# Observational Signatures of Modified Bardeen Black Hole: Shadow and Strong
Gravitational Lensing

Niyaz Uddin Molla<EMAIL_ADDRESS>Department of Mathematics, Indian
Institute of Engineering Science and Technology, Shibpur, Howrah-711 103,
India,  Amna Ali<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata-700032, India,  Ujjal Debnath<EMAIL_ADDRESS>Department of Mathematics, Indian Institute of Engineering Science and
Technology, Shibpur, Howrah-711 103, India,  Saraswathy Shamini Gunasekaran
<EMAIL_ADDRESS>Institute Of Informatics $\&$ Computing In Energy
(IICE),Universiti Tenaga Nasional (UNITEN), Malaysia.

###### Abstract

This study is dedicated to exploring the distinctive observational features
affected by the modified Bardeen black hole through meticulous analysis of its
shadow and strong gravitational lensing observations. We thoroughly
investigated the impact of the black hole’s parameters q, g, and $\mu$ on its
shadow radius through numerical simulations and graphical representations.
Notably, recent observations by the Event Horizon Telescope (EHT) scrutinized
the images and shadows of supermassive black holes like $M87^{*}$ and
$SgrA^{*}$. The observed angular diameters of the shadows were
$\theta_{d}=42\pm 3$ for $M87^{*}$ and $\theta_{d}=51.8\pm 2.3$ for
$SgrA^{*}$. Utilizing the EHT data, we constrained the parameters q and $\mu$
of the modified Bardeen black hole within specific ranges:
$-0.89\leq\mu/8M^{2}\leq 0.4$ and $0\leq|q|\leq 0.185$ for $M87^{*}$; and
$-1.38\leq\mu/8M^{2}\leq 0.1$ and $0\leq|q|\leq 0.058$ for $SgrA^{*}$, while
maintaining the fixed value $g/2M=0.2$. This restriction of the ($\mu$, $q$)
parameter space by the EHT findings illustrates the viability of modified
Bardeen black holes as astrophysical candidates.

Additionally, we investigate gravitational lensing in the strong field limit
for the modified Bardeen black hole, comparing its behavior to other
astrophysical black holes such as the Schwarzschild ($\mu=\&q=0$) and regular
Bardeen ($\mu=0$) black holes. Our investigation elucidated how the parameters
of the modified Bardeen black hole influenced various observables associated
with strong gravitational lensing. By examining the astrophysical
ramifications through strong gravitational lensing, considering supermassive
black holes in diverse galaxies, we uncovered that the modified Bardeen black
hole exhibits distinct characteristics, offering a quantitative distinction
from other black holes such as the Schwarzschild and regular Bardeen black
holes. These findings in astrophysical consequences provide a promising
pathway to differentiate the modified black hole from its counterparts in the
realm of general relativity.

Keywords: Shadow, Gravitational lensing, Null geodesics, Bardeen black hole.

##  I Introduction

Black holes, postulated several decades ago within the framework of general
relativity, stand as enigmatic and compact entities in our universe, pivotal
to the foundation of astrophysics and high-energy physics [1, 2, 3]. These
cosmic phenomena continue to captivate researchers across diverse scientific
domains, owing to their profound implications on thermodynamics, quantum
effects, and gravitational interactions within curved spacetime.

Recent years have witnessed a surge in both theoretical conjectures and
substantial advancements in observational and experimental investigations
surrounding black holes. Observational breakthroughs, notably the measurement
of black hole spin in X-ray binaries, detection of gravitational waves from
binary black hole mergers by LIGO [4, 5, 6], and the unprecedented imaging of
a black hole at the core of galaxy M87 by the Event Horizon Telescope (EHT)
collaboration [7], have fortified our comprehension of these cosmic enigmas.
Additionally, discoveries like the identification of wide star-black hole
binary systems through radial velocity measurements [8]have substantially
contributed to reinforcing our understanding of black holes.

Beyond their significance in astronomy and astrophysics, black holes have
spurred investigations in various branches of physics, particularly in
addressing the singularity problem inherent in general relativity. The
conception of regular black holes as solutions without singularities but with
a horizon has paved the way for extensive studies on modified versions of
classical black hole solutions. The Bardeen black hole, proposed as a regular
alternative to the classical Schwarzschild black hole, addresses the
singularity problem by replacing it with a de Sitter core [9]. Subsequent
explorations led to variations like the Bardeen-anti-de Sitter and Bardeen-de
Sitter black holes [10, 11], serving as crucial alternatives that extend our
understanding beyond conventional paradigms.

The evolution of the Bardeen black hole has not been confined to its original
formulation. Researchers have introduced modifications encompassing various
physical effects and theoretical challenges [12, 13, 14], involving additional
fields like scalar or electromagnetic fields and alterations to the theory of
gravity itself, including higher-order curvature terms. Moreover, within 4D
Einstein-Gauss-Bonnet (EGB) gravity, a modified Bardeen black hole presents
fresh insights into regular black holes and their astrophysical implications
[15]. Identifying a modified Bardeen black hole would profoundly impact our
understanding of gravity, astrophysics, and fundamental physics, opening new
avenues for exploration and enriching our knowledge of black holes’ nature and
properties.

Astronomical and astrophysical aspects, such as black hole parameter
estimation, shadows, gravitational lensing, quasinormal modes, and particle
motion around black holes, have been extensively studied for various types of
black holes [16, 17, 18, 19, 20, 21, 22, 23, 13].These investigations, carried
out over the last few decades, have expanded our knowledge and understanding
of black hole physics.

In this paper, we aim to build upon previous analyses conducted on
Schwarzschild, modified regular, and regular Bardeen black holes (explored in
the literature by [24, 16, 25, 26, 27, 18, 15] and apply these insights to the
modified Bardeen black hole. Our focus revolves around observing shadows and
strong gravitational lensing, particularly investigating the astrophysical
implications of the modified Bardeen black hole in comparison to other
astrophysical black holes, such as the Schwarzschild and regular Bardeen black
holes.

The black hole shadow, an intriguing astrophysical feature observed through
strong gravitational lensing, has gained immense importance in recent years.
The remarkable images of supermassive black holes in $M87^{*}$ and $SgrA^{*}$
released by the Event Horizon Telescope (EHT) collaboration [7, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37] provide pivotal evidence for the existence of
black holes. These images reveal a distinct region at the heart of black
holes, known as the black hole shadow. The concept of light rays bending due
to gravitational lensing leads to the observation of a sharply defined
boundary region of brightness on the distant image plane, offering robust
tools for estimating black hole parameters [38, 39, 40] and probing general
relativity and its alternatives [41, 42, 43, 44, 45, 46, 47, 48].

Gravitational lensing, categorized into weak and strong regimes, serves as a
powerful tool for understanding various properties of spacetime, the
distribution of matter in the universe, and phenomena related to dark matter,
dark energy, and cosmological parameters [49, 50, 51, 52, 53, 54, 55, 56, 57,
58]. Our specific focus lies on strong gravitational lensing, which manifests
as multiple images, arcs, and Einstein rings when a source is almost directly
aligned behind a massive object.

The study of strong gravitational lensing holds immense potential for testing
general relativity and alternative theories of gravity. Pioneering works by
Bozza et al. [59, 24] laid the groundwork for obtaining the deflection angle
in strong gravitational fields, shedding light on various types of black
holes, naked singularities, and wormholes [60, 61, 39, 62, 63, 64, 65, 66, 67,
68, 69, 70, 71, 72, 73]. These studies have extensively discussed
astrophysical consequences, including the angular position, separation,
magnification, Einstein rings, and time delays of relativistic images for
rotating and non-rotating black holes [74, 75, 76, 77, 78, 79, 80, 81, 82, 83,
84].

Observationally, gravitational lensing phenomena by black holes have garnered
significant attention, motivating our work to study the observational
signatures of the modified Bardeen black hole through shadow and strong
lensing observations. We intend to delve into various astrophysical
consequences, including the black hole shadow, angular position, separation,
Einstein ring, and time delays of relativistic images concerning the modified
Bardeen black hole, and compare these with other astrophysical black holes,
such as the Schwarzschild black hole and the ordinary regular Bardeen black
hole.

The structure of this paper is organized as follows: In Section II, we provide
a concise overview of the modified Bardeen black hole, analyzing its null
geodesics along the equatorial plane. Section III delves into studying the
shadow of the modified Bardeen black hole and constraining its observables
using observational data from $M87^{*}$ and $SgrA^{*}$. Following this,
Section IV focuses on investigating strong gravitational lensing by the
modified Bardeen black hole and exploring its observables, such as angular
image position, separation, magnification, Einstein ring, and time delays of
relativistic images. We also briefly overview the astrophysical implications
of the modified Bardeen black hole. Subsequently, Section V estimates the
strong lensing observables for the supermassive black hole BH $NGC4649$ and
compares these results with those obtained for the Schwarzschild black hole
and the regular Bardeen black hole. Finally, in Section VI, we discuss and
summarize the outcomes of our study.

##  II  Modified Bardeen Black holes and Null geodesics

This paper explores the modified Bardeen black hole, initially conceptualized
in [12]. This adaptation introduces alterations to the rotating iteration of
the Bardeen black hole, exploring its potential as a mechanism for particle
acceleration. Investigations into modified Bardeen black holes have been
driven by diverse motivations, encompassing:

  1. 1.

Exploring the dynamics of particle acceleration involving two colliding
particles in close proximity to the horizon of these black holes.

  2. 2.

Analyzing changes in the observable characteristics caused by the Bardeen
black hole, particularly in relation to shadow effects and strong
gravitational lensing.

  3. 3.

