hybrid star would be converted into a strange star
in a finite amount of time [55, 56]. To accomplish this task, we use
$B^{1/4}=158$ MeV. We also define $G_{V}~{}\equiv~{}(g/m_{V})^{2}$ and
$X_{V}~{}\equiv~{}(g_{sV}/g_{uV})$. The $X_{V}$ is then taken as $X_{V}=0.4$
once its value was calculated based on symmetry group arguments (see reference
[18] for additional details). For the $G_{V}$, we use two different values to
produce a weak and a strong interaction between the quarks, $G_{V}=0.40$, and
$0.42$ fm2. The Dirac sea contribution is taken into account, assuming a small
value for $b_{4}\ (=0.5$). Within such a small value, we are still able to
produce massive stars, with $M\geq 2.0M_{\odot}$, while assuring that the
quark matter is energetically favorable at the limit of higher density.
Finally, the pressure is easily obtained by thermodynamic relations:
$p=\sum_{i}\mu_{i}n_{i}-\varepsilon$.

###  II.4 Maxwell construction and phase transition

To identify the point where the hadronic matter becomes deconfined quark
matter, we use the so-called Maxwell construction. In this approach, the
transition pressure is the one where the Gibbs free energy per baryon
$G/n_{B}$ of both phases intersect, being the energetically preferred phase,
the one with a lower $G/n_{B}$ [57]. The Gibbs free energy per baryon
coincides with the baryon chemical potential. Therefore, we call the
intersection point critical pressure and critical chemical potential. The
Maxwell criteria read:

$\mu_{H}=\mu_{Q}=\mu_{0},\quad\mbox{at}\quad p_{H}=p_{Q}=p_{0}.$ (17)

Hadron | Quark | $\mu_{0}$ | $p_{0}$ | $\epsilon_{H}$ | $\epsilon_{Q}$
---|---|---|---|---|---
Nucleonic | Weak | 1214 | 85 | 470 | 507
Nucleonic | Strong | 1302 | 129 | 557 | 609
Hyperonic | Weak | 1228 | 92 | 506 | 528
Hyperonic | Strong | 1489 | 253 | 860 | 904
Table 2: The critical chemical potential (in MeV), critical pressure (in
MeV/fm3), and the energy densities (in MeV/fm3) for hadron and quark phases,
within Maxwell construction.

The critical pressure ($p_{0}$), therefore, marks the onset of the quark
phase. We have a pure hadronic star for a neutron star whose central pressure
is lower than $p_{0}$, and a hybrid star if $p_{c}~{}>~{}p_{0}$. When
$p_{c}=p_{0}$ we have the minimum neutron star mass that presents quarks in
the core.

We display in Tab. 2 the critical chemical potential and the critical pressure
for beta-stable matter with only Nucleons (Nucleonic) and with nucleons and
hyperons (Hyperonic), as well with the quark matter with $G_{V}=0.40$ fm2
(Weak) and $G_{V}$ = 0.42 fm2 (Strong). We also display the correspondent
energy density for the hadronic and quarkyonic phases at the transition point.
We can notice that by increasing the value of $G_{V}$, we also increase the
critical chemical potential, pushing the hadron-quark phase transition to
higher densities and pressures. In the same sense, the presence of hyperons
softens the hadron EOS, making the hadron-quark phase transition more
difficult. Indeed, for even higher values of $G_{V}$, we are not able to
produce stable hybrid stars with hyperons once their critical chemical
potentials are beyond those reached in the maximum mass. These results are in
accordance with the discussion presented in ref. [33].

