# Quasinormal modes, thermodynamics and shadow of black holes in Hu-Sawicki
$\boldsymbol{f(R)}$ gravity theory

Ronit Karmakar  0000-0002-9531-7435<EMAIL_ADDRESS>Department of
Physics, Dibrugarh University, Dibrugarh 786004, Assam, India  Umananda Dev
Goswami  0000-0003-0012-7549<EMAIL_ADDRESS>Department of Physics,
Dibrugarh University, Dibrugarh 786004, Assam, India

###### Abstract

We derive novel black hole solutions in a modified gravity theory, namely the
Hu-Sawicki model of $f(R)$ gravity. After obtaining the black hole solution,
we study the horizon radius of the black hole from the metric and then analyse
the dependence of the model parameters on the horizon. We then use the 6th-
order WKB method to study the quasinormal modes of oscillations (QNMs) of the
black hole perturbed by a scalar field. The dependence of the amplitude and
damping part of the QNMs are analysed with respect to variations in model
parameters and the errors associated with the QNMs are also computed. After
that we study some thermodynamic properties associated with the black hole
such as its thermodynamic temperature as well as greybody factors. It is found
that the black hole has the possibility of showcasing negative temperatures.
We also analyse the geodesics and derive the photon sphere radius as well as
the shadow radius of the black hole. The photon radius is independent of the
model parameters while the shadow radius showed a fair amount of dependence on
the model parameters. We tried to constrain the parameters with the help of
Keck and VLTI observational data and obtained some bounds on $m$ and $c_{2}$
parameters.

Hu-Sawicki gravity; Black holes; Quasinormal modes; Thermodynamics; Shadow

##  I Introduction

General relativity (GR) has undoubtedly been successful in accounting for
observational results in the solar system and beyond [1, 2]. GR theory
predicts with great accuracy the precession of the perihelion of planet
Mercury [1, 3] and bending of light due to gravitational field [4] in the
local as well as distant observations. GR has predicted the existence of black
holes and gravitational waves (GWs) which has been recently experimentally
verified by the LIGO-Virgo collaboration [5, 6, 7, 8, 9]. The recent direct
images of the black hole shadows published by the Event Horizon Telescope
(EHT) group [10, 11, 12, 13, 14, 15] also back GR in terms of experimental
verification of the theory. In spite of these successes, GR fails to address
recent observations like the accelerated expansion of the Universe [16, 17,
18]. Indeed, it does not provide any insights regarding the dark components of
the Universe [19, 20]. Thus, to overcome these issues, physicists worked on
modified theories of gravity, the most common among them includes the
$\Lambda$CDM [21] model, $f(R)$ gravity theory [22, 23, 24, 25], $f(R,T)$
gravity theory [26], $f(Q)$ gravity theory [27, 28, 29, 30] and so on (see
[31]). These theories can compensate for the effects of dark components [32],
explain galactic rotation curves [33], accelerated expansion of the Universe
[34, 35] and are well constrained by modern observations. Rastall gravity
proposed in 1972 is arguably a unique modified theory of gravity which does
not follow from an established Lagrangian formalism [36]. It advocates the
violation of conservation of energy-momentum tensor $T_{\mu\nu}$ and equates
it to the derivative of the Ricci scalar $R$. In recent times, a number of
$f(R)$ gravity models have been proposed, some of them include Starobinsky
[37], Hu-Sawicki [38], Sujikawa [40] and other two models mentioned in Refs.
[39, 41] to name a few. These models have been extensively studied in the
literature regarding various aspects like their cosmological and astrophysical
implications [33, 42, 43, 44], dynamical system analysis [45, 46], early
Universe mysteries [47] and so on. Similarly, $f(Q)$ gravity has also
attracted a lot of attention in the recent times and many cosmological studies
have been carried out in this theory, for instance see Refs. [46, 48, 49, 50]
and references therein.

The first vacuum solution of the Einstein field equations leading to a black
hole was given by Schwarzschild in 1916 [51]. Since then, a number of black
hole solutions have been proposed from time to time in various frameworks of
gravity. Black holes are often studied with an engulfing field around them.
These fields may include quintessence fluid [52, 53, 54, 55], matter in the
form of dust, radiation [56], plasma [57], dark matter halo [58] and so on.
These surrounding fields have impacts on various thermodynamic properties,
quasinormal modes (QNMs), shadow radius etc. of black holes and have been
extensively studied in the literature. In a recent paper [53], GUP-improved
Schwarzschild-type solution, its thermodynamic properties and quasinormal
modes have been studied. In another work [59], a Schwarzschild-type black hole
in Bumblebee gravity has been considered and its thermodynamics and shadow
have been studied.

Black hole solutions have been derived in the framework of $f(R)$ gravity in
many recent papers. In Ref. [39], Saffari and Rahvar derived novel black hole
solutions in the $f(R)$ framework and also proposed a novel $f(R)$ form that
is feasible in both local and galactic scales. In Ref. [60], the authors
derived black hole solutions in various $f(R)$ models. In a recent work [61],
novel black hole solutions were derived in various $f(R)$ models and the
authors studied topological and thermodynamic properties of the solutions
obtained. Motivated by these ongoing researches, we derive novel black hole
solutions in Hu-Sawicki gravity. Here, we intend to study various properties
relating to the black hole solutions obtained. Our solution is unique in the
sense that the black hole solution for the Hu-Sawicki model of $f(R)$ gravity
has not been worked out before to the best of our knowledge and thus we are
motivated to study its properties including QNMs, thermodynamics as well as
shadow radius and greybody factors.

Further, the black hole’s shadow also has gained a fair amount of attention,
credits to the recently released data and images of the black holes at the
center of M87 galaxy and Sgr A. This has opened up a new window to constrain
various theories of gravity and parameter values as well. Recently in Ref.
[62], different parameters of modified theories of gravity have been
constrained using modern shadow radius data. Another work [63] constraints
regular black hole parameters with shadow data of the EHT. Recent works
regarding black hole shadows have gained momentum as shadow provides
interesting new insights and data to constrain black hole physics [64, 65, 66,
67, 68, 69, 70, 71, 72].

QNMs of oscillations of a perturbed black hole hold promise of constraining
physics at the extreme regimes of black holes. The QNMs are basically complex
frequencies linked to GWs produced when a black hole is perturbed by some
external means. There has been an upsurge of research regarding various
aspects of QNMs, new techniques of computing QNMs, their relationships with
shadow data and so on. The WKB method of computing QNMs is the most widely
used technique, though many spectral and analytical techniques are often used
in combination. There has been a wide range of applications of QNMs in
understanding various phenomena such as testing the No-Hair theorem [73] and
constraining theories of modified gravity [74]. QNMs can also be used to study
the stability of background spacetime when it is acted upon by a minute
perturbation [75]. The relation between shadow radius and QNMs has been dealt
with in Ref. [72]. QNMs and Hawking radiation sparsity for GUP-corrected black
holes with topological defects have been studied in Ref. [76]. A brief account
of various methods employed in recent times to compute the QNMs can be found
in the Refs. [76, 77, 78, 79, 80].

