# The key role of Lagrangian multiplier in mimetic gravitational theory in the
frame of isotropic compact star

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
Center for Space Research, North-West University, Potchefstroom 2520, South
Africa

(August 28, 2024)

###### Abstract

Recently, the mimetic gravitational theory has gained much attention in the
frame of cosmology as well as in the domain of astrophysics. In this study, we
show that in the frame of mimetic gravitation theory we are not able to derive
an isotropic model. As a result, our focus shifts towards combining mimetic
gravitational theory with the Lagrangian multiplier. The field equations of a
static isotropic gravitational system that controls the geometry and dynamics
of star structure are studied in the frame of mimetic theory coupled with a
Lagrangian multiplier using a non-linear equation of state. An energy density
is assumed from where all the other unknowns are fixed and a new isotropic
model is derived. The physical analysis of this model is studied from
different viewpoints and consistent results compatible with a realistic
isotropic star are investigated analytically and graphically. Ultimately, we
demonstrate the stability of the model in question by employing the adiabatic
index technique.

Mimetic gravitational theory coupled with Lagrangian multiplier, isotropic
solution, stability.

###### pacs:

04.50.Kd, 04.25.Nx, 04.40.Nr

##  I Introduction

Recently, conclusive evidence developed proposing that Einstein’s theory of
general relativity (GR), is needy to be modified.The justifications for this
observation stem from the challenges in renormalizing general relativity and
the uncertain behavior exhibited in high gravity regions, such as the exterior
of black holes and neutron stars. Additionally, the confirmed accelerated
expansion of our universe can not be explained by GR. To be able to use GR to
explain the phenomena of accelerated expansion of our universe an assumption
of the presence of exotic matter fields like dark matter and dark energy must
be taken into account. Up to date no experimental support for these was
imminent. Another method is to try to amend Einstein’s GR somehow so that we
keep its basic gains. Despite these issues in GR, it still submits success in
the solar system. The success of the Event Horizon Telescope in capturing the
image of a black hole’s shadow in M87 Akiyama  _et al._ (2019) and the
advancements made in detecting gravitational waves Abbott  _et al._ (2016)
have significantly bolstered the position of General Relativity as the
preeminent gravitational theory compared to other theories of gravity.
However, aspects that GR’s have not clarified must also face. We advocate the
perspective that a modification of the governing field equations within the
geometric sector holds the key to addressing these unresolved concerns.

