# Weak Gravitational Lensing in Dark Matter and Plasma Mediums for Wormhole-
like Static Aether Solution

Wajiha Javed<EMAIL_ADDRESS>Department of Mathematics, Division of
Science and Technology, University of Education, Lahore-54590, Pakistan
Sibgha Riaz<EMAIL_ADDRESS>Department of Mathematics, Division of
Science and Technology, University of Education, Lahore-54590, Pakistan
Reggie C. Pantig<EMAIL_ADDRESS>Physics Department, De La Salle
University, 2401 Taft Avenue, Manila, 1004 Philippines  Physics Department,
Mapúa University, 658 Muralla St., Intramuros, Manila 1002, Philippines  Ali
Övgün<EMAIL_ADDRESS>Physics Department, Eastern Mediterranean
University, Famagusta, 99628 North Cyprus via Mersin 10, Turkey.

###### Abstract

In this paper, we study the deflection angle for wormhole-like static aether
solution by using Gibbons and Werner technique in non-plasma, plasma, and dark
matter mediums. For this purpose, we use optical spacetime geometry to
calculate the Gaussian optical curvature, then implement the Gauss-Bonnet
theorem in weak field limits. Moreover, we compute the deflection angle by
using a technique known as Keeton and Petters technique. Furthermore, we
analyze the graphical behavior of the bending angle $\psi$ with respect to the
impact parameter $b$, mass $m$ as an integration constant, and parameter $q$
in non-plasma and plasma mediums. We examine that the deflection angle is
exponentially increasing as direct with charge. Also, we observe that for
small values of $b$, $\psi$ increases, and for large values of $b$ the angle
decreases. We also considered analysis to the shadow cast of the wormhole
relative to an observer at various locations. Comparing it the Schwarzschild
shadow, shadow cast is possible for wormhole as $r<2m$. At $r>2m$, the
Schwarzschild is larger. As $r\to\infty$, we have seen that the behavior of
the shadow, as well as the weak deflection angle, approaches that of the
Schwarzschild black hole. Overall, the effect of plasma tends to decrease the
value of the observables due to the wormhole geometry.

General Relativity; Gravitational Lensing; Wormhole-like Static Aether
Solution; Gauss-Bonnet Theorem; Plasma and Non-Plasma Mediums; Dark Matter;
Modified Gravity.

###### pacs:

95.30.Sf, 98.62.Sb, 97.60.Lf

##  I Introduction

Einstein’s theory of general relativity (GR) looked into the physical
existence of black holes (BHs) in the universe Einstein (1916). The American
astronomer John Wheeler invented the word “ BH ”. The study of BHs have
received a lot of attention since the Event Horizon Telescope obtained the
first images of the Messier $87$ BH and then Sagittarius A* BH Akiyama et al.
(2019, 2022). Black hole emits Hawking radiation as a complete thermal
spectrum by incorporating quantum theory. Stellar, intermediate, super-massive
and microscopic BHs are four main types of BHs. The outer horizon, inner event
horizon as well as the singularity are the three “layers” of a BH. A BH’s
event horizon is the boundary around a BH and beyond which light cannot
escape. Singularity, is a region in space where the density of the existing
mass is infinite.

Wormholes (WHs), just like BHs, can be expressed as solution of Einstein field
equations. Schwarzschild BH solution is the simplest solution of Einstein
field equations. Flamm first proposed the concept of a WH, after the discovery
of Schwarzschild’s BH solution. A WH is a hypothetical spacetime that connects
two separate regions of universe and give a shortcut through them. Einstein
and Rosen Einstein and Rosen (1935) proposed the existence of WH-like objects,
often known as Einstein-Rosen bridges. Misner and Wheeler Wheeler (1957)
formulated the concept of a “WH” Misner and Wheeler (1957). Wormholes have not
been yet physically demonstrated. After that, Wheeler Wheeler (1955) explained
that WHs are unstable and non-traversable even by a photon. Morris and Thorne
Morris and Thorne (1988) invented the term traversable WH. However, Morris,
Thorne and Yurtsever Morris and Thorne (1988) explained how to convert a WH
through traversing space into traversing-time. They demonstrated that by
computing the Einstein field equations, we get the solution showing WH-
geometry in a terms of a static spherically symmetric line-element. After
that, by following the Morris-Thorne papers, a lot of physicists looked into
WHs from a different point of views Damour and Solodukhin (2007); Bueno et al.
(2018); Hawking (1988); Övgün et al. (2019); Halilsoy et al. (2014); Ovgun and
Halilsoy (2016). Later on, another form of traversable WH were introduced by
Matt Visser Visser (1995), that is known as thin-shell WH in which the path
through the WH can be formed in such a way that traversing path does not cross
the region of exotic matter. Although, exotic matter causes the problem to
create a stable WH. Recently, it has been explained that WHs also play an
important part in explaining quantum entanglement Marolf and Polchinski
(2013).

The concept of gravitational lensing (GL) due to its gravitational effects
occurs when a huge object distorts the space around it and twisting the
direction of light passes through it. Gravitational lensing is a strong
astrophysical tool for determining the mass of galaxies and clusters, as well
as detecting dark matter (DM) Massey et al. (2010). There are three types of
GL; strong GL, weak GL and micro GL Bartelmann and Schneider (2001); Cunha et
al. (2015). The strong GL enables us to compute the area and intensity of a
BH. Moreover, the impact of “weak GL” is actually weaker but yet observable
analytically. Micro GL isn’t the same as strong and weak GL. In this kind of
lensing, the lens is small in comparison to weak and strong GL.

Gibbons and Werner Gibbons and Werner (2008) proposed the method to calculate
the angle of deflection by various BHs in the weak field limits. The bending
angle $\psi$ can be computed by using asymptotically flat spacetime Gibbons
and Werner (2008) such as:

$\psi=-\int\int_{p_{\infty}}\mathcal{K}dS.$ (1)

Here, $\mathcal{K}$ is the Gaussian curvature, $dS$ is the surface component
and ${p_{\infty}}$ stands for infinite domain of the space.

Numerous writers have used GBT to examine the angle of deflection for various
BHs and WHs Gibbons and Warnick (2009); Gibbons et al. (2009); Gibbons and
Vyska (2012); Bloomer (2011); Werner (2012); Övgün (2019); Javed et al.
(2019a, b, c); Pantig and Rodulfo (2020a); Pantig and Rodulfo (2020b); Pantig
Reggie C. and Ali (2022); Pantig and Övgün (2022a, b); Kuang and Övgün (2022);
Uniyal et al. (2022); Kumaran and Övgün (2020, 2021, 2022) Javed et al. Javed
et al. (2020, 2021) calculated the weak GL by stringy BH and tidal charged BH.
He and Lin He and Lin (2016) examined the GL for Kerr-Newman BH having
arbitrary uniform velocity. Crisnejo and Gallo Crisnejo and Gallo (2018)
looked into the deflection angle of light in the existence of plasma medium.
Nakajima and Asada Nakajima and Asada (2012) studied GL by Ellis WH.
Deflection angle for static and axisymmetric rotating Teo WH was examined by
Jusufi and Ovgun Jusufi and Övgün (2018). Ovgun Övgün (2018) worked on the
light deflection by Damour-Solodukin WHs using GBT.