Investigating the thermodynamic attributes inherent in static Modified Bardeen
black holes, specifically within the framework of Anti-de Sitter spacetime.

These studies collectively contribute to a comprehensive understanding of
modified Bardeen black holes, shedding light on their multifaceted
implications across various scientific disciplines. This discussion focuses on
the static version of the modified Bardeen black hole, as referenced in [13],
where we specifically examine the scenario with $a=0$. The static, spherically
symmetric spacetime of this modified Bardeen black hole is mathematically
described as presented in [13].

$ds^{2}=-f(r)dt^{2}+\frac{1}{h(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (1)

where

$f(r)=\left(1-\frac{2Mr^{2}}{\left(q^{2}+r^{2}\right)^{3/2}}\right)\left(1-\frac{\mu
M}{\left(g^{2}+r^{2}\right)^{3/2}}\right)$ (2)

$h(r)=\left(1-\frac{2Mr^{2}}{\left(q^{2}+r^{2}\right)^{3/2}}\right)$ (3)

The metric is defined by the magnetic charge $q$ and mass parameter $M$, while
the parameters $\mu$ and $g$ serve to modify the spacetimes of the modified
Bardeen black hole. This metric adheres to the following conditions:

i) It maintains Schwarzschild-like behavior as r becomes large. ii) It
incorporates the 1-loop quantum correction. iii) It permits a finite time
dilation between the central region and infinity.

When the parameter $\mu$ is absent, the metric (1) simplifies to the regular
Bardeen black hole. Additionally, setting $\mu=0$ and $q=0$ results in the
Schwarzschild black hole.

The motion of photons around the modified Bardeen black hole is elucidated
through the use of the Lagrangian formalism:
$\mathcal{L}=-\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$.

Without losing generality, we restricted the photon trajectory around the
modified Bardeen black hole to the equatorial plane, where
$\theta=\frac{\pi}{2}$. For the metric of the modified Bardeen spacetime (1),
the equation governing the motion of photons around the black hole is
expressed as the Lagrangian equation:

$\begin{split}&\mathcal{L}=-\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\\\
&=\frac{1}{2}(f(r)dt^{2}-\frac{1}{h(r)}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}))=\delta\\\ \end{split}$ (4)

where $\dot{x}^{\mu}$ denotes the four-velocity of a photon, and the dot
signifies differentiation with respect to the affine parameter $\tau$. The
parameter $\delta$ takes on values of $-1$, $0$, or $1$, denoting spacelike,
null, and timelike geodesics, respectively. Due to the metric coefficients in
(4), the coordinates $t$ and $\theta$ cannot be explicitly determined, leading
to two conserved quantities: $E$ and $L$. These represent the energy and
angular momentum of the particle, respectively, and can be defined as follows
[85, 15]:

$E=-g_{\mu\nu}t^{\mu}\dot{x}^{\nu}=f(r)\dot{t}$ and
$L=g_{\mu\nu}\phi^{\mu}\dot{x}^{\nu}=r^{2}\dot{\phi}$.

The photon travels along null geodesics around the modified Bardeen black
hole, implying $\delta=0$. The null geodesics derived from equation (4) are as
follows:

$\dot{t}=\frac{dt}{d\tau}=\frac{E}{\left(1-\frac{2Mr^{2}}{\left(q^{2}+r^{2}\right)^{3/2}}\right)\left(1-\frac{\mu
M}{\left(g^{2}+r^{2}\right)^{3/2}}\right)}$ (5)

$\dot{\phi}=\frac{d\phi}{d\tau}=\frac{L}{r^{2}}$ (6)

$\dot{r}=\frac{dr}{d\tau}=\pm\sqrt{h(r)\biggr{(}\frac{E^{2}}{f(r)}-\frac{L^{2}}{r^{2}}\biggr{)}}$
(7)

The symbol $\pm$ in $\dot{r}$ signifies the direction of radial motion,
distinguishing between ingoing and outgoing trajectories, respectively. The
functions $f(r)$ and $h(r)$ are derived from Eqns. (2) and (3).

Equation (7) can be expressed as

$\biggr{(}\frac{dr}{d\tau}\biggr{)}^{2}+V_{eff}=0$ (8)

We can proceed without loss of generality by setting $E=1$. The effective
potential function, denoted as $V_{\text{eff}}$, is described by:

$V_{eff}=h(r)\biggr{(}\frac{L^{2}}{r^{2}}-\frac{1}{f(r)}\biggr{)}$ (9)

The critical photon ring orbit is characterized by the effective potential
function, $V_{\text{eff}}(r)$, satisfying specific conditions:

$V_{\text{eff}}(r)=\frac{dV_{\text{eff}}(r)}{dr}=0,$
$\frac{d^{2}V_{\text{eff}}(r)}{dr^{2}}>0\text{ (for stable)},$
$\frac{d^{2}V_{\text{eff}}}{dr^{2}}<0\text{ (for unstable)}\text{ circular
orbit}.$

In the case of both the modified Bardeen and the ordinary regular Bardeen
black hole, it’s observed that
$\frac{d^{2}V_{\text{eff}}}{dr^{2}}|_{r_{\text{ph}}}<0$, corresponding to an
unstable circular photon orbit (refer to Fig.1). Consequently, photon rays
arriving from infinity toward the vicinity of the modified Bardeen black hole,
with a minimum impact parameter at the closest distance $r_{0}$, follow
unstable circular orbits around the black hole, forming a photon sphere with a
radius of $r_{\text{ph}}$.

##  III Shadows of modified Bardeen black hole

The black hole’s shadow stands as a crucial signature of the spacetime
geometry, representing a dark area encompassed by circular orbits of photons
forming what’s known as a photon sphere. This phenomenon provides insight into
the black hole’s properties, optically revealing details related to
gravitational lensing effects on nearby radiation. Several comprehensive
reviews have delved into black hole shadows and their observables, offering
extensive insights (see, for instance, [86, 45, 87, 88, 89, 90, 91]).

Notably, the Event Horizon Telescope (EHT) collaboration has utilized black
hole shadow properties to detect images of black holes [7, 31, 32]), garnering
significant attention. In this section, we aim to explore the shadow of a
modified Bardeen black hole and its observables, focusing on supermassive
black holes like $M87^{*}$ and $SgrA^{*}$ positioned at the centers of
galaxies.

The black hole’s shadow is intimately linked to the critical impact parameter
of the photon orbit. Utilizing the conditions discussed in the previous
section, one can define the critical impact parameter:

$u_{cr}=\frac{L}{E}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}$ (10)

where the photon sphere radius $r_{ph}$ is the largest real root of the
equation

$2f(r_{ph})-r_{ph}f^{\prime}(r_{ph})=0$ (11)

The black hole shadow radius, denoted as $r_{\text{sh}}$, while the observer
is positioned far away from the black hole, can be expressed in terms of
celestial coordinates (X, Y) as follows:

$r_{sh}=\sqrt{X^{2}+Y^{2}}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}$ (12)

where the celestial co-ordinate ($X$, $Y$ ) at the boundary curve of black
hole shadow defined as

$X=\lim_{r_{0}\rightarrow\infty}(r_{0}^{2}\sin\theta_{0})\frac{d\phi}{dr}$
(13)

$Y=\lim_{r_{0}\rightarrow\infty}(r_{0}^{2}\frac{d\theta}{dr})$ (14)

Here, $r_{0}$ represents the radial distance between the black hole and the
observer, while $\theta_{0}$ denotes the inclination angle between the
observer and the black hole.

To express the shadow radius in terms of dimensional quantities, a set of
transformations is applied: $t\rightarrow\frac{t}{2M}$,
$r\rightarrow\frac{r}{2M}$, $q\rightarrow\frac{q}{2M}$,
$g\rightarrow\frac{g}{2M}$, and $\mu\rightarrow\frac{\mu}{8M^{2}}$ within the
function $f(r)$. This transformation defines the shadow radius as follows:

$r_{sh}=r_{ph}\left[\left(1-\frac{r^{2}}{\left(q^{2}+r^{2}\right)^{3/2}}\right)\left(1-\frac{\mu}{\left(g^{2}+r^{2}\right)^{3/2}}\right)\right]^{-1/2}$
(15)

Table 1: Estimation of the photon sphere radius and the shadow radius for different values of black hole parameters . $\mu$ |  $g$ |  Photon sphere radius $\mathbf{r_{ph}}$ |  Shadow radius $\mathbf{r_{sh}}$
---|---|---|---
|  |  $|q|=0.0$ $|q|=0.1$ $|q|=0.2$ $|q|=0.4$ |  $|q|=0.0$ $|q|=0.1$ $|q|=0.2$ $|q|=0.4$
0 |  0 |  $1.50000$ $1.48309$ $1.42899$ $1.09915$ |  $2.59808$ $2.58054$ $2.52504$ $2.22043$
1 |  0.2 |  $1.74892$ $1.73991$ $1.71281$ $1.60617$ |  $2.95742$ $2.94772$ $2.91837$ $2.79827$
1 |  1.2 |  $1.58109$ $1.56574$ $1.51702$ $1.24412$ |  $2.79269$ $2.77813$ $2.73253$ $2.50259$
3 |  0.2 |  $2.19323$ $2.18977$ $2.17956$ $2.14160$ |  $3.50634$ $3.50177$ $3.48813$ $3.43500$
3 |  1.2 |  $1.84867$ $1.83975$ $1.81263$ $1.69950$ |  $3.21602$ $3.20768$ $3.18241$ $3.07755$
5 |  0.2 |  $2.51788$ $2.5158$ $2.50965$ $2.48650$ |  $3.90482$ $3.90178$ $3.89272$ $3.85741$
5 |  1.2 |  $2.15933$ $2.15468$ $2.14087$ $2.08791$ |  $3.60475$ $3.59975$ $3.58479$ $3.52573$
7 |  0.2 |  $2.77523$ $2.77374$ $2.76930$ $2.75243$ |  $4.22334$ $4.22334$ $4.21412$ $4.18715$
7 |  1.2 |  $2.43328$ $2.43041$ $2.42189$ $2.38949$ |  $3.93403$ $3.93056$ $3.92019$ $3.87958$

(a) (b)

Figure 1: Variation of the effective potential $V_{eff}$ for regular Bardeen
(left panel) and modified Bardeen (right panel)black holes as a function of
radial coordinate $r$.

