# Weak Deflection Angle, Hawking Radiation, Greybody Bound and Shadow Cast for
Static Black Hole in the Framework of $f(R)$ Gravity

Surajit Mandal111Corresponding author

###### Abstract

In this work, we probe the weak gravitational lensing by a static spherically
symmetric black hole in view of $f(R)$ gravity in the background of the non-
plasma medium (vacuum). We provide a discussion on a light ray in a static
black hole solution in $f(R)$ gravity. To adore this purpose, we find the
Gaussian optical curvature in weak gravitational lensing by utilizing the
optical geometry of this black hole solution. Furthermore, we find the
deflection angle in non-plasma medium up to the leading order by employing the
Gauss-Bonnet theorem (considering Gibbons and Werner’s technique). We present
the graphical analysis of the deflection angle with respect to the various
parameters that govern the black hole. Further, we calculate the Hawking
temperature for this black hole via a topological method and compare it with a
standard method of deriving the Hawking temperature. We also analyze the
Schr$\ddot{o}$dinger-like Regge-Wheeler equation and derive bound on the
greybody factor for a static black hole in the framework of $f(R)$ gravity and
graphically inquire that bound converges to 1. We also investigate the
silhouette or shadow generated by this static $f(R)$ black hole. Moreover, we
constrain the non-negative real constant (linear dark-matter-related parameter
for considered black hole) from the observed angular diameters of
M$87^{\star}$ and Sgr $A^{\star}$ released by the EHT. Finally, we probe how
cosmological constant, non-negative real constant and mass affected the radius
of shadow.

##  1 Introduction

In the theory of general relativity (henceforth GR), black holes are one of
the most attractive and engrossing objects in our universe. Black holes
provide a cabalistic tool to test and verify the essential laws of the
universe. Initially, the image of a black hole was only in simulations, but
later the Event Horizon Telescope (EHT) collaboration [1, 2, 3, 4] exposed the
first picture of the black hole giving confirmation of the existence of it. In
the universe, a black hole is a massive object having exorbitant gravity that
nothing (including light) can escape from its pull. The appearance of
gravitational lensing and gravitational waves as a consequence of GR
discovered by Einstein in 1916 [5]. In addition, the gravitational waves from
black holes and neutron stars merger have been determined via the Laser
Interferometer Gravitational-Wave Observatory (LIGO), which corroborated the
theoretical prospect that nicely impels with experimental studies [6, 7]. A
spacious variety of gravity theories encounter many flaws after the
identification of gravitational waves, meanwhile, the identification of
gravitational waves also seized the deliberation in the field of gravitational
lensing (abbreviated as GL) [8]. In 1801, Soldner first introduced the concept
of GL within the context of Newton’s theory of gravitation [9]. GL is an
enormously useful way of appreciating the knowledge of galaxies and the
universe. During the period of solar eclipse, the bending within the starlight
appears as a consequence of the Sun’s gravitational field. After this, they
noticed that they got findings similar to the results predicted by Einstein.
As a result, it played a documentary role in the experiential warrant of
Einstein’s prediction. Gravitational lensing denotes the bending of light
around extensive objects which is predicted by GR. Three types of GL can be
found in literature such as strong GL, weak GL, and micro GL which have been
documented in Refs. [10, 11, 12, 13, 14, 15].

Weak lensing is a feasible way for the identification of massive astronomical
object (including black hole) which has huge radii and mass without knowing
about their structures. Gravitation lensing also studies the expansion of the
universe. Nowadays, the study of GL has become an interesting area for
researchers. GL plays an important role in investigating many astrophysical
objects like black holes, naked singularities, and wormholes (see e.g. Refs.
[10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]). Many investigations, after
the confirmation of Gibbons and Werner’s about the feasible manner to derive
the deflection angle from black holes, have connected GL with the Gauss-Bonnet
theorem (henceforth GBT). The deflection angle is calculated by integrating
the Gaussian optical curvature of the black hole [22], which is articulated as
follows:

$\boldsymbol{\tilde{\delta}}=-\iint_{\mathcal{D}_{\infty}}\mathcal{K}dS$ (1.1)

here $\tilde{\delta}$, $\mathcal{K}$, and $dS$ are the deflection angle, the
Gaussian optical curvature, and the optical surface respectively. Also,
$\mathcal{D}_{\infty}$ denotes the infinite domain encompassed by the photon
ray. Thus, the Gibbons and Werner methodology has addressed in a unique
prospect for black holes and wormholes [23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39].

The shadow is essentially illustrated as the two-dimensional ($2D$) dark
region occurs in the celestial plane. The concept of the black-hole shadow
appears when there exists a geometrically thick and optically thin emission
region around the event horizon of a black hole [40]. The size of the shadow
radius depends on its contour and on the intrinsic parameters of the black
hole. Furthermore, the shadow size is estimated by the instability (orbital)
of the photon rays from the photon sphere, which merely seems to be a dark,
$2D$ region for a faraway observer exposed by its uniform, bright surrounding
[1, 41, 42, 43, 44, 45, 46, 47, 48].

In the framework of quantum field theory, Hawking’s discovery tells us that
particles can escape from black holes known as radiation [49]. This becomes
one of the exigent findings of Hawking and later this radiation is familiar as
Hawking radiation. Two distinct ideas, such as GR and quantum mechanics were
united by his work [49, 50]. In view of quantum field theory, there is a
possibility to create and annihilate the particles. When pair production
occurs in the vicinity of a black hole’s horizon, it is noteworthy that one of
the particles from that pair production escapes from the black hole resulting
in Hawking radiation, while the rest of the particles fall back into the black
hole. In GR, spacetime may bend around massive objects (like a black hole),
producing a gravitational potential inside which the particles can move. The
black hole can transmit some of the radiation and goes to infinity and the
remaining are reflected back into the black hole. As a result, the observed
Hawking radiation which passes through the gravitational potential and which
has not been crossed through the gravitational potential is different. The
difference has been investigated by the greybody factor.

Several methods have been established for calculating Hawking radiation [51,
52, 53, 54, 55]. Hawking and Gibbons [51] have derived Hawking temperature by
utilizing the Euclidean path integral for the gravitational field. Later,
Robson, Villari, and Biancalana (RVB) [56, 57] proposed that Hawking
temperature of a black hole can also be attained topologically. This
topological technique requires the invariants of the topology like Euler
characteristic222The Euler characteristic of Euclidean geometry is represented
by the parameter $\chi$ and it can give the number of the Killing horizons.
Moreover, it is related to the structure of the manifold and it is a
topological invariant [56, 57]. and GBT. One can interpret the Hawking
radiation in a crystallized way for the 2-dimensional spacetime of Euclidean
geometry without losing the information of 4-dimensional spacetime under the
consideration of topological technique. In the framework of the topological
method, Zhang et. al [58] studied the Hawking temperature of the BTZ black
hole. Moreover, few studies to derive the Hawking temperature for black holes
by employing the topological strategy can be found in Refs. [59, 60]. In the
context of non-linear electrodynamics, the exploration of the Hawking
temperature for a magnetically charged black hole through surface gravity and
horizon was performed in Ref. [61].

The greybody factor can be estimated by using several methods such as the
matching technique [62, 63, 64], and WKB approximation [65, 66]. People
developed a new technique to compute the greybody factor with accuracy. This
new method includes the computation of the rigorous bound on the greybody
factor. Many researchers, with the help of this new technique, have estimated
the bounds on the greybody factor for the scenario of different black holes
[67, 68, 69]. The greybody factor of a charged massive BTZ black hole is
discussed in Ref. [69]. Apart from these studies, there are several
significant studies dealing with rigorous bound on the greybody factor for
different black hole systems [70, 71, 72, 73, 74, 75, 76, 77, 78]. Motivated
by these prior researches, this work aims to study the greybody bound for the
static black hole in $f(R)$ gravity.

In GR, the limitation of the cosmological constant includes the discrepancies
of flat galactic rotation curves, accelerated expansion of the universe, anti-
lensing, the observed anisotropies on the cosmic microwave background
radiation (CMB), and the problem of coincidence [79, 80, 81, 82, 83, 84].
Since the last decades, people have been making an effort to solve these
problems with the support of modified theories of gravity. Now, we deal with
the $f(R)$ theory of gravity [85] which is modified gravity among the various
modified theories of gravity. This is due to the fact that GR heeds gravity
with high exactitude when the curvature is small [86], however, there does not
subsist any such evidence for very broad values of curvature. In this
connection, black holes become the norm places to look for modified GR [87]. A
proper $f(R)$ action model has been developed in Refs. [88, 89, 90] which is
consistent with cosmological as well as galactic scales.

