#  Dark Energy Compact Stars in Extended Teleparallel Gravity

Allah Ditta<EMAIL_ADDRESS>Department of Mathematics, Shanghai
University and Newtouch Center for Mathematics of Shanghai University,
Shanghai,200444, P.R.China.  Xia Tiecheng<EMAIL_ADDRESS>Department of
Mathematics, Shanghai University and Newtouch Center for Mathematics of
Shanghai University, Shanghai,200444, P.R.China.  G. Mustafa
<EMAIL_ADDRESS>Department of Physics, Zhejiang Normal University,
Jinhua 321004, People’s Republic of China  Institute of Fundamental and
Applied Research, National Research University TIIAME, Kori Niyoziy 39,
Tashkent 100000, Uzbekistan  Değer Sofuoğlu<EMAIL_ADDRESS>Department
of Physics, Istanbul University, 34134, Vezneciler, Fatih, Istanbul, Turkey
Asif Mahmood<EMAIL_ADDRESS>College of Engineering, Chemical Engineering
Department, King Saud University Riyadh, Saudi Arabia

###### Abstract

This paper presents the study of dark-energy compact stars in the context of
modified Rastall teleparallel gravity. It is the first time that dark energy
celestial phenomena have been explored in this modified gravitational theory.
Employing the torsion-based functions, $f(T)$ and $h(T)$, we analyzed their
effects in a spherically symmetric spacetime chosen as the interior geometry,
while using the Schwarzschild geometry as an outer spacetime. In this study,
we explored various dark energy stellar properties, including dark energy
pressure components, energy conditions, and equation of state components. Our
findings reveal that the observed negative behavior of these stellar
properties served as compelling evidence, validating the presence of dark
energy in stellar configurations. Detailed investigations of the energy
conditions, pressure profiles, sound speeds, TOV equation, adiabatic index,
gradients, mass function, compactness, and redshift function forecasts a
comprehensive assessment, affirming the acceptability and realism of the
investigated stellar configuration.
Keywords: Anisotropic spheres; Dark energy stars; Modified Rastall
Teleparallel Gravity; $f(T)$ gravity.

##  I Introduction

A revolutionary finding about the cosmos was made in 1998 Riess  _et al._
(1998a): it was expanding faster than observers had predicted. Scientists
became quite curious about what was causing this expansion as a result.
Eventually, they discovered that the source of this expansion was dark energy,
also referred to as ($\Lambda$) the cosmological constant, which is the
negative pressure fluid. But it disagreed with the energy conditions, which
made many scientists question whether General theory of Relativity (GR) could
deal with it satisfactorily. Local gravity is well explained by GR theory,
which is dependent upon the symmetric and torsion-less Levi-Civita connection
Aldrovandi and Pereira (2012). When considering gravity on a larger, global
scale, it is insufficient. In order to overcome this weakness, scientists
began to think about altering GR. A large number of these modifications aimed
at expanding the geometry of GR, among which the most successful were $f(R)$
theories. The Lagrangian function in such hypotheses is written as $R$, the
Ricci scalar Sotiriou and Faraoni (2010); Bamba  _et al._ (2012).

Because it closely resembles General Relativity (GR), teleparallel gravity, or
TEGR, has become increasingly popular. TEGR is a curvature-free theory based
on torsion. It is a popular alternative to general relativity (GR) in some
circumstances since its field equations are almost the same as GR’s. These two
ideas, albeit similar, are not understood mathematically in the same way. The
$f(T)$ gravity theory is a more generalized variant of TEGR, and it is closely
related to $f(R)$ theory. In contrast to General Relativity, this theory is
curvature-free and has non-zero torsion since it is founded on the Weitzenbock
connection Aldrovandi and Pereira (2012).Torsion served as the foundation for
Einstein’s original definition of space-time Einstein (1925). Unlike the
metric function, the tetrad is important in TEGR for setting up the field
equations. Selecting the appropriate tetrad is essential to releasing the
function ”$f$” from constraints because different tetrads can result in
distinct field equations. This framework offers a basis for TEGR modification.
On the other hand, a theory which is merely established by Birkhoff’s Theorem
can be produced by utilizing a diagonal tetrad; this specific theory cannot be
solved by the Schwarzschield metric. However, the expressions of ”$f$” and
”$T$” are not constrained in any way by an off-diagonal tetrad. That’s why, in
the current research, we employ an off-diagonal tetrad.

The conservation law of the energy-momentum tensor (EMT) connects spacetime
geometry and matter in Einstein’s General Relativity (GR). But this
conservation is only valid under certain circumstances, leaving GR open to
modify. While Rastall challenges the standard assumption in GR,
$\Theta^{\nu}{}{\mu;\nu}=a_{\mu}$, where $a_{\mu}$ nullifies in flat spacetime
Rastall (1972), the standard assumption in GR is $\Theta^{\nu}{}{\mu;\nu}=0$.
Spacetime curvature is caused by matter, which also causes gravity, and vice
versa. Matter is impacted by tidal gravitational forces, which are caused by
curvature. Let $\Theta^{\nu}{}{\mu;\nu}=0$ in flat spacetime. With Rastall’s
adjustment, the non-minimal coupling between geometry and matter is measured
by a coupling parameter, $\lambda$. For $\lambda=0$, it reduces to GR. Rastall
gravity has been modified to incorporate solutions for unique black holes,
such as $\nabla\mu T^{\mu\nu}=\nabla_{\nu}(\lambda^{\prime}R)$ Moradpour  _et
al._ (2017) and $\nabla_{\mu}T^{\mu\nu}=\lambda\nabla^{\nu}f(R)$ Lin and Qian
(2019). In the references Saaidi and Nazavari (2020); Nazavari  _et al._
(2023), Rastall-based adjustments are extended to teleparallel gravity. Our
focus in this research is teleparallel gravity with Rastall effects Nazavari
_et al._ (2023)

It is evident that GR has been effective in analyzing the majority of research
projects pertaining to different astrophysical phenomena along with stellar
models up to this point. However, the credibility of GR on broad cosmological
scales has been criticized in the late time accelerating phase of the
Universe, according to various observational data models Perlmutter  _et al._
(1998); Bahcall  _et al._ (1999); Riess  _et al._ (1998b); Tegmark  _et al._
(2004) Riess  _et al._ (1998a); de Bernardis  _et al._ (2000); Peebles and
Ratra (2003). A negative pressure, or gravitationally opposed, concealed
source of energy known as ”dark energy” (DE) is estimated to account for a
large proportion of energy sources, or roughly $70\%$ of total energy, in
order to explain the Universe’s accelerating tendency Ade  _et al._ (2016). In
the cosmology community, there is still disagreement over the precise nature
and genesis of DE, nevertheless. Currently, multiple GR alternations have been
devised to address this undesirable state of the universe. Modified theories
of gravity (MTG) are the term most generally used to refer to these
alternative theories. some modified theories are discussed in the refrences
Capozziello and Faraoni (2010); Capozziello and De Laurentis (2011); Nojiri
and Odintsov (2007); Bamba  _et al._ (2012); Koyama (2016); Sotiriou and
Faraoni (2010); Saaidi and Nazavari (2020); Nazavari  _et al._ (2023).

Early research indicates that the matter organization in spherically symmetric
framework depends on perfect fluid. Coincidence of the tangential component of
pressure and radial component of the pressure enables the implementation of
the isotropic criteria ($p_{r}=p_{t}$) on EFEs. For an improved comprehension
of the dispersion of matter within the celestial objects, Jeans Jeans (1922)
proposed in 1922 that peculiar criterion predominate within the innermost
regions of stellar bodies, suggesting the participation of an anisotropic
factor. Anisotropy is only a measure of the departure from isotropy and is
expressed as $\Delta(p_{t}-p_{r})$. The literature Herrera and Santos (1997);
Mak and Harko (2002) has a significant amount of comprehensive information to
explore the impact of an anisotropy on star formations due to spherical
symmetry. Anisotropy in the relativistic star phenomenon results from the
presence of several types of fluids, including superfluids, external or
magnetic fields, phase transitions, rotary motions, and other fluids.

Scientists are intrigued by the evolution of heavily dense matter in extreme
conditions, viewing compact stars as potential final stages in regular star
life cycles. Compact stellar objects like pulsars, characterized by high
densities and strong magnetic fields, play a crucial role in astrophysics
Lattimer (2010). At first, these things were thought to be well described by a
spherically symmetric matter dispersion in an isotropic fluid, where radial
pressure and tangential pressure proved equal. Nonetheless, Jeans argued in
1992 that anisotropy was required to explain matter dispersion in a stellar
body’s special circumstances Jeans (1992). Anisotropy, denoted as
$\Delta=p_{t}-p_{r}$, indicating deviation from isotropic conditions, can
result from various factors like superfluids, fluid mixtures, solid cores,
phase transitions, viscosity, and the magnetic fields inside the star Jeans
(1992); Lemaître (1933); Ruderman (1972); Bowers and Liang (1974); Herrera and
Santos (1997); Weber (1999); Sokolov (1980). Some of the recent literature
Ditta  _et al._ (2023a, b, c); Bhar (2023); Sedrakian  _et al._ (2023); Maurya
_et al._ (2024); Lohakare  _et al._ (2023); Maurya  _et al._ (2023a) discuss
some acceptable models of stellar configurations. Some more studies enriched
with sound knowledge are present in the references Nojiri and Odintsov (2011);
Nojiri  _et al._ (2017); Astashenok  _et al._ (2022, 2021, 2020); Astashenok
and Odintsov (2020a, b); Astashenok  _et al._ (2015); Odintsov and Oikonomou
(2023).

DE is typically seen as an origin of energy that defies energy conditions such
as the Strong Energy Condition (SEC), which is defined as $\rho+3p\geq 0$, in
contrast to the anisotropic fluid, which has tangential pressure $p_{t}$ and
radial pressure $p_{r}$, which is $\rho+p_{r}+2p_{t}\geq 0$. The primary
reason for this infraction is because DE has a negative, pressured nature. The
EoS parameter in cosmology is described as a dimensionless number that,
assuming a perfect fluid, is a relation of pressure $p$ to energy density
$\rho$.Taking into account the range of the EoS parameter $\omega$, several
types of DE have been thought of up to this point. The cosmological constant,
represented by the conventional symbols $\Lambda$, is the most straightforward
DE applicant among them and it satisfies the EoS $p=-\rho$. In terms of
physics, $\Lambda$ is a uniformly filled vacuum and static type energy. On the
other hand, scalar fields like quintessence or moduli are active variables
that can change over time in space. EoS $p=\omega\rho$ defines the
quintessential DE, with a potential range of $-1<\omega<-1/3$. The DE
candidate also exhibits Phantom (or ghost)-like DE when $\omega<-1$. The
possibility of DE as a black hole substitute and internal component of a
compact relativistic entity opens up new avenues for cosmological inquiry.
Gravastars’ absence of an event horizon and center singularity is a crucial
characteristic. A few research studies on gravastars can be reviewed to learn
more about the solutions and physical properties of gravastars in relation to
GR and MTGs Usmani  _et al._ (2011); Rahaman  _et al._ (2012a); Das  _et al._
(2017); Ghosh  _et al._ (2017); Sengupta  _et al._ (2020). Subsequently, some
writers hypothesized a compact relativistic object known as a ”dark energy
star” that is composed of a mixture of non-interacting DE and regular matter,
based on the stable physical structure of gravastars. Some recent literature
on the study of DE stars is available Pretel (2023); Astashenok  _et al._
(2023); Bhar and Pretel (2023); Saklany  _et al._ (2023) and so on.

