# Testing Strong Gravitational Lensing Effects of Supermassive Black Holes
with String-Inspired Metric, EHT Constraints and Parameter Estimation

Amnish Vachher<EMAIL_ADDRESS>Centre for Theoretical Physics,
Jamia Millia Islamia, New Delhi 110025, India  Shafqat Ul Islam
<EMAIL_ADDRESS>Astrophysics and Cosmology Research Unit, School of
Mathematics, Statistics and Computer Science, University of KwaZulu-Natal,
Private Bag 54001, Durban 4000, South Africa  Sushant G. Ghosh
<EMAIL_ADDRESS>Centre for Theoretical Physics, Jamia Millia Islamia, New
Delhi 110025, India  Astrophysics and Cosmology Research Unit, School of
Mathematics, Statistics and Computer Science, University of KwaZulu-Natal,
Private Bag 54001, Durban 4000, South Africa

###### Abstract

We examine and compare the gravitational lensing, in the strong field limit,
for the spherically symmetric string-inspired Euler-Heisenberg black holes,
characterized by additional parameters $Q^{2}$ and $\alpha-\beta$,
representing magnetic charge and coupling constant, respectively. Our analysis
reveals a reduction in the photon sphere radius $x_{ps}$, critical impact
parameter $u_{ps}$ and angular position $\theta_{\infty}$ with increasing
magnitude of $Q^{2}$ and $\alpha-\beta$. Consequently, the value of these
quantities is consistently lower than that of its GR equivalents. Further, the
ratio $r_{mag}$ of the flux of the first image to all others decreases with
$Q^{2}$ and $\alpha-\beta$. Unlike Schwarzschild black holes, string-inspired
Euler-Heisenberg black holes have a smaller deflection angle $\alpha_{D}$,
which decreases even more as $Q^{2}$ increases. Moreover, the time delay for
Sgr A* and M87* can reach up to $~{}11.302$ and $~{}17085.1$ minutes,
respectively, at $Q^{2}=0.1$ and $\alpha-\beta=-1$, deviating from
Schwarzschild black holes by $~{}0.194$ and $~{}293.6$ minutes which are not
very significant. For Sgr A* and M87*, we determine $\theta_{\infty}$ to range
within $(23.81,26.28)~{}\mu as$ and $(17.89,19.78)~{}\mu as$ respectively,
with angular separations $s$ ranging from $(3.33-5.67)~{}nas$ for Sgr A* and
$(2.51-4.26)~{}nas$ for M87*. EHT bounds on the $\theta_{sh}$ of Sgr A* and
M87* within the $1\sigma$ region, bound the parameters $Q^{2}$ and
$\alpha-\beta$ as: for Sgr A* $0.29278\leq Q^{2}\leq 0.60778$ and for M87*
$0<Q^{2}\leq 0.08473$, but in both the cases we found no bound on the
parameter $\alpha-\beta$. Our investigation reveals that a significant part of
the parameter space of string-inspired Euler-Heisenberg black holes, falling
within the $1\sigma$ range, is consistent with findings from the Event Horizon
Telescope (EHT) concerning M87* and Sgr A*. This suggests that these black
holes, inspired by string theory, could be strong contenders for astrophysical
black holes. We also estimate the parameters $\alpha-\beta$ and $Q^{2}$
associated with string-inspired Euler-Heisenberg black holes using the EHT
observation results of Sgr A* and M87*.

##  I Introduction

The validation of Einstein’s general relativity (GR) has been significantly
strengthened by recent observations made by the Event Horizon Telescope (EHT),
which has provided remarkable insights into the shadows cast by supermassive
black holes M87* EventHorizonTelescope:2019dse  and the SgrA*
EventHorizonTelescope:2022wkp . The observations made by the EHT provide
strong support for the validity of GR and offer valuable insights into the
intriguing phenomenon of strong gravitational lensing. As predicted by
Einstein’s general relativity, gravitational lensing is a crucial tool for
exploring spacetime and studying astrophysical objects, especially black
holes. It has been extensively applied across various astrophysical scenarios,
aiding in understanding galaxy cluster dynamics and revealing intricate
structures around supermassive black holes in galactic centres. Gravitational
lensing manifests as the strong gravitational pull of a black hole bends
light, leading to the emergence of a distinct ”black hole shadow” at its core
and a luminous ”photon ring” encircling it EventHorizonTelescope:2019dse ;
EventHorizonTelescope:2022wkp . This phenomenon arises directly from the
profound light bending induced by the black hole’s gravity, providing a
distinctive avenue for investigating gravitational effects on a grand scale.

