# Investigation of Traversable Wormhole Solutions in Modified $f(R)$ Gravity
with Scalar Potential

Adnan Malik<EMAIL_ADDRESS><EMAIL_ADDRESS>School of
Mathematical Sciences, Zhejiang Normal University,
Jinhua, Zhejiang, China.  Department of Mathematics, University of Management
and Technology,
Sialkot Campus, Lahore, Pakistan  Tayyaba Naz<EMAIL_ADDRESS>National
University of Computer and Emerging Sciences,
Lahore Campus, Pakistan.  Abdul Qadeer<EMAIL_ADDRESS>Department of
Mathematics, University of Management and Technology,
Sialkot Campus, Lahore, Pakistan  M. Farasat Shamir<EMAIL_ADDRESS>National University of Computer and Emerging Sciences,
Lahore Campus, Pakistan.  Zeeshan Yousaf<EMAIL_ADDRESS>Department of
Mathematics, University of the Punjab,
Quaid-i-Azam Campus, Lahore-54590, Pakistan.

###### Abstract

Abstract

The objective of this manuscript is to investigate the traversable wormhole
solutions in the background of the $f(R,\phi)$ theory of gravity, where $R$ is
the Ricci scalar and $\phi$ is the scalar potential respectively. For this
reason, we use the Karmarkar criterion for traversable static wormhole
geometry to create a wormhole shape function. The suggested shape function
creates wormhole geometry that links two asymptotically flat spacetime regions
and meets the necessary requirements. The embedding diagram in three-
dimensional Euclidean space is also discussed in order to demonstrate the
wormhole configurations. For our current analysis, we choose the suitable
values of free parameters for $f(R,\phi)$ gravity models to discuss the
wormhole geometry. It can be observed that our proposed shape function
provides the wormhole solutions with less amount of exotic matter. It can be
noticed that energy conditions especially null energy conditions are violated
for all considered models. The violation of energy conditions indicates the
existence of exotic matter and wormhole geometry. It is concluded that the
shape function acquired through the Karmarkar technique yields validated
wormhole configurations with even less exotic matter correlating to the chosen
$f(R,\phi)$ gravity models.
Keywords: Traversable wormholes geometry; Karmarker condition; $f(R,\phi)$
theory of gravity; Energy constraints.

##  I Introduction

Exploring such types of astonishing objects as wormholes have been a
fascinating topic of discussion in cosmological literature. A wormhole can be
defined as a fictitious topological feature channel that connects two distinct
spacetimes in that manner; time travel will be reduced. A wormhole must have
exotic fluid under the application of the General theory of relativity (GR),
which tends to interrupt the null energy condition. In 1935, the well-known
physicist Einstein along with Nathan Rosen suggested the concept of “bridges”
101 , which are known as the wormholes or Einstein-Rosen bridges. The goal of
these bridges is to help link two disparate places in space-time, hence
shortening the distance between them. It is more important to note that a
wormhole is the solution of Einstein field equations in which gravity acts as
“tunnels”. Wormhole is made up of two parts: a “mouth” and a “throat”. The
mouth is the entrance of the wormhole and the wormhole throat that connects
the mouth. While discussing the mathematical answer for black holes, the first
wormhole-like solutions were discovered. These specific answers lead to an
expanded version, in which the geometric explanation is loaned to two copies
of the black hole geometry. These black holes are linked by a “throat”: the
Einstein-Rosen bridge or wormhole, where the throat is described as a
dynamical entity dedicated to the two holes that pinch off extraordinarily and
fast into a limited connection between them. A wormhole may have connected to
very enormous measurements i.e. billions of light-years or more, the smallest
distance being a handful of meters, connecting various places or universes in
space. There are different types of wormholes mentioned over here:
Schwarzschild and Kerr wormholes (Event Horizons), Morris-Thorne wormholes
(Horizon-free), and Reissner-Nordstrom wormhole (Electrically charged).
Modified theories of gravities play a very important role in the investigation
of wormhole geometry. Mishra, et al. 2 studied the traversable wormhole
geometry in $f(R)$ theory gravity, and these wormhole solutions for some
assumed $f(R)$ functions are presented. The assumption of $f(R)$ is based on
the fact that its behavior changed with an assumed parameter rather than the
deceleration parameter. Sharif and Zahra 3 explored the wormhole solutions for
anisotropic and isotropic fluids as well as the barotropic equation of state
with radial pressure. Benedictis and Horvat 4 investigated the presence and
features of wormhole throats while discussing the wormhole solutions in
modified $f(R)$ theory of gravity theory __. Samanta and Godani 5 calculated
the range of the radius of the throat of the wormhole, where the energy
conditions are satisfied. The dynamical wormhole solutions were investigated
within the context of the $f(T)$ theory of gravity with anisotropic fluid,
assuming a generic dynamical spherically symmetric wormhole spacetime with a
specified shape function and scale factor 6 . Jamil, et al. 7 derived some new
exact solutions of static wormholes in $f(T)$ gravity and discussed
independent cases of the pressure components including isotropic and
anisotropic pressure. By giving an equation of state for the matter field and
applying the flaring out condition at the throat, Azizi 8 was able to get the
form function of the wormhole in the context of $f(R,T)$ gravity. Naz et al. 9
examined the geometry of static wormholes with anisotropic matter distribution
in the context of modified $f(G)$ gravity and considered the well-known
Noether and conformal symmetries to investigate the geometry of the wormhole.
By studying three distinct types of fluids, Malik and Nafees 10 explored the
spherically symmetric spacetime to debate the presence of wormhole geometry
for certain actual places. Shamir and Zia 11 investigated some feasible
regions for the existence of traversable wormhole geometries in the background
of the $f(R,G)$ theory of gravity. Samanta et al. 12 have investigated a
capacitative analysis of wormholes by discussing the energy conditions,
equation-of-state, and anisotropy parameter are analyzed in $f(R)$ gravity,
$f(R,T)$ gravity and general relativity. Sharma et al. 13 investigated the
solutions of traversable wormholes with normal matter in the throat within the
framework of symmetric teleparallel gravity $f(Q)$, where $Q$ is the non-
metricity scalar that defines the gravitational interaction. Sharif and Ikram
14 formulated the explicit expressions for matter variables and evaluate
wormhole solutions either specifying $f(G,T)$ model to construct the shape
function or taking a specific form of the shape function to determine $f(G,T)$
model and found that null energy condition for the effective energy-momentum
tensor is violated throughout the evolution in both cases while physically
acceptable wormhole solutions exist only for a considered $f(G,T)$ model.

