# Realistic Anisotropic Neutron Stars: Pressure Effects

L. M. Becerra<EMAIL_ADDRESS>Grupo de Investigación en
Relatividad y Gravitación, Escuela de Física, Universidad Industrial de
Santander A. A. 678, Bucaramanga 680002, Colombia  E. A. Becerra-Vergara
<EMAIL_ADDRESS>Grupo de Investigación en Relatividad y
Gravitación, Escuela de Física, Universidad Industrial de Santander A. A. 678,
Bucaramanga 680002, Colombia  F. D. Lora-Clavijo<EMAIL_ADDRESS>Grupo de
Investigación en Relatividad y Gravitación, Escuela de Física, Universidad
Industrial de Santander A. A. 678, Bucaramanga 680002, Colombia

###### Abstract

In this paper, we study the impact of anisotropy on neutron stars with
different equations of state, which have been modeled by a piecewise
polytropic function with continuous sound speed. Anisotropic pressure in
neutron stars is often attributed to interior magnetic fields, rotation, and
the presence of exotic matter or condensates. We quantify the presence of
anisotropy within the star by assuming a quasi-local relationship. We find
that the radial and tangential sound velocities constrain the range of
anisotropy allowed within the star. As expected, the anisotropy affects the
macroscopic properties of stars, and it can be introduced to reconcile them
with astrophysical observations. For instance, the maximum mass of anisotropic
neutron stars can be increased by up to 15% compared to the maximum mass of
the corresponding isotropic configuration. This allows neutron stars to reach
masses greater than $2.5M_{\odot}$, which may explain the secondary compact
object of the GW190814 event. Additionally, we propose a universal relation
for the binding energy of an anisotropic neutron star as a function of the
star’s compactness and the degree of anisotropy.

††preprint: APS/123-QED

##  I  Introduction

Neutron stars (NS) are extremely dense and compact objects that have
traditionally been modeled as isotropic configurations [1, 2, 3, 4]. Recently,
researchers have focused on using anisotropic fluids to provide a more
accurate description of their internal structure, composition, and to account
for various physical phenomena associated with them [5, 6, 7, 8, 9, 10, 11,
12, 13, 14]. For example, phenomena such as phase transitions [15], pion
condensation [16], the presence of very high magnetic field [17, 18], and
relativistic nuclear interactions [19] could produce anisotropy in the stellar
interior, mainly where nuclear matter reaches densities greater than $10^{15}$
g/cm3 [see 20, 5, 21, 14, for a complete review of these mechanisms and of the
anisotropic compact stellar solution in general relativity].

The study of anisotropic pressure effects on NS can shed light on the behavior
of matter under extreme conditions and provide insights into the equation of
state (EOS), which describes the relationship between the pressure, density,
and composition of matter. The anisotropic distribution of nuclear matter can
reveal the presence of exotic particles [22] such as hyperons, kaon
condensates, or a deconfined phase of strange matter [23], as well as phase
transitions [24], the crystallization of the core [25], superfluid core [26],
and the nature of the strong force interactions [27, 28, 12]. This knowledge
is crucial for understanding the behavior of matter in the early universe,
nuclear physics, and astrophysical processes involving high densities.

The first work exploring the impact of anisotropic pressure on stable NS
configurations was carried out by Bowers and Liang [6]. They requested that
the anisotropy vanishes at the origin, and assumed that it exhibits a non-
linear dependence on radial pressure, and is induced by gravity. Their
findings indicated that this anisotropy has non-negligible effects on the mass
and redshift of an NS. Since then, numerous publications have extensively
studied solutions of Einstein’s equations for spherically symmetric static
arrangements, considering anisotropic pressure [8, 7, 29, 30, 31, 21, 32].
These works reveal that anisotropy has remarkable effects on the structure and
properties of NS, including observable properties such as the mass-radius
ratio [6, 33], moment of inertia [34], redshift [35], tidal deformability [36,
37, 38], maximum mass [32, 39], and non-radial oscillation [40]. Furthermore,
anisotropy can stabilize stellar configurations that would otherwise be
unstable [30, 21, 41]. For example, Biswas and Bose [38] noted that certain
EOS ruled out by gravitational waves and electromagnetic observations could
become viable if the star attains a substantial degree of anisotropy.

The influence of pressure anisotropy on stellar properties varies depending on
the selected model and the amount of anisotropy present. However, these
factors can be constrained by the analysis of observational data. Silva  _et
al._ [42] suggested that binary pulsar observations could constrain the range
of anisotropy. Rahmansyah and Sulaksono [43] used a relativistic mean-field
model to show that an anisotropic NS, based on the Horvat  _et al._ [30]
model, is consistent with the constraints imposed by NS multimessenger
observations. Furthermore, the range of pressure anisotropy inside stars was
constrained in [38] using tidal deformability data from GW170817. Similarly,
Das [44] and Ranjan Mohanty  _et al._ [45] constrained the canonical moment of
inertia and f-mode frequency for different degrees of anisotropy using the
GW170817 and GW190814 events. Additionally, the inferred mass and equatorial
radius from the NICER observations of the pulsars PSR J0030+045 [46, 47] and
PSR J0740+662 [48, 49] could further constrain the anisotropy range allowed
within a NS.

