# Impact of Ricci Inverse Gravity on Hybrid Star Model

Adnan Malik<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>School of Mathematical Sciences, Zhejiang Normal
University,
Jinhua, Zhejiang, China.  Department of Mathematics, University of Management
and Technology,
Sialkot Campus, Lahore, Pakistan  Amjad Hussain<EMAIL_ADDRESS>National University of Computer and Emerging Sciences,
Chiniot-Faisalabad Campus, Pakistan  M. Farasat Shamir
<EMAIL_ADDRESS>National University of Computer and Emerging
Sciences,
Lahore Campus, Pakistan.  Ayesha Almas<EMAIL_ADDRESS>Department of
Mathematics, University of Management and Technology,
Sialkot Campus, Pakistan.

###### Abstract

The objective of our current study is to explore novel aspects of a stationary
anisotropic relativistic hybrid compact star that consists of quark matter
(QM) in its core and ordinary baryonic matter (OBM) in its crust. This study
has been done by adopting separate equations of states (EoSs) for quark matter
and baryonic matter. The MIT bag model equation of state
$p_{q}=\frac{1}{3}(\rho_{q}-4B)$ has been used to demonstrate a correlation
between the density and pressure of weird quark matter in the interior of the
star. In addition, we present a simple linear equation of state
$p_{r}=\beta_{1}\rho-\beta$ that links radial pressure and matter density for
OBM. The stellar model was formulated within the context of
$f(\mathcal{R},\mathcal{A})$ gravity, utilizing a linear correlation between
Ricci tensor $\mathcal{R}$ and anticurvature scaler $\mathcal{A}$. To solve
the field equations of this novel alternative gravity, we employ the Krori and
Barua approach to the metric potentials. The validity of our suggested model
is assessed using the graphical approach, while ensuring that the conditions
are physically feasible. Our focus is specifically on the tiny celestial
object known as LMC X-4
[$\text{M}=(1.04^{+0.09}_{-0.09})M_{\odot};\text{R}=8.301^{+0.2}_{-0.2}\text{km}$],
which we consider a viable candidate for a strange quark star. We want to
clarify the model’s physical validity by examining a variety of physical
assessments, including dynamical equilibrium, energy conditions, compactness
factor, mass function, and surface redshift. The resulting outcome confirms
the authenticity of the hybrid star model under analysis.

Keywords: Ricci inverse gravity theory, Hybrid stars, Quark matter, Ordinary
baryonic matter.

##  I INTRODUCTION

Once a star exhausts its fuel, it usually undergoes a series of changes that
result in the creation of a dense object such as a white dwarf, neutron star,
or black hole, which is determined by its mass. Neutron stars are famous for
their exceptional compactness and density. The compact stars have a mass
ranging from $1\sim 2$ $M_{\odot}$ and a very small radius of $10\sim 13$ km.
As a result, the average density of matter within these dense stars is around
$10^{14}$ $g/cm^{3}$. Through the use of electromagnetic spectrum and
gravitational wave observations, we have gathered a wealth of information
about these dense celestial objects in recent years. Neutron stars are created
when huge stars have supernova explosions and gravitational collapse, causing
their cores to compress to densities similar to atomic nuclei. They themselves
can take various forms due to the high density in the core; for instance they
could be composed of normal matter mixed with hyperons and/or condensed
mesons. At such high densities, the matter inside the star has the potential
to convert into quark matter. Numerous studies have been conducted to support
the hypothesis that massive neutron stars could contain quark matter PR8 ; PR9
; PR10 ; PR11 ; PR12 ; PR13 . Observations have shown that the existence of
free quark matter could potentially give rise to a new class of compact stars
that have a core made up of QM surrounded by an outer layer made up of OBM
Implications-of-supermassive-neutron-star36 ; Implications-of-supermassive-
neutron-star37 ; Implications-of-supermassive-neutron-star38 . These stars are
named hybrid stars, and they are different from conventional neutron stars,
which consist solely of hadronic matter Implications-of-supermassive-neutron-
star39 ; Implications-of-supermassive-neutron-star40 ; Implications-of-
supermassive-neutron-star41 ; Implications-of-supermassive-neutron-star42 ;
Implications-of-supermassive-neutron-star43 . Researchers have investigated
the possible existence of these hybrid stars, which have masses similar to the
mass of the second object in GW190814, and explored the properties of such
massive hybrid stars Implications-of-supermassive-neutron-star44 ;
Implications-of-supermassive-neutron-star45 ; Implications-of-supermassive-
neutron-star46 ; Implications-of-supermassive-neutron-star47 ; Implications-
of-supermassive-neutron-star48 .

The inquiry about the rapid expansion of the cosmos in General Relativity (GR)
can be addressed by modifying the lagrangian of GR, leading to the formulation
of modified theories of gravity (MTG). These MTG extend beyond GR as they
introduce additional degrees of freedom and higher-order derivatives in the
field equations. Although GR has achieved success in demonstrating gravity on
various scales S1 ; S2 , the need for modification arises due to its limited
capacity to comprehensively account for phenomena such as dark energy and the
universe’ accelerating expansion. So in the frame work of MTG, researchers aim
to bridge the gap left by GR for comprehending the dynamics of the cosmos.
Reference S10 provides a historical review of the early efforts to broaden the
scope of GR, driven by the various motivations. For a thorough understanding
of MTG, see references S11 ; S12 . This study focuses on Ricci-inverse gravity
S13 as an alternative theory of gravity in relation to the investigation of
hybrid stars. A unique class of fourth-order MTG model is introduced in this
framework by the generalization of the Einstein-Hilbert action. The novel
alternative gravity proposed by Amendola et al. S13 involves the
generalization of Einstein-Hilbert action. The reference S14 provides a
generalization of Ricci inverse gravity and establishes two distinct classes
of Ricci inverse gravity: Class I and Class II. In Class I, the Lagrangian
incorporates an arbitrary function $f$ that is dependent on the Ricci scaler
$\mathcal{R}$ and the anticurvature scalar $\mathcal{A}$, defined as the trace
of the Ricci-inverse tensor $A_{\mu\nu}$. It is important to remember that the
anti-curvature scalar $\mathcal{A}$ does not correspond to the inverse of the
Ricci scalar $\mathcal{R}$. On the other hand, in Class II, the function takes
the form $f(\mathcal{R},A^{\mu\nu}A_{\mu\nu})$, constituting a function of the
Ricci scalar and the square of the anti-curvature tensor. Researchers have
investigated Ricci-inverse gravity in several contexts. The utilization of
Ricci inverse gravity to investigate the compact star structure in the context
of spherically symmetric spacetime was initially introduced by Shamir et al.
M40 . An analysis has been conducted on the matter-antimatter asymmetry
through baryogenesis in the realm of $f(\mathcal{R},\mathcal{A})$ S16 .
Additionally, a study has been carried out on a non-relativistic static and
spherically symmetric cosmic structure embedded into a de Sitter cosmologyS17
. Furthermore, researchers have investigated a no-go theorem for inflation
within an extended Ricci-inverse gravity model S18 ; S19 , emphasizing the
various uses and consequences of MTG in cosmological and astrophysical
scenarios. The study by Souza et al. M42 calculated two different axially
symmetric spacetimes, and they showed that Ricci inverse gravity allows the
existence of closed time like curves (CTCs).

There is no astrophysical entity that is entirely composed of perfect fluid.
Therefore, the incorporation of pressure anisotropy is crucial in the
formulation of accurate models for ultra dense stars, since it significantly
impacts their structure and characteristics. The incorporation of anisotropy
has a substantial impact on observable properties, including the ratio of mass
to radius, moment of inertia, redshift, and stability of star configurations.
Ruderman PR56 proposed through theoretical studies that stellar matter can
display anisotropy when its density exceeds $10^{15}$ $gm/cm^{3}$. For such
high-dense stellar structures, there can be a disparity between radial
pressure $p_{r}$ and tangential pressure $p_{t}$, and both components are
functions of radial coordinate r due to the spherical symmetry. The primary
causes that contribute to pressure anisotropy in the distribution of matter
are viscosity, pion condensation PR62 , phase transitions in superfluids PR63
, or other related physical processes. These factors can induce anisotropy in
the pressure within super dense stars, impacting their internal structure and
properties. Mak and Harko PR65 conducted computational simulations to model
celestial bodies with anisotropy using the principles of general relativity.
In this context, PR68 ; PR69 ; PR70 ; PR71 ; PR72 encompass further relevant
studies concerning anisotropic compact stellar objects.

Building on the observations we’ve already made, our goal is to do a full
investigation that gives a complete picture of how the physical features of
relativistic hybrid neutron stars change when anisotropy is present. Our
current work is expected to provide significant findings for the examination
of compact stellar entities, which will also enabling us to study the
exploration of matter’s response under intense gravitational forces. The
thermodynamic bag model (tdBag) and the Nambu-Jona-Lasino type (NJL) models
are the two most often used effective quark matter models in astrophysics PR74
; PR75 ; PR76 ; PR77 . The thermodynamic bag model (tdBAG) is widely used in
the field of astrophysics compared to the other models. The recently
introduced vBag model PR78 have been shown to be a useful model for conducting
astrophysical investigations. In this paper, we use the MIT bag model as an
EoS for QM. Numerous researchers have effectively utilized this MIT bag EoS
model to simulate strange quark stars (SQS) PR79 ; PR80 ; PR81 ; PR82 ; PR83 ;
PR84 ; PR85 . This phenomenological model is the initial successful
representation of the phase transition from the quark gluon plasma (QGP) state
to the hadronic phase. The pressure-density relation for QM in the MIT bag EoS
is expressed as $p_{q}=\frac{1}{3}(\rho_{q}-4B_{g})$, where $\rho_{q}$ and
$p_{q}$ represent the density and pressure of quark matter, respectively. The
bag constant $B_{g}$ in this context is defined as the difference between the
densities of perturbative and non-perturbative quantum chromo dynamical (QCD)
vacuums. The fundamental model is well-suited for studying the stable
arrangement of a cosmic compact object made up of u, d, and s quarks. Usually,
the $B_{g}$ falls within the range of 58.9-91.5 $MeV/fm^{3}$ for massless
strange quarks, while for massive quarks $B_{g}$ ranges of 56-78 $MeV/fm^{3}$
PR86 .