Using Eqns. (11) and (15), we estimated both the photon sphere radius and the
shadow radius across various values of black hole parameters: $\mu=0,1,3,5,7$;
$g=0.2,1.2$; and $|q|=0,0.1,0.2,0.4$. These findings are summarized in Table
1. Notably, while keeping the parameters $\mu$ and $g$ constant, the photon
sphere radius and the shadow radius of the black hole exhibit a decreasing
trend as the parameter $q$ increases.

###  III.1 Observational Constraints using M87* and Sgr A* observations data

We’ve previously investigated how the parameters $\mu$, $g$, and $q$ influence
the photon sphere radius and the shadow radius. In this section, our goal is
to determine the values of these parameters, $\mu$, $g$, and $q$, based on the
observed angular diameter of the shadow. To simplify the parameter estimation
process, we opt to fix the parameter $g=0.2$ and subsequently determine the
potential range for the parameters $\mu$ and $q$.

For a distant observer, the measurement of a black hole’s shadow image is
consistently represented by the angular diameter $\theta_{d}$ [45]. This
angular diameter is given by:

$\theta_{d}=\frac{2u_{ph}}{D_{ol}}$ (16)

where $D_{ol}$ is the distance of the black hole to the observer. The above
equation can be expressed as

$\theta_{d}(\mu as)=\biggr{(}\frac{6.191165\times
10^{6}}{\pi}\biggr{)}(\frac{\gamma}{D_{ol}/Mps})\biggr{(}\frac{2u_{ph}}{M}\biggr{)}$
(17)

where $\gamma$ denotes the mass ratio of a black hole to the sun, and
$u_{ph}=u_{cr}$ is obtained from Eqn. (10).

Figure 2: The angular diameter of the shadow for the modified Bardeen black
hole is presented as a function of the parameters $\frac{\mu}{8M^{2}}$ and
$\frac{q}{2M}$. In the left panel, the red solid curve corresponds to
$\theta_{d}=39.4615\mu as$ for the modified Bardeen black hole, the blue solid
curve represents $\theta_{d}=38.80\mu as$ for the Bardeen black hole, and the
green solid curve corresponds to $\theta_{d}=39.9265\mu as$ for the
Schwarzschild black hole. These values fall within the $1\sigma$ region of the
measured angular diameter, $\theta_{d}=42\pm 3\mu as$, for $M87^{*}$. In the
right panel, for $SgrA^{*}$, the red solid curve represents
$\theta_{d}=52.15\mu as$ for the modified Bardeen black hole, the blue solid
curve corresponds to $\theta_{d}=49.15\mu as$ for the Bardeen black hole, and
the green solid curve corresponds to $\theta_{d}=52.77\mu as$ for the
Schwarzschild black hole. These values also fall within the $1\sigma$ region
of the measured angular diameter, $\theta_{d}=51.8\pm 2.3\mu as$.

By utilizing Eqns. (16) and (17), we’ve studied the angular diameter of the
black hole shadow as a function of the parameters $\frac{\mu}{8M^{2}}$ and
$\frac{q}{2M}$, presented in Fig.2. Our investigation into the angular
diameter of the black hole shadow focuses on supermassive black holes,
specifically $M87^{*}$ with mass and distance from Earth approximately
$M\approx 6.5\times 10^{9}\dot{O}$ and $D_{ol}\approx 16.8\text{ Mpc}$
respectively, and $SgrA^{*}$ with mass and distance from Earth approximately
$M\approx 4.28\times 10^{6}\dot{O}$ and $D_{ol}\approx 8.32\text{ kpc}$
respectively. The measured shadow angular diameters are $\theta_{d}=42\pm 3$
for $M87^{*}$ and $\theta_{d}=51.8\pm 2.3$ for $SgrA^{*}$ [33, 34].

We’ve constrained the modified black hole parameters $q$ and $\mu$ while
keeping the value of $g$ fixed based on EHT collaboration data for the angular
shadow diameter of $M87^{*}$ and $SgrA^{*}$. Our observations indicate
constrained ranges for the parameters $\mu$ and $q$ of the modified Bardeen
black hole: $-0.89\leq\frac{\mu}{8M^{2}}\leq 0.4$ and $0\leq|q|\leq 0.185$ for
$M87^{*}$; and $-1.38\leq\frac{\mu}{8M^{2}}\leq 0.1$ and $0\leq|q|\leq 0.058$
for $SgrA^{*}$, while maintaining the fixed value of $\frac{g}{2M}=0.2$. These
additional parameters $\mu$, $g$, and $q$ characterize the modified Bardeen
black holes akin to the supermassive black holes $M87^{*}$ and $SgrA^{*}$.
Notably, our results suggest that the modified Bardeen black hole complies
with the EHT constraint, signifying its potential detectability and
differentiation from other astrophysical black holes in future observations.

##  IV Strong gravitational lensing and it’s observable

We aim to explore the effects of strong gravitational lensing caused by the
modified Bardeen black hole, investigating how its parameters—$\mu$, $q$, and
$g$—influence several astrophysical observations. Our focus includes analyzing
effects such as changes in angular position, separation, magnification, the
formation of Einstein’s rings, and time delays of relativistic images. We’ll
compare these observations with both an ordinary regular Bardeen black hole
($\mu=0$) and a standard Schwarzschild black hole ($\mu=0$, $q=0$).

In our investigation, we’re particularly interested in determining the strong
deflection angle of photon rays within the equatorial plane
($\theta=\frac{\pi}{2}$) due to the modified Bardeen black hole. To compute
this angle, we apply dimensionless operations on metric (1):
$t\rightarrow\frac{t}{2M}$, $r\rightarrow\frac{r}{2M}$,
$q\rightarrow\frac{q}{2M}$, $g\rightarrow\frac{g}{2M}$, and
$\mu\rightarrow\frac{\mu}{8M^{2}}$. This allows us to better understand the
impact of these parameters on the gravitational lensing effects near the black
hole. Rewriting the metric:

$d\bar{s}^{2}=-A(x)dt^{2}+B(x)dr^{2}+C(x)d\phi^{2}$ (18)

where

$A(r)=\left(1-\frac{r^{2}}{\left(q^{2}+r^{2}\right)^{3/2}}\right)\left(1-\frac{\mu}{\left(g^{2}+r^{2}\right)^{3/2}}\right)$

$B(r)=\frac{1}{\left(1-\frac{r^{2}}{\left(q^{2}+r^{2}\right)^{3/2}}\right)}$

and

$C(r)=r^{2}$

When particles approach the closest distance $r=r_{0}$ to the central black
hole, signifying $\frac{dr}{d\tau}=0$, we can define the minimum impact
parameter $u_{0}$ in relation to the closest distance $r_{0}$ [24] as:

$u_{0}=\frac{r_{0}}{\sqrt{A(r_{0})}}$ (19)

(a)

Figure 3: The behaviour of the photon sphere radius $\mathit{r_{ph}}$ vs
parameter $q$ with the different values of $\mu$ for fixed value of $g=0.2$
(left panel) ;and the photon sphere radius $\mathit{r_{ph}}$ as a function of
both the parameters $\mu$ and $q$ for the fixed value of $g=0.2$ (right panel)
.

(a) (b)

Figure 4: The behaviour of the minimum impact parameter
$\mathit{u_{ph}/R_{sh}}$ vs parameter $q$ with the different values of $\mu$
for fixed value of $g=0.2$ (left panel) ;and the minimum impact parameter
$\mathit{u_{ph}}/R_{sh}$ as a function of both the parameters $\mu$ and $q$
for the fixed value of $g=0.2$ (right panel) .

The behavior of the photon sphere radius, $r_{ph}$, has been depicted as a
function of the parameter $q$ in Fig.3(a), and as a function of the parameters
$\mu$ and $q$ in Fig.3(b). Notably, it has been observed that, while keeping
the parameters $\mu$ and $g$ fixed, the photon radius $r_{ph}$ slightly
decreases with the parameter $q$. Conversely, for fixed parameters $g$ and
$q$, the photon radius $r_{ph}$ tends to increase with the parameter $\mu$.

Moreover, in Fig.3(a), the photon radius $r_{ph}$ corresponding to the
modified Bardeen black hole is observed to exceed that of the ordinary regular
Bardeen black hole (red solid line). Additionally, this $r_{ph}$ value for the
modified Bardeen black hole surpasses the value of $r_{ph}=1.5$, which
corresponds to the Schwarzschild black hole (yellow horizontal line) [24].

When $r_{0}\rightarrow r_{ph}$, deflection angle becomes divergent and for
$r_{0}>r_{ph}$, it becomes finite only. The photon having impact parameter
$u<u_{ph}$ falls into the black hole and for the case when $u>u_{ph}$, it
reaches the closest distance $r_{0}$ near the black hole; while for the impact
parameter $u=u_{cr}=u_{ph}$, photon revolved in unstable orbit around the
black hole.