The significance of accounting for anisotropy within NSs is undeniable. This
phenomenon holds the potential to guide us towards a more profound
comprehension of the intricate interplay between the internal structure and
observable properties inherent to these astrophysical entities. Therefore, our
paper focuses on investigating the impact of anisotropy on the macroscopic
properties of NSs, such as mass, radius, compactness, and binding energy. We
obtain anisotropic NS configurations by solving Einstein’s field equations for
spherically symmetric matter. In this context, we characterize the anisotropy
of pressure within the star by adopting the quasi-local relation proposed by
Horvat  _et al._ [30]. According to this relation, the difference between the
radial and tangential pressure is assumed to be proportional to the star’s
local compactness and radial pressure, ensuring that anisotropy vanishes at
the center of the star. Furthermore, we study different nuclear EOSs covering
different models and particle compositions of the star. We parameterize them
using the Generalized Piecewise Polytropic (GPP) fit proposed by [50], which
guarantees continuity in radial pressure and sound speed within the star.

The paper is organized as follows. In Sec. II, we present the equations of
structure for anisotropic spherically symmetric stars in General Relativity
(Sec. II.1 ). We also introduce the parameterization used to model various
EOSs for the NS matter with the GPP fit (Sec. II.2), and give a brief
description of the numerical code used to integrate the equations (Sec. II.3).
The macroscopic properties of anisotropic NSs, including mass-radius relation,
compactness, and binding energy, are presented in Sec. III. In this section,
we discuss the essential criteria that the NS configurations must meet to be
considered physically realistic, and the constraints imposed by observations
(see Sec. III.1). Furthermore, in Sec. III.2, we propose a fit for the binding
energy of the star with anisotropy. Finally, we give our concluding remarks in
Sec. IV.

##  II  Neutron Star Matter

###  II.1 Tolman-Oppenheimer-Volkoff Equations

Let us consider an anisotropic fluid in a spherically symmetric spacetime,
whose line element is given in terms of the components of the metric
$g_{\alpha\beta}$ tensor by

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-c^{2}\alpha^{2}\mathrm{d}\mathrm{t}^{2}+\left(1-\dfrac{2Gm}{c^{2}r}\right)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega,$
(1)

being $\alpha=\alpha(r)$, $m=m(r)$,
$\mathrm{d}\Omega=\mathrm{d}\theta^{2}+\sin^{2}\theta\ \mathrm{d}\phi^{2}$,
$G$ the gravitational constant and $c$ the speed of light. The general energy-
momentum tensor for a static and spherically symmetric fluid can be written as
[51]

$\displaystyle T_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(\epsilon+P_{\perp})u_{\alpha}u_{\beta}+P_{\perp}g_{\alpha\beta}+(P-P_{\perp})n_{\alpha}n_{\beta},$
(2)

where $P$ is the radial pressure and $P_{\perp}$ is the tangential pressure.
It is worth mentioning that $u^{\alpha}$ and $n^{\alpha}$, given by the
following expressions

$\displaystyle u^{\alpha}$ $\displaystyle=$
$\displaystyle\left[\dfrac{1}{c\alpha},0,0,0\right],$ (3) $\displaystyle
n^{\alpha}$ $\displaystyle=$
$\displaystyle\left[0,\left(1-\dfrac{2Gm}{c^{2}r}\right)^{1/2},0,0\right],$
(4)

correspond to unitary time-like and space-like vectors, that is,
$u^{\alpha}u_{\alpha}=-1$ and $n^{\alpha}n_{\alpha}=1$.

By solving the Einstein field equations and matter equations, a general
expression for an anisotropic spherically symmetric compact star is obtained

$\displaystyle\dfrac{\mathrm{d}m}{\mathrm{d}r}$ $\displaystyle=$
$\displaystyle 4\pi r^{2}\epsilon,$ (5)
$\displaystyle\dfrac{\mathrm{d}P}{\mathrm{d}r}$ $\displaystyle=$
$\displaystyle-\dfrac{\left(\epsilon+\dfrac{P}{c^{2}}\right)\left(m+\dfrac{4\pi
r^{3}P}{c^{2}}\right)}{\dfrac{r^{2}}{G}\left(1-\dfrac{2Gm}{rc^{2}}\right)}+\frac{2}{r}(P_{\perp}-P),$
(6) $\displaystyle\dfrac{1}{\alpha}\dfrac{\mathrm{d}\alpha}{\mathrm{d}r}$
$\displaystyle=$ $\displaystyle\dfrac{G}{c^{2}r^{2}}\left(m+\dfrac{4\pi
r^{3}P}{c^{2}}\right)\left(1-\dfrac{2Gm}{rc^{2}}\right)^{-1}.$ (7)

Notice that Eq.(6) is the only one that contains the contribution of the
radial and tangential pressure by the difference $P-P_{\perp}$. The assumption
of spherical symmetry holds for static matter sources where the energy-
momentum tensor exhibits the specific condition: $\lvert
T_{\theta}^{\theta}-T_{\phi}^{\phi}\rvert<<T_{\theta}^{\theta}$. This
assumption works in diverse physical scenarios, including those characterized
by a solid core, or the state of pion condensation.