Researchers have shown significant attention to investigating the stellar
structures by employing the Krori-Barua metric (KB metric) measure. Bhar F35
uses KB metric potentials to study a unique hybrid star model that
incorporates OBM and QM in the interior region of stars. In their study,
Rahaman et al. F36 investigated the feasibility of utilizing the KB model for
the purpose of characterizing ultra-compact objects, such as strange stars.
They also conducted an analysis of the implications of employing a
mathematical representation for the purpose of modeling these strange stellar
phenomena. Biswas et al. F37 explored the anisotropic charged strange stars by
utilizing the KB spacetime in the framework of the
$f(\mathcal{R},\mathcal{T})$ theory of gravity. Sharif and Waseem F38 studied
the behavior of anisotropic compact stars under the $\mathcal{R}+\alpha
R_{\mu\nu}\mathcal{T}^{\mu\nu}$ gravity theory model. The KB metric was used
for the investigation and determined the parameters’s value by utilizing the
observed masses and radii of compact stars such as Her X-1, SAX J 1808.43658,
and 4U182030. Using the Gauss-Bonnet MTG, also referred to as $f(\mathcal{G})$
MTG, Abbas et al. F39 explored the development of compact stars in the
presence of anisotropy. This theory is widely regarded as a leading candidate
for explaining the rapid expansion of the cosmos. They utilized an analytical
solution obtained from the KB metric, which was applied to the Einstein field
equations. Their analysis utilized an anisotropic form of matter and a power
law representation of $f(\mathcal{G})$ gravity. Considering the utilization of
KB metric potentials , Shamir and Malik F40 examined the characteristics of
anisotropic compact stars within the framework of the $f(\mathcal{R},\phi)$
theory of gravity. In this context, $\mathcal{R}$ represents the Ricci scalar,
while $\phi$ represents the scalar field. Malik et al. F41 ; F42 ,
investigated the behavior of compact stars in the presence of charge in the
modified $f(\mathcal{R})$ theories of gravity. They utilized the KB metric
potentials and presented a singularity-free solution for a charged fluid
sphere. Incorporating the Chaplygin EoS in $f(\mathcal{R},\mathcal{T})$ MTG,
Bhar F43 studied the spherically symmetric spacetime along with anisotropic
fluid distribution in the presence of an electric field. Hossein et al. F44
employed the KB metric potentials to model the formation of anisotropic
compact star. They achieved this by considering the analytical solution of the
metric potentials. Subsequently, Rehman et al. F45 investigated the
feasibility of utilizing the Krori and Barua model to characterize ultra-
compact objects such as strange stars. They also established some limitations
on the model parameters to ensure its stability when studying strange stars.

Our paper is structured as follows: Section II introduces the fundamental
concepts of Ricci inverse gravity theory, along with the model used for
$f(\mathcal{R},\mathcal{A})$. Section III provides a detailed explanation of
the energy-momentum tensor components, field equations, and EoSs for both OBM
and QM. Section IV details the analytic expressions of $\rho$, $p_{r}$,
$p_{t}$, $\rho_{q}$, and $p_{q}$ using KB metric potentials. This section also
discusses the methodology for determining the unknown parameters present in
the metric potentials by lining up the interior spacetime with the exterior
Schwarzschild line element. Section V delves into the graphical analysis of
various physical characteristics of the hybrid star model, such as energy
density, radial and tangential pressure components, anisotropy, energy
conditions, and EOS parameters, mass function, compactness factor, and
redshift function. Section VI focuses on assessing our model’s stability by
examining the speed of sound, adiabatic index, and equilibrium condition of
all forces. Finally, section VII summarizes our findings.

##  II Basic formalism of Ricci Inverse Gravity

This section presents a concise summary of the basic principles of Ricci
inverse gravity, a newly formulated alternative theory of gravity. The inverse
of the Ricci tensor $\mathcal{R}_{\mu\nu}$ is referred to as the anticurvature
tensor $\mathcal{A}^{\mu\nu}$. The relationship between the anticurvature
tensor and the Ricci tensor can be expressed as

$\mathcal{A}^{\mu\nu}\mathcal{R}_{\nu\sigma}=\delta^{\mu}_{\sigma},$ (1)

where $\delta^{\mu}_{\sigma}$ is called Kronecker delta. The anti-curvature
scalar $\mathcal{A}$ which is based on the anti-curvature tensor can be
defined as

$\mathcal{A}=g_{\mu\nu}\mathcal{A}^{\mu\nu},$ (2)

(obviously, $\mathcal{A}\neq\mathcal{R}^{-1}$). It should be noted that in the
case, where the determinant of Ricci tensor is zero, there is no way to invert
$\mathcal{R_{\mu\nu}}$. Hence the inclusion of
$\text{det}(\mathcal{R}_{\mu\nu})\neq 0$ is an essential prerequisite to
investigate a metric as a potential solution of Ricci-inverse gravity.

To formulate the field equations of modified Ricci inverse gravity, one can
start by considering the following action:

$S=\int d^{4}x\sqrt{-g}f(\mathcal{R},\mathcal{A})+\int
d^{4}x\sqrt{-g}\mathcal{L}_{m},$ (3)

here, $f(\mathcal{R},\mathcal{A})$ represents a function that can take any
values for $\mathcal{R}$ and $\mathcal{A}$, $\mathcal{L}_{m}$ refers to the
matter langrangian, while $g$ represents the determinant of the metric tensor.
The field equations are obtained by calculating the variation of the action
defined in Eq. (3) with respect to the metric tensor, resulting in the
following set of equations Amendola .

$f_{\mathcal{R}}\mathcal{R}^{\mu\nu}-f_{\mathcal{A}}\mathcal{A}^{\mu\nu}-\frac{1}{2}fg^{\mu\nu}+g^{\gamma\mu}\nabla_{\delta}\nabla_{\gamma}f_{\mathcal{A}}\mathcal{A}^{\delta}_{\zeta}\mathcal{A}^{\nu\zeta}-\frac{1}{2}\nabla^{2}(f_{\mathcal{A}}\mathcal{A}^{\mu}_{\zeta}\mathcal{A}^{\nu\zeta})-\frac{1}{2}g^{\mu\nu}\nabla_{\delta}\nabla_{\eta}(f_{\mathcal{A}}\mathcal{A}^{\delta}_{\zeta}\mathcal{A}^{\eta\zeta})-\nabla^{\mu}\nabla^{\nu}f_{\mathcal{R}}+g^{\mu\nu}\nabla^{2}f_{\mathcal{R}}=\kappa\mathcal{T}^{\mu\nu},$
(4)

where $f_{\mathcal{R}}=\frac{\partial f}{\partial R}$,
$f_{\mathcal{A}}=\frac{\partial f}{\partial A}$ , $\nabla_{\alpha}$ represents
covarient derivative, $\nabla^{2}=\nabla_{\mu}\nabla^{\mu}$ is the d’Alembert
operator, and $\mathcal{T}^{\mu\nu}$ implies energy momentum tensor the energy
momentum tensor defined as

$\mathcal{T}^{\mu\nu}=\frac{1}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g_{\mu\nu}}$ (5)

###  II.1 Linear Model of Ricci Inverse Gravity

In this research, to keep things simple, we consider the simplest linear model
for $f(\mathcal{R},\mathcal{A})$ gravity defined as Amendola

$f(\mathcal{R},\mathcal{A})=\mathcal{R}+\alpha\mathcal{A},$ (6)

where $\alpha$ is the coupling parameter. So in view of Eq. (6), Eq. (4) can
be written as

$\mathcal{R}^{\mu\nu}-\frac{1}{2}\mathcal{R}g^{\mu\nu}-\alpha\mathcal{A}^{\mu\nu}-\frac{1}{2}\alpha\mathcal{A}g^{\mu\nu}+\frac{\alpha}{2}\bigg{[}2g^{\gamma\mu}\nabla_{\delta}\nabla_{\gamma}\mathcal{A}^{\delta}_{\zeta}\mathcal{A}^{\nu\zeta}-\nabla^{2}\mathcal{A}^{\mu}_{\zeta}\mathcal{A}^{\nu\zeta}-g^{\mu\nu}\nabla_{\delta}\nabla_{\eta}\mathcal{A}^{\delta}_{\zeta}\mathcal{A}^{\eta\zeta}\bigg{]}=\kappa\mathcal{T}^{\mu\nu}.$
(7)

For simplicity, Eq. (7) can be written as

$\mathcal{R}^{\mu\nu}-\frac{1}{2}\mathcal{R}g^{\mu\nu}+M^{\mu\nu}=\kappa\mathcal{T}^{\mu\nu},$
(8)

where $M^{\mu\nu}$ contains the terms that modify the field equations of
general relativity and is given by

$M^{\mu\nu}=-\alpha\mathcal{A}^{\mu\nu}-\frac{1}{2}\alpha\mathcal{A}g^{\mu\nu}+\frac{\alpha}{2}\bigg{[}2g^{\gamma\mu}\nabla_{\delta}\nabla_{\gamma}\mathcal{A}^{\delta}_{\zeta}\mathcal{A}^{\nu\zeta}-\nabla^{2}\mathcal{A}^{\mu}_{\zeta}\mathcal{A}^{\nu\zeta}-g^{\mu\nu}\nabla_{\delta}\nabla_{\eta}\mathcal{A}^{\delta}_{\zeta}\mathcal{A}^{\eta\zeta}\bigg{]},$
(9)

the term $M^{\mu\nu}$ arises due to the inclusion of anti-curvature scalar
$\mathcal{A}$ in the action defined in Eq. (3).

In addition, the four divergence of the field Eq. (7) yields:

$\nabla_{\mu}\mathcal{T}^{\mu\nu}=-\frac{\alpha}{2}g^{\mu\nu}\nabla_{\mu}\mathcal{A}-\alpha\nabla_{\mu}(\mathcal{A}^{\mu\nu})+\alpha
g^{\gamma\mu}\nabla_{\mu}\bigg{[}\nabla_{\delta}\nabla_{\gamma}(\mathcal{A}^{\delta}_{\zeta}\mathcal{A}^{\nu\zeta})\bigg{]}-\frac{\alpha}{2}\nabla_{\mu}\bigg{[}\nabla^{2}(\mathcal{A}^{\mu}_{\zeta}\mathcal{A}^{\nu\zeta})\bigg{]}-\frac{\alpha}{2}g^{\mu\nu}\nabla_{\mu}\bigg{[}\nabla_{\delta}\nabla_{\eta}(\mathcal{A}^{\delta}_{\zeta}\mathcal{A}^{\eta\zeta})\bigg{]}.$
(10)

##  III Interior space-time and Field equations

The interior space-time of a static spherically symmetric compact object can
be characterized by examining the interior line element in the conventional
form, defined by:

$ds^{2}=-e^{2\psi}dt^{2}+e^{2\lambda}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\phi^{2},$ (11)

where the metric potential functions $\psi$ and $\lambda$ are solely dependent
on the radial coordinate $r$. The stress energy-momentum tensor for an
anisotropic two-fluid matter configuration can be formulated as follows

$\text{Energy-Momentum Tensor }\begin{cases}T^{0}_{\mkern
8.0mu0}=-(\rho+\rho_{q}),\\\ T^{1}_{\mkern 8.0mu1}=(p_{r}+p_{q}),\\\
T^{2}_{\mkern 8.0mu2}=(p_{t}+p_{q}).\end{cases}$

The non-zero components of the stress energy-momentum tensor defined above
provide information about the anisotropic distribution of matter inside the
formation of compact objects. These objects are composed of two forms of
matter: ordinary baryonic matter (OBM) and quark matter (QM). In this context,
$\rho$ represents the matter-energy density, $p_{r}$ represents the radial
pressure component, and $p_{t}$ represents the transverse pressure component
of the OBM. Similarly, $\rho_{q}$ represents the matter-energy density, and
$p_{q}$ represents the pressure associated with the QM.