Discussion of wormhole geometry utilizing the Karmarkar condition is a very
fascinating topic of interest among researchers. Karmarkar 15 was the first
who illustrated the compulsory condition for a static spherically symmetric
spacetime to be of embedding 1. That condition is known as the Karmarkar
condition which is very helpful to find the exact solutions of field
equations. Fayyaz and Shamir 16 constructed a wormhole shape function by using
the Karmarkar condition and observed that the proposed shape function connects
two asymptotically flat regions. Samanta and Godani 17 investigated the
traversable wormhole solutions within the background of the $f(R)$ theory of
gravity with a particular viable case and discussed the energy conditions
using the shape function. The same authors 18 analyzed the traversable
wormhole geometry and energy conditions with two different shape functions in
$f(R)$ theory of gravity. Golchin and Mehdizadeh 19 verified the standard
energy conditions for the asymptotically spherical, flat, and hyperbolic
wormhole solutions in the context of $f(R)$ modified gravity. Harko, et al. 20
presented the most general conditions in the context of modified gravity, in
which the matter threading the wormhole throat satisfies all of the energy
conditions, and it is the higher order curvature terms, which may be
interpreted as a gravitational fluid, that support these nonstandard wormhole
geometries. Kuhfittig 21 generalized a previous result based on the well-
established embedding theorems that connect the classical theory of relativity
to spacetimes of higher dimensions. The same author 22 discussed the idea of
embedding class one, which is applied to two diverse models, a complete
solution for a charged wormhole admitting a one-parameter group of conformal
motions and a new model to explain the flat rotation curves in spiral galaxies
without the need for dark matter. Malik along with his collaborators 23
examined the traversable wormhole models in the $f(R)$ theories of gravity by
applying the Karmarkar condition and spherically symmetric space-time. Tello-
Ortiz and Contreras 24 employed the class I approach to obtain wormhole
solutions in the framework of general relativity in two different ways and
proposed a suitable red-shift function to find its associated shape function.
Gul and Sharif 25 used the Noether symmetry approach to examine the viable and
stable traversable wormhole solutions in the framework of the $f(R,T^{2})$
theory by considering a specific model of this modified theory to obtain the
exact solutions of the Noether equations. Rahaman et al. 27 investigated a new
wormhole solution inspired by noncommutative geometry with the additional
condition of allowing conformal Killing vectors.

Modified theories of gravities may give a better explanation of the
accelerating expansion of the universe. One of these modified gravities is the
$f(R,\phi)$ theory of gravity, which is the mixture of the Ricci scalar $R$
and scalar potential $\phi$ 28 . Malik along with his collaborators 28a
discussed some dark energy cosmological models in the $f(R,\phi)$ theory of
gravity, which gives a better explanation for expanding the universe. Shamir
and Malik 29 considered the Friedmann-Robertson-Walker spacetime for finding
some exact solutions by using different values of the equation of state
parameters for discussion of nature of universe. Malik et al. 30 formulated
the inequalities in energy constraints and use the Hubble, deceleration, jerk,
and snap parameters to evaluate the feasibility of the models in the
$f(R,\phi)$ theory of gravity. Malik and Shamir 31 investigated the modified
field equations by using anisotropic and perfect fluid distributions for the
Godel-type universe inthe $f(R,\phi)$ theory of gravity. Malik et al., 32
studied three different Bianchi type lines elements, like Bianchi type-I,
Bianchi type-III, and Kantowski Sachs space-time in the framework of
$f(R,\phi)$ theory of gravity and found the exact solution of vacuum field
equations by taking the valuable assumption that the expansion scalar is
proportional to the shear scalar. Malik 33 discussed the cylindrical symmetric
solutions, especially levia-civita solutions in the background of $f(R,\phi)$
theory of gravity and examined the energy conditions in all cases. Myrzakulov
et al. 34 studied a class of models that can support early-time acceleration
with the emerging of an effective cosmological constant at high curvature in
the $f(R,\phi)$ theory of gravity. Stabile and Capozziello 35 investigated the
possibility of explaining theoretically the galaxy rotation curves by a
gravitational potential in the total absence of dark matter. Nozari and
Pourghasemi 36 studied the possible crossing of the phantom divide the line in
a DvaliGabadadzePorrati-inspired $f(R,\phi)$ braneworld scenario and showed
that there are appropriate regions of the parameter space which account for
late-time acceleration and admit crossing of the phantom divide line. Panda et
al. 37 studied a constant-roll inflationary model in the Palatini formalism
and calculated the tensor-to-scalar ratio and the spectral index using the
slow-roll parameters and the results obtained are matched with the Planck 2018
data in $f(R,\phi)$ theory of gravity.

Inspired by the above past relevant work, we investigated the traversable
wormhole solutions in $f(R,\phi)$ gravity using the Karmarkar condition. The
arrangement of this paper is as follows: Section III is dedicated to
presenting the Class I methodology. In section III, we presented the embedding
diagram for further analysis of wormhole geometry. Some basic formalism of the
$f(R,\phi)$ theory of gravity has been investigated in Section IV. In section
V, we considered four compatible models of $f(R,\phi)$ gravity and discussed
the energy conditions with graphical analysis. The comparison of our current
research work with past related work has been discussed in Section VI. Lastly,
we have concluded our work in Section VII.

##  II FORMATION OF TRAVERSABLE WORMHOLE SOLUTION FUNCTION

We have taken the spherically symmetric space-time as,

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

The non-zero Riemannian curvature components of above line element Eq. (1) are
given as,

$\\\
\begin{split}R_{1313}=R_{3131},~{}~{}~{}~{}~{}~{}R_{1212}=R_{2121},~{}~{}~{}~{}~{}~{}R_{1221}=R_{2112},\\\
R_{1331}=R_{3113},~{}~{}~{}~{}~{}~{}R_{2332}=R_{3223},~{}~{}~{}~{}~{}~{}R_{2323}=R_{3232},\\\
R_{1414}=R_{4141},~{}~{}~{}~{}~{}~{}R_{4242}=R_{2424},~{}~{}~{}~{}~{}~{}R_{4114}=R_{1441},\\\
R_{4224}=R_{2442},~{}~{}~{}~{}~{}~{}R_{4334}=R_{3443},~{}~{}~{}~{}~{}~{}R_{4343}=R_{3434}.\end{split}$
(2)

Now, we define the famous known Karmarkar constraint as

$R_{1414}=\frac{R_{1212}R_{3434}+R_{1221}R_{4334}}{R_{2323}}.$ (3)

By putting the values of all non-zero Riemannian curvature components in the
above Karmarkar relation, we get the following equation as

