# Confront $f(R,T)=\mathcal{R}+\beta T$ modified gravity with the massive
pulsar ${\textit{P}SRJ0740+6620}$

G. G. L. Nashed *  —<EMAIL_ADDRESS>Centre for Theoretical Physics, The
British University in Egypt, P.O. Box 43, El Sherouk City, Cairo 11837, Egypt

(August 28, 2024)

###### Abstract

Many physically inspired general relativity (GR) modifications predict
significant deviations in the properties of spacetime surrounding massive
neutron stars. Among these modifications is $f(\mathcal{R},{\mathbb{T}})$,
where $\mathcal{R}$ is the Ricci scalar, ${\mathbb{T}}$ represents the trace
of the energy-momentum tensor, the gravitational theory that is thought to be
a neutral extension of GR. Neutron stars with masses above 1.8 $M_{\odot}$
expressed as radio pulsars are precious tests of fundamental physics in
extreme conditions unique in the observable universe and unavailable to
terrestrial experiments. We obtained an exact analytical solution for
spherically symmetric anisotropic perfect-fluid objects in equilibrium
hydrostatic using the frame of the form of
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta{\mathbb{T}}$ where $\beta$ is a
dimensional parameter. We show that the dimensional parameter $\beta$ and the
compactness, $C=\frac{2GM}{Rc^{2}}$ can be used to express all physical
quantities within the star. We fix the dimensional parameter $\beta$ to be at
most111 Here ${\mathrm{\kappa}^{2}}$ is the coupling constant of Einstein
which is figured as ${\mathrm{\kappa}^{2}=\frac{8\pi G}{c^{4}}}$, the
Newtonian constant of gravitation is denoted as ${\mathrm{F}}$ while
${\mathrm{c}}$ represents the speed of light.
$\beta_{1}=\frac{\beta}{\kappa^{2}}=0.1$ in positive values through the use of
observational data from NICER and X-ray Multi-Mirror telescopes on the pulsar
${\textit{P}SRJ0740+6620}$, which provide information on its mass and radius.
The radius and mass of the pulsar ${\textit{P}SRJ0740+6620}$ were determined
by analyzing data obtained from NICER and X-ray Multi-Mirror telescopes. It is
important to mention that no assumptions about equations of state were made in
this research. Nevertheless, the model demonstrates a good fit with linear
patterns involving bag constants. Generally, when the dimensional parameter
$\beta$ is positive, the theory predicts that a star of the same mass will
have a slightly larger size than what is predicted by GR. It has been
explained that the hydrodynamic equilibrium equation includes an additional
force resulting from the coupling between matter and geometry. This force
partially reduces the effect of gravitational force. As a result, we compute
the maximum compactness allowed by the strong energy condition for
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta{\mathbb{T}}$ and for GR, which
are $C=0.757$ and $0.725$, respectively. These values are approximately 3%
higher than the prediction made by GR. Moreover, we calculate the maximum mass
to be approximately $4.26M_{\odot}$ at a radius of around $15.9$ km for a
boundary density of $\rho_{\text{nuc}}=2.7\times 10^{14}$ g/cm3.

Massive stars, Non-standard theories of gravity, Stellar structures

##  I Introduction

Neutron stars (NSs) provide a special environment for studying matter at high
densities, which are several times higher than the nuclear saturation density
of $\rho_{\text{nuc}}\approx 2.7\times 10^{14}$ g/cm3. Such high-density
matter cannot be studied on Earth. Recent observations of X-ray and
gravitational wave signals provide novel opportunities for determining the
masses and radii of NSs. The primary objective of the NICER mission revolves
around ascertaining the mass and radius of millisecond pulsars (MSPs), and
consequently their Equation of State (EoS). This is accomplished by examining
the deflection of light due to gravity and studying the patterns of X-ray
emission from the rotating heated regions found on the surfaces of millisecond
pulsars (MSPs) (Bogdanov:2019ixe, ; Bogdanov:2019qjb, ). NICER has been
monitoring some MSPs with masses around 1.5 solar masses, including PSR
J0030+0451, which has a mass of $M=1.44^{+0.15}{-0.14}M\odot$ according to
(Miller:2019cac, ), and $M=1.34^{+0.15}{-.16}M\odot$ according to
(Raaijmakers:2019qny, ). Other MSPs such as ${\textit{P}SRJ0437-4715}$, with a
mass of $M=1.44\pm 0.07M_{\odot}$ according to (Reardon:2015kba, ), are also
being examined (2014HEAD…1411607B, ). The radius of the isolated pulsar
${\textit{P}SRJ0030+0451}$ has been determined using two separate analysis of
data from NICER. The first analysis yielded a radius of
$R=13.02_{-1.06}^{+1.24}$ km according to (Miller:2019cac, ), while the second
analysis produced a radius of $R=12.71^{+1.14}_{-1.19}$ km
(Raaijmakers:2019qny, ). Although different assumptions are made regarding the
characteristics of the hot spots in these studies, both approaches yield
consistent results when it comes to measuring the radius of the pulsar.

Moreover, a few MSP with mass $\sim 2M_{\odot}$, for
example${\textit{P}SRJ1614-2230}$ with mass $M=1.908\pm 0.016M_{\odot}$
(Demorest:2010bx, ; Fonseca:2016tux, ; NANOGRAV:2018hou, ),
${\textit{P}SRJ0348+0432}$ with $M=2.01\pm 0.04M_{\odot}$ (Antoniadis:2013pzd,
) and ${\textit{P}SRJ0740+6620}$ with mass $M=2.08\pm 0.07M_{\odot}$ using the
relativistic Shapiro time delay (NANOGrav:2019jur, ; Fonseca:2021wxt, ), are
particularly intriguing because the measurements obtained fall within the
maximum limit for a NS . The pulsar ${\textit{P}SRJ0740+6020}$, is in a binary
structure that offers individual readings of mass. As a result, the mass can
be determined with greater accuracy, though the lower NICER count rate
presents a challenge when compared to ${\textit{P}SRJ0348+0432}$. The radius
of PSR J0740+6020 was determined using data from both the NICER and X-ray
Multi-Mirror (XMM) Newton observatories. In a study by Miller et al. (2021),
the radius was calculated as $R=13.7_{-1.5}^{+2.6}$ km. Another study by Riley
et al. (2021) independently derived a radius of $R=12.39_{-0.98}^{+1.30}$ km
at a $68\%$ confidence level. Additionally, combining data from NICER and XMM
using a Gaussian process and non-parametric equation of state (EoS) method,
Legred et al. (2021) obtained a mass of $M=2.07\pm 0.11M_{\odot}$ and a radius
of $R=12.34^{+1.89}_{-1.67}$ km. These findings are consistent with the
results presented by Landry et al. (2020) at a $68\%$ confidence level.

Numerous solutions to the field equations for spherically symmetric fluid
spheres in general relativity (GR) have been well-documented in the literature
Stephani:2003tm . Nevertheless, it was identified that only nine (out of 127)
fulfill the basic criteria for physical plausibility Delgaty:1998uy . The
Buchdahl 1967ApJ…147..310B , Nariai IV 1999GReGr..31..945N , and the Tolman
VII (T-VII) Tolman:1939jz , Krori-Barua 1975JPhA….8..508K  solutions are
frequently used. Krori and Barua’s (KB) 1975JPhA….8..508K  singularity space-
time metric meets the physical requirements of a real star. The main
motivation for considering KB spacetime is to investigate gravitational
theories or models that deviate from the standard general relativity. Krori
and Barua spacetime, was proposed by the physicists Krori and Barua,
represents an alternative metric that can be used to describe the geometry of
spacetime in specific scenarios. By studying this spacetime, researchers aim
to explore its implications and potential applications in understanding
gravity, cosmology, and other related phenomena. The motivation lies in
expanding our understanding of gravitational theories beyond the conventional
framework and potentially uncovering new insights into the nature of spacetime
and the universe. As a consequence, the KB model has garnered considerable
attention in the literature, with numerous intriguing properties highlighted
in Roupas:2020mvs ; Nashed:2023pxd ; Rahaman:2011cw ; MonowarHossein:2012ec ;
Kalam:2012sh ; Rahaman:2010mr .

To address these unanswered concerns, a variety of modifications of the
gravitational lagrangian of Einstein-Hilbert (EH) have been studied
Nojiri:2017ncd . Notably, Einstein made his attempt to change the Lagrangian
of EH when he proposed the cosmological constant222The impact of the repulsive
cosmological constant within the cosmological framework was summarized in
Carroll:2000fy , specifically, there have been particular considerations about
accretion events in compact objects and dark matter halos in the work of
Stuchlik:2020rls ; Boehmer:2004nu ; Stuchlik:2016xiq ; Novotny:2021zlq , and
the motion of relative galaxies in Stuchlik:2011zz . to enable his field
equations to have a static universe solution Einstein:1917ce ;
ORaifeartaigh:2017uct . One can consider a later modification of the gravity
sector by treating the EH term as an elevated linear function of the Ricci
scalar $\mathcal{R}$. One of the simplest and most effective extensions of GR
is achieved by the replacement of the EH expression with arbitrary function of
$\mathcal{R}$ in $f(\mathcal{R})$ theory Buchdahl:1970ynr ; Sotiriou:2008rp .
This theory allows for the inclusion of the accelerated phases of the universe
Copeland:2006wr ; DeFelice:2010aj ; Nojiri:2010wj . Additionally, compact
object solutions have been found within this theory, as seen in (see
DeFelice:2010aj, ) and the references therein.

The so-called $f(\mathcal{R},{\mathbb{T}})$ gravity Harko:2011kv , which
includes the trace of the energy-momentum tensor (EMT) ${\mathbb{T}}\equiv
g^{\mu\nu}{\mathbb{T}}_{\mu\nu}$ in the construction of the modified gravity
sector, is another generalization of $f(\mathcal{R})$ gravity that has gained
significant interest in recent years. The generalized curvature-matter
coupling theories, which include the $f(\mathcal{R},{\mathbb{T}})$ theory of
gravity, produce intriguing phenomenology when used in cosmological contexts
Harko:2014gwa ; Goncalves:2021vci ; Goncalves:2022ggq . Yet, in
$f(\mathcal{R},{\mathbb{T}})$ gravity, solutions to relativistic compact stars
have also been studied in Hansraj:2018jzb ; Bhar:2021uqr ; Kumar:2021vqa ;
Feng:2022bvk . In the frame of $f(\mathcal{R},{\mathbb{T}})$ an anticipation
of the presence of quark stars composed of quark matter in the color-flavor-
locked phase of color superconductivity has been studied in Tangphati:2022arm
. Moreover, the properties of quark stars have been investigated in the
context of $f(\mathcal{R},{\mathbb{T}})$ Tangphati:2022mur .