Black hole thermodynamics has gained momentum and attracted a lot of attention
following the path-breaking work of Bekenstein and Hawking [81, 82, 83]. Their
idea led to the development of four laws of black hole thermodynamics.
Recently a number of research works have been carried out in this field.
Schwarzschild black holes with quantum corrections have been investigated for
scattering and absorption cross-section [84]. In Ref. [85], the authors
studied absorption and scattering by a black hole with a global monopole in
$f(R)$ gravity. Recently, thermodynamic properties of extended GUP-corrected
black holes has been carried out in Ref. [86]. Thermodynamics of static
dilaton black holes have been studied in Ref. [87].

In this work, we derive black hole solutions in the Hu-Sawicki model of $f(R)$
gravity and study its thermodynamic properties along with its QNMs using the
6th-order WKB method. We compute the shadow radius and present the plots of
its variation with respect to different model parameters. The primary
motivation for choosing the Hu-Sawicki model is that black hole solutions have
not been worked out in this model, and thus it is really intriguing to study
the properties of such a solution. The Hu-Sawicki model is a viable choice as
it is observationally consistent in cosmological scales [88, 89]. It is
consistent with the solar system tests and thus shows viability in the local
scales as well [90]. The choice of this model is thus motivated by viability
and ongoing research works that utilize this theory to study various aspects
of astrophysics and cosmology [38, 91, 92]. The plan of the paper is as
follows. In the second section, we introduce the field equations in the $f(R)$
gravity framework and briefly discuss the method of solving the equations.
After attaining the black hole solution, we move to the third section where we
compute QNMs of the black hole. Then in the fourth section, we discuss the
thermodynamic properties including temperature, entropy and heat capacity
along with greybody factors. Then in the fifth section, we compute the shadow
radius and plot it for variations in parameters. Finally, we conclude the work
with a brief summary and future scopes.

##  II Field equations in $\boldsymbol{f(R)}$ gravity theory

The field equations for the $f(R)$ gravity theory will be presented here in
the spherically symmetric spacetime by adopting the metric formalism of the
theory, in which the variation of action is done with respect to the metric
only. The $f(R)$ gravity field equation can be obtained from an action in
which the Ricci scalar $R$ in the Einstein-Hilbert action is replaced by some
function $f(R)$ of $R$. Thus the generic action of the $f(R)$ gravity theory
can be written as [60]:

$S=\frac{1}{2\kappa}\int d^{4}x\,\sqrt{-g}f(R)+S_{m},$ (1)

where $\kappa=8\pi Gc^{-4}$ and $S_{m}$ is the matter part of the action. As
mentioned already, taking the variation of the above action (1) with respect
to the metric $g_{\mu\nu}$, one can obtain the field equations of $f(R)$
gravity as

$FR_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}-\left(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square\right)F=\kappa\,T_{\mu\nu},$
(2)

where $F=df(R)/dR$ and $\square=\nabla_{\alpha}\nabla^{\alpha}$. Taking the
trace of this (2), we can write the function $f(R)$ as

$f(R)=\frac{1}{2}\left(3\,\square F+FR-\kappa T\right).$ (3)

The derivative of this Eq. (3) with respect to the radial coordinate $r$ leads
to an equation in terms of $F$ and $R$ as given by

$F^{\prime}R-FR^{\prime}+3(\square F)^{\prime}=\kappa T^{\prime},$ (4)

where the prime denotes the derivative with respect to the radial coordinate
$r$. This equation will serve as a consistency relation for the function $F$
that any solution for $F$ must satisfy this relation in order to be a solution
of the field equations, Eq. (2). Further, using Eq. (3) in Eq. (2), the field
equations can be expressed in terms $F$ instead of $f(R)$ as

$R_{\mu\nu}-\frac{1}{4}\,g_{\mu\nu}R=\frac{\kappa}{F}\Big{(}T_{\mu\nu}-\frac{1}{4}\,g_{\mu\nu}T\Big{)}+\frac{1}{F}\Big{(}\nabla_{\mu}\nabla_{\nu}F-\frac{1}{4}\,g_{\mu\nu}\square
F\Big{)}.$ (5)

Considering the case of the vacuum where the energy-momentum tensor and its
trace vanish, we can rewrite the above equation as

$FR_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}F=\frac{1}{4}\,g_{\mu\nu}\Big{(}FR-\square
F\Big{)}.$ (6)

Since we are interested in the solution of this time-independent spherically
symmetric vacuum field equations following the procedure adopted in Ref. [60],
we consider a generic spherically symmetric metric in the form:

$g_{\mu\nu}=\begin{pmatrix}-N(r)&0&0&0\\\ 0&M(r)&0&0\\\ 0&0&r^{2}&0\\\
0&0&0&r^{2}\sin^{2}\theta\\\ \end{pmatrix},$ (7)

where $N(r)$ and $M(r)$ are metric coefficients to be determined, associated
with the time and space components of the metric respectively which are indeed
functions of $r$. For this spherically symmetric metric, both sides of Eq. (6)
become diagonal and accordingly, we can define an index independent parameter
from this equation as

$P_{\mu}\equiv\frac{FR_{\mu\mu}-\nabla_{\mu}\nabla_{\mu}F}{g_{\mu\mu}}.$ (8)

As this quantity $P_{\mu}$ is independent of indices, we can have
$P_{\mu}-P_{\nu}=0$ for all $\mu$ and $\nu$ values and hence from this
property one can obtain the following expressions:

$\displaystyle
2F\frac{X^{\prime}}{X}+rF^{\prime}\frac{X^{\prime}}{X}-2\,rF^{\prime\prime}$
$\displaystyle=0,$ (9) $\displaystyle
N^{\prime\prime}+\left(\frac{F^{\prime}}{F}-\frac{X^{\prime}}{2X}\right)N^{\prime}-\frac{2}{r}\left(\frac{F^{\prime}}{F}-\frac{X^{\prime}}{2X}\right)N-\frac{2}{r^{2}}N+\frac{2}{r^{2}}$
$\displaystyle=0.$ (10)

Here $X=MN$. In this work, our solution is considered to have constant
curvature for the sake of simplicity. Hence the terms $F^{\prime}$ and
$F^{\prime\prime}$ vanish and the field Eqs. (9) and (10) take the forms:

$\displaystyle NM^{\prime}+N^{\prime}M$ $\displaystyle=0,$ (11) $\displaystyle
1-M+\frac{r}{2}\left(\frac{N^{\prime}}{N}+\frac{M^{\prime}}{M}\right)\left(\frac{r}{2}\frac{N^{\prime}}{N}-1\right)-\frac{r^{2}N^{\prime\prime}}{2N}$
$\displaystyle=0.$ (12)