The discovery of DM Epps and Hudson (2017) by weak deflection is an important
topic, as it can assist us to understand the massive structure of the universe
Bartelmann and Schneider (2001). Zwicky was the $1$st astronomer who proposed
the DM. Dark matter is a type of matter which cannot visualize directly. It
does not release any light or energy, that’s why standard instrument and
detectors cannot detect it. Dark Matter consists of $27\%$ of the total mass
energy of the universe Hinshaw et al. (2013). Dark matter can be detected by
gravitational interaction and possesses electromagnetic interactions Latimer
(2013). Super-interacting massive particles, weakly interacting massive
particles, sterile neutrinos and axions are the types of dark matter
candidates. Refractive index used in dark matter maintains the propagation
speed. The DM medium’s refractive index is defined as Latimer (2013):

$n(\omega)=1+\beta{A_{0}}+{A_{2}}{\omega^{2}}.$ (2)

It’s important to remember that $\beta=\frac{\rho_{o}}{4m^{2}\omega^{2}}$,
${\rho_{o}}$ is the mass density of scattered DM particles,
$A_{o}=-2\epsilon^{2}e^{2}$ and $A_{2j}\geq 0$. The polarizability of the DM
candidate is connected to $\mathcal{O}\left({\omega^{2}}\right)$ and higher
terms. The charged DM candidate has an order of $\omega^{-2}$, while the
neutral DM candidate has an order of $\omega^{2}$. Furthermore, if the parity
and charge parity inequalities are present, then the linear term appears in
$\omega$.

Oost, Mukohyama and Wang Oost et al. (2021a) obtained the exact stable
solution in Einstein-aether theory. The solution is asymptotically smooth,
expressed in the isotropic coordinates and specified by two parameters: mass
$m$ is an integration constant and $c_{14}$ is a combined coupling parameter.
For $c_{14}=2$, metric reduces to Schwarzschild solution in Einstein theory in
isotropic coordinates and for $c_{14}\neq 2$, the solution illustrates finite
size throat that is slightly trapped but smoothly connects the two untrapped
patches: one of the patch has a singularity at finite proper distance and
other patch is asymptotically flat at infinite proper distance. The aether
configuration and spacetime geometry are similar to static WH aether solutions
Penrose (1969). The WH-like static aether solution is physically undesirable,
according to the cosmic censorship conjecture Eling and Jacobson (2006).
Moreover, Zhu et al. studied the shadows and deflection angle of charged and
slowly rotating black holes in Einstein-Æther theory Zhu et al. (2019).

One of the goal of this paper is to find the bending angle for WH-like static
aether solution by using GBT and Keeton and Petters method. Moreover, we will
study the impact of plasma, non-plasma and DM mediums on the deflection angle
of given WH. After that, we analyze graphically, the behaviour of $q$, $b$ and
$m$ on the bending angle $\psi$. In addition to these goals, while exploring
the effects of plasma, we also examined the behavior of the shadow radius of
the wormhole based on the different observer locations. To this aim, we have
used the methods pioneered by Perlick et al. (2015), which not only been
applied to BH shadows, but as well as wormholes. Since then, the interest
regarding the shadow of wormholes have risen Guerrero et al. (2022); Zhu and
Wang (2021); Rahaman et al. (2021); Bouhmadi-López et al. (2021); Bugaev et
al. (2021); Peng et al. (2021); Guerrero et al. (2021); Wielgus et al. (2020);
Wang et al. (2020); Gyulchev et al. (2019); Övgün et al. (2018); Övgün and
Sakallı (2020); Övgün et al. (2020); Çimdiker et al. (2021).

This paper is organized as follows: in section 2, we discuss WH-like static
aether solution. In section 3, we compute the bending angle of WH in non-
plasma medium. Section 5, consists of the calculations of the deflection angle
in plasma medium. In section 6, we discuss the graphical behaviour of the
deflection angle in plasma and non-plasma mediums. Section 7 is based on the
study of the effects of DM on the deflection angle. In section 8, we find the
bending angle by using Keeton and Petters method. In section 9, we conclude
our results.

##  II Wormhole-like Static Aether Solution

The Einstein aether theory is a vector tensor theory which violates Lorentz
invariance by connecting the metric to a unit time-like vector field (the
aether field) at every point in space. According to the Einstein-aether
theory, WH solutions have been studied. The spherically symmetric WH-like
static aether solution of the Einstein field equations connects two separate
regions of space and give a shortcut through them. Exact solution to Einstein-
aether theory is exposing analytically in isotropic coordinates
$(t,r,\phi,\theta)$, given as follows Zhu and Wang (2021):

$ds^{2}=-\left({\frac{1-\frac{m}{2r}}{1+\frac{m}{2r}}}\right)^{q}dt^{2}+\frac{({1+\frac{m}{2r}})^{q+2}}{({1-\frac{m}{2r}})^{q-2}}(dr^{2}+r^{2}d\Omega^{2}),$
(3)
$\text{where}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q=2\left({\frac{2}{2-c_{14}}}\right)^{1/2}\geq
2.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Note that $m$ is an integration constant and $c_{14}$ is small nonnegative
parameter. The static spherically symmetric spacetime for WH-like static
aether solution can also be written as Zhu and Wang (2021)

$~{}~{}~{}~{}~{}~{}~{}~{}ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+D(r)d\Omega^{2},$ (4)
$\text{where}~{}~{}~{}d\Omega^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}~{}~{}~{}~{}~{}\text{and}\\\ $
$~{}~{}~{}~{}A(r)=\left({\frac{1-\frac{m}{2r}}{1+\frac{m}{2r}}}\right)^{q},~{}~{}~{}~{}~{}~{}B(r)=\frac{D(r)}{r^{2}}=\frac{({1+\frac{m}{2r}})^{q+2}}{({1-\frac{m}{2r}})^{q-2}}.\\\
$

Here, mass $m$ is an integration constant, $r$ is radial coordinate and $q$ is
parameter. It is noted that $q>2$, the above metric has a curvature
singularity and a marginally trapped throat Oost et al. (2021b).

##  III Deflection Angle $\psi$ in Non-Plasma Medium

In this section, we determine the bending angle of WH in non-plasma medium by
using GBT. When source and observer are both in the tropical region and null
photon also in the same region, so one can infer that
$(\theta=\frac{\pi}{2})$. In order to get optical metric, we put $ds^{2}=0$ in
Eq.(4) and get;

$dt^{2}=\frac{B(r)}{A(r)}{dr^{2}}+\frac{B(r)r^{2}}{A(r)}{d\phi^{2}}.$ (5)

The non-zero Christoffel symbols for the above metric can be obtained as:

$\Gamma^{0}_{00}=\frac{1}{2}\left(\frac{-A^{\prime}(r)}{A(r)}+\frac{B^{\prime}(r)}{B(r)}\right),~{}~{}~{}~{}\Gamma^{1}_{10}=\frac{1}{r}-\frac{A^{\prime}(r)}{2A(r)}+\frac{B^{\prime}(r)}{2B(r)},$
$\Gamma^{0}_{11}=\frac{1}{2}r\left(-2+\frac{rA^{\prime}(r))}{A(r)}-\frac{rB^{\prime}(r)}{B(r)}\right),$

where $0$ and $1$ are showing the $r$-coordinate and $\phi$-coordinate,
respectively and the optical metric’s Ricci scalar is calculated as:

$\displaystyle\mathcal{R}$ $\displaystyle=$
$\displaystyle\frac{1}{rA(r)B(r)^{3}}\left(-rB(r)^{2}{A^{\prime}(r)^{2}}+{A(r)}{B(r)^{2}}({A^{\prime}(r)}+{rA^{\prime\prime}(r)})\right.$
(6) $\displaystyle-$
$\displaystyle\left.{A(r)^{2}}(-r{B^{\prime}(r)^{2}}+{B(r)}({B^{\prime}(r)}+{rB^{\prime\prime}(r)}))\right).$

The Gaussian curvature is defined as:

$\mathcal{K}=\frac{\mathcal{R}}{2}=-\frac{64mr^{3}\left(1-\frac{m}{2r}\right)^{q}\left(\frac{m}{2r}+1\right)^{-q}\left(-\frac{m-2r}{m+2r}\right)^{q}\left(m^{2}q-4mr+4qr^{2}\right)}{\left(m^{2}-4r^{2}\right)^{4}}.$
(7)

For the given WH, using Eq.(6), the Gaussian curvature is computed as:

$\mathcal{K}\simeq\frac{-qm}{r^{3}}+\mathcal{O}\left({m^{2}}\right).$ (8)

In the region of non-singular domain $\mathcal{H}_{e}$, the deflection angle
for WH-like static aether solution by using GBT, can be obtained by using the
following formula;

$\int\int_{\mathcal{H}_{e}}\mathcal{K}dS+\oint_{\partial\mathcal{H}_{e}}kdt+\sum_{i}\epsilon_{i}=2\pi\xi(\mathcal{H}_{e}),$
(9)

in the above expression, $k$ indicates geodesic curvature, stated as
$k=\bar{g}(\nabla_{\dot{\eta}}\dot{\eta},\ddot{\eta})$ and
$\bar{g}(\dot{\eta},\dot{\eta})=1$, $\ddot{\eta}$ denotes unit acceleration
vector and $\epsilon_{i}$ expresses the exterior angle at the ith vertex. As
$e\rightarrow\infty$, the corresponding jump angles reduce into $\pi/2$ and we
obtain $\theta_{O}+\theta_{S}\rightarrow\pi$. Euler characteristic is
$\xi(\mathcal{H}_{e})=1$. So,

$\int\int_{\mathcal{H}_{e}}\mathcal{K}dS+\oint_{\partial\mathcal{H}_{e}}kdt+\epsilon_{i}=2\pi\xi(\mathcal{H}_{e}),$
(10)

here, $\epsilon_{i}=\pi$ represents jump angle. As $e\rightarrow\infty$, the
geodesic curvature is obtained as

$k(D_{e})=\mid\nabla_{\dot{D}_{e}}\dot{D}_{e}\mid.$ (11)

Since the radial component of geodesic curvature is;

$(\nabla_{\dot{D}_{e}}\dot{D}_{e})^{r}=\dot{D}^{\phi}_{e}\partial_{\phi}\dot{D}^{r}_{e}+\Gamma^{0}_{11}(\dot{D}^{\phi}_{e})^{2}.$
(12)

For large value of $e$, $D_{e}:=r(\phi)=e=const$, then the result is;

$(\nabla_{\dot{D}^{r}_{e}}\dot{D}^{r}_{e})^{r}\rightarrow\frac{1}{e}.$ (13)

The geodesic curvature does not have a topological defect so,
$k(D_{e})\rightarrow e^{-1}$. However, by using the optical metric Eq.(5), it
can be expressed as follows: $dt=ed\phi$. As a result, we have:

$k(D_{e})dt=d\phi.$ (14)

Now, using previous expression one can obtain the following equation;

$\int\int_{\mathcal{H}_{e}}\mathcal{K}ds+\oint_{\partial\mathcal{H}_{e}}kdt\overset{h\rightarrow\infty}{=}\int\int_{T_{\infty}}\mathcal{K}dS+\int^{\pi+\psi}_{0}d\phi.$
(15)

The $0$th order light ray in the weak field limits is calculated as
$r(t)=b/\sin\phi$. Using Eqs.(9) and (15), the bending angle can be obtained
as

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\psi=-\int^{\pi}_{0}\int^{\infty}_{b/\sin\phi}\mathcal{K}\sqrt{det\bar{g}}~{}drd\phi,$
(16)

where

$\sqrt{det\bar{g}}=r+{2qm}+\mathcal{O}\left({m^{2}}\right).$

Using the Gaussian curvature upto the leading order terms and angle of
deflection is calculated as

$\displaystyle\psi$ $\displaystyle\thickapprox$
$\displaystyle\frac{2mq}{b}+\mathcal{O}\left({m^{2}}\right).$ (17)

The first term of the obtained deflection angle $\psi$ (17) is depending on
the first order of mass $m$, $q$ and $b$. While, the higher order terms of
$\psi$ are depending upon higher orders of $m$, $q$ and $b$. For the sake of
simplicity, we consider the only first order term of the mass $m$. The
obtained bending angle in non-plasma medium converts into the deflection angle
of Schwarzschild BH after putting $q=2$.

##  IV Deflection Angle $\psi$ in Plasma Medium

This section is based on the computation of the deflection angle for WH-like
static aether solution in plasma medium. The refractive index $n(r)$ for WH-
like solution is calculated as

$n^{2}\left(r,\omega(r)\right)=1-\frac{\omega_{e}^{2}(r)}{\omega_{\infty}^{2}(r)}{A(r)},$

which can also be represented as:

$n(r)=\sqrt{{1-\frac{\omega_{e}^{2}}{\omega_{\infty}^{2}}\left(A(r)\right)}},$
(18)

where electron plasma frequency is denoted by $\omega_{e}$, while
$\omega_{\infty}$ denotes photon frequency calculated at infinity by observer,
then the corresponding optical metric can be defined as;

$dt^{2}=g^{opt}_{lm}dx^{l}dx^{m}=n^{2}\left(\frac{B(r)}{A(r)}{dr^{2}}+\frac{B(r)r^{2}}{A(r)}{d\phi^{2}}\right).$
(19)

For our metric, we can write the above values as:

$A(r)=\left({\frac{1-\frac{m}{2r}}{1+\frac{m}{2r}}}\right)^{q},~{}~{}~{}~{}~{}B(r)=\frac{D(r)}{r^{2}}=\frac{({1+\frac{m}{2r}})^{q+2}}{({1-\frac{m}{2r}})^{q-2}}.\\\
$

The Gaussian optical curvature can be defined as:

$\displaystyle\mathcal{K}=\frac{A^{\prime\prime}(r)}{2B(r)n(r)^{2}}-\frac{A^{\prime}(r)B^{\prime}(r)}{4B(r)^{2}n(r)^{2}}+\frac{A^{\prime}(r)D^{\prime}(r)}{4B(r)D(r)n(r)^{2}}-\frac{A^{\prime}(r)^{2}}{2A(r)B(r)n(r)^{2}}$
$\displaystyle+\frac{A(r)B^{\prime}(r)D^{\prime}(r)}{4B(r)^{2}D(r)n(r)^{2}}-\frac{A(r)D^{\prime\prime}(r)}{2B(r)D(r)n(r)^{2}}+\frac{A(r)D^{\prime}(r)^{2}}{4B(r)D(r)^{2}n(r)^{2}}.$
(20)