The exploration of gravitational lensing has a rich history, dating back to
the inception of General Relativity (GR) itself Perlick:2010zh . The study of
gravitational lensing has proven to be invaluable in understanding the
structure of spacetime Einstein:1936llh ; Liebes:1964zz ; Mellier:1998pk ;
Bartelmann:1999yn ; Schmidt:2008hc ; Barnacka:2013lfa . From a strong-
gravitational lensing perspective, light rays near black holes can curve
multiple times before reaching an observer. Notably, Darwin unknown-author-no-
date  was among the first to employ gravitational lensing concepts in the
context of Schwarzschild black holes and further developed Luminet:1979nyg .
Recent theoretical studies on strong-field gravitational lensing, mainly
credited to Virbhadra and Ellis Virbhadra:1999nm ; Virbhadra:2002ju ,
encompass discussions on the formation and position of the ring and the
magnification of relativistic images around Schwarzschild black holes. Later,
Frittelli, Killing, and Newman Frittelli:1999yf  provided more rigorous
analytical descriptions of the lens equation and its solution, comparing their
findings with those of Virbhadra and Ellis Virbhadra:1999nm ; Virbhadra:2002ju
. They Virbhadra:1999nm  found relativistic images on each side of the optical
axis alongside the primary and secondary images in Schwarzschild black hole
lensing. Subsequently, Bozza Bozza:2001xd ; Bozza:2002zj ; Bozza:2007gt ;
Bozza:2002af  and Tsukamoto Tsukamoto:2016jzh ; Tsukamoto:2017fxq  introduced
novel methods for studying strong lensing in various spherically symmetric
static spacetimes. Torres Eiroa:2002mk  derived analytical expressions for the
positions and magnifications of relativistic images in Reissner-Nordström
black hole lensing. The strong-field gravitational lensing by black holes is
also used as a means to test the predictions of general relativity
Virbhadra:1999nm ; Frittelli:1999yf ; Bozza:2001xd ; Bozza:2002zj ;
Bozza:2002af ; Bozza:2010xqn ; Eiroa:2002mk ; Iyer:2006cn ; Tsukamoto:2016jzh
; Virbhadra:2007kw ; Shaikh:2019jfr . Notably, numerical methods developed by
Virbhadra and Ellis Virbhadra:1999nm , along with analytical approaches by
Bozza Bozza:2001xd ; Bozza:2002zj ; Bozza:2002af ; Bozza:2010xqn  and
Tsukamoto Tsukamoto:2016jzh , have significantly contributed to our
understanding of strong-field gravitational lensing. These methods have gained
general acceptance in the scientific community and are frequently applied to
investigate strong-gravitational lensing by black hole in modified gravity
theories Ghosh:2020spb ; Whisker:2004gq ; Abbas:2021whh ; Eiroa:2005ag ;
Gyulchev:2006zg ; Ghosh:2010uw ; Gyulchev:2012ty ; Molla:2023hog ;
Grespan:2023cpa ; Kumar:2022fqo ; KumarWalia:2022ddq ; Kumar:2021cyl ;
Lu:2021htd ; Ali:2021psk ; Hsieh:2021scb . The strong gravitational field
remains a focal point of research, with recent works exploring lensing effects
from miscellaneous black holes Chen:2009eu ; Sarkar:2006ry ; Javed:2019qyg ;
Shaikh:2019itn  and modifications to Schwarzschild geometry Eiroa:2010wm ;
Ovgun:2019wej ; Panpanich:2019mll ; Bronnikov:2018nub ; Shaikh:2018oul ;
Lu:2021htd ; Babar:2021nst , including in the higher curvature gravity
Kumar:2020sag ; Islam:2020xmy ; Narzilloev:2021jtg . In particular, strong-
gravitational lensing near black holes and compact objects unveils phenomena
such as shadows, photon rings, and relativistic images Synge:1966okc ;
Gralla:2019xty ; darwin1959gravity ; Cunha:2018acu ; Bozza:2010xqn ;
Bozza:2001xd ; Bozza:2002zj ; Bozza:2002af .

The strong field gravitational lensing effects caused by a charged black hole
within the framework of heterotic string theory (GHS black
holes)Gibbons:1987ps ; Garfinkle:1990qj  was performed by Bhadra Bhadra:2003zs
and compared with the Reissner-Nordström black hole. Their findings suggested
that no notable string-related influences were observed in the gravitational
lensing characteristics under conditions of strong gravitational fields.
Sharif and Iftikhar Sharif:2017abc  studied the strong-gravitational lensing
of charged black holes in string theory and emphasized the significant
influence of the charge parameter. In the low-energy limit of string theory,
Younes et al. Younesizadeh:2022czv  investigated the gravitational lensing and
shadow of charged black holes. They found that the dilaton parameter
significantly increases the size of the black hole shadow and the position of
the innermost image. Molla et al. Molla:2024lpt  examined the gravitational
lensing behaviour of spherically symmetric $\alpha$-corrected Reissner-
Nordström black holes derived from heterotic superstring effective field
theory compactified on $T^{6}$. The parameter $\alpha$ acts as a correction
term for the Reissner-Nordström black hole, representing the Regge slope
parameter within the effective action of the heterotic superstring, as
discussed in Cano et al. (cano2020alpha, ). Their analysis uncovers
significant distinctions compared to Reissner-Nordström’s black hole case
Molla:2024lpt .

Inspired by the aforementioned context, this study focuses on the
gravitational lensing phenomena produced by the string-inspired Euler-
Heisenberg black holes Bakopoulos:2024hah  and compares and contrasts these
phenomena with those predicted by the GHS black holes Bhadra:2003zs  and the
Schwarzschild black holes Bozza:2002zj . Furthermore, by considering
supermassive black holes such as Sgr A* and M87* as the gravitational lens, we
analyze the positions, separations, magnifications, and time delays in forming
relativistic images. Specifically, we examine how the parameter $\alpha-\beta$
affects various gravitational lensing observables and the time delay observed
between relativistic images. In addition to investigating the impact of the
black hole charge $Q$, we also intend to explore the influence of the
parameter $\alpha-\beta$ on black hole lensing characteristics.

The paper is organized as follows: The spacetime structure of String-inspired
Euler-Heisenberg Black holes is briefly reviewed for completeness, with a
particular focus on the parameter space and black hole horizons in Sec. II.
The framework for gravitational lensing, which encompasses the lens equation,
deflection angle, and coefficients for strong lensing, is outlined in Sec.
III. We analyze strong-gravitational lensing observables in Section IV, which
include the position of the innermost image ($\theta_{\infty}$), the
separation between images ($s$), the flux ratio of the first image to all
others ($r_{mag}$), and the time delay between the first and second images for
supermassive black holes such as Sgr A* and M87* acting as lenses. Section V
focuses on constraining and estimating parameters for the string-inspired
Euler-Heisenberg black hole using observational data from EHT regarding the
angular shadow diameter of Sgr A* and M87*.

We conclude our findings in Section VI, wrapping up the paper. We have
employed units where the speed of light and the gravitational constant are set
to $8\pi G=c=1$; however, these values are reinstated in the tables.

##  II String-inspired Euler-Heisenberg Black holes

Here, we briefly reviewed the String-inspired Euler-Heisenberg black holes the
metric, which is constructed as a modification from the metric of the GHS
black hole with additional parameter $\alpha-\beta$ apart from mass $M$ and
charge $Q$. Here, we start with a string-loop corrected effective action,
which, in the Einstein frame, is given by

$\displaystyle\mathcal{S}=\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}\Big{[}\mathcal{R}-2\nabla^{\mu}\phi\nabla_{\mu}\phi-e^{-2\phi}\mathcal{F}^{2}-$
(1) $\displaystyle
f(\phi)\big{(}2\alpha\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}-\beta\mathcal{F}^{4}\big{)}\Big{]}~{}.$

