# Constraints on Quasinormal modes from Black Hole Shadows in regular non-
minimal Einstein Yang-Mills Gravity

Dhruba Jyoti Gogoi  0000-0002-4776-8506<EMAIL_ADDRESS>Department of
Physics, Moran College, Moranhat, Charaideo 785670, Assam, India  Theoretical
Physics Division, Centre for Atmospheric Studies, Dibrugarh University,
Dibrugarh 786004, Assam, India  Supakchai Ponglertsakul<EMAIL_ADDRESS>Strong Gravity Group, Department of Physics, Faculty of Science, Silpakorn
University, Nakhon Pathom 73000, Thailand

###### Abstract

This work deals with the scalar quasinormal modes using higher order WKB
method and black hole shadow in non-minimal Einstein Yang-Mills theory. To
validate the results of quasinormal modes, time domain profiles are also
investigated. We found that with an increase in the magnetic charge of the
black hole, the ring-down gravitational wave increases non-linearly and
damping rate decreases non-linearly. The presence of magnetic charge also
results in a decrease in the black hole shadow non-linearly. It is found that
for large values of the coupling parameter, the black hole changes to a
solitonic solution and the corresponding ring-down gravitational wave
frequency increases slowly with a decrease in the damping rate. For the
solitonic solutions, the shadow is also smaller. The constraints on the model
parameters calculated using shadow observations of M87* and Sgr A* and an
approximate analytic relation between quasinormal modes and shadow at the
eikonal limit is discussed.

Quasinormal modes; Yang-Mills gravity; Black hole shadow; Time domain profile

##  I Introduction

Black holes are one of the most fundamental objects in the universe. They
originate from collapsing stars and are governed by Einstein’s theory of
general relativity (GR). Regardless of complex interior structure of dying
stars, once formed, black holes become one of the simplest celestial objects.
According to the no-hair theorem, black holes are uniquely described by three
physical parameters i.e, mass, angular momentum and electric charge. However,
there are several scenarios where the no-hair theorem can be evaded. For
instance, in the Einstein-Yang-Mills (EYM) theory and its variants, several
numerical black holes with Yang-Mills hair are studied [1, *Aichelburg:1992st,
*Donets:1992zb, *Kleihaus:1997rb, *Winstanley:1998sn, *PhysRevD.93.064064,
*PhysRevD.47.2242, *Ponglertsakul:2016fxj]. When asymptotic structure of
spacetime is modified to boxlike boundary, various hairy black holes are found
[9, *Ponglertsakul:2016wae, *Ponglertsakul:2016anb, *PhysRevLett.116.141101,
*PhysRevLett.116.141102, *Sanchis-Gual:2016tcm, *Basu:2016srp, *Peng:2017squ,
*Dias:2018yey].

Moreover, there has been much interest in gravitational theories coupled with
nonlinear electrodynamics sources. Such marriage leads to a novel type of
black hole with event horizon but possesses no essential singularity or
regular black hole [18, *PhysRevD.70.047504, *PhysRevD.90.124045, *Ma:2015gpa,
*Bronnikov:2000vy, *Fan:2016hvf]. Regular black holes are also studied in
modified theories of gravity, for instance, $f(R)$ gravity [24, 25], Gauss-
Bonnet gravity [25, 26, 27]. An exact spherically symmetric Wu-Yang monopole
solution is derived and studied [28, 29, 30]. This is a non-minimally extended
of Einstein-Yang-Mills equation with $SU(2)$ Wu-Yang ansatz. Later, this
regular magnetic black hole is generalized to include a cosmological constant
[31].

As predicted by GR, any accelerated massive objects create disturbances in
spacetime around them. The disturbances propagate through spacetime in the
form of gravitational waves (GWs). These waves contain crucial information
about their astrophysical sources. When two black holes collide and merge into
a single massive black hole, during this process GW is emitted continuously.
After the merger, a single black hole emits GWs signal which is prominently
dominated by a ringdown mode. The black hole ringdown is generally described
by quasinormal modes (QNMs) [32, 33, 34, 35]. An associated complex frequency
is determined by black hole’s mass and angular momentum. The real component of
the frequency is the emission frequency while the imaginary component denotes
exponential decay. Thus, studying QNMs of black holes is crucial since they
allow us to attain various characteristics of black holes.

Remarkably, the detection of GWs [36] in 2015 has opened a new window to
testing theories of gravity. Vast number of works have been devoted to study
the QNMs of various black holes. QNMs of nonminimally coupled scalar field in
three dimensional spacetime are explored in [37]. In addition, QNMs of
nonminimally coupled scalar field in a five dimensional Einstein-Power-Maxwell
background are studied via the Wentzel–Kramers–Brillouin (WKB) method and
pseudospectral Chebyshev method [38]. Optical properties and QNMs of black
holes dependence on the Yang-Mill charges are investigated in Einstein-Power-
Yang-Mills theory [39]. QNMs of black holes within the framework of
generalized uncertainty principle (GUP) are analyzed [40, 41, 42, 43]. Due to
variety of black hole solutions in modified theories of gravity, there are
many works considering QNMs of black holes in various models e.g., $f(Q)$
gravity [44], $f(R,T)$ gravity [45], Horndeski gravity [46, 47], Rastall
gravity [48, 49, 50] and de Rham-Gabadadze-Tolley (dRGT) massive gravity [51,
52, 53, 54, 55]. Very recently, QNMs of black hole in emergent gravity
framework are computed via the WKB and asymptotic iteration methods [56].