By assuming the gravitational units $G=c=1$ and with the use of the non-zero
components of the energy-momentum tensor defined above, we can derive a set of
the following independent equations as a result of solving Eq. (7)

$\displaystyle
8\pi(\rho+\rho_{q})=\mathcal{R}^{00}e^{2\psi}+\frac{\mathcal{R}}{2}-\alpha
e^{2\psi}\mathcal{A}^{00}+\frac{\alpha\mathcal{A}}{2}+\frac{\alpha}{2}\bigg{[}\bigg{(}\frac{4}{r^{2}}+2\lambda^{\prime
2}+2\lambda^{\prime\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+2re^{-2\lambda}(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}-e^{2\psi}e^{-2\lambda}\bigg{(}\frac{2\psi^{\prime}}{r}+$
$\displaystyle\qquad 2\psi^{\prime
2}+2\psi^{\prime\prime}-2\lambda^{\prime}\psi^{\prime}\bigg{)}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})+3\lambda^{\prime}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}-e^{2\psi}e^{-2\lambda}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}\bigg{]},$ (12)

$\displaystyle
8\pi(p_{r}+p_{q})=\mathcal{R}^{11}e^{2\lambda}-\frac{\mathcal{R}}{2}-\alpha
e^{2\lambda}\mathcal{A}^{11}-\frac{\alpha\mathcal{A}}{2}+\frac{\alpha}{2}\bigg{[}-\bigg{(}\frac{4\lambda^{\prime}}{r}+2\lambda^{\prime}\psi^{\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})-2re^{-2\lambda}(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}-4e^{-2\lambda}(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})$
$\displaystyle\qquad-\bigg{(}\frac{2}{r}+\psi^{\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+\psi^{\prime}e^{2\psi}e^{-2\lambda}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}+2\psi^{\prime
2}e^{2\psi}e^{-2\lambda}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})\bigg{]},$ (13) $\displaystyle
8\pi(p_{t}+p_{q})=r^{2}\mathcal{R}^{22}-\frac{\mathcal{R}}{2}-\alpha
r^{2}\mathcal{A}^{22}-\frac{\alpha
A}{2}+\frac{\alpha}{2}\bigg{[}-\bigg{(}\frac{2}{r^{2}}+2\psi^{\prime
2}+2\lambda^{\prime 2}+2\lambda^{\prime\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+e^{-2\lambda}\bigg{(}-6r-r^{2}\psi^{\prime}+r^{2}\lambda^{\prime}\bigg{)}$
$\displaystyle\qquad\times(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}+e^{-2\lambda}\bigg{(}-4+2r\lambda^{\prime}-2r\psi^{\prime}\bigg{)}(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})+\psi^{\prime}e^{2\psi}e^{-2\lambda}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}-r^{2}e^{-2\lambda}(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime\prime}-3\lambda^{\prime}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}$
$\displaystyle\qquad-(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}\bigg{]},$ (14)

where the values of $\mathcal{R}^{00}$, $\mathcal{R}^{11}$,
$\mathcal{R}^{22}$, $\mathcal{R}$, $\mathcal{A}^{00}$, $\mathcal{A}^{11}$,
$\mathcal{A}^{22}$, $\mathcal{A}$, $\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00}$, $\mathcal{A}^{1}_{\mkern 8.0mu1}\mathcal{A}^{11}$,
and $\mathcal{A}^{2}_{\mkern 8.0mu2}\mathcal{A}^{22}$ are given in the
appendix section.

To solve the above set of Eqs. (III)-(III) for normal baryonic matter, we have
to close the system. For this purpose we can use the relationship between
matter density $\rho$ and radial pressure $p_{r}$ defined as

$p_{r}=\beta_{1}\rho-\beta,$ (15)

where $\beta_{1}$ and $\beta$ are positive constants and $\beta_{1}$ lies in
the range $0<\beta_{1}<1$ with $\beta_{1}\neq\frac{1}{3}$. Let us also assume
that, for QM, the relationship between pressure $p_{q}$ and matter density
$\rho_{q}$ is described by the following MIT Bag EoS model MITEoS

$p_{q}=\frac{1}{3}(\rho_{q}-4B),$ (16)

where $B$ describes the Bag constant.

##  IV Solution of Field Equations

In order to determine the physical attributes of our model, we have to solve
field Eqs. (III)-(III) along with the EoS describe in Eq. (15) and Eq. (16).
For this, we can Utilize the parametrization for the metric potentials
proposed by Krori and Barua PR54 (known as KB ansatz) which is defined as:

$\text{KB metric }\begin{cases}e^{2\psi(r)}=e^{Hr^{2}+P},\\\
e^{2\lambda(r)}=e^{Gr^{2}},\\\ \end{cases}$

here, $G$ and $H$ are constant parameters measured in the units of $km^{-2}$,
with $P$ being a dimensionless constant. These constants can be evaluated
numerically by using the smooth matching of interior and exterior space-times.
We are primarily motivated to introduce this collection of functions for the
static metric goes back to the singularity free structure within the star’s
boundary.

With the choice of linear model for Ricci inverse gravity defined in Eq. (6)
together with the choice of KB metric potentials and the EoSs for both OBM and
QM defined in Eqs. (15)-(16), the solution of the Eqs. (III)-(III) yields the
values of $\rho$, $p_{r}$, $p_{t}$, $\rho_{q}$, and $p_{q}$, for both OBM and
QM. The mathematical expressions are given in the appendix section.

###  IV.1 Regularity of our chosen metric

This subsection focuses on the behavior of the KB metric potentials that we
have chosen. We note that $e^{2\psi}|_{r=0}=e^{P}>0$ and
$e^{2\lambda}|_{r=0}=1$, which identifies that both metric potentials are
finite at the center and have regularity throughout the domain $r<R$.
Moreover, the derivatives of the metric coefficients are:
$\big{[}\frac{d}{dr}(e^{2\lambda})\big{]}_{r=0}=2Gre^{Gr^{2}}|_{r=0}=0$ and
$\big{[}\frac{d}{dr}(e^{2\psi})\big{]}_{r=0}=2Hre^{Hr^{2}+P}|_{r=0}=0$. These
values define that, at the core of the star, the derivatives of the metric
potentials are equal to zero. Hence, the selection of the metric potentials in
this context guarantees that the metric functions exhibit non-singularity and
continuity within the star’s interior, and these metric potentials are
perfectly aligned to the external Schwarzschild line element. On a physical
basis, this is considered a desirable characteristic for any well-behaved
model.

###  IV.2 Calculation of Metric Potentials from Junction Conditions

To solve the field equations outlined in Eq.(III)-(III), we establish the
values of the parameters $G$, $H$, and $P$ appearing in the KB metric. For
this purpose, we match the interior spacetime defined in Eq. (11) to the
exterior spacetime under the restricted boundary conditions at $r=R$. The
exterior metric of the star, which is described by the following Schwarzschild
solution given by

$ds^{2}=-\bigg{(}1-\frac{2M}{r}\bigg{)}dt^{2}+\bigg{(}1-\frac{2M}{r}\bigg{)}^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\phi^{2},$ (17)

here, $M$ represents the total mass enclosed within the boundary of the
compact star. Ensuring the continuity of the metric coefficients $g_{tt}$,
$g_{rr}$, and $\frac{\partial g_{tt}}{\partial r}$ across the boundary surface
$r=R$, separating the interior and exterior regions yields the following set
of relationships:

$1-\frac{2M}{R}=e^{Hr^{2}+P},$ (18)
$\bigg{(}1-\frac{2M}{R}\bigg{)}^{-1}=e^{Gr^{2}},$ (19)
$\frac{M}{R^{2}}=HRe^{HR^{2}+P}.$ (20)

Upon solving the Eqs. (18)-(20) yields the following values of constants $G$,
$H$, and $P$ in terms of the total mass $M$ and radius $R$, and are given by

$G=-\frac{1}{R^{2}}ln\bigg{(}1-\frac{2M}{R}\bigg{)},$ (21)
$H=\frac{M}{R^{3}}\bigg{(}1-\frac{2M}{R}\bigg{)}^{-1},$ (22)
$P=ln\bigg{(}1-\frac{2M}{R}\bigg{)}-\frac{M}{R}\bigg{(}1-\frac{2M}{R}\bigg{)}^{-1}.$
(23)

Furthermore, at star’s boundary where radius of star is indicated as $R$, the
pressure in radial direction vanishes, denoted mathematically as
$p_{r}(r=R)=0$. This condition leads to the expression for $\beta$ as provided
in the appendix.

As a result, we have successfully determined the values of $G$, $H$, and $P$
involved in the KB metric coefficients in terms of mass $M$ and radius $R$.
Using the observed values of various candidates of compact stars, we can find
the numerical values of the constants $G$, $H$, and $P$.

##  V Physical attributes of present model

In this section, we graphically analyze the physical attributes of our present
hybrid star model. Focusing especially on the star LMC-X4 with observed mass
$M=(1.04^{+0.09}_{-0.09})M_{\odot}$ and observed radius
$R=8.301^{+0.2}_{-0.2}\text{ km}$. The entire analysis has been conducted with
the choice of $\beta_{1}=0.3$.

###  V.1 Nature of fluid components associated with OBM and QM

In this subsection, we examine the behaviors of energy density $\rho$, radial
pressure $p_{r}$, and tangential pressure $p_{t}$ in the context of outer
baryonic matter OBM, as well as $\rho_{q}$ and $p_{q}$ for the QM.
Additionally, we explore the gradients of $\rho$, $p_{r}$, and $p_{t}$. Due to
the very complicated analytic forms, we will examine their behaviors through
the graphical representations. In Fig. (1), we have plotted the OBM matter-
energy density and pressure components. The graphical analysis shows the
consistent decreasing behaviors of $\rho$, $p_{r}$, and $p_{t}$ against radius
$r$ for different values of coupling parameter $\alpha$, as shown in the
figure. The graphical visualization shows that $\rho$, $p_{r}$, and $p_{t}$
have the maximum value at the core of the star. Also, $\rho$, $p_{r}$, and
$p_{t}$ are non-negative inside the star. As anisotropic matter distribution
has been taken into consideration, indicating a difference between pressure
components $(p_{r},p_{t})$. Therefore, $\Delta=p_{t}-p_{r}$ is used to define
the anisotropy factor. The anisotropic force, denoted by $\frac{2\Delta}{r}$,
will be replusive in nature if $\Delta>0$ and attractive otherwise. The bottom
right panel of Fig. (1) shows the profile for the anisotropy factor. In Fig.
(2), we have graphed the energy density $\rho_{q}$ and pressure $p_{q}$ for
the QM . This Fig. shows that both display a negative nature inside the
compact stellar object.

In our present model, Fig. (3) shows the variation of density and pressure
gradients due to OBM. The graphical illustration shows that the gradient of
density and pressure due to OBM stays negative throughout the fluid sphere,
which is expected for a realistic physical model.

Figure 1:  Profile of baryonic matter-energy density $\rho$, pressure
components $(p_{r},p_{t})$, and anisotropic factor $\Delta(r)$ against $r$.

Figure 2:  Profile of QM density $\rho_{q}$ and QM pressure $p_{q}$ against
$r$.

Figure 3:  Profiles of $\frac{d\rho}{dr}$, $\frac{dp_{r}}{dr}$, and
$\frac{dp_{t}}{dr}$ against $r$ inside the stellar interior.

###  V.2 Energy Conditions

In order to ensure the physical realism and feasibility of the matter field,
it is necessary for the stress-energy tensor to adhere to certain mathematical
contstraints. We generally referred to these constraints as energy conditions
(Ecs). The analysis of ECs is essential in comprehending the properties of
ordinary and unusual nature of matter within the stellar structure. The ECs
are depicted as follows hybrid_star_fG_PR_90 ; hybrid_star_fG_PR_91 ;
hybrid_star_fG_PR_92 ; hybrid_star_fG_PR_93 ; hybrid_star_fG_PR_94 :

  * •

Null Energy Condition (NEC) $\rho+p_{i}\geq 0$,

  * •

Weak Energy Condition (WEC) $\rho\geq 0$, $\rho+p_{i}\geq 0$,

  * •

Strong Energy Condition (SEC) $\rho+p_{i}\geq 0$, $\rho+p_{r}+2p_{t}\geq 0$,

  * •

Dominant Energy Condition (DEC) $\rho-p_{i}>0$, $\rho\geq 0$

where $i=r,t$.