Using Eq.(20), the Gaussian optical curvature can be obtained as:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{K}\simeq-\frac{qm}{r^{3}}-\frac{3qm}{2r^{3}}\frac{\omega_{e}^{2}}{\omega_{\infty}^{2}}+\mathcal{O}\left({m^{2}}\right).$
(21)

We compute the bending angle by using the GBT, for this purpose we apply
straight line approximation $r=\frac{b}{sin\phi}$ at 0th order and obtain the
deflection angle as;

$\psi=-\int_{0}^{\pi}\int_{\frac{b}{\sin\phi}}^{\infty}\mathcal{K}dS,$ (22)

where $dS$=$\sqrt{-g}drd\phi$ and

$dS=r-{r}\frac{\omega_{e}^{2}}{\omega_{\infty}^{2}}+\left(2q-q\frac{\omega_{e}^{2}}{\omega_{\infty}^{2}}\right){m}+\mathcal{O}\left({m^{2}}\right)drd\phi.$
(23)

Using Eq.(22), the deflection angle in plasma medium can be obtained as:

$\displaystyle\psi$ $\displaystyle\thickapprox$
$\displaystyle\frac{2mq}{b}+\frac{mq}{b}\frac{\omega_{e}^{2}}{\omega_{\infty}^{2}}.$
(24)

Figure 1: Plot of $\psi$ versus $b$.

The obtained deflection angle (24) depends on $m$, $b$ and $q$. The above
results presents that photon rays are travelling into a homogenous plasma
medium. We examined that Eq.(24) converted into Eq.(17), if the effect of
plasma is neglected. For $q=2$, the obtained angle (24) also reduces into the
Schwarzschild angle. $\psi$ versus $b$

Here, we use $p=\frac{\omega_{e}}{\omega_{\infty}}$=$10^{-1}$ and examine the
graphical behaviour of angle $\psi$ w.r.t $b$ and fixed $m=1$ which are
discussed below.

Figure 1 shows the relationship between $\psi$ and $b$ by varying $q$ and
fixing $m$. It consists of small values of $b$ and $m$. When the values of
impact parameter $b$ approaches to zero, the deflection angle $\psi$
approaches towards the infinity at $m=1$. As $b$ increases i.e ;
$b\rightarrow+\infty$, the $\psi$ approaches to zero. It is to be noted that
the graphical behaviour of bending angle in the non-plasma case after
neglecting the plasma effect will be similar with the plasma case. The impact
of the plasma on the bending angle is negligible, one can obtain the same
graphs by taking same values in both plasma Eq.(24) and non-plasma mediums
Eq.(17).

##  V Deflection Angle in Dark Matter Medium

This section is devoted to study the impact of the DM medium on deflection
angle. For this purpose, we use DM medium’s refractive index (2):

$n(\omega)=1+\beta{A_{0}}+{A_{2}}{\omega^{2}}.$

The WH’s optical geometry in 2-dimensional is:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}dt^{2}=n^{2}\left(\frac{B(r)}{A(r)}{dr^{2}}+\frac{B(r)r^{2}}{A(r)}{d\phi^{2}}\right),$
(25)

where

$A(r)=\left({\frac{1-\frac{m}{2r}}{1+\frac{m}{2r}}}\right)^{q},~{}~{}~{}~{}B(r)=\frac{D(r)}{r^{2}}=\frac{({1+\frac{m}{2r}})^{q+2}}{({1-\frac{m}{2r}})^{q-2}}.$

The Gaussian curvature is expressed as

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{K}\simeq-\frac{qm}{r^{3}(1+\beta{A_{0}}+{A_{2}}{\omega^{2}})^{2}}+\mathcal{O}\left({m^{2}}\right).$
(26)

The deflection angle can be obtained as:

$\psi=-\int_{0}^{\pi}\int_{\frac{b}{\sin\vartheta}}^{\infty}\mathcal{K}\sqrt{det\bar{g}}~{}drd\phi,$
(27)

For the given line-element one can be determined as;

$\sqrt{det\bar{g}}=r(1+\beta{A_{0}}+{A_{2}}{\omega^{2}})^{2}+2qm(1+\beta{A_{0}}+{A_{2}}{\omega^{2}})^{2}+\mathcal{O}\left({m^{2}}\right).$
(28)

The deflection angle for WH-like static aether solution in DM medium can be
calculated as;

$\displaystyle\psi$ $\displaystyle\thickapprox$
$\displaystyle\frac{2mq}{b(1+\beta{A_{0}}+{A_{2}}{\omega^{2}})^{6}},$ (29)

which is depending upon $m$, $q$ and $b$. If $q=2$, then the obtained bending
angle reduces into bending angle of the Schwarzschild BH upto first order of
$m$ in dark matter medium. We also observe that the bending angle in case of
dark matter medium is larger than in general. This expression simplifies to
the vacuum case in the absence of the DM medium.

##  VI Deflection Angle by Keeton and Petters Method

The calculations of the deflection angle $(\psi)$ of the WH-like static aether
solution by using the Keeton and Petters technique are discussed in this
section. The PPN (post-post-Newtonian) framework is a direct method to deal
with all kinds of gravity ideas for which the weak-deflection limit is stated
as in a single variable $m$ in a series expansion. The concept was extended to
the $3rd$-order Sarmiento (1982). However, Keeton and Petters modified that
technique to make it more compatible with their approach and offers new-
results Keeton and Petters (2005). The spacetime geometry is supposed to be
stable, non-linear spherically symmetric and asymptotically flat:

$\displaystyle ds^{2}=-\bar{A}(r)dt^{2}+\bar{B}(r)dr^{2}+r^{2}d\Omega^{2}.$
(30)

The coefficients of Eq.(30) should be written in PPN series upto the 3rd-order
Keeton and Petters (2005) as follows:

$\displaystyle\bar{A}(r)=1+2a_{1}\left(\frac{\phi}{c^{2}}\right)+2a_{2}\left(\frac{\phi}{c^{2}}\right)^{2}+2a_{3}\left(\frac{\phi}{c^{2}}\right)^{3}+......$
(31)
$\displaystyle\bar{B}(r)=1-2b_{1}\left(\frac{\phi}{c^{2}}\right)+4b_{2}\left(\frac{\phi}{c^{2}}\right)^{2}-8b_{3}\left(\frac{\phi}{c^{2}}\right)^{3}+......,$
(32)

where $\phi$ is a 3-dimensional Newtonian potential;

$\displaystyle\frac{\phi}{c^{2}}=-\frac{m}{r}.$ (33)

The deflection angle in a series form is defined as;

$\displaystyle\psi=A_{1}\left(\frac{m}{b}\right)+A_{2}\left(\frac{m}{b}\right)^{2}+A_{3}\left(\frac{m}{b}\right)^{3}+\mathcal{O}\left(\frac{m}{b}\right)^{4},$
(34)

where

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle 2(a_{1}+b_{1}),$
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\left(2a_{1}^{2}-a_{2}+a_{1}b_{1}-\frac{b_{1}^{2}}{4}+b_{2}\right)\pi,$
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\left(35a_{1}^{3}+15a_{1}^{2}b_{1}-3a_{1}(10a_{2}+b_{1}^{2}-4b_{2})+6a_{3}\right.$
(35) $\displaystyle+$
$\displaystyle\left.b_{1}^{3}-6a_{2}b_{1}-4b_{1}b_{2}+8b_{3}\right).$