In recent times, the visual representation of black holes has garnered
significant attention, driven by observations of celestial objects like M87⋆
[57, 58, 59, 60, 61, 62] and Sgr A⋆ [63]. These observations have sparked
interest in the realms of both GR and modified gravity (MOG), offering an
intriguing opportunity to depict and analyze hypothetical images associated
with these observed phenomena. This avenue allows us to engage in a geometric
and topological exploration, comparing the theoretical depiction of black hole
images with the actual images of M87⋆ and Sgr A⋆. Consequently, the
visualization of black holes assumes a captivating optical dimension,
particularly as we endeavour to uncover the relationships between optical
features, such as the shadows they cast, through the application of
thermodynamics [64, 65] and quasi-normal modes [66, 67]. Additionally, when
considering the shape data, the shadow images can take the form of either
distorted geometries on the image plane [68, 69, 70, 71, 72] or a pattern of
concentric circles, contingent on the presence of a rotation parameter within
the spacetime. To gain a deeper understanding of these shadow behaviours, the
Hamilton-Jacobi formalism [73] emerges as a compatible analytical framework.
It is based on the concept that massless photons in the vicinity of a black
hole generate specific orbits around the region, a phenomenon referred to as
the geodesic null background. In the realm of black hole imaging, ongoing
research aims to provide a comparative analysis of observed images, such as
those of M87⋆ and Sgr A⋆, in terms of their size and shape.

Motivated by these studies, in this work, we aim to investigate the QNMs and
black hole shadows in the framework of regular non-minimally EYM theory in
four-dimensional spacetime. We also introduce constraints on the model
parameters from black hole shadow observations and try to relate the shadow
behaviour with the QNMs of the black hole spacetime.

This paper is organized as follows. In Section II, we discuss the EYM theory
and the associated black hole solution in brief. In Section III, massless
scalar perturbations on the black hole spacetime have been investigated and
the QNMs spectrum is analysed. Section IV deals with the evolution of scalar
perturbation. In Section V, we investigate the black hole shadow. In Section
VI, we discuss how QNMs are linked with shadows of the black hole. Finally in
Section VII, we provide a concluding remark of our work.

Throughout the manuscript, we consider $c=\hbar=8\pi G=1$.

##  II Regular non-minimal Einstein Yang-Mills Black Hole

In this section, we shall discuss the theory and the black hole solution in
brief. The associated action of the $4$-D regular non-minimally EYM theory is
[28, 31]

$S=\frac{1}{8\pi}\int
d^{4}x\sqrt{-g}\left[R+\frac{1}{2}F_{\mu\nu}^{\left(a\right)}F^{\mu\nu{\left(a\right)}}+\frac{1}{2}\bar{R}^{\alpha\beta\mu\nu}F_{\alpha\beta}^{\left(a\right)}F_{\mu\nu}^{\left(a\right)}\right],$
(1)

where $R$ stands for the Ricci curvature scalar. The indices labeled with
Greek letters range from 0 to 3, while those labeled with Latin letters range
from 1 to 3. Additionally, the notation $F_{\mu\nu}^{\left(a\right)}$
represents the Yang-Mills (YM) tensor and is connected to the YM potential
$A_{\mu}^{\left(a\right)}$ through the equation:

$F_{\mu\nu}^{\left(a\right)}=\nabla_{\mu}A_{\nu}^{\left(a\right)}-\nabla_{\nu}A_{\mu}^{\left(a\right)}+f_{\left(b\right)\left(c\right)}^{\left(a\right)}A_{\mu}^{\left(b\right)}A_{\nu}^{\left(c\right)},$
(2)

where $\nabla{\mu}$ represents the covariant derivative and the symbols
$f{\left(b\right)\left(c\right)}^{\left(a\right)}$ denote the real structure
constants of the three-parameter YM gauge group $SU\left(2\right)$. The tensor
$\bar{R}^{\alpha\beta\mu\nu}$ is defined as [31]

$\displaystyle\bar{R}^{\alpha\beta\mu\nu}$
$\displaystyle=\frac{\xi_{1}}{2}R\left(g^{\alpha\mu}g^{\beta\nu}-g^{\alpha\nu}g^{\beta\mu}\right)$
$\displaystyle~{}~{}~{}~{}~{}+\frac{\xi_{2}}{2}\left(R^{\alpha\mu}g^{\beta\nu}-R^{\alpha\nu}g^{\beta\mu}+R^{\beta\nu}g^{\alpha\mu}-R^{\beta\mu}g^{\alpha\nu}\right)$
$\displaystyle~{}~{}~{}~{}~{}+\xi_{3}R^{\alpha\beta\mu\nu}.$ (3)

Here, $R^{\alpha\beta}$ and $R^{\alpha\beta\mu\nu}$ represent the Ricci and
Riemann tensors, respectively, and $\xi_{i}$ $(i=1,2,3)$ denote the non-
minimal coupling parameters between the YM field and the gravitational field.
Assuming the gauge field is described by the Wu-Yang ansatz and taking
$\xi_{1}=\xi,\xi_{2}=4\xi,\xi_{3}=-6\xi$ with $\xi>0$, a regular, static, and
spherically symmetric black hole solution was discovered [28, 31]. This black
hole solution is described by the metric [28, 31],

$ds^{2}=-f\left(r\right)dt^{2}+f^{-1}(r)dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (4)

where the metric function has the following form:

$f\left(r\right)=1+\left(\frac{r^{4}}{r^{4}+2\xi
Q^{2}}\right)\left(\frac{Q^{2}}{r^{2}}-\frac{2M}{r}\right),$ (5)

with $M$ representing the mass of the black hole and $Q$ being the magnetic
charge.