$\frac{1}{2}a^{\prime\prime}e^{2}+\frac{1}{4}a^{\prime
2}e^{a}-\frac{1}{4}a^{\prime}b^{\prime}e^{b}=\frac{\frac{1}{2}(rb^{\prime})(\frac{1}{2}ra^{\prime}e^{a-b}\sin^{2}\theta)+\frac{1}{2}(rb^{\prime})(\frac{1}{2}ra^{\prime}e^{a-b}\sin^{2}\theta)}{-e^{b}r^{2}\sin^{2}\theta+r^{2}\sin^{2}\theta}.$
(4)

Here, $a$ and $b$ are functions of the radial coordinate $r$. By simplifying,
we get the following relation as

$b^{\prime}a^{\prime}-a^{\prime\prime}2-a^{\prime
2}=\frac{2b^{\prime}a^{\prime}}{-e^{b}+1},$

where $e^{b}$ is not equal to 1. Now by interpreting the above equation, we
get

$e^{b}=Be^{a}a^{\prime 2}+1,$ (5)

Here $B$ is a constant of integration. To construct wormhole shape function,
we have taken the Morris-Thorne metric as

$ds^{2}=-e^{a(r)}dt^{2}+\frac{1}{1-\frac{\varepsilon}{r}}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.$ (6)

By comparing Eqs. (1) and (6), we obtain

$e^{b}=\frac{1}{1-\frac{\varepsilon}{r}}.$ (7)

For further calculation, we take a redshift function $a(r)$ 43 as

$a(r)=-\frac{2\eta}{r},$ (8)

where $\eta$ is a constant. Using the above Eqs. (5) ,(7) and (8), we get the
wormhole shape function as

$\varepsilon(r)=r-\frac{r^{5}}{r^{4}+4B\eta^{2}e^{\frac{-2\eta}{r}}}.$ (9)

By Morris and Thorne, wormhole shape function must fulfill the following
important conditions:

  * •

At the throat: $r=\varepsilon(r)$ at $r=r_{o}$ .

  * •

The essential condition
$\frac{\varepsilon(r)-r\varepsilon^{\prime}(r)}{\varepsilon^{\prime}(r)}>0$
must be satisfy at $r=r_{o}$.

  * •

$\varepsilon(r)$ must satisfy $\varepsilon^{\prime}(r)<1$.

  * •

For the maintenance of the asymptotical flatness of space time geometry,
$\frac{\varepsilon(r)}{r}\to 0$ as $r\to\infty.$

Here, $r$ represents the radial coordinate and $r_{o}$ is the throat radius,
and we have $r_{o}\leq r\leq\infty$. By evaluating Eq. (9) at the throat
condition, $\varepsilon(r_{o})-r_{o}=0$, we get trivial solutions at
$r=r_{o}$. To deal with this issue, we will add an arbitrary constant C in Eq.
(9) as

$\varepsilon(r)=r-\frac{r^{5}}{r^{4}+4B\eta^{2}e^{\frac{-2\eta}{r}}}+C.$ (10)

In the above equation, we apply the condition $r=\varepsilon(r)$ at $r=r_{o}$
and after some simplification, we get

$B=\frac{r_{o}^{4}(r_{o}-C)}{4C\eta^{2}e^{\frac{-2\eta}{r}}}.$ (11)

By using the above value of $B$ in Eq.(10), we get

$\varepsilon(r)=r-\frac{r^{5}}{r^{4}+r_{o}^{4}(r_{o}-C)}+C.$ (12)

We can observe that the behavior of $\varepsilon(r)$ is positive through the
graphical representation in Fig. (1). The graphical analysis of
$\varepsilon(r)-r$ is initially positive and then becomes negative with the
movement of radial coordinate $r$, as shown in the second plot of Fig. (1).
Furthermore, it is clear from the third plot of Fig. (1) that the
$\frac{\varepsilon(r)}{r}\to 0$ as $r\to\infty$. Moreover, left plot of Fig.
(2) shows the satisfaction of essential condition
$\frac{\varepsilon(r)-r\varepsilon^{\prime}(r)}{\varepsilon^{\prime}(r)}>0$
and it can be observed that $\varepsilon^{\prime}(r)<1$ is satisfied from the
right plot of Fig. (2).

Figure 1:  Graphical behavior of $\varepsilon(r)$, $\varepsilon(r)-r$ and
$\varepsilon(r)/r$ from left to right.

##  III EMBEDDING DIAGRAM FOR WORMHOLE GEOMETRY

In this section, we have used an embedding diagram to illustrate the behavior
of wormhole geometry. To understand the working of gravity in our universe,
the embedding diagram is very essential for it. Now we can set
$\theta=\frac{\pi}{2}$ and $t=const$ in spherically symmetric space-time.
Using this assumption in Eq. (6), then it comes as

$ds^{2}=\frac{r}{r-\varepsilon}dr^{2}+r^{2}d\phi^{2}.$ (13)

Now, we can write the above equation in three dimensional cylindrical
coordinates as,

$ds^{2}=dr^{2}+dh^{2}+r^{2}d\phi^{2},ds^{2}=dr^{2}(1+(\frac{dh}{dr})^{2})+r^{2}d\phi^{2}.$
(14)

After comparing Eq. (13) and Eq. (14), we obtain

$\frac{r}{r-\varepsilon}=1+(\frac{dh}{dr})^{2}.$

After simplification, we get

$\frac{dh}{dr}=(1-\frac{\varepsilon}{r})^{\frac{-1}{2}}.$ (15)

An example of an embedding diagram for the bottom and top universes can be
seen in Fig. (8) and Fig. (8), by utilizing $\theta=\frac{\pi}{2}$ and
$t=const$. Moreover, Eq. (15) indicates that the embedded surface at the
throat is vertical because $\frac{dh}{dr}\to\infty$. Furthermore, we have
studied that how space is spherically flat apart from the throat as
$\frac{dh}{dr}\to\infty$ at the time $r\to 0$. The embedded diagram is shown
in Fig. (8) and Fig. (8) for the upper and lower universes, respectively, in
radial coordinates with $2\pi$ rotation about the h axis.

##  IV $f(R,\phi$) Gravity

The action of $f(R,\phi)$ theory of gravity is given as 38 -42 ,

$S=\frac{1}{2K}\int
d^{4}x\sqrt{-g}(f(R,\phi)+w(\phi)\phi_{;}\phi^{;\alpha}+L_{m}),$ (16)

where,

  * •

$\textit{L}_{m}$ is lagrangian matter,

  * •

$\kappa=8\pi G$,

  * •

$g$ represents the determinant of $g_{\mu v}$,

  * •

$f(R,\phi)$ is an analytic function of Ricci scalar and scalar potential.