To accommodate the transition between the isotropic and anisotropic regimes,
it becomes necessary to introduce a functional form for $P-P_{\perp}$,
considering that the specific relationship between energy density and radial
and tangential pressures is unknown due to its dependence on microscopic
factors. Following the work of Horvat  _et al._ [30], the tangential pressure
can be written as

$P_{\perp}=P\left(1+\lambda_{a}~{}\frac{2Gm}{c^{2}r}\right)\,,$ (8)

where $\lambda_{a}$ is a parameter controlling the degree of anisotropy. When
the anisotropy is due to a condensate phase of pions [16],
$0\leq\left(P-P_{\perp}\right)/P\leq 1$, therefore we could expect that the
maximum value for the pressure difference is of the order of unity.
Considering that the compactness of NSs ranges from $0.05$ to $0.3$ more or
less, the range of values for the anisotropy parameter we will be using for
this work is $-2\leq\lambda_{a}\leq 2$ [30, 40]. Among various models in the
literature, relation (8) introduces the effects of pressure anisotropy in a
phenomenological manner [see e. g. 52, for a covariant relation between $P$
and $P_{\perp}$]. Additionally, there are also works that introduce pressure
anisotropy self-consistently as the models presented in [53] of stars with
elastic matter [see also 54, for a review].

###  II.2  Equation of State

We parameterize the NS EOS using the Generalized Piecewise Polytropic (GPP)
fit presented in O’Boyle  _et al._ [50]. For the mass density, $\rho$, in the
interval between $[\rho_{i},\rho_{i+1}]$, the pressure, $P$, and energy
density, $\epsilon$, are given by:

$\displaystyle P(\rho)$ $\displaystyle=$ $\displaystyle
K_{i}\rho^{\Gamma_{i}}+\Lambda_{i}$ (9) $\displaystyle\epsilon(\rho)$
$\displaystyle=$
$\displaystyle\frac{K_{i}}{\Gamma_{i}+1}\rho^{\Gamma_{i}}+(1+a_{i})\rho-\Lambda_{i}\,,$
(10)

where the parameters $K_{i}$, $\Lambda_{i}$ and $a_{i}$ are determined by
enforcing continuity and differentiability of the energy density and pressure
across the dividing densities:

$\displaystyle K_{i+1}$ $\displaystyle=$ $\displaystyle
K_{i}\frac{\Gamma_{i}}{\Gamma_{i+1}}\rho^{\Gamma_{i}-\Gamma_{i+1}}$ (11)
$\displaystyle\Lambda_{i+1}$ $\displaystyle=$
$\displaystyle\Lambda_{i}+\left(1-\frac{\Gamma_{i}}{\Gamma_{i+1}}\right)K_{i}\rho^{\Gamma_{i}}$
(12) $\displaystyle a_{i+1}$ $\displaystyle=$ $\displaystyle
a_{i}+\Gamma_{i}\frac{\Gamma_{i+1}-\Gamma_{i}}{(\Gamma_{i+1}-1)(\Gamma_{i})-1}K_{i}\rho^{\Gamma_{i}-1}$
(13)

This paper explores a broad range of nuclear EOS from different models and
composition. All the EOS are listed in Table 2 in Appendix A and were obtained
using Compose [55]. For the low-density crust, we use the fit of the Sly(4)
EOS given in [50]. More details on the best-fits made with the GPP framework
can be found in Appendix A. We use these fits to compute anisotropic NS
configurations.

###  II.3  Numerical Calculations

All the simulations are computed using a fourth-order Runge-Kutta integrator
in a 1D spherical grid extending from $r=0M$ to the outer domain boundary,
$r_{max}=100M$. To avoid the singular behaviour at $r=0$, we follow the
procedure shown in [56], where a Taylor expansion is performed around this
point. The resulting approximate regular equations are programmed at least for
the first mesh point located at $r=\Delta r$, being $\Delta r$ the uniform
spatial resolution of the grid. On the other hand, the radius $R_{\rm NS}$ of
the surface of the star is defined as the radius $r=R_{\rm NS}$, where the
mass density $\rho(R_{\rm NS})=10^{4}$ g cm-3. In this sense, the mass of the
configuration is $M=m(R_{\rm NS})$. In particular, this code has been used to
constrain several configurations of neutron and quark stars using the GW170817
observation [57]. In addition, we have used this code to study anisotropic
quark stars with an interacting quark equation of state, where the
contribution of the fourth-order corrections parameter a4 of the QCD
perturbation on the radial and tangential pressure generates significant
effects on the mass-radius relation and the stability of the quark star [12].

##  III  Neutron Stars Properties

###  III.1  Mass-Radius Relation

Figure 1: Mass-radius relations for different equations of states with the GPP
parameterization (see Table 2) The color scale corresponds to the value of the
anisotropic parameter, $\lambda_{a}$. Positive values support more massive
configurations. Configurations on the left of the solid black line are stable
against radial perturbations. Configurations on the right of the dashed black
lines satisfy the causality conditions for radial sound speed, while the
configurations inside the region enclosed by the dotted black line satisfy it
for tangential sound speed.  Figure 2: Fraction between the maximum mass for
an anisotropic configuration, $M_{\rm NS,max}$, and the isotropic one,
$M^{\lambda_{a}=0}_{\rm NS,max}$, as a function of the anisotropic parameter.
Configuration with $M<M_{\rm NS,max}$ satisfies the stability and causality
conditions. Colored lines correspond to the results of the EOS used, and black
line is the fit given in Eq. 15.

For the EOS presented in Table 2, we solve equations (5)-(7) together with
equation (8), for different values of the anisotropic parameter $\lambda_{a}$
and the central mass density $\rho_{c}$. The resulting mass-radius relations
are shown in Fig. 1. In general, positive (negative) values of the anisotropic
parameter support more (less) massive configurations compared with the
isotropic model ($\lambda_{a}=0$). This is expected since positive (negative)
$\lambda_{a}$ produces an outward (inward) force that is opposite (aligned) to
the gravitational force (see Eq. 6).