The electromagnetic field four-potential for the regular non-minimal magnetic
black hole is given by:

$A_{\nu}=\left(0,0,0,Q\cos\theta\right).$ (6)

It’s evident that when $Q=0$, the metric (4) simplifies to the standard
Schwarzschild black hole metric, while when $\xi=0$, it corresponds to the
Reissner-Nordström black hole metric but with a magnetic charge instead of an
electric charge. The quantity of event horizons of the non-minimal EYM black
hole depends on both the non-minimal parameter and the magnetic charge,
potentially resulting in multiple horizons. In this work, we shall use the
black hole line element (4) to study the massless scalar QNMs and the black
hole shadow.

##  III Massless Scalar Quasinormal modes

In this section, we shall investigate the QNMs associated with massless scalar
perturbation. In this case, we shall assume that the test field has negligible
impact or influence on the black hole spacetime defined in the previous
section and the associated scalar field is massless in nature. We obtain a
Klein-Gordon-type equation associated with the QNMs while considering the
pertinent conservation relations associated with the spacetime under
consideration. We shall implement Padé averaged 6th order WKB approximation
method to obtain the QNMs numerically.

Taking into consideration of axial perturbation only, it is possible to
express the perturbed metric in the following form [74, 44]:

$\displaystyle ds^{2}=$
$\displaystyle-\,|g_{tt}|\,dt^{2}+r^{2}\sin^{2}\\!\theta\,(d\phi-
p_{1}(t,r,\theta)\,dt$ $\displaystyle-p_{2}(t,r,\theta)\,dr-
p_{3}(t,r,\theta)\,d\theta)^{2}+g_{rr}\,dr^{2}$
$\displaystyle+r^{2}d\theta^{2}.$ (7)

In this context, the parameters $p_{1}$, $p_{2}$, and $p_{3}$ are intricately
linked to the perturbation occurring in the spacetime surrounding the black
hole. These parameters play a crucial role in characterizing the modifications
induced by the perturbation. Specifically, they contribute to the altered
dynamics and structure of the black hole spacetime.

Now, for the massless scalar field, as it is considered that the effect of the
field on the spacetime is minimal, the perturbed metric Eq. (III) takes the
following form:

$ds^{2}=-\,|g_{tt}|\,dt^{2}+g_{rr}\,dr^{2}+r^{2}d\Omega^{2}.$ (8)

The Klein-Gordon equation in curved spacetime in this case is written as

$\square\Phi=\dfrac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi)=0.$
(9)

Eq. (9) describes the QNMs associated with the scalar perturbation in the
considered black hole spacetime.

Utilising spherical harmonics, it is possible to express the wave function of
the scalar field in the following decomposed form:

$\Phi(t,r,\theta,\phi)=\sum_{l,m}\dfrac{\psi_{l}(r)}{r}Y_{lm}(\theta,\phi)e^{-i\omega
t}.$ (10)

In the above equation, $Y_{lm}(\theta,\phi)$ is the spherical harmonics part
of the wave function and the parameters $l$ and $m$ are the associated
parameters of spherical harmonics. Now, one can use this decomposed form of
the wave function in the Eq. (10) to obtained a reduced wave equation given by

$\partial^{2}_{r_{*}}\psi_{l}(r_{*})+\omega^{2}\psi_{l}(r_{*})=V_{s}(r)\psi_{l}(r_{*}).$
(11)

This is a standard differential equation where the differentiation is done
with respect to the tortoise coordinate $r_{*}$. The tortoise coordinate is
connected with the normal coordinate $r$ by the following definition:

$\dfrac{dr_{*}}{dr}=\sqrt{g_{rr}\,|g_{tt}^{-1}|}.$ (12)

In the Eq. (11), parameter $V_{s}(r)$ is the effective potential for the
massless scalar perturbation and it is given by:

$V_{s}(r)=|g_{tt}|\left(\dfrac{l(l+1)}{r^{2}}+\dfrac{1}{r\sqrt{|g_{tt}|g_{rr}}}\dfrac{d}{dr}\sqrt{|g_{tt}|g_{rr}^{-1}}\right).$
(13)

In this context, the symbol $l$ is employed to signify the multipole moment
associated with the QNMs of the black hole. The variation of the scalar
potential $V_{s}(r)$ with respect to the multipole moment $l$ is shown in Fig.
1. The overall effective potential increases with $\ell$.

Figure 1: Variation scalar potential with respect to $r$ using $M=1$, $Q=0.3$
and $\xi=0.1$.

Figure 2: Variation scalar potential with respect to $r$ using $M=1$ and
$l=3$. On the first panel, we have used $\xi=0.1$ and on the second $Q=0.7$.
For $\xi=0.9$ and $\xi=1.3$, these configurations are not black holes but
represent solitonic solutions.

The other model parameters $Q$ and $\xi$ also have noticeable impacts on the
potential $V_{s}(r)$ which are depicted in Fig. 2. It is seen that with an
increase in the black hole charge $Q$, the peak value of the potential
increases non-linearly and the corresponding $r$ shifts towards the black hole
event horizon. Similarly, an increase in the value of $\xi$ also increases the
peak value of the potential. However, the variation is comparatively smaller
at the peak position and the peak value seems to change uniformly with respect
to $\xi$. This suggests that the model parameter $Q$ might have a non-linear
impact on the spectrum of QNMs and $\xi$ might have an almost linear impact on
QNMs. The details will be investigated in the next part of the study.

###  III.1 The Padé averaged WKB approximation method

In this study, we implement a well-known method called WKB approximation
method to calculate the QNMs numerically for the massless scalar perturbation
in the above-mentioned black hole spacetime. We consider Padé averaged 6th-
order corrections to the WKB method [75, 76, 77, 78] to obtain QNMs with
higher accuracy.