Throughout this investigation, our focus is directed towards the KB spacetime
1975JPhA….8..508K , a model proven to be pertinent in exploring the interior
structure of neutron stars (NSs). We derive an analytical extension within the
linear approximation of
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta{\mathbb{T}}$ gravity for this
model and examine the influence of the dimensional parameter $\beta$ on the
internal structure of the stellar object. Furthermore, we employ numerical
methods to solve the modified Tolman-Oppenheimer-Volkoff (TOV) system of
equations with a uniform density profile, emphasizing the impact of the extra
force arising from this theory.

Anisotropic spacetime is relevant in alternative theories of gravity beyond
general relativity. Various modified gravity theories propose modifications to
the Einstein field equations, allowing for anisotropic contributions to the
gravitational field. By studying anisotropic spacetime, researchers can test
and refine these alternative theories, exploring their implications and
potential observational signatures.

The arrangement of this article is as follows: In Sec. II we give a summary of
the construction of $f(\mathcal{R},{\mathbb{T}})$ gravitational theory, where
we present the generalized form of the field equations. As a result, in
Sec.III, we adopt KB’s approach (1975JPhA….8..508K, ) to derive an analytic
solution for the field equations within the framework of linear
$f(\mathcal{R},{\mathbb{T}})$ gravitational theory. Additionally, in Sec. IV,
we utilize the observations of the PSR ${\textit{P}SRJ0740+6620}$ made by the
X-ray telescopes NICER and XMM to determine the values of the parameters used
in the model.Furthermore, we examine the intrinsic characteristics of the
pulsar PSR ${\textit{P}SRJ0740+6620}$ and evaluate its stability by
considering the limitations imposed by the existing model. In Sec. V, we
compare our model to additional data from pulsars, which is presented in
Tables 1 and 3. In Sec. VI, we calculate the highest permissible compactness
based on physical constraints and subsequently generate curves of mass versus
radius for various surface densities. For each category, the maximum
attainable mass for a stable structure serves as its representation. In Sec.
VII, we present the conclusions of this study.

Nomenclature and notation: We adopt the metric signature $(-,+,+,+)$
throughout this work. Unless explicitly mentioned, primes denote derivatives
w.r.t, the radial coordinate.

##  II Brief of $f(\mathcal{R},{\mathbb{T}})$ gravitational theory

In the following section, we will present a brief summary of
$f(\mathcal{R},{\mathbb{T}})$ gravity within the framework of metric
construction, using the Levi-Civita connection as the basis, as outlined in
Harko:2011kv . It’s important to note that in the frame of Palatini
construction, the connection is considered as independent variable through the
process of the variation of the Lagrangian (see Wu:2018idg, ). The action for
$f(\mathcal{R},{\mathbb{T}})$ theory takes the following form:

$S=\int{\text{d}^{4}x\sqrt{-g}}\left[\frac{f(\mathcal{R},{\mathbb{T}})}{\kappa^{2}}+\mathcal{L}_{m}\right]\,,$
(1)

where ${\mathrm{\kappa}^{2}}$ represents the Einstein coupling constant,
determined as ${\mathrm{\kappa}^{2}=\frac{8\pi G}{c^{4}}}$, with
${\mathrm{G}}$ denoting the Newtonian gravitational constant and
${\mathrm{c}}$ representing the speed of light. If we assume $\mathcal{L}_{m}$
to depend on the metric, then energy-momentum tensor of matter is defined by:

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

The EMT of an anisotropic fluid takes the following form:

${\mathbb{T}{{}^{\mu}}{{}_{\nu}}=(p_{t}+\rho
c^{2})w{{}^{\mu}}w{{}_{\nu}}+p_{t}\delta^{\mu}_{\nu}+(p_{r}-p_{t})v{{}^{\mu}}v{{}_{\nu}}}\,.$
(3)

In this particular scenario, the symbol ${\mathrm{\rho}=\rho(r)}$ denotes the
density of the fluid. The term $\mathrm{{p}_{r}={p}_{r}(r)}$ refers to the
pressure exerted in the radial direction (i.e., in alignment with the time-
like four-velocity ${\mathrm{w}^{\alpha}}$’s trajectory), while
$\mathrm{{p}{t}={p}{t}(r)}$ denotes the tangential pressure exerted orthogonal
to ${\mathrm{w}^{\alpha}}$. Moreover, ${\mathrm{v}^{\alpha}}$ represents the
normalized spatial vector aligned radially. Consequently, the energy-momentum
tensor can be written in a diagonal form as
${\mathrm{T}{{}^{\alpha}}_{\beta}=diag(-\rho c^{2},\,p_{r},\,p_{t},\,p_{t})}$.
By varying the metric with respect to Eq. (1), the resulting equations
correspond to the modified Einstein Eqs. Harko:2011kv :

$f_{\mathcal{R}}\mathcal{R}_{\mu\nu}-\frac{1}{2}f\,g_{\mu\nu}+S_{\mu\nu}f_{\mathcal{R}}=\kappa^{2}T_{\mu\nu}-f_{T}\left(T_{\mu\nu}+\Theta_{\mu\nu}\right)\equiv{\mathbb{T}}_{\mu\nu}\,,$
(4)

where $f_{\mathcal{R}}\equiv\frac{\partial
f(\mathcal{R},{\mathbb{T}})}{\partial\mathcal{R}}$,
$f_{\mathbb{T}}\equiv\frac{\partial
f(\mathcal{R},{\mathbb{T}})}{\partial{\mathbb{T}}}$, $S_{\mu\nu}$ stands for
$S_{\mu\nu}\equiv\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)$ and

$\Theta_{\mu\nu}\equiv
g^{\alpha\beta}\frac{\delta{\mathbb{T}}_{\alpha\beta}}{\delta
g^{\mu\nu}}=-2{\mathbb{T}}_{\mu\nu}+g_{\mu\nu}\mathcal{L}_{m}-2g^{\alpha\beta}\frac{\partial^{2}\mathcal{L}_{m}}{\partial
g^{\mu\nu}\partial g^{\alpha\beta}}\,.$ (5)

Following the approach given by Harko _et al._ Harko:2011kv , we assume
$\mathcal{L}m=p$ (see Harko:2011kv , $\mathcal{L}m=-p$). Following such
assumption, we get $\Theta{\mu\nu}=-2{\mathbb{T}}{\mu\nu}+pg{\mu\nu}$. The
trace of Eq. (4) yields:

$f_{\mathcal{R}}\mathcal{R}-2f+3\Box
f_{\mathcal{R}}=\kappa^{2}{\mathbb{T}}\,.$ (6)

In $f(\mathcal{R},{\mathbb{T}})$ theory, Equation (6) becomes nonlinear in
$\mathcal{R}$, resulting in a second-order differential equation, which is not
algebraic as in the case of pure General Relativity (GR). Consequently,
nonlinear forms of $f(\mathcal{R},{\mathbb{T}})$ lead to a non-vanishing of
$\mathcal{R}$. This is because of the non-vanishing value of the trace of EMT
in the action of $f(\mathcal{R},{\mathbb{T}})$ gravity. As a result, the
-divergence of EMT is given by BarrientosO:2014mys

$\nabla^{\mu}{\mathbb{T}}_{\mu\nu}=\frac{f_{\mathbb{T}}}{\kappa^{2}-f_{\mathbb{T}}}\left[\left({\mathbb{T}}_{\mu\nu}+\Theta_{\mu\nu}\right)\nabla^{\mu}\ln{f_{\mathbb{T}}}+\nabla^{\mu}\Theta_{\mu\nu}-\frac{1}{2}\nabla_{\nu}{\mathbb{T}}\right]\,.$
(7)

The non-conservation of the energy-momentum tensor (EMT) of matter in
$f(\mathcal{R},{\mathbb{T}})$ theory leads to an additional force. This extra
force could have effects for gravitational beyond those predicted by GR and
could be significant at galactic scales and in the solar system Harko:2011kv .

The conservation of the EMT is verified when
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}$. In such cases, the form of
$f(\mathcal{R},{\mathbb{T}})$ can be constructed based on this condition (see
also Pretel:2021kgl, ). However, the corresponding field equations can only be
solved in a numerical way. Since our main focus in this study is on obtaining
analytical solutions, we will not delve into discussing these specific models
of $f(\mathcal{R},{\mathbb{T}})$ gravitational theory.

It is appropriate to recast Eq. (4) into an equivalent form of GR, where the
left-hand side contains the Einstein tensor $G_{\mu\nu}$, and the right-hand
side combines the effective EMT taking into account contributions from both
matter fields and curvature terms, given by:

$G_{\mu\nu}=\frac{1}{f_{\mathcal{R}}}\left[{\mathbb{T}}_{\mu\nu}+\frac{f-\mathcal{R}\,f_{\mathcal{R}}}{2}g_{\mu\nu}-S_{\mu\nu}f_{\mathcal{R}}\right]\equiv{\mathbb{T}}^{(\text{eff})}_{\mu\nu}\,.$
(8)

In general, for an arbitrary $f(\mathcal{R},{\mathbb{T}})$ theory, the
components of ${\mathbb{T}}^{(\text{eff})}_{\mu\nu}$ will be quite complex,
making it exceedingly challenging to derive an exact analytical solution in
this theory. It should be noted that for the special model

$f(\mathcal{R},T)=\mathcal{R}+h(T)\,,$ (9)

that is the specific case where the $f(\mathcal{R},{\mathbb{T}})$ theory is
linear in the Ricci scalar and does not involve mixing terms between
$\mathcal{R}$ and ${\mathbb{T}}$, the contributions of curvature fluids in the
effective energy-momentum tensor (EMT) simplify considerably. As a result, the
formulation of the field equations becomes more manageable and takes the form:

$G_{\mu\nu}=\kappa^{2}{\mathbb{T}}_{\mu\nu}+\frac{h}{2}g_{\mu\nu}+h_{\mathbb{T}}\left({\mathbb{T}}_{\mu\nu}-p\,g_{\mu\nu}\right)\,.$
(10)

In this study and in order to explain the minimal extension of Einstein GR in
the framework of $f(\mathcal{R},{\mathbb{T}})$ gravity, and to avoid overly
complicated field equations that could be prevent the search of an exact
analytic solution, we limit our discussion to the case of linear
$f(\mathcal{R},{\mathbb{T}})$ gravity in which $f(\mathcal{R},{\mathbb{T}})$
takes the form:

$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta{\mathbb{T}}\,.$ (11)

In this linear case, the field equations take a more straightforward form,
with $\beta$ representing a dimensional constant parameter of the theory. For
this specific linear scenario, an analytic solution can be easily derived for
the field equations ((see also Hansraj:2018jzb, ; Bhar:2021uqr, ;
Pretel:2020oae, ; Pretel:2021kgl, )).

However, it is important to note that for the more complicated non-linear
forms of $f(\mathcal{R},{\mathbb{T}})$ gravity, the field equations generally
become more complex. One way to understand the complexity is by exploring the
dynamical equivalence of scalar-tensor representations of the theory and
employing the Palatini construction (see also Rosa:2021teg, ; Rosa:2022cen, ).