The spacetime metric for WH-like static aether solution already defined by
Eq.$(\ref{AH0})$;

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+D(r)d\Omega^{2},$
$\text{with}~{}~{}~{}~{}d\Omega^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
$
$A(r)=\left({\frac{1-\frac{m}{2r}}{1+\frac{m}{2r}}}\right)^{q},~{}~{}~{}~{}~{}B(r)=\frac{({1+\frac{m}{2r}})^{q+2}}{({1-\frac{m}{2r}})^{q-2}}.\\\
$

Dividing the right hand side of the metric with $B(r)$, where
$B(r)=D(r)/r^{2}$. Then the standard form of metric is written as;

$\displaystyle ds^{2}=-\frac{A(r)}{B(r)}dt^{2}+dr^{2}+r^{2}d\Omega^{2},$ (36)
$\text{where}~{}~{}~{}~{}~{}~{}~{}~{}G(r)=\frac{A(r)}{B(r)}=1-\frac{2qm}{r}+\frac{(1+4q^{2})m^{2}}{2r^{2}}+\frac{(-7q-8q^{3})m^{3}}{6r^{3}}+\mathcal{O}\left({m^{4}}\right).$
(37)
$\text{and}~{}~{}~{}~{}~{}~{}~{}~{}H(r)=1,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Now, we compare the $G(r)$ with $\bar{A}(r)$ and $\bar{B}(r)$ with $H(r)$ and
write the PPN coefficients as;

$\displaystyle~{}~{}a_{1}=q,~{}~{}~{}~{}a_{2}=q^{2}+\frac{1}{4},~{}~{}~{}~{}a_{3}=\frac{7q+8q^{3}}{12},~{}~{}~{}~{}~{}~{}~{}b_{1}=b_{2}=b_{3}=0.$

After putting all above coefficient into Eq.(35) we get;

$\displaystyle
A_{1}=2q,~{}~{}~{}~{}A_{2}=\left(q^{2}-\frac{1}{4}\right)\pi,~{}~{}~{}~{}~{}~{}A_{3}=6q^{3}-\frac{8q}{3}.$

Hence, the bending angle for WH-like static aether solution by Keeton and
Petters method can be computed as;

$\displaystyle\psi=2q\left(\frac{m}{b}\right)+\left(q^{2}-\frac{1}{4}\right)\pi\left(\frac{m}{b}\right)^{2}+\left(6q^{3}-\frac{8q}{3}\right)\left(\frac{m}{b}\right)^{3}+\mathcal{O}\left(\frac{m}{b}\right)^{4}.$
(38)

The obtained deflection angle depends on $m$, $q$ and $b$. The obtained angle
(38) reduces to the deflection angle of Schwarzschild by using Keeton and
Petters technique when $q=2$.

##  VII Photon ring and wormhole shadow

In this section, let us examine the photonsphere and the shadow produced by
the wormhole considered in this study. There have been various studies of the
shadow of black holes and shadow of wormholes Kuang and Övgün (2022); Uniyal
et al. (2022); Khodadi et al. (2020); Vagnozzi et al. (2022); Roy et al.
(2022); Vagnozzi and Visinelli (2019); Allahyari et al. (2020); Atamurotov et
al. (2013); Abdujabbarov et al. (2015); Wei et al. (2019a, b); Abdolrahimi et
al. (2015a); Adair et al. (2020); Abdolrahimi et al. (2015b); Herdeiro et al.
(2021); Cunha et al. (2020, 2019); Cunha and Herdeiro (2018); Cunha et al.
(2017); Afrin et al. (2021); Jha and Rahaman (2021); Khodadi et al. (2021).
But first time here we will include the influence of a non-magnetized cold
plasma with electron plasma frequency $\omega_{e}(r)$, which can be done
through by means of deriving the equations of motion (EoS) through the
Hamiltonian Perlick et al. (2015) to wormhole spacetime.

$H=\frac{1}{2}g^{ik}p_{i}p_{k}=\frac{1}{2}\left(-\frac{p_{t}^{2}}{A(r)}+\frac{p_{r}^{2}}{B(r)}+\frac{p_{\phi}^{2}}{C(r)}+\omega_{e}(r)^{2}\right).$
(39)

We only considered motion along the equatorial plane, thus, $D(r)=C(r)$. We
can then derive the EoS through

$\dot{x}^{i}=\frac{\partial H}{\partial
p_{i}},\quad\quad\dot{p}_{i}=-\frac{\partial H}{\partial x^{i}},$ (40)

which enables us to extract the two constants of motion:

$E=A(r)\frac{dt}{d\lambda},\quad L=C(r)\frac{d\phi}{d\lambda}.$ (41)

Also, using this, we can define the impact parameter as

$b\equiv\frac{L}{E}=\frac{C(r)}{A(r)}\frac{d\phi}{dt}.$ (42)

Going back to the metric, null geodesics requires that $ds^{2}=0$, and the
orbit equation can then be expressed as

$\displaystyle\left(\frac{dr}{d\phi}\right)^{2}=\frac{C(r)}{B(r)}\left(\frac{h(r)^{2}}{b^{2}}-1\right).$
(43)

Following methods in Ref. Perlick et al. (2015), the orbit equation allows one
to define the function

$h(r)^{2}=\frac{C(r)}{A(r)}n(r)^{2}=\frac{C(r)}{A(r)}\left(1-\frac{\omega_{e}^{2}}{\omega_{0}^{2}}A(r)\right),$
(44)

under the assumption that the homogeneous plasma is non-gravitating Crisnejo
and Gallo (2018). It is also easy to see how the above reduces to the standard
case if $n(r)=0$. The photonsphere can then be sought off by solving $r$ in
$h^{\prime}(r)=0$. Depending how complicated the expression for the metric
coefficients whether one can obtain an analytic expression or not. One can
determine the photonsphere radii via

$\left(\frac{\omega_{e}^{2}}{\omega_{0}^{2}}A(r)^{2}-A(r)\right)C^{\prime}(r)+C(r)A^{\prime}(r)=0,$
(45)

and for the case of $n(r)=0$, we get

$r_{\text{ph}}=\frac{m}{2}\omega^{\pm},$ (46)

where we write first $\omega^{\pm}=q\pm\sqrt{q^{2}-1}$ for brevity Epps and
Hudson (2017). We note that there is a third solution $r_{\text{ph}}=m/2$, but
it does not produce any shadow cast due to the wormhole.