In higher-order WKB approximation method, the oscillation frequency $\omega$
of ring-down GWs is expressed as:

$\omega=\sqrt{-\,i\left[(n+1/2)+\sum_{k=2}^{6}\bar{\Lambda}_{k}\right]\sqrt{-2V_{0}^{\prime\prime}}+V_{0}}.$
(14)

Here, the parameter $n$, representing the overtone number, takes on discrete
values starting from zero $(n=0,1,2,\ldots)$. Additionally, $V_{0}$ represents
the value of the potential function $V(r)$ at a specific radial coordinate
$r_{\text{max}}$, while $V_{0}^{\prime\prime}$ signifies the second derivative
of $V(r)$ with respect to $r$ evaluated at the same position. The pivotal
location $r_{\text{max}}$ corresponds to where the potential function achieves
its maximum value, rendering this radial coordinate particularly significant.

Integral to our computational framework is the correction terms
$\bar{\Lambda}_{k}$, which play a crucial role in refining the precision of
our calculations. These correction terms are instrumental in capturing subtle
nuances in the determination of the oscillation frequency, contributing to the
overall accuracy of the framework.

To elaborate on the procedural aspects of our approach, we draw upon
established references in the field. Specifically, Ref.s [75, 76, 77, 78]
provide comprehensive insights into the mathematical formulations of the
correction terms and elaborate on the intricacies of the Padé averaging method
applied in our study.

By incorporating these computational techniques, our research aims to uncover
the nuanced behaviour of ring-down GWs, making significant contributions to
the broader understanding of astrophysical phenomena in such types of theories
of gravity.

Table 1: QNMs with different values of multipole moment $l$ calculated using $6$th order Padé averaged WKB approximation method. In this table we have chosen $n=0,M=1,Q=0.3$ and $\xi=0.1$.  $l$ | Padé averaged WKB | $\Delta_{rms}$ | $\Delta_{6}$
---|---|---|---
$1$ | $0.297616-0.098004i$ | $1.18328\times 10^{-9}$ | $0.0000209617$
$2$ | $0.491302-0.097122i$ | $3.04799\times 10^{-7}$ | $2.38347\times 10^{-6}$
$3$ | $0.686034-0.096870i$ | $4.63924\times 10^{-8}$ | $4.55756\times 10^{-7}$
$4$ | $0.881104-0.096766i$ | $1.04331\times 10^{-8}$ | $1.20843\times 10^{-7}$
$5$ | $1.076325-0.096713i$ | $3.41801\times 10^{-9}$ | $4.44961\times 10^{-8}$
$6$ | $1.271628-0.096682i$ | $1.36926\times 10^{-9}$ | $1.88337\times 10^{-8}$
$7$ | $1.466978-0.096663i$ | $6.75515\times 10^{-10}$ | $9.20866\times 10^{-9}$

Table 1 showcases the QNMs across a range of multipole moment values, with a
specific focus on instances where the overtone number is set to $n=0$. The
second column of the table delineates the QNMs obtained through the
application of the 6th-order Padé averaged WKB approximation method. Within
this tabular representation, the term $\Delta_{rms}$ assumes significance as
it quantifies the root mean square error associated with the 6th-order Padé
averaged WKB approximation.

Moreover, the term $\Delta_{6}$ is introduced to gauge the error between two
adjacent order approximations. This error term is computed by assessing the
absolute difference between $\omega_{7}$, representing QNMs computed using the
Padé averaged 7th order WKB approximation method, and $\omega_{5}$, denoting
QNMs obtained from the Padé averaged 5th order WKB approximation method.
Mathematically, $\Delta_{6}$ is expressed as follows:

$\Delta_{6}=\dfrac{|\omega_{7}-\omega_{5}|}{2}.$ (15)

A notable trend within the table is the observed reduction in error associated
with QNMs as the multipole moment $l$ increases. This behaviour aligns with
the characteristic limitations of the WKB approximation method, particularly
evident when the overtone number $n$ surpasses the multipole moment $l$. The
diminishing error with increasing multipole moment is a distinctive feature of
the WKB method, signifying its challenges in providing accurate results when
overtone numbers exceed the corresponding multipole moment values [41, 79, 44,
80, 81, 82, 43]. By keeping this fact in mind, in this investigation, we have
considered $n<l$ to calculate QNMs.

Figure 3: Variation of real and imaginary QNMs using $M=1$, $l=3$ and
$\xi=0.1$.

Figure 4: Variation of real and imaginary QNMs using $M=1$, $l=3$ and $Q=0.3$.

We have shown the variation of the massless scalar QNMs in Fig. 3 and 4 with
respect to $Q$ and $\xi$, respectively. In Fig. 3, one can see that with an
increase in the charge of the black hole $Q$, the real QNMs increase non-
linearly. On the other hand, the damping rate or decay rate of ring-down GWs
increases very slowly initially up to around $Q=0.6$ and beyond this the
damping rate decreases drastically showing a highly non-linear pattern.

In Fig. 4, the initial panel intricately illustrates the nuanced shift in real
quasinormal frequencies concerning the parameter $\xi$, while the second panel
meticulously delineates the corresponding alteration in the damping rate of
ring-down GWs with respect to $\xi$. Evidently, as the parameter $\xi$
experiences an incremental surge, the real quasinormal frequencies manifest a
discernible linear escalation. In contrast, a rise in the value of $\xi$
induces a linear reduction in the damping rate. It is noteworthy, however,
that this variation is comparatively modest when juxtaposed with the scenario
involving the parameter $Q$.