In $f(R)$ modified gravity models, at large distances, the geometry of the
space-time should be different from that of standard GR. Hence, we need to
introduce a method that could observationally discriminate and test the
possible deviations from GR. The study of weak deflection angle, greybody
factor, and shadow cast by the static spherically symmetric $f(R)$ black hole
could be one of the possible methods. In 2016, S. Soroushfar et al. [91]
studied the thermodynamical behavior, stability conditions, and phase
transition of static, static-charged, and rotating-charged black holes in
$f(R)$ gravity. Also, they extend their study to various thermodynamic
geometry methods such as Ruppeiner, Weinhold, and GTD. Later (2019), the study
of thermal corrections and phase transition considering the static black hole
in $f(R)$ gravity has been carried out by M. Rostami et al. [92]. The
investigation of accretion disc for a static spherically symmetric black hole
in $f(R)$ gravity can be found in Ref. [93]. The Geodesic study [94] and study
of perturbed thermodynamics and thermodynamic geometry [95] of a static
charged black hole Spacetime in the framework of $f(R)$ Gravity has been
established. A brief study on the features of static and spherically symmetric
solutions with a horizon in $f(R)$ theories can be found in Ref. [96].
Recently (2022), the study of the trajectory of massive particles in the
vicinity of a static black hole in $f(R)$ gravity was accomplished by S.
Mandal et al. and found in Ref. [97]. In this continuity, here in the present
study, we intend to investigate the Weak deflection angle by utilizing the GBT
in a non-plasma medium, Hawking radiation, and the bound on greybody factor
for a static black hole in the environments of $f(R)$ gravity. We also study
the shadow cast by this black hole. Our study also focuses on how the
cosmological constant and non-zero real constant (introduced in static
spherically symmetric $f(R)$ black hole) affect the deflection angle, Hawking
temperature, greybody bound, and silhouette cast by static spherically
symmetric $f(R)$ black hole.

This paper is presented in nine parts. In section 2, we give a brief outline
of a static black hole solution in $f(R)$ gravity, and in the framework of
weak gravitational lensing, we derive corresponding optical metric and
Gaussian curvature. In section 3, we study GBT in detail and derive the
deflection angle of such a black hole in the presence of a non-plasma medium.
In section 4, by doing a graphical analysis, we focus mainly on how the
calculated deflection angle depends on various parameters that govern our
considered black hole. We explore the Hawking temperature based on GBT and a
graphical interpretation is also accomplished to see the dependency of Hawking
temperature on various parameters in section 5. In section 6, we estimated the
potential and rigorous bounds of the greybody factor (transmission
probability) of the static $f(R)$ black hole. The graphical seasoning of the
potential and the corresponding bound on the greybody factor are addressed in
section 7. We have a discussion on null geodesics for the static black hole in
the environments of $f(R)$ model in section 8. In section 9, we present
deliberation related to black hole shadow in the framework of a non-plasma
medium. The shape of the silhouette is executed with the employment of the
geodesic equations of a test particle near our considered black hole. In this
section, we also addressed a graphical analysis to observe the dependency of
shadow radius on various parameters. Also, we did a comparative plot for the
angular diameter of the static $f(R)$ black hole with observed angular
diameter of M$87^{\star}$ and Sgr $A^{\star}$ by the EHT and constrain the
parameter $\beta$ of the spacetime. Finally, Section 10 is devoted to
concluding the results obtained from our analysis and to making final remarks.
We consider the metric signature ($-,+,+,+$) throughout the paper.

##  2 Static black hole in $f(R)$ gravity

In this section, we introduce the metric structure of the static spherically
symmetric black hole in the context of $f(R)$ gravity. To embark on our
investigation in 4-dimensional spacetime recalling $f(R)$ gravity, the generic
action based on the Ricci scalar is articulated as [98]

$\mathcal{I}=\frac{1}{2}\int d^{4}x\sqrt{-g}f(R)+\mathcal{I}_{m}$ (2.1)

where $f(R)$ is a function of the Ricci scalar ($R$), and the matter part of
the above action is denoted by $\mathcal{I}_{m}$. Here, we set the natural
unit $G=c=\hbar=\kappa=1$. Recapitulating the variation principle of the
action $\mathcal{I}$ presented in (2.1) with respect to metric leading to the
following field equations :

$f^{\prime}(R)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f(R)-(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square)f^{\prime}(R)=T_{\mu\nu}$
(2.2)

with $\square=\nabla_{\alpha}\nabla^{\alpha}$, $f^{\prime}=\frac{df(R)}{dR}$,
and $T_{\mu\nu}$ is the energy -momentum tensor. The proposed vacuum solution
to the field equation 2.2 for a 4-dimensional static spherically symmetric
black hole in f(R) gravity is given by the line element [98]

$ds^{2}=-B(r)dt^{2}+\frac{dr^{2}}{B(r)}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (2.3)

in the conventional Schwarzschild coordinates $x^{\mu}=(t,r,\theta,\phi)$. The
model chosen for $f(R)$ gravity is given by

$f(R)=R+\Lambda+\frac{R+\Lambda}{\frac{R}{R_{0}}+\frac{2}{\alpha}}ln\frac{R+\Lambda}{R_{c}}$
(2.4)

where $R_{0}=\frac{6\alpha^{2}}{d^{2}}$ in which $\alpha$ and $d$, represent
free parameters of the action, $\Lambda$ denotes the cosmological constant,
and $R_{c}$ is a constant extracting from performing the integration of the
action 2.1. Here, $\alpha$ has no dimension whereas $[d]=m$. The cosmological
constant takes the value of $|\Lambda|\leq 10^{-52}m^{-2}$ as suggested in
Refs. [99, 97]. A reader can see the detailed derivation of the metric
solution [98, 100]. In the sequel, the metric function (or, lapse function)
designed up to the first order in the free parameters of the mentioned action
2.1 becomes

$B(r)=1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda r^{2}$ (2.5)

where $M$ is a constant pertained with the mass of the black hole,
$\beta=\frac{\alpha}{d}\geq 0$ denotes a real constant [98, 99, 97, 101] which
is related to the dark-matter. However, one should note that non-zero
cosmological constant has intricate consequences in various perspectives of
static and spherically symmetric black holes as documented in Refs. [102,
103]. The conception of the static radius is significant in an extending
universe subservient by a cosmological constant as it can personate a natural
boundary of gravitationally bound systems, as addressed in various situations
[104, 105, 106, 107, 108, 109]. However, in the scenario when $\beta=0$, one
can impeccably retrieve the space-time of the Schwarzschild AdS black hole. It
is noteworthy to mention that performing the limit $\beta=\Lambda=0$ in 2.5
leads to the reduction of the well-known Schwarzschild solution.

It is worth mentioning that the static spherically symmetric vacuum solution
to the first order in free parameters in $f(R)$ gravity describes an uncharged
black hole which corroborates with the Mannheim-Kazanas (MK) vacuum solution
to the fourth-order Weyl conformal gravity [110]. In the MK vacuum solution,
the linear term $\beta r$ plays the character of an extra potential recoup for
the galactic rotation curves (flat). In the same manner, for a typical galaxy,
the model written in Eq. 2.5 raises that low values of $\alpha$ can give the
flat galactic rotation curves. However, for strong curvature, the model
mentioned in 2.4 is reduced to

$f(R)=R+R_{0}\ln\frac{R}{R_{c}},$ (2.6)

in which $R>>\Lambda$ and $\frac{R}{R_{0}}>>\frac{2}{\alpha}$ which is
affiliated to the case of stellar black holes. Nevertheless, in the
cosmological scales (where $R\simeq R_{0}\simeq\Lambda$ and $\alpha<<1$), the
model shortened to $f(R)=R+\Lambda$ which can respond to the Einstein-Hilbert
action along with a cosmological constant that portrayed the accelerated
expansion of the universe. Thus, the low-valued non-negative free parameter
$\beta$ given in Eq. 2.5 can veil both the small and large-scale phenomena in
the universe.

The horizon structure of this black hole has been presented in Refs. [93, 97].
Here, it is worth mentioning that, due to small values of $\Lambda$ and
positive values of $\beta$ (wide range), for the sake of simplicity, all the
graphs are scaled with $\Lambda$ ranging $1\leq\Lambda<50$ and $\beta$ ranging
$0<\beta<60$ whereas $M$ taking values in the range $1\leq M<40$. This
consideration is to make the graphical analysis more simpler and it has no
other physical meaning.

###  2.1 Optical metric and Gaussian curvature in weak gravitational lensing

This section focuses on null geodesics deflected by the above-mentioned black
hole. It is well-established that light satisfies the null geodesic (i.e.
$ds^{2}=0$). This null geodesic is meticulously designed to define the optical
metric that delineates Riemannian geometry followed by light. Now, imposing
the null condition unveils the following optical metric (in new coordinate
system $\tilde{r}$):

$dt^{2}=\bar{g}_{ij}dx^{i}dx^{j}=d\tilde{r}^{2}+B(\tilde{r})^{2}d\phi^{2}$
(2.7)

where

$d\tilde{r}=\frac{dr}{(1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda r^{2})},$
(2.8) $B(\tilde{r})=\frac{r}{\sqrt{1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda
r^{2}}}.$ (2.9)

It is remarkable that the previous system $(i,j)$ is recast into a new system
$(r,\phi)$ giving the determinant value as
$det(\bar{g}_{ij})=\frac{r^{2}}{B(\tilde{r})^{3}}$. It is indisputable that
the equatorial plane in the optical metric offers a prospective surface of
revolution. The non-vanishing Christoffel symbols affiliated to metric (2.7)
are computed as

$\displaystyle\Gamma^{\tilde{r}}_{\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{r(rB^{\prime}(\tilde{r})-2B(\tilde{r}))}{2},$ (2.10)
$\displaystyle\Gamma^{\phi}_{\tilde{r}\phi}$ $\displaystyle=$
$\displaystyle\frac{2B(\tilde{r})-rB^{\prime}(\tilde{r})}{2},$ (2.11)
$\displaystyle\Gamma^{\tilde{r}}_{\tilde{r}\tilde{r}}$ $\displaystyle=$
$\displaystyle-\frac{B^{\prime}(\tilde{r})}{B(\tilde{r})}.$ (2.12)