It is the simplified version of the Einstein-frame action functional,
resulting from a non-diagonal reduction of the Gauss-Bonnet action
Charmousis:2012dw . An exact solution within this framework is the GMGHS or
GHS black hole Gibbons:1987ps ; Garfinkle:1990qj  when $f(\phi)=0$. In the
provided action (1), key components include $\mathcal{R}$ representing the
Ricci scalar,
$\mathcal{F}^{2}\equiv\mathcal{F}{\mu\nu}\mathcal{F}^{\mu\nu}\sim{E}^{2}-{B}^{2}$
as the Faraday scalar, and
$\mathcal{F}^{4}\equiv\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}\mathcal{F}_{\alpha\beta}\mathcal{F}^{\alpha\beta}$,
where $\mathcal{F}_{\mu\nu}$ denotes the field strength
$\mathcal{F}_{\mu\nu}=\partial{\mu}\mathcal{A}{\nu}-\partial{\nu}\mathcal{A}_{\mu}$,
and $\alpha,\beta$ are coupling constants with dimensions of (length)2,
treated phenomenologically. The scalar field $\phi$ and the associated scalar
function $f(\phi)$ are dimensionless. The field equations emanating from (1)
are of the following form

$\displaystyle G_{\mu\nu}=2\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\partial^{\alpha}\phi\partial_{\alpha}\phi+2e^{-2\phi}(\mathcal{F}_{\mu}^{~{}\alpha}\mathcal{F}_{\nu\alpha}-$
(2)
$\displaystyle\frac{1}{4}g_{\mu\nu}\mathcal{F}^{2})+f(\phi)(8\alpha\mathcal{F}_{\mu}^{~{}\alpha}\mathcal{F}_{\nu}^{~{}\beta}\mathcal{F}_{\alpha}^{~{}\eta}\mathcal{F}_{\beta\eta}-$
$\displaystyle\alpha
g_{\mu\nu}\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}r-4\beta\mathcal{F}_{\mu}^{~{}\xi}\mathcal{F}_{\nu\xi}\mathcal{F}^{2}+\frac{1}{2}g_{\mu\nu}\beta\mathcal{F}^{4})~{},$

$\displaystyle
4\square\phi=-2e^{-2\phi}\mathcal{F}^{2}+\frac{df(\phi)}{d\phi}\left(2\alpha\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}-\beta\mathcal{F}^{4}\right)~{},$
(3)

$\displaystyle\partial_{\mu}\Big{\\{}\sqrt{-g}\Big{[}4\mathcal{F}^{\mu\nu}\left(2\beta
f(\phi)\mathcal{F}^{2}-e^{-2\phi}\right)-16\alpha\mathcal{F}^{\mu}{}_{\kappa}\mathcal{F}^{\kappa}{}_{\lambda}\mathcal{F}^{\nu\lambda}\Big{]}\Big{\\}}=0$
(4)

By incorporating the higher-order electromagnetic invariants $\mathcal{F}^{4}$
and
$\mathcal{F}^{\alpha}{~{}\beta}\mathcal{F}^{\beta}{\gamma}\mathcal{F}^{\gamma}{~{}\delta}\mathcal{F}^{\delta}{\alpha}$,
we aim to expand upon the GMGHS solution Gibbons:1987ps ; Garfinkle:1990qj .
It involves introducing the general spherically symmetric metric ansatz, which
can be rewritten as

$ds^{2}=-F(r)dT^{2}+\frac{dr^{2}}{F(r)}+[R(r)]^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (5)

The functions $F(r)$ and $R(r)$ are unspecified and will be determined using
the field equations. We also include electric and magnetic charges using the
compatible spherical symmetric four-vector:
$\mathcal{A}_{\mu}=(V(r),0,0,Q\cos\theta)$ with $Q$ representing the magnetic
charge. This choice of $\mathcal{A}_{\mu}$ satisfies the $\varphi$ component
of the Maxwell equations, assuming that the scalar field $\phi$ only depends
on $r$. The values of the coupling constants $\alpha$ and $\beta$ determine
whether the contribution of the non-linear electromagnetic terms is
significant. When $\alpha=\beta$, the higher-order electromagnetic term has no
effect; however, for $\alpha\neq\beta$, the behaviour of the non-linear
electromagnetic terms depends on the signs of $\alpha$ and $\beta$: they can
either attract or repel.

$\displaystyle
2\alpha\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}-\beta\mathcal{F}^{4}=\frac{4(\alpha-\beta)Q^{4}}{[R(r)]^{8}}+\frac{8\beta
Q^{2}[V^{\prime}(r)]^{2}}{[R(r)]^{4}}+$ $\displaystyle
4(\alpha-\beta)[V^{\prime}(r)]^{4}~{},$

where the prime notation signifies differentiation for $r$. Let’s consider a
non-trivial coupling function $f(\phi)$

$f(\phi)=-[3\cosh(2\phi)+2]\equiv-\frac{1}{2}\left(3e^{-2\phi}+3e^{2\phi}+4\right)~{}.$
(6)

The coupling function $f(\phi)$ includes the dilatonic coupling $e^{2\xi\phi}$
with $\xi=\pm 1$ and a constant term. The exponential dilaton terms in the
coupling function (6) can be expressed as
$f(\phi)=-\frac{3}{2}(g_{s}^{-2}+g_{s}^{2})-2$, where $g_{s}=\exp(\phi)$
represents the string coupling, $g_{s}^{-2}$ which is tree-level dilaton-
Maxwell term coupling Gross:1986mw ; Green:2012oqa ; Green:2012pqa , and
$g_{s}^{2}$ denotes corrections from two-string-loop. To find a solution to
the field equations (1), taking into account (6), an exact solution for a
magnetically charged black hole is given by Bakopoulos:2024hah  \- namely
String-inspired Euler-Heisenberg black holes as

$\displaystyle
F(r)=1-\frac{2M}{r}-\frac{2(\alpha-\beta)Q^{4}}{r^{3}\left(r-\frac{Q^{2}}{M}\right)^{3}}~{},$
(7) $\displaystyle[R(r)]^{2}=r\left(r-\frac{Q^{2}}{M}\right).$ (8)

It is noteworthy that when $\alpha=\beta$, the metric (5) contains the GHS
solution Gibbons:1987ps ; Garfinkle:1990qj  and the Schwarzschild solution
Schwarzschild:1916abc  when $Q^{2}=0$. Note that when the radial coordinate
$r\in(Q^{2}/M,+\infty)$, then $R\in(0,+\infty)$. The sign of coupling
constants $\alpha-\beta$ is also important in determining the contribution of
the higher-order electromagnetic terms. The parameter space
[$(\alpha-\beta)/M^{2},Q^{2}/M$] for the string-inspired Euler-Heisenberg
black holes is illustrated in Fig. 1. The solid line separating the black hole
region from the no-black hole region corresponds to an extremal string-
inspired Euler-Heisenberg BH with degenerate horizons. The light blue (black
hole) region corresponds to the values of parameters as $Q^{2}\in(0,2)$ and
$(\alpha-\beta)\in(-\infty,0)$.