The critical impact parameter for the unstable photon orbit, $u_{ph}$, is
determined by the equation:

$u_{ph}=\frac{r_{ph}}{\sqrt{A(r_{ph})}}$ (20)

Its behavior has been graphically represented as a function of the parameter
$q$ in Fig.4(a), and as a function of the parameters $\mu$ and $q$ in
Fig.4(b). Upon examination, it has been noted that for fixed parameters $\mu$
and $g$, the critical impact parameter $u_{ph}$ experiences a slight decrease
with the parameter $q$. Conversely, for fixed parameters $g$ and $q$, $u_{ph}$
tends to increase with the parameter $\mu$ (refer also to Table.3).

In Fig.4(a), it’s further observed that the critical impact parameter
$u_{ph}/R_{s}$ corresponding to the modified Bardeen black hole surpasses that
of the ordinary regular Bardeen black hole (red solid line). Additionally,
this $u_{ph}/R_{s}$ value for the modified Bardeen black hole exceeds the
value of $u_{ph}/R_{sh}=2.59808$, which corresponds to the Schwarzschild black
hole (yellow horizontal line) [24].

The strong deflection angle for the modified Bardeen black hole spacetime, as
a function of the closest approach distance $r_{0}$, can be read as: [92, 93,
79])

$\alpha_{D}(r_{0})=I(r_{0})-\pi=2\int_{r_{0}}^{\infty}\frac{\sqrt{B(r)}dr}{\sqrt{C(r)\sqrt{\frac{A(r_{0})C(r)}{A(r)C(r_{0})}-1}}}dr-\pi$
(21)

The strong deflection angle $\alpha_{D}(r_{0})$ depends upon the relation
between $r_{0}$ and $r_{ph}$ and while $r_{0}\approx r_{ph}$, it is increased.
So, we define a new variable z as:

$z=1-\frac{r_{0}}{r}$ (22)

For, $r_{0}\approx r_{ph}$, the strong deflection angle becomes:

$\alpha_{D}(u)=-\bar{a}~{}log\left(\frac{u}{u_{ph}}-1\right)+\bar{b}+\mathcal{O}((u-u_{ph})log(u-u_{ph}))$
(23)

where

$\bar{a}=\sqrt{\frac{2A(r_{ph}B(r_{ph})}{A(r_{ph})C^{\prime\prime}(r_{ph})-A^{\prime\prime}(r_{ph}C(r_{ph})}}$
(24)

and

$\bar{b}=-\pi+I_{R}(r_{ph}+\bar{a}~{}log\biggr{[}r_{ph}^{2}\biggr{(}\frac{C^{\prime\prime}_{ph}}{c_{ph}}-\frac{A^{\prime\prime}_{ph}}{A_{ph}}\biggr{)}\biggr{]},$
(25)

Here,
$I_{R}(r_{ph})=2\int_{0}^{1}\Bigg{(}r_{ph}\biggr{[}\sqrt{\frac{B(z)}{C(Z)}}\biggr{(}\frac{A(r_{ph})}{C(r_{ph})}\frac{C(z)}{A(z)}-1\biggr{)}\frac{1}{(1-z)^{2}}\biggr{]}-\frac{\bar{a}}{z~{}r_{ph}}\Bigg{)}dz$,
which is obtained numerically.

Figure 5: The behaviour of the deflection limit coefficient $\mathit{\bar{a}}$
vs parameter $q$ with the different values of $\mu$ for fixed value of $g=0.2$
(left panel) ;and the deflection limit coefficient $\mathit{\bar{a}}$ as a
function of both the parameters $\mu$ and $q$ for the fixed value of $g=0.2$
(right panel) .

Figure 6: The behaviour of the deflection limit coefficient $\mathit{\bar{b}}$
vs parameter $q$ with the different values of $\mu$ for fixed value of $g=0.2$
(left panel) ;and the deflection limit coefficient $\mathit{\bar{b}}$ as a
function of both the parameters $\mu$ and $q$ for the fixed value of $g=0.2$
(right panel) . Table 2: Estimation of strong lensing coefficients with the
different value of black hole parameters $\mu=0,1,3$ ;$g=0.2,1.2$; and
$|q|=0,0.05,0.1,0.2,0.4$ . Strong Lensing Coefficients
---
$\mu$ |  $g$ |  $|q|$ |  $\mathbf{\bar{a}}$ |  $\mathbf{\bar{b}}$ |  $\mathbf{u_{ph}/R_{sh}}$
0 |  |  $0$ |  1 |  $-0.40023$ |  $2.59808$
|  |  $0.05$ |  1.0028 |  $-0.401611$ |  $2.59808$
0 |  |  $0.1$ |  1.101151 |  $-0.406055$ |  $2.59373$
|  |  $0.2$ |  1.05179 |  $-0.429858$ |  $2.58054$
|  |  $0.4$ |  1.55181 |  $-1.18175$ |  $2.22043$
|  0.2 |  $0.05$ |  0.921272 |  $-0.477287$ |  $2.955$
|  0.2 |  $0.1$ |  0.923386 |  $-0.476401$ |  $2.94772$
|  0.2 |  $0.2$ |  0.931409 |  $-0.470916$ |  $2.91837$
1 |  0.2 |  $0.4$ |  0.949693 |  $-0.399493$ |  $2.79827$
|  1.2 |  $0.05$ |  1.03705 |  $-0.516087$ |  $2.78908$
|  1.2 |  $0.1$ |  1.04578 |  $-0.52368$ |  $2.77813$
|  1.2 |  $0.2$ |  1.08555 |  $-0.561458$ |  $2.73253$
|  1.2 |  $0.4$ |  1.46926 |  $-1.19512$ |  $2.50259$
|  0.2 |  $0.05$ |  0.761981 |  $-0.384548$ |  $3.5052$
|  0.2 |  $0.1$ |  0.761421 |  $-0.381893$ |  $3.50177$
|  0.2 |  $0.2$ |  0.75901 |  $-0.37101$ |  $3.48813$
3 |  0.2 |  $0.4$ |  0.746808 |  $-0.324242$ |  $3.435$
|  1.2 |  $0.05$ |  0.986331 |  $-0.641022$ |  $3.21394$
|  1.2 |  $0.1$ |  0.989931 |  $-0.644732$ |  $3.20768$
|  1.2 |  $0.2$ |  1.00474 |  $-0.660005$ |  $3.18241$
|  1.2 |  $0.4$ |  1.06929 |  $-0.7222588$ |  $3.07755$

(a) (b) (c) (d)

Figure 7: The behaviour of the deflection angle $\mathit{\alpha_{D}}$ vs
parameter $q$ with the different values of $\mu$ for fixed value of $g=0.2$ (
panel (a)) ; the deflection angle $\mathit{\alpha_{D}}$ as a function of both
the parameters $\mu$ and $q$ for the fixed value of $g=0.2$ (panel (b)) ;the
deflection angle $\mathit{\alpha_{D}}$ as a function of impact parameter $u$
with different value of $q$ for the fixed value of parameters $\mu$ and $q$
(panel (c)); and the deflection angle $\mathit{\alpha_{D}}$ as a function of
impact parameter $u$ with different value of $\mu$ for the fixed value of
parameters $\mu$ and $q$ (panel (d)).The solid red line corresponds to the
case of regular Bardeen and the solid yellow line corresponds to the case of
modified Bardeen black hole. The dots in panels (c)&(d) indicate the values of
the impact parameter $u=u_{ph}$, where the deflection angle
$\mathit{\alpha_{D}}$ becomes divergent.

We’ve obtained the numerical values of the lensing coefficients $\bar{a}$ and
$\bar{b}$ alongside $u_{ph}/R_{sh}$ for various parameters of the modified
Bardeen black hole: $\mu=0,1,3$, $g=0.2,1.2$, and $q=0,0.05,0.1,0.2,0.4$
(refer to Table.2). Observations from this table indicate that, for fixed
parameters $g=0.2,1.2$ and $\mu=0,1,3$, the lensing coefficient $\bar{a}$
increases as the parameter $q$ grows in magnitude, while the lensing
coefficient $\bar{b}$ decreases. However, this behavior doesn’t hold for the
case when $g=0.2$ and $\mu=3$. Specifically, when $\mu=0$ and $q=0$, the
values of the lensing coefficients are $\bar{a}=1$ and $\bar{b}=-0.40023$,
aligning with the characteristics of the Schwarzschild black hole [24].

The behaviors of the lensing coefficients $\bar{a}$ and $\bar{b}$ are visually
displayed in Figs.5 and 6, respectively.

Moreover, the behavior of the deflection angle $\alpha_{D}$ around the
modified Bardeen black hole is presented in Fig.7. Observations from Figs.7(a)
and 7(b) indicate that $\alpha_{D}$ increases with the increasing magnitude of
the charge parameter $q$ for small values of $\mu$ while keeping $g$ fixed.
Conversely, $\alpha_{D}$ decreases with an increasing value of $\mu$, assuming
other parameters remain constant. Furthermore, the deflection angle
$\alpha_{D}$ for the modified Bardeen black hole lies between the values for
the Schwarzschild black hole ($\mu=0,q=0$) and the ordinary regular Bardeen
black hole ($\mu=0$).

Lastly, in Fig.7(c), $\alpha_{D}$ decreases with the critical impact parameter
$u$ for various values of the parameter $\mu$ while $q$ and $g$ are fixed. In
Fig.7(d), $\alpha_{D}$ decreases with different magnitude values of parameter
$q$ for fixed values of $\mu$ and $g$.