Here, prime (dash) replicates the derivative with respect to $r$. The above
Christoffel symbols offer a unique Riemann tensor for optical curvature that
takes non-zero value: $R_{\tilde{r}\phi\tilde{r}\phi}$=-$kB^{2}(\tilde{r})$. A
particularly noteworthy connection route that manifests the naive interplay
between the Gaussian optical curvature $\mathcal{K}$ and the Ricci Scalar
$R_{icciScalar}$ as

$\mathcal{K}=\frac{R_{icciScalar}}{2}=-\frac{1}{B(\tilde{r})}\left[\frac{dr}{d\tilde{r}}\frac{d}{dr}\left(\frac{dr}{d\tilde{r}}\right)\frac{dB}{dr}+\frac{d^{2}B}{dr^{2}}\left(\frac{dr}{d\tilde{r}}\right)^{2}\right].$
(2.13)

By reminiscing Eqs. (2.8) and (2.9), in the sequel, Gaussian optical curvature
can be written in the following explicit form:

$\displaystyle\mathcal{K}$ $\displaystyle=$
$\displaystyle-\Big{(}\frac{1}{3}\Lambda+\frac{\beta^{2}}{4}\Big{)}+\frac{2M\Lambda}{r}-\frac{3M\beta}{r^{2}}-\frac{2M}{r^{3}}+\frac{3M^{2}}{r^{4}}$
(2.14) $\displaystyle+$ $\displaystyle O(M^{3},\Lambda^{2},\beta^{2}).$

Here, it is evident that the Gaussian optical curvature has a dependency on
various parameters such as mass $M$, cosmological constant $\Lambda$, and real
constant $\beta$.

##  3 Deflection angle of a static black hole in $f(R)$ gravity in non-plasma
medium

This section aims to elucidate the process of calculating the deflection angle
of black holes resembling the static black hole within the framework of $f(R)$
gravity and this study is in the non-plasma medium by employing the GBT. The
GBT, which plays a crucial role in connecting the (intrinsic) geometry of
metric with the underlying topology in the region $\mathcal{V}_{\cal{R}}$ with
boundary $\partial\mathcal{V}_{\cal{R}}$, as elegantly enunciated below [111,
69]

$\iint_{\mathcal{V}_{\cal R}}\mathcal{K}dS+\oint_{\partial\mathcal{V}_{\cal
R}}kdt+\sum_{z}\alpha_{z}=2\pi{\Xi}\left(\mathcal{V}_{\cal{R}}\right),$ (3.1)

in which $\mathcal{V}_{\cal R}\subset S$ is a regular domain of a simple two-
dimensional surface $S$ having closed, regular, and positive oriented boundary
$\partial\mathcal{V}_{\cal R}$. Here, $k$ denotes the geodesic curvature of
$\partial\mathcal{V}_{\cal R}$ define as
$k=\bar{g}\left(\nabla_{\dot{\gamma}}\dot{\gamma},\ddot{\gamma}\right)$, in
which $\bar{g}(\dot{\gamma},\dot{\gamma})=1$ and $\ddot{\gamma}$ indicating
unit acceleration vector. At the $z^{\mbox{th}}$ vertex$,\alpha_{z}$
represents the exterior angle. Also, here in Eq. 3.1,
$\Xi\left(\mathcal{V}_{\cal{R}}\right)$ refers to the Euler characteristic
number. If we consider the limit $R\rightarrow\infty$, we can acquire the jump
angle with value $\pi/2$ so that $\theta_{0}+\theta_{S}=\pi$ [112] and Euler
the characteristic number takes a unit value. Consequently, one can expect the
following result:

$\iint_{\mathcal{V}_{\cal R}}\mathcal{K}dS+\oint_{\partial\mathcal{V}_{\cal
R}}kdt+\alpha_{z}=2\pi{\Xi}\left(\mathcal{V}_{\cal{R}}\right),$ (3.2)

where the total angle of jumps is denoted by $\alpha_{z}=\pi$, and since
$S\rightarrow\infty$, the geodesic curvature can be expressed as
$k\left(E_{\cal R}\right)=|\nabla_{\dot{E}_{\cal R}}\dot{E}_{\cal R}|$. The
radial component of geodesic curvature can be acquired as [111, 69]

$\left(\nabla_{\dot{E}_{\cal R}}\dot{E}_{\cal R}\right)^{r}=\dot{E}_{\cal
R}^{\phi}\partial_{\varphi}\dot{E}_{\cal
R}^{r}+\Gamma_{\varphi\varphi}^{\tilde{r}}\left(\dot{E}_{\cal
R}^{\phi}\right)^{2}.$ (3.3)

At large ${\cal R}$, we have $E_{\cal R}:=r(\phi)={\cal R}=$ constant
resulting in $\left(\dot{E}_{R}^{\phi}\right)^{2}=\frac{1}{f^{2}(\tilde{r})}$.
Recalling
$\Gamma_{\varphi\phi}^{\tilde{r}}=\frac{r\left(rf^{\prime}(\tilde{r})-2f(\tilde{r})\right.}{2}$,
geodesic curvature obtained as [111]

$\left(\nabla_{\dot{E}_{\cal R}^{r}}\dot{E}_{{\cal
R}}^{r}\right)^{r}\rightarrow\frac{1}{\cal R}.$ (3.4)

This indicates that $k(E_{\cal R})\rightarrow\frac{1}{\cal R}$. We can have
$dt={\cal R}d\phi$ by employing optical metric (2.7). In the sequel, one might
get

$k(E_{\cal R})dt=\lim_{{\cal R}\to\infty}[k(E_{\cal R})dt]=\lim_{{\cal
R}\to\infty}\left[\frac{1}{2\sqrt{\bar{g}_{rr}\bar{g}_{\phi\phi}}}(\frac{\partial\bar{g}_{\phi\phi}}{\partial
r})\right]d\phi=d\phi.$ (3.5)

Considering all the above results into account, the GBT becomes

$\iint_{\mathcal{V}_{{\cal R}}}\mathcal{K}dS+\oint_{\partial\mathcal{V}_{{\cal
R}}}kdt\stackrel{{\scriptstyle{\cal
R}\rightarrow\infty}}{{=}}\iint_{S_{\infty}}\mathcal{K}dS+\int_{0}^{\pi+\tilde{\delta}}d\phi.$
(3.6)

In the framework of weak deflection limit, the light ray for the zeroth order
apprehends by a straight line approximation as $r(t)=\frac{b}{sin\varphi}$,
where $b$ is related to the impact parameter. By utilizing this formula, the
deflection angle can be calculated as

$\boldsymbol{\tilde{\delta}}=-\int_{0}^{\pi}\int_{b/\sin\varphi}^{\infty}\mathcal{K}dS=-\int_{0}^{\pi}\int_{b/\sin\varphi}^{\infty}\mathcal{K}\sqrt{\operatorname{det}\bar{g}}drd\phi=-\int_{0}^{\pi}\int_{b/\sin\varphi}^{\infty}\mathcal{K}\frac{r}{f(r)^{\frac{3}{2}}}drd\phi$
(3.7)

In the environments of the non-plasma medium, for the mentioned metric
function of a static black hole in $f(R)$ gravity, the deflection angle is
simplified to

$\displaystyle\boldsymbol{\tilde{\delta}}$ $\displaystyle=$
$\displaystyle-\int_{0}^{\pi}\int_{b/\sin\varphi}^{\infty}\left[-\Big{(}\frac{1}{3}\Lambda+\frac{9}{2}M\Lambda\beta-\frac{\beta}{4}\Big{)}r+\frac{1}{2}\Lambda\beta
r^{2}+\frac{1}{r}\Big{(}6\Lambda
M^{2}+\frac{3}{2}M^{2}\Lambda\Big{)}+\frac{1}{r^{2}}\Big{(}-\frac{27}{2}M^{2}\beta-2M\Big{)}\right.$
(3.8) $\displaystyle-$
$\displaystyle\left.\frac{6M^{2}}{r^{3}}+O(M^{3},\Lambda^{2},\beta^{2})\right]d\tilde{r}\
d\phi.$ $\displaystyle\boldsymbol{\tilde{\delta}}$ $\displaystyle=$
$\displaystyle-\Big{(}\frac{1}{6}\Lambda+\frac{9}{4}M\Lambda\beta-\frac{\beta}{8}\Big{)}b^{2}\int_{0}^{\pi}\left.\csc^{2}\phi\
d\phi+\frac{\Lambda\beta b^{3}}{6}\int_{0}^{\pi}\csc^{3}\phi\
d\phi+\Big{(}6\Lambda
M^{2}+\frac{3}{2}M^{2}\Lambda\Big{)}\\{\pi\log(b)\right.$ (3.9)
$\displaystyle-$ $\displaystyle\left.\int_{0}^{\pi}\log(\sin\phi)\
d\phi\\}+\frac{(\frac{27}{2}M^{2}\beta+2M)}{b}\int_{0}^{\pi}\sin\phi\
d\phi+\frac{3M^{2}}{b^{2}}\int_{0}^{\pi}\sin^{2}\phi\ d\phi\right.$