Figure 1: The parameter space [$(\alpha-\beta)/M^{2},Q^{2}/M$] for the string-
inspired Euler-Heisenberg black holes. The solid line separates the black hole
region from the no-black hole region.

For the discussion on gravitational lensing, we rewrite the metric (5) by
redefining the following quantities $r,T,\alpha-\beta,Q^{2}$ in the unit of M
as

$d{\widetilde{s}}^{2}=-A(x)dt^{2}+\frac{1}{A(x)}dx^{2}+C(x)d\Omega^{2}~{},$
(9) $\displaystyle
A(x)=1-\frac{2}{x}+\frac{2(\alpha-\beta)Q^{4}}{x^{3}\left(x-Q^{2}\right)^{3}},$
(10) $\displaystyle C(x)=x\left(x-Q^{2}\right),$

The horizons of the string-inspired Euler-Heisenberg black holes can be
obtained from $g^{rr}=A(x)=0$, that is

$A(x)=1-\frac{2}{x}+\frac{2(\alpha-\beta)Q^{4}}{x^{3}\left(x-Q^{2}\right)^{3}}=0.$
(11)

The elementary analysis of $A(x)=0$ suggests that, depending on the values of
parameters $Q^{2}$ and $\alpha-\beta$, $A(x)$ admits up to two real positive
roots. The sign of $\alpha-\beta$ for coupling constants decides whether the
black holes have two horizons (for $(\alpha-\beta)<0$) or one horizon (for
$(\alpha-\beta)>0$) Bakopoulos:2024hah . The event horizon ($x_{+}$) decreases
slowly with the parameter $Q^{2}$ increasing. Still, the Cauchy horizon
($x_{-}$) increases rapidly with increasing $Q^{2}$ and at a certain value, it
merges with the event horizon. (c.f. Fig. 2).

Figure 2: Position of the event horizon ($x_{+}$) represented by solid lines
and Cauchy horizon ($x_{-}$) represented by dashed lines for string-inspired
Euler-Heisenberg black holes along with parameter $Q^{2}$ at different values
of the parameter $\alpha-\beta$.

##  III Strong gravitational lensing by Black hole

The deflection angle and lens equation govern the strong gravitational lensing
by a black hole Bozza:2002zj ; Bozza:2001xd . The null geodesic equation
satisfies the equation reads as

$\left(\frac{dx}{d\tau}\right)^{2}\equiv\dot{x}^{2}={\mathcal{E}}^{2}-\mathcal{L}^{2}\frac{A(x)}{C(x)}.$
(12)

where energy $\mathcal{E}=-p_{\mu}\xi_{t}^{\mu}$, and angular momentum
$\mathcal{L}={p_{\mu}\xi}_{\phi}^{\mu}$ such that $\xi_{t}^{\mu}$ and
$\xi_{\phi}^{\mu}$ are killing vectors, associated with time translation
symmetry and axial rotational symmetry, respectively. The path of photons
orbiting around a black hole can be described by radial effective potential
Pugliese:2010ps , for which the string-inspired Euler-Heisenberg black holes
reads

$V_{\text{eff}}(x)=-1+\frac{u^{2}}{x(x-Q^{2})}\left(1-\frac{2}{x}-\frac{2(\alpha-\beta)Q^{4}}{x^{3}(x-Q^{2})^{3}}\right)$
(13)

For a photon to remain in a stable circular orbit, the effective potential
should exhibit a local maximum, indicating a stable equilibrium point. At this
maximum potential, photons can orbit the black hole at that specific radius
without being drawn into it or escaping to infinity. Nevertheless, even a
minor disturbance can disrupt this equilibrium, leading the photons to fall
into the black hole or escape to infinity. This scenario characterizes the
circular orbit as an unstable photon sphere Bozza:2002zj ; Virbhadra:1999nm .

|
---|---
Figure 3: Variation of the effective potential $V_{eff}$ for string-inspired Euler-Heisenberg black holes at $Q^{2}=0.5$ and $\alpha-\beta=-2$ with radial coordinate $x$. The photons with black solid lines make unstable circular orbits around black holes with critical impact parameters ($u_{ps}$) (left). The trajectory of the light ray for $Q^{2}=0.5$ and $\alpha-\beta=-2$, for different impact parameters $u$ in polar coordinates ($r,\phi$). The black line corresponds to the $u\approx u_{ps}$ value. Note that here we have shown the black hole as a solid disk and the unstable photon sphere as a dashed black circle (right). |
---|---
Figure 4: Variation of the effective potential $V_{eff}$ for string-inspired
Euler-Heisenberg black holes at $Q^{2}=0.65$ and $\alpha-\beta=-1$ with radial
coordinate $x$. The photons with black solid lines make unstable circular
orbits around black holes with critical impact parameters ($u_{ps}$) (left).
The trajectory of the light ray for $Q^{2}=0.65$ and $\alpha-\beta=-1$, for
different impact parameters $u$ in polar coordinates ($r,\phi$). The black
line corresponds to the $u\approx u_{ps}$ value. We have shown the black hole
as a solid disk and the unstable photon sphere as a dashed black circle
(right.

The conditions for the existence of a photon sphere around a black hole are
determined by the properties of the spacetime metric, particularly the radial
effective potential $V_{\text{eff}}(x)$. A light ray exists in the region
where $V_{\text{eff}}\leqslant 0$ (refer to Figures 3 and 4). Additionally,
one can define an unstable (or stable) circular orbit that satisfies
$V_{\text{eff}}=V^{\prime}{\text{eff}}(x)=0$ and
$V^{\prime\prime}{\text{eff}}(x_{\text{ps}})<0$ (or
$V^{\prime\prime}{\text{eff}}(x{\text{ps}})>0$). In the case of string-
inspired Euler-Heisenberg black hole spacetime, we observe that
$V^{\prime\prime}{\text{eff}}(x{\text{ps}})<0$, indicating unstable photon
circular orbits (refer to Figures 3 and 4).

The radius of photon sphere($x_{ps}$) which is the largest positive root of
Claudel:2000yi ; Virbhadra:2002ju  can be described by:

$\frac{A^{\prime}(x)}{A(x)}=\frac{C^{\prime}(x)}{C(x)}$ (14)

which is depicted in Fig. 5. The photon sphere radii $x_{ps}$ decreases with
the parameters $Q^{2}$ and $\alpha-\beta$.