We consider the obtained configurations to represent physical objects if they
satisfy the two following conditions:

  * •

Stability condition: the central density $\rho_{c}$ should be smaller than
$\rho_{c}(M_{\rm max})$, where $M_{\rm max}$ corresponds to the configuration
with the maximum mass. Models with $\rho_{c}>\rho_{c}(M_{\rm max})$ are
unstable to radial perturbation and collapse into black holes [58]. In Fig. 1,
configurations located to the left of the solid black line are stable to
radial perturbations.

  * •

Causality condition: The radial and tangential speeds of sound, denoted by:

$c_{s,r}^{2}=\frac{\partial P}{\partial\epsilon}\qquad{\rm and}\qquad
c_{s,t}^{2}=\frac{\partial P_{\perp}}{\partial\epsilon},$ (14)

respectively, must not exceed the speed of light, $c$. In Fig. 1,
configurations to the left of the dashed black lines have superluminal radial
sound speeds. While, the configurations inside the region bounded by the
dotted black line satisfies the condition $0<c_{\rm s,t}^{2}<c^{2}$. It is
important to clarify that in a spherically symmetric anisotropic
configuration, up to five independent wave speed modes could be identified
[59, 54]. While in this work we define and analyze the two main wave modes in
the radial and tangential directions, it would be extremely interesting to
explore, in future research, whether these configurations exhibit all wave
speeds and if subluminal speeds persist.

Further conditions of acceptability for self-gravitating stellar models have
been compiled in other works [see e.g. 60, 61]. But, we only checked for the
two above conditions because we considered them to be the minimum set of
conditions that a model would have to meet in order to be physically feasible.

As expected, EOS models built with the relativistic mean field theory, such as
GM1Y6, XMLSL, DDH, DD2Y, QHC19, QHC21, and CMF8-EOS, naturally satisfy the
casualty condition for the radial speed of sound. This is also the case for
the SKI2, SKI3, SKI4, SKI5, SKI6 and BsK-25-EOS. For the BHF, RS, SKMP, SKOP,
SLY230a and SLY9-EOS, we found configurations with superluminal radial
velocities, but they are unstable to radial perturbations. This is also the
case for the remaining EOS studied here, when the anisotropic parameter is
positive, $\lambda_{a}>0$ (i.e. tangential pressure is greater than the radial
one). For the opposite case, when $\lambda_{a}<0$ (i.e. radial pressure is
greater than the tangential one), we found stable configurations with radial
sound speed greater than the speed of light. It is particular the case of the
APR-EOS because we found this last situation but for all the values of the
anisotropic parameter considered here. For this reason, we discard the APR-EOS
and do not consider it in the following analysis.

The maximum value of the anisotropic parameter for all EOS is limited by the
tangential sound speed. Configurations with superluminal tangential sound
speeds occur generally for $\lambda_{a}>1.0$. In the case of the relativistic
EOS, GM1Y6, DDH, XML5L, DD2Y, and CMF8, there are no restrictions on the
minimum value of the anisotropic parameter. However, for the remaining EOS,
configurations with negative anisotropic parameter can be stable to radial
perturbations, but do not satisfy the causality condition of the tangential
sound speed, and thus the maximum stable mass is restricted.

For a given value of the anisotropic parameter, the maximum mass is the mass
of the most massive configuration that satisfies both the stability and
causality conditions for the radial and tangential sound speeds. Fig. 2
displays the ratio between the maximum mass for an anisotropic configuration
and the one of the isotropic model ($\lambda_{a}=0$) as a function of the
anisotropy parameter, $\lambda_{a}$, for each EOS. The maximum mass increases
with the anisotropic parameter until $\lambda_{a}\sim 1.0$, where it becomes a
decreasing function. In general, for negative values of the anisotropic
parameter or values greater than one, the maximum mass is defined by the
causality condition for the tangential sound speed, while for positive values
and less than one, it is defined by the stability condition. We can fit the
maximum mass with the following relation, independent of the EOS:

$M_{\rm NS,max}=\begin{cases}M^{\lambda_{a}=0}_{\rm
NS,max}(1.0+0.16\lambda_{a})&\lambda_{a}<1.0\\\ &\\\ M^{\lambda_{a}=0}_{\rm
NS,max}(1.0-0.5\lambda_{a})&\lambda_{a}>1.0\end{cases}$ (15)

This relation reproduced the maximum allowed mass for an anisotropic NS
configuration with $1-\chi^{2}\approx 0.015$ for $\lambda_{a}<1$ and
$1-\chi^{2}\approx 0.038$ for $\lambda_{a}>1$.