The following is the plan that will guide us as we move forward with the next
stage of our study: We shall review the principles of MRTG and evaluate the
FEs with an off-diagonal tetrad in Section-II. Furthermore, we use the
Karmarkar condition to find the generalised solutions in this section. We
shall match the outer and inner geometries in Section-III in order to
ascertain the constant parameters that we used in our stellar modelling. After
that, we will explain our findings in Section-IV, and then we conclude up in
Section-V.

##  II Basic formalism of MODIFIED Rastall TELEPARALLEL GRAVITY(MTRG) and our
generalized solutions

A basic knowledge of the internal functioning of stars is essential to
comprehending their structure. Spherically symmetric spacetime is a useful
model in this situation. This spacetime has consistent features in all
directions, making it amenable to mathematical representation using a variety
of models, including the Schwarzschild metric. Scientists may learn a great
deal about the behavior and formation of stars by analyzing spherically
symmetric spacetimes. These discoveries have important ramifications for a
variety of fields in astronomy. This spherically symmetric spacetimes is given
as:

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dt^{2}+r^{2}sin^{2}\theta
d\phi^{2}+d\theta^{2}.$ (1)

The Minkowski metric $\eta_{ij}=\operatorname{diag}(-1,1,1,1)$ and the tetrad
fields $e_{\mu}^{i}$ can be used to describe the metric tensor $g_{\mu\nu}$
described on a manifold:

$g_{\mu\nu}=\eta_{ij}e_{\mu}^{i}e_{\nu}^{j}$ (2)

wherein the Latin letter $(i,j,\ldots=0,1,2,3)$ provides tangent space indices
and the Greek alphabet $(\mu,\nu,\ldots=0,1,2,3)$ provides space-time indices.
In the realm of mathematics, the Weitzenbock connection can be summed up as:

$\Gamma_{\mu\nu}^{\mu}=e_{i}^{\alpha}\partial_{\nu}e_{\mu}^{i}=-e_{\mu}^{i}\partial_{\nu}e_{i}^{\alpha}.$
(3)

The teleparallel hypothesis makes use of a particular kind of link that has
zero curvature but non-zero torsion. The definition of the torsion tensor,
which can be expressed as follows, depends heavily on these connections:

$T_{\mu\nu}^{\sigma}\equiv\Gamma_{\nu\mu}^{\sigma}-\Gamma_{\mu\nu}^{\sigma}=e_{i}^{\sigma}\left(\partial_{\mu}e_{\nu}^{i}-\partial_{\nu}e_{\mu}^{i}\right)$
(4)

The subsequent relation connects the Levi-Civita connection denoted as
$\bar{\Gamma}_{\mu\nu}^{\sigma}$, to the Weitzenbock connection:

$\Gamma_{\mu v}^{\sigma}=\Gamma_{\mu\nu}^{\sigma}-K_{\mu v}^{\sigma},$ (5)

in this case, the contorsion tensor, denoted by $K{}_{\mu\nu}$, is provided
by:

$K_{\mu\nu}^{\sigma}=\frac{1}{2}\left(T_{\mu}^{\sigma}\sigma_{\nu}+T_{\nu}^{\sigma}\sigma_{\mu}-T_{\mu\nu}^{\sigma}\right).$
(6)

The torsion scalar expression can be understood as:

$T=S^{\sigma\mu\nu}T_{\sigma\mu\nu}$ (7)

When the super-potential $S^{\sigma\mu\nu}$ equation is provided by:

$S^{\sigma\mu\nu}=-S^{\sigma\nu\mu}=\frac{1}{2}\left(K^{\mu\nu\sigma}-g^{\sigma\nu}T^{\alpha\mu\mu}+g^{\sigma\mu}T^{a\nu}\right).$
(8)

According to Nazavari  _et al._ (2023), the amended teleparallel gravity
action is as follows:

$S=S_{G}+S_{m}=\frac{1}{4\kappa}\int ef(T)d^{4}x+\int eL_{m}d^{4}x$ (9)

wherein $e$ is the representation of the tetrad field $e^{a}{{}_{F}}$
determinant and the function $f(T)$ is torsion based. Moreover, the matter
Lagrangian is represented by $L_{m}$. The associated field equation can be
obtained by calculating the action’s variation with respect to the tetrad
field:

$S_{i}^{\mu\nu}f_{TT}\partial_{\mu}T+e^{-1}\partial_{\mu}\left(eS_{i}^{\mu\nu}\right)f_{T}-T^{\alpha}{}_{\mu
i}S_{\sigma}^{\nu\mu}f_{T}\\\
-\frac{1}{4}e_{i}{}^{\nu}f=-\xi\Theta_{\mu}^{\nu}$ (10)

while the normal EMT of the ideal fluid is denoted by $\Theta_{\mu}^{v}$. It
is demonstrated that:

$\left(S_{i}^{\mu
v}f_{TT}\partial_{\mu}T+e^{-1}\partial_{\mu}\left(eS_{i}{}^{\mu\nu}\right)f_{T}-T^{\sigma}{}_{\mu
i}S_{\sigma}^{\nu\mu}f_{T}\right.\\\
\left.-\frac{1}{4}e_{i}^{v}f\right)_{;v}=0,$ (11)

In which the covariant derivative in a teleparallel foundation is indicated by
a semicolon:

$V_{;\nu}^{\mu}=\partial_{\nu}V^{\mu}+\left(\Gamma_{\lambda\nu}^{\mu}-K_{\lambda\nu}^{\mu}\right)V^{\lambda},$
(12)

for any vector $V^{\mu}$ in space-time. The EMT’s covariant derivative
likewise vanishes, because of the calculation above:

$\Theta^{\nu}{}_{\mu;\nu}=0.$ (13)

The law of conservation for EMT is the same for our revised teleparallel
gravity theory and Einstein’s theory. But Rastall’s new equation,
$T^{\mu}{\nu;\mu}=\lambda R{,\nu}$, called into question Einstein’s theory’s
conservation equation. This formula indicates a relationship between matter
and geometry and provides a modified field equation, implying an interesting
interaction between the two. We assume a similar thing in our MRTG theory,
which is motivated by Rastall’s idea. By connecting geometry and matter via
the scalar torsion of geometry, we create a relation in which the divergence
of the torsion scalar corresponds to the divergence of the EMT,
$\Theta^{\nu}{}{\mu}$.

$\Theta^{\nu}{}_{\mu;\nu}=\lambda{h(T)_{,\mu}},$ (14)

here $h(T)$ is an analytical function of torsion and $\lambda$ is an arbitrary
real valued constant. Next is the rewrite of the field Eq. (11):

$S_{u}^{\mu\nu}f\gamma\tau\partial_{\mu}T+e^{-1}\partial_{\mu}\left(S_{i}^{\mu\nu}\right)f_{T}-T_{\mu}^{\sigma}S_{\sigma}^{\nu\mu}f_{T}-\frac{1}{4}e_{i}^{\nu}f-\delta_{\mu}^{\nu}\xi\lambda
h(T)=-\xi\Theta_{\mu\nu}$ (15)

The value of the gravitational parameter in Rastall’s theory, $\xi$, is
represented as follows in relation to Newtonian gravity:

$\xi=\frac{4\gamma-1}{6\gamma-1}\xi_{G},$ (16)

where the Einstein coupling constant $\xi_{C}=4\pi G$ is represented by the
notions $\gamma=\lambda\xi$ and $\xi_{G}$.

Fluid’s total EMT, which describes the compact star’s core, is expressed by:

$\Theta_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}_{m}\right)}{\delta
g^{\mu\nu}}.$ (17)

Assuming for the moment that the inner matter component is composed of a
mixture of two non-interacting fluids, such as isotropic regular matter and
anisotropic type unidentified matter (referred to as DE). Thus, the subsequent
tensor may be utilized for expressing all energy sources together Harko and
Lobo (2011); Rahmansyah and Sulaksono (2021).

$\Theta_{\mu\nu}^{eff}=\underbrace{\left(\rho^{D}+p_{t}^{D}\right)W_{\mu}W_{\nu}+p_{t}^{D}g_{\mu\nu}+\left(p_{r}^{D}-p_{t}^{D}\right)V_{\mu}V_{\nu}}_{\text{dark
energy }}+\underbrace{(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu}}_{\text{ordinary
matter }}.$ (18)

In this case, the DE applicant’s energy density, radial pressure, and
tangential pressure are denoted by the appropriate symbols
$\rho^{D},\;p_{r}^{D}$, and $p_{t}^{D}$, while the energy density and pressure
component of the normal matter are represented by $\rho$ and $p$,
respectively. Consequently, the mixed EMT $\Theta_{v}$ non-vanishing component
can be expressed in the following form:

$\displaystyle\rho_{eff}=\Theta_{0}^{0}$ $\displaystyle=$
$\displaystyle\rho+\rho^{D},$ (19) $\displaystyle p_{reff}=\Theta_{1}^{1}$
$\displaystyle=$ $\displaystyle-\left(p+p_{r}^{D}\right),$ (20) $\displaystyle
p_{teff}=\Theta_{2}^{2}$ $\displaystyle=$
$\displaystyle\Theta_{3}^{3}=-\left(p+p_{t}^{D}\right),$ (21)
$\displaystyle\Theta_{0}^{1}$ $\displaystyle=$
$\displaystyle\Theta_{1}^{0}=0.$ (22)

The teleparallel technique in GR includes tetrad fields ($e^{i}_{\mu}$)
representing coordinates and frames through Greek and Latin indices,
respectively. The combination of these indices forms the tetrad matrix,
satisfying $e_{i}^{\mu}e^{\nu}_{i}=\delta^{\mu}_{\nu}$ and
$e^{\mu}_{i}e^{\mu}_{j}=\delta_{ij}$. This technique extends the manifold to
include torsion alongside curvature, with the Riemannian curvature tensor
expected to be zero. Torsion-free geometry or tetrad-induced torsion can
explain gravity. Tamanini and Bohmer Tamanini and Boehmer (2012) discussed the
concept of a ”good tetrad” for studying a wider aspect of cosmologies in
$f(T)$ theory. Bohmer et al. Boehmer  _et al._ (2011) investigated
relativistic system of compact stars in $f(T)$ gravity, favoring off-diagonal
tetrad for spherically symmetric solutions due to issues with the diagonal
tetrad. Numerous studies on spherically symmetric spacetime within $f(T)$
modified gravity are available in the literature, including references Li  _et
al._ (2011); Ruggiero and Radicella (2015); Bahamonde  _et al._ (2019);
Hohmann  _et al._ (2019); Krššák  _et al._ (2019); Nashed and Saridakis
(2019), among others.