Figure 5: The photon sphere radius ($x_{ps}$) along with parameter $Q^{2}$ for
different values of parameter $\alpha-\beta$. The black solid line corresponds
to the GHS black holes.

By solving $V_{\text{eff}}(x_{0})=0$ we get the expression for impact
parameter $u$ in terms of the closest approach distance $x_{0}$ as
Bozza:2002zj

$u\equiv\left|\frac{\mathcal{L}}{\mathcal{E}}\right|=\sqrt{\frac{C(x_{0})}{A(x_{0})}}.$
(15)

As the $x_{o}$ approaches $x_{ps}$ the deflection angle becomes unboundedly
large and the impact parameter $u$ becomes the critical impact parameter
$u_{ps}$ Virbhadra:1999nm

The deflection angle as a function of the closest approach distance in the
strong field limit can be described as Virbhadra:1999nm  as

$\alpha_{D}(x_{0})=I(x_{0})-\pi,$ (16)

where

$I(x_{0})=2\int_{x_{0}}^{\infty}\frac{d\varphi}{dx}dx,$ (17)

here

$\frac{d\varphi}{dx}=\frac{1}{\sqrt{A(x)C(x)}\sqrt{(A(x_{0})/C(x))(A(x)/C(x_{0}))-1}}.$
(18)

Now let us define a new variable $z=1-x_{0}/x$ Jha:2023qqz ; Islam:2021dyk .
Using this, we can rewrite the Eq. 17 as

$I(x_{0})=\int_{0}^{1}{R(z,x_{0})f(z,x_{0})}dz.$ (19)

such that $R(z,x_{0})$ is

$R\left(z,x_{0}\right)=\frac{2x_{0}}{\left(1-z\right)^{2}}\frac{\sqrt{C\left(x_{0}\right)}}{C\left(x\right)},$
(20)

and $f(z,x_{0})$ as

$f(z,x_{0})=\frac{1}{\sqrt{A(x_{0})-A(x)\frac{C(x_{0})}{C(x)}}}.$ (21)

where $R(z,x_{0})$ is regular, but the function $f(z,x_{0})$ diverges when
$z=0$. Therefore, to avoid the divergence, we do a Taylor series expansion for
$f(z,x_{0})$ at $z=0$ to approximate the function $f(z,x_{0})\approx
f_{0}(z,x_{0})$

$f_{0}(z,x_{0})=\frac{1}{\sqrt{\Gamma(x_{0})z+\gamma(x_{0})z^{2}}},$ (22)

Now, the integral can be divided into two parts as

$I(x_{0})=I_{D}(x_{0})+I_{R}(x_{0}),$ (23)

where $I_{D}(x_{0})$ is the divergent part described as

$I_{D}(x_{0})=\int_{0}^{1}{R(0,x_{ps})f_{0}(z,x_{0})}dz,$ (24)

and $I_{R}(x_{0})$ is the regular part

$\displaystyle I_{R}(x_{0})=\int_{0}^{1}\text{g}(z,x_{0})dz,$ (25)
$\displaystyle\text{g}(z,x_{0})=R(z,x_{0})f(z,x_{0})-R(0,x_{ps})f_{0}(z,x_{0})$

By solving both the integrals (24) and (25), the deflection angle as a
function of impact parameter $u$ which can be written in terms of angular
position of image as $u\approx\theta D_{OL}$, near $u\to u_{ps}$ can be
described as Bozza:2002zj ; Bozza:2003cp

$\alpha_{D}(u)=-\bar{a}\log\left(\frac{u}{u_{ps}}-1\right)+\bar{b}+O(u-u_{ps}).$
(26)

where $\bar{a}$ and $\bar{b}$ are the strong lensing coefficients described as

$\bar{a}=\frac{R(0,x_{ps})}{2\sqrt{\gamma\left(x_{ps}\right)}},$ (27)
$\bar{b}=-\pi+I_{R}(x_{ps})+\bar{a}\log\left(\frac{2\gamma(x_{ps})}{A(x_{ps})}\right).$
(28)

We have characterized the relationship between the strong gravitational
lensing coefficients and parameters of string-inspired Euler-Heisenberg BH
through numerical methods. The deflection coefficient $\bar{a}$ gradually
increases with parameter $Q^{2}$ and the $(\alpha-\beta)$ magnitude. In
contrast, the deflection coefficient $\bar{b}$ gradually decreases as
parameters $Q^{2}$ as well as with $(\alpha-\beta)$ (see Table 6). The
critical impact parameter also gradually decreases as the parameter $Q^{2}$
increases and parameter $\alpha-\beta$ decreases (see Table 6). Note when the
parameters $Q^{2}$ and $\alpha-\beta$ vanish, the string-inspired Euler-
Heisenberg black hole becomes Schwarzschild BH, such that $\bar{a}=1$,
$\bar{b}=-0.40023$ (see Table 6). Also, parameters $Q^{2}\neq 0$ and
$\alpha=\beta$ correspond to GHS black holes (see Table 6). Figure 6,
illustrates that the deflection angle diverges at specific impact parameter
values $u=u_{ps}$ for different $Q^{2}$. The deflection angle of the string-
inspired Euler-Heisenberg black hole gradually decreases as the parameter
$Q^{2}$ increases for the fixed value of parameter $\alpha-\beta$ (cf. Fig.
6).