Now the field equation of $f(R,\phi)$ gravity can be obtained by varying the
action in Eq. (16) with respect to the metric tensor is given as,

$f_{R}R_{\mu v}-\frac{1}{2}[f+w(\phi)\phi_{;}\phi^{;\alpha}]g_{\mu
v}+w(\phi)\phi_{;\mu}\phi_{;v}-f_{r;\mu v}+g_{\mu v}\square f_{R}=kT_{\mu v}.$
(17)

Here, $w$ is an arbitrary function of $\phi$,
$\square=\nabla_{\mu}\nabla^{v}$, $\nabla_{\mu}$ denotes the covariant
derivative and $f_{R}\equiv\frac{\partial f}{\partial R}$. The energy momentum
tensor is given by,

$T^{m}_{\eta\xi}=(\rho+p_{t})u_{\eta}u_{\xi}-p_{t}g_{\eta\xi}+(p_{r}-p_{t})v_{\eta}v_{\xi},$
(18)

Figure 2:  Graphical behavior of
$\frac{\varepsilon(r)-r\varepsilon^{\prime}(r)}{\varepsilon^{2}(r)}$ and
$\varepsilon^{\prime}(r)$ from left to right.

where $\rho$ , $p_{r}$ and $p_{t}$ represent the energy density, radial and
tangential components of pressure respectively. We have
$u_{\eta}=e^{a/2}\delta^{0}_{\eta}$ and $v_{\xi}=e^{b/2}\delta^{1}_{\xi}$.
Furthermore, we have considered the spherically symmetric spacetime as

$ds^{2}=-e^{a(r)}dt^{2}+e^{b(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\theta^{2}.$ (19)

By using line element Eq. (19), along with energy momentum tensor Eq. (18) in
Eq.(17), we get the following field equations as

$\rho=-f/2+e^{-b}\bigg{[}-f_{R}^{\prime\prime}-f_{R}^{\prime}\bigg{(}\frac{2}{r}+\frac{b^{\prime}(r)}{2}\bigg{)}+\frac{1}{2}w(\phi)\phi^{\prime}(r)^{2}+\frac{1}{4}f_{R}\bigg{(}\frac{4a^{\prime}(r)}{r}+a^{\prime}(r)^{2}-a^{\prime}(r)b^{\prime}(r)+2a^{\prime\prime}(r)\bigg{)}\bigg{]},$
(20)
$p_{r}=f/2+e^{-b}\bigg{[}f_{R}^{\prime}\bigg{(}\frac{2}{r}+\frac{a^{\prime}(r)}{2}+b^{\prime}(r)\bigg{)}+\frac{1}{2}w(\phi)\phi^{\prime}(r)^{2}-\frac{1}{4}f_{R}\bigg{(}a^{\prime}(r)^{2}-\frac{4b^{\prime}(r)}{r}-a^{\prime}(r)b^{\prime}(r)+2a^{\prime\prime}(r)\bigg{)}\bigg{]},$
(21)
$p_{t}=f/2+e^{-b}\bigg{[}-f_{R}^{\prime\prime}-f_{R}^{\prime}\bigg{(}\frac{1}{r}+\frac{a^{\prime}(r)}{2}+\frac{b^{\prime}(r)}{2}\bigg{)}-\frac{1}{2}w(\phi)\phi^{\prime}(r)^{2}+\frac{1}{2}f_{R}\bigg{(}\frac{-2}{r^{2}}+\frac{2e^{b(r)}}{r^{2}}-\frac{a^{\prime}(r)}{2}+\frac{b^{\prime}(r)}{2}\bigg{)}\bigg{]}.$
(22)

Here, we have $w(\phi)=w_{o}$ $\phi^{m}$ and $\phi=r^{\beta}$, where $m,w_{o}$
and $\beta$ are arbitrary non-zero constants. For the further analysis and
calculation, we have used the famous software MATHEMATICA.

##  V Discussion of Energy Conditions for Different Models

In this section we have taken four different models of $f(R,\phi)$ gravity and
investigated the nature of these constraints via graphical representation.
Moreover, we explain different energy constraints and the method of predicting
the existence of wormholes by using these constraints. Energy conditions are
very important and helpful for explaining traversable wormhole geometry
because the violation of energy conditions may predict the presence of
wormholes. The energy conditions, we utilize in our study can be explained as:

$\displaystyle NEC$ $\displaystyle:$ $\displaystyle\rho+p_{r}\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho+p_{t}\geq 0,$ $\displaystyle WEC$
$\displaystyle:$ $\displaystyle\rho\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho+p_{r}\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho+p_{t}\geq 0,$ $\displaystyle SEC$
$\displaystyle:$ $\displaystyle\rho+p_{r}\geq
0,~{}~{}~{}~{}~{}~{}\rho+p_{t}\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho+p_{r}+2p_{t}\geq 0,$ $\displaystyle DEC$
$\displaystyle:$ $\displaystyle\rho\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho-|p_{r}|\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}\rho-|p_{t}|\geq 0.$

Here, NEC, WEC, SEC and DEC stand for null energy conditions, werak energy
conditions, strong energy conditions and dominant energy conditions
respectively. Now we will discuss the four different compatible models and
investigates the nature of these energy constraints via graphical
representation.

## A. Model-I

The first model, we have considered for our analysis is,

$f(R,\phi)=(R+\alpha R^{2})*\phi(r),$ (23)

where $\alpha$ is free parameter and $\phi=\phi(r)$. Using this model in Eqs.
(20)-(22), we have

$\rho=\frac{1}{4}e^{-b}r^{-2+\beta}\bigg{[}-2(e^{b}r^{2}R(1+R\alpha)+\beta(2+2\beta-(r^{\beta})^{1+m}wo\beta+4R\alpha(1+\beta)))+r(1+2R\alpha)(-2\beta
b^{\prime}+a^{\prime}(4+ra^{\prime}-rb^{\prime})+2ra^{\prime\prime})\bigg{]},$
(24)
$p_{r}=\frac{1}{4}e^{-b}r^{-2+\beta}\bigg{[}2(e^{b}r^{2}R(1+R\alpha)+\beta(4+8R\alpha+(r^{\beta})^{1+m}wo\beta))+r(1+2R\alpha)(-ra^{\prime
2}+4b^{\prime}(1+\beta)+(2\beta+b^{\prime}r)a^{\prime}-2ra^{\prime\prime})\bigg{]},$
(25)
$p_{t}=\frac{1}{4}e^{-b}r^{-2+\beta}\bigg{[}e^{b}(2+4R\alpha+r^{2}R(1+R\alpha))-(r^{\beta})^{1+m}wo\beta^{2}+2(1+2R\alpha)(-1+\beta^{2})+r(1+2R\alpha)((-1+\beta)a^{\prime}+(1+\beta)b^{\prime})\bigg{]}.$
(26)