In this work to evaluate field equations, we will use the off-diagonal tetrad
matrix provided in Boehmer  _et al._ (2011); Tamanini and Boehmer (2012);
Daouda  _et al._ (2012):

$e^{a}{}_{\mu}=\left(\begin{array}[]{cccc}e^{\nu/2}&0&0&0\\\
0&e^{\lambda/2}\sin\theta\cos\phi&r\cos\theta\cos\phi&-r\sin\theta\sin\phi\\\
0&e^{\lambda/2}\sin\theta\sin\phi&r\cos\theta\sin\phi&r\sin\theta\cos\phi\\\
0&e^{\lambda/2}\cos\theta&-r\sin\theta&0\end{array}\right)\text{. }$ (23)

in which $g_{rr}=e^{\lambda(r)}$ and $g_{tt}=e^{\nu(r)}$ . In the tetrad field
above, $e=det\left(e^{a}{}_{\mu}\right)=r^{2}\sin\theta e^{(\nu+\lambda)/2}$
is the determimant. Using Eqs. (4), (7) and (8), the torsion scalar takes the
follows form:

$T(r)=\frac{2e^{-\lambda}}{r^{2}}\left(e^{\frac{\lambda}{2}}-1\right)\left(e^{\frac{\lambda}{2}}-1-r\nu^{\prime}\right).$
(24)

After substituting Eqs. (30, 32, and 23) in the field Eq. (1), the EMT’s
nonzero components can be interpreted as follows:

$\displaystyle\xi\left(\rho+\rho^{D}\right)$ $\displaystyle=$
$\displaystyle\frac{e^{-\lambda/2}}{r}\left(1-e^{-\lambda/2}\right)f_{T}^{\prime}-\left(\frac{T}{4}-\frac{1}{2r^{2}}\right)f_{T}+\frac{e^{-\lambda}}{2r^{2}}\left(r\lambda^{\prime}-1\right)f_{T}+\frac{f}{4}+\gamma
h(T),$ (25) $\displaystyle\xi\left(p+p_{r}^{D}\right)$ $\displaystyle=$
$\displaystyle{\left[\frac{T}{4}-\frac{1}{2r^{2}}+\frac{e^{-\lambda}}{2r^{2}}\left(1+r\nu^{\prime}\right)\right]f_{T}-\frac{f}{4}-\gamma
h(T)},$ (26) $\displaystyle\xi\left(p+p_{t}^{D}\right)$ $\displaystyle=$
$\displaystyle\frac{e^{-\lambda}}{2}\left(\frac{\nu^{\prime}}{2}+\frac{1}{r}-\frac{e^{\lambda/2}}{r}\right)f_{T}^{\prime}-\frac{f}{4}-\gamma
h(T)+f_{T}\Big{[}\frac{T}{4}+\frac{e^{-\lambda}}{2r}\Big{[}\left(\frac{1}{2}+\frac{r\nu^{\prime}}{4}\right)\left(\nu^{\prime}-\lambda^{\prime}\right)$
(27) $\displaystyle+$
$\displaystyle\frac{r\nu^{\prime\prime}}{2}\Big{]}\Big{]}.$

The conduct of the results are notably affected by the existence of the term
$\gamma h(T)$ connected with the Rastall”s coefficient $\gamma$, and the
coefficient $\xi$ in FEs, which may change the energy conditions.

A variety of suppositions for the available options in literature for the
functions $f(T)$ and $h(T)$ must be taken into account to generate solutions
for compact objects in literature. We use the off-diagonal tetrad in our
investigation, which increases the physicality of the analysis without adding
any limitations Tamanini and Boehmer (2012); Boehmer  _et al._ (2011) on
functions that rely on torsion, that is, $f(T)$ and $h(T)$. In pursuit of
broader solutions, we select non-linear models such as the power law form of
the $f(T)$ function Bamba  _et al._ (2012); Zubair and Abbas (2016) and the
logarithmic model for the $h(T)$ function Ditta  _et al._ (2023a):

$\displaystyle f(T)$ $\displaystyle=$ $\displaystyle\beta T^{n},$ (28)
$\displaystyle h(T)$ $\displaystyle=$ $\displaystyle\psi\log\left(\phi
T^{\chi}\right),$ (29)

wherein $\beta,\;n,\;\psi,\;\phi\;\&\;\chi$ are all arbitrary real constants.

Eisenhart class-I $(n+1)$-dimensional space $V^{n+1}$ can be embedded into a
$(n+2)$-dimensional Euclidean space $E^{n+2}$, as shown by Eisenhart Eisenhart
(1925), as long as a symmetric tensor exists satisfying the Gauss-Codazzi
equations:

$R_{stpq}=2ea_{s[p{}a_{q}]t}$ and
$a_{s[t;p]}-\Gamma^{q}{}_{[tp]}a_{sq}+\Gamma^{q}_{s[t{}a_{p}]q}=0$,

where, the Riemannian curvature tensor is denoted by $R_{stpq}$, $e=\pm 1$,
and anti-symmetrization is indicated by square brackets. It is widely proposed
in the scientific community that the Karmarkar condition can be used as a
handy and simple way to compute the components $e^{\nu(r)}$ and
$e^{\lambda(r)}$ of a spherically symmetric spacetime. A prerequisite that is
enough for the embedding of a class-I metric, is given by this condition. Here
is how the Karmarkar condition can be described:

$R_{0101}R_{2323}=R_{0202}R_{1313}-R_{1202}R_{1303}.$ (30)

Regarding metric (3), the non-zero elements of the Riemannian tensor are as
follows:

$\displaystyle R_{0101}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{\nu(r)}\left(-a^{\prime}(r)\lambda^{\prime}(r)+\nu^{\prime
2}(r)+2\nu^{\prime\prime}(r)\right),\quad
R_{2323}=-r^{2}\sin^{2}\theta\left(1-e^{-\lambda(r)}\right),\quad$
$\displaystyle R_{0202}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}r\nu^{\prime}(r)e^{\nu(r)-\lambda(r)},R_{1313}=-\frac{1}{2}\lambda^{\prime}(r)r\sin^{2}\theta,\quad
R_{1202}=0,\quad R_{1303}=0.$

By just putting these Riemannian tensor constituent values into Eq. (30), the
differential equation that follows can be constructed with ease:

$\left(\lambda^{\prime}(r)-\nu^{\prime}(r)\right)\nu^{\prime}(r)e^{\lambda(r)}+2\left(1-e^{\lambda(r)}\right)\nu^{\prime\prime}(r)+\nu^{\prime
2}(r)=0.$ (31)

The fact that class-I embedding solutions can be obtained from Eq. (31) and
inserted into a 5-dimensional Euclidean space is an intriguing point to make.
Eq. (31) for $e^{a}$ can be integrated to obtain its expression in terms of
$e^{b}$, as follows:

$e^{\nu(r)}=\left(A+B\int\sqrt{e^{\lambda(r)}-1}\,dr\right)^{2},$ (32)

where A and B are constants for integration. The metric potentials
$e^{\nu(r)}$ and $e^{\lambda(r)}$ are shown to be connected to one another
from Eq. (32). Consequently, this connection can be used to compute the other
metric component if one of them is known. In order to build a realistic
anisotropic model, we take into consideration a well-known model Gedela  _et
al._ (2019) for the $g_{rr}$ component of the metric, which is provided by:

$\displaystyle\lambda(r)$ $\displaystyle=$
$\displaystyle\log\left(1+ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}\right)$
(33)

where $\kappa$, $b$, $c$, and $a$ notions the arbitrary constants. Reference
Gedela  _et al._ (2019) makes clear that, for a given range of parameters
related to this $g_{rr}$ metric component, the solution given in Eq. (33) is
regular, correct, and yields a decent estimation of the neutron model. Eq.
(33) can be substituted into Eq. (32) to yield:

$\displaystyle\nu(r)$ $\displaystyle=$
$\displaystyle\log\left(\frac{B\left(\kappa\sqrt{1-\left(br^{2}+c\right)^{2}}+2br^{2}+2c\right)\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}}{b\left(\kappa^{2}+4\right)r}+A\right)^{2},$
(34)

here the values of the variables $A$, $B$, $\kappa$, $b$, $c$, and $a$ are
selected for defining this spacetime geometry according to various physical
considerations. According to literature Biswas  _et al._ (2021); Zubair  _et
al._ (2021); Rahaman  _et al._ (2012b); Roupas and Nashed (2020); Maurya  _et
al._ (2023b), this geometry has demonstrated efficacy in modeling self-
gravitating stellar structures for both GR and modified gravity. Employing
Eqs. (33-34) along with Eqs. (28-29) into Eqs. (25-27), we yield:

$\displaystyle\rho+\rho^{D}$ $\displaystyle=$
$\displaystyle,\frac{(6\gamma-1)}{4\pi(4\gamma-1)r^{2}}\Big{[}-\frac{\beta
2^{n-2}n\left(1-\mathbb{Z}_{8}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{3}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\beta
2^{n-2}r^{2}\left(\mathbb{Z}_{4}(r)\right){}^{n}$ (35) $\displaystyle+$
$\displaystyle\beta
2^{n-2}(1-n)nrr\left(-\mathbb{Z}_{9}\right)\left(1-\mathbb{Z}_{9}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-2}+\gamma
r^{2}\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]},$
$\displaystyle p+p_{r}^{D}$ $\displaystyle=$
$\displaystyle\frac{6\gamma-1}{4\pi(4\gamma-1)}\Big{[}-\frac{\beta
2^{n-1}n\left(\mathbb{Z}_{4}(r)\right){}^{n-1}\left(\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\mathbb{Z}_{5}(r)-1\right)}{r^{2}\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1\right)}-\beta
2^{n-2}\left(\mathbb{Z}_{4}(r)\right){}^{n}$ (36) $\displaystyle-$
$\displaystyle\gamma\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]},$
$\displaystyle p+p_{t}^{D}$ $\displaystyle=$
$\displaystyle\frac{6\gamma-1}{4\pi(4\gamma-1)}\Big{[}\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{r^{2}\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1\right)}\Big{[}\frac{1}{4}r\Big{[}\frac{4bB\left(\kappa^{2}+4\right)\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}\right)^{3/2}}{r\left(\mathbb{Z}_{1}(r)\right){}^{2}\sqrt{1-\left(br^{2}+c\right)^{2}}}\Big{[}B\Big{[}b^{2}\kappa
r^{4}$ (37) $\displaystyle-$ $\displaystyle
2br^{2}\mathbb{Z}_{10}(r)-c^{2}\kappa+2c\mathbb{Z}_{10}(r)+\kappa\Big{]}\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}+Ab\left(\kappa^{2}+4\right)r\left(b\kappa
r^{2}+r\mathbb{Z}_{10}\right)\Big{]}+\left(\mathbb{Z}_{6}(r)+2\right)$
$\displaystyle\times$
$\displaystyle\Big{[}\frac{2abB\left(\kappa^{2}+4\right)r^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}{\mathbb{Z}_{1}(r)}-\mathbb{Z}_{7}(r)\Big{]}\Big{]}+\left(\mathbb{Z}_{2}(r)\right)\left(\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}-1\right)\Big{]}$
$\displaystyle-$ $\displaystyle\frac{\beta
2^{n-4}(1-n)n\left(\mathbb{Z}_{4}(r)\right){}^{n-2}\left(-2\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\mathbb{Z}_{6}(r)+2\right)}{r\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1\right)}-\beta
2^{n-2}\left(\mathbb{Z}_{4}(r)\right){}^{n}$ $\displaystyle-$
$\displaystyle\gamma\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]},$