##  III Interior spherically symmetric solution

In this section we are going to derive a spherically symmetric interior
solution in the frame of
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta\,{\mathbb{T}}$. For this aim we
use the general spherically symmetric geometry and static which is by the
following line element Nashed:2004pn ; Nashed:2006yw ; Nashed:2020kjh :

$ds^{2}=-e^{\alpha(r)}dt^{2}+e^{\alpha_{1}(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\,,$ (12)

where $\alpha(r)$ and $\alpha_{1}(r)$ are functions of the radial coordinate.

As Eq.(11) represents a particular form of the theories described by Eq.(9),
when we substitute Eqs.(3) and (12) into Eq. (10), we obtain the following
system of differential equations:

$\displaystyle\mathrm{\kappa^{2}\rho c^{2}}$ $\displaystyle=$
$\displaystyle\mathrm{\frac{1}{1+3\,\beta_{1}}\left[\frac{\alpha^{\prime}_{1}r+{e^{\alpha_{1}}-1}}{{r}^{2}{e^{\alpha_{1}}}}+\frac{5\kappa\beta_{1}}{3}\left(p_{r}+2\,p_{t}\right)\right]}\,,$
$\displaystyle\mathrm{\kappa^{2}p_{r}}$ $\displaystyle=$
$\displaystyle\mathrm{\frac{3}{3+11\,\beta_{1}}\left[\frac{\alpha^{\prime}r-{e^{\alpha_{1}}+1}}{{r}^{2}{e^{\alpha_{1}}}}+\frac{\kappa\beta_{1}}{3}\left(3\rho
c^{2}-10p_{t}\right)\right]}\,,$ $\displaystyle\mathrm{\kappa^{2}p_{t}}$
$\displaystyle=$
$\displaystyle\mathrm{\frac{3}{3+16\,\beta_{1}}\left[\frac{2\alpha^{\prime\prime}r+\alpha^{\prime}[\alpha^{\prime}-\alpha^{\prime}_{1}]+2}{4{r}{e^{\alpha_{1}}}}+\kappa\beta_{1}\left(\rho
c^{2}-\frac{5}{3}p_{r}\right)\right]}\,.$ (13)

Here, the prime symbol (′) indicates differentiation with respect to the
radial coordinate, i.e., $r$ and $\beta_{1}=\frac{\beta}{\kappa^{2}}$. By
utilizing the aforementioned equations, we can derive the anisotropy
parameter, denoted as $\mathrm{\Delta(r)}$, which is defined as the difference
between the tangential pressure $\mathrm{p_{t}}$ and the radial pressure
$\mathrm{p_{r}}$ as:

$\mathrm{\Delta(r)=\frac{2\alpha^{\prime\prime}{r}^{2}+\alpha^{\prime
2}{r}^{2}-\left[{\alpha^{\prime}_{1}}{r}-2\right]\,r\alpha^{\prime}+4e^{\alpha_{1}}-2{\alpha_{1}}r-4}{4\kappa\left(2\beta_{1}+1\right){r}^{2}e^{\alpha_{1}}}}.$
(14)

Surprisingly, the connection between matter and geometry that results from
trace component does not effect the anisotropy as described by Eq. (14),
unless we suppose that the spacetime configuration is spherically symmetric
(Nashed:2022zyi, ). Nonetheless, if $\beta_{1}\neq 0$, we can anticipate a
minor alteration in anisotropy. Hence, different anisotropic effects are
unable to conceal discrepancies from GR that are produced by the connection
between matter and geometry. If the dimensionless parameter $\beta_{1}$ is
equal to zero, the differential equations (13) will be the same as the field
equations of GR theory for a spherically symmetric interior spacetime (c.f.,
Roupas:2020mvs, ).

The system of differential equations (13) has three independent non-linear
differential equations and five unknowns $\mathrm{\alpha}$,
$\mathrm{\alpha_{1}}$, $\mathrm{\rho}$, $\mathrm{p_{r}}$ and $\mathrm{p_{t}}$.
Therefore, we required two extra conditions to limit the system. Therefore, we
will introduce the Krori-Barua (KB) ansatz as 1975JPhA….8..508K :

$\alpha(r)=\frac{\mu\,r^{2}}{R^{2}}+\nu\equiv\mu\,x^{2}+\nu,\,\qquad\qquad\qquad{\alpha_{1}}(r)=\frac{\lambda\,r^{2}}{R^{2}}\equiv\lambda\,x^{2},$
(15)

where the dimensionless radius $x$ is defined as

$0\leq\frac{r}{R}=x\leq 1\,.$

In this context, $R$ corresponds to the stellar radius. Furthermore, we
establish the values of the dimensionless coefficients $\mu,\nu,\lambda$ by
enforcing matching criteria at the stellar’s surface. Moreover, we present
dimensionless quantities to simplify the analysis.

$\widetilde{\rho}(r)=\frac{\rho(r)}{\rho_{\ast}}\,,\qquad\widetilde{p}_{r}(r)=\frac{p_{r}(r)}{\rho_{\ast}c^{2}}\,,\qquad\widetilde{p}_{t}(r)=\frac{p_{t}(r)}{\rho_{\ast}c^{2}}\,,\qquad\widetilde{\Delta}(r)=\frac{\Delta(r)}{\rho_{\ast}c^{2}}\,,$
(16)

where $\rho_{\ast}$ is the characteristic density defined as:

$\rho_{\ast}=\frac{1}{\kappa^{2}c^{2}R^{2}(1+10\beta_{1}+16\beta_{1}{}^{2})}\,.$
(17)

Therefore, the equation of motions (13) can be expressed as follows:

$\displaystyle\widetilde{\rho}$ $\displaystyle=$
$\displaystyle\frac{e^{-\lambda x^{2}}}{x^{2}}(e^{\lambda x^{2}}-1+2\lambda
x^{2})$ $\displaystyle+$
$\displaystyle\frac{2\beta_{1}}{x^{2}}\left[\left(5\mu(\mu-\lambda)x^{4}+(16\lambda+15\mu)x^{2}-8\right)e^{-\lambda
x^{2}}+8\right],$ $\displaystyle\widetilde{p}_{r}$ $\displaystyle=$
$\displaystyle\frac{e^{-\lambda x^{2}}}{x^{2}}(1-e^{\lambda x^{2}}+2\mu
x^{2})$ $\displaystyle-$
$\displaystyle\frac{2\beta_{1}}{x^{2}}\left[\left(5\mu(\mu-\lambda)x^{4}-(9\lambda-8\mu)x^{2}-8\right)e^{-\lambda
x^{2}}+8\right],$ $\displaystyle\widetilde{p}_{t}$ $\displaystyle=$
$\displaystyle e^{-\lambda x^{2}}(2a_{0}-\lambda+\mu(\mu-\lambda)x^{2})$
$\displaystyle-$
$\displaystyle\frac{2\beta_{1}}{x^{2}}\left[\left(7\mu(\mu-\lambda)x^{4}-(9\lambda-4\mu)x^{2}+8\right)e^{-\lambda
x^{2}}-8\right],$

and anisotropy parameter (14) takes the following form:

$\widetilde{\Delta}=\frac{e^{-\lambda
x^{2}}}{x^{2}}(1+8\beta_{1})\left[e^{\lambda
x^{2}}-1+\mu(\mu-\lambda)x^{4}-\lambda x^{2}\right].$ (19)

The parameter characterizing the cumulative mass contained inside a given
radius $r$ is referred to as the function of mass, and its definition is as
follows:

$\mathbb{M}(r)=4\pi\int_{0}^{r}\rho(\widetilde{r})\,\widetilde{r}{{}^{2}}d\widetilde{r}.$
(20)

Substitute the density given by Eq. (LABEL:eq:Feqs2) in Eq. (20) we get:

$\mathbb{M}(x)=\frac{M}{C(1+10\beta_{1}+16\beta_{1}{}^{2})}e^{-\lambda
x^{2}}\left[x(e^{\lambda x^{2}}-1)+\beta_{1}\,\eta(x)\right].$ (21)

Here, $C$ denotes the compactness and $\eta(x)$ is defined as:

$\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{2GM}{c^{2}R},$
$\displaystyle\eta(x)$ $\displaystyle=$
$\displaystyle\frac{\lambda^{-\frac{9}{2}}}{3}\left\\{\left(16x\lambda^{\frac{9}{2}}+\frac{15}{4}\mu\sqrt{\pi}\lambda^{2}(\lambda+\mu)\text{erf}(\sqrt{\lambda}x)\right)e^{\lambda
x^{2}}\right.$ $\displaystyle-$
$\displaystyle\left.x\left[\left(\frac{15}{2}\mu\lambda^{\frac{5}{2}}+5\lambda^{\frac{7}{2}}(\mu
x^{2}+\frac{3}{2})\right)\mu\lambda^{\frac{5}{2}}-(5\mu
x^{2}-16)\lambda^{\frac{9}{2}}\right]\right\\}\,,$

where $erf$ is the error function defined as

$erf(y)=\frac{2}{\sqrt{\pi}}\int_{t=0}^{t=y}e^{-t^{2}}dt\,.$

Equations (21) reduces to the model of GR when $\beta_{1}=0$ (Roupas:2020mvs,
).

###  III.1 Matching conditions

The fact that both Einstein’s General Relativity and the modified theory
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ share
identical vacuum solutions is widely recognized. In the case of the exterior
solution, it is described by Schwarzschild’s solution, which can be expressed
as follows:

$ds^{2}=-\left(1-\frac{2GM}{c^{2}r}\right)c^{2}dt^{2}+\frac{dr^{2}}{\left(1-\frac{2GM}{c^{2}r}\right)}+r^{2}d\theta^{2}+r^{2}\,\sin^{2}\theta
d\phi^{2}.$ (23)

Remembering the interior spacetime given by Eq. (LABEL:eq:Feqs2), therefore,
we assume

$\alpha(x=1)=\ln(1-{C}),\,\alpha_{1}(x=1)=-\ln(1-{C}).$ (24)

Furthermore, we assume that $p_{r}$ becomes zero at the surface of the
star,i.e.,

$\bar{p}_{r}(x=1)=0.$ (25)

Using the KB assumption (15), the radial pressure (LABEL:eq:Feqs2), and the
boundary conditions mentioned above, we obtain:

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{9-5\ln\mathfrak{C}}{10}+\frac{1}{10\mathfrak{C}^{1/2}{\beta_{1}}}\left(5\left\\{2{\beta_{1}}^{2}\mathfrak{C}\left(\ln\mathfrak{C}\right)-10\,\beta_{1}\,\mathfrak{C}\left({\beta_{1}}+{\frac{3}{25}}\ln\mathfrak{C}\right)-{\frac{81}{25}}\left(3C-1\right){\beta_{1}}^{2}-{\frac{6}{25}}\left(16C-9\right)\beta_{1}+{\frac{9}{25}}\mathfrak{C}\right\\}^{1/2}\right.$
$\displaystyle\left.-3\mathfrak{C}^{1/2}\right)\,,$ $\displaystyle\nu$
$\displaystyle=$
$\displaystyle-\frac{1}{10{\sqrt{\mathfrak{C}}\beta_{1}}}\left\\{\beta_{1}\sqrt{\mathfrak{C}}[9-15\,\beta_{1}\,\ln\mathfrak{C}]+\left[\mathfrak{C}[9-30\,\beta_{1}\,\ln\mathfrak{C}+50{\beta_{1}}^{2}\ln\mathfrak{C}]-84\,\beta_{1}\,C-250\mathfrak{C}\,{\beta_{1}}^{2}\ln\mathfrak{C}-241\,{\beta_{1}}^{2}C+54\,\beta_{1}\right.\right.$
(26)
$\displaystyle\left.\left.+81\,{\beta_{1}}^{2}\right\\}^{1/2}+3\mathfrak{C}^{1/2}\right]\,,\qquad\lambda=-\ln\mathfrak{C}\,,\qquad\mbox{where}\qquad\mathfrak{C}=(1-C)\,.$