Figure 3: Mass-radius relation of the isotropic configuration for all the EOS summarized in Table 2. In blue are the configurations composed of npem matter, in red the ones with hyperons and in black the ones with quarks. Colors bands indicate observational constrains given by the pulsar masses of PSR J0348+0432 [62] and PSR J0952-0607, the mass and radius NICER constraints for the pulsars PSR J0030+0451 [46, 47] and PSR J0740+662 [48, 49], and the mass of the secondary compact object of the GW190814 event [63]. The dashed gray line corresponds to the faster observed pulsar, PSR J1748-2446ad [64], and the dotted gray line to the Buchdahl limit for the star compactness [65].  Figure 4: Same as Fig. 3 but for anisotropic NSs with, $\lambda_{a}$ $\epsilon$ $(-2,2)$ for two selected EOS: BHF and GM1Y6. Only models satisfying the stability and causality conditions are displayed. EoS |  PSR J0348+0432 |  PSR J0952-0607 |  PSR J0030+0451 |  PSR J0740+6620 |  GW190814
---|---|---|---|---|---
| $M>1.97M_{\odot}$ | $M>2.17M_{\odot}$ | NICER | NICER | $M>2.5M_{\odot}$
BHF | $\lambda_{a}>0.24$ | $\lambda_{a}>0.52$ | ✗ | ✗ | ✗
KDE0v | $\lambda_{a}>0.08$ | $\lambda_{a}>0.88$ | $\lambda_{a}>0.24$ | ✗ | ✗
KDE1v | $\lambda_{a}>0.08$ | $\lambda_{a}>0.84$ | $\lambda_{a}>-0.36$ | ✗ | ✗
RS | $\lambda_{a}>-0.40$ | $\lambda_{a}>0.28$ | ✓ | $\lambda_{a}>-0.12$ | ✗
SK255 | $\lambda_{a}>-0.40$ | $\lambda_{a}>0.20$ | ✓ | $\lambda_{a}>0.04$ | ✗
SK272 | $\lambda_{a}>-0.52$ | $\lambda_{a}>-0.04$ | ✓ | $\lambda_{a}>-0.43$ | ✗
SkI2 | $\lambda_{a}>-0.56$ | $\lambda_{a}>0.12$ | ✓ | $\lambda_{a}>-0.40$ | ✗
SkI3 | $\lambda_{a}>-0.76$ | $\lambda_{a}>-0.12$ | ✓ | $\lambda_{a}>-0.63$ | $\lambda_{a}>0.84$
SkI4 | $\lambda_{a}>-0.56$ | $\lambda_{a}>0.08$ | ✓ | $\lambda_{a}>-0.24$ | ✗
SkI5 | $\lambda_{a}>-0.72$ | $\lambda_{a}>-0.12$ | ✓ | $\lambda_{a}>-0.63$ | $\lambda_{a}>0.88$
SkI6 | $\lambda_{a}>-0.60$ | $\lambda_{a}>0.04$ | ✓ | $\lambda_{a}>-0.32$ | ✗
SkMp | $\lambda_{a}>-0.36$ | $\lambda_{a}>0.32$ | ✓ | $\lambda_{a}>0.04$ | ✗
SkOp | $\lambda_{a}>0.08$ | $\lambda_{a}>0.8$ | $\lambda_{a}>-1.68$ | $\lambda_{a}>0.72$ | ✗
SLy230a | $\lambda_{a}>-0.32$ | $\lambda_{a}>0.32$ | $\lambda_{a}>-0.76$ | $\lambda_{a}>0.40$ | ✗
SLy2 | $\lambda_{a}>-0.20$ | $\lambda_{a}>0.52$ | $\lambda_{a}>-0.72$ | $\lambda_{a}>0.76$ | ✗
SLy4 | $\lambda_{a}>-0.16$ | $\lambda_{a}>0.52$ | $\lambda_{a}>-0.52$ | $\lambda_{a}>0.80$ | ✗
SLy9 | $\lambda_{a}>-0.44$ | $\lambda_{a}>0.16$ | ✓ | $\lambda_{a}>-0.12$ | ✗
BSk25 | $\lambda_{a}>-0.84$ | $\lambda_{a}>-0.20$ | ✓ | $\lambda_{a}>-0.72$ | $\lambda_{a}>0.72$
GM1Y6 | $\lambda_{a}>-0.92$ | $\lambda_{a}>-0.32$ | ✓ | $\lambda_{a}>-0.80$ | $\lambda_{a}>0.56$
DDH | $\lambda_{a}>-0.44$ | $\lambda_{a}>0.16$ | ✓ | $\lambda_{a}>-0.32$ | ✗
DD2Y | $\lambda_{a}>-0.20$ | $\lambda_{a}>0.43$ | ✓ | $\lambda_{a}>-0.08$ | ✗
XMLSL | $\lambda_{a}>-0.60$ | $\lambda_{a}>0.04$ | ✓ | $\lambda_{a}>-0.48$ | $\lambda_{a}>0.96$
QHC19 | $\lambda_{a}>0.16$ | $\lambda_{a}>0.84$ | $\lambda_{a}>0.20$ | $\lambda_{a}>0.84$ | ✗
QHC21 | $\lambda_{a}>-0.56$ | $\lambda_{a}>0.08$ | ✓ | $\lambda_{a}>-0.32$ | ✗
CMF8 | $\lambda_{a}>-0.20$ | $\lambda_{a}>0.44$ | ✓ | $\lambda_{a}>-0.08$ | ✗
Table 1:  Limiting values for the anisotropic parameter from which the
observational constraints are satisfied. When the ( ✗) symbol is present, none
of the values within the range $[-2,1]$ satisfy the given constraint.
Conversely, when the ( ✓) symbol is present, all configurations within the
same range satisfy the constraint. It is worth mentioning that the maximum
value of the anisotropic parameter for all EOS is limited by the tangential
sound speed. Configurations with superluminal tangential sound speeds occur
generally for $\lambda_{a}>1.0$.