Since the integral of $csc\phi$ does not converge on $\\{0,\pi\\}$ and it is
notified from the above Eq. (3.9) that the first two-term do not contribute to
the deflection angle. Therefore, the explicit expression for the deflection
angle for a static black hole in $f(R)$ gravity in the weak limit becomes

$\boldsymbol{\tilde{\delta}}=\Big{(}6\Lambda
M^{2}+\frac{3}{2}M^{2}\Lambda\Big{)}\Big{\\{}\pi\log(b)-\pi\log(2)\Big{\\}}+\frac{1}{b}\Big{(}27M^{2}\beta+4M\Big{)}+\frac{3M^{2}\pi}{2b^{2}}+O(M^{3},\Lambda^{2},\beta^{2})$
(3.10)

Here, it is evident that the deflection angle of the static black hole in
$f(R)$ gravity depends on the various parameters such as impact parameter $b$,
the mass of black hole $M$, real constant $\beta$ and cosmological constant
$\Lambda$. So, it is expected that there may be a significant change in
$\boldsymbol{\tilde{\delta}}$ because of the different parameters that govern
the black hole. It is remarkable that by overriding the term that contains
$\beta$ in the above calculated deflection angle, one can attain the
deflection angle of Schwarzschild Ads black hole. Nevertheless, one can
achieve the deflection angle for Schwarzschild black hole up to the second
order of mass when both $\Lambda=\beta=0$.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr1.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr2.eps}\end{@subfigure}\\\
\begin{@subfigure}\includegraphics[width=216.81pt]{fr3.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr4.eps}\end{@subfigure}\\\
\begin{@subfigure}\includegraphics[width=216.81pt]{fr5.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr6.eps}\end{@subfigure}\end{array}$

Figure 1: The graph of deflection angle ($\boldsymbol{\tilde{\delta}}$) with
respect to impact parameter ($b$). 1 and 1 : for varying $\Lambda$ but fixed
$\beta=1$. 1 and 1 : for varying $\beta$ but fixed $\Lambda=1$. 1 and 1 : for
varying $M$ and for $\Lambda=\beta=1$. We set $M=1$ in the first two panels.
Inset shows the behavior of $\boldsymbol{\tilde{\delta}}$ for lower $b$.

##  4 Graphical analysis of deflection angle for non-plasma medium

In this section, we study the behavior of deflection angle and their
dependence on various parameters.

###  4.1 Effect of impact parameter ($b$) on deflection angle
($\boldsymbol{\tilde{\delta}}$)

We plot the behavior of the deflection angle and its dependence on the impact
parameter ($b$) for changing $\Lambda$, $\beta$, and $M$ in Fig. 1. Here, from
Figs. 1 and 1, it is clear that the deflection angle decreases with the impact
parameter for a lower (positive) value of the cosmological constant and then
shows the increasing behavior softly. Here, $\boldsymbol{\tilde{\delta}}$
takes positive value only. In contrast, for a higher (positive) valued
cosmological constant, $\tilde{\delta}$ goes to the negative region by
illustrating the decreasing nature for smaller $b$ but then increases
gradually and takes positive values only. Moreover, the deflection angle
increases with increasing cosmological constant. In Fig. 1, the deflection
angle shows an increasing tendency with impact parameter $b$ after performing
a certain decline nature in the lower region of $b$ for small values of
$\beta$ but remains always positively valued. However, in Fig. 1,
$\boldsymbol{\tilde{\delta}}$ shows asymptotically decreasing behavior for
large $\beta$ but remains positively valued. Here also
$\boldsymbol{\tilde{\delta}}$ increases when $\beta$ takes a higher value.
However, in Figs. 1 and 1, for both small and large $M$,
$\boldsymbol{\tilde{\delta}}$ decreases initially and finally deliberates
increasing demeanor. Here, $\boldsymbol{\tilde{\delta}}$ takes only positive
values.

###  4.2 Effect of mass ($M$) on deflection angle
($\boldsymbol{\tilde{\delta}}$)

The variation of defection angle ($\boldsymbol{\tilde{\delta}}$) with mass
parameter ($M$) for different $\Lambda$ and $\beta$ is depicted in Fig. 2. The
deflection angle is an increasing function of mass parameter $M$. Moreover,
for the larger value of $\Lambda$ and $\beta$, the deflection angle for
massive black holes increases.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr7.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr8.eps}\end{@subfigure}\end{array}$

Figure 2: In 2, the behavior of deflection angle
($\boldsymbol{\tilde{\delta}}$) with respect to mass $M$ by varying $\Lambda$
with fixed $\beta=1$. In 2, the behavior of deflection angle
($\boldsymbol{\tilde{\delta}}$) with respect to mass $M$ by varying $\beta$
with fixed $\Lambda=1$. Here, $b=1$.

###  4.3 Effect of cosmological constant ($\Lambda$) on deflection angle
($\tilde{\delta}$)

We present the effect of cosmological constant ($\Lambda$) on deflection angle
($\boldsymbol{\tilde{\delta}}$) in Fig. 3. In this plot, the deflection angle
increases sharply. In Fig.3, the value of $\boldsymbol{\tilde{\delta}}$
decreases when $b$ increases. In figure 3, for increasing $\beta$,
$\boldsymbol{\tilde{\delta}}$ increases and remains positively valued.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr9.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr10.eps}\end{@subfigure}\end{array}$

Figure 3: In 3, the behavior of $\boldsymbol{\tilde{\delta}}$ with respect to
$\Lambda$ by changing impact parameter $b$ with fixed $\beta=1$. In 3, the
behavior of $\boldsymbol{\tilde{\delta}}$ with respect to $\Lambda$ by
changing $\beta$ with fixed $b=1$.

###  4.4 Effect of real constant ($\beta$) on deflection angle
($\boldsymbol{\tilde{\delta}}$)

The impact of real constant ($\beta$) on deflection angle
($\boldsymbol{\tilde{\delta}}$) is depicted in Fig. 4. The plot tells us that
$\boldsymbol{\tilde{\delta}}$ is an increasing function of real constant
$\beta$. Here, $\boldsymbol{\tilde{\delta}}$ decreases and keep increasing for
increasing $b$ and increasing $\Lambda$ respectively.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr11.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr12.eps}\end{@subfigure}\end{array}$

Figure 4:  In 4, plot of $\boldsymbol{\tilde{\delta}}$ with respect to $\beta$
by changing impact parameter $b$ but fixed $\Lambda=1$. In 4, plot of
$\boldsymbol{\tilde{\delta}}$ with respect to $\beta$ by changing $\Lambda$
but $b=1$.

The blue dotted curve in all the figures shows the behavior of the deflection
angle for the Schwarzschild black hole case when we take the condition as
$\Lambda=\beta=0$. The above graphical analysis resembles that the deflection
angle is seen to be vividly influenced by parameters like $\Lambda$, and
$\beta$. Due to the presence of these parameters, the deviating curve
illustrates that the deflection angle for a static black hole in $f(R)$
gravity will be more than the Schwarzschild black hole case. Therefore, the
presence of these extra parameters appears to be augmenting the deflection
angle.

##  5 Hawking radiation

This section aims to explore the calculation of the Hawking temperature of the
static black hole in $f(R)$ gravity by employing a topological technique based
on GBT. This topological technique helps us to calculate the black hole
temperature by using 2-dimensional Euler characteristic $\boldsymbol{\chi}$
and GBT [56, 57]. For this purpose, the line element for a 4-dimensional
static spherically symmetric black hole in $f(R)$ gravity is given by [98,
113]

$ds^{2}=-B(r)dt^{2}+\frac{dr^{2}}{B(r)}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$ (5.1)

By using the Wick rotation($\tau=it$, $\theta=\pi/2$), we can get the
2-dimensional Euclidean metric from the 4-dimensional spherically symmetric
metric as [114]

$ds^{2}=-B(r)dt^{2}+\frac{dr^{2}}{B(r)}$ (5.2)

The event horizon radius of a static black hole in $f(R)$ gravity is estimated
as

$r_{h}=\frac{\beta}{\Lambda}+\frac{2^{\frac{1}{3}}(-9\beta^{2}-9\Lambda)}{3\Lambda\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}+\frac{\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}{3\times
2^{\frac{1}{3}}\Lambda}$ (5.3)

where $\psi=54\beta^{3}+81\beta\Lambda-162M\Lambda^{2}$. The formula to
compute the Hawking temperature of this black hole is written as follows [56,
60]

$T_{H}=\frac{1}{4\pi\chi}\int_{r_{h}}\sqrt{g}\mathcal{R}dr$ (5.4)

where, $g$ denotes the determinant of Eq. 5.3 and its value is 1, $\chi=1$ is
the Euler characteristic ans $r_{h}$ is the radius of the event horizon. The
Ricci scalar $\mathcal{R}$ is obtained as [58],

$\mathcal{R}=\frac{4M}{r^{3}}+\frac{2}{3}\Lambda$ (5.5)

After plugging the values of $g$, $R$ and then integrating Eq. 5.3 along the
event horizon, we get the Hawking temperature $T_{H}$ of the static black hole
in $f(R)$ gravity as

$\displaystyle T_{H}=\frac{M}{2\pi r_{h}^{2}}-\frac{\Lambda}{6\pi}r_{h}$ (5.6)

However, the standard form of the Hawking temperature of a static black hole
in $f(R)$ gravity using horizon function
($T_{H}=\frac{f^{\prime}(r_{h})}{4\pi}$) is

$\displaystyle T_{H}=\frac{M}{2\pi
r_{h}^{2}}-\frac{\Lambda}{6\pi}r_{h}+\frac{\beta}{4\pi}$ (5.7)

which is consistent with the result documented in Ref. [92]. Comparing 5.6 and
5.7, one will find that the temperature calculated from GBT is quite different
from the standard form of Hawking temperature for a static black hole in
$f(R)$ gravity. Therefore, it seems that the GBT method can not determine the
correct Hawking temperature for the static black hole in $f(R)$ gravity.
However, it is worth mentioning that there exists an integral constant when
one performs the integral 5.4. We can add, by keeping this in mind, an
integral constant $\frac{\beta}{4\pi}$ into Eq. 5.6. Consequently, we will get
the correct Hawking temperature 5.7 for the static black hole in $f(R)$
gravity.