where,

$\displaystyle\mathbb{Z}_{1}(r)$ $\displaystyle=$ $\displaystyle
Ab\left(\kappa^{2}+4\right)\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}+aBr\left(\kappa\sqrt{-b^{2}r^{4}-2bcr^{2}-c^{2}+1}+2br^{2}+2c\right)e^{\kappa\sin^{-1}\left(br^{2}+c\right)},$
$\displaystyle\mathbb{Z}_{2}(r)$ $\displaystyle=$
$\displaystyle-\frac{2abB\left(\kappa^{2}+4\right)r^{3}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}{\mathbb{Z}_{1}(r)}+\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}-1,$
$\displaystyle\mathbb{Z}_{3}(r)$ $\displaystyle=$
$\displaystyle\frac{2abB\left(\kappa^{2}+4\right)r^{3}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}\left(\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}-1\right)}{\mathbb{Z}_{1}(r)}+2\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}-1,$
$\displaystyle\mathbb{Z}_{4}(r)$ $\displaystyle=$
$\displaystyle\frac{\left(\mathbb{Z}_{2}(r)\right)\left(\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}-1\right)}{r^{2}\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1\right)},\;\mathbb{Z}_{6}(r)=\frac{2abB\left(\kappa^{2}+4\right)r^{3}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}{\mathbb{Z}_{1}(r)},$
$\displaystyle\mathbb{Z}_{5}(r)$ $\displaystyle=$
$\displaystyle\frac{abB\left(\kappa^{2}+4\right)r^{3}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}\left(\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}-2\right)}{\mathbb{Z}_{1}(r)},$
$\displaystyle\mathbb{Z}_{7}(r)$ $\displaystyle=$
$\displaystyle\frac{2are^{\kappa\sin^{-1}\left(br^{2}+c\right)}+\frac{2ab\kappa
r^{3}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}{\sqrt{1-\left(br^{2}+c\right)^{2}}}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1},\;\mathbb{Z}_{8}(r)=\frac{2ar^{2}\left(\frac{b\kappa
r^{2}}{\sqrt{1-\left(br^{2}+c\right)^{2}}}+1\right)e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1},$
$\displaystyle\mathbb{Z}_{9}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}},\;\mathbb{Z}_{10}(r)=\sqrt{-b^{2}r^{4}-2bcr^{2}-c^{2}+1}.$

Eqs (35-37) can be solved to yield the expressions of pressures and energy
density pertaining to both dark energy and ordinary matter. Assume for the
moment that the dark energy density, $\left(\rho^{D}\right)$, is proportional
to the radial pressure associated with it:

$p_{r}^{D}=-\rho^{D}$

Where the typical baryonic matter density is proportionate to the energy
density associated with dark energy, that is,

$\rho^{D}=\omega\rho$

The earlier research work listed in references Ghezzi (2011); Bhar and Pretel
(2023); Das  _et al._ (2023) served as inspiration for this strategy. In this
case, $\omega$ is a non-zero constant that is taken as free parameter. The
energy density and pressure for typical baryonic matter are now derived by
solving the aforementioned set of equations that are listed below:

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{6\gamma-1}{4\pi(4\gamma-1)r^{2}(\omega+1)}\Big{[}-\frac{\beta
2^{n-2}n\left(1-\mathbb{Z}_{8}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{3}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}$
(38) $\displaystyle+$ $\displaystyle\beta
2^{n-2}r^{2}\left(\mathbb{Z}_{4}(r)\right){}^{n}+\beta
2^{n-2}(1-n)nr\left(-\mathbb{Z}_{9}(r)\right)\left(1-\mathbb{Z}_{9}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-2}+\gamma
r^{2}\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]},$
$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{(6\gamma-1)\omega}{4\pi(4\gamma-1)r^{2}(\omega+1)}\Big{[}-\frac{\beta
2^{n-2}n\left(1-\mathbb{Z}_{8}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{3}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}$
(39) $\displaystyle+$ $\displaystyle\beta
2^{n-2}r^{2}\left(\mathbb{Z}_{4}(r)\right){}^{n}+\beta
2^{n-2}(1-n)nr\left(-\mathbb{Z}_{9}(r)\right)\left(1-\mathbb{Z}_{9}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-2}+\gamma
r^{2}\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]}$
$\displaystyle+$
$\displaystyle\frac{(6\gamma-1)}{4\pi(4\gamma-1)}\Big{[}-\frac{\beta
2^{n-1}n\left(\mathbb{Z}_{4}(r)\right){}^{n-1}\left(\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\mathbb{Z}_{5}(r)-1\right)}{r^{2}\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1\right)}-\beta
2^{n-2}\left(\mathbb{Z}_{4}(r)\right){}^{n}$ $\displaystyle-$
$\displaystyle\gamma\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]},$
$\displaystyle\rho^{D}$ $\displaystyle=$
$\displaystyle\frac{6\gamma-1}{4\pi(4\gamma-1)r^{2}}\Big{[}-\frac{\beta
2^{n-2}n\left(1-\mathbb{Z}_{8}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{3}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\beta
2^{n-2}r^{2}\left(\mathbb{Z}_{4}(r)\right){}^{n}$ (40) $\displaystyle+$
$\displaystyle\beta
2^{n-2}(1-n)nr\left(-\mathbb{Z}_{9}(r)\right)\left(1-\mathbb{Z}_{9}(r)\right)\left(\mathbb{Z}_{4}(r)\right)^{n-2}+\gamma
r^{2}\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right)^{\chi}\right)\Big{]}-\frac{6\gamma-1}{4\pi(4\gamma-1)r^{2}(\omega+1)}$
$\displaystyle\times$ $\displaystyle\Big{[}-\frac{\beta
2^{n-2}n\left(1-\mathbb{Z}_{8}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{3}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\beta
2^{n-2}r^{2}\left(\mathbb{Z}_{4}(r)\right)^{n}$ $\displaystyle+$
$\displaystyle\beta
2^{n-2}(1-n)nrr\left(-\mathbb{Z}_{9}\right)\left(1-\mathbb{Z}_{9}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-2}+\gamma
r^{2}\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]},$
$\displaystyle p_{r}^{D}$ $\displaystyle=$
$\displaystyle-\frac{(6\gamma-1)\omega}{4\pi(4\gamma-1)r^{2}(\omega+1)}\Big{[}-\frac{\beta
2^{n-2}n\left(1-\mathbb{Z}_{8}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{3}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}$
(41) $\displaystyle+$ $\displaystyle\beta
2^{n-2}r^{2}\left(\mathbb{Z}_{4}(r)\right){}^{n}+\beta
2^{n-2}(1-n)nr\left(-\mathbb{Z}_{9}(r)\right)\left(1-\mathbb{Z}_{9}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-2}+\gamma
r^{2}\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]},$
$\displaystyle p_{t}^{D}$ $\displaystyle=$
$\displaystyle\frac{(6\gamma-1)}{4\pi(4\gamma-1)}\Big{[}\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{r^{2}\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1\right)}\Big{[}\frac{1}{4}r\Big{[}\frac{4bB\left(\kappa^{2}+4\right)\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}\right)^{3/2}}{r\left(\mathbb{Z}_{1}(r)\right){}^{2}\sqrt{1-\left(br^{2}+c\right)^{2}}}\Big{[}B\Big{[}b^{2}\kappa
r^{4}$ (42) $\displaystyle-$ $\displaystyle
2br^{2}\mathbb{Z}_{10}(r)-c^{2}\kappa+2c\mathbb{Z}_{10}(r)+\kappa\Big{]}\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}+Ab\left(\kappa^{2}+4\right)r\left(b\kappa
r^{2}+\right)\Big{]}$ $\displaystyle+$
$\displaystyle\left(\mathbb{Z}_{6}(r)+2\right)\Big{[}\frac{2abB\left(\kappa^{2}+4\right)r^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}}{\mathbb{Z}_{1}(r)}-\mathbb{Z}_{7}(r)\Big{]}\Big{]}+\left(\mathbb{Z}_{2}(r)\right)\left(\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}-1\right)\Big{]}$
$\displaystyle-$ $\displaystyle\frac{\beta
2^{n-4}(1-n)n\left(\mathbb{Z}_{4}(r)\right){}^{n-2}\left(-2\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\mathbb{Z}_{6}(r)+2\right)}{r\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1\right)}-\beta
2^{n-2}\left(\mathbb{Z}_{4}(r)\right){}^{n}$ $\displaystyle-$
$\displaystyle\gamma\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]}-\frac{(6\gamma-1)\omega}{4\pi(4\gamma-1)r^{2}(\omega+1)}\Big{[}-\frac{\beta
2^{n-2}n\left(1-\mathbb{Z}_{8}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}$
$\displaystyle+$ $\displaystyle\frac{\beta
2^{n-2}n\left(\mathbb{Z}_{3}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-1}}{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\beta
2^{n-2}r^{2}\left(\mathbb{Z}_{4}(r)\right){}^{n}+\beta
2^{n-2}(1-n)nr\left(-\mathbb{Z}_{9}(r)\right)\left(1-\mathbb{Z}_{9}(r)\right)\left(\mathbb{Z}_{4}(r)\right){}^{n-2}$
$\displaystyle+$ $\displaystyle\gamma
r^{2}\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]}-\frac{(6\gamma-1)}{4\pi(4\gamma-1)}\Big{[}-\frac{\beta
2^{n-1}n\left(\mathbb{Z}_{4}(r)\right){}^{n-1}\left(\sqrt{ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1}+\mathbb{Z}_{5}(r)-1\right)}{r^{2}\left(ar^{2}e^{\kappa\sin^{-1}\left(br^{2}+c\right)}+1\right)}$
$\displaystyle-$ $\displaystyle\beta
2^{n-2}\left(\mathbb{Z}_{4}(r)\right){}^{n}-\gamma\psi\log\left(2^{\chi}\phi\left(\mathbb{Z}_{4}(r)\right){}^{\chi}\right)\Big{]}.$

Figure 1: Evolution of $\rho_{eff},\;p_{reff},\;p_{teff}$ along radial
coordinate $r$ for range of $\kappa=1-2$ for compact star candidate
$Her\;X-1$. Other fixed constant parameters are mentioned in Table 1.

Figure 2: central densities $\rho_{effc}$, central pressures
$p_{reffc}\;\&\;p_{teffc}$ along radial coordinate $r$ for compact stars
candidates.
$Her\;X-1(\text{black}),\;LMC\;X-4(\text{purple}),\;Cen\;X-3(\text{Red}),\;PSR\;J1614-2230(\text{blue}),\;PSR\;J0740+6620(\text{green})\;\&\;GW190814(\text{black})magenta$.

Figure 3: Evolution of density $\rho$ and pressure $p$ along radial coordinate
$r$ for range of $\kappa=1-2$ for compact star candidate $Her\;X-1$. Other
fixed constant parameters are mentioned in Table 1.