Consequently, when contemplated through the prism of QNMs, the influences
exerted by $Q$ and $\xi$ exhibit notable disparities. Looking forward, as we
anticipate a wealth of data from state-of-the-art GW detectors such as Laser
Interferometer Space Antenna (LISA), there emerges the exciting prospect of
establishing stringent constraints on the Yang-Mills field by harnessing
insights derived from the meticulous examination of QNMs.

##  IV Time Evolution of Scalar Perturbations

In the previous section, we conducted numerical computations to unravel the
characteristics of QNMs and scrutinized how they hinge on the model
parameters. Shifting our attention to the subsequent section, our exploration
now centers on the temporal behaviours of massless scalar perturbations. To
elucidate these profiles, we adopt the time domain integration framework
delineated by Gundlach et al. [83, 84]

To progress further, we denote $\psi(r^{*},t)=\psi(i\Delta r^{*},j\Delta
t)=\psi_{i,j}$, and $V(r(r^{*}))=V_{i}$. Here $r^{*}$ represents the tortoise
coordinate and $t$ denotes time. Subsequently, we express the scalar wave
equation (9) as:

$\displaystyle\dfrac{\psi_{i+1,j}-2\psi_{i,j}+\psi_{i-1,j}}{\Delta
r_{*}^{2}}-$
$\displaystyle\dfrac{\psi_{i,j+1}-2\psi_{i,j}+\psi_{i,j-1}}{\Delta t^{2}}$
$\displaystyle-V_{i}\psi_{i,j}=0.$ (16)

The initial conditions are specified as
$\psi(r_{*},t)=\exp\left[-\dfrac{(r_{*}-k_{1})^{2}}{2\sigma^{2}}\right]$ and
$\psi(r_{*},t)|_{t<0}=0$, where $k_{1}$ and $\sigma$ represent the median and
width of the initial wave-packet, respectively. Employing these initial
conditions, we compute the time evolution associated with the scalar
perturbation using the iterative scheme:

$\psi_{i,j+1}=-\,\psi_{i,j-1}+\left(\dfrac{\Delta t}{\Delta
r_{*}}\right)^{2}(\psi_{i+1,j+\psi_{i-1,j}})+\left(2-2\left(\dfrac{\Delta
t}{\Delta r_{*}}\right)^{2}-V_{i}\Delta t^{2}\right)\psi_{i,j}.$ (17)

We implement this iterative scheme to obtain the time domain profiles by
keeping a fixed value of $\frac{\Delta t}{\Delta r_{*}}$. However, it is
important to mention that one must consider $\frac{\Delta t}{\Delta r_{*}}<1$
to satisfy the Von Neumann stability condition throughout the numerical
procedure. Throughout our numerical calculations, we have considered this
ratio around $0.7$ to obtain the time evolution of the massless scalar
perturbations on the black hole spacetime. We have set $k_{1}=0$ and
$\sigma=5$ for the numerical calculations of time domain profiles.

Figure 5: Time domain profiles of scalar perturbation with $M=1$. On the first
panel, $Q=0.3$ and $\xi=0.5$; on the second panel $l=1$ and $Q=0.3$ and on the
third panel $l=1$ and $\xi=0.5$.

In Fig. 5, we have shown the time domain profiles in the units of $M$. One may
observe that the variation of time domain profiles with respect to multiple
moment $l$ stands in agreement with our Table 1. From the second and third
panel, it is evident that the impact of the coupling parameter $\xi$ on the
QNMs spectrum is very small in comparison to the charge parameter $Q$ as
predicted earlier. These results are also in agreement with the results
obtained from Padé averaged WKB approximation method.

##  V Optical Behaviour of the Black hole: Shadow

The investigation into supermassive black holes (SMBHs) holds significant
importance in astrophysics, given their mysterious nature and their central
positions in galaxies, where they exert exceptional gravitational forces with
event horizons preventing light escape. Recent advances in observational
techniques, exemplified by the Event Horizon Telescope (EHT), have offered
unprecedented insights into these colossal cosmic entities.

In this part, we explore the mysterious concept of black hole shadows,
uncovering the hidden areas within the complex structure of space and time
that cover the edge of these cosmic objects. Black holes have incredibly
strong gravity that forces even light to be pulled in when it crosses the
boundary called the event horizon [85, 86]. Consequently, the outer boundary
of a black hole manifests as an indistinct yet solemn circle, projecting its
inscrutable silhouette against the surrounding cosmic tableau of matter or
luminous radiance. The dimensions and intricate contours of this umbral figure
furnish profound insights into the fundamental characteristics of black holes
and the essence of gravity itself. Recent advancements in both astronomical
observation and technological capabilities have conferred upon us the capacity
to capture and visually scrutinize these elusive black hole shadows,
representing a paradigmatic leap in our pursuit to comprehend these enigmatic
celestial phenomena [87]. As we peer into these cosmic shadows, we not only
unveil the visual facets of black holes but also unlock new strata of
scientific comprehension, shedding light on the intricacies veiled within the
expansive cosmic tapestry. These advancements herald a gateway to a deeper
understanding of the universe’s intricacies, underscoring the remarkable
synergy between cutting-edge technology and our relentless scientific
inquisitiveness about the cosmos. Such a study will extensively contribute
towards our understanding of different modified gravity theories and the
validity of GR in light of current observational aspects.

The Euler-Lagrange equation is given by

$\frac{d}{d\tau}\\!\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)-\frac{\partial\mathcal{L}}{\partial
x^{\mu}}=0,$ (18)

where the Lagrangian is expressed as

$\mathcal{L}(x,\dot{x})=\frac{1}{2}\,g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}.$
(19)

For static and spherically symmetric black hole, the Lagrangian becomes

$\mathcal{L}(x,\dot{x})=\frac{1}{2}\left[-f(r)\,\dot{t}^{2}+\frac{1}{f(r)}\,\dot{r}^{2}+r^{2}\left(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\right)\right].$
(20)

Here the dot over the variables denotes the derivative with respect to the
proper time $\tau$.