##  III Spherically symmetric hybrid stars

###  III.1 Hydrostatic equilibrium

In this section, we briefly review the theory of static equilibrium
distribution of matter, which is spherically symmetric but whose stress tensor
is, in general, locally anisotropic, as originally introduced in ref. [15]. In
the isotropic case, the stress-energy tensor reads: $T_{\mu\nu}$ =
diag($\rho,-p,-p,-p$). We now introduce anisotropies without breaking the
spherical symmetry by assuming the following stress-energy tensor:
$T_{\mu\nu}$ = diag($\rho,-p_{r},-p_{t},-p_{t}$). Spherical symmetry implies
that (in canonical coordinates) the stress-energy tensor $T_{\mu\nu}$ is
diagonal, and moreover, $p_{\phi}=p_{\theta}=p_{t}$ [16].

We can now redefine the anisotropic energy-momentum tensor as done in refs.
[58, 59, 16]:

$\displaystyle
T_{\mu\nu}=(\rho+p_{t})u_{\mu}u_{\nu}+(p_{r}-p_{t})k_{\mu}k_{\nu}+p_{t}g_{\mu\nu},$
(18)

where $\rho$, $p_{r}$, and $p_{t}$ are the energy density, radial pressure,
and tangential pressure, respectively. $k_{\mu}$ is the unit radial vector
($k^{\mu}k_{\mu}=1$) with $u^{\mu}k_{\mu}=0$. The Schwarzschild metric for
this type of star having the spherically symmetric and static configuration is
defined as

$\displaystyle
ds^{2}=-e^{2\nu}dt^{2}+e^{2\lambda}dr^{2}+r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\varphi^{2}\,,$ (19)

where $r$, $\theta$, and $\phi$ are the Schwarzschild coordinates.

Applying Einstein’s field equations, we obtain:

$\displaystyle\lambda^{\prime}(r)=\frac{8\pi
r^{2}e^{2\lambda(r)}\rho(r)-e^{2\lambda(r)}+1}{2r},$ (20)
$\displaystyle\nu^{\prime}(r)=\frac{8\pi
r^{2}p_{r}(r)e^{2\lambda(r)}+e^{2\lambda(r)}-1}{2r},$ (21)

and the contracted Bianchi identities give us the following:

$\displaystyle\frac{dp_{r}}{dr}=-(\rho+p_{r})\frac{\nu^{\prime}}{2}+\frac{2}{r}(p_{t}-p_{r}).$
(22)

Finally, by isolating the $\nu^{\prime}$ in Eq. 21 and replacing it in Eq. 22,
we can write the equilibrium equations in the Tolman-Oppenheimer-Volkoff form
[58]:

$\displaystyle\frac{dp_{r}}{dr}=-\frac{\left(\rho+p_{r}\right)\left(m+4\pi
r^{3}p_{r}\right)}{r\left(r-2m\right)}+\frac{2\sigma}{r}\,,$ (23)
$\displaystyle\frac{dm}{dr}=4\pi r^{2}\rho\,,$ (24)

where $\sigma=p_{t}-p_{r}$ is the anisotropy parameter, ‘$m$’ is the mass
enclosed within the radius $r$. The radial pressure is then obtained from a
pre-determined EOS. On the other hand, for the case of the transverse
pressure, we use the BL model in the following [15]:

$\displaystyle p_{t}=p_{r}+\frac{\lambda_{\rm
BL}}{3}\frac{(\rho+3p_{r})(\rho+p_{r})r^{2}}{1-2m/r}\,,$ (25)

where the factor $\lambda_{\rm BL}$ measures the degree of anisotropy in the
fluid. There are some boundary conditions required to solve the above Eqs.
(23-25) as done in Refs. [60, 14]. Also, different fluid conditions must be
satisfied for the anisotropic stars such as (i) $p_{r},p_{t}$, and $\rho>0$,
(ii) $0<c_{\rm s,t}^{2}<1$ (where $c_{\rm s,t}=dp_{t}/d\rho$), (iii)
$p_{r}=p_{t}$ for $r=0$, etc. Other conditions are mentioned in ref. [14].