A static observer at infinity can construct the following relation,

$\tan(\alpha_{\text{sh}})=\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta
x}=\left(\frac{C(r)}{B(r)}\right)^{1/2}\frac{d\phi}{dr}\bigg{|}_{r=r_{\text{obs}}},$
(47)

which can be simplified into

$\sin^{2}(\alpha_{\text{sh}})=\frac{b_{\text{crit}}^{2}}{h(r_{\text{obs}})^{2}}$
(48)

with the help of the orbit equation. The critical impact parameter can be
sought off under the condition $d^{2}r/d\phi^{2}=0$ and we find

$b_{\text{crit}}^{2}=\frac{h(r_{\text{ph}})}{\left[B^{\prime}(r_{\text{ph}})C(r_{\text{ph}})-B(r_{\text{ph}})C^{\prime}(r_{\text{ph}})\right]}\Bigg{[}h(r_{\text{ph}})B^{\prime}(r_{\text{ph}})C(r_{\text{ph}})-h(r_{\text{ph}})B(r_{\text{ph}})C^{\prime}(r_{\text{ph}})-2h^{\prime}(r_{\text{ph}})B(r_{\text{ph}})C(r_{\text{ph}})\Bigg{]},$
(49)

where the derivatives with respect to $r$ are evaluate at $r\to
r_{\text{ph}}$. The analytic expression is quite lengthy with the inclusion of
plasma, but for the case without its influence, we can obtained two solutions:

$b_{\text{crit}}^{2}=\frac{m^{2}\sqrt{q^{2}-1}(\omega^{\pm}\mp
1)^{1-2q}(\omega^{\pm}\pm 1)^{2q+1}}{2\omega^{\pm}}.$ (50)

This will be used to the calculation of the shadow, which gives us the exact
analytical formula of

$R_{\text{sh}}=\left[\frac{8r_{\text{obs}}^{4}m^{2}\sqrt{q^{2}-1}(\omega^{\pm}\mp
1)^{2q+1}(\omega^{\pm}\pm
1)^{1-2q}(2r_{\text{obs}}-m)^{2(q-1)}(2r_{\text{obs}}+m)^{-2(q+1)}}{\omega^{\pm}}\right]^{1/2}$
(51)

for the case $n(r)=0$. For the case with plasma, we plot it numerically. See
Fig. 2.

Figure 2: Plot of the shadow radius of the wormhole at varying locations of a
static observer. Here, we compared the shadow behavior between the
Schwarzschild case, and the wormhole with and without the influence of plasma.
We set $m=1$ and the plasma parameter $p=10^{-1}$.

We can then see how the shadow radius behave in relation to the location of
the static observer with respect to the wormhole. We can see that the
Schwarzschild case behave as it is. But, when the wormhole is considered, we
can see that the zero radius shadow is nearer to the black hole. The lower
left inset plot reveals that the effect of plasma is not that evident. But we
can see that the shadow radius of the wormhole slightly increases, then
decreases again. The intersection point, near $r=2m$, indicates that the
angular radius of the shadow is $\theta=\pi/2$. In this location, the observer
will see that half of the sky is dark. After, we can see the obvious
deviations at farther distances. Notably, the plasma’s effect is to increase
the wormhole shadow. At very large distances, we can see that the rate of the
change in the shadow radius levels-off near the Schwarzschild case. It
indicates that the shadow can be useful to detect the imprints of plasma.
Finally, we remark that in this plot, we have only used the upper sign in Eq.
(2). Using the lower sign, one can verify that using $q>0$ does not produce
any shadow, and using $q<0$ gives infinitely large shadow near the wormhole,
which is unphysical. However, we found out that at very large distances, the
effect of the second solution is nearly the same as the one in the upper right
inset plot.

Let us now use the DM refractive index $n(\omega)$ in Eq. (2) for the next
case. We then find that the location of the photonsphere is independent of
$n(\omega)$:

$n(\omega)^{2}[C^{\prime}(r)A(r)-A^{\prime}(r)C(r)]=0,$ (52)

which yields the same expression for the photonsphere in Eq. (46). For the
critical impact parameter, we find

$b_{\text{crit}}^{2}=\pm\frac{n(\omega)^{2}m^{2}\sqrt{q^{2}-1}(\omega^{\pm}\mp
1)^{2(1-q)}(\omega^{\pm}\pm 1)^{2(q+1)}}{4\omega^{\pm}(\omega^{\pm}q-1)},$
(53)

where we are only interested in using the upper sign. With Eq. (48), we can
get an analytical expression for the shadow radius as

$R_{\text{sh}}=4mn(\omega)(-1)^{q}r_{\text{obs}}^{2}(m-2r_{\text{obs}})^{q-1}(m+2r_{\text{obs}})^{-(q+1)}\left[\pm\frac{\sqrt{q^{2}-1}(\omega^{\pm}\mp
1)^{2(1-q)}(\omega^{\pm}\pm
1)^{2(q+1)}}{4\omega^{\pm}(\omega^{\pm}q-1)}\right]^{1/2},$ (54)

which is quite a worked out equation. Interestingly, for static observers in a
remote location from the wormhole, we can apply Taylor expansion to get a
simplified and approximated equation:

$R_{\text{sh}}=n(\omega)m\left[\pm\frac{\sqrt{q^{2}-1}(\omega^{\pm}\mp
1)^{2(1-q)}(\omega^{\pm}\pm
1)^{2(q+1)}}{4\omega^{\pm}(\omega^{\pm}q-1)}\right]^{1/2}.$ (55)

In a case where $q=2$, we find

$R_{\text{sh}}=3\sqrt{3}mn(\omega),$ (56)

where we could see clearly the influence of dark matter to the shadow radius.
Furthermore, in this remote region, we saw again that the effect of the
wormhole mimics the Schwarzschild case.

##  VIII Conclusion

In this paper, we have discussed WH-like static aether solution and derived
deflection angle in the non-plasma, plasma and DM mediums. Also, we have found
the deflection angle by using Keeton and Petters technique. For this purpose,
we have used an optical metric to determine Gaussian optical curvature and
then applied GBT to examine the deflection angle.

We have examined that deflection angle $(\ref{P1})$ in the non-plasma medium,
plasma medium $(\ref{P2})$, DM medium $(\ref{sib14})$ and by Keeton and
Petters technique $(\ref{sib28})$ is depending upon the parameter $q$, mass
$m$ as a integrating constant and impact parameter $b$.

The graphical behaviour of bending angle in the plasma medium has examined in
such a way that when we make a relation between $\psi$ and $b$ and vary the
value of $q$. We have seen that the deflection angle increases when the values
of $q$ increases. Moreover, when we make a relation between $\psi$ and $m$ and
varying $b$, we have noticed that the angle is decrease. Afterwards, we have
noticed that when we make a relation between $\psi$ and $q$ and varying $m$,
the angle is increases. It is to be mentioned here that the plots in non-
plasma have shown the same behaviour as plasma medium.

We have observed that if the plasma effect is ignored as
$(\frac{\omega_{e}}{\omega_{\infty}}\rightarrow 0)$, then the bending angle
$(\ref{P2})$ has reduced into the bending angle $(\ref{P1})$. In case of DM
medium, we have observed that if we removed the effect of DM medium then this
obtained angle converts into the angle obtained in $(\ref{P1})$. We have also
examined that the obtained deflection angles by plasma, non-plasma, DM and
Keeton and Petters technique reduces to the Schwarzschild deflection angle
upto $1$st order term of $m$ by taking $q=2$.

The results we have obtained for WH-like static aether solution in the
presence of different mediums i.e plasma and non-plasma shows that deflection
angle $\psi$ has direct relation with mass $m$ and a parameter $q$ which means
that WH with greater mass has greater gravitational pull and bends the light
passing by it at large angle. Whereas, WH with smaller mass deflect the light
at smaller angle. We also notice that deflection angle $\psi$ has inverse
relation with impact parameter $b$, which shows that smaller value of impact
parameter has larger deflection angle and vice versa.