Solving these two Eqs. (11) and (12), one can obtain:

$M(r)=\frac{s_{1}}{N(r)},\;\;\;\text{and}\;\;\;N(r)=s_{1}+\frac{s_{2}}{r}+s_{3}\,r^{2},$
(13)

where $s_{1}$, $s_{2}$ and $s_{3}$ are constants of integration. In order to
get these coefficients, we follow the procedure in Refs. [60, 61] and compare
the second solution with the standard Schwarzschild-de Sitter solution. The
standard Schwarzschild-de Sitter black hole metric coefficient is [60]

$C(r)=1-\frac{2M}{r}-\frac{\Lambda r^{2}}{3}.$ (14)

Again, the relationship between the scalar curvature and cosmological constant
is [60]

$R=-\,4\Lambda.$ (15)

Now, comparing the second solution in Eq. (13) with Eq. (14), we have

$s_{1}=1,s_{2}=-\,2M,s_{3}=-\,\frac{\Lambda}{3}=\frac{R}{12}.$

From (3), considering the vacuum case and constant curvature $R_{0}$, we have

$R_{0}=\frac{2f(R_{0})}{F(R_{0})}.$ (16)

As mentioned earlier, the $f(R)$ gravity model we employed in our work is the
Hu-Sawicki model [38], which is given by

$f(R)=-\,m^{2}\frac{c_{1}(\frac{R}{m^{2}})^{n}}{c_{2}(\frac{R}{m^{2}})^{n}+1},$
(17)

where $m$, $n\,(>0)$, $c_{1}$ and $c_{2}$ are the model parameters. Here
$c_{1}$ and $c_{2}$ are dimensionless and $m$ represents the mass (energy)
scale [32]. For this model, we solve Eq. (16) to get the constant curvature

$R_{0}=12s_{3}=m^{2}\bigg{(}\frac{n-2}{2\,c_{2}}\bigg{)}^{\frac{1}{n}}\\!\\!\\!.$
(18)

Thus, we arrive at our black hole solution for the Hu-Sawicki model as

$N(r)=1-\frac{2M}{r}+\frac{m^{2}}{12}\bigg{(}\frac{n-2}{2c_{2}}\bigg{)}^{\frac{1}{n}}r^{2}.$
(19)

It is clear that our black hole solution is independent of the Hu-Sawicki
model parameter $c_{1}$. Fig 1 shows the metric function versus radial
distance for the other two Hu-Sawicki model parameters $m$ and $c_{2}$, while
taking the parameter $n=1$ for simplicity (this is considered for the whole
study if we do not mention otherwise). In the plots, it is seen that the black
hole solution (19) has two horizons for a range of parameter values. The first
plot shows that with increasing $m$, the outer horizon moves closer to the
inner horizon, while the second plot shows that for higher values of $c_{2}$,
the outer horizon increases. After a certain higher value of the parameter $m$
and the lower value of the parameter $c_{2}$, the black hole appears to be a
horizonless singularity for the given values of the other parameters.

Figure 1: Black hole metric function versus radial distance $r$ for different
values of parameters. In the left plot, we use $c_{2}=3$ and $n=1$ while in
the right plot, we use $m=0.5$ and $n=1$. The red dashed line shows the ideal
Schwarzschild case.

##  III Quasinormal modes of the Black Hole

In this section, we compute the QNMs of the black hole (19) using the most
common method, the 6th-order WKB approximation method. To this end, we apply a
perturbation to the black hole in the form of a probe coupled minimally to a
scalar field $\Phi$ and having the equation of motion [54]:

$\frac{1}{\sqrt{-g}}\,\partial_{\alpha}\Big{(}\sqrt{-g}g^{\alpha\beta}\partial_{\beta}\Big{)}\Phi=\mu^{2}\Phi,$
(20)

where $\mu$ is the mass of the scalar field, which for our convenience will be
taken as a massless scalar field with $\mu=0$. We can express the scalar field
$\Phi$ in terms of spherical harmonics of the form [54]:

$\Phi(t,r,\theta,\phi)=e^{-i\omega
t}\frac{\Psi(r)}{r}\,Y_{l}^{p}(\theta,\phi).$ (21)

Here $\Psi(r)$ represents the radial part of the wave and $Y_{l}^{p}$
represents the spherical harmonic part. Employing Eq. (21) in Eq. (20), we get
a Schrödinger-type equation, as given below:

$\frac{d^{2}\Psi}{dx^{2}}+\left[\omega^{2}-V(x)\right]\Psi=0,$ (22)

with the new variable, the tortoise coordinate, which is defined as

$x=\int\frac{dr}{N(r)}.$ (23)

The effective potential in Eq. (22) can be expressed as

$V(r)=N(r)\bigg{(}\frac{N^{\prime}(r)}{r}+\frac{l(l+1)}{r^{2}}\bigg{)}.$ (24)

It is necessary to apply the appropriate boundary conditions to Eq. (22) for
the physical consistency both at the black hole horizon and at infinity. For
spacetime which is flat asymptotically the following quasinormal criteria have
to be satisfied:

$\Psi(x)\rightarrow\begin{cases}Ae^{+i\omega
x}\;\;\;\text{if}\;\;x\rightarrow-\infty,\\\ Be^{-i\omega
x}\;\;\;\text{if}\;\;x\rightarrow+\infty\end{cases}.$ (25)

Here, the coefficients $A$ and $B$ represent the amplitudes of the waves.
These ingoing and outgoing waves are in accordance with the physical
requirements that nothing can escape from the black hole horizon and no
radiation comes from the infinity respectively. Further, these make sure of
the existence of an infinite set of discrete complex numbers, usually known as
the QNMs.

To study the behaviour of the potential (24) before calculating the QNMs of
the black hole (19), we plot the potential versus $r$ for different variations
of model parameters in Fig. 2. As seen from the left plot of Fig. 2, the peak
of the potential decreases for higher $m$ values. From the middle plot, one
can see that increasing values of parameter $c_{2}$ enhances the peak of the
potential. A similar trend is seen with mutipole $l$ values, where peaks are
found to increase for higher $l$ values.

Figure 2: Behaviours of black hole potential with respect to radial distance
$r$ for different model parameters. The left plot uses values of $c_{2}=2$,
$n=1$ and multipole $l=1$. The middle plot uses $m=0.5$, $n=1$ and multipole
$l=1$. The right plot uses $m=0.1$, $n=1$ and $c_{2}=2$.

The QNMs have been calculated utilising the 6th-order WKB method in the form
of their amplitude and damping varying with the model parameters. As seen from
Fig. 3, the general trend of amplitude and damping of QNMs is that both
decrease with the parameter $m$ for all values of multiple $l$. On the other
hand, Fig. 4 shows that both amplitude and damping increase slightly with the
parameter $c_{2}$ for all $l$ values. In both cases, the effect of $l$ is more
dominating on the amplitude than that on the damping. Moreover, in both cases
of $m$ and $c_{2}$ variations, the amplitude increases, while the damping
decreases with the increasing value of $l$.