In Fig. 1, we display the mass-radius relation for different values of
$\lambda_{\rm BL}$ with some neuron star configurations: the neutron stars
that present hyperons are on the right, while those that do not present
hyperons are on the left. From top to bottom, there are pure hadronic stars,
hybrid stars with the weak quark-quark interaction, and hybrid stars with the
strong quark-quark interaction.

|
---|---
|
|
Figure 1: Mass-radius relations for different neutron stars’ configurations. The anisotropic parameter lies $-2.0~{}<~{}\lambda_{\rm BL}~{}<~{}+2.0$.  |
---|---
|
Figure 2: Hybrid star masses as a function of the central density. The
anisotropic parameter lies $-2.0~{}<~{}\lambda_{\rm BL}~{}<~{}+2.0$.

We follow [27] and begin our analyses with the symmetric case. For
$\lambda_{\rm BL}$ = 0, we see that the maximum mass is 2.30$M_{\odot}$ for a
pure Nucleonic star, while its central number density is around 0.94 fm-3. A
maximum mass of 2.13$M_{\odot}$, with a central density around 0.99 fm-3, was
found for a hadronic neutron star with nucleons and hyperons. In the case of
hybrid neutron stars, the maximum mass is 2.06 $M_{\odot}$ for the weak quark-
quark interaction and 2.09 $M_{\odot}$ for the strong quark-quark interaction.
It is interesting to point out that we obtain the same maximum mass regardless
of whether the hadronic EOS presents hyperons. Such a feature was first
pointed out in ref. [33]. Concerning their central densities, we see that for
the weak quark-quark interaction, we have a central density around 0.95 fm-3
for a pure Nucleonic hybrid star and around 0.92 fm-3 for a hybrid star with
nucleons and hyperons. For a strong quark-quark interaction, we have exactly
the opposite: a central density around 0.92 fm${-3}$ for a pure Nucleonic
hybrid star and around 0.95 fm-3 for a hybrid star with nucleons and hyperons.
Nevertheless, although the presence of hyperons does not affect the maximum
mass of hybrid stars, their presence makes the phase transition more
difficult, pushing it to higher densities. This can more easily seen from Fig.
2, where we display the hybrid star mass as a function of the central density
$n$. The onset of the quark matter can be identified by the gap in the number
density (represented by small horizontal lines, indicating that the mass does
not change). The first hybrid star for a weak quark-quark interaction has a
mass of 1.63 $M_{\odot}$, and a density around 0.48 fm-3 for a Nucleonic EOS
and $M=1.68~{}M_{\odot}$ with $n=0.50$ fm-3 for a Hyperonic hybrid star. In
the case of the strong quark-quark interaction, we have the first hybrid star
with a mass of 1.84 $M_{\odot}$ with $n=0.57$ fm-3 for a Nucleonic hybrid star
and $M=$ 2.05 $M_{\odot}$ with $n=0.78$ fm-3 for a Hyperonic one. Now we
discuss the radius of the 1.4 $M_{\odot}$ canonical star. For the isotropic
case, we found that all stars have the same radius of 12.82 km, as neither
hyperons nor quarks are present at such low mass.

We now discuss the role of anisotropy on the neutron star properties. For a
pure Nucleonic star, the maximum mass drops to 1.98 $M_{\odot}$ for
$\lambda_{\rm BL}$ = -2.0 while peaks 2.74 for$\lambda_{\rm BL}$ = +2.0. On
the other hand, the central density is as high as 1.12 fm-3 or $\lambda_{\rm
BL}$ = -2.0 and drops to 0.77 fm-3 for $\lambda_{\rm BL}$ = +2.0. For a pure
Hyperonic neutron star we have $M=$ 1.83 $M_{\odot}$ for $\lambda_{\rm BL}$ =
-2.0, and $M=$ 2.56 for $\lambda_{\rm BL}$ = +2.0. In the same sense, we have
$n=1.20$ fm-3 for $\lambda_{\rm BL}$ = -2.00, and $n=0.79$ fm-3
for$\lambda_{\rm BL}$ = +2.0. As can be seen, the presence of a positive
anisotropy increases the maximum mass but reduces the central density and
vice-versa.

