# Impact of Rastall gravity on mass, radius and sound speed of the pulsar PSR
J0740+6620

Waleed El Hanafy Centre for Theoretical Physics, The British University in
Egypt, P.O. Box 43, El Sherouk City, Cairo 11837, Egypt

(Accepted September 19, 2022)

###### Abstract

Millisecond pulsars are perfect laboratories to test possible matter-geometry
coupling and its physical implications in light of recent Neutron Star
Interior Composition Explorer (NICER) observations. We apply Rastall field
equations of gravity, where matter and geometry are non-minimally coupled, to
Krori-Barua interior spacetime whereas the matter source is assumed to be
anisotropic fluid. We show that all physical quantities inside the star can be
expressed in terms of Rastall, $\epsilon$, and compactness, $C=2GM/Rc^{2}$,
parameters. Using NICER and X-ray Multi-Mirror (XMM-Newton) X-ray
observational constraints on the mass and radius of the pulsar PSR J0740+6620
we determine Rastall parameter to be at most $\epsilon=0.041$ in the positive
range. The obtained solution provides a stable compact object, in addition the
squared sound speed does not violate the conjectured sound speed
$c_{s}^{2}\leq c^{2}/3$ unlike the general relativistic treatment. We note
that there is no equations of state are assumed, the model however fits well
with linear patterns with bag constants. In general, for $\epsilon>0$, the
theory predicts a slightly larger size star in comparison to general
relativity for the same mass. This has been explained as an additional force,
due to matter geometry coupling, in the hydrodynamic equilibrium equation
contributes to partially diminish the gravitational force effect.
Consequently, we calculate the maximal compactness as allowed by the strong
energy condition to be $C=0.735$ which is $\sim 2\%$ higher than general
relativity prediction. Moreover, for surface density at saturation nuclear
density $\rho_{\text{nuc}}=2.7\times 10^{14}$ g/cm3 we estimate the maximum
mass $M=4M_{\odot}$ at radius $R=16$ km.

Massive stars (732) — Millisecond pulsars (1062) — Neutron star cores (1107) —
Non-standard theories of gravity (1118) — Stellar structures (1631)

##  1 Introduction

Neutron stars (NSs) provide a unique laboratory to test matter at extremely
high density conditions, a few times the nuclear saturation density
$\rho_{\text{nuc}}\approx 2.7\times 10^{14}$ g/cm3, which are terrestrially
inaccessible. The nature of their cores are poorly understood due to lacking
of knowledge of the equation of state (EoS) at these densities (Özel & Freire,
2016). Many EoSs are proposed to describe the neutron star cores by including
baryonic or exotic matter or possible combinations. One way to test these EoSs
is to simultaneously determine the NS mass and radius. This represents the
primary aim of several astrophysical observations nowadays. The pulsar timing
measurement– depends on the determination of radio pulse times of arrival
(TOAs) in comparison with a stable reference clock– of pulses is being used
for many decades to determine the pulsars’ masses. The pulse period for
millisecond pulsars (MSP) ranges from 1.39 to 30 ms, slow down rate $\leq
10^{-19}$ s/s with a characteristic age $10^{9}$ yrs. Those are very useful to
test relativistic theories (c.f., Stairs, 2003; Reardon et al., 2016).

Recently X-ray and gravitational wave signal observations open new windows to
determine NSs masses and radii. NICER mission is devoted to measure mass and
radius (and consequently EoS) of MSPs by analyzing the gravitational light
bending and the X–ray light curves produced by rotating hot spots on their
surfaces (Bogdanov et al., 2019a, b). Some MSP with masses $\sim 1.5M_{\odot}$
have been observed and analyzed by NICER, e.g. PSR J0030+0451 with mass
$M=1.44^{+0.15}_{-0.14}M_{\odot}$ (Miller et al., 2019) and independently by
(Raaijmakers et al., 2019) where $M=1.34^{+0.15}_{-.16}M_{\odot}$, while
others, e.g. PSR J0437-4715 with mass $M=1.44\pm 0.07M_{\odot}$ (Reardon et
al., 2016), are in progress (Balsamo et al., 2014). Notably, for the isolated
pulsar PSR J0030+0451, two independent analyses of NICER data have been done
to measure its radius, $R=13.02_{-1.06}^{+1.24}$ km (Miller et al., 2019) and
$R=12.71^{+1.14}_{-1.19}$ km (Raaijmakers et al., 2019). Although these
studies assume different patterns of the hot spots, both agree on the measured
value of the pulsar radius. Additionally, some MSP with mass $\sim
2M_{\odot}$, e.g. PSR J1614-2230 with mass $M=1.908\pm 0.016M_{\odot}$
(Demorest et al., 2010; Fonseca et al., 2016; Arzoumanian et al., 2018a), PSR
J0348+0432 with mass $M=2.01\pm 0.04M_{\odot}$ (Antoniadis et al., 2013) and
PSR J0740+6620 with mass $M=2.08\pm 0.07M_{\odot}$ using the relativistic
Shapiro time delay (Cromartie et al., 2019; Fonseca et al., 2021), are of a
great interest since they approach the upper limit of a NS or the lower limit
of a Black Hole (BH). In particular, the pulsar PSR J0740+6020 is in a binary
system which provides an independent measurements of mass and inclination. So
its mass can be determined with higher precision, however its low NICER count
rate relative to PSR J0348+0432 represents a challenge. Notably both NICER and
XMM-Newton datasets have been used to determine the radius of the pulsar PSR
J0740+6020 by Miller et al. (2021), $R=13.7_{-1.5}^{+2.6}$ km, and
independently by Riley et al. (2021), $R=12.39_{-0.98}^{+1.30}$ km (68%
credible level), while from NICER+XMM based on Gaussian process applying a
non-parameteric equation of state approach, the mass and radius are given as
$M=2.07\pm 0.11M_{\odot}$ and $R=12.34^{+1.89}_{-1.67}$ km (Legred et al.,
2021) which is in agreement with Landry et al. (2020) at 68% credible level.

Gravitational wave signals provide another window to test matter at high
density. This research is led by Laser Interferometer Gravitational-Wave
Observatory (LIGO) and Virgo collaboration (Abbott et al., 2016). For example,
GW190814, a binary compact system merger with a companion of mass $\sim
2.6M_{\odot}$ (Abbott et al., 2020b), provides the first observed compact
object in the hypothesized lower mass gap 2.5–5 $M_{\odot}$. This may point
out some kind of modified gravity as suggested by Moffat (2020). Also,
GW170817 (Abbott et al., 2017, 2018) and GW190425 (Abbott et al., 2020a)
signals indicate no tidal deformation in the observed patterns of their
gravitational waves, which in turn inform that the EoS is not too stiff. This
is unlike what is usually assumed for high mass compact objects.

Remarkably, NICER observations of PSR J0030+0451 and PSR J0740+6020 provide an
evidence against more squeezable models. The latter has much more mass than
the former while both are having almost same size. Then it is reasonable to
assume some mechanisms to justify the non-squeezability of a NS when it gains
more mass. On the other hand, existence of high-mass pulsars $\sim 2M_{\odot}$
such as PSR J0740+6020 is known to favor violation of the upper sound speed
conformal limit $c_{s}^{2}=c^{2}/3$ which represents another challenge for
theoretical models even at low density cases as shown by Bedaque & Steiner
(2015) (see also Cherman et al., 2009; Landry et al., 2020). For the pulsar
PSR J0740+6020, in particular, Legred et al. (2021) in their study concluded
that the conformal sound speed is strongly violated at the NS core whereas
$c_{s}^{2}=0.75c^{2}$ with density $\sim 3.60\,\rho_{\text{nuc}}$.

When a NS gains more mass, one would expect stronger gravity and then the NS
collapses to smaller size (consequently higher density). However, the NICER
observations of the pulsars PSR J0030+0451 and PSR J0740+6020 do not agree
with this mechanism. We argue that at high mass NS, one expect the presence of
high density to form anisotropy (Herrera & Santos, 1997), whereas radial,
$p_{r}$, and tangential, $p_{t}$, pressures are different. The induced
anisotropic force becomes repulsive when $p_{t}>p_{r}$, subsequently acts
against the attractive gravitational force as can be seen from
Tolman–Oppenheimer–Volkoff (TOV) equation (Bowers & Liang, 1974). This in
return supports the NS to hold its size. On the other hand, we suggest that a
non-minimal coupling between matter and geometry inside compact objects (high
curvatured spacetime) to play a crucial role to hold the conjectured sound
speed limit from the NS core to its surface unlike the GR case.

In fact, the null covariant divergence of the energy-momentum tensor is one of
the fundamental assumptions of the General Relativity (GR). On the contrary,
Rastall attempted to modify GR by dropping this assumption assuming that the
covariant derivative of the energy-momentum tensor to be proportional to Ricci
scalar derivative, which directly reflects the non-minimal coupling between
matter and geometry in curved spacetimes Rastall (1972, 1976). In flat
spacetimes the interaction term vanishes and GR would be recovered indeed.
Rastall theory (RT) could be strongly tested within stellar structure models
in the light of the recently unprecedented improvements of mass–radius
observations of astrophysical objects as discussed above.

We arrange the present study as follows: In Sec. 2, we review the modification
of the general relativity as proposed by Rastall. In Sec. 3, we setup the main
assumptions of the present study. In Sec. 4, we use the X-ray NICER and XMM
observations of the PSR J0740+6620 to constrain the model parameters. Also, we
discuss the physical features of the pulsar and its stability as obtained by
the present model. In Sec. 5, we confront the model to more pulsars data. In
Sec. 6, we estimate the maximum compactness as allowed by physical conditions.
Consequently, we plot the mass-radius curves for different choices of surface
densities showing the maximal mass in each case for a stable configuration. In
Sec. 7, we conclude the work.