Figure 3: Variation of amplitude and damping of QNMs with respect to parameter
$m$ for three values of multipole $l$. Here we use $n=1$, $M=1$ and $c_{2}=4$
to represent the features of QNMs.

Figure 4: Variation of amplitude and damping of QNMs with respect to parameter
$c_{2}$ for three values of multipole $l$. Here parameters $n=1$, $M=1$ and
$m=1$ have been used.

We compute the error associated with the WKB QNMs with a prescribed formula
that has been used extensively in the literature. This error estimating
formula for the WKB method is as follows [54, 76, 77]:

$\Delta_{6}=\frac{\big{|}W\\!K\\!B_{7}-W\\!K\\!B_{5}\big{|}}{2},$ (26)

where $W\\!K\\!B_{5}$ and $W\\!K\\!B_{7}$ are respectively the QNMs obtained
from the 5th and 7th order WKB method. In Table 1, we present the 6th-order
WKB QNMs along with the associated errors for various values of the model
parameters along with the multipole number $l$. It is clear that the errors
are reduced for higher multipole numbers $l$. Similar trends in variations of
QNMs with respect to different parameters as seen in Figs. 3 and 4 are
displayed in the tabulated data. The estimated errors in most of the cases lie
around $10^{-4}-10^{-5}$.

Table 1: 6th-order WKB QNMs of the black hole specified by the metric function (19) for the multipoles $l=1,2,3$ with $n=0$ and for different values of the model parameters. The estimated errors associated with the WKB results have also been shown. | Multipole | $m$ | $c_{2}$ | 6th-order QNMs | $\Delta_{6}$ |
---|---|---|---|---|---|---
| $l=1$ | $0.1$ | $4.0$ | 0.29242 - 0.097672i | $0.9871\times 10^{-4}$ |
| $0.3$ | $4.0$ | 0.288479 - 0.096937i | $0.9483\times 10^{-4}$ |
| $0.5$ | $4.0$ | 0.280478 - 0.0953844i | $0.9076\times 10^{-4}$ |
| $1.0$ | $4.0$ | 0.240646 - 0.0863446i | $0.6341\times 10^{-4}$ |
|  | $1.0$ | $6.5$ | 0.261572 - 0.0913734i | $0.7809\times 10^{-4}$ |
| $l=2$ | $0.1$ | $4.0$ | 0.482906 - 0.0966493i | $0.8355\times 10^{-5}$ |
| $0.3$ | $4.0$ | 0.476986 - 0.0957027i | $0.7845\times 10^{-5}$ |
| $0.5$ | $4.0$ | 0.46497 - 0.0937446i | $0.7326\times 10^{-5}$ |
| $1.0$ | $4.0$ | 0.405024 - 0.0832849i | $0.5689\times 10^{-5}$ |
|  | $1.0$ | $6.5$ | 0.436572 - 0.0889274i | $0.6718\times 10^{-5}$ |
| $l=3$ | $0.1$ | $4.0$ | 0.674374 - 0.096375i | $0.1534\times 10^{-5}$ |
| $0.3$ | $4.0$ | 0.666393 - 0.0953604i | $0.1534\times 10^{-5}$ |
| $0.5$ | $4.0$ | 0.65018 - 0.0932773 i | $0.1527\times 10^{-5}$ |
| $1.0$ | $4.0$ | 0.56893 - 0.0824456i | $0.5235\times 10^{-6}$ |
|  | $1.0$ | $6.5$ | 0.611771 - 0.0882334i | $0.1033\times 10^{-5}$ |

##  IV Thermodynamic Charactistics of the black hole

As mentioned earlier, a black hole as a thermodynamic system was first
conceptualised in the ground-breaking work of Hawking and Bekenstein in the
early 1970s. In this paper, we analyse the black hole temperature and the grey
body factors which are important properties that give useful insights in this
regard. The temperature of a black hole is an important property that is
associated with the quantum particles created near its horizon. It is
inversely related to the size or mass of the black hole, that is a larger
black hole will have a lower temperature. Hawking conceptualised the
temperature of black hole in the form of radiation, which is today referred to
as Hawking radiation. It remains a challenge to detect such radiation
experimentally. We can theoretically compute the black hole temperature from
the metric solution (19), by employing the simple relation:

$T_{BH}=\frac{N^{\prime}(r)}{4\pi}=\frac{1}{4\pi
r_{H}^{2}}\bigg{[}2M+\frac{m^{2}}{12}\Big{(}\frac{n-2}{c_{2}}\Big{)}^{\frac{1}{n}}r_{H}^{3}\bigg{]}.$
(27)

It can also be calculated using the First law of black hole thermodynamics as
follows:

$T_{BH}=\frac{dM}{dS}=\frac{dM^{\prime}}{dS^{\prime}}=\frac{1}{4\pi
r_{H}^{2}}\bigg{[}2M+\frac{m^{2}}{12}\Big{(}\frac{n-2}{c_{2}}\Big{)}^{\frac{1}{n}}r_{H}^{3}\bigg{]}.$
(28)

This confirms our computation of the black hole temperature from the first
law. We plot the thermodynamic temperature (27) with respect to the horizon
radius $r_{H}$ in Fig. 5. Here, we see clearly that with horizon radius
$r_{H}$, the temperature of the black hole is always in the decreasing trend.
The left plot shows the temperature variations with respect to $r_{H}$ for
three different values of parameter $m$. It is seen that higher values of $m$
lead to negative temperatures. While for the parameter $c_{2}$, lower values
lead to negative temperatures as can be seen from the right panel of Fig. 5.
Though the negative temperature seems unphysical, this has been encountered in
the literature and explained as a possible state of formation of ultra-cold
black holes [54, 80].

Figure 5: Variation of temperature versus $r_{H}$ for three different values
of $m$ on the left plot and for three different values of $c_{2}$ on the right
plot. For the left plot, we use $n=M=1$ and $c_{2}=2$, while for the right
plot, we use $M=n=1$ and $m=0.5$.