Figure 4: evolution of dark energy density $\rho^{D}$, dark energy radial
pressure $p_{r}^{D}$, and dark energy tangential pressure $p_{t}^{D}$ along
radial coordinate $r$ for
$\kappa=1.0(\text{black}),\;\kappa=1.2(\text{purple}),\;\kappa=1.4(\text{red}),\;\kappa=1.6(\text{blue}),\;\kappa=1.0(\text{green})\;\&\;\kappa=1.0(\text{magenta})$
for compact star candidate $Her\;X-1$. Other fixed constant parameters are
mentioned in Table 1.

Figure 5: evolution of anisotropy left graph along radial coordinate $r$ for
$\kappa=1.0(\text{black}),\;\kappa=1.2(\text{purple}),\;\kappa=1.4(\text{red}),\;\kappa=1.6(\text{blue}),\;\kappa=1.0(\text{green})\;\&\;\kappa=1.0(\text{magenta})$
for compact star candidate $Her\;X-1$. Gradients middle graph and adiabatic
index right graph along radial coordinate $r$ for range of $\kappa=1-2$. Other
fixed constant parameters are mentioned in Table 1.

Figure 6: Evolution of energy conditions along radial coordinate $r$ for range
of $\kappa=1-2$ for compact star candidate $Her\;X-1$. Other fixed constant
parameters are mentioned in Table 1.

Figure 7: Equation of state components for normal matter upper two right
graphs, and dark matter lower two graphs along radial coordinate $r$ for range
of $\kappa=1-2$ for compact star candidate $Her\;X-1$. Evolution of TOV forces
along radial coordinate $r$ for
$\kappa=1.0(\text{black}),\;\kappa=1.2(\text{purple}),\;\kappa=1.4(\text{red}),\;\kappa=1.6(\text{blue}),\;\kappa=1.0(\text{green})\;\&\;\kappa=1.0(\text{magenta})$
for compact star candidate $Her\;X-1$. Other fixed constant parameters are
mentioned in Table 1.

Figure 8: Evolution of sound speeds and Abreu condition along radial
coordinate $r$ for
$\kappa=1.0(\text{black}),\;\kappa=1.2(\text{purple}),\;\kappa=1.4(\text{red}),\;\kappa=1.6(\text{blue}),\;\kappa=1.0(\text{green})\;\&\;\kappa=1.0(\text{magenta})$
for compact star candidate $Her\;X-1$. Other fixed constant parameters are
mentioned in Table 1.

Figure 9: Evolution of redshift, compactness, and mass function along radial
coordinate $r$ for range of $\kappa=1-2$ for compact star candidate
$Her\;X-1$. Other fixed constant parameters are mentioned in Table 1.

##  III Matching Conditions

An essential step in the study of compact celestial objects is the evaluation
of constant parameters. In the present article, these unknowns are assessed by
the uniform matching of inner space time Eq. (1), and outer vacuum space time
as provided by Schwarzschild solution Schwarzschild (1916):

$\displaystyle\mathrm{d}s^{2}=-\left(1-\frac{2M}{R}\right)\mathrm{d}t^{2}+\left(1-\frac{2M}{R}\right)^{-1}\mathrm{~{}d}r^{2}-r^{2}\left(\mathrm{~{}d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right).$
(43)

The following renowned Israel-Darmois junction criteria Israel (1966); Darmois
(1927) must be met for uniform and consistent matching:

$\displaystyle\left(\mathrm{d}s^{2}\right)^{+}=\left(\mathrm{d}s^{2}\right)^{-},\left(\kappa_{ij}\right)^{+}=\left(\kappa_{ij}\right)^{-},$
(44)
$\displaystyle\left.e^{2\lambda^{+}}\right|_{r=R}=\left.e^{2\lambda^{+}}\right|_{r=R},\left.e^{2v^{+}}\right|_{r=R}=\left.e^{2v^{+}}\right|_{r=R},$
(45) $\displaystyle\left.\frac{\partial e^{2v^{+}}}{\partial
r}\right|_{r=R}=\left.\frac{\partial e^{2v^{+}}}{\partial
r}\right|_{r=R},\left.\&p_{r}\right|_{r=R}=0.$ (46)

Thus, if we consider the aforementioned circumstances, we obtain the
subsequent set of equations:

$\displaystyle\Big{[}A+\frac{B\left(n\sqrt{1-\left(bR^{2}+c\right)^{2}}+2bR^{2}+2c\right)\sqrt{aR^{2}e^{n\sin^{-1}\left(bR^{2}+c\right)}}}{b\left(n^{2}+4\right)R}\Big{]}^{2}$
$\displaystyle=$ $\displaystyle 1-\frac{2M}{R},$ (47) $\displaystyle
aR^{2}e^{n\sin^{-1}\left(bR^{2}+c\right)}+1$ $\displaystyle=$
$\displaystyle\frac{1}{1-\frac{2M}{R}},$ (48) $\displaystyle
2aBRe^{n\sin^{-1}\left(bR^{2}+c\right)}\Big{[}\frac{AR}{\sqrt{aR^{2}e^{n\sin^{-1}\left(bR^{2}+c\right)}}}+\frac{B\left(2bR^{2}+2c+n\mathbb{Z}(R)_{10}\right)}{b\left(n^{2}+4\right)}\Big{]}$
$\displaystyle=$ $\displaystyle\frac{2M}{R^{2}}.$ (49)

The subsequent formulations for the constants $a,\;A,\;B$ are obtained from
the solution of above equations:

$\displaystyle a$ $\displaystyle=$
$\displaystyle-\frac{2Me^{-n\sin^{-1}\left(bR^{2}+c\right)}}{R^{2}(2M-R)},$
(50) $\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{b\left(n^{2}+4\right)R}\Big{[}bR\Big{[}-2BR\sqrt{aR^{2}e^{n\sin^{-1}\left(bR^{2}+c\right)}}+n^{2}\sqrt{1-\frac{2M}{R}}+4\sqrt{1-\frac{2M}{R}}\Big{]}$
(51) $\displaystyle-$ $\displaystyle
B\left(n\sqrt{-\left(bR^{2}+c-1\right)\left(bR^{2}+c+1\right)}+2c\right)\sqrt{aR^{2}e^{n\sin^{-1}\left(bR^{2}+c\right)}}\Big{]},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{M}{R^{2}\sqrt{2-\frac{4M}{R}}\sqrt{-\frac{M}{2M-R}}}.$
(52)

The constant parameters’ specific values are provided in Table-1.

Table 1: Values of constants compact star candidate $Her\;X-1\;(M=0.85\;\&\;R=8.5)$, by fixing $n=1,\;\beta=5,\;\gamma=1,\;\phi=-2,\;\psi=-2.0\times 10^{-8},\;b=0.001,\;c=0.001,\;\chi=5\;\&\;\omega=0.13$. $\kappa$ | a | A | B |  $\frac{p_{reffc}}{\rho_{effc}}$ ($r=0$) | $\rho_{effc}$ | $p_{reffc}$ | $p_{teffc}$ | $\rho_{c}$ | $p_{c}$ | $\rho^{D}_{c}$ | $p_{rc}^{D}$ | $p_{tc}^{D}$
---|---|---|---|---|---|---|---|---|---|---|---|---
1.0 | 0.00321 | 0.540428 | 0.0263 | $<1$ | 0.00320 | 0.000111 | 0.000111 | 0.00283 | 0.000479 | 0.000368 | -0.000368 | -0.000368
1.2 | 0.003168 | 0.439978 | 0.0263 | $<1$ | 0.00315 | 0.000228 | 0.000228 | 0.00279 | 0.000591 | 0.000363 | -0.000363 | -0.000363
1.4 | 0.003122 | 0.349802 | 0.0263 | $<1$ | 0.00311 | 0.000364 | 0.000364 | 0.00275 | 0.000722 | 0.000357 | -0.000357 | -0.000357
1.6 | 0.003077 | 0.270756 | 0.0263 | $<1$ | 0.00306 | 0.000521 | 0.000521 | 0.00271 | 0.000874 | 0.000352 | -0.000352 | -0.000352
1.8 | 0.003032 | 0.202694 | 0.0263 | $<1$ | 0.00302 | 0.000698 | 0.000698 | 0.00267 | 0.001046 | 0.000347 | -0.000347 | -0.000347
2.0 | 0.002988 | 0.144856 | 0.02637 | $<1$ | 0.00297 | 0.000896 | 0.000896 | 0.00263 | 0.001239 | 0.000342 | -0.000342 | -0.000342
Table 2: Comparison of densities and pressures of different compact stars candidate, by fixing $\kappa=1.5$ $n=1,\;\beta=5,\;\gamma=1,\;\phi=-2,\;\psi=-2.0\times 10^{-8},\;b=0.001,\;c=0.001,\;\chi=5\;\&\;\omega=0.13$. Compact star | $M_{O}$ | $R\;(km)$ | $\frac{p_{reffc}}{\rho_{effc}}$ | $\rho_{effc}$ | $p_{reffc}$ | $p_{teffc}$ | $\rho_{c}$ | $p_{c}$ | $\rho^{D}_{c}$ | $p_{rc}^{D}$ | $p_{tc}^{D}$
---|---|---|---|---|---|---|---|---|---|---|---
$Her\;X-1$ | 0.85 | 8.5 | $<1$ | $3.0884\times 10^{-3}$ | $4.4052\times 10^{-4}$ | $4.4052\times 10^{-4}$ | 0.0027 | 0.0007 | 0.00035 | -0.00035 | -0.00035
$LMC\;X-4$ | 1.29 | — | $<1$ | $5.3836\times 10^{-3}$ | $9.3519\times 10^{-4}$ | $9.3519\times 10^{-4}$ | 0.0047 | 0.0015 | 0.00061 | -0.00061 | -0.00061
$Cen\;X-3$ | 1.49 | — | $<1$ | $6.6688\times 10^{-3}$ | $1.2865\times 10^{-3}$ | $1.2865\times 10^{-3}$ | 0.0043 | 0.0014 | 0.00057 | -0.00057 | -0.00057
$PSR\;J1614-2230$ | 1.97 | — | $<1$ | $1.0673\times 10^{-2}$ | $2.8107\times 10^{-3}$ | $2.8107\times 10^{-3}$ | 0.0040 | 0.0015 | 0.00052 | -0.00052 | -0.00052
$PSR\;J0740+6620$ | 2.14 | — | $<1$ | $1.2528\times 10^{-2}$ | $3.7932\times 10^{-3}$ | $3.7932\times 10^{-3}$ | 0.0040 | 0.0015 | 0.00052 | -0.00052 | -0.00052
$GW190814$ | 2.50 | — | $<1$ | $1.7646\times 10^{-2}$ | $7.8450\times 10^{-3}$ | $7.8450\times 10^{-3}$ | 0.0036 | 0.0015 | 0.00047 | -0.00047 | -0.00047

##  IV Discussion and evaluation of calculated results

We investigate the DE stellar models in this section of our case study by
selecting the appropriate functions $f(T)$ and $h(T)$, respectively, from the
literature Ditta  _et al._ (2023a); Bamba  _et al._ (2012). We select the
compact star model $Her\;X-1$ for the whole graphical analysis. We then
compare the pressures and densities of DE compact stellar candidates, as shown
in table-2 and 2, including
$Her\;X-1,\;LMC\;X-4,\;Cen\;X-3,\;PSR\;J1614-2230,\;PSR\;J0740+6620\;\&\;GW190814$.
The following is a synopsis of our acquired results:

###  IV.1 Energy densities and pressure components of dark energy stars models

Physical validity is crucial for studying compact stars because any study that
isn’t physically admissible wouldn’t be worth doing. One tool used to
guarantee the physical affirmation of the study is the density parameters and
the pressure components. The behaviour for density and pressure patterns in
compact designs with DE should be as follows:

  * •

In normal matter distribution density and pressure profiles should be
positive: $\rho_{eff}>0,\;p_{reff}\geq 0,\;p_{teff}>0,\;\rho>0\;\&\;p>0$.