##  2 Rastall Gravity

The contracted Bianchi identity, in Riemannian geometry, implies a covariant
divergence free Einstein tensor

$\nabla{{}_{\alpha}}\mathfrak{G}_{\alpha\beta}=\nabla{{}_{\alpha}}(\mathfrak{R}_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\mathfrak{R})\equiv
0,$ (1)

where $\mathfrak{G}_{\alpha\beta}$ denotes Einstein tensor,
$\mathfrak{R}_{\alpha\beta}$ denotes Ricci tensor and
$\mathfrak{R}=g^{\alpha\beta}\mathfrak{R}_{\alpha\beta}$ denotes Ricci
invariant. On the other hand, applying the minimal coupling procedure one
requires a divergence free energy momentum

$\nabla_{\alpha}\mathfrak{T}{{}^{\alpha}}{{}_{\beta}}=0.$ (2)

Therefore, the GR field equations, as formulated by Einstein, are given

$\mathfrak{G}_{\alpha\beta}=\kappa_{E}\mathfrak{T}_{\alpha\beta},$ (3)

where the Einstein coupling constant $\kappa_{E}=8\pi G/c^{4}$ with $G$ is the
Newtonian gravitational constant and $c$ is the speed of light.

In Rastall’s gravity the minimal coupling assumption (2) has been replaced by
(Rastall, 1972, 1976)

$\nabla_{\alpha}\mathfrak{T}{{}^{\alpha}}{{}_{\beta}}=\frac{\epsilon}{\kappa}\,\partial_{\beta}\mathfrak{R}.$
(4)

Clearly the non-divergence free energy momentum assumption–proportional to
Ricci scalar derivative– reflects a non-minimal coupling between matter and
geometry, where $\epsilon/\kappa$ is a constant of the proportionality. The
local violation of the conservation law, as assumed by Rastall, is a
manifestation of an interaction between matter and curved spacetime, which no
longer exists when spacetime is being flat. This, accordingly, led Rastall to
obtain a some kind of generalized field equations

$\mathfrak{G}_{\alpha\beta}=\mathfrak{R}_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\mathfrak{R}=\kappa(\mathfrak{T}_{\alpha\beta}-\frac{\epsilon}{\kappa}g_{\alpha\beta}\mathfrak{R}).$
(5)

For vacuum solutions, obviously, the GR theory is restored. By contracting Eq.
(5) we write

$\mathfrak{R}=-\frac{\kappa}{1-4\epsilon}\mathfrak{T},\quad(\epsilon\neq\frac{1}{4})$
(6)

where $\mathfrak{T}=g^{\alpha\beta}\mathfrak{T}_{\alpha\beta}$ is the trace of
the energy-momentum tensor. Alternatively, the field equations (5) can be
written as

$\mathfrak{G}_{\alpha\beta}=\kappa\widetilde{\mathfrak{T}}_{\alpha\beta},$ (7)

where the effective energy-momentum tensor is given by

$\widetilde{\mathfrak{T}}_{\alpha\beta}=\mathfrak{T}_{\alpha\beta}+\frac{\epsilon}{1-4\epsilon}g_{\alpha\beta}\mathfrak{T}.\qquad(\epsilon\neq\frac{1}{4})$
(8)

The Newtonian limit of Rastall gravity requires a rescaled effective
gravitational constant (c.f. Rastall, 1972; Moradpour et al., 2017a, b)

$\kappa=\eta\,\kappa_{E}\text{ and
}\eta=\frac{1-4\epsilon}{1-6\epsilon},\qquad(\epsilon\neq\frac{1}{6})$ (9)

where $\kappa$ acquires the dimension of $\kappa_{E}$ and $\kappa=\kappa_{E}$
only if $\epsilon=0$ (i.e. $\eta=1$). In practice, the deviations from GR are
given in terms of the dimensionless Rastall parameter $\epsilon$ via the
effective energy-momentum tensor $\widetilde{\mathfrak{T}}_{\alpha\beta}$ as
well as the effective gravitational constant $\kappa$. For $\epsilon=0$ the
conservation law is restored and the GR version of gravity is recovered. In
other words, RT generalizes GR by dropping the divergence free energy momentum
assumption and replaces it by assuming that the conservation law can be
violated in curved spacetime due to existence of a non-minimal coupling
between matter and geometry. It has been argued that RT is a reproduction of
GR and both theories are completely equivalent as noted by Visser (2018). On
the contrary, Darabi et al. (2018) have investigated that claim showing that
the matter-geometry coupling term in RT has been misinterpreted which led
finally to a wrong conclusion. A counter example has been given by applying
Visser’s argument to $f(R)$ gravity, this led to an equivalence between $f(R)$
and GR as well which is not true. In this sense, stellar structure models are
important applications to examine RT whereas the presence of matter
(consequently curved spacetime) plays a crucial role to determine matter-
geometry coupling, if any.

##  3 The Model

We consider a static spherically symmetric line element for a 4-dimensional
spacetime which can be expressed in a spherical polar coordinates
($t,r,\theta,\phi$)

$ds^{2}=-e^{\alpha(r)}c^{2}dt^{2}+e^{\beta(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(10)

where $\alpha(r)$ and $\beta(r)$ be the metric potentials. We further assume
the energy-momentum tensor for a anisotropic fluid with spherical symmetry,
i.e.

$\mathfrak{T}{{}^{\alpha}}{{}_{\beta}}=(p_{t}+\rho
c^{2})U{{}^{\alpha}}U{{}_{\beta}}+p_{t}\delta^{\alpha}_{\beta}+(p_{r}-p_{t})V{{}^{\alpha}}V{{}_{\beta}},$
(11)

where $\rho=\rho(r)$ is the fluid energy density, $p_{r}=p_{r}(r)$ its radial
pressure (in the direction the time-like four-velocity $U_{\alpha}$),
$p_{t}=p_{t}(r)$ its tangential pressure (perpendicular to $U_{\alpha}$) and
$V{{}^{\alpha}}$ is the unit space-like vector in the radial direction. Then,
the energy-momentum tensor takes the diagonal form
$\mathfrak{T}{{}^{\alpha}}{{}_{\beta}}=diag(-\rho
c^{2},\,p_{r},\,p_{t},\,p_{t})$. Applying Rastall’s field equations (7) to the
spacetime (10) where the matter sector is as given by (11), we obtain the
following

$\displaystyle\kappa\rho c^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{-\beta}}{r^{2}}(e^{\beta}+\beta^{\prime}r-1)-{\epsilon\kappa\over
1-4\epsilon}(\rho c^{2}-p_{r}-2p_{t})\,,$ $\displaystyle\kappa p_{r}$
$\displaystyle=$
$\displaystyle\frac{e^{-\beta}}{r^{2}}(1-e^{\beta}+\alpha^{\prime}r)+{\epsilon\kappa\over
1-4\epsilon}(\rho c^{2}-p_{r}-2p_{t})\,,$ $\displaystyle\kappa p_{t}$
$\displaystyle=$
$\displaystyle\frac{e^{-\beta}}{4r}\left[(2\alpha^{\prime\prime}-\alpha^{\prime}\beta^{\prime}+\alpha^{\prime
2})r+2(\alpha^{\prime}-\beta^{\prime})\right]$ (12)
$\displaystyle+{\epsilon\kappa\over 1-4\epsilon}(\rho c^{2}-p_{r}-2p_{t})\,,$

where the prime denotes the derivative with respect to the radial coordinate.
Consequently, we obtain the anisotropy parameter, $\Delta(r)=p_{t}-p_{r}$, as
follows

$\Delta(r)=\frac{e^{-\beta}}{4\kappa
r^{2}}\left[(2\alpha^{\prime\prime}-\alpha^{\prime}\beta^{\prime}+\alpha^{\prime
2})r^{2}-2(\alpha^{\prime}+\beta^{\prime})r+4(e^{\beta}-1)\right].$ (13)

Remarkably, the matter-geometry coupling due to the trace $\mathfrak{T}$ does
not contribute to the anisotropy once the spherically symmetric spacetime
configuration is assumed (Nashed & El Hanafy, 2022). However, for
$\epsilon\neq 0$ ($\kappa\neq\kappa_{E}$) slight change in anisotropy is
expected. Therefore, deviations from GR due to matter-geometry coupling cannot
be spoiled with various anisotropic effects. For the $\epsilon=0$ case, the
differential equations (12) coincide with Einstein field equations of an
interior spherically symmetrical spacetime (c.f., Roupas & Nashed, 2020).

###  3.1 Krori-Barua ansatz

We introduce Krori & Barua (1975) ansatz (hearafter KB),

$\alpha(r)=a_{0}x^{2}+a_{1},\,\beta(r)=a_{2}x^{2},$ (14)

where the dimensionless radius $0\leq x=r/R\leq 1$ with $R$ being the radius
of the star. Additionally, the set of constants {$a_{0},a_{1},a_{2}$} are
dimensionless to be determined by matching conditions on the boundary surface
of the star. We further define the dimensionless variables

$\bar{\rho}(r)=\frac{\rho(r)}{\rho_{\star}},\,\bar{p}_{r}(r)=\frac{p_{r}(r)}{\rho_{\star}c^{2}},\,\bar{p}_{t}(r)=\frac{p_{t}(r)}{\rho_{\star}c^{2}},\,\bar{\Delta}(r)=\frac{\Delta(r)}{\rho_{\star}c^{2}},$
(15)

where $\rho_{\star}$ denotes a characteristic density

$\rho_{\star}=\frac{1}{\kappa_{E}c^{2}R^{2}}.$ (16)