Ordinary matter always satisfies these ECs due to its inherent positive energy
density and pressure. Fig. (4) shows the graphical visualization of
$\rho+{p_{r}}$, $\rho+{p_{t}}$, $\rho-{p_{r}}$, $\rho-{p_{t}}$, and
$\rho+{p_{r}}+2p_{t}$ for different values of $\alpha$, and this Fig. visually
confirm the validity of the ECs. The ECs mentioned earlier are all met by the
proposed hybrid star model in Ricci Inverse gravity suggesting our model is
physically acceptable.

Figure 4:  Behavior of all energy conditions against $r$ inside the stellar
interior for LMC-X4.

###  V.3 Equation of State

The equation of state (EoS) is a fundamental relation that establishes the
relationship between radial pressure and energy density, as well as tangential
pressure and energy density. Creating two separate EoSs is advantageous, with
one dedicated to radial pressure and the other to tangential pressure. In our
study, we establish the EoS by employing the following equations:

$p_{r}=\omega_{r}\rho,~{}~{}~{}~{}~{}p_{t}=\omega_{t}\rho,$ (24)

The two dimensionless parameters, denoted as $\omega_{r}$ and $\omega_{t}$,
are commonly referred to as EoS parameters that describe the relation between
matter density and pressure, and these parameters also give information about
the state of matter. The graphical profiles of $\omega_{r}$ and $\omega_{t}$
are depicted in Fig. (5) for different variations in $\alpha$. The findings
indicate that the two features were most beneficial closer to the star’s
center and fell within the ranges $0<\omega_{r}<1$ and $0<\omega_{t}<1$.

Figure 5:  Profiles for $w_{r}$ and $w_{t}$ against $r$.

###  V.4 MASS FUNCTION, COMPACTNESS FACTOR, SURFACE REDSHIFT

In this subsection, we will look at some key parameters of our model: mass
function $m(r)$, compactness factor $u(r)$, and surface redshift $Z_{s}$. The
function $m(r)$ plays a significant role as it represents the distribution of
mass within the star as a function of radial distance $r$ and is defined by
the following formula:

$m(r)=\int_{0}^{r}4\pi(\rho+\rho_{q})\zeta^{2}d\zeta.$ (25)

after applying the KB potentials on Eq. (25), we finally get
hybrid_star_fG_PR_86 ; hybrid_star_fG_PR_87

$m(r)=\frac{r}{2e^{2\lambda}}\bigg{(}-1+e^{2\lambda}\bigg{)}=\frac{r}{2}\bigg{(}1-e^{-Gr^{2}}\bigg{)}$
(26)

Fig. (6) displays the variation of $m(r)$ against $r$. It is evident from the
Fig. that there is no singularity in the function $m(r)$ and possesses
monotonically increasing behavior and having maximum value at the surface of
the star. Also graphical illustration shows that when $r\rightarrow 0$,
$m(r)\rightarrow 0$. The compactness factor $u(r)$ plays a crucial role in
understanding the gravitational properties and structural stability of these
dense astronomical objects. It is defined as the ratio of mass of the object
to its radius. Mathematically,

$u(r)=\frac{m(r)}{r}$ (27)

The graphical evolution of $u(r)$ has been displayed in Fig. (6) and shows the
monotonically increasing behavior against $r$ and having maximum value at the
surface of the star. The surface redshift $Z_{s}$ for the present compact star
candidate can be obtained by using the following expression:

$Z_{s}=(1-2u(r))^{-\frac{1}{2}}-1$ (28)

The graphical evolution of $Z_{s}$ for different values of $\alpha$ is also
shown in Fig. (6). This graph shows the increasing behavior of $Z_{s}$ and its
maximum value at the star’s surface.

Figure 6:  Variations of mass function $m(r)$, compactness factor $u(r)$, and
surafce redshift $Z_{s}$ are shown against radius $r$.

##  VI Stability Analysis

In this section, we will evaluate the stability of our model by examining the
causality condition, adiabatic index, and equilibrium condition.

###  VI.1 Velocity of sound

Analyzing sound velocity can provide valuable information about the properties
of dense matter in celestial bodies like hybrid stars or quark stars. The
speed of sound within a stellar object is a crucial determinant that impacts
its overall stability. In order to establish a physically accurate model, it
is imperative to verify the causality condition, which implies that the
velocity of sound within the compact object must be less than the speed of
light. The subsequent formulas for the radial sound speed associated with the
radial pressure denoted by $v_{r}^{2}$ and tangential sound speed denoted by
$v_{t}^{2}$ can be utilized, which are expressed as.

$v_{r}^{2}=\frac{dp_{r}}{d\rho},~{}~{}~{}v_{t}^{2}=\frac{dp_{t}}{d\rho}$ (29)

The stability of a viable system is significantly shaped by the concept
introduced by Herrera Amina91 . Based on this theoretical work, the speed of
sound associated to both the radial and transverse pressure components must
adhere to the causality condition, where the speed of sound falls within the
range of [0,1] i.e., $0\leq v_{r}^{2}\leq 1$ and $0\leq v_{t}^{2}\leq 1$.
Abreu Amina92 introduced a significant concept for examining potential stable
and unstable configurations of celestial objects. The constraints imposed on
$v_{r}^{2}$ and $v_{t}^{2}$ lead to the following condition:

$\begin{cases}-1\leq v_{t}^{2}-v_{r}^{2}\leq 0~{}~{}~{}~{}\text{indicating
potential stability},\\\ ~{}~{}0\leq v_{t}^{2}-v_{r}^{2}\leq
1~{}~{}~{}~{}\text{indicating potential instability}.\end{cases}$

Due to the complexities in the expressions of $v_{r}^{2}$ and $v_{t}^{2}$, we
analyzed these terms graphically. Fig. (7) shows the graphical behavior of
both $v_{r}^{2}$ and $v_{t}^{2}$ , revealing that they are both restricted
within the range $[0,1]$ within the boundary of stellar object. This condition
is commonly referred as the causality condition.

Figure 7:  Plotting of squares of radial sound velocity $v_{r}^{2}$ and
transverse sound velocity $v_{t}^{2}$ against $r$.

###  VI.2 Adiabatic Index

Here, we delve into the analysis of a crucial ratio defined as the adiabatic
index denoted by $\Gamma$ to assess the stability region of the hybrid star
model. Chen et al. adnan102 introduced the concept of adiabatic index for an
isotropic fluid sphere, however, Chandrasekhar adnan103 was one of the
pioneers in utilizing the adiabatic index to examine the stability zone of a
spherical star. The following expression can be used for the adiabatic index
in the presence of anisotropy in the pressure:

$\Gamma_{r}=\frac{\rho+p_{r}}{p_{r}}\frac{dp_{r}}{d\rho},$ (30)

where $\Gamma_{r}$ is the adiabatic index for radial pressure. According to
the study conducted by Heintzmann and Hillebtanddt adnan104 , the stellar
object satisfies the conditions of stability when the values of the
aforementioned expression $\Gamma_{r}$ exceed 4/3. Due to the intricate nature
of the expressions, it is unfeasible to analytically prove this requirement.
The profiles of $\Gamma_{r}$ for different values of $\alpha$ have been
plotted for our model, and we see that $\Gamma_{r}$ is an increasing function
of $r$ and $\Gamma_{r}>\frac{4}{3}$ everywhere inside the boundary of the star
as depicted in Fig. (8). The graph illustrates that $\Gamma_{r}$ exhibits
values exceeding 4/3 throughout the fluid sphere, confirming the complete
satisfaction of the stability condition. The right panel of Fig. (8) shows the
graphical illustration of $|v_{t}^{2}-v_{r}^{2}|$, confirming the potential
stability of our considered Ricci Inverse gravity.

Figure 8:  Behavior of $\Gamma_{r}$ and stability factor
$|v_{t}^{2}-v_{r}^{2}|$ against $r$ .

###  VI.3 Equilibrium Condition

In this subsection, we will look at the model’s equilibrium with respect to
various forces acting on the system. The equilibrium equation involves four
main forces: hydrostatic force $(F_{h})$, gravitational force $(F_{g})$,
anisotropic force $(F_{a})$, and the force associated with quark matter
$(F_{q})$. Additionally, there are some extra terms arises due to the
modification the gravitational theory and we have labeled these terms as
$F_{\text{ext}}$.

The Tolman-Oppenheimer-Volkoff (TOV) equation serves as a crucial framework
for comprehending this equilibrium. To assess the stability of our model, we
formulate the modified version of the TOV equation within the context of the
$f(\mathcal{R},\mathcal{A})$ gravity theory. So Eq. (10) yields the following
equation

$\displaystyle-\frac{dp_{r}}{dr}-\bigg{(}\rho+p_{r}\bigg{)}\psi^{\prime}+\frac{2\Delta}{r}+\alpha\bigg{[}-\bigg{(}\psi^{\prime}+2\lambda^{\prime}+\frac{2}{r}\bigg{)}e^{2\lambda}\mathcal{A}^{11}-\psi^{\prime}e^{2\psi}\mathcal{A}^{00}+2r\mathcal{A}^{22}\bigg{]}-\frac{\alpha}{2}(\mathcal{A})^{\prime}-\alpha
e^{2\lambda}(\mathcal{A}^{11})^{\prime}+$
$\displaystyle\qquad\frac{\alpha}{2r^{3}}\bigg{[}\bigg{(}-10r+8r^{2}\lambda^{\prime}+4r^{3}\lambda^{\prime}\psi^{\prime}-6r^{3}\psi^{\prime
2}-r^{3}\psi^{\prime\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+\bigg{(}2r^{2}+2r^{3}\psi^{\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}+e^{-2\lambda}\bigg{(}2r^{3}+4r^{4}\lambda^{\prime}\bigg{)}$
$\displaystyle\qquad\times(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}-2r^{4}e^{-2\lambda}(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime\prime}+r^{3}\psi^{\prime}e^{-2\lambda}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}+\bigg{(}-2r^{3}\psi^{\prime}\lambda^{\prime}+r^{3}\psi^{\prime\prime}\bigg{)}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}+\bigg{(}16+8r^{2}\lambda^{\prime 2}+$
$\displaystyle\qquad 4r^{3}\psi^{\prime}\lambda^{\prime
2}+4r^{2}\lambda^{\prime\prime}+4r^{3}\psi^{\prime
3}+2r^{3}\psi^{\prime}\lambda^{\prime\prime}-4r^{3}\psi^{\prime}\psi^{\prime\prime}-12r\lambda^{\prime}-8r^{3}\lambda^{\prime}\psi^{\prime
2}-2r^{3}\lambda^{\prime}\psi^{\prime\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})\bigg{]}-(\rho_{q}+p_{q})\frac{\psi^{\prime}}{2}-$
$\displaystyle\qquad\frac{d}{dr}(p_{q})=0,$ (31)

here, prime indicates the derivative concerning the radial coordinate $r$. The
above equation can be viewed as