According to the graphical analysis of the above Eqs. (24-26), it can be seen
that the behavior of $\rho$ is positive and decreasing, which can be seen in
Fig. (3). Furthermore, the graphical nature of $\rho+p_{t}$ is positive but
the nature of $\rho+p_{r}$ is negative, as shown in the right and middle plot
of Fig. (3). The negative trends of $\rho+p_{r}$ are the justification that
NEC is violated. As NEC is linked with WEC, so we can also conclude that WEC
is violated. Moreover, it can be observed in Fig. (9) that the behavior of
$\rho-p_{r}$ and $\rho-p_{t}$ is positive, so DEC is satisfied. The graphical
behavior of $\rho+2p_{t}+p_{r}$ is also negative which means that SEC is
violated as seen in Fig. (9). Hence, the violation of these energy constraints
especially WEC and NEC indicates the presence of exotic matter, which may
justify wormhole existence in the $f(R,\phi)$ gravity model.

Figure 3:  Shows graph of $\rho$(first plot), $(\rho+p_{r})$(second plot) and
$(\rho+p_{t})$(third plot) from left to right.

## B. Model-II

In this case we have taken another compatible model for the investigation of
wormholes geometry. Our second model is given as

$f(R,\phi)=\phi(r)^{M}*R.$ (27)

Using this model Eq. (27) in Eq.(20)-(22), we have:

$\rho=\frac{e^{-b}}{4r^{2}}\big{[}2r^{2\beta}r^{\beta M}wo\beta^{2}-r^{\beta
M}(2e^{b}r^{2}R+4M\beta(M\beta+1)+r(2M\beta
b^{\prime}+a^{\prime}(-4-ra^{\prime}+rb^{\prime})-2ra^{\prime\prime}))\big{]},$
(28) $p_{r}=\frac{e^{-b}}{4r^{2}}\big{[}2r^{2\beta}r^{\beta
M}wo\beta^{2}+r^{\beta M}(2e^{b}r^{2}R+8M\beta+r(-ra^{\prime
2}+4(1+M\beta)b^{\prime}+a^{\prime}(2M\beta+rb^{\prime})-2ra^{\prime\prime}))\big{]},$
(29) $p_{t}=\frac{e^{-b}}{2r^{2}}\big{[}-r^{2\beta}r^{\beta
M}wo\beta^{2}+r^{\beta
M}(e^{b}(2+r^{2}R)+r(-1+M\beta)a^{\prime}+r(1+M\beta)(-2+2M\beta+rb^{\prime}))\big{]}.$
(30)

According to the graphical analysis of the above Eqs. (28-30), it can be seen
that the nature of $\rho$ is positive but decreasing towards the boundary as
can be seen in the first plot of Fig. (4). Furthermore, the nature of
$\rho+p_{t}$ is positive and $\rho+p_{r}$ is negative, which can be seen in
the right and middle plot of Fig. (4), which represent the disturbance of NEC.
We can also notice that not only NEC but WEC is also disobeyed. Besides this,
it can be seen through the extreme left and middle plot of Fig. (10) that the
nature of $\rho-p_{r}$ and $\rho-p_{t}$ is positive, so DEC is not violated in
this case. The behavior of $\rho+2p_{t}+p_{r}$, which can be seen in Fig.
(10), is also negative which indicates the violation of SEC. Hence, the
violation of these energy constraints specially WEC and NEC indicates the
presence of wormholes in this model.

Figure 4:  Shows graph of $\rho$(first plot), $(\rho+p_{r})$(second plot) and
$(\rho+p_{t})$(third plot) from left to right.

## C. Model-III

Here we have considered another special model which is given as,

$f(R,\phi)=(1+k^{2}\phi(r)^{2}\xi)*R,$ (31)

where $k,\beta$, and $\xi$ are arbitrary constants. By using the above model
Eq. (31) in Eq. (20)-(22) as a result we obtain the following equations as:

$\rho=\frac{1}{4r^{2}}\big{[}-2r^{2}R(1+k^{2}r^{2\beta}\xi)+e^{-}b(r(a^{\prime}(4+ra^{\prime}-rb^{\prime}+2ra^{\prime\prime}+r^{2\beta}(2r^{\beta
m}wo\beta^{2}+k^{2}\xi(-8\beta(1+2\beta)+r(-4\beta
b^{\prime}+a^{\prime}(4+ra^{\prime}-rb^{\prime}+2ra^{\prime\prime}))))\big{]},$
(32)
$p_{r}=\frac{1}{4r}\big{[}2r^{2}R(1+k^{2}r^{2\beta}\xi)+e^{-}b(2r^{-1+2\beta}r^{\beta
m}wo\beta^{2}+4k^{2}r^{-1+2\beta}\beta\xi(4+ra^{\prime}+2rb^{\prime})+(1+k^{2}r^{2\beta}\xi)(-ra^{\prime
2}+4b^{\prime}+ra^{\prime}b^{\prime}-2ra^{\prime\prime}))\big{]},$ (33)
$p_{t}=\frac{1}{2r^{2}}\big{[}e^{-}b(-2+2e^{b}+e^{b}r^{2}R-ra^{\prime}+rb^{\prime}+r^{2\beta}(-r^{m\beta}wo\beta^{2}+k^{2}(-2+e^{b}(2+r^{2}R)+8\beta^{2})\xi+k^{2}r\xi((-1+2\beta)a^{\prime}+(1+2\beta)b^{\prime})))\big{]}.$
(34)

By analyzing the graphical analysis of the above equations, we can predict the
presence of wormholes for this model. With the help of the graphical analysis
of the above Eqs. (28)-(30), we can demonstrate that the nature of $\rho$ is
positive but decreasing, which is given in the extreme left plot of Fig. (5).
Furthermore, for the disobeying nature of NEC, we must need at least
$\rho+p_{r}<0$ or $\rho+p_{t}<0$. As the behavior of $\rho+p_{t}$ is positive
and $\rho+p_{r}$ is negative, can be seen in the right and middle plot of Fig.
(5) that shows the disobeying nature of NEC. We can also exhibit that not only
NEC but WEC is also disobeyed because both are linked. It is shown in Fig.
(11) that the nature of $\rho-p_{t}$ and $\rho-p_{r}$ is positive, so we can
observe that DEC is not disobeyed in this case. The behavior of
$\rho+2p_{t}+p_{r}$, which is given in Fig.(11), is also negative which
indicates the disobeying behavior of SEC. Hence, the disturbance of these
energy constraints may justify the existence of a wormhole in this model.