Choosing the equatorial plane where $\theta=\pi/2$, the conserved energy
$\mathcal{E}$ and angular momentum $L$ can be computed by utilizing the
killing vectors $\partial/\partial\tau$ and $\partial/\partial\phi$ as given
by

$\mathcal{E}=f(r)\,\dot{t},\quad L=r^{2}\dot{\phi}.$ (21)

In the case of photon, we can write the geodesic equation as,

$-f(r)\,\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}\,+r^{2}\dot{\phi}^{2}=0.$ (22)

Utilizing Eq. (22) in conjunction with the conserved quantities, namely,
$\mathcal{E}$ and $L$, the orbital equation for a photon can be derived as.
[87]

$\left(\frac{dr}{d\phi}\right)^{2}=V_{eff},$ (23)

where $V_{eff}$ is defined as

$V_{eff}=r^{4}\left[\frac{\mathcal{E}^{2}}{L^{2}}-\frac{f(r)}{r^{2}}\right].$
(24)

Expressing Eq. (23) in radial form yields,

$V_{r}(r)=\dfrac{1}{\zeta^{2}}-\dot{r}^{2}/L^{2}.$ (25)

In this expression, $\zeta$ represents the impact parameter and is defined as
$\zeta=L/\mathcal{E}$. Additionally, the reduced potential $V_{r}(r)$ can be
expressed as:

$V_{r}(r)=\frac{f(r)}{r^{2}}.$ (26)

To scrutinize the shadows of black holes, our focus is directed towards a
specific point along the trajectory denoted as $r_{ph}$. This particular point
corresponds to the turning point, signifying the location of the light ring
that encircles the black hole. It can also be interpreted as the radius of the
photon sphere, holding significant relevance in the analysis of the black
hole’s shadow characteristics. At this pivotal turning point of a black hole
[88, 85, 89, 90, 91, 92, 87],

$\left.V_{eff}\right|_{r_{ph}}\\!\\!\\!\\!=0,\;\;\text{and}\;\;\left.V_{eff}\right|_{r_{ph}}\\!\\!=0.$
(27)

It is feasible to determine the impact parameter $\zeta$ at the turning point
by

$\frac{1}{\zeta_{crit}^{2}}=\frac{f(r_{ph})}{r_{ph}^{2}}.$ (28)

Therefore, the radius of the photon sphere, denoted as $r_{ph}$, can be
determined using[87]:

$\left.\frac{d}{dr}\,\mathcal{A}(r)\right|_{r_{ph}}\\!\\!\\!\\!\\!\\!\\!=0.$
(29)

This equation can be reformulated as

$\frac{f^{\prime}(r_{ph})}{f(r_{ph})}-\frac{h^{\prime}(r_{ph})}{h(r_{ph})}=0,$
(30)

here we have $\mathcal{A}(r)=h(r)/f(r)$ with $h(r)=r^{2}$.

Now, to derive the shadow of the black hole, we express Eq. (23) using Eq.
(28) in terms of the function $\mathcal{A}(r)$ as

$\left(\frac{dr}{d\phi}\right)^{\\!2}=h(r)f(r)\left(\frac{\mathcal{A}(r)}{\mathcal{A}(r_{ph})}-1\right).$
(31)

Utilizing Eq. (31), one can calculate the shadow radius of the black hole. In
the scenario where a static observer is positioned at a distance $r_{0}$ from
the black hole, we can ascertain the angle $\alpha$ between the light rays
originating from the observer and the radial direction of the photon sphere.
This angle can be computed as [87]

$\cot\alpha=\frac{1}{\sqrt{f(r)h(r)}}\left.\frac{dr}{d\phi}\right|_{r\,=\,r_{0}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!.$
(32)

In conjunction with Eq. (31), the aforementioned equation can be represented
as

$\cot^{2}\\!\alpha=\frac{\mathcal{A}(r_{0})}{\mathcal{A}(r_{ph})}-1.$ (33)

Once more, the above equation can be reformulated using the relation
$\sin^{2}\\!\alpha=1/(1+\cot^{2}\\!\alpha)$ as

$\sin^{2}\\!\alpha=\frac{\mathcal{A}(r_{ph})}{\mathcal{A}(r_{0})}.$ (34)

By substituting the expression for $\mathcal{A}(r_{ph})$ from Eq. (28) and
using $\mathcal{A}(r_{0})=r_{0}^{2}/f(r_{0})$, the shadow radius of the black
hole for a static observer at $r_{0}$ can be approximated as [87]

$R_{s}=r_{0}\sin\alpha=\sqrt{\frac{r_{ph}^{2}f(r_{0})}{f\left(r_{ph}\right)}}.$
(35)

Once more, as $r_{0}\rightarrow\infty$, i.e., for a static observer at a large
distance, $f(r_{0})\rightarrow 1$. Consequently, for such an observer, the
shadow radius $R_{s}$ becomes,

$R_{s}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}.$ (36)

Finally, through the stereographic projection of the shadow from the black
hole’s plane to the observer’s image plane with coordinates $(X,Y)$, the
apparent form of the shadow can be determined. These coordinates are defined
as [87, 93, 42]

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

In this context, $\theta_{0}$ denotes the angular orientation of the observer
in relation to the plane of the black hole.

Figure 6: Stereographic projection of black hole shadow using $M=1$. On the
first panel, we have used $Q=0.3$ and on the second panel $\xi=0.5$.