Modified gravitational theories have made significant progress and performance
in explaining some of the unsolved issues in GR. There are many modified
theories of gravity that can overcome the shortcomings of GR. One of the
modification of GR is to add a scalar field. In the frame of a scalar field
coupled with Ricci scalar a static neutron star perspective using two types of
cosmological inflationary attractor theories, i.e., the induced inflationary
attractors and the quadratic inflationary attractors are investigated
Oikonomou (2023a). Among these modification is the $f(R)$ gravitational theory
(see for exmaple Nojiri  _et al._ , 2019; Odintsov and Oikonomou, 2019). In
the frame of $f(R)$ a study of the neutron star phenomenology of $R^{p}$
attractor theories in the Einstein frame have been analyzed Oikonomou (2023b).
In this study we are interested in another modification of GR, i.e., we are
focused on the gravitational mimetic theory which has been suggested as a
fresh approach to studying the problem of dark matter.Chamseddine and Mukhanov
(2013). This concerns the introduction of a mimetic scalar field denoted as
$\eta$, which, despite lacking dynamics in its construction, plays a crucial
role in imparting dynamism to the longitudinal degree of freedom within the
gravitational system. The gravitational system’s dynamic longitudinal degree
serves as an analogue to pressureless dark matter, mimicking its properties
Chamseddine and Mukhanov (2013). The problem of cosmological singularities
Chamseddine and Mukhanov (2017a) and the singularity at the core of a black
hole Chamseddine and Mukhanov (2017b) can be effectively tackled through the
altered variant of mimetic gravitational theory. Furthermore, the
gravitational theory of mimetic has substantiated that the propagation of
gravitational waves at the speed of light aligns in perfect accordance with
the discoveries made from the event GW170817 and the corresponding optical
observations Casalino  _et al._ (2018, 2019); Sherafati  _et al._ (2021).
Furthermore, it has been demonstrated that mimetic theory can explore the
coherent rotational patterns observed in spiral galaxies without relying on
the existence of dark matter particles Vagnozzi (2017); Sheykhi and Grunau
(2021). Lately, there has been a significant surge of enthusiasm surrounding
the cosmological framework due to the emergence of the mimetic theory
Chamseddine  _et al._ (2014); Baffou  _et al._ (2017); Dutta  _et al._ (2018);
Sheykhi (2018); Abbassi  _et al._ (2018); Matsumoto (2016); Sebastiani  _et
al._ (2017); Sadeghnezhad and Nozari (2017); Gorji  _et al._ (2019, 2018a);
Bouhmadi-Lopez  _et al._ (2017); Gorji  _et al._ (2018b); Chamseddine  _et
al._ (2019a); Russ (2021); de Cesare  _et al._ (2020); Cárdenas  _et al._
(2021); Hosseini Mansoori  _et al._ (2021); Arroja  _et al._ (2018) and black
holes physics Deruelle and Rua (2014); Myrzakulov and Sebastiani (2015);
Myrzakulov  _et al._ (2016); Ganz  _et al._ (2019); Chen  _et al._ (2018);
Nashed (2018b); Ben Achour  _et al._ (2018); Brahma  _et al._ (2018); Zheng
_et al._ (2017); Shen  _et al._ (2019); Nashed  _et al._ (2019); Chamseddine
_et al._ (2019b); Gorji  _et al._ (2020); Sheykhi (2020, 2020); Nashed and
Nojiri (2021); Chamseddine (2021); Zheng (2021); Bakhtiarizadeh (2021); Nashed
(2021). The theory has further extended its scope to include $f(R)$ mimetic
gravity, incorporating additional insights and explanations Nojiri and
Odintsov (2014); Odintsov and Oikonomou (2016a); Oikonomou (2016a, b, c);
Myrzakulov and Sebastiani (2016); Odintsov and Oikonomou (2015, 2016b, 2016c);
Nojiri  _et al._ (2017); Odintsov and Oikonomou (2018); Bhattacharjee (2021);
Kaczmarek and Szczesniak (2021); Chen  _et al._ (2021) and mimetic Gauss-
Bonnet gravity Astashenok  _et al._ (2015); Oikonomou (2015); Zhong and
Elizalde (2016); Zhong and Sáez-Chillón Gómez (2018); Paul  _et al._ (2020).
Specifically, a comprehensive framework combining early inflation and late-
time acceleration within the context of mimetic $f(R)$ gravity was formulated
Nojiri  _et al._ (2016). It has been stressed that within the context of
mimetic $f(R)$ gravity, the period of inflation can be identified Nojiri  _et
al._ (2016).

Nowadays, the mimetic theory is one of the most compelling theories of
gravity, which without introducing any additional field of matter, represents
the dark side of the universe which is represented as a geometric effect. The
observations ensure that approximately 26$\%$ of the energy content of the
universe is related to the dark matter sector, while approximately 69$\%$
construct the dark energy Yang and Gong (2020). Many facts ensure the presence
of dark matter and dark energy Seljak  _et al._ (2006). Dark energy, which has
gained significance in recent times, is believed to be a smooth element
characterized by negative pressure. It possesses an anti-gravity
characteristic and is actively propelling the universe, playing a key role in
the accelerated expansion of the cosmos Oks (2021). Dark matter performs two
crucial functions in the development of the universe: Firstly, it provides the
necessary gravitational force for the rotation of spiral galaxies and galaxy
clusters. Secondly, it plays a significant role as an essential component in
the amplification of disturbances and the formation of structures during the
early stages of the universe. As a result, dark matter begins to condense into
an intricate system of dark matter halos, whereas regular matter undergoes
collapse due to photon radiation, eventually settling into the well-formed
potential of dark matter. In the absence of dark matter, the process of galaxy
formation in the universe would be significantly more extensive than what has
been observed.

The structure of the current investigation is outlined as: In Sec. II, we
introduce the fundamental principles of the mimetic theory combined with the
Lagrangian multiplier. In Sec. III, we list the necessary conditions that must
be obeyed by any realistic isotropic model. Also, in Sec.n III we show that
the model under consideration satisfies all the necessary conditions,
analytically and graphically, that must possess by any realistic isotropic
star. In Section IV we study the stability of the model presented in this
study using the adiabatic index. Final Section is devoted to discussing the
main results of the present study.