As the parameter $\beta_{1}$ approaches zero, the set of constants (26)
becomes identical to the corresponding values in GR (Roupas:2020mvs, )

$\mu=\frac{C}{2\mathfrak{C}}\,,\qquad\nu=\ln\mathfrak{C}-\frac{C}{2\mathfrak{C}}\,,\qquad\lambda=-\ln\mathfrak{C}\,.$
(27)

It is fascinating that all the measurable properties of the KB spacetime in
the star, where $0\leq x\leq 1$, can be rephrased using the dimensionless
equations that involve $\beta_{1}$ and the compactness parameters ${C}$. These
non-dimensional expressions include $\widetilde{\rho}(\beta_{1},C)$,
$\widetilde{p}_{r}(\beta_{1},C)$, and $\widetilde{p}_{t}(\beta_{1},C)$.
Generally, the data of observation fixes a particular numerical value for the
compactness parameter, which indirectly limits the non-minimal coupling
proposed by $\beta_{1}$. Furthermore, this presents a chance to set maximum
limit on the allowable value of $C$ for NS. Consequently, the upper numerical
value mass can achieved. A detailed examination of this subject will be
presented in Section VI.

###  III.2 Radial derivatives

In this subsection, we compute the radial changes in the density and pressures
of the fluid, denoted by:

$\displaystyle\widetilde{\rho}^{\prime}=\frac{2e^{-a_{2}x^{2}}}{x^{3}}\left\\{1-e^{\lambda
x^{2}}-2\lambda^{2}x^{4}+\lambda x^{2}-\frac{2\beta_{1}}{3}\left[8e^{\lambda
x^{2}}-8-5\mu
a_{2}(\lambda-\mu)x^{6}+[5(4\lambda-\mu)\mu-16a_{2}^{2}]x^{4}-8\lambda
x^{2}\right]\right\\}\,,$
$\displaystyle\widetilde{p}^{\prime}_{r}=\frac{-2e^{-\lambda
x^{2}}}{x^{3}}\left\\{1-e^{\lambda x^{2}}+2\mu\lambda x^{4}+\lambda
x^{2}-\frac{2\beta_{1}}{3}\left[8-e^{\lambda
x^{2}}-5\mu\lambda(\lambda-\mu)x^{6}-[4(2\lambda-\mu)\mu-5\mu^{2}]x^{4}-8\lambda
x^{2}\right]\right\\}\,,$ (29)
$\displaystyle\widetilde{p}^{\prime}_{t}=\frac{2e^{-\lambda
x^{2}}}{x^{3}}\left\\{\mu\lambda(\lambda-\mu)x^{6}+[\lambda(\lambda-3\mu)+\mu^{2}]x^{4}+\frac{2\beta_{1}}{3}\left[4-4e^{\lambda
x^{2}}+7\mu\lambda(\lambda-\mu)x^{6}-[(16\lambda-7\mu)\mu-4\mu^{2}]x^{4}\right.\right.$
$\displaystyle\left.\left.+4\lambda x^{2}\right]\right\\}\,,$

where ${}^{\prime}\equiv\frac{d}{dx}$. Equations (III.2), (29) and (III.2)
describe the gradients of the components of the EMT. These equations play a
crucial role in demonstrating the physical stability of the compact object.

##  IV Observational limits and the equilibrium conditions

In the present analysis, we apply the pulsar ${\textit{P}SRJ0740+6620}$, the
radius and mass determined from NICER+XMM based on a non-parametric equation
of state method. The stellar system under consideration has a radius of
$R=12.34^{+1.89}{-1.67}$ km and a mass of $M=2.07\pm 0.11M\odot$ according to
a study by Legred Legred:2021hdx , with the solar mass denoted as
$M_{\odot}=1.9891\times 10^{30}$ kg. To be considered physically legitimate, a
star model must comply with a group of requirements called (I) to (XIII):

###  IV.1 Matter sector

(a) The energy-density

(b) The radial pressure

(c) The tangential pressure

(d) The gradients (GR)

(e) The gradients f(R,T)

(f) The anisotropy (GR)

(g) The anisotropy f(R,T)

Figure 1: The patterns of the components of energy-momentum tensor as
described by Eqs. (LABEL:eq:Feqs2) of the pulsar ${\textit{P}SRJ0740+6620}$
are plotted in Figs. 0(a)–0(c). Figures 0(d) and 0(e) illustrate the radial
profiles of the gradients of these quantities in both the General Relativity
(GR) and $f(\mathcal{R},{\mathbb{T}})$ scenarios. Figures 0(f) and 0(g)
display the variation of $\Delta$ for both the $f(\mathcal{R},{\mathbb{T}})$
and GR scenarios, as described by Eq. (19). All the graphs presented in Fig. 1
verify the fulfillment of the constraints indicated from (I) to (IV).

Constrain (I): The components of the EMT, energy content, radial and
tangential stress, within the stellar fluid ($0<r/R<1$), should maintain a
positive outlook, specifically ${\mathrm{\rho}(0<r/R<1)>0}$,
${\mathrm{p}_{r}(0<r/R<1)>}0$ and ${\mathrm{p}_{t}(0<r/R<1)>0}$.

Constrain (II): Interior solutions must have a regular, non-singular
pattern.Hence, it is required that the density, radial stress, and tangential
stress investigated in this study remain regular at the core. Furthermore,
these quantities must exhibit regular behavior throughout the star’s interior,
i.e.,

  * A-

${\mathrm{\rho}(r=0)>0}$, ${\mathrm{\rho}^{\prime}=0}$,
${\mathrm{\rho}^{\prime\prime}<0}$, and ${\mathrm{\rho}^{\prime}(0<x=r/R\leq
1)<0}$ ,

  * B-

${\mathrm{p}_{r}(r=0)>0}$, ${\mathrm{p}^{\prime}_{r}(r=0)=0}$,
${\mathrm{p}^{\prime\prime}_{r}(r=0)<0}$, and
${\mathrm{p}^{\prime}_{r}(0<x=r/R\leq 1)<0}$ ,

  * C-

${\mathrm{p}_{t}(r=0)>0}$, ${\mathrm{p}^{\prime}_{t}(r=0)=0}$,
${\mathrm{p}^{\prime\prime}_{t}(r=0)<0}$, and
${\mathrm{p}^{\prime}_{t}(0<x=r/R\leq 1)<0}$ .

Constrain (III): At the star’s boundary, the radial pressure must be zero,
denoted as ${\mathrm{p}_{r}=0}$. However, it is not necessary for $p_{t}$ to
be zero at the boundary.

The constraints (I)–(III) are confirmed for ${\textit{P}SRJ0740+6620}$, as
depicted in Fig. 1. As displayed in Fig. 1 0(a), the present study
approximates a NS core density ${\mathrm{\rho}_{\text{core}}\approx 5.84\times
10^{14}}$ g/cm${}^{3}\approx 3.26\rho_{\text{nuc}}$ for the pulsar
${\textit{P}SRJ0740+6620}$. Thus, the present model does not rule out the
potential existence of neutrons in the core of the pulsar. Moreover, the value
of the energy density at the center reinforces the hypothesis that the fluid
adopts an anisotropic configuration.

Constrain (IV): The anisotropy parameter, $\Delta$, must reach a value of zero
at the core of the pulsar, meaning that ${\mathrm{p}_{r}=p_{t}}$. Moreover,
$\Delta$ must have an increasing value at the boundary of the pulsar, i.e.
$\Delta^{\prime}(0\leq r/R\leq 1)>0$. Thus, the anisotropic force
${\mathrm{F}_{a}=2\frac{\Delta}{r}}$ should be vanishing at the core. As shown
in Eq. (19), taking the limit $r\to 0$ gives $\Delta\to 0$.

The constraint (IV) has been verified for PSR J0740+6620, as demonstrated in
Figs. 10(f) and 10(g).

We can also demonstrate that ${\mathrm{\Delta}(r>0)>}0$ ensuring
$\mathrm{p_{t}>p_{r}}$. This condition is crucial to make the anisotropy force
repulsive allowing larger NS sizes than for an isotropic fluid. Nevertheless,
when considering $\beta_{1}>0$, it indicates that $\Delta$ in
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ is marginally
lower than in GR.

###  IV.2 The condition of Zeldovich

Constraint (V): Based on the study by 1971reas.book…..Z , the radial pressure
at the core should be:

$\mathrm{\frac{\widetilde{p}_{r}(0)}{\widetilde{\rho}(0)}\leq 1.}$ (31)

At the center of the pulsar, we derive equations for the density, the pressure
in the radial direction, and the pressure in the tangential direction as:

$\displaystyle\mathrm{{\widetilde{\rho}}(x=0)=3\lambda+2\beta_{1}(8\lambda+5\mu)}\,,$
$\displaystyle\mathrm{\widetilde{p}_{r}(x=0)={\widetilde{p}}_{t}(x=0)={2\mu(1+3\beta_{1})-\lambda}}\,.$
(32)

We calculate the compactness parameter of the pulsar PSR J0740+6620 to be
$C=0.491+\pm 0.0547087977$. Thus, we can use the Zeldovich condition (31) to
determine a valid range of values for the dimensionless parameter $\beta_{1}$,
which is $0\leq\beta_{1}\leq 0.1$. In this interval, the value of $\beta_{1}$
is expected to be consistent with that of GR when $\beta_{1}=0$.

###  IV.3 The radius and mass observational limits of PSR J0740+6620

(a) The Mass function

(b) The Compactness parameter

Figure 2: We have generated plots for the mass and the compactness,
represented by Eqs. (21) and (III), respectively, pertaining to PSR
J0740+6620. The resulting curves are in excellent harmony with the data of
($M=2.07\pm 0.11M_{\odot}$ and $R=12.34\pm 1.89$ km) Legred:2021hdx . For
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1},{\mathbb{T}}$ case, we
assign the following values to the parameters: $\beta_{1}=0.1$,
$\kappa=2.302\times 10^{-43},N^{-1}$, $\mu=0.4248802917$, $\nu=-1.109209$,
$\lambda=0.6843287084$. In the case of GR, we set $\beta_{1}=0$, and the
corresponding values for the parameters are $\mu=0.4912202964$,
$\nu=-1.175549005$.