The greybody factor or the transmission coefficient is a measure of the
probability that a particle created by quantum processes near the event
horizon of a black hole will escape to infinity or get absorbed inside the
black hole. Greybody factor ($T^{2}$) equal to $1$ means that all the
particles that are created are able to escape the black hole while lower
values of it mean that some of them end up inside it. If $T^{2}=0$, it means
that the black hole is completely dark and absorbs every single particle. The
greybody factor has been extensively studied in the literature in various
scenarios. We can express the reflection and transmission of the particles
hitting the black hole barrier potential in the following form [93]:

$\displaystyle\psi(x)$ $\displaystyle=T(\omega)\exp(-i\omega
x),\;\;x\rightarrow-\infty,$ (29) $\displaystyle\psi(x)$
$\displaystyle=\exp(-i\omega x)+R(\omega)\exp(i\omega
x),\;\;x\rightarrow+\infty,$ (30)

where $R(\omega)$ and $T(\omega)$ are respectively reflection and transmission
coefficients and are functions of frequency $\omega$. WKB approximation
formula is used to get to the computational form of these two coefficients
which are presented below [93]:

$\displaystyle|R(\omega)|^{2}$ $\displaystyle=\frac{1}{1+\exp(-2\pi i\tau)},$
(31) $\displaystyle|T(\omega)|^{2}$ $\displaystyle=\frac{1}{1+\exp(2\pi
i\tau)},$ (32)

where the parameter $\tau$ is defined in the WKB method as the following [93]:

$\tau=\frac{i(\omega^{2}-V_{0})}{\sqrt{-2V_{0}^{{}^{\prime\prime}}}}-\Lambda_{j}.$
(33)

Here double primes represent the double derivative of the maximum of the
effective potential $V_{0}$ with respect to $x$ and $\Lambda_{j}$ can be
obtained from the WKB formula found in Ref. [79]. In Fig. 6, we plot the
greybody factors with respect to frequency $\omega$ for three values of the
model parameters $m$ considering $n=1$, $c_{2}=1$ and $M=1$ with multipole
$l=1$ (left plot) and $l=2$ (middle plot). It is seen that for higher $m$
values the grebody factor increases faster with respect to $\omega$ than that
for smaller $m$ values. Also, for a smaller $l$ value ($l=1$), the greybody
factor increase is more rapid and begins from a smaller $\omega$ value as
compared to a higher $l$ value ($l=2$). It needs to be mentioned that the
values of the parameter $c_{2}$ are found to be insensitive in the variation
of the greybody factors with respect to $\omega$ as shown in the right plot of
Fig. 6.

Figure 6: Greybody factors versus frequency $\omega$ for different values of
$m$ with $l=1$ (left plot) and $l=2$ (middle plot). The right plot shows that
the greybody factor is independent of parameter $c_{2}$. We have used $M=n=1$
and $c_{2}=1$ in the first two plots and $m=0.1$ and $l=1$ for the third plot.

##  V Shadow of the black hole

Black hole shadow has been extensively studied in the literature as it
provides good scope to test various theories of gravity and black hole physics
in extreme gravity regimes. The recent observational data of black hole shadow
radius has provided the scientific community an opportunity to constrain model
parameters using these data. In this section, we compute the photon sphere and
the shadow radius expression and plot the same for analysing its dependence on
various model parameters. We also try to constrain the parameter space with
observational data of the EHT group.

The simple condition to determine the photon sphere radius of a black hole in
spherical symmetry consideration is given by the following relation [71, 72]:

$2-\frac{rN^{\prime}(r)}{N(r)}=0.$ (34)

Using the form of $N(r)$ from Eq. (19), we solve the Eq. (34) for $r$ to get
the photon sphere radius as

$r_{ph}=3M.$ (35)

From the photon radius, we can derive the shadow radius as follows:

$r_{sh}=\frac{r_{ph}}{\sqrt{N(r)}|_{r\rightarrow
r_{ph}}}=\frac{3M}{\sqrt{\frac{3}{4}m^{2}M^{2}\left(\frac{n-2}{\text{2\,c2}}\right)^{1/n}+\frac{1}{3}}}.$
(36)

Obviously, the shadow radius depends on all three model parameters associated
with $N(r)$. It is also evident that when model parameters $m=n=c_{2}=0$, we
recover the standard $r_{sh}=3\sqrt{3}M$ which is the shadow radius for the
Schwarzschild black hole. Now, for the 2-D stereoscopic projection of shadow
radius, we define celestial coordinates $X$ and $Y$ as given by [71, 72]

$\displaystyle X$
$\displaystyle=\lim_{r_{0}\rightarrow\infty}\left(-\,r_{0}^{2}\sin\theta_{0}\left.\frac{d\phi}{dr}\right|_{r_{0}}\right),$
(37) $\displaystyle Y$
$\displaystyle=\lim_{r_{0}\rightarrow\infty}\left(r_{0}^{2}\left.\frac{d\theta}{dr}\right|_{(r_{0},\theta_{0})}\right).$
(38)

Here $\theta_{0}$ is the observer’s angular position with regards to the plane
of the black hole. In Fig.7, we show the variation of the shadow radius with
parameters $m$ and $c_{2}$. It is seen from the left plot that with an
increase in $m$, the shadow radius increases. In the right plot, it is evident
that the shadow radius decreases with increasing $c_{2}$ values. Thus, the
parameters $m$ and $c_{2}$ have opposite influences on the shadow radius.

Figure 7: Stereoscopic projection of shadow radius in terms of celestial
coordinates. The left plot is for the variation of $m$ with parameters
$c_{2}=M=n=1$ and the right plot is for the variation of $c_{2}$ with $M=n=1$
and $m=0.1$

.

In order to constrain the parameters of the model, we shall employ the
technique mentioned in Ref. [71]. We briefly present some important steps in
this direction. The main point of the methodology is that we compare the
observed angular radius of the Sgr A* black hole as captured by the EHT group
recently with the theoretically calculated shadow radius from the expression
(36) by constraining the model parameters. This requires the prior value of
the mas-to-distance ratio for Sgr A*. Another feature that is required for
this method is the calibration factor that correlates the observed to the
calculated shadow radius. This method has been used to constrain model
parameters in the literature [94, 95, 96] and we shall follow the same route.

A new parameter $\delta$ defined by the EHT group to refer to the fractional
deviation between the observed shadow radius $r_{s}$ and shadow radius of a
Schwarzschild black hole $r_{sch}$ is [71]

$\delta=\frac{r_{s}}{r_{sch}}-1=\frac{r_{s}}{3\sqrt{3}M}-1.$ (39)

This parameter was estimated by the Keck and VLTI measurements as [71]

$\displaystyle\textrm{Keck}:\delta=-\,0.04^{+0.09}_{-0.10}$
$\displaystyle\textrm{VLTI}:\delta=-\,0.08^{+0.09}_{-0.09}$

For simplification, we shall adopt the mean of the two observations as
considered in Ref. [71] in the rest of the work, which is

$\delta=-\,0.060\pm 0.065.$ (40)

This leads to $\delta$ parameter’s 1$\sigma$ and $2\sigma$ intervals as

$\displaystyle-\,0.125\lesssim$ $\displaystyle\,\delta\lesssim
0.005\,(1\sigma),$ (41) $\displaystyle-\,0.190\lesssim$
$\displaystyle\,\delta\lesssim 0.070\,(2\sigma).$ (42)

It is found that the bounds (41) and (42) when imposed upon Eq. (39) give the
bounds on $r_{sh}$ as follows [71]:

$\displaystyle 4.55\lesssim$ $\displaystyle\,r_{sh}/M\lesssim
5.22\,(1\sigma),$ (43) $\displaystyle 4.21\lesssim$
$\displaystyle\,r_{sh}/M\lesssim 5.56\,(2\sigma).$ (44)

We plot the shadow radius with the bounds imposed by the observations of Keck
and VLTI in Fig. 8. The left plot shows that the shadow radius increases with
increasing $m$ values as found in Fig. 7. It shows that for smaller values of
$c_{2}$, the shadow radius quickly moves to the forbidden region. With
increasing $c_{2}$ values, the shadow radius within the allowed region
increases. In the right plot, the shadow radius is plotted versus $c_{2}$
which shows that the shadow radius decreases with increasing $c_{2}$ values as
observed earlier. The plots are within the 2$\sigma$ allowed region in this
case with the exception of larger $m$ and smaller $c_{2}$ values as clearly
visible from the plot. It is quite evident that the constraints imposed are
not rigid but depend on the range of values of the parameters of the model.

This method of constraining parameters of a theory has been adopted in the
literature [94, 95] and by the EHT group themselves [96] and provides a robust
way of constraining parameters. But in cases of model parameters exceeding
one, we need some supporting constraining methods so that one parameter can be
cornered and rigorous constraints can be obtained. However, we leave this as a
future extension of the work.

Figure 8: Shadow radius versus parameter $m$ and $c_{2}$ have been plotted in
the background of Keck and VLTI constrains [71] from observations of Sgr A*.
We have chosen $M=1$ and $n=1$ for these plots. The red portion represents the
zone forbidden by Keck-VLTI observation.

##  VI Summary and Conclusion

In this work, we derive novel black hole solutions in the framework of Hu-
Sawicki gravity. We plot the metric function versus $r$ for various values of
model parameters and encountered two horizons of the black hole. It is seen
that higher $m$ values cause the horizon radius to shrink while the opposite
trend is observed for the parameter $c_{2}$. We then analyse the QNMs of the
novel black hole solution using the 6th-order WKB approximation. The amplitude
increases with an increase in $c_{2}$ values while it decreases with $m$. The
damping decreases with increasing $m$ while it increases slightly with
$c_{2}$. This trend can be realised from the tabulated QNM data in Table 1. It
is evident that the QNM frequencies are affected by the model parameters. The
associated error is found to be around $10^{-4}$ to $10^{-5}$ in some cases.
The thermodynamic temperature associated with the black hole is investigated
and it is found to decrease with the black hole radius $r_{H}$ in all cases.
The temperature can also become negative, suggesting the possibility of the
formation of an ultra-cold black hole. The greybody factors are also computed,
especially the transmission coefficients with respect to frequency $\omega$
and the dependence of the model parameter $m$ is studied. Higher $m$ results
in a swifter increase in the greybody factors towards the saturation value of
$1$. It is noteworthy that increasing the multipole $l$ lowers the rate of
increase of the greybody factors and saturation is achieved at higher
$\omega$.

The photon radius and the shadow radius associated with the spherically
symmetric black hole spacetime are then studied. We presented the
stereographic projection of the shadow in the celestial coordinate system and
using the contour-type feature, showed the variation of the shadow radius with
increasing model parameters $m$ and $c_{2}$. Taking into consideration the
already established constraints on $r_{sh}$ by Keck and VLTI observations, we
constrain our model parameters using a well-proven scheme. The parameter $m$
is roughly constrained to be less than $\sim 0.5$ while parameter $c_{2}$ is
constrained to be greater than $\sim 2$, as can be seen from Figure 8.

The recent technical advancements made in the fields of astrophysics and
observational astronomy have made the present era very suitable for
theoretical physicists to constrain and test fundamental theories and models,
which was not possible until a decade back. With the ground-breaking leaps in
the form of the LIGO-Virgo team’s observation of GWs in 2015 along with the
first-ever image of the black hole M87* and later that of Sgr A*, scientists
plan to further enhance the sensitivity of the present detectors as well as
new ambitious projects like the space-based LISA project and the Einstein
Telescope are already in the planning stages. As a future scope of this work,
we can analyse other viable models of gravity like $f(R,T)$ and $f(Q)$, and
work on new black hole solutions as well as rotating Kerr-type solutions that
can also be explored. Further, the study of black hole shadows surely holds a
lot of potential in constraining fundamental physics and it certainly deserves
further investigation.

## Acknowledgements

UDG is thankful to the Inter-University Centre for Astronomy and Astrophysics
(IUCAA), Pune, India for awarding the Visiting Associateship of the institute.