Now, plugging the value of 5.3 in 5.7, we get

$\displaystyle T_{H}$ $\displaystyle=$
$\displaystyle\frac{M}{2\pi}\Bigg{(}\frac{\beta}{\Lambda}+\frac{2^{\frac{1}{3}}(-9\beta^{2}-9\Lambda)}{3\Lambda\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}+\frac{\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}{3\times
2^{\frac{1}{3}}\Lambda}\Bigg{)}^{-2}$ (5.8) $\displaystyle-$
$\displaystyle\frac{\Lambda}{6\pi}\Bigg{(}\frac{\beta}{\Lambda}+\frac{2^{\frac{1}{3}}(-9\beta^{2}-9\Lambda)}{3\Lambda\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}+\frac{\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}{3\times
2^{\frac{1}{3}}\Lambda}\Bigg{)}$ $\displaystyle+$
$\displaystyle\frac{\beta}{4\pi}$

where $\psi=54\beta^{3}+81\beta\Lambda-162M\Lambda^{2}$. It is evident that
the Hawking temperature calculated in Eq. 5.8 depends on parameters like the
mass of the black hole $M$, cosmological constant $\Lambda$, and real constant
$\beta$. If we take the limit $\Lambda\rightarrow 0$, $\beta\rightarrow 0$ in
Eq. 5.8, the calculated Hawking temperature of this black hole reduces to the
Hawking temperature of the Schwarzschild black hole ($T_{H}=\frac{1}{8M\pi}$)
[115].

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{th.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{th1.eps}\end{@subfigure}\end{array}$

Figure 5: In 5, plot of $T_{H}$ with respect to $r_{h}$ for varying $\Lambda$.
In 5, plot of $T_{H}$ with respect to $r_{h}$ for varying $\beta$. Here,
$M=1$.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr14.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr15.eps}\end{@subfigure}\end{array}$

Figure 6: Plot of $T_{H}$ with respect to $\Lambda$ for fixed
$\beta=\Lambda=1$. Here, we consider $M=1$.

To see the nature of Hawking temperature graphically, we present the graph of
Hawking temperature $T_{H}$ with respect to $r_{h}$ (Eq. 5.7) in Figure 5 and
the graph of $T_{H}$ with respect to $\Lambda$ and $\beta$ (Eq. 5.8) is
presented in Figure 6. Fig. 5 illustrates that $T_{H}$ decreases for an
increasing $r_{h}$. In addition to that, $T_{H}$ increases for decreasing
$\Lambda$ and increasing $\beta$. However, we noticed from Fig. 6 that $T_{H}$
is an increasing function of $\Lambda$ for $\beta=1$ and is a decreasing
function of $\beta$ for $\Lambda=1$. In both plots, $T_{H}$ takes negative
values only.

The first-order corrected temperature of this black hole due to small
statistical fluctuations around equilibrium and also the effect of correction
parameter on Hawking temperature was studied in Ref. [92].

##  6 Bound on greybody factor of static black hole in $f(R)$ gravity

This section is obedient to inspecting the greybody bound for static
spherically symmetric $f(R)$ black hole. According to the GR, black holes can
absorb all radiation and matter in the periphery of the event horizon and
ejaculate Hawking radiation that depends on the mass of the black hole
sustaining the quantum information of the devouring particles [49, 116]. The
descending aspects of a radiating black hole are measured by the “greybody
factor”: the terminology emphasizes the transit from the black body behavior.
Moreover, it is a function of quasinormal mode frequency ($\omega$) and
angular momentum ($\l$). However, the amount of deviation in the emission
spectrum of a black hole from the spectrum of a perfect black body can be
determined by the greybody factor. The main libellee for this inconsequence is
the scattering of the radiation due to the geometry of the black hole itself
connecting to the radiated (captured) particles that decline into the extreme
force. As a consequence, the spectrum emitted by the black hole will deviate
and can be rendered as the “greybody factor” [117]. There are many strategies
to find the “greybody factor” like the matching technique and WKB
approximation [118]. In this present work, we will consider the technique that
doesn’t put such approximations, for example, a rigorous lower bound on the
“greybody factor” [119, 120, 121, 122, 123, 124, 125, 126, 127].

In other words, the “greybody factor” or the transmission probability in black
hole physics is a quantity related to the quantum nature of a black hole which
corrects the Planckian spectrum. This “greybody factor” can determine the
emissivity of the given black hole solution (non-perfect blackbody). Here, we
calculate the rigorous bound of the transmission probability of the static
black hole in $f(R)$ gravity. Now, the general bounds of the “greybody factor”
can be stated as [128, 129]

$T\geq\operatorname{sech}^{2}\left(\frac{l}{2\omega}\int_{-\infty}^{\infty}\mathcal{V}(r)dr_{*}\right),$
(6.1)

where,
$\mathcal{V}(r)$=$\frac{\sqrt{(h^{\prime})^{2}+(\omega^{2}-V-h^{2})^{2}}}{2h}$,
$r_{*}$ is the tortoise coordinate and $\omega$ denotes the quasinormal mode
frequency. Now, for $h=\omega$, Eq. 6.1 takes the form

$T\geq\operatorname{sech}^{2}\left(\frac{l}{2\omega}\int_{-\infty}^{\infty}V(r)dr_{*}\right),$
(6.2)

Now, we study the Schr$\ddot{o}$dinger-like Regge-Wheeler equation for angular
momentum $\l$ and calculate rigorous bounds on the greybody factors. Let us
introduce the Regge-Wheeler equation, which dominates the modes for a wave
function $\psi(r)$ anticipating the position of an electron to its wave
amplitude, as

$\left(\frac{d^{2}}{dr_{*}^{2}}+\omega^{2}-V(r)\right)\psi(r)=0,$ (6.3)

where,

$dr_{*}=\frac{1}{B(r)}dr,$ (6.4)

in which $r_{*}$ is known as tortoise coordinate333The purpose of tortoise
coordinate is to evolve to infinity in a way that palmist the singularity of
the metric under perscrutation. and potential $V(r)$ in $4D$ is given as

$V(r)=B(r)\Big{[}\frac{B^{\prime}(r)}{r}+\frac{l(l+1}{r^{2}}\Big{]}.$ (6.5)

The potential for a static black hole in $f(R)$ gravity is derived as

$V(r)=\Big{(}1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda
r^{2}\Big{)}\Bigg{[}\frac{\Big{(}\frac{2M}{r^{2}}+\beta-\frac{2}{3}\Lambda
r\Big{)}}{r}+\frac{l(l+1}{r^{2}}\Bigg{]}.$ (6.6)

This reincarnates the potential energy of the particle as long as there
present an external force (or, field). The information about the scattering
states, the wavefunction, the rigorous bound, etc. can be evolved from this
quantity.

In order to discuss the bound value of the greybody factor, we first write the
expression of 6.2 with the help of 6.4 as

$T\geq\operatorname{sech}^{2}\left(\frac{l}{2\omega}\int_{-\infty}^{\infty}\frac{V(r)dr}{B(r)}\right).$
(6.7)

The lower bound value of the greybody factor (transmission probability) $T$
can be written as

$T\geq\operatorname{sech}^{2}\left[\frac{1}{2\omega}\int_{r_{h}}^{\infty}\left(\frac{B^{\prime}(r)}{r}+\frac{l(l+1)}{r^{2}}\right)dr\right].$
(6.8)

This expression for $T$ is the measurement of the probability of a particle
having specific angular momentum ($\l$) and energy runaway the black hole and
being identified at infinity. Here, $r_{h}$ is the event horizon radius of the
black hole which affects the radiation spectrum emitted by the black hole.
Because of the angularity of solving for the polynomial of order three, we
derive $r_{h}$ numerically in Eq. 5.3. Moreover, the ample shape of the
spectrum of the radiation emitted by a black hole is particularized by $\l$.