In addition to the conditions mentioned above, we also compare our models to
astrophysical observations. Fig. 3 shows the mass-radius relation of the
isotropic configuration for all the EOS summarized in Table 2. In this figure,
we have included the following observational constraints:

  * •

Observation of pulsar masses: PSR J0348+0432 with $2.01\pm 0.04~{}M_{\odot}$
[62] and PSR J0952-0607 with $2.35\pm 0.17M_{\odot}$ [66].

  * •

NICER constraints for the mass-radius relation of PSR J0030+0451 [46, 47] and
PSR J0740+662 [48, 49].

  * •

The mass of the secondary compact object of the merger in GW190814 [63], which
is between $2.5$ and $2.7~{}M_{\odot}$.

As can be seen in Fig. 3, no isotropic model could explain the massive
secondary object of GW190814 as a non-rotating NS. On the other hand, BHF,
KDE0v, KDE1v-EOS, SKOP and QHC19-EOS would be ruled out by the constraints of
pulsar observations. However, anisotropic configurations have the potential to
satisfy some of these constraints for a given EOS, when the isotropic model
does not. For example, Fig. 4 shows the mass-radius relation with different
values of the anisotropic parameter, $\lambda_{a}$, for the BHF and GM1Y6-EOS.
While the isotropic configurations with the BHF-EOS do not satisfy any of the
observational constraints, the anisotropic configurations with
$\lambda_{a}>0.24$ satisfy the mass constraint given by the pulsar PSR
J0348+0432, and the configuration with $\lambda_{a}>0.52$ could model the
pulsar PSR J0030+0451. On the other hand, GM1Y6-EOS satisfies nearly all
observational constraints, and a configuration with $\lambda_{a}>0.56$ of the
anisotropic model could describe the secondary object of GW190814.

In Table 1 we report the value range of the anisotropic parameter that
satisfies the observational constraint for each EOS considered here. If the
boundary value of the anisotropic parameter is negative, it means that the
corresponding isotropic model satisfies the given constraint. Notably, NICER
observations of PSR J0030+0451 rule out BHF, KDE0v, and KDE1v-EOS, even when
anisotropic configurations are taken into consideration. The secondary object
observed in the GW190814 event could potentially be explained as an
anisotropic NS with SKI3, SKI5, GM1Y6, BsK25, and XMLSL-EOS.

###  III.2  Compactness and Binding-Energy

For a non-rotating, spherically symmetric NS, its total gravitational mass is
defined as:

$M_{\rm NS}=\int_{0}^{R_{\rm NS}}4\pi r^{2}\epsilon\,dr\,$ (16)

while, its rests mass is given by:

$M_{B}=m_{B}\mathcal{N}$ (17)

where $\mathcal{N}$ is the total number of baryons:

$\mathcal{N}=\frac{1}{m_{B}}\int_{0}^{R_{\rm NS}}4\pi
r^{2}\rho\left(1-\frac{2Gm}{rc^{2}}\right)^{-1/2}dr\,,$ (18)

Then, the binding energy of the star, $BE$, is:

${\rm BE}=(M_{\rm NS}-M_{B})c^{2}\,,$ (19)

which can be understood as the amount of energy needed to bring $\mathcal{N}$
baryons together from infinity to form a stable star. During a core collapse,
approximately 99% of the star’s binding energy is released through the
emission of neutrinos [67].

The binding energy depends on the internal structure of the star and can be
calculated using Eqs. (5)-(7). Fig. 5 shows the specific binding energy as
function of the compactness of the star, $\mathcal{C}=GM_{\rm NS}/R_{\rm
NS}c^{2}$, for the different EOS and values of the anisotropic parameter.
Positive values for the anisotropic parameter increase the binding energy,
while, negative values, decrease it. It can be clearly seen that, for the EOS
considered here, the specific binding energy is a universal function of the
compactness of the star.

In fact, in [67] is proposed a relatively accurate universal relation of the
binding energy as:

$\dfrac{\rm BE}{M_{\rm NS}c^{2}}(\mathcal{C})\approx\frac{(0.6\pm
0.05)\mathcal{C}}{1-0.5\mathcal{C}}.$ (20)

In this work, we propose the following extended relation that includes
anisotropic configurations:

$\dfrac{\rm BE}{M_{\rm
NS}c^{2}}(\mathcal{C},\lambda_{a})=\dfrac{\alpha(\lambda_{a})\
\mathcal{C}}{1-\beta(\lambda_{a})\mathcal{C}}\,,$ (21)

where

$\alpha(\lambda_{a})=0.5213-0.0197\lambda_{a}+0.0059\lambda_{a}^{2}\,,$
$\beta(\lambda_{a})=0.5732+0.6024\lambda_{a}-0.1942\lambda_{a}^{2}\,.$

The above relation is plotted in Fig. 6. The error reported is calculated as:

${\rm error}=\frac{|{\rm BE}-{\rm BE}_{\rm fit}|}{\rm BE}.$ (22)

The differences between the individual EOSs and the universal fits are in the
order of a few percent. This universal fit performs better for compactness
values greater than $0.1$, with an error less than 10%.

Figure 5: Specific binding energy as a function of star compactness for all
EOS summarized in Table 2. The color scale corresponds to the value of the
anisotropic parameter. Only models satisfying the stability and causality
conditions are displayed. Figure 6: Fit for the specific binding energy as a
function of star compactness (see equation 21). It is worth mentioning that
all fits were done up to $\mathcal{C}=M_{max}/R_{max}$.