Thus the field equations (12) read

$\displaystyle\eta\bar{\rho}$ $\displaystyle=$
$\displaystyle\frac{e^{-a_{2}x^{2}}}{x^{2}}(e^{a_{2}x^{2}}-1+2a_{2}x^{2})$
$\displaystyle+$
$\displaystyle\frac{2\epsilon}{x^{2}}\left[\left(a_{0}(a_{0}-a_{2})x^{4}-(2a_{2}-3a_{0})x^{2}+1\right)e^{-a_{2}x^{2}}-1\right],$
$\displaystyle\eta\bar{p}_{r}$ $\displaystyle=$
$\displaystyle\frac{e^{-a_{2}x^{2}}}{x^{2}}(1-e^{a_{2}x^{2}}+2a_{0}x^{2})$
$\displaystyle-$
$\displaystyle\frac{2\epsilon}{x^{2}}\left[\left(a_{0}(a_{0}-a_{2})x^{4}-(2a_{2}-3a_{0})x^{2}+1\right)e^{-a_{2}x^{2}}-1\right],$
$\displaystyle\eta\bar{p}_{t}$ $\displaystyle=$ $\displaystyle
e^{-a_{2}x^{2}}(2a_{0}-a_{2}+a_{0}(a_{0}-a_{2})x^{2})$ $\displaystyle-$
$\displaystyle\frac{2\epsilon}{x^{2}}\left[\left(a_{0}(a_{0}-a_{2})x^{4}-(2a_{2}-3a_{0})x^{2}+1\right)e^{-a_{2}x^{2}}-1\right],$

and the anisotropy factor (13) becomes

$\eta\bar{\Delta}=\frac{e^{-a_{2}x^{2}}}{x^{2}}\left[e^{a_{2}x^{2}}-1+a_{0}(a_{0}-a_{2})x^{4}-a_{2}x^{2}\right].$
(18)

The mass content within a radius $r$ is given by the mass function

$\mathfrak{M}(r)=4\pi\int_{0}^{r}\rho(\bar{r})\,\bar{r}{{}^{2}}dr.$ (19)

Substitute the density (LABEL:eq:Feqs2) we obtain

$\mathfrak{M}(x)=\frac{M}{C}e^{-a_{2}x^{2}}\left[x(e^{a_{2}x^{2}}-1)+\epsilon\,\zeta(x)\right].$
(20)

where the compactness parameter $C$ and the function $\zeta(x)$ are

$\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{2GM}{c^{2}R},$
$\displaystyle\zeta(x)$ $\displaystyle=$ $\displaystyle
a_{2}^{-\frac{9}{2}}\left\\{\left(2xa_{2}^{\frac{9}{2}}-\frac{3}{4}a_{0}\sqrt{\pi}a_{2}^{2}(a_{2}+a_{0})\text{erf}(\sqrt{a_{2}}x)\right)e^{a_{2}x^{2}}\right.$
$\displaystyle+$
$\displaystyle\left.x\left[\left(\frac{3}{2}a_{0}a_{2}^{\frac{5}{2}}+a_{2}^{\frac{7}{2}}(a_{0}x^{2}+\frac{3}{2})\right)a_{0}-(a_{0}x^{2}+2)a_{2}^{\frac{9}{2}}\right]\right\\}.$

We note that the mass function (20) reduces to the GR version where
$\epsilon=0$ (Roupas & Nashed, 2020).

###  3.2 Matching conditions

Since vacuum solutions of both GR and RT are equivalent, the exterior solution
is nothing rather Schwarzschild’s one,

$ds^{2}=-\left(1-\frac{2GM}{c^{2}r}\right)c^{2}dt^{2}+\frac{dr^{2}}{\left(1-\frac{2GM}{c^{2}r}\right)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (22)

Recalling the interior spacetime (10), we thus take

$\alpha(x=1)=\ln(1-C),\,\beta(x=1)=-\ln(1-C).$ (23)

In addition, we make the radial pressure to vanish on the boundary, i.e.

$\bar{p}_{r}(x=1)=0.$ (24)

Using KB ansatz (14) and the radial pressure (LABEL:eq:Feqs2) along with the
above the boundary conditions, we derive

$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}-\frac{1}{2}\ln(1-C)-\frac{\sqrt{\chi}+(1-C)}{2\epsilon(1-C)},$
$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\frac{3}{2}+\frac{3}{2}\ln(1-C)+\frac{\sqrt{\chi}+(1-C)}{2\epsilon(1-C)},$
$\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle-\ln(1-C),$ (25)

where

$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\epsilon^{2}(1-C)^{2}\ln(1-C)^{2}-2\epsilon(1+\epsilon)(1-C)^{2}\ln(1-C)$
(26) $\displaystyle-$
$\displaystyle(1-C)[(5C-9)\epsilon^{2}-2(2C-3)\epsilon-(1-C)].$

Taking the limit $\epsilon\to 0$, the set of constants (25) reduces to the GR
version (Roupas & Nashed, 2020)

$a_{0}=\frac{C}{2(1-C)},\,a_{1}=\ln(1-C)-\frac{C}{2(1-C)},\,a_{2}=-\ln(1-C).$
(27)

Interestingly all physical quantities of KB spacetime within a given star,
$0\leq x\leq 1$, can be written as dimensionless forms in terms of Rastall and
the compactness parameters, i.e. $\bar{\rho}(\epsilon,C)$,
$\bar{p}_{r}(\epsilon,C)$ and $\bar{p}_{t}(\epsilon,C)$. In practice the
compactness parameter is determined by observational data which sets a direct
constraint to estimate the non-minimal coupling between matter and geometry as
assumed in Rastall gravity. On the other hand it gives a chance to set an
upper bound of the allowed compactness for anisotropic neutron star and
consequently the maximum mass. This will be discussed in details in Sec. 6.

###  3.3 Radial gradients

We derive the radial gradients of the fluid density, pressures as obtained by
Eqs. (LABEL:eq:Feqs2).

$\displaystyle\bar{\rho}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2e^{-a_{2}x^{2}}}{x^{3}R}\left\\{1-e^{a_{2}x^{2}}-2a_{2}^{2}x^{4}+a_{2}x^{2}-2\epsilon\left[1-e^{a_{2}x^{2}}\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.a_{0}a_{2}(a_{2}-a_{0})x^{6}-[2(a_{2}-a_{0})^{2}-a_{0}^{2}]x^{4}+a_{2}x^{2}\right]\right\\},$
$\displaystyle\bar{p}^{\prime}_{r}$ $\displaystyle=$
$\displaystyle\frac{-2e^{-a_{2}x^{2}}}{x^{3}R}\left\\{1-e^{a_{2}x^{2}}+2a_{0}a_{2}x^{4}+a_{2}x^{2}-2\epsilon\left[1-e^{a_{2}x^{2}}\right.\right.$
(29) $\displaystyle-$
$\displaystyle\left.\left.a_{0}a_{2}(a_{2}-a_{0})x^{6}-[2(a_{2}-a_{0})^{2}-a_{0}^{2}]x^{4}+a_{2}x^{2}\right]\right\\},$
$\displaystyle\bar{p}^{\prime}_{t}=\frac{2e^{-a_{2}x^{2}}}{x^{3}R}\left\\{a_{0}a_{2}(a_{2}-a_{0})x^{6}+[a_{2}(a_{2}-3a_{0})+a_{0}^{2}]x^{4}+2\epsilon\,\times\right.$
$\displaystyle\left.\left[1-e^{a_{2}x^{2}}-a_{0}a_{2}(a_{2}-a_{0})x^{6}-[2(a_{2}-a_{0})^{2}-a_{0}^{2}]x^{4}+a_{2}x^{2}\right]\right\\},$

where ${}^{\prime}=R^{-1}d/dx$. Those gradients are of a great importance
since they determine the physical stability of the compact object as will be
discussed in the following section.

##  4 Observational Constraints and Stability Conditions

In this section we use the precise X-ray mass and radius constraints on the
pulsar PSR J0740+6620 from NICER+XMM based on Gaussian process applying a non-
parameteric equation of state approach, the mass and radius are given as
$M=2.07\pm 0.11M_{\odot}$ and $R=12.34^{+1.89}_{-1.67}$ km Legred et al.
(2021) where the solar mass $M_{\odot}=1.9891\times 10^{30}$ kg. For a stellar
model to be physically well behaved, it needs to satisfy the conditions as
labeled from (i) to (xii) as follows:

###  4.1 Zeldovich condition and Rastall parameter

Condition (i): According to Zeldovich & Novikov (1971) the central radial
pressure must be less than or equal to the central density, then we write

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

We obtain the density, the radial and the tangential pressure at the center

$\displaystyle\bar{\rho}(x=0)$ $\displaystyle=$ $\displaystyle
6(a_{0}-a_{2})\epsilon+3a_{2}$ $\displaystyle\bar{p}_{r}(x=0)$
$\displaystyle=$ $\displaystyle-6(a_{0}-a_{2})\epsilon-
a_{2}+2a_{0}=\bar{p}_{t}(x=0).\quad$ (32)

For the pulsar PSR J0740+6620 we evaluate its compactness parameter
$C=0.510\pm 0.0999$. Then, by applying Zeldovich condition (31) we choose a
reasonable valid interval of Rastall parameter $0\leq\epsilon\leq 0.278\pm
0.0067$ where $\epsilon$ is expected to be close to the GR version
($\epsilon=0$).

###  4.2 Mass and radius observational constraints from PSR J0740+6620

Figure 1: The plot of the mass function (20) of the pulsar PSR J0740+6620. The
shaded area represents the feasible region as imposed by NICER+XMM
observational data ($M=2.07\pm 0.11M_{\odot}$ and $R=12.34^{+1.89}_{-1.67}$
km) Legred et al. (2021). For Rastall case we use {$\epsilon=0.041$,
$\kappa=2.302\times 10^{-43}\,N^{-1}$, $C=0.491$, $a_{0}=0.438$,
$a_{1}=-1.114$, $a_{2}=0.676$}. For GR case ($\epsilon=0$) the corresponding
values are {$\kappa=\kappa_{E}=2.077\times 10^{-43}\,N^{-1}$, $a_{0}=0.491$,
$a_{1}=-1.176$, $a_{2}=0.684$}.