$F_{h}+F_{g}+F_{a}+F_{\text{ext}}+F_{q}=0.$ (32)

Eq. (VI.3) establishes the equilibrium condition of the configuration,
considering the mentioned forces, which can be expressed as follows:

$\displaystyle F_{h}=-\frac{dp}{dr},$ (33) $\displaystyle
F_{a}=\frac{2}{r}(p_{t}-p_{r})=\frac{2}{r}\Delta,$ (34) $\displaystyle F_{g}=$
$\displaystyle-\bigg{(}\rho+p_{r}\bigg{)}\psi^{\prime}$ (35) $\displaystyle
F_{q}=$
$\displaystyle-\bigg{(}\rho_{q}+p_{q}\bigg{)}\frac{\psi^{\prime}}{2}-\frac{d}{dr}(p_{q})$
(36) $\displaystyle F_{\text{ext}}=$
$\displaystyle\alpha\bigg{[}-\bigg{(}\psi^{\prime}+2\lambda^{\prime}+\frac{2}{r}\bigg{)}e^{2\lambda}\mathcal{A}^{11}-\psi^{\prime}e^{2\psi}\mathcal{A}^{00}+2r\mathcal{A}^{22}\bigg{]}-\frac{\alpha}{2}(\mathcal{A})^{\prime}-\alpha
e^{2\lambda}(\mathcal{A}^{11})^{\prime}+$
$\displaystyle\qquad\frac{\alpha}{2r^{3}}\bigg{[}\bigg{(}-10r+8r^{2}\lambda^{\prime}+4r^{3}\lambda^{\prime}\psi^{\prime}-6r^{3}\psi^{\prime
2}-r^{3}\psi^{\prime\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+\bigg{(}2r^{2}+2r^{3}\psi^{\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}+e^{-2\lambda}\bigg{(}2r^{3}+$
$\displaystyle\qquad 4r^{4}\lambda^{\prime}\bigg{)}(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}-2r^{4}e^{-2\lambda}(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime\prime}+r^{3}\psi^{\prime}e^{-2\lambda}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}+\bigg{(}-2r^{3}\psi^{\prime}\lambda^{\prime}+r^{3}\psi^{\prime\prime}\bigg{)}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}+$
$\displaystyle\qquad\bigg{(}16+8r^{2}\lambda^{\prime
2}+4r^{3}\psi^{\prime}\lambda^{\prime
2}+4r^{2}\lambda^{\prime\prime}+4r^{3}\psi^{\prime
3}+2r^{3}\psi^{\prime}\lambda^{\prime\prime}-4r^{3}\psi^{\prime}\psi^{\prime\prime}-12r\lambda^{\prime}-8r^{3}\lambda^{\prime}\psi^{\prime
2}-2r^{3}\lambda^{\prime}\psi^{\prime\prime}\bigg{)}(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})\bigg{]}.$ (37)

The Fig. (9) illustrates the varying behaviors of all the forces $F_{g}$,
$F_{h}$, $F_{a}$, $F_{\text{ext}}$, and $F_{q}$ acting on the model under
concern for different variations in $\alpha$. This figure also confirms that
the collective impact of all forces ensures the stability of our model.

Figure 9:  Variations of ${\color[rgb]{0,1,0}\bigstar}$ gravitational,
${\color[rgb]{0,1,1}\bigstar}$ hydrostatic, ${\color[rgb]{1,0,1}\bigstar}$
anisotropic, ${\color[rgb]{.5,.5,.5}\bigstar}$ quark, and
${\color[rgb]{0,0,1}\bigstar}$ $F_{\text{ext}}$ forces against $r$ for
$\alpha=-2\times 10^{-6}$, $\alpha=-2.5\times 10^{-6}$, $\alpha=-3\times
10^{-6}$, $\alpha=-3.5\times 10^{-6}$, and $\alpha=-4\times 10^{-6}$.

##  VII CONCLUDING REMARKS

The challenge of determining an appropriate model to depict the intricate
geometry of compact objects has garnered attention in the field of GR and
extended theories of gravity. In this investigation, we have developed a
method to examine the potential development of a hybrid star model within the
framework of modified Ricci inverse gravity $f(\mathcal{R},\mathcal{A})$, an
extension to GR. Modeling such astrophysical objects poses significant
challenges using $f(\mathcal{R},\mathcal{A})$. Due to the complexities of the
field equation of this modified gravity, we opted for the linear model:
$f(\mathcal{R},\mathcal{A})=\mathcal{R}+\alpha\mathcal{A}$. The stellar model
has been developed based on the choice of metric potentials known as Krori
Barua metric potentials that involve the arbitrary constant $G$, $H$, and $P$.
For our analysis, we have considered the compact star LMC-X4, and all
investigation has been done by considering a hybrid stellar model with an
anisotropic matter distribution comprising both OBM and QM. By using the
observed values of mass and radius of this star, we have calculated the values
of the aforementioned arbitrary constants by matching interior space-time with
Schwarzschild space-time. The physical analysis of our results reveals that
this anisotropic hybrid stellar model in $f(\mathcal{R},\mathcal{A})$ gravity
exhibits the following conclusive properties:

  * •

The chosen KB metric demonstrates a consistent and well-behaved description of
the gravitational field by maintaining regularity throughout the stellar
interior. The absence of singularities within the stellar interior underscores
the reliability of the KB metric in characterizing gravitational dynamics.

  * •

The behavior of $\rho$ and pressure components $(p_{r},p_{t})$ corresponding
to the OBM is positive and decreasing across the interior domain of the star
and having maximum values at the core as depicted in Fig. (1).

  * •

Considering the presence of anisotropic matter, the difference between $p_{t}$
and $p_{r}$, which is denoted by $\Delta$, is non-zero. The bottom right panel
of Fig. (1) visually represents the behavior of $\Delta$. Its positive
behavior signifies that anisotropic force is regenerative in nature.

  * •

The energy density $\rho_{q}$ and pressure profiles $p_{q}$ exhibit negative
behaviors due to QM within the compact stellar object, as demonstrated in Fig.
(2).

  * •

The pressure and density gradients corresponding to OBM remain negative
throughout the fluid sphere, which one would anticipate from a physically
accurate model.

  * •

The mass function $m(r)$ is directly proportional to the radial distance $r$
and exhibits positive and increasing behavior in the interior of the star,
having maximum value at the surface.

  * •

The compactness factor and surface redshift exhibit a continuous increase as
the radial distance $r$ grows.

  * •

The causality condition is satisfied as both radial and transverse sound
speeds, represented as $v_{r}^{2}$ and $v_{t}^{2}$ respectively, remain within
the bound $[0,1]$ throughout the interior of the stellar object. Furthermore,
our model adheres to Herrera’s cracking criterion, indicating its physical
consistency and potential stability throughout the stellar distribution in
$f(\mathcal{R},\mathcal{A})$ gravity.

  * •

All ECs show positive behavior throughout the interior of the star, showing
the presence of ordinary matter, so this proves the existence a realistic
matter content.

Various stellar solutions have been derived by using diverse modified theories
of gravity, confirming the reliability of these types of frameworks. As a
result of all the notable findings, it is evident that we can build a viable,
stable, and singularity-free generalized hybrid stellar model within the
interior fluid distribution in this specific $f(\mathcal{R},\mathcal{A})$
gravity framework. Hence, it is possible to analyze the physical
characteristics of strange star entities using theoretical and astrophysical
methods through the study of an exceptionally dense and compact celestial body
consisting of QM. All the analysis leads to the conclusion that the hybrid
star model presented in this work adequately clarifies the physical
characteristics in the framework of Ricci inverse gravity.

##  VIII Appendix

$\mathcal{R}^{00}=\Big{(}4Hr^{2}(H-G)+6H\Big{)}e^{-2\left(r^{2}(G+H)+P\right)}$
$\mathcal{R}^{11}=e^{-4Gr^{2}}\Big{(}4Hr^{2}(G-H)+4G-2H\Big{)}$
$\mathcal{R}^{22}=\frac{e^{-2Gr^{2}}\left(2r^{2}(G-H)-1\right)+1}{r^{4}}$
$\mathcal{R}=\frac{2e^{-2Gr^{2}}\left(4Hr^{4}(G-H)+2r^{2}(2G-3H)+e^{2Gr^{2}}-1\right)}{r^{2}}$
$\mathcal{A}^{00}=\frac{e^{-2\left(r^{2}(H-G)+P\right)}}{2H\left(2r^{2}(H-G)+3\right)}$
$\mathcal{A}^{11}=\frac{1}{4Hr^{2}(G-H)+4G-2H}$
$\mathcal{A}^{22}=\frac{1}{e^{-2Gr^{2}}\left(2r^{2}(G-H)-1\right)+1}$
$\mathcal{A}=\frac{2r^{2}e^{2Gr^{2}}}{2r^{2}(G-H)+e^{2Gr^{2}}-1}+\frac{e^{2Gr^{2}}}{4Hr^{2}(G-H)+4G-2H}-\frac{e^{2Gr^{2}}}{4Hr^{2}(H-G)+6H}$

$\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00}=-\frac{e^{2Gr^{2}-2\left(r^{2}(H-G)+P\right)}}{2H\left(2r^{2}(H-G)+3\right)\left(4Hr^{2}(H-G)+6H\right)}$
$\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11}=\frac{e^{2Gr^{2}}}{\left(4Hr^{2}(G-H)+4G-2H\right)^{2}}$
$\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22}=\frac{r^{2}e^{2Gr^{2}}}{\left(2r^{2}(G-H)+e^{2Gr^{2}}-1\right)\left(e^{-2Gr^{2}}\left(2r^{2}(G-H)-1\right)+1\right)}$
$\displaystyle\rho=-4B+\frac{1}{16\pi}\Bigg{[}4e^{-2Gr^{2}}H\left(-2Gr^{2}+2Hr^{2}+3\right)+\frac{2}{r^{2}}{e^{-2Gr^{2}}\left(4(G-H)Hr^{4}+2(2G-3H)r^{2}+e^{2Gr^{2}}-1\right)}+e^{2Gr^{2}}\times$
$\displaystyle\qquad\left(\frac{2r^{2}}{2(G-H)r^{2}+e^{2Gr^{2}}-1}+\frac{1}{4(G-H)Hr^{2}+4G-2H}-\frac{1}{4H(H-G)r^{2}+6H}\right)\alpha+\alpha\Bigg{\\{}-4e^{2\left((H-G)r^{2}+P\right)}H\times$
$\displaystyle\qquad\left(-2Gr^{2}+4Hr^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})+4\left(2G^{2}r^{2}+G+\frac{1}{r^{2}}\right)(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+\frac{2}{r}{e^{2\left((H-G)r^{2}+P\right)}\left(Gr^{2}-4Hr^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}}$
$\displaystyle\qquad+6Gr(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+2e^{-2Gr^{2}}r(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}-e^{2\left((H-G)r^{2}+P\right)}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}+\Bigg{\\}}-\frac{e^{2Gr^{2}}\alpha}{H\left(-2Gr^{2}+2Hr^{2}+3\right)}\Bigg{]}-\frac{3}{1-3\text{$\beta_{1}$}}\times$
$\displaystyle\qquad\Bigg{[}\beta+4B\text{$\beta$1}-\frac{1}{16\pi}(\frac{2e^{2Gr^{2}}\alpha
r^{2}}{2(G-H)r^{2}+e^{2Gr^{2}}-1}-8e^{2\left((H-G)r^{2}+P\right)}H^{2}\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})r^{2}-2e^{2\left((H-G)r^{2}+P\right)}H\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}r+$ $\displaystyle\qquad
2H\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}r+2e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}r-8e^{-2Gr^{2}}H+\frac{3e^{2Gr^{2}}\alpha}{4(G-H)Hr^{2}+4G-2H}+8G\left(Hr^{2}+1\right)\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+$ $\displaystyle\qquad
4e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})-\frac{2e^{-2Gr^{2}}}{r^{2}}+\frac{2}{r^{2}}-\frac{e^{2Gr^{2}}\alpha}{4H(H-G)r^{2}+6H}+\frac{2\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}}{r})-\frac{\text{$\beta_{1}$}}{16\pi}\Bigg{(}4e^{-2Gr^{2}}H\left(-2Gr^{2}+2Hr^{2}+3\right)$
$\displaystyle\qquad+\frac{2}{r^{2}}{e^{-2Gr^{2}}\left(4(G-H)Hr^{4}+2(2G-3H)r^{2}+e^{2Gr^{2}}-1\right)}{r^{2}}+e^{2Gr^{2}}\Bigg{(}\frac{2r^{2}}{2(G-H)r^{2}+e^{2Gr^{2}}-1}$
$\displaystyle\qquad+\frac{1}{4(G-H)Hr^{2}+4G-2H}-\frac{1}{4H(H-G)r^{2}+6H}\Bigg{)}\alpha+\alpha\Big{(}-4e^{2\left((H-G)r^{2}+P\right)}H\left(-2Gr^{2}+4Hr^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})$
$\displaystyle\qquad+4\left(2G^{2}r^{2}+G+\frac{1}{r^{2}}\right)(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+\frac{2}{r}{e^{2\left((H-G)r^{2}+P\right)}\left(Gr^{2}-4Hr^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}}{r}+6Gr(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+2e^{-2Gr^{2}}r\times$
$\displaystyle\qquad(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}-e^{2\left((H-G)r^{2}+P\right)}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}+(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}\Big{)}-\frac{e^{2Gr^{2}}\alpha}{H\left(-2Gr^{2}+2Hr^{2}+3\right)}\Bigg{)}\Bigg{]},$
(38)