Figure 7: Variation of $R_{s}^{2}$ as a function of black hole charge $Q$ (the
left panel) and non-minimal coupling constant $\xi$ (the right panel). We have
used $M=1$.

Figure 8: Constraint on black hole shadow radius from Sgr A*. On the left
panel, we have used $\xi=0.1$ and on the right panel, we have used $Q=0.35$.
The excluded (allowed) regions are denoted by the red (grey) shaded areas. The
black dotted horizontal line is the radius of the shadow of the Schwarzschild
black hole.

Figure 9: Constraint on black hole shadow radius from M87*. On the left panel, we have used $\xi=0.1$ and on the right panel, we have used $Q=0.35$. The excluded (allowed) regions are denoted by the red (grey) shaded areas. The black dotted horizontal line is the radius of the shadow of the Schwarzschild black hole. | constraint on $Q$ using $\xi=0.1$ | constraint on $\xi$ using $Q=0.35$
---|---|---
$1\sigma$(upper/lower) from M87* | none /$0.8912$ |  $-138.11$/none
$1\sigma$(upper/lower) from Sgr A* | none/$0.789$ |  $-8.695$/none
Table 2: Variation of shadow with $1\sigma$ ranges of $Q$ and $\xi$ based on
the shadow radii of Sgr. A* and M87* as depicted in Figs. 8 and 9.

The investigation presents stereographic projections of the black hole shadow
for the specific black hole under examination. Fig. 6 illustrates these
projections, offering visual depictions of the shadow’s characteristics
corresponding to various model parameters. In the initial panel of Fig. 6, the
stereographic projections of the black hole are displayed for different values
of the model parameters $Q$ and $\xi$. It is evident that an increase in the
model parameter $Q$ results in a decrease in the black hole shadow. The second
panel illustrates the black hole shadow for various values of the parameter
$\xi$, revealing a gradual decrease in the shadow with an increase in $\xi$.
For enhanced visualization, Fig. 7 presents the square of the black hole
shadow radius concerning the model parameters $Q$ and $\xi$. This
representation indicates that $\xi$ has a minimal impact on the black hole
shadow. Therefore, the investigation demonstrates that both model parameters
exert distinct influences on the black hole shadow.

A notable feature of SMBHs is the presence of a shadow cast by their event
horizons, a phenomenon arising from the gravitational bending of light. The
size and shape of this shadow furnish valuable information about the SMBH and
its surrounding environment. To comprehend and quantify the shadow’s
dimensions, the spatial separation $D$ between the SMBH and the galactic
center must be considered.

A conventional method for determining the classical diameter of the black hole
shadow involves the application of the arclength equation:

$d_{\text{sh}}=\frac{D\theta_{\text{sh}}}{M},$ (39)

where $d_{\text{sh}}$ denotes the shadow’s diameter, $\theta_{\text{sh}}$
represents the angular size of the shadow, and $M$ corresponds to a specific
unit of measurement.

For M87*, one of the extensively studied SMBHs, the measurement of the shadow
diameter has been realized through EHT observations [57, 63]. The reported
diameter of the M87* shadow is $d^{\text{M87*}}_{\text{sh}}=(11\pm 1.5)M$,
serving as a crucial data point for scrutinizing the properties of this SMBH.

However, it is imperative to consider the uncertainties associated with these
measurements for a more accurate comprehension of the physical parameters. The
determination of such uncertainties requires meticulous analysis, and in the
case of the M87* shadow diameter, Ref.s [94, 95] offer detailed insights into
the employed methodology.

This study aims to delve deeper into the implications of the measured M87* and
Sgr A* shadow diameters by incorporating uncertainties from the aforementioned
references. Our specific focus is on constraining the potential values of the
black hole charge $Q$ by calculating the $1\sigma$ limits based on results
obtained for M87* and Sgr A*. By considering these constraints, we aspire to
augment our understanding of the model behaviour and the model parameters from
the physical properties of M87* and Sgr A* and contribute to the broader
comprehension of SMBHs as well as possible constraints on QNMs from them.

In Figs. 8 and 9, we have shown the constraints on the model parameters $Q$
and $\xi$ from the observed shadow radii of Sgr A* and M87* by following Ref.s
[94, 95]. One can see that the observational data do not put a strong
constraint on the coupling parameter $\xi$. However, the charge parameter $Q$
is well constrained by the observed shadow radii of Sgr A* and M87*.

##  VI Connection between quasinormal modes and shadow of the black hole

In this section, we shall briefly discuss the approximate analytical
connection of QNMs with the black hole shadow radius. One may note that from
the numerical results of QNMs and black hole shadow, there is a possible
correspondence between them.

In this scenario, we shall consider only $3^{rd}$ order WKB expansion given by

$\displaystyle\omega=\Bigg{\\{}V+\frac{V_{4}}{8V_{2}}\left(\nu^{2}+\frac{1}{4}\right)-\left(\frac{7+60\nu^{2}}{288}\right)\frac{V_{3}^{2}}{V_{2}^{2}}+i\nu\sqrt{-2V_{2}}\left[\frac{1}{2V_{2}}\left[\frac{5V_{3}^{4}(77+188\nu^{2})}{6912V_{2}^{4}}\right.\right.$
$\displaystyle\left.\left.-\frac{V_{3}^{2}V_{4}(51+100\nu^{2})}{384V_{2}^{3}}+\frac{V_{4}^{2}(67+68\nu^{2})}{2304V_{2}^{2}}+\frac{V_{5}V_{3}(19+28\nu^{2})}{288V_{2}^{2}}+\frac{V_{6}(5+4\nu^{2})}{288V_{2}}\right]-1\right]\Bigg{\\}}^{1/2}_{r=r_{ph}}.$
(40)

One may note that even we consider higher order expansion such as $6$th order
or $12$th order expansion, in the eikonal limit, the first significant terms
are identical with those obtained by the $3$rd order WKB approximation method.
The term $V_{i}$ stands for the i-th derivative of the potential $V$. We also
considered $\nu=n+\frac{1}{2}$, where $n$ is the overtone number of QNMs.