Taking into account the restrictions imposed by the mass and radius of
${\textit{P}SRJ0740+6620}$, we find that the mass function given by Eq. (21)
predicts a total mass of $M=2.12M_{\odot}$ at $R=13.65$ km, corresponding to
${C}=0.3214945055$, when $\beta_{1}$ is set to $0.1$. This is consistent with
the values of ($M=2.07\pm 0.11M_{\odot}$ and $R=12.34\pm 1.89$ km)
Legred:2021hdx . This determines the values of the set of constants {
$\mu=0.4248802917$, $\nu=-1.109209$, $\lambda=0.6843287084$,
$\kappa=2.302\times 10^{-43}\,N^{-1}$}. These values unequivocally confirm the
fulfillment of Eq. (31). In Figs. 2 1(a) and 1(b), we present the trends of
the mass function and compactness, respectively, to demonstrate the model’s
agreement with the observed mass and radius of the pulsar. In comparison to
GR, the model $f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1},{\mathbb{T}}$
with a positive $\beta_{1}$ predicts the same mass but at a larger radius.
Consequently, the
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1},{\mathbb{T}}$ anticipates a
higher compactness for a given mass. This demonstrates that
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1},{\mathbb{T}}$ has the
capacity to accommodate higher masses or greater compactness values while
still adhering to stability conditions.

###  IV.4 Geometric sector

Constrain (VI): The metric potentials ${\textit{g}_{rr}}$ and
${\textit{g}_{tt}}$ should have no physical or coordinate singularities within
the stellar interior $0\leq x\leq 1$. The metric (15) satisfies this
constraint because: At the core, ${\mathrm{g}_{tt}(x=0)=e^{\nu}}$ and
${\mathrm{g}_{rr}(x=0)=1}$, i.e, they remain finite.

Constrain (VII): The solutions for the metric potentials in both the exterior
and interior regions must match smoothly at the star’s boundary.

Figure 3: The metric components, $g_{rr}$ and $g_{00}$, of the star
${\textit{P}SRJ0740+6620}$ have finite values within its interior of the
pulsar and seamlessly match with the Schwarzschild vacuum solution outside.
The above plots provide confirmation of these conditions.

It is evident from Fig. that conditions (VI) and (VII) are fulfilled using PSR
J0740+6620.

Constrain (VIII): The red-shift of the metric (15) is figured as:

$Z=\frac{1}{\sqrt{-g_{tt}}}-1=\frac{1}{\sqrt{e^{\mu\,x+\nu}}}-1.$ (33)

The gravitational redshift must possess the following characteristics: it
should be positive ($Z>0$), have a finite value inside the stellar body, and
decrease as one moves towards the stellar surface ($Z^{\prime}<0$).

Figure 4: The red-shift of Eq. (33) of PSRJ0740 is illustrated in Fig. 4,
confirming the satisfaction of condition (VIII).

It is evident from Fig. 4 that condition (VIII) is fulfilled using PSR
J0740+6620.

At center: Gravitational red-shift factor is $Z(0)\approx 0.7412521838$ (the
value of $Z(0)$ is below the corresponding value in GR, which is approximately
$0.7999781079$.). This central value is the maximum red-shift within the star.
At the stellar surface: The red-shift factor is $Z_{R}\approx 0.4079916880$
(around the same value as in GR). This surface value is below the upper bound
of $Z_{R}=2$ Buchdahl:1959zz , which has been explored in other studies, such
as (Ivanov:2002xf, ; Barraco:2003jq, ) for anisotropic analyses and
Boehmer:2006ye  in the context of a cosmological constant. The maximum red-
shift limit does not provide a tight constraint and cannot properly bound
compactness alone, as discussed in other works. Less restrictive constraints
are needed (c.f., Ivanov:2002xf, ; Barraco:2003jq, ; Boehmer:2006ye, ).
Applying energy conditions to the matter sector yields considerably more
stringent constraints, as demonstrated in (see, Roupas:2020mvs, ).

###  IV.5 Energy conditions

In GR, focusing theorems relate the positivity of the tidal tensor trace to
physical phenomena, ${\mathrm{R}_{\alpha\beta}w^{\alpha}w^{\beta}\geq 0}$ and
${\mathrm{R}_{\alpha\beta}\ell^{\alpha}\ell^{\beta}\geq 0}$, where,
${\mathrm{w}^{\alpha}}$ refers to a vector with a timelike component that can
take any value, while ${\mathrm{\ell}^{\alpha}}$ represents any null vector
that is directed towards the future. These theorems impose four restrictions
on the energy-momentum tensor (EMT) ${\mathbb{T}}^{\alpha\beta}$. These
constraints are commonly referred to as the energy conditions. The energy
conditions can be extended or generalized to encompass modified gravitational
theories beyond GR. They provide limits on the matter sector based on
geometric properties. Specifically, for the theory
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$: The energy
conditions can be expressed using the (EMT) as:
${{\mathbb{T}}{{}^{\alpha}}{{}_{\beta}}}=\mathrm{diag(-\rho
c^{2},p_{r},p_{t},p_{t})}$.

(a) WEC & NEC (radial)

(b) WEC &NEC (tangential)

(c) SEC

(d) DEC (radial)

(e) DEC (tangential)

Figure 5: he figures presented in Fig. 5 for PSR J0740+6620 provide evidence
that the energy conditions (IX) are satisfied.

Constrain (IX): Summary of the necessary energy conditions for a physically
acceptable stellar model:

  * a-

The fulfillment of the weak energy condition (WEC) requires the inequalities
to hold true, i.e., ${\mathrm{\rho}\geq 0}$, ${\mathrm{p}_{r}+\rho c^{2}>0}$,
${\mathrm{\rho}c^{2}+p_{t}>0}$ ,

  * b-

The null energy condition (NEC) needs to adhere to the following inequalities:
${\mathrm{\rho}c^{2}+p_{r}\geq 0}$, ${\mathrm{p}_{t}+\rho c^{2}\geq 0}$ ,

  * c-

The strong energy condition (SEC) must adhere to the following inequalities:
${\mathrm{\rho}c^{2}-p_{r}-2p_{t}\geq 0}$, ${\mathrm{p}_{r}+\rho c^{2}\geq
0}$, ${\mathrm{\rho}c^{2}+p_{t}\geq 0}$ ,

  * d-

The dominant energy conditions (DEC) must meet the following inequalities:
${\mathrm{\rho}\geq 0}$, ${\mathrm{\rho}c^{2}-p_{r}\geq 0}$, and
${\mathrm{\rho}c^{2}-p_{t}\geq 0}$ .

Now let us reformulate the field equations (13) as:

$\displaystyle{\mathrm{\rho}c^{2}}$ $\displaystyle=$
$\displaystyle{\mathrm{\rho}_{GR}c^{2}+\frac{5\beta_{1}}{3(1+3\,\beta_{1})}\left(p_{r}+2\,p_{t}\right)},$
$\displaystyle{\mathrm{p}_{r}}$ $\displaystyle=$
$\displaystyle{\mathrm{p_{r}}_{GR}+\frac{\beta_{1}}{3+11\beta_{1}}\left(3\rho
c^{2}-10p_{t}\right)},$ $\displaystyle{\mathrm{p}_{t}}$ $\displaystyle=$
$\displaystyle{\mathrm{p_{t}}_{GR}+\frac{3\beta_{1}}{3+16\,\beta_{1}}\left(\rho
c^{2}-\frac{5}{3}p_{r}\right)}\,.$ (34)

When the dimensionless parameter $\beta_{1}=0$ we can recover GR. It is time
to analysis constrain (IX) in analytic way.

By the use of constrains (I) and (II), we can demonstrate that: density and
radial/tangential stresses have positive values within the interior of the
star. This confirms the NEC is verified. Some additional implications and
proofs can shown:

$\displaystyle{\mathrm{\rho}c^{2}}+{\mathrm{p}_{r}}=({\mathrm{\rho}_{GR}c^{2}}+{\mathrm{p_{r}}_{GR}})+{\mathrm{\beta}_{1}}\frac{\left(27\,{\beta_{1}}+9\,\right){\mathrm{\rho}c^{2}}+\left(55\,{\beta_{1}}+15\,\right){\mathrm{p}_{r}}+20\,{\mathrm{\beta_{1}}}{\mathrm{p}_{t}}}{9+99\,{\beta_{1}}^{2}+60\,\beta_{1}},$
$\displaystyle{\mathrm{\rho}c^{2}}+{\mathrm{p}_{t}}=({\mathrm{\rho}_{GR}c^{2}}+{\mathrm{p_{t}}_{GR}})-\beta_{1}\,\frac{\left(10{\mathrm{p}_{t}}[3-16\beta_{1}]-125\,{\mathrm{p}_{r}}\beta_{1}+9\,{\mathrm{\rho}}c^{2}[1+3\beta_{1}]\right)}{3\left(1+3\,\beta_{1}\right)\left(3-16\,\beta_{1}\right)},$
$\displaystyle{\mathrm{\rho}}c^{2}-{\mathrm{p}_{r}}-2{\mathrm{p}_{t}}=({\mathrm{\rho}_{GR}}c^{2}-{\mathrm{p_{r}}_{GR}}-2{\mathrm{p_{t}}_{GR}})$
$\displaystyle+\beta_{1}{\frac{\left(162\,{\beta_{1}}^{2}+297{\beta_{1}}+81\right){\mathrm{\rho}c^{2}}-\left(1870{\beta_{1}}^{2}+675{\beta_{1}}+45\right){\mathrm{p}_{r}}+20{\mathrm{p}_{t}}\left(3-16\beta_{1}\right)}{27-1584\,{\beta_{1}}^{3}-663\,{\beta_{1}}^{2}+36\,\beta_{1}}},$
(35)
$\displaystyle{\mathrm{\rho}c^{2}}-{\mathrm{p}_{r}}=({\mathrm{\rho}_{GR}c^{2}}-{\mathrm{p_{r}}_{GR}})+\beta_{1}\,\frac{\left(15\,{\mathrm{p}_{r}}+55{\mathrm{p}_{r}}\beta_{1}+60\,p_{t}+200\,{\mathrm{p}_{t}}\beta_{1}-9\,{\mathrm{\rho}}c^{2}[1+3\beta_{1}]\right)}{3\left(1+3\,\beta_{1}\right)\left(3+11\,\beta_{1}\right)},$
$\displaystyle{\mathrm{\rho}c^{2}}-{\mathrm{p}_{t}}=({\mathrm{\rho}_{GR}c^{2}}-{\mathrm{p_{t}}_{GR}})+\beta_{1}{\frac{\left({\mathrm{p}_{r}}[30-35\beta_{1}]+{\mathrm{p}_{t}}[30-160{\mathrm{p}_{t}}\beta_{1}]-9{\mathrm{\rho}c^{2}}[1+\beta_{1}]\right)}{9-144{\beta_{1}}^{2}-21\,\beta_{1}}},$
$\displaystyle{\mathrm{\rho}c^{2}}-{\mathrm{p}_{r}}-2{\mathrm{p}_{t}}=({\mathrm{\rho}_{GR}c^{2}}-{\mathrm{p_{r}}_{GR}}-2{\mathrm{p_{t}}_{GR}})$
(36)
$\displaystyle+\frac{\beta_{1}\left({\mathrm{p}_{r}}[135+525\beta_{1}+110{\beta_{1}}^{2}]+{\mathrm{p}_{t}}[180-360\beta_{1}-3200\,{\beta_{1}}^{2}]-3\,{\mathrm{\rho}c^{2}}[27+99\beta_{1}+54{\beta_{1}}^{2}]\right)}{3\left(1+3\,\beta_{1}\right)\left(3+11\,\beta_{1}\right)\left(3-16\,\beta_{1}\right)}.\qquad$
(37)