Figure 5:  Shows graph of $\rho$(first plot), $(\rho+p_{r})$(second plot) and
$(\rho+p_{t})$(third plot) from left to right.

## D. Model-IV

Last model that we have considered for our investigation is given as,

$f(R,\phi)=\phi(r)+R^{4/3},$ (35)

Now by utilizing above model Eq. (35) in Eqs. (20)-(22), we have:

$\rho=\frac{1}{6r}\big{[}-3r(r^{\beta}+R^{4/3})+e^{-}b(3r^{-1+2\beta}r^{m\beta}wo\beta^{2}+2R^{2/3}(a^{\prime}(4+ra^{\prime}-rb^{\prime})+2ra^{\prime\prime}))\big{]},$
(36)
$p_{r}=\frac{1}{2}\big{(}r^{\beta}+R^{4/3})+e^{-}b(\frac{1}{2}r^{-2+2\beta}r^{m\beta}wo\beta^{2}-\frac{1}{3}R^{1/3}(a^{\prime
2}-\frac{4b^{\prime}}{r}-a^{\prime}b^{\prime}+2a^{\prime\prime})\big{)},$ (37)
$p_{t}=\frac{1}{6r^{2}}\big{[}3r(r^{\beta}+R^{4/3})+e^{-}b(-3r^{-1+2\beta}r^{m\beta}wo\beta^{2}+4R^{1/3}(-2+2e^{b}-ra^{\prime}+rb^{\prime}))\big{]}.$
(38)

With the help of the graphical analysis of the above Eqs. (36)-(38), we can
observe that the nature of $\rho$ is positive with decreasing behavior, as
given in the first plot of Fig. (6). As the nature of $\rho+p_{t}$ is positive
and $\rho+p_{r}$ is negative that can be seen through the right and middle
plot of Fig. (6), which represents the disobeying behavior of NEC. We can also
demonstrate that not only NEC but WEC is also showing disobeying behavior
because whenever NEC disobeyed WEC must be violated. Besides this, it can be
seen from the left and middle plot of Fig. (12) that the graphical analysis of
$\rho-p_{r}$ is positive and $\rho-p_{t}$ shows positive but also negative
behavior near the boundary, so we can conclude that DEC is disobeyed. The
behavior of $\rho+2p_{t}+p_{r}$, which is given in Fig.(12), is also negative
which indicates the violation of SEC. Hence, the disobeying nature of these
energy constraints, particularly NEC and WEC may justify the existence of a
wormhole in this model.

Figure 6:  Shows graph of $\rho$(first plot), $(\rho+p_{r})$(second plot) and
$(\rho+p_{t})$(third plot) from left to right.

##  VI Comparison

Shamir and fayyaz 112 constructed a wormhole shape function by employing the
Karmarkar condition for static traversable wormhole geometry and generated the
wormhole geometry that connected two asymptotically flat regions of spacetime
and satisfied the required conditions. In our work, we have discussed the
embedding diagram in three-dimensional Euclidean space to present the wormhole
configurations and investigate the traversable wormhole geometry in
$f(R,\phi)$ gravity. Moreover, the authors 112 considered three different
models to discuss the wormhole geometry in $f(R)$ gravity but we have
considered four different models in $f(R,\phi)$ gravity to investigate the
behavior of traversable wormholes, which make our work more generalized and
comprehensive than the previous investigation. Godani and Samanta aa discussed
the wormhole solutions in $f(R)$ theory of gravity by taking well defined
$f(R)$ gravity model and particular shape functions. However, in this work we
first construct the shape function by using the Karmarkar condition and then
discuss the wormhole solutions for more than one model, which is the
interesting feature of our work. Sharif and Fatima a1 evaluated the
traversable wormhole solutions through Karmarkar condition in $f(R,T)$ theory
of gravity. They considered two different $f(R,T)=f(R)+f(T)$ gravity models,
where $f(R)$ was taken as exponential gravity model and Starobinsky model but
$f(T)=\lambda T$ was fixed for the sake of simplicity. In our present work, we
investigate the traversable wormhole solutions for $f(R,\phi)$ gravity models
by taking the product of Ricci scalar $R$ and scalar potential $\phi$, which
make our work more interesting with the previous investigations. Banerjee et
al. a1 presented a class of solutions for static and spherically symmetric
wormholes geometries by considering specific choices for the $f(Q)$ form and
for constant redshift with different shape functions. We have done analysis
for wormhole solutions by constructing the wormhole geometry, and investigate
the energy conditions for all considered cases. Furthermore, we perform a more
detailed analysis by taking different values of $C$ appeared in the shape
function i.e., $C=1.9$ (color Green), $C=1.8$ (Blue color), $C=1.7$ (color
Pink), $C=1.6$ (color Red), $C=1.5$ (color Brown), $C=1.4$ (color Yellow),
$C=1.3$ (color Black). This is also a very crucial point of our research work,
which make our work different from the previous one. Mishra, et al. a3
discussed the traversable wormhole geometry by taking three shape functions
and constant redshift but our analysis is different because we choose a more
general redshift function depending upon the radial coordinate $r$.

Figure 7:  Graphical representation of embedding diagram ($h(r)>0$ and it is
for upper universe).

Figure 8:  Graphical representation of embedding diagram ($h(r)<0$ and it is
for bottom universe).

Figure 9:  Shows graph of $(\rho-p_{r})$(first plot), $(\rho-p_{t})$(second
plot) and $(\rho+2p_{t}+p_{r})$(third plot) from left to right.

Figure 10:  Shows graph of $(\rho-p_{r})$(first plot), $(\rho-p_{t})$(second
plot) and $(\rho+2p_{t}+p_{r})$(third plot) from left to right.

Figure 11:  Shows graph of $(\rho-p_{r})$(first plot), $(\rho-p_{t})$(second
plot) and $(\rho+p_{r}+2p_{t})$(third plot) from left to right.

Figure 12:  Shows graph of $(\rho-p_{r})$(first plot), $(\rho-p_{t})$(second
plot) and $(\rho+2p_{t}+p_{r})$(third plot) from left to right.