For the metric function of our considered black hole, mentioned in Eq. 2.3,
the above lower bound simplifies to

$\displaystyle T$ $\displaystyle\geq$
$\displaystyle\operatorname{sech}^{2}\left[\frac{1}{2\omega}\left\\{\frac{M}{r_{h}^{2}}-\beta\log(r_{h})+\frac{2}{3}\Lambda
r_{h}+\frac{l(l+1)}{r_{h}}\right\\}\right].$ (6.9)

Substituting the value of event horizon $r_{h}$ from equation (5.3), the bound
on the greybody factor can be evaluated as

$\displaystyle T$ $\displaystyle\geq$
$\displaystyle\operatorname{sech}^{2}\Bigg{[}\frac{M}{2\omega}\Bigg{(}\frac{\beta}{\Lambda}+\frac{2^{\frac{1}{3}}(-9\beta^{2}-9\Lambda)}{3\Lambda\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}+\frac{\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}{3\times
2^{\frac{1}{3}}\Lambda}\Bigg{)}^{-2}$ (6.10) $\displaystyle-$
$\displaystyle\frac{\beta}{2\omega}\log\Bigg{(}\frac{\beta}{\Lambda}+\frac{2^{\frac{1}{3}}(-9\beta^{2}-9\Lambda)}{3\Lambda\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}+\frac{\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}{3\times
2^{\frac{1}{3}}\Lambda}\Bigg{)}$ $\displaystyle+$
$\displaystyle\frac{\Lambda}{3\omega}\Bigg{(}\frac{\beta}{\Lambda}+\frac{2^{\frac{1}{3}}(-9\beta^{2}-9\Lambda)}{3\Lambda\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}+\frac{\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}{3\times
2^{\frac{1}{3}}\Lambda}\Bigg{)}$ $\displaystyle+$
$\displaystyle\frac{l(l+1)}{2\omega}\Bigg{(}\frac{\beta}{\Lambda}+\frac{2^{\frac{1}{3}}(-9\beta^{2}-9\Lambda)}{3\Lambda\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}+\frac{\Big{(}\psi+\sqrt{4(-9\beta^{2}-9\Lambda)^{3}+\psi^{2}}\Big{)}^{\frac{1}{3}}}{3\times
2^{\frac{1}{3}}\Lambda}\Bigg{)}^{-1}\Bigg{]}.$

where $\psi=54\beta^{3}+81\beta\Lambda-162M\Lambda^{2}$. The lower bound
derived in Eq. 6.10 depends upon various parameter such as $M$, $\Lambda$, and
$\beta$. In the limit $\Lambda=\beta=0$, one can also attain the expression of
the greybody bound of the Schwarzschild black hole [125] as expected.

##  7 Graphical analysis of greybody factor with $\omega$

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr16.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr18.eps}\end{@subfigure}\\\
\begin{@subfigure}\includegraphics[width=216.81pt]{fr17.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr19.eps}\end{@subfigure}\end{array}$

Figure 7: The Left figure represents the behavior of the potential whereas the
right figure corresponds to the relevant bound for varying $\Lambda$. The
first panel, second panel, and third panel are for $l=0$, and $l=1$
respectively. Here, we set $M=1$ for all plots.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr20.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr20a.eps}\end{@subfigure}\\\
\begin{@subfigure}\includegraphics[width=216.81pt]{fr21.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr21a.eps}\end{@subfigure}\end{array}$

Figure 8: The Left figure represents the behavior of the potential whereas the
right figure corresponds to the relevant bound for varying $\beta$. The first
panel, second panel, and third panel are for $l=0$, and $l=1$ respectively.
Here, we set $M=1$ for all plots.

In this section, we present the graphical behavior of the potential and
greybody bound of a static black hole in $f(R)$ gravity. In order to do this,
we consider the fixed values of mass $M$, and angular momentum $l=0,1$ for
varying cosmological constant $\Lambda$ and $\beta$.

Figure 7 demonstrates the graphical behavior of the greybody bound with
$\omega$, and its potential with respect to $r$ with $\Lambda$ taking values
in the range $1\leq\Lambda\leq 5$ for fixed $\beta=3$. The domain of
quasinormal mode frequency $\omega$ is taken to be $0\leq\omega\leq 500$.
However, the inset of this figure resembles the variation of rigorous bound
for lower $\omega$.

Now, in Fig. 7, we can observe the dependency of the transmission probability
on the shape of potential $V(r)$ for $\l=0$. The plot shows that the potential
is an increasing function of $r$ when $\Lambda$ changes. The potential
increases sharply for larger $\Lambda$. We notice that the bound corresponding
to $V(r)$ increases sharply and finally saturates after a certain value of
$\boldsymbol{\omega}$ and it becomes 1 as long as $\boldsymbol{\omega}$
approaches infinity as depicted in the Fig.7. Nevertheless, if the value of
$\Lambda$ increases then the corresponding greybody bound becomes lower which
in turn makes it more difficult for the waves to be transmitted through the
higher potential value as illustrated in Fig.7. It is also noted that the
potential and the greybody bound for $\l=1$ show similar kinds of behavior as
for $\l=0$ as shown in Figs.7 and 7.

The graphical behavior of potential with $r$ and the corresponding greybody
bound with $\omega$ for $\beta$ taking values in the range $1\leq\beta\leq 5$
for fixed $\Lambda=1$ is demonstrated in Fig. 8. Here also the domain of
$\omega$ is taken $0\leq\omega\leq 500$. Fig.8 indicates that, for
$r\rightarrow 0$, the value of potential becomes high, while the potential
starts decreasing from the maximum value for the large values of $r$ and
further shows increasing behavior. As in Fig.7, here the plot for $\l=0$
delineates that the rigorous bound corresponding to potential increases
sharply and display the convergent nature by converging to 1 after a certain
value of $\boldsymbol{\omega}$ as exhibited in the Fig. 8. Furthermore, the
greybody bound decreases for increasing values of $\beta$. The plot for $\l=1$
parades a similar kind of demeanor as observed in Figs.8 and 8.

##  8 Null geodesics of a static black hole in $f(R)$ gravity

This section focuses on studying the null geodesics for a static black hole in
the framework of $f(R)$ gravity with due help of the Hamilton-Jacobi method.
In order to achieve the shadow of this black hole, we will derive the
celestial coordinates of the unstable null orbits. Now, we recall the
following Lagrangian to explore the motion of the particle on the static black
hole in $f(R)$ gravity:

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}u^{\mu}u^{\nu},$ (8.1)

in which $u^{\mu}(=\frac{dx^{\mu}}{d\lambda})$ refers to the four velocity of
particle with affine parameter $\lambda$ along the geodesics.

Hence, the Lagrangian for the motion of photon of the above mentioned black
hole can be written, using 2.3, as follows :

$2\mathcal{L}=-\Big{(}1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda
r^{2}\Big{)}\dot{t}^{2}+\Big{(}1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda
r^{2}\Big{)}^{-1}\dot{r}^{2}+r^{2}\dot{\theta}^{2}+r^{2}Sin^{2}\theta\dot{\phi}^{2}$
(8.2)

Here the derivative with respect to the affine parameter $\lambda$ is denoted
by an over dot. However, the Lagrangian has no dependency on $t$ and $\phi$,
as a result, corresponding to these one can get two new constants of motion,
namely the energy $E$ and the angular momentum $L$, given by:

$E=-p_{t}=\frac{\partial{\mathcal{L}}}{\partial{\dot{t}}}=-\Big{(}1-\frac{2M}{r}+\beta
r-\frac{1}{3}\Lambda r^{2}\Big{)}\dot{t}$ (8.3)

and

$L=p_{\phi}=\frac{\partial{\mathcal{L}}}{\partial{\dot{\phi}}}=r^{2}Sin^{2}\theta\dot{\phi}$
(8.4)

To find the restrictions of the geodesic, we can write the above constants as
follows :

$\frac{dt}{d\lambda}=\dot{t}=\frac{E}{1-\frac{2M}{r}+\beta
r-\frac{1}{3}\Lambda
r^{2}}\;,\;\frac{d\phi}{d\lambda}=\dot{\phi}=\frac{L}{r^{2}Sin^{2}\theta}$
(8.5)

Now, the r-part and $\theta$-part of the momentum can be defined as :

$p_{r}=\frac{\partial{\mathcal{L}}}{d\dot{r}}=\frac{\dot{r}}{1-\frac{2M}{r}+\beta
r-\frac{1}{3}\Lambda
r^{2}}\;,\;p_{\theta}=\frac{\partial{\mathcal{L}}}{\partial{\theta}}=r^{2}\dot{\theta}$
(8.6)

In order to calculate the r-part and $\theta$-part of the geodesic equations,
we utilize the following relativistic Hamilton-Jacobi equation:

$\frac{dA}{d\lambda}=-\frac{1}{2}g_{\mu\nu}\frac{dA}{dx^{\mu}}\frac{dA}{dx^{\nu}}$
(8.7)

where $A$ is the Jacobi action. We can attain the Hamilton-Jacobi equation by
taking into account the following ansatz [130]:

$A=\frac{1}{2}m_{\star}^{2}\lambda-Et+L\phi+A_{r}(r)+A_{\theta}(\theta)$ (8.8)

where $m_{\star}$ denotes the mass of the test particle, $A_{r}(r)$ and
$A_{\theta}(\theta)$ are the functions of $r$ and $\theta$ respectively. For
photons ($m_{\star}=0$), Eq. 8.8 give us the following result:

$A=-Et+L\phi+A_{r}(r)+A_{\theta}(\theta)$ (8.9)