By expanding (40), at the eikonal regime, one can obtain [39, 96]

$\displaystyle\omega=\omega_{R}-i\omega_{I}=\left[\ell\frac{\sqrt{f(r)}}{r}\Bigg{|}_{r=r_{ph}}+\frac{\sqrt{f(r)}}{2r}\Bigg{|}_{r=r_{ph}}+\mathcal{O}(\ell^{-1})\right]_{R}-$
$\displaystyle
i\left[\frac{n+1/2}{\sqrt{2}}\frac{\sqrt{f(r)}}{r}\Bigg{|}_{r=r_{ph}}\sqrt{6rf^{\prime}(r)-6f(r)-r^{2}f^{\prime\prime}(r)-r^{2}f(r)^{-1}{f^{\prime}(r)}^{2}}\Bigg{|}_{r=r_{ph}}+\mathcal{O}(\ell^{-1})\right]_{I}.$
(41)

Comparing the imaginary part

$\omega_{I}=\frac{n+1}{\sqrt{2}}R_{s}^{-1}\sqrt{2f(r_{ph})-r_{ph}^{2}f^{\prime\prime}(r_{ph})}+\mathcal{O}(\ell^{-1}),$
(42)

and the real part

$\omega_{R}=R_{s}^{-1}\left(\ell+\frac{1}{2}+\mathcal{O}(\ell^{-1})\right).$
(43)

Further exploring the constraints on the visual representation of a black hole
i.e. shadow can be achieved through the examination of observational data
concerning its QNMs. It is essential to emphasize the direct relationship
between the features of the black hole’s silhouette and the specific values of
its QNMs as this will help us to uncover different properties of the black
hole hairs. Consequently, the availability of significant observational data
from space-based gravitational wave detectors like LISA presents an avenue for
imposing stricter constraints on the parameters of models used to understand
black holes. Moreover, this process enables the assessment of consistency
between two distinct approaches-shadow analysis and QNMs-in testing the
underlying theoretical framework.

From the approximated relations (42) and (43), at the eikonal limit, we
observe that the black hole shadow has an impact on the behaviour of the QNMs
spectrum. Most importantly, an observational constraint on the black hole
shadow will also put a constraint on the real and imaginary parts of QNMs of
the black hole at the eikonal limit. However, as evident from the table 2,
such a constraint on the QNMs will be very weak at the current stage. In the
near future, observational results from LISA might put a very strong
constraint on QNMs which may be useful along with the observational results of
black hole shadow to check for the consistencies between them and to constrain
the theory more stringently.

##  VII Concluding Remarks

In this work, we considered a charged black hole solution in non-minimally
coupled EYM theory and studied the scalar perturbation on the black hole
spacetime and the associated QNMs. From the behaviour of the potential, we
found that for higher values of the coupling parameter $\xi$, the line element
represents solitonic solutions instead of a black hole solution.

In order to obtain the QNMs, we used Padé averaged WKB approximation method.
The analysis showed that black hole charge $Q$ has a more significant impact
on the quasinormal mode spectrum. With an increase in charge, the oscillation
frequency of ring-down GWs increases non-linearly. In the case of the damping
rate of GWs, we found that initially, with an increase in $Q$, the damping or
decay rate increases slowly and beyond some threshold value around $Q=0.65$,
it starts to decrease rapidly as $Q$ approaches $1$. One may note that some
previous Ref.s [39, 49] also found similar variation trends of QNMs with
respect to black hole charge $Q$. However, due to the presence of coupling
term and non-linear charge distribution, in this case, we observe slightly
different behaviour of QNMs towards higher values of $Q$.

Unlike black hole charge $Q$, the coupling parameter $\xi$ has a linear effect
on the QNMs spectrum. With an increase in the value of $\xi$, the oscillation
frequency of QNMs increases while the damping rate decreases slowly.

We also investigated time domain profiles for different model parameters and
found that the profiles depict similar results as those found by using the WKB
method.

In the next part of our work, we considered the shadow of the black hole and
found that $Q$ has a noticeable impact on the shadow behaviour of the black
hole. For a charged black hole, the shadow is expected to be smaller in size.
A similar behaviour is observed in the case of $\xi$ also. However, the
variation of shadow with respect to $\xi$ is small. Thus the impact of the
coupling parameter as well as the presence of solitonic solutions may not be
well differentiated by utilising shadow observations. As a result,
observational data on black hole shadows, do not have a stringent constraint
on $\xi$.

We also studied the relationship between the QNMs and the shadow of the black
hole and found that the shadow observations can provide a constraint on the
QNMs spectrum. In the near future, observational data of QNMs from LISA may
provide us with scope to constrain the theory more efficiently and to check
whether the shadow observations are well consistent with it or not.

Our study has uncovered that in the standard non-minimally coupled EYM theory,
both the charge parameter $Q$ and the coupling term $\xi$ exert notable
influences on the quasinormal spectrum and shadow of the black hole. The
effects of these parameters on the quasinormal mode spectrum differ
significantly in their nature.

## Acknowledgments

DJG acknowledges the contribution of the COST Action CA21136 – “Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse)”. SP acknowledges the funding supported from the NSRF via the
Program Management Unit for Human Resources $\&$ Institutional Development,
Research and Innovation [grant number B37G 660013].

## Data Availability Statement

There are no new data associated with this article.

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