In the case of hybrid stars with a weak quark-quark interaction, we found a
maximum mass of 1.75 $M_{\odot}$ for $\lambda_{\rm BL}$ = -2.0 and $M=$ 2.51
for $\lambda_{\rm BL}$ = +2.0. Here, as in the isotropic case, the maximum
mass does not depend on the presence of hyperons or not. Indeed, we have the
same maximum mass for a fixed $\lambda_{\rm BL}$ and a fixed quark EOS. The
central density presents a small difference, $n=1.18$ fm-3 for Nucleonic+Weak
hybrid star, and $n=1.19$ fm-3 for a Hyperonic+Weak hybrid one for
$\lambda_{\rm BL}$ = -2.0. For $\lambda_{\rm BL}$ = +2.0 we have $n=0.77$ fm-3
and $n=0.76$ fm-3 for a Nucleonic and a Hyperonic hybrid star respectively.

For a strong quark-quark interaction, we again have the maximum mass
independent of the presence of hyperons for all values of $\lambda_{\rm BL}$.
The maximum mass lies between 1.77 $M_{\odot}$ for $\lambda_{\rm BL}$ = -2.0
up to 2.56 $M_{\odot}$ for $\lambda_{\rm BL}$ = +2.0. It is interesting to
notice that the maximum mass for a strong quark-quark interaction hybrid star
presents the same maximum mass as the pure hadronic Hyperonic star for
$\lambda_{\rm BL}$ = +2.0. For the other value of $\lambda_{\rm BL}$, the
maximum mass of the hybrid star is always below the one of the hadronic
Hyperonic star. In the case of the central density, we find values between
1.15 fm-3 for $\lambda_{\rm BL}$ = -2.0 and 0.74 fm-3 for $\lambda_{\rm BL}$ =
+2.0 for the case Nucleonic+Strong. For Hyperonic+Strong, we have values
between 1.16 fm-3 and 0.74 fm-3 for $\lambda_{\rm BL}$ = -2.0 and
$\lambda_{\rm BL}$ = +2.0 respectively. It is important to point out that for
$\lambda_{\rm BL}$ = +2.0, the central density is below the density at the
quark-hadron phase transition. Therefore, we do not have a stable hybrid star
for this configuration. All stable stars are pure hadronic. This is why both
Hyperonic and Hyperonic+Strong have the same mass for $\lambda_{\rm BL}$ =
+2.0.

We can also discuss what is the minimum neutron star mass that presents quarks
in their core. We see that for $\lambda_{\rm BL}$ = -2.0 we have 1.23
$M_{\odot}$ (Nucleonic+Weak), 1.27 $M_{\odot}$ (Hyperonic+Weak), 1.49
$M_{\odot}$ (Nucleonic+Strong) and 1.67 $M_{\odot}$ (Hyperonic+Strong). For
$\lambda_{\rm BL}$ = +2.0, we have 2.21 $M_{\odot}$ (Nucleonic+Weak), 2.28
$M_{\odot}$ (Hyperonic+Weak), and 2.47 $M_{\odot}$ (Nucleonic+Strong).
Hyperonic+Strong does not have a stable hybrid star.