##  IV  Discussions and Conclusions

In this paper, we study spherical anisotropic configurations of NS, adopting
the quasi-local relation for the tangential pressure proposed by [30] (see Eq.
8). The anisotropy strength is proportional to the anisotropic parameter,
$\lambda_{a}$. It should be noted that our results can be quite sensitive to
this choice. We have studied a wide range of nuclear EOS and parameterized
them using the GPP formalism [50]. We have calculated the mass-radius relation
for all of them. As expected, it is affected by the value of the anisotropic
parameter. More massive configurations can be obtained by making the
tangential pressure greater than the radial one ( i.e. for $\lambda_{a}>0$).

We consider an anisotropic configuration to be physically feasible if it is
stable to radial perturbations and satisfies the causality condition for the
radial and tangential sound speeds. We have found that the maximum value of
the anisotropic parameter for all EOS is mainly limited by the tangential
sound speed. Configurations with superluminal tangential sound speeds
generally occur for $\lambda_{a}>1.0$ and $\lambda_{a}<0$. We have found that
introducing anisotropy in the form of Eq. 8 can increase the maximum mass for
an EOS to about 15% of the maximum mass of the isotropic configuration.

In addition, we have compared the mass-radius relations obtained for each EOS
with the observational constraints: observations of pulsar masses (PSR
J0348+0432 and PSR J0952-0607), the results of NICER observations of PSR
J0030+045 and PSR J0740+662, and gravitational wave data. We conclude than an
anisotropic configuration has the potential to satisfy the observational
constraints, especially if the corresponding isotropic configuration does not
(see Table 1). For example, the massive secondary object of GW190814 could be
explained by an anisotropic NS.

Finally, we propose a universal relation for the binding energy of anisotropic
stars, given by Eq. 21. This relation is satisfied for all EOSs used here,
regardless of their particle composition. It is worth saying that this
relation, as well as Eq. 15, depends directly on the anisotropic parameter,
$\lambda_{a}$, introduced in Eq. 8. For a different function of the tangential
pressure, these fits will not apply.

###### Acknowledgements.

F.D.L-C was supported by the Vicerrectoría de Investigación y Extensión -
Universidad Industrial de Santander, under Grant No. 3703. L. M. B is
supported by the Vicerrectoría de Investigación y Extensión - Universidad
Industrial de Santander Postdoctoral Fellowship Program No. 2023000359. E. A.
B-V is supported by the Vicerrectoría de Investigación y Extensión -
Universidad Industrial de Santander Postdoctoral Fellowship Program No.
2023000354.

##  Appendix A Parameterization for the Equation of State

In this Appendix, we describe our numerical approach to obtain a GPP
parameterization for a nuclear EOS.

For the high-density core, following [50], we used a three-segment
parameterization, with the limiting densities $\rho_{1}=10^{14.87}$g cm-3 and
$\rho_{2}=10^{14.99}$g cm-3. For the low-density crust, we use the fit of the
Sly(4) EOS given in Table II of [50]. Then, in order to obtain a parameterized
model of a given EOS, we need to determine the set of four parameters:
$\\{\rho_{0},\Gamma_{1},\Gamma_{2},\Gamma_{3}\\}$, where $\rho_{0}$ is the
density separating the star’s core from its crust. The remaining parameters
($K_{i},\Lambda_{i}$ and $a_{i}$) are given by equations (11)-(13).

We perform a Markov Chain Monte Carlo (MCMC) simulation to find the set of
parameters: $\\{\rho_{0},\Gamma_{1},\Gamma_{2},\Gamma_{3}\\}$ that minimize
the difference between the tabulated EOs and the GPP parameterization. For
this we used the PyMC3 library 111https://github.com/pymc-devs/pymc of Python.
The parameters are assumed to be normal distributed. First, we found an
initial guess for the set of parameters using a
Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization algorithm, and iterate
from these values for a total of $40,000$ iterations using a No-U-Turn Sampler
(NUTS). The results are summarized in Table 2. We report the rms residual of
the fit:

$\mathrm{rms}=\sqrt{\frac{1}{N}\sum_{j}(\log P_{j}-\log(P(\rho_{j}))^{2}}\,$
(23)

where $N$ is the number of points of the tabulated EoS, $P_{j}$ is the
tabulated pressure and $P(\rho)$ is given by equation (9). For illustration,
Fig. 7 shows a comparison of the pressure and mass density relation between
the tabulated EOS and the obtained GPP parameterization.