##  II Isotropic solution in the mimetic theory combined with the Lagrange
multiplier

What is called “dark matter of mimetic” was delivered in the scientific
society in Chamseddine  _et al._ (2014). Despite of this mimetic theories had
already been discussed in Lim  _et al._ (2010); Gao  _et al._ (2011); Nashed
and Saridakis (2023); Nashed and Nojiri (2022); Nashed (2021); Nashed and
Nojiri (2021); Nashed (2018a); Nashed  _et al._ (2019); Nashed (2018b);
Capozziello  _et al._ (2010); Sebastiani  _et al._ (2017).

According to the theory of mimetic, the metric $g_{\alpha\beta}$, which is a
physical one, is linked to the metric $\bar{g}_{\alpha\beta}$, which is an
auxiliary metric, and to the mimetic scalar field $\eta$ by the conformal
transformation:

$g_{\alpha\beta}=-\left(\bar{g}^{\mu\nu}\partial_{\mu}\eta\partial_{\nu}\eta\right)\bar{g}_{\alpha\beta}\,,$
(1)

where the metric $\bar{g}^{\alpha\beta}$ undergoes conformal transformations,
specifically $\bar{g}^{\alpha\beta}\to\Omega\bar{g}^{\alpha\beta}$, the metric
$g^{\alpha\beta}$ remains invariant, meaning that it remains unchanged.

In the present study, we will coin the mimetic-like gravity coupled with the
Lagrange multiplier. The action of the gravity model resembling the mimetic
theory, in conjunction with the Lagrange multiplier $\lambda$ and the function
$\omega$, is described as follows: has the form111It was shown in Nojiri and
Nashed (2022) that the function $\omega$ is necessary in the construction of
the mimetic theory which allows us to achieve the geometric characteristics of
a black hole, including the presence of one or more horizons. Nojiri and
Odintsov (2014):

$S=\int\mathrm{d}x^{4}\sqrt{-g}\left\\{R+\lambda\left(g^{\mu\nu}\omega\partial_{\mu}\eta\partial_{\nu}\eta+1\right)\right\\}+L_{\mathrm{matt}}\,,$
(2)

where $L_{\mathrm{matt}}$ is the Lagrangian of the matter field and $\eta$ is
the mimetic scalar field. The field equations can be obtained by taking the
variation of the action (2) with respect to the metric tensor $g_{\mu\nu}$,
resulting in the following expressions:

$\displaystyle
0=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\frac{1}{2}g_{\mu\nu}\left\\{\lambda\left(g^{\rho\sigma}\omega\partial_{\rho}\eta\partial_{\sigma}\eta+1\right)\right\\}-\lambda\partial_{\mu}\eta\partial_{\nu}\eta+\frac{1}{2}T_{\mu\nu}\,,$
(3)

where $T_{\mu\nu}$ is the energy-momentum tensor corresponding to the matter
field. Additionally, differentiating the action (2) with respect to the
mimetic field $\eta$ yields:

$2\nabla^{\mu}(\lambda\omega\,\partial_{\mu}\eta)=0\,.$ (4)

When the action (2) is varied with respect to the Lagrange multiplier
$\lambda$, the following outcome is obtained:

$g^{\rho\sigma}\omega\partial_{\rho}\eta\partial_{\sigma}\eta=-1\,.$ (5)

It is the time to apply the field equations (3) and (4), using the constrains
of Eq. (5), to the following spherically symmetric spacetime

$ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f_{1}(r)}-r^{2}\left(d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right)\,,$ (6)

to derive an interior solution. The functions $f(r)$ and $f_{1}(r)$ mentioned
here are unfamiliar functions that will be determined by solving the set of
field equations.

Furthermore, we assume that $\eta$ is solely dependent on $r$. Using Eqs. (3)
and (4) to the spacetime (6), yields the following set of differential
equations. The $(t,t)$-component of the field equation (3) is:

$\displaystyle\rho(r)=$
$\displaystyle\frac{1-f_{1}-rf^{\prime}_{1}}{r^{2}}\,,$ (7)

the field equation (3) can be expressed as the $(r,r)$-component.