Recalling the X-ray mass and radius constraints on the pulsar PSR J0740+6620
from NICER+XMM, the mass function (20) estimates a total mass
$M=1.96M_{\odot}$ at a radius of $R=13.04$ km with compactness $C=0.491$ in
agreement with the observed value, ($M=2.07\pm 0.11M_{\odot}$ and
$R=12.34^{+1.89}_{-1.67}$ km) Legred et al. (2021), for a choice of Rastall
parameter $\epsilon=0.041$. This determines the coupling constant, (9), and
the set of constants, Eqs. (25), {$\kappa=2.302\times 10^{-43}\,N^{-1}$,
$a_{0}=0.438$, $a_{1}=-1.114$, $a_{2}=0.676$}. These values clearly satisfy
Zeldovich condition (31). We plot the mass function pattern in Fig. 1 to show
the agreement of the predicted mass-radius of the pulsar PSR J0740+6620 and
their observed values as obtained by NICER+XMM data. In comparison with the GR
prediction, RT with positive $\epsilon$ estimates the same mass as in GR but
within larger size star (or less mass at same size). On another word, RT
predicts compactness value lesser than GR for a given mass. This reflects the
capability of RT to allow for more masses or equivalently higher values of
compactness while the stability conditions are being still fulfilled. This
will be discussed in more details in Sec. 6. Notably, for $\epsilon<0$, the
matter-geometry coupling turns the star to slightly smaller sizes in
comparison with GR for the same mass (Nashed & El Hanafy, 2022); and therefore
we omit this choice in the present work.

###  4.3 Geometric sector

Condition (ii): For the geometric sector, the metric potentials $g_{tt}$ and
$g_{rr}$ should be free from coordinate and physical singularities within the
interior region of the star $0\leq r\leq R$, where the center (boundary) is at
$r=0$ ($r=R$) respectively. Obviously the metric (10) satisfies these
conditions whereas at the center, $g_{tt}(x=0)=-e^{a_{1}}$ and
$g_{rr}(x=0)=1$, and they are finite everywhere inside the star $0\leq x\leq
1$.

Condition (iii): The metric potentials of the interior solution and the
exterior should match smoothly at the boundary.

Figure 2: The KB potentials of the pulsar PSR J0740+6620 are finite everywhere
inside the star and match smoothly Schwarzschild exterior vacuum solution. The
plot confirms that conditions (ii) and (iii) are satisfied.

Obviously conditions (ii) and (iii) are satisfied for the pulsar PSR
J0740+6620 as obtained by Fig. 2.

Condition (iv): We define the gravitational red-shift of the metric (10)

$Z=\frac{1}{\sqrt{-g_{tt}}}-1=\frac{1}{\sqrt{e^{a_{0}x^{2}+a_{1}}}}-1.$ (33)

The redshift should be finite and positive everywhere inside the star and
decreases monotonically toward the boundary, i.e. $Z>0$ and $Z^{\prime}<0$.

Figure 3: The redshift function (33) of the pulsar PSR J0740+6620. The plot
confirms that condition (iv) is satisfied.

Obviously condition (iv) is satisfied for the pulsar PSR J0740+6620 as
obtained by Fig. 3.

At the center $Z(0)\approx 0.75$ (less than the GR value $Z(0)\approx 0.8$) is
maximum while at the surface $Z_{R}\approx 0.4$ (similar to the GR value) less
than the upper redshift bound $Z_{R}=2$ as obtained by Buchdahl (1959), see
also (Ivanov, 2002; Barraco et al., 2003) for anisotropic case and Böhmer &
Harko (2006) in presence of a cosmological constant. We note that the maximal
redshift constraint is not a very strict condition, subsequently it cannot
provide a proper way to put an upper bound on the compactness as already
concluded by some work (c.f., Ivanov, 2002; Barraco et al., 2003; Böhmer &
Harko, 2006). This conclusion is altered when using the energy conditions on
the matter sector which provide much better constraints, c.f. Roupas & Nashed
(2020).

###  4.4 Matter sector

(a) The energy-density

(b) The radial pressure

(c) The tangential pressure

(d) The energy-density

(e) The radial pressure

(f) The tangential pressure

Figure 4: The patterns of the density, radial and tangential pressures, Eqs.
(LABEL:eq:Feqs2), of the pulsar PSR J0740+6620 are given by Figs. 4(a)–4(c).
The radial gradients of these quantities are given by Figs. 4(d) and 4(e) for
GR and RT cases. 4(f) the plots of the anisotropy parameter $\Delta$, Eq.
(18). The plots confirm that conditions (v)–(vii) are satisfied.

Condition (v): For a solution to be regular, the density, radial and
tangential pressures of the fluid should be non singular everywhere inside the
star. Additionally, these physical quantities should have maximum values at
the center and monotonically decrease towards the surface of the star. i.e.

  * a.

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

  * b.

$p_{r}(r=0)>0$, $p_{r}^{\prime}(r=0)=0$, $p_{r}^{\prime\prime}(r=0)<0$ and
$p_{r}^{\prime}(0<r\leq R)<0$,

  * c.

$p_{t}(r=0)>0$, $p_{t}^{\prime}(r=0)=0$, $p_{t}^{\prime\prime}(r=0)<0$ and
$p_{t}^{\prime}(0<r\leq R)<0$.

Condition (vi): The density, radial and tangential pressures of the fluid
within the star ($0<r<R$), should be positive, i.e. $\rho(0<r<R)>0$,
$p_{r}(0<r<R)>0$ and $p_{t}(0<r<R)>0$.

Condition (vii): The radial pressure of the fluid should vanish at the
boundary surface of the star, i.e. $p_{r}(r=R)=0$. This is not necessarily for
the tangential pressure.

Obviously conditions (v)–(vii) are satisfied for the pulsar PSR J0740+6620 as
obtained by Fig. 44(a)–4(e).

We note that the model estimates a NS core density $\rho_{\text{core}}\approx
5.6\times 10^{14}$ g/cm${}^{3}\approx 2.1\rho_{\text{nuc}}$ of the pulsar PSR
J0740+6620, which does not exclude the possibility of the pulsar core being
neutrons. In addition, at this density the anaisotropic fluid would be a
realistic assumption.

Condition (vii): The anisotropy parameter $\Delta$ should vanish at the center
of the star, i.e., $p_{r}(r=0)=p_{t}(r=0)$, increasing toward the boundary,
i.e. $\Delta^{\prime}(0\leq r\leq R)>0$. Consequently the anisotropic force
$F_{a}=2\Delta/r$ vanishes at the center. Taking the limit $x\to 0$ in Eq.
(18) we obtain $\Delta\to 0$.

Obviously condition (vii) is satisfied for the pulsar PSR J0740+6620 as
obtained by Fig. 44(f).

Also, one finds that the anaistropy parameter $\Delta(r>0)>0$ (i.e.,
$p_{t}>p_{r}$) which is necessarily for the induced anaisotropic force to be
repulsive, and therefore allows for larger NS size in comparison with the
isotropic perfect fluid case. However, for $\epsilon>0$, the anisotropy in RT
is slightly less than GR due to larger coupling constant $\kappa$ of Rastall
relative to Einstein one $\kappa_{E}$ as obtained by Eq. (13).

###  4.5 Energy conditions

In the context of GR, the focusing theorem implies the positivity of the tidal
tensor trace $\mathfrak{R}_{\alpha\beta}u^{\alpha}u^{\beta}\geq 0$ and
$\mathfrak{R}_{\alpha\beta}\ell^{\alpha}\ell^{\beta}\geq 0$ in Raychaudhuri
equation, where $u^{\alpha}$ is an arbitrary timelike vector and
$\ell^{\alpha}$ is an arbitrary future directed null vector. This imposes four
conditions on the energy momentum tensor $\mathfrak{T}^{\alpha\beta}$, those
are the energy conditions. These could be extended to modified gravity. In the
particular case of Rastall’s gravity the energy conditions could be written in
terms of the effective energy momentum tensor
$\widetilde{\mathfrak{T}}{{}^{\alpha}}{{}_{\beta}}=diag(-\tilde{\rho}c^{2},\tilde{p}_{r},\tilde{p}_{t},\tilde{p}_{t})$,
since
$\mathfrak{R}_{\alpha\beta}=\kappa\left(\widetilde{\mathfrak{T}}_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\widetilde{\mathfrak{T}}\right)$.

(a) Dominance of density over radial pressure

(b) Dominance of density over tangential pressure

(c) Dominance of density over total pressure

Figure 5: The patterns of the energy conditions according to the solution (12)
of the pulsar PSR J0740+6620. The plots confirm that energy conditions (viii)
all are satisfied.

Condition (viii): A physical model should satisfy the modified energy
conditions as stated below:

  * a.

Weak energy condition (WEC): $\tilde{\rho}\geq 0$,
$\tilde{\rho}c^{2}+\tilde{p}_{r}>0$, $\tilde{\rho}c^{2}+\tilde{p}_{t}>0$,

  * b.

Null energy condition (NEC): $\tilde{\rho}c^{2}+\tilde{p}_{r}\geq 0$,
$\tilde{\rho}c^{2}+\tilde{p}_{t}\geq 0$,

  * c.

Strong energy condition (SEC):
$\tilde{\rho}c^{2}+\tilde{p}_{r}+2\tilde{p}_{t}\geq 0$,
$\tilde{\rho}c^{2}+\tilde{p}_{r}\geq 0$, $\tilde{\rho}c^{2}+\tilde{p}_{t}\geq
0$,

  * d.

Dominant energy conditions (DEC): $\tilde{\rho}\geq 0$,
$\tilde{\rho}c^{2}-\tilde{p}_{r}\geq 0$ and
$\tilde{\rho}c^{2}-\tilde{p}_{t}\geq 0$.