We now estimate the percentual amount of quark present in the maximally
massive hybrid star for each neutron star configuration. To accomplish that,
we follow refs. [7, 10] and solve the hydrostatic equilibrium equations for
the quark EoS from the density corresponding to the central density at the
maximum mass neutron stars and stop at the density corresponding to the
critical chemical potential. The results are presented in Fig. 3. We see that
the configuration that presents a higher amount of quarks in the core is
Nucleonic+Weak, since it favors the earlier onset of quark matter, but still
yields a large maximum mass. In the same sense, the lower values of quark mass
are within the Hyperonic+Strong configuration. We also see that, despite some
fluctuations, the percentage of quarks decreases with the increase of
$\lambda_{\rm BL}$. For $\lambda_{\rm BL}$ = -2.0, the total amount of quarks
are 52$\%$ for Nucleonic+Weak, 49$\%$ for Hyperonic+Weak, 38$\%$ for
Nucleonic+Strong, and 17$\%$ for Hyperonic+Strong. For $\lambda_{\rm BL}$ =
+2.0, these values drop to 46$\%$ for Nucleonic+Weak, 41$\%$ for
Hyperonic+Weak, 25$\%$ for Nucleonic+Strong, and 0$\%$ for Hyperonic+Strong,
once the maximum mass is reached before the hadron-quark phase transition. The
decrease of the percentual amount of quarks with $\lambda_{\rm BL}$ can be
explained by the reduction of the central density with $\lambda_{\rm BL}$ as
well.

Figure 3: Percentual amount of quarks in the core of the most massive hybrid
stars as a function of $\lambda_{\rm BL}$ for different hybrid star
configurations as discussed in the text.

We now discuss the radius of the canonical 1.4 $M_{\odot}$ and some
constraints related to pulsars’ observations. For all hybrid stars, the radius
of the canonical star is almost independent of internal composition. It varies
from 12.09 km for $\lambda_{\rm BL}$ = -2.0, up to 13.71 km for $\lambda_{\rm
BL}$ = +2.0, independently from both: the presence of hyperons, as well the
strength of the quark-quark interaction. Concerning the constraint related to
the canonical star, we see that we need a value between -2.0
$\leq~{}\lambda_{\rm BL}~{}\leq$+0.6 to satisfy the refined NICER constraint,
$11.80<R_{1.4}<13.10$ km [22]. About the PSR J0740+6620 with a mass of 2.08
$\pm$ 0.07 $M_{\odot}$ and a radius of 12.35 $\pm$ 0.75 km [22], we see that
all hybrid stars configurations satisfy this constraint if we assume that -0.2
$\leq~{}\lambda_{\rm BL}~{}\leq$+1.2. Such restriction is independent of the
presence of hyperons or the strength of the quark-quark interaction. Combining
both constraints, we see that -0.2 $\leq~{}\lambda_{\rm BL}~{}\leq$+0.6
produce hybrid stars that simultaneously satisfy the canonical 1.4 $M_{\odot}$
and the revised mass and radius value of the PSR J0740+6620.

About the less orthodox constraints, we see that the speculative mass of the
black widow pulsars PSR J0952-0607, $M=2.35\pm 0.17M_{\odot}$ [31], can be
reached for $\lambda_{\rm BL}~{}\geq$ +0.4 for a strong quark-quark
interaction and $\lambda_{\rm BL}~{}\geq$ +0.6 in the case of the weak quark-
quark interaction. Here again, the results are independent of the presence of
hyperons. The mass-gap object in the GW190814 event [32], whose mass was
estimated to be $2.50-2.67\ M_{\odot}$ can be reached for $\lambda_{\rm
BL}~{}\geq~{}+1.8$ for the strong case, and only within $\lambda_{\rm
BL}=+2.0$ in the weak one. Finally, the constraint related to the HESS object
[25] cannot be fulfilled for any value of $\lambda_{\rm BL}$.

###  III.2 Tidal deformation

|
---|---
|
Figure 4: Dimensionless tidal parameter for different configurations of hybrid
stars. The anisotropic parameter lies $-2.0~{}<~{}\lambda_{\rm
BL}~{}<~{}+2.0$.