  * •

in Dark matter distribution the density should be positive and pressures must
be negative: $\rho^{D}>0,\;p_{r}^{D}<0\;\&\;p_{t}^{D}<0$.

  * •

Further, at the center, the above profiles should be Maximum in numeric values
with the positive and negative signs whatever is applicable, and at the
boundary, they should tend toward the zero, mean should be minimum in numeric
values retaining their signs.

Considering the propagation of the effective pressures
$p_{reff}\;\&\;p_{teff}$ and the effective energy parameter $\rho_{eff}$, Fig.
(1) contains useful details. As per the physical standards, the energy density
and pressure components exhibit highest values near the centre (as also shown
in Tables-1-2) and uniform, positive declines throughout the star the star
from center to boundary ($0<r\leq R$). This validates the celestial body’s
physical legitimacy.

Fig. (2) compares the central effective densities and effective pressure of DE
compact stars candidates
$Her\;X-1,\;LMC\;X-4,\;Cen\;X-3,\;PSR\;J1614-2230,\;PSR\;J0740+6620\;\&\;GW190814$.
This confirms that the star candidates with more masses have higher central
effective densities and effective radial and tangential pressure. Detailed
information regarding central densities and pressure for dark matter and
normal distributions is provided in table- 2.

Moreove, energy $\rho$, and pressure $p$ are shown in Fig. (3), which also
authenticate the physical validity of normal distribution as mentioned earlier
in acceptability criteria.

Moreover, Fig. (4), indicates the dark matter distribution criteria is fully
justified in our study. One can easily extract information from these graphs
as, $\rho^{D}>0,\;p_{r}^{D}<0,\;\&\;p_{t}^{D}<0$. This fact is also mentioned
in Tables-1-2, in the case of central values.

###  IV.2 Anisotropy, gradients, and adiabatic index

Anisotropy, the repulsive force provided by $\Delta=p_{teff}-p_{reff}$
neutralizes the consequences of gradient components, significantly improving
the equilibrium balance and stability of stellar configurations. The
observable positive anisotropic conduct attests to the long-lasting advantages
of the said repulsive forces. The requirement that $\Delta>0$ within the
boundary of compact body for $p_{t}>p_{r}$, where, determines the anisotropy.
Nevertheless, $\Delta\rightarrow 0$ as $r\rightarrow 0$, where
$p_{reff}=p_{teff}$. The anisotropy $\Delta$ illustrated in Fig. (5)’s first
graph complies with the necessary behavior.

Gradients usually have a $0$ value at the center and a negative, declining
trend. The expression
$(\frac{d\rho_{eff}}{dr}=\frac{dp_{reff}}{dr}=\frac{dp_{teff}}{dr}|{r\rightarrow
0}=0)$, with the exception of the following in their graphical portrayal:
$(\frac{d\rho_{eff}}{dr},\;\frac{dp_{reff}}{dr},\;\frac{dp_{teff}}{dr})|{0<r\leq
R}<0$. The anticipated gradients fall within this range, as shown by the
results in the second graph of Fig. (5).

Chandrasekhar investigated stability under theÂ adiabatic index in
Chandrasekhar (1964). The adiabatic index stability constraint was established
by Heintzmann and Hillebrandt Heintzmann and Hillebrandt (1975) by resolving
the inequality $\Gamma\Big{|}_{0\leq r\leq R}=\frac{4}{3}$. The adiabatic
index can be expressed mathematically as follows:

$\displaystyle\Gamma=\frac{p_{reff}+\rho_{eff}}{p_{reff}}v^{2}_{r}.$ (53)

The third graph of Fig. (5) shows the graphical behavior of the adiabatic
index. It is important to note that in a stable polytropic stellar body, the
adiabatic index should be larger than $\frac{4}{3}$ by a factor that relies on
the ratio $\frac{\rho}{p_{r}}$ at the centre of the compact celestial object
Glass and Harpaz (1983). According to Harrison (1965), the equation of state
(EoS) associated with neutron star matter has a ratio $\frac{\rho}{p_{r}}$
that ranges from two to four. In this regard, the graphical analysis shows
that the adiabatic index is always bigger than $\frac{4}{3}$ throughout the
heavenly distribution; as a result, the celestial model represents a stable
configuration concerning the radial adiabatic infinitesimal perturbations.

###  IV.3 Energy conditions

The EMT in GR describes mass, momentum, and stress and depicts the arrangement
of gravitation-free fields (GFF) and matter fields in spacetime. Nevertheless,
neither the admissible GFF in the spacetime manifold nor the state of matter
are directly related to the Einstein field equations (EFEs). Energy
requirements, which validate all sorts of fluid, oppose GFF in GR, and
guarantee a feasible and physically admissible allocation of matter, are used
to establish physically legitimate solutions of the field equations. To attain
this distribution, the anisotropic behaviour of energy should stay positive
and adhere to certain constraining limits across the star body. Eqs (54-57)
express these limitations, which are also famed as the Strong Energy Condition
(SEC), the Weak Energy Condition (WEC), Null Energy Condition (NEC), and the
Dominant Energy Condition (DEC):

$\displaystyle\mathrm{SE}C$ $\displaystyle:$
$\displaystyle\rho_{eff}+p_{\gamma\\{eff\\}\\}}\geq
0,\rho_{eff}+p_{reff}+2p_{teff}\geq 0,$ (54) $\displaystyle\mathrm{WEC}$
$\displaystyle:$ $\displaystyle\rho_{eff}\geq
0,\rho_{eff}+p_{\gamma\\{eff\\}}\geq 0,$ (55) $\displaystyle\mathrm{NEC}$
$\displaystyle:$ $\displaystyle\rho_{eff}+p_{\gamma\\{eff\\}}\geq 0,$ (56)
$\displaystyle\mathrm{DEC}$ $\displaystyle:$
$\displaystyle\rho_{eff}>|p_{\gamma\\{eff\\}}|.$ (57)

Here, $r$ & $t$ stand for the radial cordinate if $\gamma=r$) and tangential
coordinates if $\gamma=t$. Our results, which are shown in Fig. (6), are
consistent with the standard parameters used in studies of compact stars.
While, in case of dark energy matter these conditions violate as shown in last
graph of Fig. (6).

###  IV.4 Equation of state profiles and forces behaviour

Whether a compact star system is made of dark matter or normal matter, its
makeup matters a great deal. The value ranges of $w_{r}$ and $w_{t}$ for
normal or byronic matter EoS must lie between $0\leq w_{r}<1$ and $0<w_{t}<1$
in order to guarantee that the system is composed of normal matter. The
phrases for EoS are provided by:

$w_{r}=\frac{p_{reff}}{\rho}\;\;\;\;\&\;\;\;\;w_{teff}=\frac{p_{teff}}{\rho},$
(58)

while in the case of dark energy stars, these equations are:

$w_{r}^{D}=\frac{p_{r}^{D}}{\rho}=\;\;\;\;\&\;\;\;\;w_{t}^{D}=\frac{p_{t}^{D}}{\rho}.$
(59)

The first two graphs in the upper panel of Fig. (7) demonstrate that these EoS
parameters meet the necessary constraining conditions, guaranteeing that
matter is dispersed uniformly realistic across the system. For the dark matter
state, EOS range should be $w_{r}^{D}\;\&\;<w_{t}^{D}<-1/3$, This criteria is
also justified in the left panel graphs of Fig. (7).

The TOV formula Tolman (1939); Oppenheimer and Volkoff (1939) approximates the
stability constraints of the stellar phenomenon. The generalised format for
the TOV equation in MTRG is as follows:

$\displaystyle\underbrace{\frac{dp_{reff}}{dr}}_{F_{g}}+\underbrace{\frac{\nu^{{}^{\prime}}(\rho_{eff}+p_{reff})}{2}}_{F_{h}}-\underbrace{\frac{2(p_{teff}-p_{reff})}{r}}_{F_{a}}-\underbrace{\frac{\gamma}{4\gamma-1}\left(\rho^{{}^{\prime}}_{eff}-p_{reff}^{{}^{\prime}}-p_{teff}^{{}^{\prime}}\right)}_{F_{r}}=0.$
(60)

Eq. (60) presents the TOV equation, which states that a stellar system comes
to equilibrium if the forces, like $F_{a}$, $F_{g}$, $F_{h}$, and $F_{r}$ are
balanced to have a net effect of zero. Maintaining this equilibrium state is
essential to prevent a unique point from forming when the system undergoes
gravitational collapse. Forces according to criteria, in this research section
are suitably balanced, as shown by the third graph of Fig. (7), which ensures
the stability of our evaluated results and prevents against any collapse.

###  IV.5 Sound speeds

Two stability parameters, $v_{r}^{2}$, the radial direction speed, and
$v_{t}^{2}$, the tangential direction speed, will be examined in this article
to assess the stability of the stellar configurations. We also need to take
into account the notion of anisotropic fluid distribution, sometimes referred
to as the Herrera Cracking idea. According to the Herrera Cracking idea, to
preserve stability, the sound speeds need to meet the following conditions:
$0<v_{r}^{2},v_{t}^{2}<1$; both speeds need to be smaller than the speed of
light, and the speed of light is $c=1$. The following is the formula for sound
speeds:

$v_{r}^{2}=\frac{dp_{reff}}{d\rho_{eff}}\;\;\&\;\;v_{t}^{2}=\frac{dp_{teff}}{d\rho_{eff}}.$
(61)

An additional stability criterion was presented by Abreu et al. Abreu  _et
al._ (2007), which states that if $v_{r}^{2}>v_{t}^{2}$ and
$v_{r}^{2}-v_{t}^{2}$ remain unaltered, the region is deemed possibly stable.
The condition was further generalised to $0<|v_{t}^{2}-v_{r}^{2}|<1$ by
Andreasson Andréasson (2008), meaning the negation of cracking depicting the
stable zone. Our results are compatible with the Abreu and Andreasson
stability criteria, suggesting, our results for compact star investigation are
stable. These results are displayed in Fig. (8). It is noteworthy that the
sound speeds $v_{r}^{2}>1\;\&\;v_{t}^{2}>1$ demonstrate the system’s
instability for $\kappa=2.0$. Our DE-based star system is stable for all other
selected values of $\kappa$.