## References

  * [1] R.M. Wald, General Relativity, The University of Chicago Press, Chicago, 1984.
  * [2] C. M. Will, Was Einstein right? A centenary assessment, arXiv:1409.7871 [gr-qc] (2004).
  * [3] C. M . Will, New General Relativistic Contribution to Mercury’s Perihelion Advance, Phys. Rev. Lett 120, 191101 (2018).
  * [4] C. M. Will, The 1919 measurement of the deflection of light, Class. Quantum Grav. 32, 124001 (2015).
  * [5] B. P. Abbott et al., Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016).
  * [6] B. P. Abbott et al., Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett. 116, 241103 (2016).
  * [7] B. P. Abbott et al., Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  * [8] B. P. Abbott et al., Observation of a Binary-Black-Hole Coalescence with Asymmetric Masses, Phys. Rev. D 102, 043015 (2020).
  * [9] R. Abbott et al., Observation of Gravitational Waves from Two Neutron Star–Black Hole Coalescences, ApJL 915, L5 (2021).
  * [10] The Event Horizon Telescope Collaboration et al., First M87 Event Horizon telescope Results. I. The Shadow of the supermassive Black Hole, ApJL 875, L1 (2019).
  * [11] The Event Horizon Telescope Collaboration et al., First M87 Event Horizon telescope Results. II. Array and Instrumentation, ApJL 875, L2 (2019).
  * [12] The Event Horizon Telescope Collaboration et al., First M87 Event Horizon telescope Results. III. Data Processing and Calibration, ApJL 875, L3 (2019).
  * [13] The Event Horizon Telescope Collaboration et al., First M87 Event Horizon telescope Results. IV. Image the Central Supermassive Black Hole, ApJL 875, L4 (2019).
  * [14] The Event Horizon Telescope Collaboration et al., First M87 Event Horizon telescope Results. V. Physical Origin of the Asymmetric Ring, ApJL 875, L5 (2019).
  * [15] The Event Horizon Telescope Collaboration et al., First M87 Event Horizon telescope Results. VI. The Shadow and Mass of the Central Black Hole, ApJL 875, L6 (2019).
  * [16] V. Faraoni and S. Capozziello, Beyond Einstein gravity: a survey of gravitational theories for cosmology and astrophysics, Fundam. Theor. Phys. 170, 1–428 (2010).
  * [17] A. G. Riess et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, The Astronomical Journal 116, 1009 (1998).
  * [18] S. Perlmutter et al., _Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae_, ApJ 517, 565 (1999)
  * [19] C. Pérez de los Heros, Status, Challenges and Directions in Indirect Dark Matter Searches, Symmetry 12, 1648 (2020).
  * [20] N. A. Bahcall, _The Cosmic Triangle: Revealing the State of the Universe_ , Science 284, 1481 (1999).
  * [21] L. Amendola and S. Tsujikawa, Dark Energy: Theory and Observations, Cambridge University Press, Cambridge, 2010.
  * [22] D. J. Gogoi and U. D. Goswami, A new f(R) gravity model and properties of gravitational waves in it, EPJC 80, 1101 (2020).
  * [23] T. P. Sotiriou and V. faraoni, f(R) theories of gravity, Rev. Mod. Phys. 82, 451 (2010).
  * [24] T. P. Sotiriou and V. faraoni, f(R) theories of gravity, Rev. Mod. Phys. 82, 451 (2010).
  * [25] A. De Felice and S. Tsujikawa, f(R) Theories, Living Rev. Relativ. 13, 3 (2010).
  * [26] T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, f(R,T) gravity, Phys. Rev. D 84, 024020 (2011).
  * [27] P. Sarmah, A. De and U. D. Goswami, Anisotropic LRS-BI Universe with f(Q) gravity theory, Phys. Dark Universe 40, 101209 (2023).
  * [28] A. De, S. Mandal, J. T. Beh, T. H. Loo and P. K. Sahoo, Isotropization of locally symmetricBianchi-I universe in f(Q)-gravity, EPJC 82, 10052 (2022).
  * [29] R. Solanki, A. De, S. Mandal and P. K. Sahoo, Accelerating expansion of the universe in modified symmetric teleparallel gravity, Phys. Dark Universe 36, 101053 (2022).
  * [30] D. J. Gogoi, A. Övgün and M. Koussour, Quasinormal Modes of Black holes in $f(Q)$ gravity, EPJC 83, 700 (2023).
  * [31] T. P. Sotiriou and V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys. 82, 451 (2010).
  * [32] N. Parbin and U. D. Goswami, Scalarons mimicking dark matter in the Hu-Sawicki model of f(R) gravity, Mod. Phys. Lett. A 36, 37 (2021).
  * [33] N. Parbin and U. D. Goswami, Galactic rotation dynamics in a new f(R) gravity model, EPJC 83, 411 (2023).
  * [34] R. Myrzakulov, Accelerating universe from F(T) gravity, EPJC 71, 1752 (2011).
  * [35] A. Mukherjee and N. Banerjee, Acceleration of the universe in f(R) gravity models, Astrophys. Space Sci. 352, 839 (2014).
  * [36] P. Rastall, Generalization of the Einstein Theory, Phys. Rev. D 6, 3357 (1972).
  * [37] A. A. Starobinsky, Disappearing cosmological constant in f(R) gravity, JETP 86, 157 (2007).
  * [38] W. Hu and I. Sawicki, Models of f(R) cosmic acceleration that evade solar system tests, Phys. Rev. D 76, 064004 (2007).
  * [39] R. Saffari and S. Rahvar, f(R) Gravity: From the Pioneer Anomaly to the Cosmic Acceleration, Phys. Rev. D 77, 104028 (2008).
  * [40] J. -Y. Cen, S. -Y. Chien, C. -Q. Geng and C. -C. Lee, Cosmological evolutions in Tsujikawa model of f(R) Gravity, Phys. Dark Universe 26, 100375 (2019).
  * [41] D. J. Gogoi and U. D. Goswami, A new f(R) gravity model and properties of gravitational waves in it, EPJC 80, 1011 (2020).
  * [42] P. Bessa, M. Campista and A. Bernui, Observational constraints on Starobinsky f(R) cosmology from cosmic expansion and structure growth data, EPJC 82, 506 (2022).
  * [43] P. V. Ky, N. T. H. Van and N. A. Ky, Gravitational radiation of a spherically symmetric source in f(R)-gravitation, EPJC 84, 298 (2024).
  * [44] J. Bora, D. J. Gogoi and U. D. Goswami, Strange stars in f(R) gravity palatini formalism and gravitational wave echoes from them, JCAP 09 057 (2022).
  * [45] L. Amendola, R. Gannouji, D. Polarski and S. Tsujikawa, Conditions for the cosmological viability of f(R) dark energy models, Phys. Rev. D 75, 083504 (2007).
  * [46] P. Sarmah and U. D. Goswami, Dynamical system analysis of LRS-BI Universe with f(Q) gravity theory, arXiv:2403.16118v1 [gr-qc]
  * [47] T. Katsuragawa, S. Matsuzaki and E. Senaha, F(R) gravity in the early Universe: electroweak phase transition and chameleon mechanism*, Chinese Physics C 43, 105101 (2019).
  * [48] R. Solanki, A. De, S. Mondal and P. K. Sahoo, Accelerating expansion of the universe in modified symmetric teleparallel gravity, Phys. Dark Universe 36, 101053 (2022).
  * [49] S. Capozziello, V. De Falco and C. Ferrara, Comparing equivalent gravities: common features and differences, EPJC 82, 856 (2023).
  * [50] D. Zhao, Covariant formulation of f(Q) theory, EPJC 82, 303 (2022).
  * [51] K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory, arXiv:physics/9905030 (1916).
  * [52] S. Fernando, Schwarzschild black hole surrounded by quintessence: null geodesics, Gen Relativ. Gravit. 44, 1857–1879 (2012).