$\displaystyle p_{r}=-\frac{1}{16\pi}\Bigg{[}\frac{2e^{2Gr^{2}}\alpha
r^{2}}{2(G-H)r^{2}+e^{2Gr^{2}}-1}-8e^{2\left((H-G)r^{2}+P\right)}H^{2}\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})r^{2}-2e^{2\left((H-G)r^{2}+P\right)}H\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}r+2H\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}r$
$\displaystyle\qquad+2e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}r-8e^{-2Gr^{2}}H+\frac{3e^{2Gr^{2}}\alpha}{4(G-H)Hr^{2}+4G-2H}+8G\left(Hr^{2}+1\right)\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+4e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})-$
$\displaystyle\qquad\frac{2e^{-2Gr^{2}}}{r^{2}}+\frac{2}{r^{2}}-\frac{e^{2Gr^{2}}\alpha}{4H(H-G)r^{2}+6H}+\frac{2\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}}{r}\Bigg{]}-\frac{1}{1-3\text{$\beta$1}}\Bigg{[}\beta+4B\text{$\beta$1}-\frac{1}{16\pi}\Bigg{\\{}\frac{2e^{2Gr^{2}}\alpha
r^{2}}{2(G-H)r^{2}+e^{2Gr^{2}}-1}-$ $\displaystyle\qquad
8e^{2\left((H-G)r^{2}+P\right)}H^{2}\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})r^{2}-2e^{2\left((H-G)r^{2}+P\right)}H\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}r+2H\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}r+2e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}r-8e^{-2Gr^{2}}H$
$\displaystyle\qquad+\frac{3e^{2Gr^{2}}\alpha}{4(G-H)Hr^{2}+4G-2H}+8G\left(Hr^{2}+1\right)\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+4e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})-\frac{2e^{-2Gr^{2}}}{r^{2}}+\frac{2}{r^{2}}-\frac{e^{2Gr^{2}}\alpha}{4H(H-G)r^{2}+6H}$
$\displaystyle\qquad+\frac{2\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}}{r}\Bigg{\\}}-\frac{1}{16\pi}\text{$\beta$1}\Bigg{[}4e^{-2Gr^{2}}H\left(-2Gr^{2}+2Hr^{2}+3\right)+\frac{2e^{-2Gr^{2}}(4(G-H)Hr^{4}+2(2G-3H)r^{2}+e^{2Gr^{2}}-1)}{r^{2}}$
$\displaystyle\qquad+e^{2Gr^{2}}\left(\frac{2r^{2}}{2(G-H)r^{2}+e^{2Gr^{2}}-1}+\frac{1}{4(G-H)Hr^{2}+4G-2H}-\frac{1}{4H(H-G)r^{2}+6H}\right)\alpha+\alpha\Bigg{(}-4e^{2\left((H-G)r^{2}+P\right)}H\times$
$\displaystyle\qquad\left(-2Gr^{2}+4Hr^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})+4\left(2G^{2}r^{2}+G+\frac{1}{r^{2}}\right)(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+\frac{2e^{2\left((H-G)r^{2}+P\right)}\left(Gr^{2}-4Hr^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}}{r}+$ $\displaystyle\qquad
6Gr(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+2e^{-2Gr^{2}}r(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}-e^{2\left((H-G)r^{2}+P\right)}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}+(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}\Bigg{)}-\frac{e^{2Gr^{2}}\alpha}{H\left(-2Gr^{2}+2Hr^{2}+3\right)}\Bigg{]}\Bigg{]},$
(39)

$\displaystyle
p_{t}=-\frac{1}{1-3\text{$\beta_{1}$}}\Bigg{[}\beta+4B\text{$\beta_{1}$}-\frac{1}{16\pi}\Bigg{\\{}\frac{2e^{2Gr^{2}}\alpha
r^{2}}{2(G-H)r^{2}+e^{2Gr^{2}}-1}-8e^{2\left((H-G)r^{2}+P\right)}H^{2}\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})r^{2}-2e^{2\left((H-G)r^{2}+P\right)}H\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}r$
$\displaystyle\qquad+2H\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}r+2e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}r-8e^{-2Gr^{2}}H+\frac{3e^{2Gr^{2}}\alpha}{4(G-H)Hr^{2}+4G-2H}+8G\left(Hr^{2}+1\right)\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+$ $\displaystyle\qquad
4e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})-\frac{2e^{-2Gr^{2}}}{r^{2}}+\frac{2}{r^{2}}-\frac{e^{2Gr^{2}}\alpha}{4H(H-G)r^{2}+6H}+\frac{2\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}}{r}\Bigg{\\}}-\frac{1}{16\pi}\text{$\beta_{1}$}\Bigg{(}4e^{-2Gr^{2}}H\left(-2Gr^{2}+2Hr^{2}+3\right)$
$\displaystyle\qquad+\frac{2}{r^{2}}{e^{-2Gr^{2}}\left(4(G-H)Hr^{4}+2(2G-3H)r^{2}+e^{2Gr^{2}}-1\right)}+e^{2Gr^{2}}\Big{(}\frac{2r^{2}}{2(G-H)r^{2}+e^{2Gr^{2}}-1}+\frac{1}{4(G-H)Hr^{2}+4G-2H}$
$\displaystyle\qquad-\frac{1}{4H(H-G)r^{2}+6H}\Big{)}\alpha+\alpha\Big{(}-4e^{2\left((H-G)r^{2}+P\right)}H\left(-2Gr^{2}+4Hr^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})+4\left(2G^{2}r^{2}+G+\frac{1}{r^{2}}\right)(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})$
$\displaystyle\qquad+\frac{2}{r}{e^{2\left((H-G)r^{2}+P\right)}\left(Gr^{2}-4Hr^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}}+6Gr(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+2e^{-2Gr^{2}}r(B22)^{\prime}-e^{2\left((H-G)r^{2}+P\right)}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}+$
$\displaystyle\qquad(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}\Big{)}-\frac{e^{2Gr^{2}}\alpha}{H\left(-2Gr^{2}+2Hr^{2}+3\right)}\Bigg{)}\bigg{]}-\frac{1}{8\pi}\Bigg{[}\left(4G^{2}r^{2}+4H^{2}r^{2}+2G+\frac{1}{r^{2}}\right)\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+\frac{1}{2}\Big{(}-2e^{-2Gr^{2}}G\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}r^{3}$
$\displaystyle\qquad+2e^{-2Gr^{2}}H\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}r^{3}-8e^{-2Gr^{2}}H^{2}r^{2}+8e^{-2Gr^{2}}GHr^{2}+\frac{4e^{2Gr^{2}}\alpha
r^{2}}{2Gr^{2}-2Hr^{2}+e^{2Gr^{2}}-1}+e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime\prime}r^{2}-$ $\displaystyle\qquad
2e^{2\left((H-G)r^{2}+P\right)}H\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}r+6G\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}r+6e^{-2Gr^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}r+4e^{-2Gr^{2}}G-8e^{-2Gr^{2}}H+$
$\displaystyle\qquad\frac{e^{2Gr^{2}}\alpha}{4G\left(Hr^{2}+1\right)-2H\left(2Hr^{2}+1\right)}-4e^{-2Gr^{2}}\left(Gr^{2}-Hr^{2}-1\right)\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})+\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}-\frac{e^{2Gr^{2}}\alpha}{4H^{2}r^{2}-4GHr^{2}+6H}\Big{)}\Bigg{]}.$
(40)

$\displaystyle
p_{q}=\frac{1}{{1-3\text{$\beta_{1}$}}}\Bigg{[}\beta+4B\text{$\beta_{1}$}-\frac{1}{{16\pi}}\bigg{\\{}-8\alpha
H^{2}r^{2}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})e^{2(r^{2}(H-G)+P)}+8\alpha G(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})\left(Hr^{2}+1\right)+4\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})e^{-2Gr^{2}}$ $\displaystyle\qquad-2\alpha
Hr(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}e^{2\left(r^{2}(H-G)+P\right)}+2\alpha
Hr(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+\frac{2}{r}{\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}}+2\alpha r(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}e^{-2Gr^{2}}+\frac{2\alpha
r^{2}e^{2Gr^{2}}}{2r^{2}(G-H)+e^{2Gr^{2}}-1}$
$\displaystyle\qquad+\frac{3\alpha
e^{2Gr^{2}}}{4Hr^{2}(G-H)+4G-2H}-\frac{\alpha
e^{2Gr^{2}}}{4Hr^{2}(H-G)+6H}-8He^{-2Gr^{2}}-\frac{2}{r^{2}}{e^{-2Gr^{2}}}+\frac{2}{r^{2}}\bigg{\\}}-\frac{1}{16\pi}\bigg{\\{}\text{$\beta_{1}$}\Bigg{(}\alpha(-4H(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})\times$
$\displaystyle\qquad\left(-2Gr^{2}+4Hr^{2}-1\right)e^{2\left(r^{2}(H-G)+P\right)}+4(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})\left(2G^{2}r^{2}+G+\frac{1}{r^{2}}\right)+\frac{2}{r}{(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}\left(Gr^{2}-4Hr^{2}-1\right)e^{2\left(r^{2}(H-G)+P\right)}}$
$\displaystyle\qquad+6Gr(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+2r(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}e^{-2Gr^{2}}-(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}e^{2\left(r^{2}(H-G)+P\right)}+(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime})+\alpha
e^{2Gr^{2}}\bigg{(}\frac{2r^{2}}{2r^{2}(G-H)+e^{2Gr^{2}}-1}$
$\displaystyle\qquad+\frac{1}{4Hr^{2}(G-H)+4G-2H}-\frac{1}{4Hr^{2}(H-G)+6H}\bigg{)}-\frac{\alpha
e^{2Gr^{2}}}{H\left(-2Gr^{2}+2Hr^{2}+3\right)}+4He^{-2Gr^{2}}\left(-2Gr^{2}+2Hr^{2}+3\right)$
$\displaystyle\qquad+\frac{2}{{r^{2}}}{e^{-2Gr^{2}}\left(4Hr^{4}(G-H)+2r^{2}(2G-3H)+e^{2Gr^{2}}-1\right)}\Bigg{)}\bigg{\\}}\Bigg{]},$
(41)