$\displaystyle p(r)=$
$\displaystyle\frac{f_{1}f^{\prime}r-f+ff_{1}-\lambda\omega(r)\eta^{\prime
2}ff_{1}r^{2}}{r^{2}f}\,,$ (8)

the components of the field equation (3) in terms of $(\theta,\theta)$ and
$(\phi,\phi)$ have the following structure:

$\displaystyle p(r)=$ $\displaystyle\frac{2\,f_{1}f^{\prime\prime}fr-f^{\prime
2}f_{1}r+f\left(2\,f_{1}+f^{\prime}_{1}r\right)f^{\prime}+2\,f^{\prime}_{1}f^{2}}{4f^{2}r}\,,$
(9)

and the field equation (4) takes the form:

$0=2\lambda^{\prime}\omega
fr+\left[\omega^{\prime}fr+\omega\left(f^{\prime}r+4\,f\right)\right]\lambda\,,$
(10)

where $f\,\equiv f(r)$, $f_{1}\,\equiv f_{1}(r)$, $\omega\,\equiv\omega(r)$,
$\lambda\,\equiv\lambda(r)$, $f^{\prime}=\frac{df}{dr}$,
$f^{\prime}_{1}=\frac{df_{1}}{dr}$, $\eta^{\prime}=\frac{d\eta}{dr}$,
$\omega^{\prime}=\frac{d\omega}{dr}$, and
$\lambda^{\prime}=\frac{d\lambda}{dr}$. Furthermore, we make the assumption
that the energy-momentum tensor of the isotropic fluid can be represented in
the following manner:

$\displaystyle{T}_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}\,,$ (11)

In this case, $\rho$ represents the energy density, $p$ denotes the pressure,
and $u^{\mu}$ is a timelike vector defined as $u^{\mu}=[1,0,0,0]$.In this
particular investigation, we consider the matter content to be characterized
by the energy density $\rho$ and the pressure $p$ respectively.222In the
entirety of this research, we will utilize geometrized units where the
constants $G$ and $c$ are set to 1..

Assuming the fluid under consideration has a perfect fluid verifying the
equation of state (EoS), $p=p\left(\rho\right)$. Using the conservation law of
matter gives:

$\displaystyle 0=\nabla^{\mu}\,T_{\mu
r}=2f\frac{dp}{dr}+f^{\prime}\left(\rho+p\right)\,.$ (12)

In this particular context, we make an assumption that the energy density and
pressure of the system exhibit variations based on the radial coordinate.

If the form of EoS $\rho=\rho(p)$ is presented, then Eq. (12) yield:

$\displaystyle\frac{1}{2}\ln
f=-\int^{r}dr\frac{\frac{dp}{dr}}{\rho+p}=-\int^{p(r)}\frac{dp}{\rho(p)+p}\,.$
(13)

In the interior of the star, Eq. (13) can be used but in the exterior, Eq.
(13) cannot be used. Nevertheless, it is possible to assume that both $f(r)$
and its derivative $f^{\prime}(r)$ exhibit continuity at the surface of the
star.

Considering the count of unknown functions and independent equations, we find
a total of six unknown functions within the compact star. To address this, we
will employ the constraint given by Eq. (5), which states that
$\eta=\frac{1}{\sqrt{-\omega f_{1}}}$. These unknown functions include two
metric potentials, namely $f(r)$ and $f_{1}(r)$ in (6), as well as the
Lagrangian multiplier, the function $\omega$, the energy density $\rho$, and
the pressure $p$ present in the action (2). However, we have 3 independent
equations, that are, the three components, Eqs. (7), (8), and (9) of the field
equation (3) of the mimetic equations, an equation of state $\rho=\rho(p)$,
and the conservation law (12). As we mentioned above, the scalar field
equation (4) can be obtained from the field equations (3) corresponding to the
mimetic equation and the conservation law $\nabla_{\mu}T^{\mu\nu}=0$ of the
matter, and therefore the scalar field equation (4) is not independent. Hence,
we have $6-5=1$ undetermined function remaining from the equations. To address
this remaining aspect, we opt for the specific configuration of energy density
$\rho=\rho(r)$ within the compact celestial object. Moreover, outside the
star, we have $\rho=p=0$, and the unknown functions, $f(r)$, $f_{1}(r)$ and
$\eta$ (or $\omega(\eta)$). We have 3 independent equations, that are, the
three components, Eqs. (7), (8), and (9). Thus there are $5-4=1$ function
undetermined from the equations, again.

If we consider a compact star like neutron star, one usually consider the EoS
as:

  1. 1.

Energy-polytrope

$\displaystyle p=k\rho^{1+\frac{1}{s}}\,,$ (14)

where $k$ and $s$ are constants. It is well known that for neutron star, $s$
lies in the interval $s\in[0.5\,,\,1]$.

  2. 2.

Mass-polytrope

$\displaystyle\rho=\rho_{m}+s_{1}p\,,\qquad\qquad
p=m_{m}\rho_{m}^{1+\frac{1}{s_{m}}}\,,$ (15)

with $\rho_{m}$ being the rest mass energy density and $m_{m}$, $s_{1}$, and
$s_{m}$ are constants.