Tidal deformability ($\Lambda$) is a quantity that measures the amount of
deformation in the shape of the star when it is in the field of its companion
star. Already, LIGO/VIRGO detected the binary neutron star deformation in
2017. It has a unique relationship with the tidal Love number ($k_{2}$) and
the compactness $(C)$ of the star [61, 62]:

$\Lambda=\frac{2}{3}k_{2}C^{-5}$ (26)

One can estimate the magnitude of $k_{2}$ by solving the following
differential equation in the Throne, and Campolattaro metric [63] for an
anisotropic star with an EOS is given by [60, 64]

$\displaystyle H^{{}^{\prime\prime}}(r)+AH^{{}^{\prime}}(r)+BH(r)=0\ \,$ (27)

where, $A$ and $B$ are represented as

$\displaystyle A=-\left(\frac{-e^{2\lambda(r)}-1}{r}-4\pi
re^{2\lambda(r)}\left(p_{r}(r)-\rho_{r}(r)\right)\right),$ (28) $\displaystyle
B=-\frac{dp_{r}}{dp_{t}}\Bigg{(}\frac{4e^{2\lambda(r)}+e^{4\lambda(r)}+1}{r^{2}}+64\pi^{2}r^{2}p_{r}^{2}(r)e^{4\lambda(r)}$
$\displaystyle+16\pi
e^{2\lambda(r)}\left\\{p_{r}(r)(e^{2\lambda(r)}-2)-p_{t}(r)-\rho(r)\right\\}-4\pi
c_{s}^{2}$ $\displaystyle\times e^{2\lambda(r)}(p_{r}(r)+\rho(r))-4\pi
e^{2\lambda(r)}(p_{r}(r)+\rho(r))\Bigg{)}\,,$ (29)

where $c_{s}^{2}=dp_{r}/d\rho$ is the speed of sound. The term $dp_{r}/dp_{t}$
can be obtained from Eq. 25 for a particular star with a fixed value of
$\lambda_{\rm BL}$. The solution of the differential equation ($y$) at the
surface of the star is given by $y=\frac{rH^{{}^{\prime}}}{H}\Big{|}_{R}$ [65,
61]. After that, one can able to obtain the tidal Love number in the following
[61, 62, 66, 67].

$\displaystyle k_{2}$
$\displaystyle=\,\frac{8}{5}C^{5}(1-2C)^{2}\big{[}2(y-1)C-y+2\big{]}$
$\displaystyle\times\Big{\\{}2C\big{[}4(y+1)C^{4}+2(3y-2)C^{3}-2(11y-13)C^{2}$
$\displaystyle+3(5y-8)C-3(y-2)\big{]}+3(1-2C)^{2}$
$\displaystyle\times\big{[}2(y-1)C-y+2\big{]}\log(1-2C)\Big{\\}}^{-1}\,.$ (30)

In Fig. 4, we depict the result for the tidal deformability as the function of
mass for four cases only for positive $\lambda_{\rm BL}$ values. We chose only
the positive values because the speed of sound becomes negative for negative
values of $\lambda_{\rm BL}$ for NSs as well as hybrid stars. The unphysical
solution of the speed of sound gives obscure solutions for $\Lambda$. The
detailed discussion can be found for NSs in refs. [60, 14].

From the figure, we observe that the neutron stars with larger radii, i.e.,
larger values of $\lambda_{\rm BL}$, actually have lower values of $\Lambda$.
This is the opposite when we compare pure isotropic within different models.
For instance, in ref. [43], the authors vary the slope of the symmetry energy
and found that when the radius of the canonical star is 12.58 km, we have
$\Lambda_{1.4}$ = 515; but when $R_{1.4}=14.30$ km, $\Lambda_{1.4}=812$, an
expected behavior. But when anisotropies are taken into account, for
$R_{1.4}=12.82$ km we have $\Lambda_{1.4}=523$ for $\lambda_{\rm BL}=0.0$,
$R_{1.4}=13.22$ km with $\Lambda_{1.4}=349$ for $\lambda_{\rm BL}=+1.0$ and
finally, $R_{1.4}=13.71$ km with $\Lambda_{1.4}=320$ for $\lambda_{\rm
BL}=+2.0$. This behavior was already pointed out in ref. [14], and it is
because the love number $k_{2}$ decreases with $\lambda_{\rm BL}$ (see fig. 5
from ref. [14]). We can also see that the values for $\Lambda_{1.4}$, as the
$R_{1.4}$, are the same for the four configurations as there are no hyperons
or quarks at this mass value. The dimensionless tidal parameter for the
maximally massive stars is almost independent of the internal composition. It
is around 15-16 for $\lambda_{\rm BL}=0.0$ and about 2-3 for $\lambda_{\rm
BL}=+2.0$.