###  IV.1 Lensing observables

We now go through the analysis of strong lensing effects caused by the
modified Bardeen black hole. In this investigation, we consider a scenario
where both the observer and the source are positioned at a significant
distance from the black hole (acting as the lens) and are nearly in alignment.
Additionally, we assume that the source lies behind the black hole.
Consequently, we can establish the lens equation [59] as:

$\beta=\theta-\frac{D_{ls}}{D_{os}}\Delta\alpha_{n}$ (26)

Here, $\Delta\alpha_{n}=\alpha_{D}(\theta)-2n\pi$ denotes the offset
deflection angle, with $n$ indicating the number of loops made by the photon
ray around the black hole. In this context, $\beta$ and $\theta$ represent the
angular separations respectively between the black hole (lens) and the source,
and between the observer and the source. Furthermore, $D_{ls}$, $D_{ol}$, and
$D_{os}$ stand for the distances of lens-source, observer-lens, and observer-
source, satisfying the condition $D_{os}=D_{ol}+D_{ls}$.

Using the Eqns. (21) and (26), the angular separation between the black
hole(lens) to the $n^{th}$ relativistic image can be expressed as

$\theta_{n}=\theta^{0}_{n}-\frac{u_{ph}e_{n}(\theta_{n}^{0}-\beta)D_{os}}{\bar{a}D_{ol}D_{ls}}$
(27)

where

$e_{n}=e^{\frac{\bar{b}-2n\pi}{\bar{a}}},$
$\theta^{0}_{n}=\frac{u_{ph}(1+e_{n})}{D_{ol}}$

. In the scenario where a photon completes $2n\pi$ loops around the black hole
(lens), $\theta^{0}_{n}$ denotes the angular position of the resultant image.

In the context of strong gravitational lensing, where surface brightness is
conserved, the magnification of the $n^{th}$ relativistic image is defined as
the ratio between the solid angle subtended by this specific image and that of
the source [16]. The expression for the magnification of the $n^{th}$
relativistic image is derived as stated in [24].

$\mu_{n}=\biggr{(}\frac{\beta}{\alpha}\frac{d\beta}{d\alpha}\biggr{)}^{-1}\biggr{|}_{\theta_{0}}=\frac{u^{2}_{ph}(1+e_{n})e_{n}D_{os}}{\beta\bar{a}D_{ls}D^{2}_{ol}}$
(28)

The equation above implies that the initial relativistic image appears as the
brightest, and its brightness diminishes exponentially with each subsequent
$n$. Consequently, the luminosity of this image predominates over the others.
Equation (28) exhibits divergence when $\beta\rightarrow 0$, indicating that
perfect alignment maximizes the probability of detecting relativistic images.

Consideration is given to the scenario where the brightest image, i.e., the
outermost image $\theta_{1}$, is observed as a solitary image, while the
remaining inner images coalesce at $\theta_{\infty}$
($\theta_{n}|_{n\rightarrow\infty}=:\theta_{\infty}$). Leveraging the
deflection angle in equation (23), we can derive essential strong lensing
parameters. These include the angular position of the image set
$\theta_{\infty}$, the angular separation between the outermost and innermost
images $S$, and the relative magnification $r_{mag}$ between the outermost
relativistic image and the grouped inner relativistic images [94, 24].

$\theta_{\infty}=\frac{u_{ph}}{d_{ol}}$ (29)
$S=\theta_{1}-\theta_{\infty}\approx\theta_{\infty}e^{\frac{(\bar{b}-2\pi)}{\bar{a}}}$
(30)

$r_{mag}=\frac{\mu_{1}}{\Sigma^{\infty}_{n=2}\mu_{n}}\approx\frac{5\pi}{\bar{a}log(10)}$
(31)

Upon measurement of the strong lensing observables—$\theta_{\infty}$, $S$, and
$r_{mag}$—from observations, the lensing coefficients $\bar{a}$, $\bar{b}$,
and the minimum impact parameter $u_{ph}$ can be readily derived by inversely
solving Eqns. (29), (30), and (31). These obtained values can then be compared
with the theoretically calculated ones. Such a comparative analysis allows for
the identification and differentiation among the modified Bardeen, ordinary
regular Bardeen, and Schwarzschild black holes based on these observations.

Taking into account the supermassive black holes $M87^{*}$, $SgrA^{*}$, and
$NGC7457$ found in nearby galaxies, we proceed to estimate the observable
quantities—$\theta_{\infty}$, $S$, and $r_{mag}$—within the framework of a
modified Bardeen black hole (Refer to Table 4). The respective mass and
distances from Earth for these black holes are as follows: For $M87^{*}$ [31,
32], the values are approximately $M\approx 6.5\times 10^{9}M_{\odot}$ and
$D_{ol}\approx 16.8$ Mpc; for $SgrA^{*}$, the values are approximately
$M\approx 4.28\times 10^{6}M_{\odot}$ and $D_{ol}\approx 8.32$ kpc [33, 34];
and for $NGC7457$, the values are approximately $M\approx 8.95\times
10^{6}M_{\odot}$ and $D_{ol}\approx 12.53$ Mpc [95].

(a) (b) (c) (d)

Figure 8: The behaviour of the angular image position
$\mathit{\theta_{\infty}}$ vs parameter $q$ with the different values of $\mu$
for the fixed value of $g=0.2$ for $M87^{*}$(upper left panel) and for
$SgrA^{*}$ (lower left panel) ;and the angular image position
$\mathit{\theta_{\infty}}$ as a function of both the parameters $\mu$ and $q$
for the fixed value of $g=0.2$ for $M87^{*}$ (lower right panel) and for
$SgrA^{*}$ (lower right panel).

(a) (b) (c) (d)

Figure 9: The behaviour of the angular image separation $\mathit{S}$ vs
parameter $q$ with the different values of $\mu$ for the fixed value of
$g=0.2$ for $M87^{*}$(upper left panel) and for $SgrA^{*}$ (lower left panel)
;and the angular image separation $\mathit{S}$ as a function of both the
parameters $\mu$ and $q$ for the fixed value of $g=0.2$ for $M87^{*}$ (lower
right panel) and for $SgrA^{*}$ (lower right panel).

Figure 10: The behaviour of the relative magnification $\mathit{r_{mag}}$ vs parameter $q$ with the different values of $\mu$ for the fixed value of $g=0.2$ (left panel) ;and the relative magnification $\mathit{r_{mag}}$ as a function of both the parameters $\mu$ and $q$ for the fixed value of $g=0.2$ (right panel) .Note that the relative magnification $\mathit{r_{mag}}$ does not the mass or distance of the black hole. Table 3: Estimation of strong lensing observables for supermassive BHs $M87^{*}$,$SgrA^{*}$, $NGC7457$ with the different value of black hole parameters $\mu=0,1,3$ ;$g=0.2,1.2$; and $|q|=0,0.05,0.1,0.2,0.4$.The observable quantity $r_{mag}$ does not depend on the mass or distance of the black hole from the observer. | parameters |  | $M87^{*}$ |  | $SgrA^{*}$ |  | $NGC7457$ |  |  $M87^{*}$,$SgrA^{*}$ , $NGC7457$
---|---|---|---|---|---|---|---|---|---
$\mu$ | $g$ | $|q|$ | $\theta_{\infty}(\mu as)$ | $S(\mu as)$ | $\theta_{\infty}(\mu as)$ | $S(\mu as)$ | $\theta_{\infty}(\mu as)$ | $S(\mu as)$ | $r_{mag}$
0 | 0 | 0 | 19.9633 | 0.024984 | 26.3315 | 0.0329538 | 0.0365211 | $4.57\times 10^{-5}$ | 6.82188
|  | 0.05 | 19.9298 | 0.025377 | 26.2874 | 0.0334722 | 0.03646 | $4.63\times 10^{-5}$ | 6.80283
0 |  | 0.1 | 19.8285 | 0.0266224 | 26.1537 | 0.0351149 | 0.0362745 | $4.87\times 10^{-5}$ | 6.74426
|  | 0.2 | 19.402 | 0.0328069 | 025.5912 | 0.0432723 | 0.0354944 | $6.0\times 10^{-5}$ | 6.48597
|  | 0.4 | 17.0615 | 0.138949 | 22.504 | 0.183274 | 0.0312125 | $2.54\times 10^{-4}$ | 4.39608
|  | 0.05 | 22.7058 | 0.0147639 | 29.9489 | 0.0194735 | 0.0415383 | $2.70\times 10^{-5}$ | 7.40485
| 0.2 | 0.1 | 22.6499 | 0.0149914 | 29.8751 | 0.0197736 | 0.041436 | $2.74\times 10^{-5}$ | 7.3879
|  | 0.2 | 22.4243 | 0.0159015 | 29.5776 | 0.020974 | 0.0410234 | $2.91\times 10^{-5}$ | 7.32426
1 |  | 0.4 | 21.5015 | 0.0189008 | 28.3604 | 0.0249302 | 0.0393352 | $3.46\times 10^{-5}$ | 7.18325
|  | 0.05 | 21.4309 | 0.0304544 | 28.2673 | 0.0401692 | 0.039206 | $5.57\times 10^{-5}$ | 6.57816
| 1.2 | 0.1 | 21.3467 | 0.0318095 | 28.1563 | 0.0419567 | 0.0390521 | $5.82\times 10^{-5}$ | 6.52325
|  | 0.2 | 20.9964 | 0.0383546 | 27.6942 | 0.0505896 | 0.038411 | $7.02\times 10^{-5}$ | 6.28426
|  | 0.4 | 19.2295 | 0.118434 | 25.3637 | 0.156215 | 0.0351788 | $2.17\times 10^{-4}$ | 4.64307
|  | 0.05 | 26.9335 | 0.00426567 | 35.5252 | 0.00562641 | 0.0492725 | $7.8\times 10^{-6}$ | 8.95282
| 0.2 | 0.1 | 26.9071 | 0.00424895 | 35.4904 | 0.00560436 | 0.0492242 | $7.77\times 10^{-6}$ | 8.95941
|  | 0.2 | 26.8023 | 0.00417578 | 35.3522 | 0.00550785 | 0.0490325 | $7.64\times 10^{-6}$ | 8.98787
3 |  | 0.4 | 26.394 | 0.00379368 | 34.8137 | 0.00500385 | 0.0482857 | $6.94\times 10^{-6}$ | 9.13472
|  | 0.05 | 24.6955 | 0.0220698 | 32.5732 | 0.02911 | 0.0451782 | $4.03\times 10^{-5}$ | 6.91642
| 1.2 | 0.1 | 24.6473 | 0.0225118 | 32.5098 | 0.029693 | 0.0450902 | $4.11\times 10^{-5}$ | 6.89127
|  | 0.2 | 24.4532 | 0.0243877 | 32.2537 | 10.0321674 | 0.044735 | $4.46\times 10^{-5}$ | 6.7897
|  | 0.4 | 23.6474 | 0.0337682 | 31.1909 | 0.0445402 | 0.043261 | $6.17\times 10^{-5}$ | 6.37982