Now, let us turn our attention to the study of the energy-polytrope case. Then
EoS (14) can be rewritten as:

$\displaystyle\rho=\tilde{k}p^{({1+\frac{1}{\tilde{s}}})}\,,\quad\tilde{k}\equiv
k^{-\frac{1}{1+\frac{1}{s}}}\,,\quad\tilde{s}\equiv\frac{1}{\frac{1}{1+\frac{1}{s}}-1}=-1-s\,.$
(16)

Eq. (13) can take the form:

$\displaystyle\frac{1}{2}\ln
f=-\int^{p(r)}\frac{dp}{\tilde{k}p^{1+\frac{1}{\tilde{s}}}+p}=\frac{c_{1}}{2}+\tilde{s}\ln\left(1+{\tilde{k}}^{-1}p^{-\frac{1}{\tilde{s}}}\right)=\frac{c_{1}}{2}-\left(1+s\right)\ln\left(1+k\rho^{\frac{1}{s}}\right)\,,$
(17)

where $c_{1}$ is a constant of integration. Using the same method of polytrope
we get for mass-polytrope (15) the function $f$ as:

$\displaystyle\frac{1}{2}\ln
f=\frac{\tilde{c}}{2}+\ln\left(1-k_{m}\rho_{m}{}^{\frac{1}{s_{m}}}\right)\,,$
(18)

where $\tilde{c}$ is a constant of integration.

To provide an illustrative example, taking into account one of the mentioned
equations of state, we can make an assumption about the profile of
$\rho=\rho(r)$ as follows.

$\displaystyle\rho=\left\\{\begin{array}[]{cc}\rho_{0}\left(1-\frac{r}{{R}}\right)&\
\mbox{when}\ r<R\\\ 0&\ \ \mbox{when}\ r\geq R\end{array}\right.\,.$ (21)

In this scenario, $\rho_{0}$ represents a fixed value expressing the energy
concentration at the core of the condensed celestial object, whereas $R$
symbolizes an additional fixed value denoting the size of the compact star’s
outer boundary. As clear from Eq. (21), the energy density $\rho$ vanishes at
the surface $r=R$. By using the energy-polytrope EoS (14) or the mass-
polytrope EoS (15), we find that the pressure $p$ also vanishes at the
surface. We have introduced the mass parameter $M$ as a constant value, which
is associated with the mass of the compact star. This parameter is defined
specifically for the polytropic equation of state (EoS) as follows:

$\displaystyle M=4\pi\int_{0}^{R}y^{2}\rho(y)dy=\frac{\pi\rho_{0}R^{3}}{3}\,.$
(22)

Then Eq. (17) gives,

$\displaystyle
f=\frac{\mathrm{e}^{c_{1}}}{\left(1+k\rho_{0}\left(1-\frac{r}{{R}}\right)\right)^{4}}\,.$
(23)

Using Eq. (21) in (7) we get:

$\displaystyle f_{1}=1-\frac{8\pi r^{2}}{3}+\frac{2r^{3}}{R}\,.$ (24)

The Lagrangian multiplier of the above model has the form

$\displaystyle\lambda(r)=$ $\displaystyle\frac{c_{2}\,\left(R+kR-
kr\right)^{2}}{{r}^{5/2}}\,.$ (25)

To finalize the determination of the remaining unknowns we assume, for
simplicity, the form of the function $\omega=c_{3}r$ and the mimetic scalar
field becomes:

$\displaystyle\eta(r)=$ $\displaystyle\sqrt{c_{3}r\left(\frac{8\pi
r^{2}}{3}-\frac{2r^{3}}{R}-1\right)}\,.$ (26)

To finalize this section we stress on the fact that if we follow the same
procedure and put $\lambda(r)=0$ in Eqs. (7), (8), and (9) we get a system
that we cannot derive from it an isotropic model.