##  VII Concluded remarks

The purpose of this research is to study the wormhole geometry by using the
Karmarkar condition in the $f(R,\phi)$ theory of gravity. For our current
analysis, we have used special four models of $f(R,\phi)$ theory of gravity
and investigated the validity of different types of energy constraints. It is
worthwhile to mention here that modified gravity theories with a scalar field
may accommodate a wide range of dark energy and modified gravity models a4 .
The addition of scalar field helps to modify the cosmological and galactic
dynamics leaving the solar system unaffected. As far as wormhole study is
concerned, the addition of scalar field supports any asymptotically well-
behaved traversable wormhole. In our work, scalar field helps in violating NEC
which is necessary for the existence of exotic matter and wormhole geometry.
The stable behavior of wormhole is also supported by the addition of scalar
field, though we have not discussed this in present work. To the best of our
knowledge, it is the first attempt to explore wormhole solutions in
$f(R,\phi)$ gravity with Karmarker condition. The detailed outcomes of our
research work are as follows:

The graphical representation of energy density is positive but decreasing
towards the boundary as can be seen in the first plot of Fig. (3). The
graphical nature of $\rho+p_{t}$ is positive but $\rho+p_{r}$ is negative, as
seen in Fig. (3). It can be noticed that NEC and WEC are violated due to the
negative behavior of the $\rho+p_{r}$ component. Moreover, the graphical
representation of $\rho-p_{r}$ and $\rho-p_{t}$ is positive, which means that
DEC is satisfied as seen in Fig. (9). The graphical trends of
$\rho+2p_{t}+p_{r}$ are negative, which is the justification that SEC is
violated as seen in Fig. (9). Hence, the violation of these energy constraints
specially WEC and NEC indicates the presence of wormholes in this model.

According to the graphical analysis for Model II, it can be seen that the
nature of energy density is positive and becomes decreasing, when we move on
the radial coordinate as plotted in Fig. (4). It can also be noticed that NEC
is violated due to the negative nature of $\rho+p_{r}$ as seen in Fig. (4). It
can also be observed that WEC is also violated because NEC is violated. For
our second model, DEC is satisfied because $\rho-p_{r}>0$ and $\rho-p_{t}>0$,
but SEC is violated due to $\rho+2p_{t}+p_{r}<0$, as seen in Fig. (10). The
significant consequences of energy condition violation, especially violation
of the NEC, may indicate the presence of exotic matter, which may justify
wormhole existence in the $f(R,\phi)$ gravity model.

In our third model, we observe that the graphical trends of energy density are
maximum near the origin and positive as seen in Fig. (5). Furthermore, it can
be seen that $p_{t}+\rho$ is positive but $p_{r}+\rho$ is negative, which is
the justification that NEC is violated. We can also conclude that not only NEC
but WEC is also violated because both are connected. It is also noticed that
the nature of $\rho-p_{r}$ and $\rho-p_{t}$ is positive, so DEC is again
satisfied in our third model. It can be seen from Fig. (11) that the nature of
$\rho+2p_{t}+p_{r}$ is negative, which is a violation of SEC.

In our fourth and last model, the behavior of energy density is also positive
with decreasing nature as shown in Fig. (6). Furthermore, the behavior of
$p_{t}+\rho$ is positive and $p_{r}+\rho$ is negative, which indicates the
violation of NEC and WEC. Besides this, it can be seen from Fig. (12) that the
graphical analysis of $\rho-p_{r}$ is positive and $\rho-p_{t}$ shows positive
but also negative behavior near the boundary, so we can conclude that DEC is
disobeyed. The behavior of $\rho+2p_{t}+p_{r}$, which is given in Fig.(12), is
also negative which indicates the violation of SEC. Hence, the disobeying
nature of these energy constraints, particularly NEC and WEC may justify the
existence of a wormhole in this model. From above discussion, we can conclude
that in above all four models these energy constraints are violated. Since
these energy constraints are very important in the study of wormholes. The
significant consequences of energy condition violation, especially violation
of the NEC, may indicate the presence of exotic matter, which may justify
wormhole existence in the $f(R,\phi)$ gravity models.

Finally, it is stated that the shape function acquired through the Karmarkar
technique yields validated wormhole configurations with even less exotic
matter correlating to the proper choice of $f(R,\phi)$ gravity models and
acceptable free parameter values. As a nutshell, the present models fulfill
all of the criterion for the presence of a wormhole. In particular, it is
concluded that the gravitational field emerging from higher order terms of
modified gravity with scalar field favor the existence of the wormhole
geometries.

## Data Availability Statement

No data was used for the research in this article. It is pure mathematics.

## Conflict of Interest

The authors declare that they have no conflict of interest.

## Contributions

We declare that all the authors have same contributions to this paper.

## ACKNOWLEDGMENTS

Adnan Malik acknowledges the Grant No. YS304023912 to support his Postdoctoral
Fellowship at Zhejiang Normal University, China.

## References

  * (1)