The behavior of strong lensing observables—angular image position
$\theta_{\infty}$, angular image separation $S$, and relative magnification
$r_{mag}$—is illustrated in Figs. 8, 9, and 10, respectively, as a function of
the parameter $q$. These figures also depict the behavior concerning both
parameters $q$ and $\mu$ for a fixed value of $g=0.2$ concerning $M87^{*}$ and
$SgrA^{*}$ (refer to Table 3 as well).

It’s notable that $\theta_{\infty}$ and $r_{mag}$ exhibit a decrease with an
increase in the magnitude of the charge parameter $q$, while $S$ displays an
increment with the rising magnitudes of both parameters $\mu$ and $q$, given
other parameters remain fixed. Moreover, $r_{mag}$ increases with higher
values of parameter $\mu$ for a constant $g$ and $q$. Additionally, the
relative magnification $r_{mag}$ for the modified Bardeen black hole surpasses
that of both the Schwarzschild ($\mu=0,q=0$) and ordinary regular Bardeen
($\mu=0$) black holes.

###  IV.2 Einstein Ring

When the source, black hole (lens), and observer are perfectly aligned i.e.,
when $\beta=0$, a black hole (lens) deflects the light rays in all direction
such that a ring-shaped image is produced, which is called an Einstein ring
[96, 97, 98, 99, 100, 101].

By simplifying the Eqn. 27 for $\beta=0$,we obtain the angular radius of
$n^{th}$ relativistic images as follows:

$\theta_{n}=\theta^{0}_{n}\biggr{(}1-\frac{u_{ph}e_{n}D_{os}}{\bar{a}D_{ls}D_{ol}}\biggr{)}$
(32)

Considering the case where the black hole (lens) is located at a half distance
between the source and receiver i.e., $D_{os}=2D_{ol}$ and taking
$D_{ol}>>u_{ph}$, thus the angular radius of the $n^{th}$ relativistic
Einstein ring in the context of a modified Bardeen black hole is given by

$\theta^{E}_{n}=\frac{u_{ph}(1+e_{n})}{D_{ol}}$ (33)

(a) (b) (c) (d)

Figure 11: The behavior of the outermost Einstein’s ring
$\mathit{\theta^{E}_{n}}$ for the case of regular Bardeen (panel a & c) for
$M87^{*}$ and $SgrA^{*}$; and the case of modified Bardeen black hole (panel b
& d) for $M87^{*}$ and $SgrA^{*}$ respectively.The solid circular ring
corresponds to the case of Schwarzschild’s black hole.

The angular radius ($\theta^{E}_{1}$) represents the outermost Einstein ring,
as depicted in Fig.11 for the supermassive black holes ( $M87^{*}$ ) (Fig.
11(a) & (b)) and ( $SgrA^{*}$ ) (Fig. 11(c) & (d)).

It’s evident that under fixed parameters $\mu$ and $g$, the outermost Einstein
rings exhibit a decrease with an increasing magnitude of the parameter $q$
concerning both the supermassive black holes $SgrA^{*}$ and $M87^{*}$.
Additionally, it’s noteworthy that the outermost Einstein rings for the
modified Bardeen black hole exceed those of the ordinary regular Bardeen black
hole.

###  IV.3 Time delay in strong field Limit

Time delay is one of the most important observable by the strong gravitational
lensing phenomenon, which is obtained by the time difference between the
formation of two relativistic images. The time difference is caused when the
photon travels in a different path around the black hole. The time travel by
the different photon paths for the different relativistic images is different
and hence, there is a time difference between the different relativistic
images. If the time signals of two relativistic images are distinguished from
the observation, one can calculate the time delay between two signals [25].
The time taken by a photon to revolve around the black hole [25] is read as

$\tilde{T}=\tilde{a}log\biggr{(}\frac{u}{u_{ph}}-1\biggr{)}+\tilde{b}+\mathcal{O}(u-u_{ph})$
(34)

With the help of the above Eq.(34), one can compute the time difference
between two relativistic images.

For spherically static symmetric black hole spacetime, the time delay between
two relativistic images, when the relativistic images are on the same side of
the black hole, are obtained as

$\Delta T_{2,1}=2\pi u_{c}=2\pi D_{ol}\theta_{\infty}$ (35)

If the time delay $\Delta T_{2,1}$ between two relativistic images is obtained
with an accuracy of $5\%$ and the critical impact parameter $u_{ph}$ with a
negligible error is measured, it becomes feasible to determine the black hole
distance with an accuracy of $5\%$.

Numerical estimations of the time delay $\Delta T_{2,1}$ for various
supermassive black holes in the context of the standard Schwarzschild
($\mu=0$, $q=0$), ordinary regular Bardeen ($\mu=0$, $q=0.3$), and modified
Bardeen ($\mu=3$, $q=0.3$) black holes have been conducted (refer to Table 4).

The findings reveal that the time delay $\Delta T_{2,1}$ between two
relativistic images associated with the modified Bardeen black hole ($\mu=3$,
$g=0.2$, $q=0.3$) is significantly greater compared to the cases of the
standard Schwarzschild ($\mu=0$, $q=0$) and ordinary regular Bardeen ($\mu=0$,
$q=0.3$) black holes.

Table 4: Estimation of time delay for some supermassive BHs in the context of
Schwarzschild ($\mu=0$,$q=0$), ordinary regular Bardeen ($q=0.3,\mu=0$) and
modified Bardeen ($\mu=3$,$g=0.2$,$q=0.3$) black hole spacetimes. Mass(M) and
distance $D_{ol}$ respectively are taken in solar mass and Mpc units [95].
Time delays $\Delta T_{2,1}$ are estimated in minutes.  Time delays
---
$Galaxy$ |  $M(M_{\odot})$ |  $D_{ol}(Mpc)$ |  $\Delta T_{2,1}$ ($\mu=0,q=0$) |  $\Delta T_{2,1}$($\mu=0,q=0.3$) |  $\Delta T_{2,1}$($\mu=3,g=0.2,q=0.3$)
M87 |  $6.5\times 10^{9}$ |  $16.68$ |  $17378.9$ |  $16186.6$ |  $21000.2$
NGC 4472 |  $2.54\times 10^{9}$ |  16.72 |  6791.12 |  6325.3 |  8206.24
NGC 4395 |  $3.6\times 10^{5}$ |  4.3 |  0.962522 |  0.896499 |  1.16309
NGC1332 |  $1.47\times 10^{9}$ |  22.66 |  3930.3 |  3660.71 |  4749.28
NGC 7457 |  $8.95\times 10^{6}$ |  12.53 |  23.9294 |  28.2880 |  28.9157
NGC 1399 |  $8.81\times 10^{8}$ |  20.85 |  2355.5 |  2193.93 |  2846.34
NGC 1374 |  $5.90\times 10^{8}$ |  19.57 |  1577.47 |  1469.26 |  1906.17
NGC 4649 |  $4.72\times 10^{9}$ |  16.46 |  12619.7 |  11754.1 |  15249.4
NGC 3607 |  $1.37\times 10^{8}$ |  22.65 |  366.293 |  341.168 |  442.62
NGC 4459 |  $6.96\times 10^{7}$ |  16.01 |  186.088 |  173.323 |  224.864
NGC 4486A |  $1.44\times 10^{7}$ |  18.36 |  38.5009 |  35.86 |  46.5236
NGC 1316 |  $1.69\times 10^{8}$ |  20.95 |  451.85 |  420.857 |  546.006
NGC 4382 |  $1.30\times 10^{7}$ |  17.88 |  34.7577 |  32.3736 |  42.0004
NGC 5077 |  $8.55\times 10^{8}$ |  38.7 |  2285.9 |  2129.9 |  2762.34
NGC 7768 |  $1.34\times 10^{9}$ |  116.0 |  3582.72 |  3336.97 |  4329.28
NGC 4697 |  $2.02\times 10^{8}$ |  12.54 |  546 |  503.036 |  652.622
NGC 5128 |  $5.69\times 10^{7}$ |  3.62 |  152.132 |  141.697 |  183.833
NGC 5576 |  $2.73\times 10^{8}$ |  25.68 |  729.912 |  679.845 |  882.009
NGC 3608 |  $4.65\times 10^{8}$ |  22.75 |  1243.26 |  1157.98 |  1502.32
M32 |  $2.45\times 10^{6}$ |  0.806 |  6.5505 |  6.10118 |  7.91547
Cygnus A |  $2.66\times 10^{9}$ |  242.7 |  7111.97 |  6624.13 |  8593.93

##  V Comparison with observation:

Commonly understood as a specialized form of modified Bardeen black holes,
standard astrophysical entities such as ordinary regular Bardeen and
Schwarzschild black holes have been extensively studied. Prior investigations
have thoroughly explored their shadow casting and gravitational lensing. In
this study, we expand on previous works conducted by researchers such as
Bozza, Virbhadra, Ellis for the Schwarzschild black hole [24, 16, 25], He et
al. [102], and Stuchik and Schee [26, 18] focusing on regular Bardeen black
holes, along with Islam et al.’s work [15] examining Bardeen black holes
within 4D Einstein’s gravity.