##  III Necessary conditions for a real physical star

Any physical reliable isotropic star model must obey the below conditions in
the interior configurations:
$\bullet$ It is essential for the metric potentials ($g_{tt}$ and $g_{r}r$) to
provide precise explanations for the density and momentum components by
establishing unambiguous definitions and displaying consistent patterns within
the core of the star and its interior.
$\bullet$ Within the interior of the star, it is required that the energy
density $\rho$ maintains a non-negative value, that is, $\rho\geq 0$.
Additionally, the energy density has a finite positive value at the central
region of the star and shows a decreasing pattern as it extends towards the
surface, characterized by the condition $\frac{d\rho}{dr}\leq 0$.
$\bullet$ Inside the fluid configuration, it is necessary for the pressure $p$
to be positive or zero ($p\geq 0$). Furthermore, within the interior of the
star, it is expected that the pressure decreases with respect to the radial
coordinate, as indicated by the condition $\frac{dp}{dr}<0$. On the outermost
boundary of the star, specifically at the surface where the distance from the
center is denoted as $r=R$, the pressure $p$ should be precisely zero. This
implies that there is no pressure exerted at the star’s outer edge.
$\bullet$ The energy conditions of an isotropic fluid sphere are given by:
(i) The null energy condition (NEC) implies that the energy density $\rho$
must be greater than zero.
(ii) According to the weak energy condition (WEC), the sum of the pressure $p$
and the energy density $\rho$ must be greater than zero, i.e., $p+\rho>0$.
(iii) In accordance with the strong energy condition (SEC), the sum of the
energy density $\rho$ and three times the pressure $p$ must be greater than
zero, i.e., $\rho+3p>0$.
$\bullet$Furthermore, to ensure a realistic model, the causality condition
must be satisfied within the interior of the star. This condition imposes a
restriction on the speed of sound, requiring it to be less than 1. In this
context, assuming the speed of light $c$ is equal to 1, the condition can be
expressed as $\frac{dp}{dr}>0$ and $1>\frac{dp}{dr}$.
$\bullet$ Finally, the adiabatic index must has a value more than
$\frac{4}{3}$.

It is our purpose to study the above conditions on the isotropic model and see
if it is real model or not.

##  IV The physical characteristics of the model

To determine whether the model described by Eqs. (17), (21), and (24)
corresponds to a realistic stellar structure, we will examine the following
aspects:

###  IV.1 Non singular model

i- The components of the metric potentials $g_{tt}$ and $g_{rr}$ fulfill the
following conditions:333We will rewrite all the physical quantities like
metric potentials, density and pressure in terms of the dimensionless quantity
$x$ where $x=\frac{r}{R}$.,

$\displaystyle f(x\rightarrow
0)=\frac{e^{c}}{(1+k\rho_{c})^{4}}\,\qquad\textrm{and}\qquad
f_{1}(x\rightarrow 0)=1.$ (27)

As a consequence of this requirement, it is necessary for the metric
potentials $g_{tt}$ and $g_{rr}$ to possess finite values at the central point
of the stellar configuration. Furthermore, their derivatives also possess
finite values at the center of the star, specifically:
$a^{\prime}(r\rightarrow 0)=0$ and $a^{\prime}_{1}(r\rightarrow
0)=\frac{4c_{1}k}{R(1+k)^{5}}$. The mentioned limitations guarantee that the
measurement remains consistent at the core and demonstrates a positive
characteristic within the inner region of the star.
ii-At center of star, the density (21) and pressure (14) take the following
form:

$\displaystyle\rho(x\rightarrow 0)=\rho_{0}\,\qquad p(x\rightarrow
0)=k\rho_{0}{}^{2}.$ (28)

By examining Eq. (28), when $\rho_{0}>0$ and $k>0$, it becomes evident that
the density and pressure in the central region of the star remain consistently
positive. Apart from that, if $\rho_{0}$ or $k$ is non-positive, the density
and pressure can become negative. These observations align with the
information depicted in Figures 1 0(a), 1 0(b), and 1 0(c), providing further
consistency to the discussed aspects.
iii-The density and pressure gradients of our model are provided in the
following manner:

$\displaystyle\rho^{\prime}=-\rho_{0},\qquad\qquad
p^{\prime}=-{2k\rho_{0}{}^{2}\,(1-x)}\,.$ (29)

Here $\rho^{\prime}=\frac{d\rho}{dx}$ and $p^{\prime}=\frac{dp}{dx}$. Equation
(29) illustrates that the components of the energy-momentum derivatives
exhibit negative values.
iv-The calculation of the speed of sound, using relativistic units where the
values of $c$ and $G$ are equal to 1, is achieved by Herrera (1992):

$\displaystyle v_{r}{}^{2}=\frac{dp}{d\rho}=2\,k\,\rho_{0}\,(1-x)\,.$ (30)

At this point, we are prepared to graphically represent the aforementioned
conditions in order to observe their behaviors. In Figure 1 0(a), we
illustrate the characteristics of the metric potentials. As depicted in Figure
1 0(a), the metric potentials take on the values $a_{1}(x\rightarrow 0)=0.4$
and $a(x\rightarrow 0)=1$ as $x$ approaches 0. This implies that within the
central region of the star, both metric potentials exhibit finite positive
values.