$\displaystyle\rho_{q}=\frac{1}{1-3\text{$\beta$1}}\Bigg{[}3\Bigg{\\{}\beta+4B\text{$\beta$1}-\frac{1}{16\pi}\Bigg{(}-8\alpha
H^{2}r^{2}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})e^{2\left(r^{2}(H-G)+P\right)}+8\alpha
G(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})\left(Hr^{2}+1\right)+4\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})e^{-2Gr^{2}}-$ $\displaystyle\qquad 2\alpha
Hr(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}e^{2\left(r^{2}(H-G)+P\right)}+2\alpha
Hr(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+\frac{2\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}}{r}+2\alpha r(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}e^{-2Gr^{2}}+\frac{2\alpha
r^{2}e^{2Gr^{2}}}{2r^{2}(G-H)+e^{2Gr^{2}}-1}$
$\displaystyle\qquad+\frac{3\alpha
e^{2Gr^{2}}}{4Hr^{2}(G-H)+4G-2H}-\frac{\alpha
e^{2Gr^{2}}}{4Hr^{2}(H-G)+6H}-8He^{-2Gr^{2}}-\frac{2e^{-2Gr^{2}}}{r^{2}}+\frac{2}{r^{2}}\Bigg{)}-\frac{1}{16\pi}\text{$\beta$1}\Bigg{(}\alpha\Bigg{(}-4H(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})$
$\displaystyle\qquad\left(-2Gr^{2}+4Hr^{2}-1\right)e^{2\left(r^{2}(H-G)+P\right)}+4(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})\left(2G^{2}r^{2}+G+\frac{1}{r^{2}}\right)+\frac{2(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}\left(Gr^{2}-4Hr^{2}-1\right)e^{2\left(r^{2}(H-G)+P\right)}}{r}$
$\displaystyle\qquad+6Gr(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+2r(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}e^{-2Gr^{2}}-(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}e^{2\left(r^{2}(H-G)+P\right)}+(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}\Bigg{)}+\alpha
e^{2Gr^{2}}\bigg{(}\frac{2r^{2}}{2r^{2}(G-H)+e^{2Gr^{2}}-1}$
$\displaystyle\qquad+\frac{1}{4Hr^{2}(G-H)+4G-2H}-\frac{1}{4Hr^{2}(H-G)+6H}\bigg{)}-\frac{\alpha
e^{2Gr^{2}}}{H\left(-2Gr^{2}+2Hr^{2}+3\right)}+4He^{-2Gr^{2}}\left(-2Gr^{2}+2Hr^{2}+3\right)+$
$\displaystyle\qquad\frac{2e^{-2Gr^{2}}\left(4Hr^{4}(G-H)+2r^{2}(2G-3H)+e^{2Gr^{2}}-1\right)}{r^{2}}\Bigg{)}\Bigg{\\}}\Bigg{]}+4B.$
(42)
$\displaystyle\beta=(3\text{$\beta_{1}$}-1)\Bigg{[}\frac{4B\text{$\beta_{1}$}}{1-3\text{$\beta_{1}$}}+\frac{1}{16\pi}\Bigg{(}\frac{2e^{2GR^{2}}\alpha
R^{2}}{2(G-H)R^{2}+e^{2GR^{2}}-1}-8e^{2\left((H-G)R^{2}+P\right)}H^{2}\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})R^{2}-2e^{2\left((H-G)R^{2}+P\right)}H\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}R$
$\displaystyle\qquad+2H\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}R+2e^{-2GR^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}R-8e^{-2GR^{2}}H+\frac{3e^{2GR^{2}}\alpha}{4(G-H)HR^{2}+4G-2H}+8G\left(HR^{2}+1\right)\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+$ $\displaystyle\qquad
4e^{-2GR^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})-\frac{2e^{-2Gr^{2}}}{R^{2}}+\frac{2}{R^{2}}-\frac{e^{2GR^{2}}\alpha}{4H(H-G)R^{2}+6H}+\frac{2\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}}{R}\Bigg{)}-\frac{1}{16\pi(1-3\text{$\beta_{1}$})}\Bigg{\\{}\frac{2e^{2GR^{2}}\alpha
R^{2}}{2(G-H)R^{2}+e^{2GR^{2}}-1}$
$\displaystyle\qquad-8e^{2\left((H-G)R^{2}+P\right)}H^{2}\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})R^{2}-2e^{2\left((H-G)R^{2}+P\right)}H\alpha(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}R+2H\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}R+2e^{-2GR^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}R-$ $\displaystyle\qquad
8e^{-2GR^{2}}H+\frac{3e^{2GR^{2}}\alpha}{4(G-H)HR^{2}+4G-2H}+8G\left(HR^{2}+1\right)\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})+4e^{-2GR^{2}}\alpha(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})-\frac{2e^{-2GR^{2}}}{R^{2}}+\frac{2}{R^{2}}-$
$\displaystyle\qquad\frac{e^{2GR^{2}}\alpha}{4H(H-G)R^{2}+6H}+\frac{2\alpha(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}}{R}\Bigg{\\}}-\frac{\beta_{1}}{16\pi(1-3\text{$\beta_{1}$})}\Bigg{(}4e^{-2GR^{2}}H\left(-2GR^{2}+2HR^{2}+3\right)+$
$\displaystyle\qquad\frac{2}{R^{2}}{e^{-2GR^{2}}\left(4(G-H)HR^{4}+2(2G-3H)R^{2}+e^{2GR^{2}}-1\right)}+e^{2GR^{2}}\bigg{(}\frac{2R^{2}}{2(G-H)r^{2}+e^{2GR^{2}}-1}+\frac{1}{4(G-H)HR^{2}+4G-2H}$
$\displaystyle\qquad-\frac{1}{4H(H-G)R^{2}+6H}\bigg{)}\alpha+\alpha\Bigg{(}-4e^{2\left((H-G)R^{2}+P\right)}H\left(-2GR^{2}+4HR^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})+4\left(2G^{2}R^{2}+G+\frac{1}{R^{2}}\right)(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})$
$\displaystyle\qquad+\frac{2}{R}{e^{2\left((H-G)R^{2}+P\right)}\left(GR^{2}-4HR^{2}-1\right)(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime}}+6GR(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime}+2e^{-2GrR2}R(\mathcal{A}^{2}_{\mkern
8.0mu2}\mathcal{A}^{22})^{\prime}-e^{2\left((H-G)r^{2}+P\right)}(\mathcal{A}^{0}_{\mkern
8.0mu0}\mathcal{A}^{00})^{\prime\prime}$
$\displaystyle\qquad+(\mathcal{A}^{1}_{\mkern
8.0mu1}\mathcal{A}^{11})^{\prime\prime}\Bigg{)}-\frac{e^{2GR^{2}}\alpha}{H\left(-2GR^{2}+2HR^{2}+3\right)}\Bigg{)}\Bigg{]}$
(43)

## Data Availability Statement

The research presented in this article is solely based on pure mathematics,
and no external data was used.

## Conflict of Interest

The authors assert that they do not have any conflicts of interest.

## Contributions

We hereby affirm that each author has contributed equally to this manuscript.

## Data Availability Statement

The authors declare that the data supporting the findings of this study are
available withinthearticle.