Islam et al. specifically investigated strong gravitational lensing by the
Bardeen black hole in 4D Einstein’s Gauss-Bonnet gravity, determining black
hole parameters using data from supermassive black holes. Their research also
highlights that the regular Bardeen black hole solution is a unique outcome of
the Bardeen black hole within 4D Einstein’s Gauss-Bonnet gravity. Recently, He
et al. [102] scrutinized the shadow and observed properties of Bardeen black
holes within diverse accretion models.

This paper focuses on discussing the shadow and strong gravitational lensing
effects of the modified Bardeen black hole, juxtaposing it with the
Schwarzschild ($\mu=0$ and $q=0$) and ordinary regular Bardeen ($\mu=0$) black
holes.

In the shadow analysis, the numerical determination of the black hole’s shadow
radius is presented in Table.1. It’s evident that the radius of the modified
Bardeen black hole’s shadow exceeds that of both regular Bardeen and
Schwarzschild black holes. The angular diameter of the modified Bardeen black
hole’s shadow, as a function of parameters ($\mu/8M^{2}$ and $q/2M$), is
visualized in Fig.2. Interestingly, the angular diameter of the modified
Bardeen black hole surpasses that of the regular Bardeen black hole,
especially in relation to supermassive black holes like $M87^{*}$ and
$SgrA^{*}$ (refer to Fig.2). Moreover, within $1\sigma$ uncertainty, specific
angular diameter values, such as $\theta_{d}=39.4615\mu as$ for the modified
Bardeen black hole, $\theta_{d}=38.80\mu as$ for the Bardeen black hole, and
$\theta_{d}=39.9265\mu as$ for the Schwarzschild black hole, align with
measured angular diameters ($\theta_{d}=42\pm 3\mu as$ for $M87^{*}$ and
$\theta_{d}=51.8\pm 2.3\mu as$ for $SgrA^{*}$).

Concerning strong gravitational lensing, we employ the method proposed by
Bozza [24] to differentiate various static spherically symmetric black holes,
analyzing astrophysical outcomes through supermassive black holes (see Table.3
& Table.5) concerning modified Bardeen, ordinary regular Bardeen, and
Schwarzschild black holes. Additionally, we estimate observable quantities
such as angular position $\theta_{\infty}$, $S$, and $r_{mag}$ for these black
holes, employing the supermassive black hole $NGC4649$ with a mass of
$M=4.72\times 10^{9}$ and a distance of $D_{OL=0.008}$ Mpc for numerical
comparisons.

Our estimations, considering supermassive black holes with similar mass and
distance, consistently reveal that the angular position of the innermost image
$\theta_{\infty}$ and angular separation $S$ consistently surpass those of
ordinary regular Bardeen and Schwarzschild black holes. Additionally, the
modified Bardeen black hole showcases larger relative magnifications.
Numerically, differences in $\theta_{\infty}$, $S$, and $r_{mag}$ between the
modified Bardeen ($\mu=3$,$g=0.2$, $q=0.3$) and ordinary regular Bardeen
($\mu=0$, $q=0.3$) are approximately $5.9\mu as$, $0.3\mu as$, and $3.1$
magnitude, respectively. Likewise, differences between the modified Bardeen
($\mu=3$,$g=0.2$, $q=0.3$) and standard Schwarzschild black hole ($\mu=0$,
$q=0$) are roughly $4.9\mu as$, $0.18\mu as$, and $2.2$ magnitude,
respectively. Noteworthy ranges include $\theta_{\infty}\in(19.59,19.82)$,
$S\in(0.00297,0.003138)$, and $r_{mag}\in(3.47,3.5)$ when $\mu=03$, $g=0.2$,
and $0\leq|q|\leq 3$. These findings suggest that the outermost image for the
modified Bardeen black hole is considerably closer to the innermost images,
potentially allowing for separation from other black hole images.

In essence, the ability to detect the outermost relativistic image could
facilitate distinguishing the modified Bardeen black hole from other standard
astrophysical black holes like Schwarzschild and ordinary regular Bardeen
black holes using technology. However, observationally, this proves
challenging, given the angular separation of the relativistic images not
exceeding $\sim 0.3\mu as$. Moreover, the Einstein’s ring
$\theta^{E}_{\infty}$ for the modified Bardeen black hole is observed to be
larger than that of other standard astrophysical black holes like
Schwarzschild and ordinary regular Bardeen black holes (refer to Fig.11).

Additionally, analysis (refer to Table.4) reveals that time delays between two
relativistic images for a modified Bardeen black hole ($\sim 15249.4$ minutes)
significantly exceed those of other astrophysical black holes such as the
Schwarzschild black hole ($\sim 12619.7$ minutes) and ordinary regular Bardeen
black holes ($\sim 11754.1$ minutes) in the context of the supermassive black
hole $NGC4649$. This discrepancy suggests that distinguishing the first and
second relativistic images from observation may offer a better chance of
differentiating the modified Bardeen black hole from other astrophysical black
holes like Schwarzschild or ordinary regular Bardeen black holes.
Consequently, a modified Bardeen black hole could be quantitatively
distinguished from a Schwarzschild or ordinary regular Bardeen black hole.

The identification and confirmation of a modified Bardeen black hole, if
achieved, would bear several substantial implications and consequences for our
understanding of black holes and general relativity. Some potential
implications include:

  * •

Providing empirical evidence for alternative theories of gravity beyond the
predictions of general relativity.

  * •

Displaying distinct astrophysical signatures and observational features
compared to standard black holes, deepening our comprehension of their
underlying physics.

  * •

Offering insights into the nature of dark matter, potentially constraining
modified gravity theories that aim to explain dark matter effects.

  * •

Challenging the no-hair theorem in general relativity if the modified Bardeen
black hole exhibits additional parameters or properties beyond mass, charge,
and angular momentum.

  * •

Pushing the boundaries of our understanding of fundamental physics, inspiring
new theoretical investigations and potentially leading to the development of
more comprehensive theories explaining black hole behavior in modified gravity
scenarios.

Table 5: Estimation of observables by taking the supermassive black hole
$NGC4649$ having mass $M=4.72\times 10^{9}M_{\odot}$ and distance
$D_{OL}=16.46Mpc$ in the context of Schwarzschild, ordinary regular
Bardeen,and modified Bardeen black hole spacetimes.  Comparison with
observables
---
|  Schwarzschild BH |  Regular Bardeen BH ($\mu=0$) |  Modified Bardeen BH ($\mu=3,g=0.2$)
|  $(\mu=0,q=0)$ |  $|q|=0.0$ $|q|=0.1$ $|q|=0.2$ $|q|=0.4$ |  $|q|=0.0$ $|q|=0.1$ $|q|=0.2$ $|q|=0.4$
$\theta_{\infty}(\mu as)$ |  14.6901 |  $14.6656$ $14.591$ $14.27$ $13.6825$ |  $19.8192$ $19.7998$ $19.7227$ $19.5959$
S$(\mu as)$ |  0.0184 |  $0.018$ $0.0196$ $0.024$ $0.037$ |  $0.00313$ $0.00312$ $0.00307$ $0.00297$
$r_{mag}$ |  6.82188 |  $6.80282$ $6.74425$ $6.48595$ $5.92957$ |  $8.95283$ $8.95941$ $8.98787$ $9.04274$
$u_{ph}/R_{sh}$ |  2.59808 |  $2.59373$ $2.58054$ $2.52504$ $2.41987$ |  $3.5052$ $3.50177$ $3.48813$ $3.4657$
$\bar{a}$ |  1.00 |  $1.0028$ $1.01151$ $1.05179$ $1.15041$ |  $0.761981$ $0.761421$ $0.75901$ $0.754404$
$\bar{b}$ |  -0.40023 |  $-0.401608$ $-0.406055$ $-0.429858$ $-0.511002$ |  $-0.384548$ $-0.381893$ $-0.37101$ $-0.35205$

##  VI Results and Conclusions

In this study, we have comprehensively explored the observational
characteristics of the modified Bardeen black hole through shadow and strong
gravitational lensing observations, contrasting these features with those of
other astrophysical black holes like the Schwarzschild black hole and the
ordinary regular Bardeen black hole. Our analysis delved into understanding
how the parameters associated with the modified Bardeen black hole influence
its observable shadow and strong lensing effects.

To commence, we derived the null geodesics pertinent to the modified Bardeen
black hole using the Hamiltonian-Jacobi action and conducted an extensive
review of the photon orbit encompassing this specific black hole. Employing
numerical techniques, we estimated both the photon sphere radius and shadow
radius. Notably, we observed a distinct trend: for fixed values of the
parameters $\mu$ and $g$, these radii exhibited a decrease with an increase in
the magnitude of the charge parameter $q$. Conversely, for fixed values of $q$
and $g$, these radii displayed an increase with a rise in the magnitude of
$\mu$. Furthermore, our investigation revealed that the shadow radius
associated with the modified Bardeen black hole is notably larger than that of
both the Schwarzschild black hole and the ordinary regular Bardeen black hole.