(a) Metric potentials, $f(r)$ and $f_{1}(r)$ given by Eq. (23) and (24)

(b) Density

(c) Pressure

Figure 1: A visual representation is provided, illustrating the relationship
between the metric potentials (23) and (24), in comparison to the
dimensionless x; 0(b) the profile of density; and 0(c) profile of pressure. We
have put $\rho_{0}=1$ and $K=0.4$.

We proceed to create graphs illustrating the density and pressure, as outlined
in Equation (21), represented in Figure 1 0(b) and 1 0(c). As depicted in
Figure 1 0(b) and 1 0(c), the components of the energy-momentum exhibit
positive values, which are consistent with predictions for a reasonable
stellar arrangement. Moreover, as depicted in Figure 1 0(b) and 0(c), the
components of the energy-momentum tensor exhibit elevated values at the core
and gradually decrease as they approach the periphery. These observed patterns
are characteristic of a plausible star.

(a) Gradient of density

(b) Gradient of pressure

Figure 2:  A graph is shown, displaying the variations in the gradient of
density and pressure from Equation (21), plotted against the dimensionless
value of $x$.

Figure 2 illustrates the presence of adverse values in the derivatives of the
components of the energy-momentum tensor, indicating a uniform decrease in
both density and pressure throughout the entirety of the star’s structure.

(a) Sound speed

(b) Relationship between mass and radius

(c) The compactness

Figure 3: A diagram is presented, depicting the behavior of various quantities
with respect to the dimensionless value of $x$. Specifically, the plot
includes the sound speed (labeled as 2(a)), the relation between mass and
radius (labeled as 2(b)), and the level of compactness exhibited by the
celestial body(labeled as 2(c)).

Figure 3 presents visual depictions of the speed of sound, the relationship
between mass and radius, and the compactness parameter. As depicted in Fig.
32(a), the speed of sound is found to be less than one, which verifies that
the causality condition is not violated within the interior of the stellar
configuration when the parameter of the equation of state (EoS) is $k<0.5$.
Furthermore, Fig. 32(c) illustrates that the compactness of our model is
restricted within the interval of $0<C<0.003$, where $C$ is defined as the
ratio of $M$ to $xR$ in the stellar arrangement.

(a) Null energy conditions

(b) Weak energy conditions

(c) Strong energy condition

Figure 4: A graph is generated to illustrate the behavior of the null, weak,
and strong energy conditions, as determined by Equation (21), plotted against
the dimensionless value of $x$.

The energy conditions are depicted in Figure 4, showcasing the characteristics
of each condition. More specifically, Fig. 4 3(a), 3(b), and 3(c) display the
presence of positive values for the NEC (Null Energy Condition), WEC (Weak
Energy Condition), and SEC (Strong Energy Condition) respectively. This
verification provides assurance that all energy conditions are met across the
entire stellar configuration, aligning with the criteria for a physically
feasible stellar model.

Figure 5: A plot is presented to visualize the equation of state (EoS) as a
function of the radial coordinate $r$, labeled as 5. Additionally, the
redshift is plotted and labeled as 5.

Figure 5 represents the plot of the Equation of State (EoS). In particular,
Figure 5 5 indicates that the EoS exhibits a linear behavior.

##  V The model’s stability

At this point, we are prepared to examine the stability concern of our model
by conducting tests involving the the index of adiabatic and the static case.

Figure 6: Plot of the gravitational, and the hydrostatic forces vs. the
dimensionless $x$.

###  V.1 An adiabatic index

To investigate the stable balance of a spacetime that possesses spherical
symmetry, an analysis of the adiabatic index can be conducted. The adiabatic
index plays a vital role in evaluating the stability requirement, serving as
an essential instrument for this purpose. Specifically, the adiabatic
perturbation, denoted as $\Gamma$, is defined as follows Chandrasekhar (1964);
Merafina and Ruffini (1989); Chan  _et al._ (1993); Nashed (2011); Nashed and
Capozziello (2021); Nashed (2011); Nashed and Capozziello (2020); Roupas and
Nashed (2020):

$\displaystyle\Gamma=\left(\frac{\rho+p(x)}{p(x)}\right)\left(\frac{dp(x)}{d\rho(x)}\right)\,.$
(31)

If the adiabatic index $\Gamma$ is greater than $\frac{4}{3}$, a Newtonian
isotropic sphere will have a stable equilibrium Heintzmann and Hillebrandt
(1975).