Now, by separating the values of $r$ and $\theta$, obtain by replacing Eq. 8.9
into Eq. 8.8, we get the Carter constant ($\pm\mathcal{C}$) [131]. By plugging
the values of the contravariant metric (i.e., $g^{\mu\nu}$) we obtain the
following equations [132]:

$\frac{1}{\sqrt{1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda
r^{2}}}\frac{dr}{d\lambda}=\pm\sqrt{\mathcal{P}(r)}$ (8.10)
$r^{2}\frac{d\theta}{d\lambda}=\pm\sqrt{\Theta(\theta)}$ (8.11)

where

$\mathcal{P}(r)=\frac{E^{2}}{1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda
r^{2}}-\frac{\mathcal{C}+L^{2}}{r^{2}}$ (8.12)
$\Theta(\theta)=\mathcal{C}-\Big{(}\frac{L^{2}}{sin^{2}\theta}\Big{)}\
cos^{2}\theta$ (8.13)

Now, equation for $A_{r}$ can be written as follows

$\Big{(}\frac{dr}{d\lambda}\Big{)}^{2}+V_{eff}=0$ (8.14)

where $V_{eff}$ represents the effective potential :

$V_{eff}=-\Bigg{(}1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda
r^{2}\Bigg{)}\mathcal{P}(r)$ (8.15)

It is evident that the effective potential depends on the mass of the black
hole $M$, real constant $\beta$, cosmological constant $\Lambda$, radius $r$,
and $R(r)$. Now, we introduce two new impact parameters (dimensionless), such
as $\xi=\frac{L}{E}$ and $\eta=\frac{\mathcal{C}}{E^{2}}$. The value of $P$ in
terms of these two new impact parameters takes the form :

$\mathcal{P}=E^{2}\Bigg{[}\frac{1}{1-\frac{2M}{r}+\beta r-\frac{1}{3}\Lambda
r^{2}}-\frac{\mathcal{\eta}}{r^{2}}\Bigg{]}$ (8.16)

##  9 Shadow of a static black hole in $f(R)$ gravity

In this section, we find the shadow of a static black hole in the environments
of $f(R)$ gravity. In order to find the shadow, we will determine the unstable
circular photon orbits [133]. For this purpose, we must follow the following
conditions:

$P=0\;and\;P^{\prime}=0$

where prime ($\prime$) is the differentiation with respect to $r$.
Substituting 8.16 into the above conditions, we get the relation for the
photon sphere as :

$\frac{B^{\prime}(r)}{B(r)}=\frac{2}{r_{p}}$ (9.1)

Considering Eq. 9.1 and the metric function for a static black hole in $f(R)$
gravity, $r_{p}$ is derived as

$r_{p}=\frac{-1+\sqrt{1+6M\beta}}{\beta}.$ (9.2)

It is observed that the above the photon sphere radius $r_{p}$ decreases when
mass $M$ decreases and $\beta$ increases and it is independent of $\Lambda$.
One can also check that

$\lim_{(\beta,M)\to(0,1)}r_{p}=3\ ,$ (9.3)

which gives the radius of an unstable photon sphere for a Schwarzschild black
hole.

For the distant observer, we can measure the celestial coordinates in the
directions perpendicular ($X$) and parallel ($Y$) to the projected rotation
axis determining the apparent angular distances of the shadow image on the
celestial sphere. In the present case, the celestial coordinates are given by
[69, 130, 134, 135]

$X=\lim_{r_{0}\to\infty}\left(-r^{2}_{0}\sin\theta_{0}\frac{d\phi}{dr}\Big{|}_{r=r_{0},\theta=\theta_{0}}\right)\;,\;Y=\lim_{r_{0}\to\infty}\left(r^{2}_{0}\frac{d\theta}{dr}\Big{|}_{r=r_{0},\theta=\theta_{0}}\right).$
(9.4)

where $(r_{0},\theta_{0})$ refers to the observer’s position in the Boyer-
Lindquist coordinate, the distance between an observer and black hole is
denoted by $r_{0}$ and the angular coordinate (inclination angle444Generally
inclination angle is being measured between the line of sight of an observer
and spinning axis of a black hole) of the larger observer is represented by
$\theta_{0}$. In the case of null geodesic (using Eqs. 8.5, 8.10, and 8.11),
this becomes

$X=-\frac{\xi}{\sin\theta_{0}}\;,\;Y=\pm\sqrt{\eta-\xi^{2}\
cot^{2}\theta_{0}}\ ,$ (9.5)

which, in turn, connects the celestial coordinates with dimensionless impact
parameters ($\xi,\eta$). As long as the observer lies on the equatorial plane
of the mentioned black hole (i.e., $\theta_{0}=\frac{\pi}{2}$), the above
celestial coordinates may take the following values:

$X=-\xi,$ (9.6) $Y=\pm\sqrt{\eta}.$ (9.7)

Finally, the radius of the silhouette of the static black hole in $f(R)$
gravity is given by

$R_{s}=\sqrt{X^{2}+Y^{2}}=\sqrt{\eta+\xi^{2}}=\frac{r_{p}}{\sqrt{B(r_{p})}}.$
(9.8)
$R_{s}=\frac{\sqrt{3}\Big{(}\sqrt{1+6M\beta}-1\Big{)}}{\sqrt{\beta^{2}\Big{(}2\sqrt{1+6M\beta}-1\Big{)}-6M\Lambda\beta+2\Big{(}\sqrt{1+6M\beta}-1\Big{)}\Lambda}}.$
(9.9)

Now, the numerical values of the shadow radius $R_{s}$ for different values of
a cosmological constant ($\Lambda$) and real constant $\beta$ of the static
black hole in $f(R)$ gravity are presented in Table 1 and Table 2,
respectively. From Table 1 and Table 2, we notice that the values of the black
hole shadow radius increases with increasing $\Lambda$ however, shadow radius
becomes smaller for larger real constant.

$\beta$ |  $r_{p}$ |  $\Lambda$ |  $R_{s}$
---|---|---|---
|  |  0.45 |  1.62617
|  |  0.55 |  1.70296
1 |  1.64575 |  0.65 |  1.79176
|  |  0.75 |  1.89609
|  |  0.85 |  2.02105
|  |  0.45 |  0.849553
|  |  0.55 |  0.974877
3 |  1.11963 |  0.65 |  1.17944
|  |  0.75 |  1.61052
|  |  0.85 |  2.13749
Table 1: Radius of the black hole shadow $R_{s}$ for changing $\Lambda$ with fixed $\beta$ and $r_{p}$. Here we set $M=1$. $\Lambda$ |  $\beta$ |  $r_{p}$ |  $R_{s}$
---|---|---|---
|  1.5 |  1.44152 |  1.19172
|  2 |  1.30278 |  0.966802
0.45 |  3 |  1.11963 |  0.725056
|  4 |  1.0 |  0.592349
|  5 |  0.913533 |  0.506489
|  1.5 |  1.44152 |  1.25249
|  2 |  1.30278 |  0.99841
0.65 |  3 |  1.11963 |  0.738105
|  4 |  0.592349 |  0.599401
|  5 |  0.913533 |  0.510876
Table 2: Radius of the black hole shadow $R_{s}$ for different $\beta$ and
$r_{p}$ with fixed $\Lambda$. Here, $M=1$.

###  9.1 Graphical analysis of shadow

We present the graph of black hole shadow for different values of cosmological
constant and real constant in Figs. 9 and 10, respectively. In Fig. 9, we fix
the parameter $\beta=1$ whereas in Fig. 9, we set $\beta=3$. However, in Fig.
10, we set $\Lambda=0.45$ and in Fig. 10, we consider $\Lambda=0.65$. The
graph shows that the shape of the shadow is perfect circle and its shadow
radius increases with an increase in cosmological constant while the shadow
radius decreases with an increase in real constant. This explains that the
gravitational field gets stronger for higher and lower values of cosmological
constant and real constant respectively. The graphical analysis also justifies
the numerical values shown in Table 1 and Table 2. The effect of parameters
such as $\Lambda$, $\beta$, and $M$ on the black hole shadow radius is clearly
presented in the next subsection.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr22.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr23.eps}\end{@subfigure}\end{array}$

Figure 9: Black hole shadow in the celestial plane for different values of
cosmological constant $\Lambda$. Here, $M=1$.

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr24.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr25.eps}\end{@subfigure}\end{array}$

Figure 10: Black hole shadow in the celestial plane for different values of
$\beta$ with $M=1$.

###  9.2 Effect of parameters on shadow radius in non-plasma medium

This subsection intends to explore the intricate interplay between the shadow
radius $R_{s}$ and its dependency on various parameters ($\Lambda$, $\beta$,
and $M$), considering the non-plasma medium. Now, in this medium, the shadow
radius for the static black hole in $f(R)$ gravity takes the following form:

$R_{s}=\frac{r_{p}}{\sqrt{B(r_{p})}}=\frac{\sqrt{3}\Big{(}\sqrt{1+6M\beta}-1\Big{)}}{\sqrt{\beta^{2}\Big{(}2\sqrt{1+6M\beta}-1\Big{)}-6M\Lambda\beta+2\Big{(}\sqrt{1+6M\beta}-1\Big{)}\Lambda}}.$
(9.10)

$\begin{array}[]{cccc}\begin{@subfigure}\includegraphics[width=216.81pt]{fr26.eps}\end{@subfigure}\begin{@subfigure}\includegraphics[width=216.81pt]{fr27.eps}\end{@subfigure}\\\
\begin{@subfigure}\includegraphics[width=346.89731pt]{fr28.eps}\end{@subfigure}\end{array}$

Figure 11: In 11: variation of the shadow radius $R_{s}$ with cosmological
constant $\Lambda$ for varying $\beta$ when $M=1$. In 11: variation of the
shadow radius $R_{s}$ with $\beta$ for changing cosmological constant
$\Lambda$ when $M=1$. In 11: variation of the shadow radius $R_{s}$ with $M$
for changing cosmological constant $\Lambda$ when $\beta=1$.