Our study also involved determining the angular diameter of the black hole
shadow concerning the parameters $\mu$ and $q$ for a fixed parameter $g$
(specifically, $g=0.2$). We specifically considered this parameter setup in
relation to supermassive black holes such as $M87^{*}$ and $SgrA^{*}$. Our
analysis vividly demonstrates that the angular diameter of the black hole
shadow attributed to the modified Bardeen black hole surpasses that of both
the Schwarzschild and regular Bardeen black holes. Moreover, employing the EHT
(Event Horizon Telescope) collaboration’s data on the angular shadow diameter
of $M87^{*}$ and $SgrA^{*}$, we were able to confine the parameter ranges of
$\mu$ and $q$ for the modified Bardeen black hole. Our observed constrained
ranges are as follows: $-0.89\leq\mu/8M^{2}\leq 0.4$ and $0\leq|q|\leq 0.185$
for $M87^{*}$; and $-1.38\leq\mu/8M^{2}\leq 0.1$ and $0\leq|q|\leq 0.058$ for
$SgrA^{*}$, maintaining the fixed value of $g/2M=0.2$.

These findings highlight the viability of modified Bardeen black holes as
potential astrophysical black hole candidates, notably incorporating
additional parameters $\mu$, $g$, and $q$ alongside the black hole’s mass $M$,
akin to the supermassive black holes $M87^{*}$ and $SgrA^{*}$. Moreover, the
constraints derived from the EHT data significantly narrow down the parameter
space ($\mu$, $q$) for the modified Bardeen black hole, adding credence to its
astrophysical relevance.

Subsequently, we went through an extensive examination of the strong
gravitational lensing phenomena caused by the modified Bardeen black hole,
shedding light on its consequential astrophysical implications. Our focus
centered on scrutinizing the influence of the parameters $\mu$, $g$, and $q$
associated with the modified Bardeen black hole on the strong deflection angle
and other crucial strong lensing observables.

Our investigative approach involved a meticulous reexamination of the null
geodesic equations, coupled with numerical estimations to ascertain the photon
radius, ultimately leading to the derivation of the lensing coefficients
$\bar{a}$, $\bar{b}$, and $u_{ph}/R_{sh}$. The insights garnered from our
analyses revealed intriguing trends: when considering fixed values of $\mu$
and $g$, we observed that $\bar{a}$ exhibited an increase while $\bar{b}$
displayed a decrease with the escalating magnitude of the charge parameter
$q$. Conversely, for fixed values of $q$ and $g$, $\bar{a}$ depicted a
decrease while $\bar{b}$ showcased an increase with the elevation of the
parameter $\mu$. Notably, our observations regarding the deflection angle
$\alpha_{D}$ unveiled a nuanced pattern wherein it initially experiences a
slight increase, achieves a maximal value, and subsequently diminishes with
the rising magnitude of the charge parameter $q$, given fixed values of $\mu$
and $g$. However, the deflection angle consistently diminishes with increasing
$\mu$, maintaining fixed values of $q$ and $g$. Notably, in comparison to
other astrophysical black holes like the Schwarzschild black hole and the
ordinary regular Bardeen black hole, the deflection angle associated with the
modified Bardeen black hole manifests as relatively smaller.

Furthermore, we conducted numerical estimations to determine the strong
lensing observables pertinent to the relativistic images, specifically
focusing on the modified Bardeen black hole within the context of supermassive
black holes such as $M87^{*}$, $SgrA^{*}$, and $NGC7457$. Our findings
unveiled notable disparities: the angular position $\theta_{\infty}$ and
magnification $r_{\text{mag}}$ associated with the relativistic images for the
modified Bardeen black hole surpassed those observed for both the
Schwarzschild black hole and the ordinary regular Bardeen black hole. However,
it’s worth noting that the angular separation $S$ between the relativistic
images concerning the modified Bardeen black hole was relatively smaller
compared to the Schwarzschild black hole and the ordinary regular Bardeen
black hole. To offer more precise insights, we outlined specific numerical
ranges for these observables, $\theta_{\infty}$ and $S$, accommodating various
values of the parameters $\mu$, $g$, and $q$, considering the gravitational
influence of the supermassive black holes $M87^{*}$, $SgrA^{*}$, and
$NGC7457`$. In the cases, where $\mu=0$ and $0\leq|q|\leq
0.4$,$\theta_{\infty}\in(17.06,19.97)\mu as$ for
$M87^{*}$,$\theta_{\infty}\in(22.5,26.3)\mu as$ for $SgrA^{*}$ and
$\theta_{\infty}\in(0.031,0.037)\mu as$ for $NGC7457$; when $\mu=1$ ,$g=0.2$
and $0<|q|\leq 0.4$,$\theta_{\infty}\in(21.5,22.8)\mu as$ for
$M87^{*}$,$\theta_{\infty}\in(28.3,30)\mu as$ for $SgrA^{*}$ and
$\theta_{\infty}\in(0.03,0.042)\mu as$ for $NGC7457$; when $\mu=1$ ,$g=1.2$
and $0<|q|\leq 0.4$,$\theta_{\infty}\in(19.2,21.44)\mu as$ for
$M87^{*}$,$\theta_{\infty}\in(25.3,28.3)\mu as$ for $SgrA^{*}$ and
$\theta_{\infty}\in(0.35,0.4)\mu as$ for $NGC7457$; when $\mu=3$ ,$g=0.2$ and
$0<|q|\leq 0.4$,$\theta_{\infty}\in(26.3,26.94)\mu as$ for
$M87^{*}$,$\theta_{\infty}\in(34.8,35.53)muas$ for $SgrA^{*}$ and
$\theta_{\infty}\in(0.048,0.05)\mu as$ for $NGC7457$; and when $\mu=1$
,$g=0.2$ and $0<|q|\leq 0.4$,$\theta_{\infty}\in(23.6,24.7)\mu as$ for
$M87^{*}$,$\theta_{\infty}\in(31.19,32.58)\mu as$ for $SgrA^{*}$ and
$\theta_{\infty}\in(0.048,0.05)\mu as$ for $NGC7457$. Moreover, the angular
separation $S\in(0.024,0.14)\mu as$ for $M87^{*}$, $S\in(0.032,0.184)\mu as$
for $SgrA^{*}$ ,$S\in(4.5\times 10^{-5},2.54\times 10^{-4})\mu as$ for NGC
$7457$ for the case when $\mu=0$ and $0\leq|q|\leq 0.4$; $S\in(0.014,0.019)\mu
as$ for $M87^{*}$, $S\in(0.019,0.025)\mu as$ for $SgrA^{*}$ ,$S\in(2.6\times
10^{-5},3.5\times 10^{-5})\mu as$ for NGC $7457$ for the case when $\mu=1$
,$g=0.2$ and $0<|q|\leq 0.4$; $S\in(0.03,0.19)\mu as$ for $M87^{*}$,
$S\in(0.04,0.16)\mu as$ for $SgrA^{*}$ ,$S\in(5.5\times 10^{-5},2.18\times
10^{-4})\mu as$ for NGC $7457$ for the case when $\mu=1$,$g=1.2$ and
$0<|q|\leq 0.4$; $S\in(0.003,0.0043)\mu as$ for $M87^{*}$,
$S\in(0.005,0.0057)\mu as$ for $SgrA^{*}$ ,$S\in(6.9\times 10^{-6},7.8\times
10^{-6})\mu as$ for NGC $7457$ for the case when $\mu=3$,$g=0.2$ and
$0<|q|\leq 0.4$; $S\in(0.022,0.025)\mu as$ for $M87^{*}$, $S\in(0.02,0.033)\mu
as$ for $SgrA^{*}$ ,$S\in(4\times 10^{-5},6.02\times 10^{-5})\mu as$ for NGC
$7457$ for the case when $\mu=3$,$g=1.2$ and $0<|q|\leq 0.4$. Considering the
supermassive black holes $M87^{*}$ and $SgrA^{*}$, the outermost Einstein’s
ring $\theta^{E}_{n}$ has been displayed in Fig.11 for the cases of modified
Bardeen ($\mu=3,g=0.2$) as well as ordinary regular Bardeen ($\mu=0$) black
holes.

Our analysis has notably unveiled that the outermost Einstein rings, denoted
as $\theta^{E}_{n}$, associated with the modified Bardeen black hole are
significantly larger in comparison to those observed with the ordinary regular
Bardeen black hole. This crucial distinction implies that the modified Bardeen
black hole showcases a notably wider angular separation between multiple
relativistic images. This increased separation not only enhances its
detectability but also serves as a distinguishing factor setting it apart from
the characteristics exhibited by the ordinary regular Bardeen black hole.

Moreover, in our exploration of various supermassive black holes, we conducted
an investigation into the time delay between the first and second-order
relativistic images pertaining to the modified Bardeen, ordinary Bardeen, and
Schwarzschild black holes. Our findings revealed a substantial contrast, with
the time delay for the modified Bardeen black hole being approximately $\sim
15249.4$ minutes. This duration notably surpasses the time delays observed for
the Schwarzschild black hole ($\sim 12619.7$ minutes) and the ordinary regular
Bardeen black holes ($\sim 11754.1$ minutes) within the context of the
supermassive black hole $NGC4649$. This substantial disparity suggests that
the modified Bardeen black hole presents distinctive temporal signatures,
offering a promising avenue for its potential detection and differentiation
from other prevalent astrophysical black holes.

## Acknowledgements

N.U.M would like to thank CSIR, Govt. of India for providing Senior Research
Fellowship (No. 08/003(0141))/2020-EMR-I).

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