We rewrite the field equations (12) in the following form

$\displaystyle\tilde{\rho}c^{2}$ $\displaystyle=$ $\displaystyle\rho
c^{2}+\frac{\epsilon}{1-4\epsilon}(\rho c^{2}-p_{r}-2p_{t}),$
$\displaystyle\tilde{p}_{r}$ $\displaystyle=$ $\displaystyle
p_{r}-\frac{\epsilon}{1-4\epsilon}(\rho c^{2}-p_{r}-2p_{t}),$
$\displaystyle\tilde{p}_{t}$ $\displaystyle=$ $\displaystyle
p_{t}-\frac{\epsilon}{1-4\epsilon}(\rho c^{2}-p_{r}-2p_{t}).$ (34)

For $\epsilon=0$ the GR is restored. It is straightforward to prove that once
the NEC is fulfilled in GR, it is also fulfilled in RT whereas
$\tilde{\rho}c^{2}+\tilde{p}_{r}=\rho c^{2}+p_{r}$ and
$\tilde{\rho}c^{2}+\tilde{p}_{t}=\rho c^{2}+p_{t}$. By virtue of conditions
(v) and (vi), explicitly the density and the pressures are positive within the
star, the NEC is satisfied. Additionally, we show that

$\displaystyle\tilde{\rho}c^{2}-\tilde{p}_{r}$ $\displaystyle=$
$\displaystyle(\rho c^{2}-p_{r})+\frac{2\epsilon}{1-4\epsilon}(\rho
c^{2}-p_{r}-2p_{t}),$ $\displaystyle\tilde{\rho}c^{2}-\tilde{p}_{t}$
$\displaystyle=$ $\displaystyle(\rho
c^{2}-p_{t})+\frac{2\epsilon}{1-4\epsilon}(\rho c^{2}-p_{r}-2p_{t}),$
$\displaystyle\tilde{\rho}c^{2}+\tilde{p}_{r}+2\tilde{p}_{t}$ $\displaystyle=$
$\displaystyle(\rho c^{2}+p_{r}+2p_{t})-\frac{2\epsilon}{1-4\epsilon}(\rho
c^{2}-p_{r}-2p_{t}).\qquad$ (35)

Given that the fraction $\frac{\epsilon}{1-4\epsilon}>0$ for $0<\epsilon<1/4$
(in our case $\epsilon=0.041$) in addition to $\rho\geq 0$, $p_{r}\geq 0$ and
$p_{t}\geq 0$; it remains to show that $\rho c^{2}-p_{r}\geq 0$, $\rho
c^{2}-p_{t}\geq 0$ and $\rho c^{2}-p_{r}-2p_{t}\geq 0$ to grantee the
verification of the energy conditions (a–d).

Obviously the energy conditions (viii) are satisfied for the pulsar PSR
J0740+6620 as obtained by Fig. 55(a)–5(c).

Finally we note that the dominance of energy density over the total pressure
is guaranteed in RT as long as it is fulfilled in GR, since
$\tilde{\rho}c^{2}-\tilde{p}_{r}-2\tilde{p}_{t}=\frac{1}{1-4\epsilon}(\rho
c^{2}-p_{r}-2p_{t})$ whereas $0\leq\epsilon<1/4$. This latter constraint has
been referred to as a strong energy condition by some authors (c.f. Kolassis
et al., 1988; Ivanov, 2017; Das et al., 2019; Roupas & Nashed, 2020). However,
we will call it as a dominant energy condition keeping in mind that imposed
constraints by this condition for matter or effective fluids are identical.

According to the above results one can see that the energy conditions for the
effective fluid are strongly related to the matter fluid in RT, but not
identical in all cases as expected, (c.f. Moradpour et al., 2017b; Li et al.,
2019). However, the DEC in the matter sector $\rho c^{2}-p_{r}-2p_{t}\geq 0$
is the core condition which allows for other conditions to be satisfied in
general whereas the matter density and pressures are positive and
$0<\epsilon<1/4$. Similar constraints on Rastall parameter motivated by
thermodynamics aspects (positivity of the horizon entropy) have been obtained
by Moradpour & Salako (2016). Furthermore, in cosmological applications, it
has been shown that the second law of thermodynamics is fulfilled in RT, if
the WEC is verified for the matter sector (Moradpour, 2016).

###  4.6 Causality and stability conditions

(a) Radial speed of sound

(b) Tangential speed of sound

(c) difference between radial and tangential speed of sounds

Figure 6: The radial and tangential sound speeds (36) of the pulsar PSR
J0740+6620. The plots confirm that the causality and stability conditions (ix)
and (x) are satisfied.

We define the square of the sound speed in radial and tangential directions,

$v_{r}^{2}=\frac{dp_{r}}{d\rho}=\frac{\tilde{p}^{\prime}_{r}}{\bar{\rho}^{\prime}},\quad
v_{t}^{2}=\frac{dp_{t}}{d\rho}=\frac{\tilde{p}^{\prime}_{t}}{\bar{\rho}^{\prime}},$
(36)

where the density and pressures radial gradients are given by Eqs.
(3.3)–(3.3).

Condition (ix): The stellar structure should be causal, i.e. satisfies the
causality condition: sound speeds should be positive and smaller than unity
everywhere inside the star ($0\leq v_{r}/c\leq 1$, $0\leq v_{t}/c\leq 1$), and
monotonically decrease toward the boundary ($v^{\prime}_{r}{{}^{2}}<0$,
$v^{\prime}_{t}{{}^{2}}<0$).

Condition (x): The stellar structure should be stable, i.e. satisfies the
stability condition $-1<(v_{t}^{2}-v_{r}^{2})/c^{2}<0$ everywhere inside the
star Herrera (1992).

Obviously the causality and stability conditions (ix) and (x) are satisfied
for the pulsar PSR J0740+6620 as obtained by Fig. 66(a)–6(c).

Although the sound speed fulfills the causality and stability conditions for
the pulsar PSR J0740+6620, the GR prediction of the sound speed within the
region $r\lesssim 10.14$ km from the center exceeds the conjectured conformal
upper bound on the sound speed $v_{r}^{2}=c_{s}^{2}\leq c^{2}/3$. This
violation is strongly suggested for NS with radii less than $\sim 11.8$ km
(Bedaque & Steiner, 2015). Also, the violation of the conformal upper limit of
the sound speed has been obtained in other models when hadronic EoS is assumed
(Cherman et al., 2009; Landry et al., 2020) or when a non-parametric EoS
approach based on Gaussian processes is applied to the pulsar PSR J0740+6620
using the X-ray NICER+XMM observations (Legred et al., 2021). On the contrary,
as seen in Fig. 66(a), the conformal bound on the sound speed for the pulsar
PSR J0740+6620 is not violated in RT from the core to the surface.

###  4.7 Adiabatic indices and hydrodynamic equilibrium

We apply two further tests to examine the stability of the obtained model
within Ratall gravity. First, we investigate the relativistic adiabatic
indices of a spherically symmetric, which define the ratio of two specific
heats (Chandrasekhar, 1964; Merafina & Ruffini, 1989; Chan et al., 1993).

$\gamma=\frac{4}{3}\left(1+\frac{F_{a}}{2|p^{\prime}_{r}|}\right)_{max},\,\Gamma_{r}=\frac{\rho
c^{2}+p_{r}}{p_{r}}v_{r}^{2},\,\Gamma_{t}=\frac{\rho
c^{2}+p_{t}}{p_{t}}v_{t}^{2}.$ (37)

For anisotropic fluid, the sphere is in a neutral (stable) equilibrium where
the adiabatic indices $\Gamma=\gamma$ ($\Gamma>\gamma$) (see Chan et al.,
1993). Clearly the adiabatic indices reduce to the isotropic sphere case where
$\gamma=4/3$ and $\Gamma_{r}=\Gamma_{t}$ (Heintzmann & Hillebrandt, 1975).
Second, we investigate the validity of TOV equation (Tolman, 1939; Oppenheimer
& Volkoff, 1939; Ponce de Leon, 1993) by assuming the sphere is in a
hydrostatic equilibrium everywhere inside the star where the acting forces
neutralize each other. The modified version of TOV equation according to the
newly introduced Rastall force $F_{R}$ can be written as

$F_{a}+F_{g}+F_{h}+F_{R}=0\,,$ (38)

where $F_{g}$ and $F_{h}$ denote the gravitational and the hydrostatic forces.
Here we have

$\displaystyle F_{a}$ $\displaystyle=$ $\displaystyle 2\Delta/r,\quad
F_{g}=-\frac{M_{g}}{r}(\rho c^{2}+p_{r})e^{\delta/2},$ $\displaystyle F_{h}$
$\displaystyle=$ $\displaystyle-p^{\prime}_{r},\quad
F_{R}=-\frac{\epsilon}{1-4\epsilon}(c^{2}\rho^{\prime}-p^{\prime}_{r}-2p^{\prime}_{t}),$
(39)

where $\delta\equiv\delta(r)=\alpha-\beta$ and the mass (energy) $M_{g}$ for
an isolated time-independent systems within 3-space $V$ ($t=$ constant) is
given by Tolman mass formula (Tolman, 1930, 1934)

$\displaystyle M_{g}(r)$ $\displaystyle=$
$\displaystyle{\int_{V}}\Big{(}-\mathfrak{T}{{}^{t}}{{}_{t}}+\mathfrak{T}{{}^{r}}{{}_{r}}+\mathfrak{T}{{}^{\theta}}{{}_{\theta}}+\mathfrak{T}{{}^{\phi}}{{}_{\phi}}\Big{)}\sqrt{-g}\,dV$
(40) $\displaystyle=$
$\displaystyle\frac{(e^{\alpha/2})^{\prime}}{e^{\alpha}}e^{\beta/2}r=\frac{\alpha^{\prime}}{2}re^{-\delta/2}.$

This reads the gravitational force $F_{g}=-\frac{a_{0}r}{R^{2}}(\rho
c^{2}+p_{r})$. Now we state explicitly the stability conditions related to the
relativistic adiabatic indices and the modified TOV equation.