$Q^{2}$ |  $(\alpha-\beta)$ |  $\bar{a}$ |  $\bar{b}$ |  $u_{ps}/R_{s}$
---|---|---|---|---
$0.0$ |  $0.0$ |  $1.00$ |  $-0.40023$ |  $5.19615$
0.1 |  0.0 |  1.0057 |  -0.40828 |  5.1085
-0.1 |  1.00577 |  -0.40828 |  5.10854
-0.5 |  1.00603 |  -0.40829 |  5.10844
-1 |  1.00635 |  -0.40829 |  5.10832
-2 |  1.00701 |  -0.40831 |  5.10808
0.3 |  0.0 |  1.0181 |  \- 0.4269 |  4.9269
-0.1 |  1.01894 |  -0.42702 |  4.9267
-0.5 |  1.02237 |  -0.42721 |  4.92553
-1 |  1.02673 |  -0.42748 |  4.92407
-2 |  1.03569 |  -0.42811 |  4.9211
0.5 |  0.0 |  1.0321 |  \- 0.45005 |  4.7356
-0.1 |  1.03574 |  -0.45037 |  4.73453
-0.5 |  1.05068 |  -0.45188 |  4.73002
-1 |  1.07076 |  -0.4544 |  4.72426
-2 |  1.1165 |  -0.46218 |  4.71229
Table 1:  Strong lensing coefficients $\bar{a}$ and $\bar{b}$ and critical impact parameter $u_{ps}$ for different values of parameter of string-inspired Euler-Heisenberg black holes are presented in the table. Note that $\alpha-\beta=0$ and $Q^{2}\neq 0$ correspond to GHS Black holes. |
---|---
Figure 6: Variation of deflection angle as a function of impact parameter $u$
for different values of parameters $Q^{2}$(left) and $\alpha-\beta$(right)
keeping the values one of the parameter as constant for String-inspired Euler-
Heisenberg black holes. Points on the horizontal line represent the values of
the critical impact parameter $u_{ps}$ at which the deflection angle diverges.
Note that the black solid line in the right figure for which $Q^{2}=0.3$ and
$\alpha-\beta=0$ corresponds to the GHS BH.

##  IV Strong gravitational lensing effect for supermassive black holes

###  IV.1 Strong lensing Observables

In the previous section III, we calculated the deflection angle for strong
gravitational lensing; now assuming that the source and observer are far from
the black hole, we deduce that asymptotically flat lens equation as
Virbhadra:1999nm ; Bozza:2001xd

$\psi=\theta-\frac{D_{LS}}{D_{OS}}\Delta\alpha_{n},$ (29)

where $D_{LS}$ is the distance between the lens and the light source, and
$D_{OS}$ is the distance between the observer and the light source. Also, the
angular positions of the source and image are $\psi$ and $\theta$,
respectively, and $\Delta\alpha_{n}=\alpha(\theta)-2n\pi$ is the offset of
deflection, where $n$ is the integer represents the number of loops. For the
position of the nth relativistic image, we solve Eq. 29 with Eq. 28
Bozza:2002zj .

$\theta_{n}=\theta_{n}^{0}+\frac{u_{ps}e_{n}(\psi-\theta_{n}^{0})D_{OS}}{\bar{a}D_{LS}D_{OL}}.$
(30)

where

$e_{n}=\exp\left(\frac{\bar{b}-2n\pi}{\bar{a}}\right).$ (31)

where $\theta_{n}^{0}$ correspond to the image position at $\alpha=2n\pi$ From
equation 30, we see that when $\psi=\theta_{n}^{0}$, then the position of the
image coincides with the source. At this moment, the position of the image
$\theta_{n}=\theta_{n}^{0}$, representing that the source and the image are on
the same side. However, to locate the $n$th image on the opposite side of the
source, this condition is extended by substituting $\psi$ with $-\psi$,
thereby determining the image’s position across the source.

The $n$th relativistic image magnification can be defined as Virbhadra:1998dy
; Bozza:2002zj ; Virbhadra:2007kw

$\mu_{n}=\left.\left(\frac{\psi}{\theta}\frac{d\psi}{d\theta}\right)^{-1}\right|_{\theta_{n}^{0}}=\frac{{u_{ps}}^{2}e_{n}(1+e_{n})D_{OS}}{\bar{a}\psi
D_{LS}{D_{OL}}^{2}},$ (32)

where the first relativistic image is the brightest, and the magnification
decreases exponentially with increasing $n$. Since the magnification is
inversely proportional to the square of the distance between the observer and
the lens thus, the relativistic images are very faint. When $\psi\to 0$, i.e.,
perfect alignment, we have the Einstein ring.

We now calculate the three lensing observables to analyze the strong-
gravitational lensing phenomena. Here, we consider the outermost image as
$\theta_{1}$ and all the other inner packed images as $\theta_{\infty}$
Bozza:2002zj . The angular position of the asymptotic relativistic image
$\theta_{\infty}$,

$\theta_{\infty}=\frac{u_{ps}}{D_{OL}},$ (33)

the angular separation $s$ between the outermost and all the other packed
images is

$s\equiv\theta_{1}-\theta_{\infty}=\theta_{\infty}\exp\left(\frac{\bar{b}-2\pi}{\bar{a}}\right),$
(34)

the flux ratio of the brightness flux between the outermost image $\theta_{1}$
and the remaining relativistic images at $\theta_{\infty}$ is

$r=\frac{\mu_{1}}{\sum_{n=2}^{\infty}\mu_{n}}=2.5\log_{10}\left[\exp\left(\frac{2\pi}{\bar{a}}\right)\right],$
(35)

Note that the flux ratio is independent of the distance between the lens and
the observer $D_{OL}$. We can obtain the coefficients $\bar{a}$, $\bar{b}$ and
the critical impact parameter $u_{ps}$, using the three lensing observables
defined above to analyze the properties specific to string-inspired Euler-
Heisenberg black holes. By comparing astronomical observations with analyzed
data, we can comprehensively understand the properties and behaviour of
string-inspired Euler-Heisenberg black holes.

###  IV.2 Strong Lensing effect for the supermassive black holes M87* and
SgrA*

We use string-inspired Euler-Heisenberg black holes to estimate the lensing
observables of the supermassive black holes M87* and Sgr A* and compare our
data with that of Schwarzschild and GHS black holes. Based on the latest
astronomical observation data, the estimated mass of M87* is $\left(6.5\pm
0.7\right)\times{10}^{9}M_{\odot}$, and its distance is $d=16.8Mpc$
EventHorizonTelescope:2019ggy . The estimated mass of SgrA* is
$4_{-0.6}^{+1.1}\times{10}^{6}M_{\odot}$, and its distance is $d=7.97kpc$
Chen:2019tdb .

When the black hole is perfectly aligned ($\psi\approx 0$) with the observer
and the lens and is at the centre of the observer and the light source, such
that $D_{OS}=D_{LS}=2D_{OL}$ Luminet:1979nyg , then equation (30) will yield
Bozza:2004kq

$\theta_{n}^{E}=\left(1-\frac{u_{ps}e_{n}D_{OS}}{\bar{a}D_{LS}D_{OL}}\right)\
\theta_{n}^{0}.$ (36)

where $\theta_{n}^{E}$ is the Einstein ring Einstein:1936llh . Now assuming
that the distance $D_{OL}$ is greater than the impact parameter i.e.
$u_{ps}(D_{OL}\gg u_{ps}$, then 36 becomes

$\theta_{n}^{E}\approx\frac{u_{ps}(1+e_{n})}{D_{OL}},$ (37)

At $n=1$, the outermost ring is formed, and the radius of the ring decreases
with increases in the value of $n$. The outermost relativistic Einstein ring
of M87* and SgrA* are shown in Figure 7. When the parameter $Q^{2}$ increases,
the Einstein ring becomes smaller for Sgr A* and M87*.