Given that $\beta_{1}>0$ (with a value of $\beta_{1}=0.1$ in this study), and
considering that ${\mathrm{\rho}\geq 0}$, ${\mathrm{p}_{r}\geq 0}$, and
${\mathrm{p}_{t}\geq 0}$, the only thing left to demonstrate is that

${\mathrm{p}_{r}}[30-35\beta_{1}]+{\mathrm{p}_{t}}[30-160p_{t}\beta_{1}]>9\,{\mathrm{\rho}c^{2}}[1+\beta_{1}]>0\,,$
$\left(162\,{\beta_{1}}^{2}+297{\beta_{1}}+81\right){\mathrm{\rho}c^{2}}+20{\mathrm{p}_{2}}{\beta_{1}}\left(3-16\beta_{1}\right)>\left(1870{\beta_{1}}^{2}+675{\beta_{1}}+45\right)p_{r}>0\,,$

and

$10{\mathrm{p}_{t}}[3-16\beta_{1}]+9{\mathrm{\rho}c^{2}}[1+3\beta_{1}]>125\,{\mathrm{p}_{r}}\beta_{1}>0\,,$

to ensure that the energy conditions (I–IV) are satisfied.

The information presented in Fig. 54(a)–4(e) indicates that conditions (IX)
are satisfied for PSR J0740+6620.

It should be noted that the condition of energy density being greater than the
sum of $p_{r}$ and $p_{t}$ pressures is guaranteed in the theory
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ just as GR,
since

${\mathrm{\rho}c^{2}}-\mathrm{p_{r}}-2\mathrm{p_{t}}=\frac{\beta_{1}\left({\mathrm{p}_{r}}[135+525\beta_{1}+110{\beta_{1}}^{2}]+p_{r}[180-360\beta_{1}-3200\,{\beta_{1}}^{2}]-3\,{\mathrm{\rho}c^{2}}[27+99\beta_{1}+54{\beta_{1}}^{2}]\right)}{3\left(1+3\,\beta_{1}\right)\left(3+11\,\beta_{1}\right)\left(3-16\,\beta_{1}\right)}.$

Some authors refer to the requirement that energy density exceeds total
pressure as the SEC (c.f. 1988CQGra…5.1329K, ; Ivanov:2017kyr, ;
2019EPJC…79..853D, ; Roupas:2020mvs, ).

From the above discussion, we understand that the energy conditions primarily
relate to and constrain the matter sector in $f(\mathcal{R},{\mathbb{T}})$
gravitational theories, though they do not necessarily require the same
formulations across all cases. However, certain key conditions are necessary
to ensure the others can also be met where density and pressures remain
positive: The dominant energy condition (DEC) stipulating
${\mathrm{\rho}c^{2}}-{\mathrm{p}_{r}}-2{\mathrm{p}_{t}}\geq 0$ is especially
crucial. This condition must be verified for the plausibility of the others.

###  IV.6  Equilibrium and causality conditions

(a) Radial speed of sound

(b) Tangential speed of sound

(c) difference between radial and tangential speed of sounds

Figure 6: The speeds of sound in the radial and tangential directions given by
Eq. (38) PSR J0740+6620. These figures assure that the conditions of causality
and stability referred to as (XII) and (XIII) have been verified.

The tangential and radial sound speeds are figured as:

${\mathrm{v}_{t}^{2}}=\frac{d{\mathrm{p}}_{t}}{d{\mathrm{\rho}}}=\frac{{\mathrm{p}^{\prime}}_{t}}{{\mathrm{\rho}^{\prime}}}\,,\qquad{\mathrm{v}_{r}^{2}}=\frac{d{\mathrm{p}}_{r}}{d{\mathrm{\rho}}}=\frac{{\mathrm{p}^{\prime}}_{r}}{{\mathrm{\rho}^{\prime}}}\,.$
(38)

Here density and derivative of radial and tangential components of pressure
are given in Eqs. (III.2)–(III.2).

Constrain (X):The star’s structure must be causal, meaning that it must
satisfy the causality condition, which requires that the speeds of sound be
positive and less than the speed of light in the stellar region
($0\leq{\mathrm{v}_{t}/c}\leq 1$, $0\leq{\mathrm{v}_{r}}/c\leq 1$),
furthermore, it can be observed that they exhibit a decline as they approach
the boundary
(${\mathrm{v}^{\prime}_{t}}{{}^{2}}<0$,${\mathrm{v}^{\prime}_{r}}{{}^{2}}<0$).

Although the speed of sound near the core of PSR J0740+6620, satisfies the
conditions of stability and causality, the speed of sound predicted by GR goes
beyond the suggested upper bound for the speed of sound, which is
$v_{r}^{2}=c_{s}^{2}\leq c^{2}/3$. Moreover, different models have exhibited
inconsistency with the conformal upper threshold for the speed of sound, often
through assumptions regarding the EoS (Cherman:2009tw, ; Landry:2020vaw, ) or
by using a non-parametric EoS approximation based on Gaussian processes in
conjunction with X-ray NICER+XMM observations of PSR J0740+6620
(Legred:2021hdx, ). In contrast,as depicted in Figure 65(a), the confining
threshold for the speed of sound, according to the indicated limit PSR
J0740+6620 is not contravened in
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1},{\mathbb{T}}$ model from
the center of the star to its boundary.

Constrain (XI ): The behavior of stellar must have a stable behavior thus it
should verify the equilibrium limit, i.e.,
$-1<({\mathrm{v}_{t}^{2}}-{\mathrm{v}_{r}^{2}})/c^{2}<0$ in the stellar
Herrera:1992lwz .

It is clear that the stability and causality conditions (X) and (XI) are
verified for PSR J0740+6620 as shown in Fig. 6.

###  IV.7 The equilibrium of hydrodynamic

Now we will employ another extra exam to test the equilibrium of the model in
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\alpha_{1}{\mathbb{T}}$. We will
examine the validity of TOV equation (Hansraj:2018jzb, ; Oppenheimer:1939ne, )
assuming that the sphere is in an equilibrium where the forces acting in the
opposite directions on a star are in equilibrium with each other. The TOV
equation can be formalized as follows to reflect the modified form caused by
the newly introduced force, $F_{T}$:

${\mathrm{F}_{a}}+{\mathrm{F}_{g}}+{\mathrm{F}_{h}}+{\mathrm{F}_{T}=0}\,,$
(39)

where the gravitational force is denoted by ${\mathrm{F}_{g}}$ and the
hydrostatic force is denoted by ${\mathrm{F}_{h}}$. In this study we figured
the various forces as follow:

$\displaystyle{\mathrm{F}_{a}}$ $\displaystyle=$
$\displaystyle\frac{2{\mathrm{\Delta}}}{\mathrm{r}},\qquad{\mathrm{F}_{g}}=-\frac{{\mathrm{M}_{g}}}{r}({\mathrm{\rho}c^{2}}+{\mathrm{p}_{r}})e^{{\mathrm{\gamma}/2}},\qquad{\mathrm{F}_{h}}=-{\mathrm{p}^{\prime}_{r}},\qquad{\mathrm{F}_{T}}=\frac{\beta_{1}}{3(1-\beta_{1})}({\mathrm{c}^{2}\rho}^{\prime}-{\mathrm{p}}^{\prime}_{r}-2{\mathrm{p}}^{\prime}_{t})\,.$
(40)

In this context, we define ${\mathrm{\gamma}}=\alpha-\alpha_{1}$, and
${\mathrm{M}_{g}}$ is the mass of the isolated systems, ${\mathrm{V}}$ (where
$t=$ con.) is determined using the mass of Tolman that has the form
(1930PhRv…35..896T, )

$\displaystyle{\mathrm{M}_{g}(r)}$ $\displaystyle=$
$\displaystyle{\int_{\mathrm{V}}}\Big{(}\mathbb{{\mathrm{T}}}{{}^{r}}{{}_{r}}+\mathbb{{\mathrm{T}}}{{}^{\theta}}{{}_{\theta}}+\mathrm{T}{{}^{\phi}}{{}_{\phi}}-\mathrm{T}{{}^{t}}{{}_{t}}\Big{)}\sqrt{-g}\,dV$
(41) $\displaystyle=$
$\displaystyle\frac{(e^{\alpha/2})^{\prime}}{e^{\alpha}}e^{\alpha_{1}/2}r=\frac{\alpha^{\prime}}{2}re^{-\gamma/2}\,,$

with ${\mathrm{F}_{g}}=-\frac{\mu
r}{R^{2}}({\mathrm{\rho}c^{2}}+{\mathrm{p}_{r}})$ being the force of gravity.
Now we ready to put the stability condition the amended TOV equation.

(a) GR

(b) $f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}\,{\mathbb{T}}$

Figure 7: The various forces (40) of the amended TOV equation given by Eq.
(39) of PSR J0740+6620. The figures assure the hydrodynamic stability
constrain (XIII) which is hold true for PSR J0740+6620.

Constrain (XII):The anisotropic star achieves the equilibrium when the total
forces verified the amended TOV (39). We employ Eqs. (LABEL:eq:Feqs2) and
(III.2)–(III.2) to compute the total forces (40) are plotted in Fig. 7 for the
$f(\mathcal{R},{\mathbb{T}})$ and GR theories. The figures indicate that the
gravitational force (which has a negative direction) compensates the other
forces to have a hydrodynamics equilibrium pattern .

It is obvious that the fulfillment of condition of hydrodynamic equilibrium
(XII) for PSR J0740+6620 is verified for as shown in Fig. 7.

###  IV.8 EoS

By the use of the data of PSR J0740+6620, through our derivation, we derived a
satisfactory result for $\beta_{1}=0.1$, which connects the relationship of
matter and geometry in the spacetime. Therefore, we evaluate the central
energy-density to be $\rho_{R}\approx 5.82\times 10^{14}$ g/cm3 but at the
surface it decreases up to $\rho_{\text{core}}\approx 3.28\times 10^{14}$
g/cm3. The value of the central density of PSR J0740+6620 assumes that core
consists of neutrons.