Concerning the gravitational wave constraints, we see that we easily satisfy
the observation related to the GW170817, $70<\Lambda_{1.4}<580$. [36], once
the tidal parameter decreases with $\lambda_{\rm BL}$. About the mass-gap
object in the GW190814 event, the constraint $458<\Lambda_{1.4}<889$ [32] can
also be fulfilled if $\lambda_{\rm BL}~{}\leq$ 0.2.

###  III.3 Moment of Inertia

|
---|---
|
Figure 5: Moment of inertia for different configurations of hybrid stars. The
anisotropic parameter lies $-2.0~{}<~{}\lambda_{\rm BL}~{}<~{}+2.0$.

The metric of the slowly rotating hybrid stars can be obtained within the
Hartle-Throne approximation, as given in refs. [68, 69, 70]:

$\displaystyle ds^{2}=$ $\displaystyle-e^{2\nu}\ dt^{2}+e^{2\lambda}\
dr+r^{2}\ (d\theta^{2}+\sin^{2}\theta d\phi^{2})$ (31)
$\displaystyle-2\omega(r)r^{2}\sin^{2}\theta\ dt\ d\phi$

For anisotropic hybrid stars, the MOI can be calculated with the methodology
developed in ref. [71]:

$\displaystyle
I=\frac{8\pi}{3}\int_{0}^{R}\frac{r^{5}J\tilde{\omega}}{r-2M}(\rho+p_{r})\left[1+\frac{\sigma}{\rho+p_{r}}\right]\,dr,$
(32)

where $J$ = $e^{-\nu}(1-2m/r)^{1/2}$, $\tilde{\omega}=\bar{\omega}/\Omega$
($\bar{\omega}$ is the frame dragging angular frequency,
$\bar{\omega}=\Omega-\omega(r)$). The final expression (using the Eq. 25) for
the MOI for the anisotropic hybrid star is in the following:

$\displaystyle I$
$\displaystyle=\frac{8\pi}{3}\int_{0}^{R}\frac{r^{5}J\tilde{\omega}}{r-2M}(\rho+p_{r})\left[1+\frac{\frac{\lambda_{\rm
BL}}{3}\frac{(\rho+3p_{r})(\rho+p_{r})r^{2}}{1-2m/r}}{\rho+p_{r}}\right]\,dr\,.$
(33)

The MOI for four scenarios is shown in Fig. 5 for $-2<\lambda_{\rm BL}<+2$
values. The behavior of the MOI as the function of the mass for all cases is
approximately the same; the only difference can be seen in their magnitudes,
as they directly depend on the mass and radius of the star. The deduced error
bars shown in the figures are taken from the ref. [38], predicted from the
universal relations for different types of neutron stars. Except for a few
curves, all lines pass through the error bars. It is observed that the higher
positive values of $\lambda_{\rm BL}$ are unfavorable for the hybrid stars
with those anisotropic configurations. For canonical stars, the MOI varies
from (in terms of $\times~{}10^{45}$ g.cm2) 1.44 for $\lambda_{\rm BL}$ = -2.0
up to 1.82 for $\lambda_{\rm BL}$ = +2.0. These values are fully independent
of the presence of hyperons as well as the strength of the quark-quark
interaction.