Figure 7: Einstein rings for supermassive black holes when considering them as
string-inspired Euler-Heisenberg black holes. Outer rings correspond to Sgr
A*, and Inner rings correspond to M87* at the values $Q^{2}=0.1$ and
$\alpha-\beta=-1$ (black solid line) and $Q^{2}=0.5$ and $\alpha-\beta=-0.8$
(red dashed line). The blue dashed line represents the GHS black holes
($\alpha-\beta=0$ and $Q^{2}=0.3$).

Clearly, as shown in Fi. 9 and Table 3, the image position $\theta_{\infty}$
and the relativistic image brightness ratio $r_{mag}$ decreases with
increasing magnitude of the parameters $Q^{2}$ as well as $(\alpha-\beta)$
(cf. Fig. 8) and the image separation $s$ to increases with increase in
parameter $Q^{2}$ and $(\alpha-\beta)$ (cf. Fig. 9). It is noteworthy that in
both M87* and SgrA*, the variation of the angular position of the relativistic
images along with parameters ranges from $19.78\mu as\geq\theta_{\infty}\geq
17.89\mu as$ for M87* and $26.28\mu as\geq\theta_{\infty}\geq 23.81\mu as$ for
Sgr A*. From Table 3 and Figure 9 we see that the deviation in the angular
position of the relativistic images $\theta_{\infty}$ and the angular
separation $s$ between the string-inspired Euler-Heisenberg black holes and
its GR counterpart is not very large (around 1$\mu$as). We have also compared
the lensing observables of string-inspired Euler-Heisenberg black holes with
the case where $\alpha=\beta$, i.e. GHS black hole (see Figure 9 and Table 3).

Figure 8: Variation of strong lensing observable $r_{mag}$ for string-inspired Euler-Heisenberg black hole with parameter $Q^{2}$ for different values of parameters $\alpha-\beta$. Note that $r_{mag}$ is independent of the black hole’s mass or distance from the observer. |
---|---
|
Figure 9: Variation of strong lensing observables $\theta_{\infty}$ (upper
panel) and $s$ (lower panel) with $Q^{2}$ for different values of
$\alpha-\beta$ for Sgr A* and M87* respectively. Black solid line in these
plots represent GHS black holes.

###  IV.3 Estimation of Time Delay for SMBHs

Time delay between different images occurs due to the different paths taken by
the photon while winding the black hole. By assuming the source to be
variable, we can numerically calculate the time delay between different
relativistic images. To calculate the time delay, we first calculate the total
time period of photon to travel from source to the observer Bozza:2003cp

$T=\tilde{T}\left(x_{0}\right)-\int_{D_{\mathrm{OL}}}^{\infty}\left|\frac{\mathrm{d}t}{\mathrm{~{}d}x}\right|\mathrm{d}x-\int_{D_{\mathrm{LS}}}^{\infty}\left|\frac{\mathrm{d}t}{\mathrm{~{}d}x}\right|\mathrm{d}x,$
(38)

When the source and the observer are far from the black hole, then the second
and third terms on the right-hand side cancel each other, then the total time
is written as

$\tilde{T}\left(x_{0}\right)=\int_{x_{0}}^{\infty}\frac{2\sqrt{C(x)A(x_{0})}}{A(x)\sqrt{A(x)}\sqrt{\frac{C(x)}{C(x_{0})}\frac{A(x_{0})}{A(x)}-1}}\mathrm{~{}d}x,$
(39)

The integral $\tilde{T}(x_{0})$ represents when the photon winds around the
black hole. In the strong deflection limit, the total time can be expanded as
a function of $u$ as

$\tilde{T}(u)=-\tilde{a}\log\left(\frac{u}{u_{ps}}-1\right)+\tilde{b}+\mathcal{O}\left(u-u_{ps}\right),$
(40)

where $\tilde{a}$ and $\tilde{b}$ are strong deflection limit coefficients and
$\tilde{a}=\bar{a}u_{ps}$ for the spacetime (11) Bozza:2003cp .
For spherically symmetric black holes, we can distinguish the case when the
two images are on the same side of the lens from the case or when the two
images are on opposite sides.
If we can at least distinguish the time signals of the outermost relativistic
image from those of the second image, then the time delay between the first
and the second relativistic image $\Delta T_{2,1}$ can be calculated by strong
deflection limit coefficients as Bozza:2003cp

$\Delta T^{s}_{2,1}=2\pi u_{ps}=2\pi D_{OL}\theta_{\infty},$ (41)

From Eq. 41, we can estimate the distance of the black hole. In Table 3, we
compare time delay $\Delta T^{s}_{2,1}$ for the supermassive black holes at
the centre of different galaxies representing Schwarzschild, string-inspired
Euler-Heisenberg and GHS black holes and also calculated their difference
$\delta T^{s}_{12}=\Delta T^{s}_{2,1}(Sch)-\Delta T^{s}_{2,1}(EH)$.