Figure 7: The adiabatic indices (37) of the pulsar PSR J0740+6620. The plots
confirm that the stability conditions (xi) are satisfied within the pulsar.
Dotted (dash-dotted) curves represent GR (RT) adiabatic indices. Figure 8: The
forces (4.7) of the modified TOV equation (38) of the pulsar PSR J0740+6620.
Dotted (dash-dotted) curves represent GR (RT) forces. The plots confirm that
the hydrodynamic stability condition (xii) is satisfied within the pulsar.

Condition (xi): The anisotropic stellar model is in a stable equilibrium
whereas the adiabatic indices satisfy the conditions $\Gamma_{r}>\gamma$ and
$\Gamma_{t}>\gamma$ everywhere inside the star. Recalling the density and
pressures Eqs. (LABEL:eq:Feqs2) and their radial gradients (3.3)–(3.3) we plot
the adiabatic indices (37) of the pulsar PSR J0740+6620 for GR and RT cases in
Fig. 7. Noting that $\gamma=4/3$ for isotropic sphere.

Obviously the adiabatic indices stability conditions (xi) are satisfied for
the pulsar PSR J0740+6620 as obtained by Fig. 7.

Condition (xii): The anisotropic stellar model is in a hydrodynamic
equilibrium whereas the forces satisfy The modified TOV equation (38). We use
Eqs. (LABEL:eq:Feqs2) and (3.3)–(3.3) to evaluate the forces (4.7) which are
plotted in Fig. 8 for GR and RT cases. The plots show that the negative
gravitational force compensates (the positive forces) for a hydrodynamics
equilibrium as required for a stable configuration. The role of Rastall force
to resize compact objects will be discussed later in Sec. 6.

Obviously the hydrodynamic equilibrium condition (xii) is satisfied for the
pulsar PSR J0740+6620 as obtained by Fig. 8.

###  4.8 Neutron core, sound speed and equation of state

Using the X-ray NICER+XMM observational data of the pulsar PSR J0740+6620, we
obtained a reasonable value of Rastall parameter $\epsilon=0.034$ which
characterizes the coupling between matter and geometry in curved spacetime.
Accordingly, we calculate the surface density which has been found as
$\rho_{R}\approx 3.2\times 10^{14}$ g/cm3 while at the center the density
increases up to $\rho_{\text{core}}\approx 5.6\times 10^{14}$ g/cm3 which is
only $\sim 2.1$ times the nuclear saturation density
$\rho_{\text{nuc}}=2.7\times 10^{14}$ g/cm3. We note that the core density in
RT with $\epsilon>0$ is less than the corresponding value in GR
($\epsilon=0$), since within a given radius the estimated mass by Rastall
gravity is less than Einsteins gravity as seen in Fig. 1. The central density
value suggests that the PSR J0740+6620 core is made of neutrons.

(a) Radial EoS

(b) Tangential EoS

Figure 9: The pressures-density relation of the pulsar PSR J0740+6620 is in a
good agreement with the linear EoS with bag constant whereas the slopes
$dp_{r}/d\rho=0.312c^{2}$ and $dp_{t}/d\rho=0.199c^{2}$ evidently confirm that
the NS matters fulfill the conjectured conformal bound on the sound speed
$c_{s}^{2}\leq c^{2}/3$ everywhere inside the star.

It is to be noted that the NS with $\sim 2M_{\odot}$ with hadronic EoS
strongly violates the conjectured conformal bound on the sound speed
$c_{s}^{2}\leq c^{2}/3$ even at low densities Bedaque & Steiner (2015). This
may point out a strongly interacting and nonconformal matter at the NS core
Cherman et al. (2009); Landry et al. (2020). A non-parametric EoS approach
based on Gaussian processes has been applied to investigate the implications
of the X-ray NICER+XMM observations of the pulsar PSR J0740+6620 in particular
on high density matter Legred et al. (2021), it has been concluded that the
conformal sound speed is strongly violated at the core whereas
$c_{s}^{2}=0.75c^{2}$. Interestingly, for the model at hand without imposing
EoS, we find that the maximum squared sound speeds at the NS core
$v^{2}_{r}\lesssim 0.332c^{2}$ (radial direction) and $v^{2}_{t}\lesssim
0.222c^{2}$ (tangential direction), which satisfy the conjectured conformal
bound on the sound speed $c_{s}^{2}\leq c^{2}/3$ at the core and also
everywhere inside the star, see Fig. 6. This unlike the GR case whereas
$c_{s}^{2}\approx 0.37c^{2}$.

Although we do not preassume a particular EoS, we find that the pressure-
density relation according to the currant model is in a good agreement with
the linear EoS with bag constant whereas $p_{r}(\rho)=0.311\rho c^{2}-9.085$
and $p_{t}(\rho)=0.198\rho c^{2}-3.412$ as seen in Figs. 9. Notably the slopes
of the fitted lines $dp_{r}/d\rho=0.311c^{2}$ and $dp_{t}/d\rho=0.198c^{2}$
are in agreement with the obtained squared sound speeds in radial and
tangential directions which also confirm the fulfillment of the conjectured
conformal bound on the sound speed $c_{s}^{2}\leq c^{2}/3$ everywhere inside
the star. We also note that the present anisotropic NS model obtains an EoS
which is not too stiff as usually found for NS models, but in agreement with
the EoS as expected from gravitational wave signals where no clear evidence on
a tidal deformation in the observed GW patterns.

##  5 Confronting the model with more pulsar data

In the previous section we confronted the model with the X-ray NICER+XMM
observational data of the pulsar PSR J0740+6620 constraining Rastall parameter
to a value of $\epsilon=0.041$. We extend our investigation to include more
pulsars with different types of observations to examine the applicability of
the model to a wide range of compact stars.

Table 1: Observed mass-radius of twenty pulsars and the corresponding model parameters ($\epsilon=0.06$). Pulsar | Ref | observed mass | observed radius | estimated mass | $a_{0}$ | $a_{1}$ | $a_{2}$
---|---|---|---|---|---|---|---
|  | ($M_{\odot}$) | [km] | ($M_{\odot}$) |  |  |
High-mass X-ray binaries
Her X-1 | Abubekerov et al. (2008) | $0.85\pm 0.15$ | $8.1\pm 0.41$ | $0.767$ | $0.199$ | $-0.570$ | $0.371$
4U 1538-52 | Gangopadhyay et al. (2013) | $0.87\pm 0.07$ | $7.866\pm 0.21$ | $0.804$ | $0.223$ | $-0.631$ | $0.407$
LMC X-4 | Rawls et al. (2011) | $1.04\pm 0.09$ | $8.301\pm 0.2$ | $0.954$ | $0.269$ | $-0.742$ | $0.472$
Cen X-3 | Naik et al. (2011) | $1.49\pm 0.49$ | $9.178\pm 0.13$ | $1.344$ | $0.417$ | $-1.070$ | $0.653$
Low-mass X-ray binaries (quiescence/thermonuclear bursts)
EXO 1785-248 | Özel et al. (2009) | $1.3\pm 0.2$ | $8.849\pm 0.4$ | $1.172$ | $0.344$ | $-0.914$ | $0.569$
M13 | Webb & Barret (2007) | $1.38\pm 0.2$ | $9.95\pm 0.27$ | $1.245$ | $0.311$ | $-0.838$ | $0.527$
X7 | Bogdanov et al. (2016) | $1.1\pm 0.35$ | $12$ | $1.073$ | $0.183$ | $-0.530$ | $0.347$
4U 1820-30 | Özel et al. (2016) | $1.46\pm 0.2$ | $11.1\pm 1.8$ | $1.317$ | $0.284$ | $-0.775$ | $0.492$
4U 1608-52 | Marshall & Angelini (1996) | $1.57\pm 0.3$ | $9.8\pm 1.8$ | $1.470$ | $0.438$ | $-1.114$ | $0.676$
KS 1731-260 | Özel et al. (2009) | $1.61\pm 0.37$ | $10\pm 2.2$ | $1.452$ | $0.410$ | $-1.055$ | $0.646$
EXO 1745-268 | Özel et al. (2009) | $1.65\pm 0.25$ | $10.5\pm 1.8$ | $1.488$ | $0.391$ | $-1.015$ | $0.624$
4U 1724-207 | Özel et al. (2009) | $1.81\pm 0.27$ | $12.2\pm 1.4$ | $1.632$ | $0.351$ | $-0.928$ | $0.577$
SAX J1748.9-2021 | Özel et al. (2009) | $1.81\pm 0.3$ | $11.7\pm 1.7$ | $1.632$ | $0.379$ | $-0.990$ | $0.611$
Millisecond Pulsars
PSR J0030+0451 | Raaijmakers et al. (2019) | $1.34\pm 0.16$ | $12.71\pm 1.19$ | $1.266$ | $0.215$ | $-0.610$ | $0.395$
PSR J0030+0451 | Miller et al. (2019) | $1.44\pm 0.16$ | $13.02\pm 1.24$ | $1.313$ | $0.219$ | $-0.620$ | $0.401$
PSR J0037-4715 | Reardon et al. (2016) | $1.44\pm 0.07$ | $13.6\pm 0.9$ | $1.387$ | $0.223$ | $-0.629$ | $0.407$
PSR J1614-2230 | Arzoumanian et al. (2018b) | $1.908\pm 0.016$ | $13\pm 2$ | $1.894$ | $0.413$ | $-1.06$ | $0.649$
PSR J0348+0432 | Antoniadis et al. (2013) | $2.01\pm 0.04$ | $13\pm 2$ | $1.975$ | $0.387$ | $-1.008$ | $0.620$
Gravitational-wave Signals
LIGO-Virgo | Abbott et al. (2020b) | $1.4$ | $12.9\pm 0.8$ | $1.400$ | $0.246$ | $-0.685$ | $0.440$
GW170817-1 | Abbott et al. (2018) | $1.45\pm 0.09$ | $11.9\pm 1.4$ | $1.263$ | $0.209$ | $-0.596$ | $0.387$
GW170817-2 | Abbott et al. (2018) | $1.27\pm 0.09$ | $11.9\pm 1.4$ | $1.210$ | $0.222$ | $-0.627$ | $0.405$

In Table 1 we list the pulsars according to different types observations with
their corresponding constraints on masses and radii. We include: (i) Four high
mass X-ray binaries. (ii) Nine low mass X-ray binaries, where the mass-radius
constraints are given by using spectral analysis techniques for two quiescent
low X-ray binaries (M13 and X7), while others are characterized by Type I
X-ray bursts thermonuclear explosions on the neutron star surface. (iii) Four
millisecond Pulsars including two of the most massive pulsars PSR J0348+0432
and PSR J1614-2230 where the mass-radius constraints are obtained by measuring
the arrival time of pulses, we further include the recent NICER observational
constraints on the two pulsars PSR J0437-4715 and PSR J0030+0451. (iv) Three
gravitational-wave signals including the first detected NS-NS merger GW170817
in addition to LIGO-Virgo constraints on NS radius from GW170817+GW190814.
Notably the companion of the GW190814 binary is the first detected compact
object in the low mass gap $M=2.6M_{\odot}$. In addition, we estimate the
values of the pulsars’ masses (corresponding to the observed/estimated mean
value of $R$) with the corresponding values of the model parameters $a_{0}$,
$a_{1}$ and $a_{2}$. As seen from the table, the estimated masses are
compatible with the corresponding observed values.