(a) Radial EoS

(b) Linear radial EoS

(c) Tangential EoS

(d) Linear tangential EoS

Figure 8: The pressures-density relationship of PSRJ0740 is in excellent
coherent with the EoS in its linear form with a bag constant. Additionally,
$dp_{1}/d\rho=0.281c^{2}$ and $dp_{2}/d\rho=0.173c^{2}$ clearly validate that
the nuclear matter within the neutron star adheres to the proposed limit on
the sound speed based on the conjecture, i.e., $c_{s}^{2}\leq c^{2}/3$
throughout the star’s interior.

In Figs. 87(a), and 7(c), when $\beta_{1}=0$ and $\beta_{1}\neq 0$, we plot
the evolution of the tangential and radial EoSs, i.e.,
$\mathrm{w_{r}(r)=p_{r}/\rho}$ and $\mathrm{w_{t}(r)=p_{t}/\rho}$ via the
radial distance from the center. The figures show that the EoSs slightly
change from the maximum at the center with a monotonically decreasing behavior
to the surface of the pulsar, where $w_{r}$ approaches to zero at the surface
as constrain (III) stated. Furthermore, we notice that the behavior of the
EoSs is more pronounced in $\mathcal{R}+\beta_{1}{\mathbb{T}}$ gravity
compared to GR. This occurrence can be ascribed to the matter-geometry
interaction, which is responsible for the observed effect. Moreover, we should
note that we did not suppose further forms of the EoS in the model under
construction, 87(a) and 7(c), nevertheless, we note that the EoS $p(\rho)$ in
the radial and tangential directions are fit well with linear models, where
the best fit are given by $p_{r}\approx 0.281\,\rho-8.56$ and $p_{t}\approx
0.173\,\rho-2.9$ for PSR J0740+6620. Surprisingly, the tangents of
$dp_{r}/d\rho=0.281=v_{r}^{2}$ and $dp_{t}/d\rho=0.173=v_{t}^{2}$ are fitted
lines. Also, the current anisotropic NS pattern generates an EoS that not
extremely stiff but is following the EoS predicted from gravitational wave,
where there isn’t any obvious indication of a tidal deformation.

##  V Extra Observational limits

Next, we will assess the model’s applicability to a range of astrophysical
observations by comparing it to data from other pulsars. Furthermore, we
present the relation between mass and radius for various boundary density
choices to demonstrate the model’s reliability in predicting masses.
Additionally, we compare the model’s predicted compactness parameter to the
Buchdahl compactness limit.

###  V.1 Pulsars’ data

Furthermore, besides PSR J0740+6620, the same discussion is applied to more 21
stars, with masses from $0.8M_{\odot}$ to $2.01M_{\odot}$ for heavy pulsars.
Assuming, the form $\mathcal{R}+\beta_{1}{\mathbb{T}}$ with $\beta_{1}=0.1$,
in Table 1, we present masses and radii, along with their model parameters
$\mu$, $\nu$, and $\lambda$. We guarantee that the pulsar masses predicted by
this study l are consistent with the observed values. We present the
determined values of relevant quantities in Table 3.

Table 1: The measured mass-radius values of twenty pulsars along with their respective model parameters ($\beta=0.1$). Star | Ref | mass | radius | evaluated mass | $\mu$ | $\nu$ | $\lambda$
---|---|---|---|---|---|---|---
|  | ($M_{\odot}$) | [km] | ($M_{\odot}$) |  |  |
X-ray binaries with high masses
Her X-1 | Abubekerov:2008inw  | $0.85\pm 0.15$ | $8.1\pm 0.41$ | $0.664$ | $0.184$ | $-0.555$ | $0.371$
4U 1538-52 | Gangopadhyay:2013gha  | $0.87\pm 0.07$ | $7.866\pm 0.21$ | $0.68$ | $0.2$ | $-0.596$ | $0.396$
LMC X-4 | Rawls:2011jw  | $1.04\pm 0.09$ | $8.301\pm 0.2$ | $0.816$ | $0.245$ | $-0.707$ | $0.462$
Cen X-3 | Naik:2011qc  | $1.49\pm 0.49$ | $9.178\pm 0.13$ | $1.182$ | $0.397$ | $-1.05$ | $0.653$
Vela X-1 | Rawls:2011jw  | $1.77\pm 0.08$ | $9.56\pm 0.08$ | $1.42$ | $0.532$ | $-1.32$ | $0.792$
X-ray binaries with low masses exhibiting quiescence and thermonuclear bursts)
EXO 1785-248 | Ozel:2008kb  | $1.3\pm 0.2$ | $8.849\pm 0.4$ | $1.026$ | $0.325$ | $-0.895$ | $0.569$
M13 | Webb:2007tc  | $1.38\pm 0.2$ | $9.95\pm 0.27$ | $1.087$ | $0.292$ | $-0.820$ | $0.527$
X7 | Bogdanov:2016nle  | $1.1\pm 0.35$ | $12$ | $0.857$ | $0.151$ | $-0.467$ | $0.316$
4U 1820-30 | Ozel:2015fia  | $1.46\pm 0.2$ | $11.1\pm 1.8$ | $1.147$ | $0.266$ | $-0.758$ | $0.492$
4U 1608-52 | 1996IAUC.6331….1M  | $1.57\pm 0.3$ | $9.8\pm 1.8$ | $1.244$ | $0.386$ | $-1.027$ | $0.641$
KS 1731-260 | Ozel:2008kb  | $1.61\pm 0.37$ | $10\pm 2.2$ | $1.277$ | $0.39$ | $-1.035$ | $0.646$
EXO 1745-268 | Ozel:2008kb  | $1.65\pm 0.25$ | $10.5\pm 1.8$ | $1.31$ | $0.371$ | $-0.995$ | $0.624$
4U 1724-207 | Ozel:2008kb  | $1.81\pm 0.27$ | $12.2\pm 1.4$ | $1.43$ | $0.331$ | $-0.908$ | $0.577$
SAX J1748.9-2021 | Ozel:2008kb  | $1.81\pm 0.3$ | $11.7\pm 1.7$ | $1.432$ | $0.36$ | $-0.97$ | $0.611$
Millisecond Pulsars
PSR J0030+0451 | Raaijmakers:2019qny  | $1.34\pm 0.16$ | $12.71\pm 1.19$ | $1.0467$ | $0.186$ | $-0.559$ | $0.373$
PSR J0030+0451 | Miller:2019cac  | $1.44\pm 0.16$ | $13.02\pm 1.24$ | $1.126$ | $0.2$ | $-0.596$ | $0.396$
PSR J0437-4715 |  Reardon:2015kba  & Gonzalez-Caniulef:2019wzi  | $1.44\pm 0.07$ | $13.6\pm 0.9$ | $1.125$ | $0.187$ | $-0.562$ | $0.375$
PSR J1614-2230 | NANOGrav:2017wvv  | $1.908\pm 0.016$ | $13\pm 2$ | $1.51$ | $0.325$ | $-0.893$ | $0.569$
PSR J0348+0432 | Antoniadis:2013pzd  | $2.01\pm 0.04$ | $13\pm 2$ | $1.59$ | $0.359$ | $-0.969$ | $0.61$
Gravitational-wave Signals
LIGO-Virgo | LIGOScientific:2020zkf  | $1.4$ | $12.9\pm 0.8$ | $1.094$ | $0.194$ | $-0.58$ | $0.387$
GW170817-1 | LIGOScientific:2018cki  | $1.45\pm 0.09$ | $11.9\pm 1.4$ | $1.137$ | $0.234$ | $-0.68$ | $0.446$
GW170817-2 | LIGOScientific:2018cki  | $1.27\pm 0.09$ | $11.9\pm 1.4$ | $0.99$ | $0.19$ | $-.568$ | $0.379$

In accordance with the criteria listed in the preceding section, we further
assess the model’s stability. Table 3 presents the essential physical
quantities of interest. The results showcase the model’s ability to predict
stable compact star structures that align with observations. It is intriguing
to observe that the speeds of sound obtained from this study for all
categories align with the conformal limit $c^{2}s\leq c^{2}/3$. It is worth
noting that the version of this model predicted $c^{2}_{s}=0.34$ at the core
above the conformal limit, as derived by Roupas:2020mvs , particularly
concerning low-mass of 4U 1608-52 and KS 1731-260. The findings presented in
Table 3 confirm that the present study ensures avoidance of sound speed
violations, aligning with our previous discourse regarding the pulsar’s sound
speed PSR J0740+6620. This underscores the possible influence of the
interaction between geometry and matter in maintaining this upper limit across
the entirety of the compact object.