Also, we have examined the impact of DM medium on bending angle of WH-like
static aether solution. The refractive index in DM medium has taken
homogeneously nonuniform. Hence it is concluded that bending angle by WH-like
static aether solution increases with increasing parameter $q$ and $m$, while
the bending angle decreases in a increasing medium of DM. It is showed that
how weak deflection angle of WH is the affected by parameter $q$ and $m$.

To broaden the scope of the study, we also examined the behavior of the shadow
radius of the wormhole, comparing it to the case where it is surrounded by
plasma. Our main result indicates that as the photons travels through the
plasma, its imprints or effects can be perceived by a static observer at
infinity through the increased shadow size. Although highly unlikely in terms
of situational applicability, our calculation also reveals that for observers
near the wormhole, the effects of plasma is rather weak, compared to an
observer at a very large distance.

###### Acknowledgements.

A. Ö. and R. P. would like to acknowledge networking support by the COST
Action CA18108 - Quantum gravity phenomenology in the multi-messenger approach
(QG-MM).

## References

  * Einstein (1916) A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916, 688 (1916).
  * Akiyama et al. (2019) K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875, L1 (2019), eprint 1906.11238.
  * Akiyama et al. (2022) K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930, L12 (2022).
  * Einstein and Rosen (1935) A. Einstein and N. Rosen, Physical Reviewv 48, 73 (1935).
  * Wheeler (1957) J. A. Wheeler, Annals of Physics 2, 604 (1957).
  * Misner and Wheeler (1957) C. W. Misner and J. A. Wheeler, Annals of physics 2, 525 (1957).
  * Wheeler (1955) J. A. Wheeler, Physical Review 97, 511 (1955).
  * Morris and Thorne (1988) M. S. Morris and K. S. Thorne, American Journal of Physics 56, 395 (1988).
  * Damour and Solodukhin (2007) T. Damour and S. N. Solodukhin, Physical Review D 76, 024016 (2007).
  * Bueno et al. (2018) P. Bueno, P. A. Cano, F. Goelen, T. Hertog, and B. Vercnocke, Physical Review D 97, 024040 (2018).
  * Hawking (1988) S. W. Hawking, in _Euclidean Quantum Gravity_ (1988), pp. 363–369.
  * Övgün et al. (2019) A. Övgün, K. Jusufi, and I. Sakallı, Phys. Rev. D 99, 024042 (2019), eprint 1804.09911.
  * Halilsoy et al. (2014) M. Halilsoy, A. Ovgun, and S. H. Mazharimousavi, Eur. Phys. J. C 74, 2796 (2014), eprint 1312.6665.
  * Ovgun and Halilsoy (2016) A. Ovgun and M. Halilsoy, Astrophys. Space Sci. 361, 214 (2016), eprint 1509.01237.
  * Visser (1995) M. Visser, _Lorentzian wormholes: From Einstein to Hawking_ (1995), ISBN 978-1-56396-653-8.
  * Marolf and Polchinski (2013) D. Marolf and J. Polchinski, Physical review letters 111, 171301 (2013).
  * Massey et al. (2010) R. Massey, T. Kitching, and J. Richard, Reports on Progress in Physics 73, 086901 (2010).
  * Bartelmann and Schneider (2001) M. Bartelmann and P. Schneider, Physics Reports 340, 291 (2001).
  * Cunha et al. (2015) P. V. Cunha, C. A. Herdeiro, E. Radu, and H. F. Rúnarsson, Physical review letters 115, 211102 (2015).
  * Gibbons and Werner (2008) G. Gibbons and M. Werner, Classical and Quantum Gravity 25, 235009 (2008).
  * Gibbons and Warnick (2009) G. Gibbons and C. Warnick, Physical Review D 79, 064031 (2009).
  * Gibbons et al. (2009) G. Gibbons, C. Herdeiro, C. Warnick, and M. Werner, Physical Review D 79, 044022 (2009).
  * Gibbons and Vyska (2012) G. W. Gibbons and M. Vyska, Classical and Quantum Gravity 29, 065016 (2012).
  * Bloomer (2011) C. Bloomer, arXiv preprint arXiv:1111.4998 (2011).
  * Werner (2012) M. Werner, General Relativity and Gravitation 44, 3047 (2012).
  * Övgün (2019) A. Övgün, Phys. Rev. D 99, 104075 (2019), eprint 1902.04411.
  * Javed et al. (2019a) W. Javed, R. Babar, and A. Övgün, Physical Review D 99, 084012 (2019a).
  * Javed et al. (2019b) W. Javed, R. Babar, and A. Övgün, Physical Review D 100, 104032 (2019b).
  * Javed et al. (2019c) W. Javed, J. Abbas, and A. Övgün, Physical Review D 100, 044052 (2019c).
  * Pantig and Rodulfo (2020a) R. C. Pantig and E. T. Rodulfo, Chinese J. Phys. 68, 236 (2020a).
  * Pantig and Rodulfo (2020b) R. C. Pantig and E. T. Rodulfo, Chin. J. Phys. 66, 691 (2020b).
  * Pantig Reggie C. and Ali (2022) R. E. T. Pantig Reggie C., Yu Paul K. and O. Ali, Annals of Physics 436, 168722 (2022).
  * Pantig and Övgün (2022a) R. C. Pantig and A. Övgün, Eur. Phys. J. C 82, 391 (2022a), eprint 2201.03365.
  * Pantig and Övgün (2022b) R. C. Pantig and A. Övgün, JCAP 2022, 056 (2022b), eprint 2202.07404.
  * Kuang and Övgün (2022) X.-M. Kuang and A. Övgün (2022), eprint 2205.11003.
  * Uniyal et al. (2022) A. Uniyal, R. C. Pantig, and A. Övgün (2022), eprint 2205.11072.
  * Kumaran and Övgün (2020) Y. Kumaran and A. Övgün, Chin. Phys. C 44, 025101 (2020), eprint 1905.11710.
  * Kumaran and Övgün (2021) Y. Kumaran and A. Övgün, Turk. J. Phys. 45, 247 (2021), eprint 2111.02805.
  * Kumaran and Övgün (2022) Y. Kumaran and A. Övgün, Symmetry 14 (2022).
  * Javed et al. (2020) W. Javed, M. B. Khadim, J. Abbas, and A. Övgün, The European Physical Journal Plus 135, 1 (2020).
  * Javed et al. (2021) W. Javed, A. Hamza, and A. Övgün, Universe 7, 385 (2021), eprint 2110.11397.
  * He and Lin (2016) G. He and W. Lin, Classical and Quantum Gravity 33, 095007 (2016).
  * Crisnejo and Gallo (2018) G. Crisnejo and E. Gallo, Physical Review D 97, 124016 (2018).