As mentioned in the introduction that NICER observations of PSR J0030+0451 and
PSR J0740+6620 represents a challenge for squeezable models, since the latter
has much more mass than the former while both are having almost same size. We
note that the present model estimates ($M=1.266M_{\odot},R=12.71$ km) and
($M=1.313M_{\odot},R=13.02$ km) for the pulsar PSR J0030+0451 in agreement
with its NICER observational constraints as reported by Raaijmakers et al.
(2019) and Miller et al. (2019) respectively. At the same time it estimates
($M=1.96M_{\odot},R=13.04$ km) for the pulsar PSR J0740+6620 in agreement with
NICER+XMM measurements (Legred et al., 2021). This is in agreement with our
claim that when a pulsar gains more mass and subsequently higher density, the
induced anisotropic force becomes relevant to hold the NS size.

We further examine the stability of the model according to the criteria given
in the previous section. We list the physical quantities of the most interest
in Table 2. The results show the capability of the model to predict stable
compact stellar structure compatible with observations. Interestingly, for all
categories the sound speeds as obtained by this model are in agreement with
the conformal bound $c^{2}_{s}\leq c^{2}/3$. It is to be noted that for the
low mass X-ray binaries 4U 1608-52 and KS 1731-260, in particular, the general
relativistic version of this study predicted $c^{2}_{s}=0.34$ at the center
above the conformal bound as obtained by Roupas & Nashed (2020). Similar to
our previous discussion on the sound speed of the pulsar PSR J0740+6620, the
results of Table 2 confirm that the sound speed violation is avoided in the
present study. This shows that the possible role of the geometry-matter
coupling to hold this upper bound everywhere inside the compact object.

Table 2: Calculated physical quantities of the most interest. Pulsar |  $\rho$ [g/cm3] | $v_{r}^{2}/c^{2}$ | $v_{t}^{2}/c^{2}$ | DEC [$dyne/cm^{2}$] | $Z_{R}$
---|---|---|---|---|---
| center | surface | center | surface | center | surface | center | surface |
High-mass X-ray binaries
Her X-1 | 7.89$\times 10^{14}$ | 5.80$\times 10^{14}$ | 0.255 | 0.228 | 0.153 | 0.118 | 5.72$\times 10^{35}$ | 4.80$\times 10^{35}$ | 0.204
4U 1538-52 | 9.19$\times 10^{14}$ | 6.55$\times 10^{14}$ | 0.264 | 0.233 | 0.161 | 0.122 | 6.49$\times 10^{35}$ | 5.38$\times 10^{35}$ | 0.226
LMC X-4 | 9.59$\times 10^{14}$ | 6.49$\times 10^{14}$ | 0.279 | 0.242 | 0.175 | 0.131 | 6.42$\times 10^{35}$ | 5.23$\times 10^{35}$ | 0.266
Cen X-3 | 1.10$\times 10^{15}$ | 6.39$\times 10^{14}$ | 0.326 | 0.271 | 0.217 | 0.158 | 6.12$\times 10^{35}$ | 4.83$\times 10^{35}$ | 0.386
Low-mass X-ray binaries (quiescence/thermonuclear bursts)
EXO 1785-248 | 1.02$\times 10^{15}$ | 6.38$\times 10^{14}$ | 0.303 | 0.257 | 0.197 | 0.145 | 6.25$\times 10^{35}$ | 4.98$\times 10^{35}$ | 0.329
M13 | 7.46$\times 10^{14}$ | 4.83$\times 10^{14}$ | 0.292 | 0.251 | 0.188 | 0.139 | 4.76$\times 10^{35}$ | 3.82$\times 10^{35}$ | 0.302
X7 | 3.36$\times 10^{14}$ | 2.51$\times 10^{14}$ | 0.250 | 0.224 | 0.147 | 0.114 | 2.48$\times 10^{35}$ | 2.10$\times 10^{35}$ | 0.189
4U 1820-30 | 5.59$\times 10^{14}$ | 3.72$\times 10^{14}$ | 0.284 | 0.245 | 0.180 | 0.134 | 3.68$\times 10^{35}$ | 2.98$\times 10^{35}$ | 0.279
4U 1608-52 | 9.91$\times 10^{14}$ | 5.70$\times 10^{14}$ | 0.332 | 0.275 | 0.222 | 0.162 | 5.41$\times 10^{35}$ | 4.27$\times 10^{35}$ | 0.402
KS 1731-260 | 9.08$\times 10^{14}$ | 5.35$\times 10^{14}$ | 0.324 | 0.270 | 0.215 | 0.157 | 5.13$\times 10^{35}$ | 4.06$\times 10^{35}$ | 0.381
EXO 1745-268 | 7.95$\times 10^{14}$ | 4.77$\times 10^{14}$ | 0.318 | 0.266 | 0.210 | 0.154 | 4.61$\times 10^{35}$ | 3.65$\times 10^{35}$ | 0.366
4U 1724-207 | 5.44$\times 10^{14}$ | 3.38$\times 10^{14}$ | 0.305 | 0.286 | 0.199 | 0.146 | 3.31$\times 10^{35}$ | 2.63$\times 10^{35}$ | 0.334
SAX J1748.9-2021 | 6.27$\times 10^{14}$ | 3.80$\times 10^{14}$ | 0.314 | 0.264 | 0.207 | 0.151 | 3.68$\times 10^{35}$ | 2.92$\times 10^{35}$ | 0.357
Millisecond Pulsars
PSR J0030+0451 | 3.41$\times 10^{14}$ | 2.46$\times 10^{14}$ | 0.260 | 0.231 | 0.158 | 0.121 | 2.43$\times 10^{35}$ | 2.03$\times 10^{35}$ | 0.218
PSR J0030+0451 | 3.30$\times 10^{14}$ | 2.37$\times 10^{14}$ | 0.262 | 0.231 | 0.159 | 0.121 | 2.34$\times 10^{35}$ | 1.95$\times 10^{35}$ | 0.222
PSR J0037-4715 | 3.07$\times 10^{14}$ | 2.19$\times 10^{14}$ | 0.263 | 0.233 | 0.160 | 0.122 | 2.17$\times 10^{35}$ | 1.80$\times 10^{35}$ | 0.225
PSR J1614-2230 | 5.39$\times 10^{14}$ | 3.17$\times 10^{14}$ | 0.325 | 0.270 | 0.216 | 0.158 | 3.04$\times 10^{35}$ | 2.40$\times 10^{35}$ | 0.383
PSR J0348+0432 | 4.45$\times 10^{14}$ | 2.67$\times 10^{14}$ | 0.317 | 0.266 | 0.209 | 0.153 | 2.59$\times 10^{35}$ | 2.05$\times 10^{35}$ | 0.364
Gravitational-wave Signals
LIGO-Virgo | 3.69$\times 10^{14}$ | 2.56$\times 10^{14}$ | 0.271 | 0.238 | 0.168 | 0.126 | 2.54$\times 10^{35}$ | 2.09$\times 10^{35}$ | 0.246
GW170817-1 | 3.24$\times 10^{14}$ | 2.35$\times 10^{14}$ | 0.259 | 0.230 | 0.156 | 0.120 | 2.32$\times 10^{35}$ | 1.94$\times 10^{35}$ | 0.213
GW170817-2 | 3.99$\times 10^{14}$ | 2.85$\times 10^{14}$ | 0.263 | 0.233 | 0.160 | 0.122 | 2.82$\times 10^{34}$ | 2.34$\times 10^{34}$ | 0.225

##  6 Maximum Compactness and Mass-Radius Relation

As is shown in the above sections the physical quantities of KB interior
solution ($0\leq x\leq 1$) are expressed in terms of Rastall and compactness
parameters. On the other hand, observational constraints on masses and radii
allow us to estimate Rastall parameter to be at most $\epsilon=0.041$. In this
section we follow Roupas & Nashed (2020) utilizing the energy dominance
condition111We note that any imposed constraints from the DEC for matter or
effective fluids are identical as discussed in Sec. 4.5., namely $\rho
c^{2}-p_{r}-2p_{t}\geq 0$ with $\epsilon=0.041$, to set an upper bound on the
compactness value for the model at hand showing the role of Rastall parameter
to support higher compactness values in comparison to GR predictions. In
addition, we plot the mass-radius relation for some boundary conditions on the
surface density compatible with nuclear density showing the maximum mass and
radius for each case in comparison to GR predictions.