  * [53] R. Karmakar, D. J. Gogoi and U. D. Goswami, Quasinormal modes and thermodynamic properties of GUP-corrected Schwarzschild black hole surrounded by quintessence, IJMPA 37, 2250180 (2022).
  * [54] D. J. Gogoi, R. Karmakar and U. D. Goswami, Quasinormal Modes of Non-Linearly Charged Black Holes surrounded by a Cloud of Strings in Rastall Gravity, IJGMMP 20, 2350007 (2023).
  * [55] S. Chen, B. Wang and R. Su, Hawking radiation in a d-dimensional static spherically symmetric black hole surrounded by quintessence, Phys. Rev. D 77, 124011 (2008).
  * [56] Y. Heydarzade and F. Darabi, Black Hole Solutions Surrounded by Perfect Fluid in Rastall Theory, Phys. Lett. B 771, 365 (2017).
  * [57] F. Atamurotov, A. Abdujabbarov and W. -B. Han, Effect of plasma on gravitational lensing by a Schwarzschild black hole immersed in perfect fluid dark matter, Phys. Rev. D 104, 084015 (2021).
  * [58] R. A. Konoplya, Shadow of a black hole surrounded by dark matter, Phys. Lett. B 795, 1 (2019).
  * [59] R. Karmakar, D. J. Gogoi and U. D. Goswami, Thermodynamics and Shadows of GUP-corrected Black Holes with Topological Defects in Bumblebee Gravity, Phys. Dark Universe 41, 101249 (2023).
  * [60] T. Multamäki and I. Vilja, Spherically symmetric solutions of modified field equations in f(R) theories of gravity, Phys. Rev. D 74, 064002 (2006).
  * [61] B. Hazarika and P. Phukon, Thermodynamic Topology of Black Holes in f(R) Gravity, PTEP 2024, Issue 4, 043E01 (2024).
  * [62] V. Prokopov, S. Alexeyev and O.Zenin, Black Hole Shadows Constrain Extended Gravity, J. Exp. Theor. Phys. 135, 91–99 (2022).
  * [63] K. Jafarzade, M. K. Zangeneh and F. S. N. Lobo, Observational optical constraints of regular black holes, Annals of Physics 446, 169126 (2022).
  * [64] İ. Çimdiker, D. Demir, and A. Övgün, Black Hole Shadow in Symmergent Gravity, Physics of the Dark Universe 34, 100900 (2021).
  * [65] S. Haroon, K. Jusufi and M. Jamil, Shadow Images of a Rotating Dyonic Black Hole with a Global Monopole Surrounded by Perfect Fluid, Universe 6 (2),23 (2020).
  * [66] M. Okyay and A. Övgün, Nonlinear electrodynamics effects on the black hole shadow, deflection angle, quasinormal modes and greybody factors, JCAP 01, 009 (2022).
  * [67] A. Belhaj and Y. Sekhmani, Shadows of rotating quintessential black holes in Einstein–Gauss–Bonnet gravity with a cloud of strings, Gen. Relativ Gravit. 54 (2021).
  * [68] R. Roy, S. Vagnozzi and L. Visinelli, Superradiance evolution of black hole shadows revisited, Phys. Rev. D 105, 083002 (2022).
  * [69] S. Vagnozzi, C. Bambi and L. Visinelli, Concerns regarding the use of black hole shadows as standard rulers, Class. Quantum Grav. 37, 087001 (2020).
  * [70] K. Jusufi et al., Black hole surrounded by a dark matter halo in the M87 galactic center and its identification with shadow images, Phys. Rev. D 100, 044012 (2019).
  * [71] S. Vagnozzi et al., Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A*, Class. Quantum Grav. 40, 165007 (2023).
  * [72] K. Jusufi, Quasinormal Modes of Black Holes Surrounded by Dark Matter and Their Connection with the Shadow Radius, Phys. Rev. D 101, 084055 (2020).
  * [73] E. Berti, V. Cardoso, and C. M. Will, On gravitational- wave spectroscopy of massive black holes with the space interferometer LISA, Phys. Rev. D 73, 064030 (2006).
  * [74] G. Franciolini, L. Hui, R. Penco, L. Santoni and E. Trincherini, Effective Field Theory of Black Hole Quasinormal Modes in Scalar-Tensor Theories, JHEP 02, 127 (2019).
  * [75] A. Ishibashi and H. Kodama, Stability of higher dimensional Schwarzschild black holes, Prog. Theor. Phys 110, 901 (2003).
  * [76] D. J. Gogoi and U. D. Goswami, Quasinormal Modes and Hawking Radiation Sparsity of GUP corrected Black Holes in Bumblebee Gravity with Topological Defects, JCAP 06, 029 (2022).
  * [77] R. A. Konoplya, Quasinormal behavior of the D-dimensional schwarzschild black hole and the higher order WKB approach, Phys. Rev. D 68, 024018 (2003).
  * [78] X. Zhang, M. Wang and J. Jing, Quasinormal modes and late time tails of perturbation ﬁelds on a schwarzschild-like black hole with a global monopole in the einstein-bumblebee theory, Sci. China Phys. Mech. Astron. 66 100411 (2023).
  * [79] R. A. Konoplya, A. Zhidenko and A. F. Zihhailo, Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations, Class. Quantum Grav. 36, 155002 (2019).
  * [80] R. Karmakar and U. D. Goswami, Quasinormal modes, temperatures and greybody factors of black holes in a generalized Rastall gravity theory, Phys. Scr. 99, 055003 (2024).
  * [81] S. W. Hawking, Particle creation by black holes, Commun. Math. 43, 199 (1975).
  * [82] S. W. Hawking, D.N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. 87, 577 (1983).
  * [83] J. M. Bardeen, B. Carter, S. W. Hawking, The four laws of black hole mechanics, Commun. Math. 31, 161 (1973).
  * [84] M. A. Anacleto, F. A. Brito, J. A. V. Campos and E. Passos, Quantum-corrected scattering and absorption of a Schwarzschild black hole with GUP, Phys. Lett. B 810, 135830 (2020).
  * [85] M. A. Anacleto, F. A. Brito, S. J. S. Ferreira and E. Passos, Absorption and scattering of a black hole with a global monopole in $f(R)$ gravity, Phys. Lett. B 788, 231-237 (2019).
  * [86] H. Su and C. -Y. Long, Thermodynamics of the black holes under the extended generalized uncertainty principle with linear terms, Commun. Theor. Phys. 74, 055401 (2022).
  * [87] J. Ji-liang, Thermodynamics of the black holes under the extended generalized uncertainty principle with linear terms, Chinese Physics Letters 14, 81 (1997).
  * [88] R. T. Hough, A. Abebe and S. E. S. Ferreira, Viability tests of f(R)-gravity models with Supernovae Type 1A data, EPJC 80, 787 (2020).
  * [89] M. Martinelli and A. Melchiorri,  Cosmological constraints on the Hu-Sawicki modified gravity scenerio, Phys. Rev. D 79, 123516 (2009).
  * [90] J. Q. Guo, SOLAR SYSTEM TESTS OF f(R) GRAVITY, IJMPD 23, 1450036 (2014).
  * [91] S. Nojiri, S. D. Odinstov and V. K. Oikonomou, Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution, Physics Reports 692, 1-104 (2017).
  * [92] .F. Cardone, S. Camera and A. Diaferio, An updated analysis of two classes of f(R) theories of gravity, JCAP 2012, 030 (2012).
  * [93] S. Dey and S. Chakrabarty, A note on electromagnetic and gravitational perturbations of the Bardeen de Sitter black hole: quasinormal modes and greybody factors, EPJC 79, 504 (2019).
  * [94] T. Johannsen et al., Testing General Relativity with the Shadow Size of SGR A*, Phys. Rev. Lett. 116, 031101 (2016).
  * [95] D. Psaltis, Testing General Relativity with the Event Horizon Telescope, Gen. Relativ. Gravit.  51, 137 (2019).
  * [96] P. Kocherlakota et al., (Event Horizon Telescope), Phys. Rev. D 103, 104047 (2021).