  * Nakajima and Asada (2012) K. Nakajima and H. Asada, Physical Review D 85, 107501 (2012).
  * Jusufi and Övgün (2018) K. Jusufi and A. Övgün, Physical Review D 97, 024042 (2018).
  * Övgün (2018) A. Övgün, Physical Review D 98, 044033 (2018).
  * Epps and Hudson (2017) S. D. Epps and M. J. Hudson, Monthly Notices of the Royal Astronomical Society 468, 2605 (2017).
  * Hinshaw et al. (2013) G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. Bennett, J. Dunkley, M. Nolta, M. Halpern, R. Hill, N. Odegard, et al., The Astrophysical Journal Supplement Series 208, 19 (2013).
  * Latimer (2013) D. C. Latimer, Physical Review D 88, 063517 (2013).
  * Oost et al. (2021a) J. Oost, S. Mukohyama, and A. Wang, Universe 7, 272 (2021a).
  * Penrose (1969) R. Penrose, Nuovo Cimento Rivista Serie 1, 252 (1969).
  * Eling and Jacobson (2006) C. Eling and T. Jacobson, Classical and Quantum Gravity 23, 5625 (2006).
  * Zhu et al. (2019) T. Zhu, Q. Wu, M. Jamil, and K. Jusufi, Phys. Rev. D 100, 044055 (2019), eprint 1906.05673.
  * Perlick et al. (2015) V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D 92, 104031 (2015).
  * Guerrero et al. (2022) M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, and D. Gómez Sáez-Chillón, Phys. Rev. D 105, 084057 (2022), eprint 2202.03809.
  * Zhu and Wang (2021) Y. Zhu and T. Wang, Phys. Rev. D 104, 104052 (2021), eprint 2109.08463.
  * Rahaman et al. (2021) F. Rahaman, K. N. Singh, R. Shaikh, T. Manna, and S. Aktar, Class. Quant. Grav. 38, 215007 (2021), eprint 2108.09930.
  * Bouhmadi-López et al. (2021) M. Bouhmadi-López, C.-Y. Chen, X. Y. Chew, Y. C. Ong, and D.-h. Yeom, JCAP 10, 059 (2021), eprint 2108.07302.
  * Bugaev et al. (2021) M. A. Bugaev, I. D. Novikov, S. V. Repin, and A. A. Shelkovnikova, Astron. Rep. 65, 1185 (2021), eprint 2106.03256.
  * Peng et al. (2021) J. Peng, M. Guo, and X.-H. Feng, Phys. Rev. D 104, 124010 (2021), eprint 2102.05488.
  * Guerrero et al. (2021) M. Guerrero, G. J. Olmo, and D. Rubiera-Garcia, JCAP 04, 066 (2021), eprint 2102.00840.
  * Wielgus et al. (2020) M. Wielgus, J. Horak, F. Vincent, and M. Abramowicz, Phys. Rev. D 102, 084044 (2020), eprint 2008.10130.
  * Wang et al. (2020) X. Wang, P.-C. Li, C.-Y. Zhang, and M. Guo, Phys. Lett. B 811, 135930 (2020), eprint 2007.03327.
  * Gyulchev et al. (2019) G. Gyulchev, P. Nedkova, V. Tinchev, and Y. Stoytcho, AIP Conf. Proc. 2075, 040005 (2019).
  * Övgün et al. (2018) A. Övgün, I. Sakallı, and J. Saavedra, JCAP 10, 041 (2018), eprint 1807.00388.
  * Övgün and Sakallı (2020) A. Övgün and I. Sakallı, Class. Quant. Grav. 37, 225003 (2020), eprint 2005.00982.
  * Övgün et al. (2020) A. Övgün, I. Sakallı, J. Saavedra, and C. Leiva, Mod. Phys. Lett. A 35, 2050163 (2020), eprint 1906.05954.
  * Çimdiker et al. (2021) I. Çimdiker, D. Demir, and A. Övgün, Phys. Dark Univ. 34, 100900 (2021), eprint 2110.11904.
  * Oost et al. (2021b) J. Oost, S. Mukohyama, and A. Wang, Universe 7, 272 (2021b), eprint 2106.09044.
  * Sarmiento (1982) A. Sarmiento, General Relativity and Gravitation 14, 793 (1982).
  * Keeton and Petters (2005) C. R. Keeton and A. Petters, Physical Review D 72, 104006 (2005).
  * Khodadi et al. (2020) M. Khodadi, A. Allahyari, S. Vagnozzi, and D. F. Mota, JCAP 09, 026 (2020), eprint 2005.05992.
  * Vagnozzi et al. (2022) S. Vagnozzi, R. Roy, Y.-D. Tsai, and L. Visinelli (2022), eprint 2205.07787.
  * Roy et al. (2022) R. Roy, S. Vagnozzi, and L. Visinelli, Phys. Rev. D 105, 083002 (2022), eprint 2112.06932.
  * Vagnozzi and Visinelli (2019) S. Vagnozzi and L. Visinelli, Phys. Rev. D 100, 024020 (2019), eprint 1905.12421.
  * Allahyari et al. (2020) A. Allahyari, M. Khodadi, S. Vagnozzi, and D. F. Mota, JCAP 02, 003 (2020), eprint 1912.08231.
  * Atamurotov et al. (2013) F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D 88, 064004 (2013).
  * Abdujabbarov et al. (2015) A. A. Abdujabbarov, L. Rezzolla, and B. J. Ahmedov, Mon. Not. Roy. Astron. Soc. 454, 2423 (2015), eprint 1503.09054.
  * Wei et al. (2019a) S.-W. Wei, Y.-C. Zou, Y.-X. Liu, and R. B. Mann, JCAP 08, 030 (2019a), eprint 1904.07710.
  * Wei et al. (2019b) S.-W. Wei, Y.-X. Liu, and R. B. Mann, Phys. Rev. D 99, 041303 (2019b), eprint 1811.00047.
  * Abdolrahimi et al. (2015a) S. Abdolrahimi, R. B. Mann, and C. Tzounis, Phys. Rev. D 91, 084052 (2015a), eprint 1502.00073.
  * Adair et al. (2020) C. Adair, P. Bueno, P. A. Cano, R. A. Hennigar, and R. B. Mann, Phys. Rev. D 102, 084001 (2020), eprint 2004.09598.
  * Abdolrahimi et al. (2015b) S. Abdolrahimi, R. B. Mann, and C. Tzounis, Phys. Rev. D 92, 124011 (2015b), eprint 1510.03530.
  * Herdeiro et al. (2021) C. A. R. Herdeiro, A. M. Pombo, E. Radu, P. V. P. Cunha, and N. Sanchis-Gual, JCAP 04, 051 (2021), eprint 2102.01703.
  * Cunha et al. (2020) P. V. P. Cunha, N. A. Eiró, C. A. R. Herdeiro, and J. P. S. Lemos, JCAP 03, 035 (2020), eprint 1912.08833.
  * Cunha et al. (2019) P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Universe 5, 220 (2019), eprint 1909.08039.
  * Cunha and Herdeiro (2018) P. V. P. Cunha and C. A. R. Herdeiro, Gen. Rel. Grav. 50, 42 (2018), eprint 1801.00860.
  * Cunha et al. (2017) P. V. P. Cunha, C. A. R. Herdeiro, B. Kleihaus, J. Kunz, and E. Radu, Phys. Lett. B 768, 373 (2017), eprint 1701.00079.
  * Afrin et al. (2021) M. Afrin, R. Kumar, and S. G. Ghosh, Mon. Not. Roy. Astron. Soc. 504, 5927 (2021), eprint 2103.11417.
  * Jha and Rahaman (2021) S. K. Jha and A. Rahaman (2021), eprint 2111.02817.
  * Khodadi et al. (2021) M. Khodadi, G. Lambiase, and D. F. Mota, JCAP 09, 028 (2021), eprint 2107.00834.