###  IV.6 Behaviors of Mass, compactness, and redshift parameters.

One important indicator of a star’s compactness is the $\frac{m(r)}{r}$
compactness ratio. The formula that follows can be used for determining the
mass:

$m(r)=4\pi\times\int\left(r^{2}\times\rho\right)dr,$ (62)

an alternative formula is given as:

$m(r)=\frac{r}{2}\left(1-e^{-\lambda(r)}\right).$ (63)

We may obtain formulas for the redshift function $z_{s}$ and the compactness
parameter $u(r)$ through the inclusion of the effects of Eq. (63):

$\displaystyle u$ $\displaystyle=\frac{m(r)}{r},$ (64) $\displaystyle z_{s}$
$\displaystyle=e^{-\frac{\nu(r)}{2}}-1.$ (65)

Buchdahl Buchdahl (1959) determined the highest value of
$u=\frac{m(R)}{R}<\frac{4}{9}$ for the compactness criterion. In Abreu  _et
al._ (2007), this criterion was extended to anisotropic matter configurations.
Additionally, Buchdahl established a maximum value criteria for $z_{s}\leq
4.77$, the redshift parameter Bowers and Liang (1974). The third graph of
Figs. (6) displays the smooth and regular mass function that our investigation
produced. The compactness and redshift parameters obtained from our research,
meet the physical admissibility requirements for the star system, as shown by
the first and second graphs in Fig. (6).

##  V conclusion

The foundation of this research on compact bodies system is a simple
modification of $f(T)$, namely the MRTG theory, that is distinct from both the
Rastall theory and $f(T)$ gravity since it incorporates Rastall’s term and
uses a torsion-based function. We include the vacuum case of the Schwarzschild
solution to serve as an outer solution and the Karmarkar ansatz based upon the
spherical symmetric spacetime as an inside solution to achieve results which
are physically admissible. We note that the results are strongly affected by
the Rastall parameter, as one can retrieve the $f(T)$ theory of gravity be
setting the Rastall parameter equal to zero. We investigate MTRG for dark
energy stellar structures by selecting the particular modifications of gravity
based on the functions $f(T)$ and $h(T)$. To examine the anisotropic conduct
of the DE field in detail, we thoroughly analyzed the results for compact
stellar candidate $Her\;X-1$ for different values of metric constant $\kappa$
(see Figs. 1-9). In particular, we also determine the central densities and
pressures for candidates
$Her\;X-1,\;LMC\;X-4,\;Cen\;X-3,\;PSR\;J1614-2230,\;PSR\;J0740+6620\;\&\;GW190814$
(see table-2 and Fig. (2)). Some Works on compact stellar modeling in Rastall-
based modifications of torsional theory have previously been done in
literature Ditta  _et al._ (2023d, a); Ditta and Xia (2022); Ashraf  _et al._
(2023). But this work is different from the previous literature, as we first
time studied dark energy stellar models in different forms of MTRG functions
$f(T)\;\&\;h(T)$. The following is a summary of our study’s primary findings:

We accurately analyzed all parameters for dark energy compact models to show
their anisotropic behavior. Dark matter parameter $\rho^{D}$ has positive
conduct, while both pressure components $p_{r}^{D}$ and $p_{t}^{D}$ are
negative for dark matter. Our results for $\rho_{eff}$, $p_{reff}$,
$p_{teff}$, $\rho$, and $p$ contain the positive conduct depicting the real
nature of matter configurations. Furthermore, $\Delta$, the anisotropy
parameter, exhibits smooth behavior with negative gradients from the center to
the boundary. Energy conditions exhibit positive behavior throughout the
stellar configurations, whereas, they are negative in the case of dark energy
density and pressures; negative EoS parameters also validate the behaviour of
the DE; the speed of sound indicates stability for the chosen range of
$\kappa$, except $\kappa=2.0$; and the causality limits satisfy the necessary
conditions. The system is stable because of TOV forces, and the mass function,
compactification, and redshift functions behave as they should according to
the acceptable criteria.

This research significantly unravels the interplay between modified Rastall
teleparallel gravity and dark energy within compact stars. By investigating
the celestial mystery of dark energy in the modified gravitational scenario,
our study enhances the understanding of the cosmos and motivates future
studies in the modified theories of gravity.

## Acknowledgement

Allah Ditta and Xia Tiecheng acknowledge this paper to be funded by the
National Natural Science Foundation of China 11975145. Asif Mahmood would like
to acknowledge the Researcher’s Supporting Project Number (RSP2024R43), King
Saud University, Riyadh, Saudi Arabia,

## Conflict Of Interest statement

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data Availability Statement

This manuscript has no associated data, or the data will not be deposited.
(There is no observational data related to this article. The necessary
calculations and graphic discussion can be made available on request.)

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# Temperature-driven reorganization of electronic order in CsV3Sb5

Q. Stahl  Institute of Solid State and Materials Physics, TU Dresden, 01069
Dresden, Germany  D. Chen  Max Planck Institute for Chemical Physics of
Solids, Dresden, Germany  T. Ritschel  Institute of Solid State and Materials
Physics, TU Dresden, 01069 Dresden, Germany  C. Shekhar  Max Planck Institute
for Chemical Physics of Solids, Dresden, Germany  C. Felser  Max Planck
Institute for Chemical Physics of Solids, Dresden, Germany  Würzburg-Dresden
Cluster of Excellence ct.qmat, Technische Universität Dresden, 01062 Dresden,
Germany  J. Geck<EMAIL_ADDRESS>Institute of Solid State and
Materials Physics, TU Dresden, 01069 Dresden, Germany  Würzburg-Dresden
Cluster of Excellence ct.qmat, Technische Universität Dresden, 01062 Dresden,
Germany

###### Abstract

We report a temperature dependent x-ray diffraction study of the electronic
ordering instabilities in the kagome material CsV3Sb5. Our measurements
between 10 K and 120 K reveal an unexpected reorganization of the three-
dimensional electronic order in the bulk of CsV3Sb5: At 10 K, a 2x2x2
superstructure modulation due to electronic order is observed, which upon
warming changes to a 2x2x4 superstructure at 60 K. The electronic order-order
transition discovered here involves a change in the stacking of electronically
ordered V3Sb5-layers and agrees perfectly with anomalies previously observed
in magneto-transport measurements. This implies that the temperature dependent
three-dimensional electronic order plays a decisive role for transport
properties, which are related to the Berry-curvature of the V-bands. Our data
also show that the bulk electronic order in CsV3Sb5 breaks the 6-fold
rotational symmetry of the underlying $P6/mmm$ lattice structure.

The physics of quantum materials containing layers of corner-sharing triangles
is full of surprises. Not only can coupled magnetic moments on such a kagome
lattice exhibit unconventional magnetic ground states Sachdev (1992). As it is
now becoming more and more evident, electrons on a kagome lattice also provide
a truly fascinating platform for various other collective quantum phenomena,
such as nontrivial topological matter Guo and Franz (2009); Bilitewski and
Moessner (2018), chiral superconductivity Yu and Li (2012); Kiesel and Thomale
(2012), bond and charge ordering Wang  _et al._ (2013); Kiesel  _et al._
(2013) or complex intertwined order Yu  _et al._ (2021); Zhao  _et al._
(2021).

In this regard the kagome materials $A$V3Sb5 ($A=$K, Rb, Cs) have become
particularly famous, generating a real surge of research into their electronic
properties Ortiz  _et al._ (2020); Denner  _et al._ (2021); Ortiz  _et al._
(2021); Jiang  _et al._ (2021); Wang  _et al._ (2021); Zhao  _et al._ (2021);
Chen  _et al._ (2021a); Liang  _et al._ (2021). One reason for the large
interest certainly is the appearance of superconductivity in a topological
$Z_{2}$ kagome metal Ortiz  _et al._ (2020). Superconductivity in layered
kagome materials is very rare to start with. But of course the coexistence of
superconductivity and non-trivial electronic topology may support elusive and
highly sought after quasiparticles like Majorana bound states Liang  _et al._
(2021). In addition to this, intriguing electronic ordering instabilities have
been observed in $A$V3Sb5. Very recent scanning tunneling microscopy (STM)
studies revealed the presence of electronic orders on the surfaces of KV3Sb5
and CsV3Sb5 Jiang  _et al._ (2021); Wang  _et al._ (2021); Zhao  _et al._
(2021).

Figure 1: (a), (b): Illustration of the three-dimensional room temperature
$P6/mmm$ structure of CsV3Sb5 (left) and V3Sb5-layer with its a kagome
sublattice of V-sites (right). (c): Representative distribution of the XRD-
intensity in the ($hk\bar{1}$) - plane measured at room temperature. The color
scale highlights the low intensities up to 250 cps. The highest intensity at
(2,2,-1) is of the order of $10^{4}$ cps.

Focusing on the case of CsV3Sb5, electronic order sets in below 95 K Ortiz
_et al._ (2020); Wang  _et al._ (2021), resulting a 2x2 superstructure, which
in fact shows signatures of chirality Wang  _et al._ (2021). The latter are
indeed in perfect agreement with previous theoretical studies of the kagome
Hubbard-model where such an instability has been predicted Kiesel and Thomale
(2012); Kiesel  _et al._ (2013). Importantly, the electronic order in CsV3Sb5
is three-dimensional (3D), i.e. there are also correlations between the kagome
layers along the c-axis (c.f. Fig. 1) Liang  _et al._ (2021); Ortiz  _et al._
(2021); Ratcliff  _et al._ (2021). These correlations along $c$ remain,
however, controversial: while some studies find clear indications for a 2x2x2
order Liang  _et al._ (2021); Ratcliff  _et al._ (2021), a very recent x-ray
diffraction (XRD) study revealed a disordered 2x2x4 modulation Ortiz  _et al._
(2021). Also, it remains an open question as to whether these correlations
along $c$ play any significant role at all for the electronic properties of
CsV3Sb5.

Figure 2: Temperature dependent XRD-data for CsV3Sb5, revealing the appearance
of two distinct 3D superlattice modulations with different out-of-plane
correlations along $c$. Data were taken with increasing temperature. (a)
Temperature dependent integrated intensity of the (0.5,2.5,-0.5) and
(0.5,2.5,-0.25) superstructure reflections, which signal the presence of a
2x2x2 and a 2x2x4 superlattice, respectively. The red and blue lines serve as
guides to the eye. Inset: Reconstructed reciprocal space map parallel to the
($hk\bar{\frac{1}{2}}$) - plane for the 2x2x2 phase at 10 K. The chirality of
the XRD-intensity is similar to that observed by STM Wang  _et al._ (2021).
$(h3l)$ \- planes for the undistorted 1x1x1 host phase (b) at 300 K, the 2x2x4
phase (c) at 75 K and the 2x2x2 phase (d) at 10 K. The indexation always
corresponds to the undistorted 1x1x1 host phase ($P6/mmm$).

Here we study the 3D electronic order in the bulk of CsV3Sb5 single crystals
using XRD. We show that, besides the electronic order-disorder transition at
about 95 K, there is an additional order-order transition at 60 K. This
transition agrees perfectly with previously reported anomalies in the magneto-
transport Chen  _et al._ (2021a); Xiang  _et al._ (2021), implying that these
changes in the electronic order play a decisive role for the topological band
structure of CsV3Sb5.