Table 2: The computed essential physical parameters Star |  $\rho$ [g/cm3] | $v_{r}^{2}/c^{2}$ | $v_{t}^{2}/c^{2}$ | DEC [$dyne/cm^{2}$] | $Z_{R}$
---|---|---|---|---|---
| core | boundary | core | boundary | core | boundary | core | boundary |
X-ray binaries with high-mass companions
Her X-1 | 7.15$\times 10^{14}$ | 5.22$\times 10^{14}$ | 0.221 | 0.201 | 0.119 | 0.09 | 7.53$\times 10^{34}$ | 5.24$\times 10^{34}$ | 0.204
4U 1538-52 | 8.1$\times 10^{14}$ | 5.79$\times 10^{14}$ | 0.226 | 0.204 | 0.124 | 0.094 | 8.62$\times 10^{34}$ | 5.86$\times 10^{34}$ | 0.219
LMC X-4 | 8.54$\times 10^{14}$ | 5.77$\times 10^{14}$ | 0.239 | 0.213 | 0.137 | 0.1 | 9.37$\times 10^{34}$ | 5.98$\times 10^{34}$ | 0.26
Cen X-3 | 10$\times 10^{14}$ | 5.79$\times 10^{14}$ | 0.28 | 0.24 | 0.176 | 0.129 | 1.2$\times 10^{35}$ | 6.44$\times 10^{34}$ | 0.386
Vela X-1 | 11.3$\times 10^{14}$ | 5.86$\times 10^{14}$ | 0.314 | 0.262 | 0.205 | 0.152 | 14.5$\times 10^{34}$ | 6.92$\times 10^{34}$ | 0.486
Binaries with low-mass companions, exhibiting quiescence and thermonuclear
bursts
EXO 1785-248 | 9.32$\times 10^{14}$ | 5.77$\times 10^{14}$ | 0.261 | 0.227 | 0.159 | 0.116 | 10.7$\times 10^{34}$ | 6.22$\times 10^{34}$ | 0.329
M13 | 6.81$\times 10^{14}$ | 4.36$\times 10^{14}$ | 0.252 | 0.222 | 0.15 | 0.111 | 7.69$\times 10^{34}$ | 4.63$\times 10^{34}$ | 0.302
X7 | 2.76$\times 10^{14}$ | 2.11$\times 10^{14}$ | 0.211 | 0.194 | 0.108 | 0.084 | 2.84$\times 10^{34}$ | 2.08$\times 10^{34}$ | 0.171
4U 1820-30 | 5.09$\times 10^{14}$ | 3.36$\times 10^{14}$ | 0.245 | 0.217 | 0.143 | 0.106 | 5.66$\times 10^{34}$ | 3.52$\times 10^{34}$ | 0.279
4U 1608-52 | 8.61$\times 10^{14}$ | 5.03$\times 10^{14}$ | 0.277 | 0.238 | 0.173 | 0.127 | 10.3$\times 10^{34}$ | 5.57$\times 10^{34}$ | 0.378
KS 1731-260 | 8.33$\times 10^{14}$ | 4.85$\times 10^{14}$ | 0.278 | 0.239 | 0.174 | 0.128 | 9.95$\times 10^{34}$ | 5.38$\times 10^{35}$ | 0.381
EXO 1745-268 | 7.29$\times 10^{14}$ | 4.32$\times 10^{14}$ | 0.273 | 0.235 | 0.17 | 0.124 | 8.62$\times 10^{34}$ | 4.75$\times 10^{34}$ | 0.366
4U 1724-207 | 4.97$\times 10^{14}$ | 3.06$\times 10^{14}$ | 0.263 | 0.229 | 0.16 | 0.117 | 5.75$\times 10^{34}$ | 3.31$\times 10^{34}$ | 0.334
SAX J1748.9-2021 | 5.74$\times 10^{14}$ | 3.44$\times 10^{14}$ | 0.27 | 0.233 | 0.167 | 0.122 | 6.74$\times 10^{34}$ | 3.76$\times 10^{34}$ | 0.357
Millisecond Pulsars
PSR J0030+0451 | 2.92$\times 10^{14}$ | 2.13$\times 10^{14}$ | 0.221 | 0.201 | 0.119 | 0.091 | 3.08$\times 10^{34}$ | 2.14$\times 10^{34}$ | 0.205
PSR J0030+0451 | 2.96$\times 10^{14}$ | 2.11$\times 10^{14}$ | 0.262 | 0.204 | 0.124 | 0.094 | 3.15$\times 10^{34}$ | 2.14$\times 10^{34}$ | 0.219
PSR J0437-4715 | 2.56$\times 10^{14}$ | 1.86$\times 10^{14}$ | 0.222 | 0.202 | 0.120 | 0.091 | 2.71$\times 10^{34}$ | 1.88$\times 10^{34}$ | 0.206
PSR J1614-2230 | 4.31$\times 10^{14}$ | 2.67$\times 10^{14}$ | 0.261 | 0.227 | 0.158 | 0.116 | 4.97$\times 10^{34}$ | 2.88$\times 10^{34}$ | 0.329
PSR J0348+0432 | 4.65$\times 10^{14}$ | 2.78$\times 10^{14}$ | 0.27 | 0.233 | 0.167 | 0.122 | 5.46$\times 10^{34}$ | 3.04$\times 10^{34}$ | 0.357
Gravitational-wave Signals
LIGO-Virgo | 2.94$\times 10^{14}$ | 2.12$\times 10^{14}$ | 0.224 | 0.203 | 0.122 | 0.093 | 3.12$\times 10^{34}$ | 2.14$\times 10^{34}$ | 0.213
GW170817-1 | 4$\times 10^{14}$ | 2.74$\times 10^{14}$ | 0.235 | 0.211 | 0.134 | 0.1 | 4.36$\times 10^{34}$ | 2.83$\times 10^{34}$ | 0.25
GW170817-2 | 3.38$\times 10^{14}$ | 2.45$\times 10^{14}$ | 0.222 | 0.202 | 0.121 | 0.092 | 3.58$\times 10^{34}$ | 2.47$\times 10^{34}$ | 0.208
Table 3: The speed of sound the center of 4U 1608-52 and KS 1731-260 aligns
with the proposed limit on the conformal sound speed, $c_{s}^{2}\leq c^{2}/3$,
in contradistinction to the version in GR (Roupas:2020mvs, ). This observation
validates the significance of the coupling of matter-geometry in verified the
upper limit for the conformal sound speed.

##  VI Mass-Radius relation

We derive boundary densities $1.86\times 10^{14}\lesssim\rho_{R}\lesssim
5.86\times 10^{14}$ g/cm3 as shown in Table 3; and therefore, we select three
boundary densities as initial conditions: $\rho_{R}=\rho_{\text{nuc}}=2\times
10^{14}$ g/cm3 at nuclear saturation density, $\rho_{R}=3.5\times 10^{14}$
g/cm3 and $\rho_{R}=5.5\times 10^{14}$ g/cm3 to cover the density of nuclear
solidification. Therefore, with $\beta_{1}=0.1$, we establish a relationship
between ss $C$ and $R$ for any surface constraint on the boundary density,
utilizing Eq. (LABEL:eq:Feqs2), where $\rho(r=R)=\rho_{R}$. In Figs. 9 and 10,
we draw the relation between compactness and radius curves using certain
surface density for the GR case and
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ and get $C=1$
and $0.868$ respectively.

(a) Figure of Compactness and Radius relation of the GR case

(b) Figure of Mass and Radius relation of the GR case

(c) LIGO-Virgo and NICER restrictions of the GR case

Figure 9: 9(a) The compactness-radius curves of the GR case. 9(b) The mass-
radius curves of the GR case. Green circles represent MSP, solid red circles
represent GW signals, and blue circles represent X-ray binaries as given from
Table 1. 9(c) A closer look at some of the most interesting pulsars.

(a) Figure of Compactness and Radius relation

(b) Figure of Mass and Radius relation

(c) LIGO-Virgo and NICER restrictions

Figure 10: 9(a) The relation between compactness and radius. Fig. 9(b), the
relation between mass and radius curves. In Fig. 9(c), we focus on some of the
most intriguing pulsars in the context of GR.

Next, the mass-radius curves for every class are then shown, with its the
relevant observational data using Table 1 as fixed by Figs. 9 8(b) and 10 9(b)
for the GR situation, and for
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ respectively.
With the help of the DEC condition (shown by line), we determine the highest
permitted masses $M_{\text{max}}=5.9M_{\odot}$ and
$M_{\text{max}}=4.86M_{\odot}$ at maximum radii of $R_{\text{max}}=19.6$ km
and $R_{\text{max}}=18.3$ km, respectively, for the surface density
$\rho_{\text{nuc}}=2\times 10^{14}$ g/cm3 for the GR case and
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ respectively
(Roupas:2020mvs, ). With a positive parameter value of $\beta_{1}=0.1$,
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ predicts
roughly the same mass inside a $\sim 1.3$ km smaller size than GR predictions.
Similar to this, for the surface densities $\rho_{R}=5.5\times 10^{14}$ g/cm3
and $\rho_{R}=5.5\times 10^{14}$ g/cm3, respectively, we ascertain the utmost
masses and radii in the subsequent manner:
($M_{\text{max}}=3.57M\odot,R_{\text{max}}=13.7$ km) and
($M_{\text{max}}=2.83M_{\odot},R_{\text{max}}=10.85$ km). These values align
with the nuclear solidification density, demonstrating compatibility. It is
noteworthy that the present study can yield neutron stars (NS) in the mass
range of $2.83$ to $4.86M_{\odot}$. In this particular scenario, the model
predicts that the NS possesses a boundary density consistent at the density of
nuclear saturation and follows a linear equation of state (EoS).

In Fig. 10 8(c), we focus on the investigation of the most intriguing pulsars.
The stares PSR J0740+6620 and PSR J0030+0451 are found to fix the
$\rho_{\text{nuc}}$-curve in a precise manner.

##  VII Conclusion

According to Harko:2011kv , we looked into the effects of the non-minimal
connection between matter and geometry, for
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ gravitational
theory, on the relationship between mass and radius for compact objects. In
presence of curved spacetime, the theory pre supposes a local violation of the
law of conservation of energy; otherwise, it degenerates into GR. Considering
the substantial curvature of spacetime, it becomes crucial to thoroughly
investigate this effect on the structures of compact objects such as neutron
stars.The accurate measurements of the radius and mass of PSR J0740+6620,
obtained through NICER observations, have been instrumental in determining a
robust estimate for the parameter $\beta_{1}$.

Our demonstration illustrated that $\Delta$ in
$\mathcal{R}+\beta_{1}{\mathbb{T}}$ gravity is equivalent to that in GR . The
equivalence between the physical quantities simplifies the detection of
deviations from general relativity (GR) with regards to matter-geometry
coupling. To represent all the relevant physical quantities, we utilized the
KB ansatz and incorporated the parameters $\beta_{1}$ $C$. Due to the accurate
observational constraints on radius and mass obtained for PSR J0740+6620, we
determined the parameter $\beta_{1}$ with a fixed positive value of
$\beta=0.1$. In contrast to GR, this situation implies a larger size for a
specific mass in the framework of $\mathcal{R}+\beta_{1}{\mathbb{T}}$. We have
demonstrated that, for a specific mass, the additional force from the
gravitational theory
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ assists in
partially offsetting the gravitational force, allowing for the existence of
larger compact stars than would be possible otherwise. Moreover, we have shown
that the Schwarzschild radius approaches the limit of the compactness i.e.,
$C=1$ for the isotropic case, slightly surpassing the limit of $C=8/9$ for the
GR case which provided by Buchdahl, as depicted in Fig. 98(a). The highest
allowed compactness for
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ is found to be
$C_{\text{max}}=0.868$, which is still lower than the Buchdahl limit, i.e.,
$C=8/9$.

Remarkably, even without imposing any specific EoS, the model conforms a
linear EoS characterized by a bag constant. What’s even more intriguing is
that we have discovered $c_{s}$ of NS at the center to be
$v^{2}_{r}=0.332c^{2}$ (radial direction) and $v^{2}_{t}=0.222c^{2}$
(tangential direction). Unlike GR, these values comply while adhering to the
suggested confining threshold for the sound speed, guaranteeing $c_{s}^{2}\leq
c^{2}/3$ within the core and interior of the neutron star

When considering a surface density corresponding to the saturation nuclear
energy $\rho_{\text{nuc}}=2\times 10^{14}$ g/cm3, the model allows mass of
$M=4.86M_{\odot}$ with radius $R=18.3$ km. This results in a smaller
compactness value compared to the GR prediction, as evident from Figs. 98(a)
and 109(a). Consequently, this finding leaves the possibility open for the
companion of $GW190814$ to be an anisotropic NS with necessitating any
assumptions about exotic matter sources

The current investigations demonstrate that the matter-geometry coupling in
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ theory enables
the speed of sound to match the conformal upper bound, a scenario not observed
in GR. This crucial finding, along with similar studies, confirms the
disparity between
$f(\mathcal{R},{\mathbb{T}})=\mathcal{R}+\beta_{1}{\mathbb{T}}$ and GR.

In summary, we have successfully derived a realistic stellar model within the
framework of $f(R,T)$ theory. Now, the question arises whether the same
procedure can be applied to $f({\mathcal{T}},T)$ theory ElHanafy:2014tzj ,
where ${\mathcal{T}}$ represents the scalar torsion. Can we utilize a similar
approach in $f({\mathcal{T}},T)$ theory, incorporating the scalar torsion
${\mathcal{T}}$, to derive a realistic stellar model?

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