However, for the maximally massive hybrid stars, we have the opposite trend.
The results depend on both, the presence of hyperons or not, as well the
quark-quark interaction, although the strength of the quark-quark interaction
is more relevant. For instance, for $\lambda_{\rm BL}$ = -2.0, in the weak
quark-quark interaction, the MOI is (in terms of $\times~{}10^{45}$ g.cm2)
1.60 and 1.61 for Hyperonic and Nucleonic stars respectively. The same
behavior is present for the strong quark-quark interaction, and the MOI is now
1.67 and 1.68 for Hyperonic and Nucleonic stars, respectively. For the
isotropic case, we have the MOI equal to 2.31 (Hyperonic) and 2.32 (Nucleonic)
in the weak quark-quark interaction and 2.41 (Hyperonic) and 2.44 (Nucleonic)
for the strong one. Finally, for $\lambda_{\rm BL}$ = +2.0, the values are
3.64 (Hyperonic+Weak), 3.65 (Nucleonic+Weak), 3.75 (Hyperonic+Strong), and
3.87 (Nucleonic+Strong).

##  IV Conclusions

In this work, we study the influence of anisotropies on the properties of
hybrid stars. We use two configurations for the hadronic EOS (with/without
hyperons), as well as two strengths for the quark-quark interaction. The
anisotropy is taken into account via the so-called BL model. Our main results
can be summarized as:

  * •

The presence of hyperons pushes the hadron-quark phase transition to higher
densities. In the same sense, increasing the quark-quark interaction ($G_{V}$)
also makes the phase transition more difficult.

  * •

The maximum mass is independent of the presence of hyperons. Depending only on
the value of $G_{V}$. This is true for the isotropic case [33], as well as for
each fixed value of $\lambda_{\rm BL}$.

  * •

Positive values of $\lambda_{\rm BL}$ increase the maximum mass but reduce the
central density. Negative values of $\lambda_{\rm BL}$ play the opposite role.
As a consequence, the minimum neutron star mass that presents deconfined quark
in the core increases with $\lambda_{\rm BL}$, but the percentual amount of
quark mass in the neutron stars’ core decreases with $\lambda_{\rm BL}$.

  * •

For a fixed $\lambda_{\rm BL}$, the value of $R_{1.4}$ is independent of the
presence of hyperons and the strength of the quark-quark interaction.

  * •

The constraint related to the radius of the canonical star [22] can be
satisfied assuming -2.0 $\leq~{}\lambda_{\rm BL}~{}\leq$+0.6. About the PSR
J0740+6620, we must have -0.2 $\leq~{}\lambda_{\rm BL}~{}\leq$+1.2. The range
-0.2 $\leq~{}\lambda_{\rm BL}~{}\leq$+0.6 satisfies both constraints
simultaneously.

  * •

The constraints related to the speculative mass of the black widow pulsars PSR
J0952-0607 [31], as well the mass-gap object in the GW190814 event [32] can
also be fulfilled for higher values of, $\lambda_{\rm BL}$. The constraint
related to the HESS object [25] cannot be satisfied within the model employed
in this work.

  * •

The dimensionless tidal parameter $\Lambda$ decreases with $\lambda_{\rm BL}$,
indicating an unexpected behavior, as it predicts that models within larger
radii have lower values of $\Lambda$. All positive values of $\lambda_{\rm
BL}$ are in agreement with the GW171817 constraint. Negative values of
$\lambda_{\rm BL}$ have unphysical solutions (see ref. [60]).

  * •

The MOI increases with $\lambda_{\rm BL}$. Moreover, although most of the
physical quantities related to the maximum mass are independent of the
presence of hyperons, this is not true for the MOI. Nucleonic EOS presents
higher values of MOI. Furthermore, inferences about the MOI from universal
relations disfavor higher values of $\lambda_{\rm BL}$. Future measurements of
MOI can put constraints on the degree of anisotropy.

Acknowledgements:

L.L.L. was partially supported by CNPq Universal Grant No. 409029/2021-1.

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