## REFERENCES

## References

  * (1) Buballa, Michael, et al. ”EMMI rapid reaction task force meeting on quark matter in compact stars.” Journal of Physics G: Nuclear and Particle Physics 41.12 (2014): 123001.
  * (2) Weber, Fridolin, et al. ”Properties of high-density matter in neutron stars.” Modern Physics Letters A 29.23 (2014): 1430022.
  * (3) Orsaria, M., et al. ”Quark deconfinement in high-mass neutron stars.” Physical Review C 89.1 (2014): 015806.
  * (4) Drago, Alessandro, Andrea Lavagno, and Giuseppe Pagliara. ”Can very compact and very massive neutron stars both exist?.” Physical Review D 89.4 (2014): 043014.
  * (5) Alvarez-Castillo, D., et al. ”New class of hybrid EoS and Bayesian M-R data analysis.” The European Physical Journal A 52 (2016): 1-11.
  * (6) Alford, Mark G., and Sophia Han. ”Characteristics of hybrid compact stars with a sharp hadron-quark interface.” The European Physical Journal A 52.3 (2016): 62.
  * (7) Alford, Mark G., Sophia Han, and Madappa Prakash. ”Generic conditions for stable hybrid stars.” Physical Review D 88.8 (2013): 083013.
  * (8) Alford, Mark G., and Sophia Han. ”Characteristics of hybrid compact stars with a sharp hadron-quark interface.” The European Physical Journal A 52.3 (2016): 62.
  * (9) Blaschke, David, and Nicolas Chamel. ”Phases of dense matter in compact stars.” The Physics and Astrophysics of Neutron Stars (2018): 337-400.
  * (10) Blaschke, David, et al. ”Astrophysical aspects of general relativistic mass twin stars.” Topics on Strong Gravity: A Modern View on Theories and Experiments. 2020. 207-256.
  * (11) Glendenning, Norman K., and Christiane Kettner. ”Possible third family of compact stars more dense than neutron stars.” Astronomy and Astrophysics, v. 353, p. L9-L12 (2000) 353 (2000): L9-L12.
  * (12) Glendenning, Norman K., and Norman K. Glendenning. ”Compact stars: From dwarfs to black holes.” Compact Stars: Nuclear Physics, Particle Physics and General Relativity (1997): 55-123.
  * (13) Zacchi, Andreas, Matthias Hanauske, and Jargen Schaffner-Bielich. ”Stable hybrid stars within a SU (3) Quark-Meson-Model.” Physical Review D 93.6 (2016): 065011.
  * (14) Li, Jia Jie, Armen Sedrakian, and Mark Alford. ”Relativistic hybrid stars with sequential first-order phase transitions and heavy-baryon envelopes.” Physical Review D 101.6 (2020): 063022.
  * (15) D. Blaschke, M. Cierniak, Astron. Nachrichten 342, 227-233 (2021).
  * (16) Christian, Jan-Erik, and Jagen Schaffner-Bielich. ”Supermassive neutron stars rule out twin stars.” Physical Review D 103.6 (2021): 063042.
  * (17) Drischler, Christian, et al. ”Limiting masses and radii of neutron stars and their implications.” Physical Review C 103.4 (2021): 045808.
  * (18) Bozzola, Gabriele, et al. ”Maximum mass and universal relations of rotating relativistic hybrid hadron-quark stars.” The European Physical Journal A 55 (2019): 1-18.
  * (19) Zhang, Na, Dehua Wen, and Houyuan Chen. ”Imprint of the speed of sound in nuclear matter on global properties of neutron stars.” Physical Review C 99.3 (2019): 035803.
  * (20) Will, Clifford M. ”The confrontation between general relativity and experiment.” Living reviews in relativity 17.1 (2014): 1-117.
  * (21) Turyshev, Slava G. ”Experimental tests of general relativity: recent progress and future directions.” Physics-Uspekhi 52.1 (2009): 1.
  * (22) Hess, P. O. ”Alternatives to Einstein’s general relativity theory.” Progress in Particle and Nuclear Physics 114 (2020): 103809.
  * (23) Clifton, Timothy, et al. ”Modified gravity and cosmology.” Physics reports 513.1-3 (2012): 1-189.
  * (24) Shankaranarayanan, S., and Joseph P. Johnson. ”Modified theories of gravity: Why, how and what?.” General Relativity and Gravitation 54.5 (2022): 44.
  * (25) Amendola, Luca, Leonardo Giani, and Giorgio Laverda. ”Ricci-inverse gravity: a novel alternative gravity, its flaws, and how to cure them.” Physics Letters B 811 (2020): 135923.
  * (26) Das, Indranil, Joseph P. Johnson, and S. Shankaranarayanan. ”Can we bypass no-go theorem for Ricci-inverse gravity?.” The European Physical Journal Plus 137.11 (2022): 1-24.
  * (27) Shamir, M. Farasat, et al. ”Ricci inverse anisotropic stellar structures.” Chinese Journal of Physics 81 (2023): 51-63.
  * (28) Jawad, Abdul, and Abdul Malik Sultan. ”Analysis of baryon to entropy ratio in Ricci inverse gravity.” Europhysics Letters 138.2 (2022): 29001.
  * (29) Scomparin, Mattia. ”Cosmic structures in Ricci-inverse theories of gravity.” arXiv preprint arXiv:2102.04676 (2021).
  * (30) Do, Tuan Q. ”No-go theorem for inflation in Ricci-inverse gravity.” The European Physical Journal C 81.5 (2021): 431.
  * (31) Do, Tuan Q. ”No-go theorem for inflation in an extended Ricci-inverse gravity model.” The European Physical Journal C 82.1 (2022): 15.
  * (32) de Souza, J. C. R., and A. F. Santos. ”An axially symmetric spacetime with causality violation in Ricci-inverse gravity.” The European Physical Journal C 83.9 (2023): 834.
  * (33) Ruderman, Malvin. ”Pulsars: structure and dynamics.” Annual Review of Astronomy and Astrophysics 10.1 (1972): 427-476.
  * (34) Sawyer, Raymond F. ”Condensed $\pi$ phase in neutron-star matter.” Physical Review Letters 29.6 (1972): 382.
  * (35) Sokolov, A. I. ”Phase transitions in a superfluid neutron liquid.” Sov. Phys. JETP 52.4 (1980): 575.
  * (36) Mak, M. K., and T. Harko. ”Anisotropic stars in general relativity.” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 459.2030 (2003): 393-408.
  * (37) Usov, Vladimir V. ”Electric fields at the quark surface of strange stars in the color-flavor locked phase.” Physical review D 70.6 (2004): 067301.
  * (38) Varela, Victor, et al. ”Charged anisotropic matter with linear or nonlinear equation of state.” Physical Review D 82.4 (2010): 044052.
  * (39) Rahaman, Farook, et al. ”A comparison of Hoavae Lifshitz gravity and Einstein gravity through thin-shell wormhole construction.” Classical and Quantum Gravity 28.15 (2011): 155021.
  * (40) Kalam, Mehedi, et al. ”Anisotropic strange star with de Sitter spacetime.” The European Physical Journal C 72 (2012): 1-7.
  * (41) Deb, Debabrata, et al. ”A new model for strange stars.” General Relativity and Gravitation 50 (2018): 1-18.
  * (42) Nambu, Yoichiro, and Giovanni Jona-Lasinio. ”Dynamical model of elementary particles based on an analogy with superconductivity. I.” Physical review 122.1 (1961): 345.
  * (43) Nambu, Yoichiro, and Giovanni Jona-Lasinio. ”Dynamical model of elementary particles based on an analogy with superconductivity. II.” Physical review 124.1 (1961): 246.
  * (44) Klevansky, Sandra P. ”The Nambu Jona-Lasinio model of quantum chromodynamics.” Reviews of Modern Physics 64.3 (1992): 649.
  * (45) Buballa, Michael. ”NJL-model analysis of dense quark matter.” Physics Reports 407.4-6 (2005): 205-376.
  * (46) K., Thomas, and Tobias Fischer. ”Vector interaction enhanced bag model for astrophysical applications.” The Astrophysical Journal 810.2 (2015): 134.
  * (47) Brilenkov, Maxim, et al. ”Dark matter and dark energy from quark bag model.” Journal of cosmology and Astroparticle Physics 2013.08 (2013): 002.
  * (48) Maharaj, S. D., J. M. Sunzu, and S. Ray. ”Some simple models for quark stars.” The European Physical Journal Plus 129 (2014): 1-10.
  * (49) Paulucci, Laura, and Jorge Ernesto Horvath. ”Strange quark matter fragmentation in astrophysical events.” Physics Letters B 733 (2014): 164-168.
  * (50) Panda, R., K. K. Mohanta, and K. Sahu. ”Radial modes of oscillations of slowly rotating magnetized compact hybrid stars.” Journal of Physics: Conference Series. Vol. 599. No. 1. IOP Publishing, 2015.
  * (51) Isayev, A. A. ”Stability of magnetized strange quark matter in the MIT bag model with a density dependent bag pressure.” Physical Review C 91.1 (2015): 015208.
  * (52) Abbas, G., Shahid Qaisar, and Abdul Jawad. ”Strange stars in f (T) gravity with MIT bag model.” Astrophysics and Space Science 359.2 (2015): 57.
  * (53) Lugones, German, and Josa DV Arbail. ”Compact stars in the braneworld: A new branch of stellar configurations with arbitrarily large mass.” Physical Review D 95.6 (2017): 064022.
  * (54) Stergioulas, Nikolaos. ”Rotating stars in relativity.” Living Reviews in Relativity 6.1 (2003): 3.
  * (55) Bhar, Piyali. ”A new hybrid star model in Krori-Barua spacetime.” Astrophysics and Space Science 357 (2015): 1-10.
  * (56) Rahaman, Farook, et al. ”Strange stars in Krori Barua space-time.” The European Physical Journal C 72 (2012): 1-9.
  * (57) Biswas, Suparna, et al. ”Anisotropic charged strange stars in Krori-Barua spacetime under f (R, T) gravity.” Annals of Physics 428 (2021): 168429.
  * (58) Sharif, M., and Arfa Waseem. ”Physical behavior of anisotropic compact stars in $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity.” Canadian Journal of Physics 94.10 (2016): 1024-1039.
  * (59) Abbas, G., et al. ”Anisotropic compact stars in $f(G)$ gravity.” Astrophysics and Space Science 357 (2015): 1-11.
  * (60) Shamir, M. Farasat, and Adnan Malik. ”Behavior of anisotropic compact stars in $f(R,\phi)$ gravity.” Communications in Theoretical Physics 71.5 (2019): 599.
  * (61) Malik, Adnan, et al. ”A study of charged stellar structure in modified $f(R,\phi,X)$ gravity.” International Journal of Geometric Methods in Modern Physics 19.11 (2022): 2250180.
  * (62) Malik, Adnan, Iftikhar Ahmad, and Kiran. ”A study of anisotropic compact stars in $f(R,\psi,X)$ theory of gravity.” International Journal of Geometric Methods in Modern Physics 19.02 (2022): 2250028.
  * (63) Bhar, Piyali. ”Charged strange star with Krori Barua potential in $f(R,T)$ gravity admitting Chaplygin equation of state.” The European Physical Journal Plus 135.9 (2020): 1-21.
  * (64) Hossein, SK Monowar, et al. ”Anisotropic compact stars with variable cosmological constant.” International Journal of Modern Physics D 21.13 (2012): 1250088.
  * (65) Rahaman, Farook, et al. ”Strange stars in Krori Barua space-time.” The European Physical Journal C 72 (2012): 1-9.
  * (66) Amendola, Luca, Leonardo Giani, and Giorgio Laverda. ”Ricci-inverse gravity: a novel alternative gravity, its flaws, and how to cure them.” Physics Letters B 811 (2020): 135923.
  * (67) Mak, M. K., and T. Harko. ”Quark stars admitting a one-parameter group of conformal motions.” International Journal of Modern Physics D 13.01 (2004): 149-156.
  * (68) Krori, K. D., and Jayantimala Barua. ”A singularity-free solution for a charged fluid sphere in general relativity.” Journal of Physics A: Mathematical and General 8.4 (1975): 508.
  * (69) Bondi, Hermann. ”Spherically symmetrical models in general relativity.” Monthly Notices of the Royal Astronomical Society 107.5-6 (1947): 410-425.
  * (70) Witten, Edward. ”A new proof of the positive energy theorem.” Communications in Mathematical Physics 80.3 (1981): 381-402.
  * (71) Visser, Matt. ”Energy conditions in the epoch of galaxy formation.” Science 276.5309 (1997): 88-90.
  * (72) Andrasson, Hkan. ”Sharp bounds on the critical stability radius for relativistic charged spheres.” Communications in Mathematical Physics 288.2 (2009): 715-730.
  * (73) Garcia, Nadiezhda Montelongo, et al. ”Energy conditions in modified Gauss-Bonnet gravity.” Physical Review D 83.10 (2011): 104032.
  * (74) Florides, Petros S. ”The complete field of charged perfect fluid spheres and of other static spherically symmetric charged distributions.” Journal of Physics A: Mathematical and General 16.7 (1983): 1419.
  * (75) Kumar, Jitendra, and Puja Bharti. ”An isotropic compact stellar model in curvature coordinate system consistent with observational data.” The European Physical Journal Plus 137.3 (2022): 330.
  * (76) Herrera, L. ”Cracking of self-gravitating compact objects.” Physics Letters A 165.3 (1992): 206-210.
  * (77) Abreu, H., Hector Hernndez, and Luis A. Nanez. ”Sound speeds, cracking and the stability of self-gravitating anisotropic compact objects.” Classical and Quantum Gravity 24.18 (2007): 4631.
  * (78) Chan, R., L. Herrera, and N. O. Santos. ”Dynamical instability for radiating anisotropic collapse.” Monthly Notices of the Royal Astronomical Society 265.3 (1993): 533-544.
  * (79) Chandrasekhar, S. ”Dynamical instability of gaseous masses approaching the Schwarzschild limit in general relativity.” Physical Review Letters 12.4 (1964): 114.
  * (80) Heintzmann, Hillebrandt, and W. Hillebrandt. ”Neutron stars with an anisotropic equation of state-mass, redshift and stability.” Astronomy and Astrophysics, vol. 38, no. 1, Jan. 1975, p. 51-55. Research supported by the Deutsche Forschungsgemeinschaft. 38 (1975): 51-55.