We see that shadow radius depends on the parameters like $\Lambda$, $\beta$,
and mass $M$ of the given black hole.

How $R_{s}$ depends on the parameters like $\Lambda$, $\beta$, and $M$ is
depicted in Fig. 11. From the plot, we observe that shadow radius is a
decreasing function of real constant and an increasing function of
cosmological constant and mass.

###  9.3 Constraint on parameters through EHT observations

This section aims to explore the constraints on the non-negative real constant
$\beta$ associated with the static black hole in $f(R)$ gravity, assigned by
the recent images of Sgr $A^{\star}$ loosened by the EHT collaboration (see
Ref. [136]). For an observer at the distance $x$ (in Mpc) from the black hole,
the shadow is identified by its angular diameter (in µas) [137]

$\Omega=6.191165\times 10^{-8}\frac{\xi R_{s}}{\pi x},$ (9.11)

in which $\xi$ represents the ratio of the mass of the black hole and the Sun,
$x$ is the distance from the observer to the black hole while $R_{s}$ is the
shadow radius mentioned in Eq. 9.10. In case of the supermassive black hole
Sgr $A^{\star}$, we have $\frac{M}{M_{\odot}}=\xi=4.14\times 10^{6}$,
$x=8.127$ kpc [136, 138] and the observed angular diameters is $51.8\pm 2.3$
µas [136] while for M $87^{\star}$, we have $\frac{M}{M_{\odot}}=\xi=6.2\times
10^{9}$, $x=16.8$ Mpc [139] and the observed angular diameters is $42\pm 3$
µas [139]

Figure 12: The black and red curves depict the $\beta$-profiles of the angular
diameter $\Omega$ for Eq. 9.11. The blue and purple regions correspond to the
observed angular diameters of M$87^{\star}$ and Sgr $A^{\star}$ respectively.

In order to add some constraints on non-negative real constant $\beta$, one
can utilize Eqs. 9.10 and 9.11. We present, with the help of Eqs. 9.10 and
9.11, the profiles of $\Omega(\beta)$ for the mentioned black hole in Fig. 12.
The plot demonstrates that the angular diameter $\Omega$ shows decreasing
nature with $\beta$ and the $\beta$-profiles intersect with the blue and
purple regions resulting in constraints on the parameter $\beta$ in the range
$0<\beta<0.023$ for M$87^{\star}$, and $0.022<\beta<0.041$ for Sgr
$A^{\star}$. Nevertheless, the constraint on $\beta$ giving the mean value
$\beta\approx 0.011$ for M$87^{\star}$, and $\beta\approx 0.031$ for Sgr
$A^{\star}$ which is consistent with the result documented in Ref. [140].

##  10 Summary and final remarks

The presence of massive compact objects (such as black holes) becomes familiar
through various astrophysical observations. Light ray moves through very
strong gravitational fields in the vicinity of black holes. Black holes in a
very special sense give amenities to investigate gravitational lensing besides
the first-order weak deflection limit (which dominates most gravitational
lensing). Moreover, the exploration of this new approach needs very strong
technical efficiency.

In this present work, we have considered a static black hole solution in the
environments of the $f(R)$ gravity model. We have calculated optical metric
with due facilitation of null geodesics in the equatorial plane. In the limit
of weak gravitational lensing, this optical metric introduces Gaussian optical
curvature. Applying the GBT, we have estimated the deflection angle by
employing the Gaussian optical curvature of the optical metric for this static
$f(R)$ black hole under the consideration of a non-plasma medium. Here, we
have noticed that the deflection angle of the mentioned static $f(R)$ black
hole relies on several parameters like impact parameter, cosmological
constant, non-negative real constant, and mass of the black hole. The presence
of these excessive parameters, compared to the Schwarzschild black hole case,
appears to be augmenting the deflection angle. To see the intricate interplay
between these various parameters and the deflection angle, we made a graphical
analysis. The graphs illustrate that the deflection angle
($\boldsymbol{\tilde{\delta}}$) decreases with the impact parameter for a
lower (positive) value of the cosmological constant and then shows the
increasing behavior softly in the positive region. On the contrary, for a
higher (positive) valued cosmological constant, $\boldsymbol{\tilde{\delta}}$
goes to the negative region by showing the decreasing nature for smaller $b$
but then increases gradually and takes positive values only.
$\boldsymbol{\tilde{\delta}}$ shows an increasing tendency with impact
parameter $b$ after performing a certain decline nature in the lower region of
$b$ for small values of $\beta$ but remains always positively valued. However,
$\boldsymbol{\tilde{\delta}}$ shows asymptotically decreasing behavior with
$b$ for large $\beta$ but remains positively valued. For both small and large
$M$, $\boldsymbol{\tilde{\delta}}$ decreases initially along with $b$ and
finally deliberates increasing demeanor by taking only positive values.
Furthermore, the deflection angle is an increasing function of the
cosmological constant and the mass of this static $f(R)$ black hole which is
consistent with the result documented in Ref. [69]. We also observed that
$\boldsymbol{\tilde{\delta}}$ shows increasing nature with non-negative real
constant $\beta$.

In the context of GBT, we have also calculated Hawking temperature for this
static $f(R)$ black hole. The resulting Hawking temperature obeys the standard
form of the Hawking Temperature ($T_{H}=\frac{f^{\prime}(r_{h})}{4\pi}$). The
Hawking temperature calculated in Eq. 5.8 depends on parameters like the mass
of the black hole $M$, cosmological constant $\Lambda$, and non-negative real
constant $\beta$. Moreover, If we consider the limit $\Lambda\rightarrow 0$,
$\beta\rightarrow 0$ in Eq. 5.8, the calculated Hawking temperature of this
black hole reduces to the Hawking temperature of the Schwarzschild black hole
($T_{H}=\frac{1}{8M\pi}$) [115]. To investigate the behavior of Hawking
temperature graphically, we present the graph of Hawking temperature $T_{H}$
with respect to $r_{h}$ in Figure 5 and the graph of $T_{H}$ with respect to
$\Lambda$ and $\beta$ is presented in Figure 6. Fig. 5 illustrates that
$T_{H}$ decreases for an increasing $r_{h}$. In addition to that, $T_{H}$
increases for decreasing $\Lambda$ and increasing $\beta$. However, Fig. 6
indicates that $T_{H}$ is an increasing function of $\Lambda$ for fixed
$\beta$ and is a decreasing function of $\beta$ for fixed $\Lambda$. In both
plots, $T_{H}$ takes negative values only.

Furthermore, we have derived the rigorous analytic bounds of the greybody
factor of the static black hole in the $f(R)$ theory of gravity and noticed
that the calculated bound in Eq. (6.10) depends upon various parameters such
as $M$, $\Lambda$, and $\beta$. In the limit $\Lambda=\beta=0$, one can also
retain the expression of the greybody bound of the Schwarzschild black hole.
From the potential and bound on greybody factor graphs, the bound
corresponding to potential increases sharply and then saturates after a
certain value of quasinormal mode frequency $\omega$ resulting in 1 as long as
$\omega$ approaches infinity. The greybody bound decreases for both increasing
$\Lambda$ and $\beta$. It is worth noting that for large values of $\Lambda$
and $\beta$, the potential becomes higher which in turn makes it difficult for
the waves to be transmitted through that potential.

Finally, We studied the shadow cast by a static $f(R)$ black hole for a
distant observer by investigating the null geodesic of this black hole. Using
the celestial coordinates ($X,Y$), we also derived the radius of the shadow
and presented its image in the outlaying observer’s sky. We see that the shape
of the silhouette is a perfect circle. We also studied the effect of
parameters such as $\Lambda$, $M$, and $\beta$ on the ceremonial of the shadow
radius in view of a non-plasma medium. Our study reveals that shadow radius is
a decreasing function of real constant and an increasing function of
cosmological constant and mass. In this regard, to constrain the dark-matter-
related non-negative parameter $\beta$, we estimate the theoretical angular
size of the black hole shadow and perform comparisons with the observed
results of EHT for M$87^{\star}$ and Sgr $A^{\star}$. We noticed that this
parameter lies in the range $0<\beta<0.023$ for M$87^{\star}$ and
$0.022<\beta<0.041$ for Sgr $A^{\star}$ and our result is consistent with the
result provided in Ref. [140].

In future research endeavors, directing our conjecture toward expected
exploration, it becomes tempting to intend the extension of these findings to
encircle the effect of both the plasma and dark matter medium on the
deflection angle and shadow. Moreover, in the framework of $f(R)$ gravity
theory, comprehensive scrutiny such as the consequences of spin parameters on
the deflection angle, shadow, and greybody bound of the black hole apprehends
the pledge for manifesting us with profound insights.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this manuscript.

## Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

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