We investigated the bulk electronic order in CsV3Sb5, using a high-performance
laboratory facility for XRD optimized for resolution and sensitivity. This
custom-made instrument is equipped with a monochromatized Mo Kβ radiation
source (beam diameter $70\mu\mathrm{m}$ on the sample) and a 300K CdTe area
detector with no read-out noise for high detection efficiency and minimum
background. The outstanding performance of the instrument is particularly well
demonstrated in Fig. 3 (d) where peak profiles with a maximum intensity of 2
count/sec are still fully resolved. The cooling of the sample down to 10 K was
accomplished by a low-vibration pulse-tube cryostat, which was itself mounted
on a specialized four-circle diffractometer.

The single crystals for the present XRD study were synthesized using the self-
flux method and well characterized prior to the XRD-experiments. The magneto-
thermal transport measurements reveal the high-quality of our CsV3Sb5 single
crystals and show a large anomalous Nernst effect. Both the Nernst and Seebeck
signals exhibit quantum oscillations, which start around 2 T Chen  _et al._
(2021b). Fig. 1 (c) shows a representative diffraction pattern of the CsV3Sb5
single crystal inside the cryostat at room temperature. The very sharp and
resolution-limited diffraction spots (typical FWHM=0.04 r.l.u.) show the
extremely high quality of the studied single crystals. The diffraction pattern
could be indexed by a hexagonal unit cell with lattice parameter a = b =
5.5159 Å and c = 9.2767 Å. In total, 145 Bragg reflections could be reached
through the windows of the cryostat, 47 of them being unique. Refinement of
the measured intensities not  yielded the expected $P6/mmm$ structure and
$z_{Sb2}$=0.742 as single atomic coordinate not fixed by a special position in
perfect agreement with previously published data Ortiz  _et al._ (2019).

Starting from room temperature, the sample was cooled down very carefully to
10 K at a rate of about 2 K per minute. After cooling down, a large number of
additional superlattice reflections was observed at 10 K. The inset of Fig. 2
(a), Fig. 2 (d) and Fig. 3 (a)–(c) show representative reciprocal space maps
and scans with typical superlattice peaks.

Figure 3:  Breaking of the 6-fold rotational symmetry due to electronic order
in the 2x2x2 and 2x2x4 phase. (a), (d): $l$-cuts $(-0.5,3.0,l)$ and
$(2.5,0.5,l)$, which are related by 6-fold symmetry, differ in both phases
(integration thickness $\delta h$ = $\delta k$ = 0.06 r.l.u.). Also the
intensity distributions in reciprocal lattice planes related by 6-fold
symmetry are distinct as demonstrated in (b), (c) (2x2x2 phase) and (e), (f)
(2x2x4). Inset: Schematic, indicating the $hk$-positions of the measurements
in (a),(d). The indexation corresponds to the undistorted 1x1x1 host phase
($P6/mmm$). The integration time for the measurements in (a) and (d) was 2160
s and 2880 s, respectively. Integration time for (b),(c) was 270 s and for
(e),(f) 360 s.

The superlattice reflections at 10 K correspond to a modulation with wave
vector (0.5,0.5,0.5), which means that the lattice period is doubled along
$a$, $b$ and $c$. We will refer to this as 2x2x2 superlattice in the
following. The superlattice reflections are extremely narrow and resolution-
limited (FWHM $\simeq 0.04$ r.l.u.) in all three directions of reciprocal
space, meaning that the structural modulation probed by XRD is long-ranged in
all three directions of direct space, i.e. we are dealing with a fully
developed 3D-order.

We note here that the electronic order at the surface is unstable and
transforms into a stripe-like $4\times 1$ surface modulation at low
temperatures Zhao  _et al._ (2021). The earlier attribution of this
instability to surface effects Wang  _et al._ (2021) is indeed supported by
the absence of corresponding superlattice peaks in our bulk sensitive XRD.

The main result of the present study concerns the temperature evolution of the
superlattice, which is displayed in Fig. 2 (a): Upon warming, the average
intensity at the $(0.5,0.5,0.5)$-positions decreases continuously up to about
55 K and then drops abruptly at about 60 K. Just above 60 K we observe
additional peaks at positions revealing a quadrupling along the $c$-direction
of the $P6/mmm$ unit cell. This reveals the formation of a modulation with
wave vector (0.5,0.5,0.25), i.e. a 2x2x4 superlattice. Note that only the
period of the modulation along $c$ is changed, while the periodicity of the
superlattice along $a$ and $b$ remains unaltered. Also the (0.5,0.5,0.25)
superlattice peaks are resolution limited, which can be observed in Figs 2 (b)
and 3 (d),(e) and (f). The 2x2x4 modulation therefore as well is long-ranged
ordered in all three directions of space. With further increasing temperature,
the decrease of the peak intensities continues, followed by an abrupt drop to
zero intensity at about 95 K.

We point out that our data excludes a coexistence of the 2x2x2 and 2x2x4 phase
at the measured temperatures between 60 K and 95 K: The intensity of
superlattice peaks with $l=0$ is always finite in the 2x2x2 phase, whereas it
vanishes completely in the 2x2x4 phase, which is incompatible with a phase
coexistence. In addition, the intensities of superlattice peaks with $l=0.5$
and $l=0.25$ above 60 K exhibit a very similar temperature dependence, as can
be seen in Fig. 2 (a), further supporting the conclusion that all superlattice
peaks have the same origin, namely the 2x2x4 phase.

The doubling along the $a$\- and $b$-direction observed in our XRD
measurements is in perfect agreement with the 2x2 superstructures observed at
the surface by STM. This surface ordering therefore corresponds to the
ordering in the bulk. The agreement between STM and XRD also provides
compelling evidence for the direct relation between the structural modulations
detected by XRD and electronic order in the V3Sb5-layers probed by STM. We
therefore conclude that also the structural transition at 60 K reflects a
change in the electronic order. More specifically, the jump
$(0.5,0.5,0.5)\rightarrow(0.5,0.5,0,25)$ of the modulation vector reveals a
reorganization of electronically ordered kagome layers along $c$, while the
period of the electronic order in the $ab$-plane remains unchanged.

We note that a previous XRD study reported a 2x2x4 phase at 15 K Ortiz  _et
al._ (2021), instead of the 2x2x2 phase observed here. However, the XRD data
in Ref. Ortiz  _et al._ , 2021 displays broad streaks along the reciprocal
$l$-direction, indicating a considerable amount of disorder. While we cannot
provide a definite explanation for this discrepancy, it might very well be
that the phase reported in Ref. Ortiz  _et al._ , 2021 is a supercooled
(0.5,0.5,0.25)-phase. Of course, this needs to be scrutinized further in
future studies.

We argue that the present results show the temperature-dependence of the
electronic order in thermal equilibrium for several reasons: Firstly, the
superlattices observed here are highly ordered in all three dimensions, which
is a very strong indication that these structures belong to the
thermodynamically stable state. Secondly, the observed transition temperatures
agree perfectly with a number of previous reports: First of all, the
transition at Tc1 =95 K is fully consistent with anomalies observed in the
magnetic susceptibility, electrical resistivity and the magnetic heat Ortiz
_et al._ (2020); Wang  _et al._ (2021). But more importantly, the transition
at Tc2 =60 K is in perfect agreement with previous magneto-transport studies
where precisely at this temperature striking anomalies were observed Chen  _et
al._ (2021a); Xiang  _et al._ (2021). It also corresponds very closely to
drastic change in the temperature dependent Seebeck coefficient Chen  _et al._
(2021b). The rearrangement of the electronic order at Tc2 is therefore
directly related to changes in the macroscopic properties.

This raises the important question as to what the microscopic structure of the
electronic order really is. Unfortunately, possible twinning of the
superstructure in our sample currently prevents a full refinement of the
superlattice modulation. There are, however, a few important conclusions that
can still be drawn: In Figs. 3 (a),(d) $l$-cuts at two $hk$-positions, which
are equivalent in $P6/mmm$, are shown. Clearly, the $l$-cuts at the two
positions differ, showing that the 6-fold symmetry of $P6/mmm$ is indeed
broken by the electronic order in both the 2x2x2 and the 2x2x4 phase. This can
also be observed very nicely in panels (b),(c) and (e),(f), which compare the
intensity distributions in reciprocal lattice planes related by the 6-fold
symmetry. Our data therefore excludes high-symmetry charge ordering models
with $P6/mmm$ symmetry discussed in the literature Christensen  _et al._
(2021). However, our data are compatible lower-symmetry models discussed in
the literature, such as star of David or inverse star of David orders as well
as a combined stacking of the two.

A very intriguing feature of the data shown in Fig. 3 (d) concerns the
symmetry of superlattice peaks with $l=0.5$ and $l=0.25$. We indeed find that
the peak intensities for $l=0.25$ still possess the 6-fold rotation symmetry,
while the ones with $l=0.5$ do not. In other words, part of the superlattice-
intensity remains six fold symmetric in the 2x2x4 phase, whereas this symmetry
is fully lost in the 2x2x2 phase. In this sense the breaking of the six-fold
symmetry gets stronger and stronger upon cooling. This observation agrees very
nicely with scenarios explaining the magneto-transport anomalies found around
60 K in terms of rotational symmetry breaking and a possible Pomeranchuk
instability Zhao  _et al._ (2021).

Now we turn to another very striking feature of the XRD-intensity
distribution, namely its chirality. As can be observed in the inset of Fig 2
(a), the intensities exhibit a chirality in the $(hk)$-plane very similar to
what has been observed by STM Jiang  _et al._ (2021); Wang  _et al._ (2021).
Indeed, chiral electronic order yielding time-reversal symmetry is found in
the kagome Hubbard model, where the chirality within a single kagome plane
results from sublattice interference Kiesel and Thomale (2012); Kiesel  _et
al._ (2013). Interestingly, a very recent $\mu$SR-study of KV3Sb5 reports the
observation of time-reversal symmetry breaking due to electronic order Mielke
III  _et al._ (2021). These $\mu$SR-data also show additional anomalies at
lower temperature, which may correspond to the transition at Tc2 =60 K.

But when interpreting the chiral XRD-intensity, care must be taken: Twinning
of a non-chiral superstructure, for example, can also yield chiral XRD-
intensity distributions. The chirality of the XRD-intensities presented in the
inset of Fig. 2 (a) can therefore not be taken as proof for chiral electronic
order. Notwithstanding, the present XRD-results are fully consistent with
chiral electronic order in CsV3Sb5.

To conclude, the present study provides new and valuable experimental insight
regarding the relation between electronic order, topological band structure
and macroscopic transport in the correlated kagome metal CsV3Sb5. But the full
microscopic architecture of the electronic order and the role played by
translational and rotational symmetry breaking remains to be fully understood.
This certainly deserves further study and will very likely yield more
surprises in the future.

## Acknowledgements

This research has been supported by the Deutsche Forschungsgemeinschaft
through the the projects C06 and C09 of the SFB 1143 (project-id 247310070)
and the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in
Quantum Matter–ct.qmat (EXC 2147, project-id 390858490). CF, DC and CS also
gratefully acknowledge additional support by the European Research Council
Advanced Grant (No. 742068) “TOPMAT” and the Deutsche Forschungsgemeinschaft
(project-id No. 258499086).

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