Figure 7:  Pressure as function of mass density for the QHC21 EoS and its corresponding GPP parameterization. The dotted vertical lines correspond to the limiting densities of the zones shown.  EoS |  Composition | $\mathbf{M_{\rm max}}$ |  $\mathbf{\log\rho_{0}}$ |  $\mathbf{\Gamma_{1}}$ |  $\mathbf{\Gamma_{2}}$ |  $\mathbf{\Gamma_{3}}$ |  $\mathbf{\rm rms}$
---|---|---|---|---|---|---|---
|  | $[\,M_{\odot}\,]$ |  $[\mathrm{g\,cm^{-3}}]$ |  |  |  |
APR [69] | npem | $2.17$ | $14.02\pm 0.02$ | $3.09\pm 0.07$ | $3.59\pm 0.17$ | $3.61\pm 0.20$ | $0.065$
BHF [70] | npem | $1.92$ | $14.08\pm 0.02$ | $3.21\pm 0.08$ | $2.58\pm 0.18$ | $2.60\pm 0.21$ | $0.068$
KDE0V [71, 72, 73] | npem | $1.97$ |  $13.95\pm 0.02$ | $2.92\pm 0.07$ |  $2.83\pm 0.19$ |  $2.85\pm 0.22$ | $0.015$
KDE1V [71, 72, 73] | npem | $1.98$ | $13.89\pm 0.02$ | $2.84\pm 0.07$ | $2.83\pm 0.19$ | $2.86\pm 0.24$ | $0.022$
RS [71, 74, 73] | npem | $2.12$ | $13.71\pm 0.03$ | $2.73\pm 0.06$ | $2.57\pm 0.28$ | $2.59\pm 0.34$ | $0.033$
SK255 [71, 75, 73] | npem | $2.15$ | $13.65\pm 0.03$ | $2.65\pm 0.05$ | $2.84\pm 0.26$ | $2.85\pm 0.33$ | $0.019$
SK272 [71, 75, 73] | npem | $2.24$ | $13.70\pm 0.04$ | $2.75\pm 0.06$ | $2.88\pm 0.26$ | $2.93\pm 0.33$ | $0.021$
SkI2 [71, 76, 73] | npem | $2.17$ | $13.65\pm 0.03$ | $2.70\pm 0.06$ | $2.49\pm 0.26$ | $2.50\pm 0.32$ | $0.044$
SkI3 [71, 76, 73] | npem | $2.25$ | $13.73\pm 0.03$ | $2.85\pm 0.06$ | $2.43\pm 0.36$ | $2.43\pm 0.42$ | $0.046$
SkI4 [71, 76, 73] | npem | $2.18$ | $13.95\pm 0.02$ | $3.16\pm 0.07$ | $2.44\pm 0.25$ | $2.43\pm 0.29$ | $0.063$
SkI5 [71, 76, 73] | npem | $2.25$ | $13.58\pm 0.04$ | $2.66\pm 0.06$ | $2.54\pm 0.39$ | $2.54\pm 0.47$ | $0.051$
SkI6 [71, 76, 73] | npem | $2.20$ | $13.94\pm 0.02$ | $3.15\pm 0.07$ | $2.42\pm 0.32$ | $2.41\pm 0.37$ | $0.054$
SkMp [71, 77, 73] | npem | $2.17$ | $13.80\pm 0.03$ | $2.83\pm 0.06$ | $2.66\pm 0.24$ | $2.67\pm 0.29$ | $0.025$
SkOp [71, 78, 73] | npem | $1.98$ | $13.77\pm 0.03$ | $2.71\pm 0.06$ | $2.62\pm 0.19$ | $2.64\pm 0.24$ | $0.009$
SLy230a [71, 79, 73] | npem | $2.11$ | $14.03\pm 0.02$ | $3.24\pm 0.07$ | $2.63\pm 0.22$ | $2.63\pm 0.26$ | $0.032$
SLy2 [71, 79, 73] | npem | $2.06$ | $13.96\pm 0.02$ | $3.02\pm 0.06$ | $2.79\pm 0.19$ | $2.80\pm 0.23$ | $0.005$
SLy4 [71, 79, 73] | npem | $2.06$ | $13.98\pm 0.02$ | $3.05\pm 0.07$ | $2.80\pm 0.19$ | $2.81\pm 0.23$ | $0.007$
SLy9 [71, 79, 73] | npem | $2.16$ | $13.90\pm 0.02$ | $3.02\pm 0.66$ | $2.66\pm 0.26$ | $2.68\pm 0.31$ | $0.011$
BSk25 [80, 81] | npem | $2.22$ | $13.90\pm 0.02$ | $3.14\pm 0.07$ | $2.26\pm 0.38$ | $2.26\pm 0.43$ | $0.046$
GM1Y6 [82, 83] | npeN | $2.29$ | $13.75\pm 0.02$ | $2.96\pm 0.05$ | $1.62\pm 0.35$ | $1.63\pm 0.39$ | $0.006$
DDH [82, 84] | npeN | $2.05$ | $14.06\pm 0.01$ | $3.52\pm 0.05$ | $1.15\pm 0.11$ | $1.14\pm 0.12$ | $0.326$
DD2Y [85, 86] | npeN | $2.04$ | $13.86\pm 0.02$ | $2.99\pm 0.06$ | $1.69\pm 0.29$ | $1.69\pm 0.32$ | $0.073$
XMLSL [87] | npemN | $2.18$ | $13.58\pm 0.03$ | $2.68\pm 0.05$ | $1.83\pm 0.47$ | $1.88\pm 0.53$ | $0.038$
QHC19 [88] | npeN-quark | $1.93$ | $14.09\pm 0.02$ | $3.38\pm 0.08$ | $1.95\pm 0.23$ | $1.98\pm 0.27$ | $0.022$
QHC21 [89] | npeN-quark | $2.20$ | $13.98\pm 0.02$ | $3.26\pm 0.07$ | $2.12\pm 0.30$ | $2.22\pm 0.37$ | $0.058$
CMF8 [90] | npeN-quark | $2.01$ | $13.78\pm 0.03$ | $2.89\pm 0.05$ | $1.29\pm 0.22$ | $1.33\pm 0.06$ | $0.098$
Table 2: GPP fit parameter for the EOS used in this paper. All EOS used here
were obtained with Compose (https://compose.obspm.fr). npem: neutrons,
protons, electrons and muons. npemN: neutrons, protons, electrons and
hyperons.

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