###  6.1 Maximum compactness

Recalling the dimensionless energy density and pressures (LABEL:eq:Feqs2)
where the model parameters {$a_{0},a_{1},a_{2}$} are given by (25), we use the
energy density dominance constraint to set an upper bound on the compactness
which has been found to be $C_{|\epsilon=0.041}\leq 0.735$ satisfying Buchdahl
compactness bound $C\leq 8/9$ (for isotropic sphere Buchdahl (1959)). In Fig.
10, we plot the energy density dominance patterns for GR ($\epsilon=0$) and RT
($\epsilon=0.041$) using different compactness values. The GR prediction of
the maximum compactness is $C=0.715$ in agreement with Roupas & Nashed (2020),
however the contribution of the matter-geometry non-minimal coupling as
assumed in RT allows $\sim 2\%$ maximum compactness higher in comparison to GR
limits.

Figure 10: A comparison between GR ($\epsilon=0$) and RT ($\epsilon=0.041$)
violation of the DEC $\bar{\rho}-\bar{p}_{r}-2\bar{p}_{t}>0$ at higher
compactness. We conclude that RT can support stable stellar configurations
with higher compactness in comparison to GR.

This result is in agreement with our earlier conclusion that RT with
$\epsilon>0$ predicts larger pulsar size relative to GR for a given mass, see
Fig. 1. Recalling TOV equation of the hydrodynamic equilibrium (38), we find
that the Rastall force contributes to oppose partially the gravitational
force, see Fig. 8, to compensate for a larger size of a compact object of a
given mass.

We further note that the obtained result of the upper bound on the compactness
is in contrast to the suggestions that Buchdahl upper bound on the compactness
can be violated in anisotropic stable compact stars (Dev & Gleiser, 2004;
Boehmer & Harko, 2006), also it sets a stronger restriction on the maximally
allowed compactness than the corresponding limits due to surface redshift
constraints (Ivanov, 2002).

###  6.2 Mass-Radius diagram

From Table 2 one obtains surface densities $2\times
10^{14}\lesssim\rho_{R}\lesssim 7\times 10^{14}$ g/cm3; and therefore we
choose four surface densities to serve as boundary conditions as follows:
$\rho_{R}=\rho_{\text{nuc}}=2.7\times 10^{14}$ g/cm3 at nuclear saturation
density, $\rho_{R}=4\times 10^{14}$ g/cm3 and $\rho_{R}=6\times 10^{14}$ g/cm3
to cover the nuclear solidification density. Then, for Rastall parameter
$\epsilon=0.041$, we use the energy density equation (LABEL:eq:Feqs2) to
obtain a relation between $C$ and $R$ for each boundary condition on the
surface density, i.e. $\rho(r=R)=\rho_{R}$. In Fig. 1111(a), we plot the
compactness-radius curves in correspondence to the chosen boundary density.
The plots confirm that the maximum compactness is $C=0.93$ just above Buchdahl
limit $C=8/9$ (represented by a horizontal dotted line) for isotropic star.
Remarkably the maximal compactness approaches a saturation-like pattern which
reflects the capability of the NSs to gain more sizes with small increase of
masses at this limit. However, as we obtained earlier in this section the
energy dominance constraint sets an upper bound on the compactness $C=0.735$
(represented by a horizontal dashed line) for stable anisotropic configuration
in KB spacetime, which consequently determines the maximal allowed radius for
each boundary density as represented by the vertical dotted lines in Fig.
1111(a).

(a) Compactness-Radius diagram

(b) Mass-Radius diagram

(c) LIGO-Virgo and NICER constraints

Figure 11: 11(a) The compactness-radius curves, at different surface
densities, show that the maximum compactness is $C=0.93$ just above Buchdahl
limit $C=8/9$ (dotted horizontal line) for isotropic sphere. However, the
strong energy condition sets a more strict constraint on the upper bound
$C=0.735$ (dashed horizontal line) for an anisotropic compact object to be
physically viable. 11(b) The mass-radius curves show that the DEC (diagonal
dashed line) obtains an upper mass of $M\approx 4M_{\odot}$ in the lower mass
gap with radius $R\approx 16$ km whereas the surface density chosen to be at
the saturation nuclear density $\rho_{\text{nuc}}=2.7\times 10^{14}$ g/cm3.
Solid gray circles represent high-mass X-ray binaries, solid blue circles
represent low-mass X-ray binaries, solid green circles represent MSP and solid
red circles represent GW signals as given from Table 1, 11(c) Close view on
some particular pulsars with most interest. LIGO-Virgo constraints on the
radius of a canonical NS, NICER constraints on the pulsar PSR J0030+0451 and
NICER+XMM on the pulsar PSR J0740+6620 all are in agreement with the
$\rho_{\text{nuc}}$-curve.

Next we give the mass-radius curves for each case in addition to the
corresponding observational data from Table 1 as obtained by Fig. 1111(b). For
the boundary density at the nuclear saturation density
$\rho_{\text{nuc}}=2.7\times 10^{14}$ g/cm3, applying the DEC constraint
(represented by a diagonal dashed line), we calculate the maximum allowed mass
$M_{\text{max}}=4M_{\odot}$ at a maximum radius of $R_{\text{max}}=16$ km.
Notably for the same boundary condition in KB spacetime the GR predicts a
maximum mass $M_{\text{max}}=4.1M_{\odot}$ at maximum radius
$R_{\text{max}}=16.8$ km (Roupas & Nashed, 2020). In comparison to GR
predictions, RT with positive parameter $\epsilon=0.041$ predicts almost same
mass within $\sim 0.8$ km smaller size than GR. Similarly, for the surface
densities $\rho_{R}=4\times 10^{14}$ g/cm3 and $\rho_{R}=6\times 10^{14}$
g/cm3 which are compatible with the nuclear solidification density, we obtain
the maximum masses and radii,
($M_{\text{max}}=3.28M_{\odot},\,R_{\text{max}}=13.18$ km) and
($M_{\text{max}}=2.68M_{\odot},\,R_{\text{max}}=10.77$ km), respectively.
Maximal radii (masses) allowed by the DEC (diagonal dashed line) are
represented by vertical (horizontal) dotted lines for each boundary density on
the mass-radius graph in Fig. 1111(b). Clearly, all pulsars lie in the
physical region as allowed by the DEC. Remarkably, the present model can
produce a NS within the mass gap $2.5-5M_{\odot}$ which does not exclude that
the companion of the binary GW190814 with mass $M=2.6M_{\odot}$ to be a NS
with a surface density compatible with nuclear saturation density with a
linear EoS. In this case the model estimates its radius to be $R=10.45$ km.

In Fig. 1111(c), we focus our discussion on some particular pulsars with most
interest. We find that the pulsars PSR J0740+6620 (NICER+XMM) and PSR
J0030+0451 (NICER) fit well with $\rho_{\text{nuc}}$-curve. Also we find that
LIGO-Virgo constraints on the radius of canonical NS to be compatible with a
surface density at $\rho_{\text{nuc}}$.

##  7 Conclusion

We investigated the impact of non-minimal coupling between matter and
geometry, as assumed by Rastall (1972), on mass-radius relation of compact
objects. The theory assumes a local violation of energy conservation in
presence of curved spacetime, otherwise it reduces to general relativity. Such
effect should be well examined by stellar structure of compact objects whereas
the spacetime is highly curvatured. The precise measurements of mass and
radius of the PSR J0740+ by NICER would help to set a strict estimation of
Rastall parameter.

For a spherically symmetric spacetime with anaisotropic matter source, we
showed that the anisotropy in RT is same as in GR, which helps to clearly
identify deviations from GR in terms of matter-geometry coupling. We applied
KB ansatz where all physical quantities are represented in terms of Rastall
parameter $\epsilon$ and the compactness parameter $C$. The precise mass-
radius observational constraints from X-ray NICER+XMM observations of PSR
J0740+6620 allowed us to estimate the Rastall parameter to be $\epsilon=0.041$
in the positive range. This case implies larger size for a given mass in
comparison to GR which in return allows for higher compactness values to be
obtained within RT framework. We showed that the additional Rastall force
contributes in the hydrodynamic equilibrium to partially compensate
gravitational force allowing for larger size compact star in comparison to GR
for a given mass. We further showed that the upper bound on the compactness
parameter is $C=0.93$ just above Buchdahl limit $C=8/9$ for isotropic case but
never approaches Schwarzschild radius bound. However we utilized the dominant
energy condition to constrain the maximum allowed compactness which has been
found to be $C_{\text{max}}=0.735$, that is $2\%$ higher than GR prediction.

Interestingly, we did not assume a particular equation of state, the model
however in a perfect agreement with linear EoS with bag constant. More
interestingly, we determined the maximum squared sound speeds at the NS core
$v^{2}_{r}=0.332c^{2}$ (radial direction) and $v^{2}_{t}=0.222c^{2}$
(tangential direction), which satisfy the conjectured conformal bound on the
sound speed $c_{s}^{2}\leq c^{2}/3$ at the core and also everywhere inside the
NS unlike the GR case. This is unlike hadronic equation of state models or
Gaussian process non-parametric equation of state (using NICER+XMM
observations of the pulsar PSR J0740+6620) approach where the speed of sound
is strongly violated $c_{s}^{2}=0.75c^{2}$.

For a surface density at saturation nuclear energy
$\rho_{\text{nuc}}=2.7\times 10^{14}$ g/cm3 the model allows for a maximum
mass $M=4M_{\odot}$ at radius $R=16$ km which indeed gives higher compactness
value in comparison to GR prediction for same surface density boundary
condition. This result keeps an open window for the companion of GW190814 to
be an anisotropic NS with no need to assume any exotic matter sources.

Based on the obtained results we conclude that Rastall’s modified gravity and
Einstein’s theory of gravity are not equivalent as argued by Visser (2018).
Our detailed calculations confirm that the matter-geometry coupling reconciles
the sound speed within high mass compact objects $\sim 2M_{\odot}$ with the
conformal upper bound which is not the case in GR. This critical results among
other similar work confirm the inequivalence between RT and GR as noted by
Darabi et al. (2018).

The author would like to thank the referee for the useful comments which
helped to improve the results.

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