Using Eq. (31), we get

$\displaystyle\Gamma=2+2k\rho_{0}(1-x).$ (32)

Figure 6 displays the adiabatic index $\Gamma$. The graph clearly indicates
that the value of $\Gamma$ remains consistently above $4/3$ throughout the
star’s interior. Therefore, we can infer that the stability requirement is
met.

###  V.2 Stability in the stationary state

Another approach to validate the stability of model (21) involves
investigating the static state proposed by Harrison, Zeldovich, and Novikov
Harrison  _et al._ (1965); Zeldovich and Novikov (1971, 1983). In a previous
study by Harrison, Zeldovich, and Novikov, it was established that a stable
configuration of a star necessitates a positive and increasing derivative of
mass with respect to the central density $\rho(x\rightarrow 0)$, denoted as
$\frac{\partial M}{\partial\rho_{0}}>0$. By applying this condition, we can
determine the specific form of the central density as follows:

$\displaystyle\rho(r\to 0)=\rho_{0}\,.$ (33)

Subsequently, utilizing Eq. (22), we can derive the mass corresponding to the
central density, denoted as:

$\displaystyle
M(\rho_{0})4\pi\int_{0}^{R}y^{2}\rho(y)dy=\frac{\pi\rho_{0}R^{3}}{3}\,.$ (34)

The behavior of the mass derivative with respect to the central density can be
described by the following pattern:

$\displaystyle\frac{\partial M(\rho_{0})}{\partial\rho_{0}}=\frac{\pi
R^{3}}{3}\,.$ (35)

Equations (34) and (35) guarantee the verification of stability condition of
our model.

##  VI Discussion and conclusions

The primary objective of this investigation is to analyze a compact star’s
static configuration, which possesses both spherical symmetry and is within
the framework of mimetic gravity theory. This theory incorporates two main
components: a scalar field and a Lagrangian multiplier. In our formulation, we
demonstrated the ability to construct a model that accurately replicates the
profile of a given spherically symmetric spacetime, regardless of the equation
of state (EoS) for matter and energy density. The shape of the density
function $\rho(x)$ is of utmost importance in determining the radius $R$ and
mass $M$ of the compact star. By manipulating the Lagrangian multiplier
$\lambda(\eta)$, it becomes feasible to establish a flexible connection
between the radius $R$ and the compact star. This creates a situation where
the Lagrangian multiplier $\lambda(x)$ and the equation of state (EoS)
describing the model exhibit a degenerate relationship. As a result, it is
clear that relying solely on the mass-radius relation is inadequate for fully
constraining the model.

To illustrate this further, we take a closer look at the polytrope equation of
state (EoS) given by (14). By selecting a specific form for the density in Eq.
(21), we construct a practical isotropic model. We then proceed to investigate
the physical characteristics of this model. Through rigorous analysis using
different analytical techniques and validations, we carefully scrutinize the
obtained analytic solution. This comprehensive examination enables us to
observe and analyze the physical behavior manifested by our solution.

It is important to highlight that the preceding discussion confirms the
satisfaction of all the physical conditions by the spherically symmetric
interior spacetime configuration considered in this study within the framework
of mimetic gravitational theory coupled with a Lagrangian multiplier. However,
it should be noted that an alternative form of mimetic gravitational theory
that does not involve the coupling with a Lagrangian multiplier may not
support the existence of the isotropic model, as evidenced by equations (7),
(8), and (9).

Moreover, the field equations governing the equilibrium of rapidly rotating
neutron stars in scalar-tensor theories of gravity, as well as representative
numerical solutions are discussed in Doneva  _et al._ (2013). New models of
slowly rotating, perfect-fluid neutron stars are constructed by extending the
classical Hartle-Thorne formalism to generic scalar-tensor theories of gravity
Pani and Berti (2014). An investigation of self-consistently slowly rotating
neutron and strange stars in R-squared gravity is investigated in Staykov  _et
al._ (2014). A study of static neutron stars in the framework of a class of
non-minimally coupled inflationary potentials have been presented in Oikonomou
(2021). Are all the previous studies presented in Doneva  _et al._ (2013);
Pani and Berti (2014); Staykov  _et al._ (2014); Oikonomou (2021) can be
discussed in the framework of the present study? This will be done elsewhere.

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