## REFERENCES

* (2) Einstein, A., and N. Rosen, “The particle problem in the general theory of relativity.” Physical Review 48 (1935): 73.
* (3) Mishra, B., et al. “Wormhole solutions in $f(R)$ gravity.” International Journal of Modern Physics D 30 (2021): 2150061.
* (4) Sharif, M., and Z. Zahra. “Static wormhole solutions in $f(R)$ gravity.” Astrophysics and Space Science 348 (2013): 275.
* (5) DeBenedictis, A., and D. Horvat. “On wormhole throats in $f(R)$ gravity theory.” General Relativity and Gravitation 44 (2012): 2711.
* (6) Samanta, G. C., and N. Godani. “Validation of energy conditions in wormhole geometry within viable $f(R)$ gravity.” The European Physical Journal C 79 (2019): 1.
* (7) Sharif, M., and S. Rani. “Dynamical wormhole solutions in $f(T)$ gravity.” General Relativity and Gravitation 45 (2013): 2389.
* (8) Jamil, M., et al. “Wormholes in a viable $f(T)$ gravity.” The European Physical Journal C 73 (2013): 1.
* (9) Azizi, T. “Wormhole geometries in $f(R,T)$ gravity.” International Journal of Theoretical Physics 52 (2013): 3486.
* (10) Naz, T., et al. “Existence of wormholes in $f(G)$ gravity using symmetries.” International Journal of Geometric Methods in Modern Physics 19 (2022): 2250100.
* (11) Malik, A., and A. Nafees. “Existence of static wormhole solutions using $f(R,\phi,X)$ theory of gravity.” New Astronomy 89 (2021): 101632.
* (12) Shamir, M. F., and S. Zia. “Existence of static wormhole solutions in $f(R,G)$ gravity.” Astrophysics and Space Science 363 (2018): 1.
* (13) Samanta, G. C., et al. “Traversable wormholes with exponential shape function in modified gravity and general relativity: A comparative study.” International Journal of Modern Physics D 29 (2020): 2050068.
* (14) Sharma, U. K., et al. “Traversable wormhole solutions with non-exotic fluid in framework of $f(Q)$ gravity.” International Journal of Geometric Methods in Modern Physics 19 (2022): 2250019.
* (15) Sharif, M., and A. Ikram. “Galactic halo traversable wormhole solutions in $f(G,T)$ gravity.” International Journal of Modern Physics D 27 (2018): 1950009.
* (16) Karmarkar, K. R. “Gravitational metrics of spherical symmetry and class one.” Proceedings of the Indian Academy of Sciences-Section A. Springer India, 1948.
* (17) Fayyaz, I., and M. F. Shamir. “MorrisThorne wormhole with Karmarkar condition.” Chinese Journal of Physics 66 (2020): 553.
* (18) Samanta, G. C., and N. Godani. “Wormhole modeling supported by non-exotic matter.” Modern Physics Letters A 34 (2019): 1950224.
* (19) Godani, N., and G. C. Samanta. “Traversable wormholes and energy conditions with two different shape functions in $f(R)$ gravity.” International Journal of Modern Physics D 28 (2019): 1950039.
* (20) Golchin, H., and M. R. Mehdizadeh. “Quasi-cosmological traversable wormholes in $f(R)$ gravity.” The European Physical Journal C 79 (2019): 1.
* (21) Harko, T., et al. “Modified-gravity wormholes without exotic matter.” Physical Review D 87 (2013): 067504.
* (22) Kuhfittig, P. K. F. “Spherically symmetric wormholes of embedding class one.” Pramana 92 (2019): 75.
* (23) Kuhfittig, P. K. F. “Two diverse models of embedding class one.” Annals of Physics 392 (2018): 63.
* (24) Malik, A,, et al. “Traversable wormhole solutions in the $f(R)$ theories of gravity under the Karmarkar condition.” Chinese Physics C 46 (2022): 095104.
* (25) Tello-Ortiz, F., and E. Contreras. “Traversable wormholes in light of class I approach.” Annals of Physics 419 (2020): 168217.
* (26) Gul, M. Z., and M. Sharif. “Traversable wormhole solutions admitting Noether symmetry in $f(R,T^{2})$ Theory.” Symmetry 15 (2023): 684.
* (27) Rahaman, F., et al. “Wormhole inspired by non-commutative geometry.” Physics Letters B 746 (2015): 73.
* (28) Shamir, M. F., and A. Malik, “Behavior of Anisotropic Compact Stars in $f(R,\phi)$ Gravity.” Communications in Theoretical Physics 71 (2019): 001.
* (29) Malik, A., et al., “Some Dark Energy Cosmological Models in $f(R,\phi)$ Gravity, New Astronomy 89 (2021): 101631.
* (30) Shamir, M. F., and A. Malik, “Investigating $f(R,\phi)$ Cosmology with Equation of State.” Canadian Journal of Physics 97 (2019): 752.
* (31) Malik, A., et al., “Energy bounds in$f(R,\phi)$ gravity with anisotropic backgrounds.” New Astronomy 79 (2020): 101392.
* (32) Malik, A., and M. F. Shamir, “The Study of Godel Type Solutions in$f(R,\phi)$ Gravity.” New Astronomy 80 (2020): 101422.
* (33) Malik, A., et al., “Some Bianchi Type Cosmological Models in$f(R,\phi)$ Gravity.” New Astronomy 81 (2020): 101418.
* (34) Malik, A., “A Study of Levi-Civita’s Cylindrical Solutions in $f(R,\phi)$ Gravity.” The European Physical Journal Plus 136 (2021): 1146.
* (35) Myrzakulov, R. et al., “Inflation in $f(R,\phi)$-theories and mimetic gravity scenario.” The European Physical Journal C 75 (2015): 1.
* (36) Stabile, A., and S. Capozziello. “Galaxy rotation curves in $f(R,\phi)$ gravity.” Physical Review D 87 (2013): 064002.
* (37) Nozari, K., and M. Pourghasemi. “Crossing the phantom divide line in a DvaliGabadadzePorrati-inspired $f(R,\phi)$ gravity.” Journal of Cosmology and Astroparticle Physics 2008 (2008): 044.
* (38) Panda, S. et al., “Constant-roll inflation in modified $f(R,\phi)$ gravity model using Palatini formalism.” The European Physical Journal C 83 (2023): 297.
* (39) Malik, A., “Analysis of Charged Compact Stars in Modified$f(R,\phi)$ Theory of Gravity.” New Astronomy 93 (2022): 101765.
* (40) Malik, A., et al., “Singularity-free Anisotropic Strange Quintessence Stars in $f(R,\phi,X)$ Theory of Gravity.” The European Physical Journal Plus 138 (2023):418.
* (41) Malik, A., et al., “A Study of Charged Stellar Structures in Modified $f(R,\phi,X)$ Theory of Gravity.” International Journal of Geometric Methods in Modern Physics 19, (2022): 2250180.
* (42) Malik, A., et al., “A Study of Levi-Civita’s Cylindrical Solutions in $f(R,\phi,X)$ Gravity.” The European Physical Journal C 82 (2022): 166.
* (43) Malik, A., et al., “A study of anisotropic compact stars in$f(R,\phi,X)$theory of gravity.” International Journal of Geometric Methods in Modern Physics 19 (2022): 2250028.
* (44) Shamir, M. F. et al., “Non-commutative Wormhole Solutions in Modified $f(R,\phi,X)$ Gravity.” Chinese Journal of Physics 73 (2021) 634-648.
* (45) Shamir, M. F., and I. Fayyaz. “Traversable wormhole solutions in $f(R)$ gravity via Karmarkar condition.” The European Physical Journal C 80 (2020): 1.
* (46) Godani, N., and G. C. Samanta. “Traversable wormholes in $f(R)$ gravity with constant and variable redshift functions.” New Astronomy 80 (2020): 101399.
* (47) Sharif, M., and A. Fatima. “Traversable wormhole solutions admitting Karmarkar condition in $f(R,T)$ theory.” The European Physical Journal Plus 138.3 (2023): 196.
* (48) Banerjee, A., et al. “Wormhole geometries in $f(Q)$ gravity and the energy conditions.” The European Physical Journal C 81 (2021): 1.
* (49) Mishra, B., et al. “Traversable wormhole models in $f(R)$ gravity.” International Journal of Modern Physics A 37 (2022): 2250010.
* (50) Bahamonde, S., et al. “Generalized $f(R,\phi,X)$ gravity and the late-time cosmic acceleration.” Universe 1 (2015): 186.