| Sgr A* | M87* |
---|---|---|---
$Q^{2}$ |  $(\alpha-\beta)$ |  $\theta_{\infty}$($\mu$as) |  $s$ ($\mu$as) |  $\theta_{\infty}$($\mu$as) |  $s$ ($\mu$as) |  $r_{mag}$
0.0 |  0.0 |  26.33 |  0.03295 |  19.78 |  0.02476 |  6.8219
0.1 |  0.0 |  25.886 |  0.03338 |  19.448 |  0.02508 |  6.7832
-0.1 |  25.8859 |  0.03339 |  19.4485 |  0.02509 |  6.7828
-1.0 |  25.8848 |  0.03352 |  19.4476 |  0.02518 |  6.7788
-4.5 |  25.8805 |  0.03402 |  19.4444 |  0.02556 |  6.7634
-9.5 |  25.8743 |  0.03476 |  19.4397 |  0.02612 |  6.7412
0.3 |  0.0 |  24.966 |  0.03427 |  18.757 |  0.02574 |  6.7006
-0.1 |  24.9645 |  0.03446 |  18.7562 |  0.02589 |  6.6951
-1 |  24.9511 |  0.03619 |  18.7462 |  0.02719 |  6.6443
-2.5 |  24.9286 |  0.03934 |  18.7292 |  0.02955 |  6.5575
-4.5 |  24.8978 |  0.04413 |  18.7061 |  0.03316 |  6.4375
0.5 |  0.0 |  23.996 |  0.03523 |  18.028 |  0.02647 |  6.6094
-0.1 |  23.9908 |  0.03602 |  18.0246 |  0.02706 |  6.5865
-0.5 |  23.9679 |  0.03942 |  18.0074 |  0.02962 |  6.4928
-1 |  23.9387 |  0.04429 |  17.9855 |  0.03328 |  6.3711
-2 |  23.878 |  0.05678 |  17.9399 |  0.04266 |  6.1101
Table 2: Estimates for Strong lensing observables for the string-inspired
Euler-Heisenberg black holes and compared with Schwarzschild black holes
($Q^{2}\to 0,(\alpha-\beta)\to 0$) in GR considering the supermassive black
holes Sgr A* and M87* as lens for different values of parameter $Q^{2}$ and
$(\alpha-\beta)$. Note that $Q^{2}\neq 0$ and $\alpha-\beta=0$ correspond to
GHS Black holes. Table 3:  Time delay estimation between the first and second
relativistic image $\Delta T^{s}_{2,1}$ for different supermassive black holes
at the centre of nearby galaxies considering string-inspired Euler-Heisenberg
black hole ($Q^{2}=0.1$ and $\alpha-\beta=-1$) and GHS black holes
($Q^{2}=0.1$ and $\alpha-\beta=0$) in comparison with Schwarzschild($Q^{2}\to
0$ and $\alpha-\beta\to 0$). Note that time delays are calculated in minutes
and $\delta T^{1}_{12}=\Delta T^{s}_{2,1}(Sch)-\Delta T^{s}_{2,1}(EH)$ and
$\delta T^{2}_{12}=\Delta T^{s}_{2,1}(Sch)-\Delta T^{s}_{2,1}(GHS)$.

Galaxy |  $M(M_{\odot})$ |  $D_{OL}$ |  $M/D_{OL}$ |  $\Delta T^{s}_{12}$ |  $\Delta T^{s}_{12}$ |  $\Delta T^{s}_{12}$ |  $\delta T^{1}_{12}$ |  $\delta T^{2}_{12}$
---|---|---|---|---|---|---|---|---
|  |  (Mpc) |  |  (Sch) |  (EH) |  (GHS) |  |
Milky Way |  $4.3\times 10^{6}$ |  $0.008$ |  $2.471\times 10^{-11}$ |  $11.4968$ |  $11.3024$ |  11.303 |  $0.1944$ |  0.1938
M87 |  $6.5\times 10^{9}$ |  $16.8$ |  $1.758\times 10^{-11}$ |  17378.7 |  17085.1 |  17085.9 |  293.6 |  292.8
NGC 4472 |  $2.54\times 10^{9}$ |  $16.72$ |  $7.246\times 10^{-12}$ |  $6791.06$ |  6676.32 |  6676.64 |  114.74 |  114.42
NGC 1332 |  $1.47\times 10^{9}$ |  $22.66$ |  $3.094\times 10^{-12}$ |  $3930.26$ |  3863.86 |  3864.04 |  66.4 |  66.2
NGC 4374 |  $9.25\times 10^{8}$ |  $18.51$ |  $2.383\times 10^{-12}$ |  $2473.12$ |  2431.34 |  2431.45 |  41.78 |  41.67
NGC 1399 |  $8.81\times 10^{8}$ |  $20.85$ |  $2.015\times 10^{-12}$ |  $2355.48$ |  2315.68 |  2315.8 |  39.8 |  39.68
NGC 3379 |  $4.16\times 10^{8}$ |  $10.70$ |  $1.854\times 10^{-12}$ |  $1112.24$ |  1093.44 |  1093.5 |  18.8 |  18.74
NGC 4486B |  $6\times 10^{8}$ |  $16.26$ |  $1.760\times 10^{-12}$ |  $1604.19$ |  1577.08 |  1577.16 |  27.11 |  27.03
NGC 1374 |  $5.90\times 10^{8}$ |  $19.57$ |  $1.438\times 10^{-12}$ |  $1577.45$ |  1550.8 |  1550.87 |  26.65 |  26.58
NGC 4649 |  $4.72\times 10^{9}$ |  $16.46$ |  $1.367\times 10^{-12}$ |  $12619$ |  12406.4 |  12407 |  212.6 |  212
NGC 3608 |  $4.65\times 10^{8}$ |  $22.75$ |  $9.750\times 10^{-13}$ |  $1243.25$ |  1222.24 |  1222.3 |  21.01 |  20.95
NGC 3377 |  $1.78\times 10^{8}$ |  $10.99$ |  $7.726\times 10^{-13}$ |  $475.909$ |  467.868 |  467.89 |  8.041 |  8.019
NGC 4697 |  $2.02\times 10^{8}$ |  $12.54$ |  $7.684\times 10^{-13}$ |  $540.077$ |  530.952 |  530.977 |  9.125 |  9.1
NGC 5128 |  $5.69\times 10^{7}$ |  $3.62$ |  $7.498\times 10^{-13}$ |  $152.131$ |  149.56 |  149.567 |  2.571 |  2.564
NGC 1316 |  $1.69\times 10^{8}$ |  $20.95$ |  $3.848\times 10^{-13}$ |  $451.816$ |  444.212 |  444.233 |  7.604 |  7.583
NGC 3607 |  $1.37\times 10^{8}$ |  $22.65$ |  $2.885\times 10^{-13}$ |  $366.265$ |  360.10 |  360.118 |  6.165 |  6.147
NGC 4473 |  $0.90\times 10^{8}$ |  $15.25$ |  $2.815\times 10^{-13}$ |  $240.628$ |  236.563 |  236.574 |  4.065 |  4.054
NGC 4459 |  $6.96\times 10^{7}$ |  $16.01$ |  $2.073\times 10^{-13}$ |  $186.086$ |  182.94 |  182.95 |  3.146 |  3.136
M32 |  $2.45\times 10^{6}$ |  $0.8057$ |  $1.450\times 10^{-13}$ |  $6.5504$ |  6.4397 |  6.44007 |  0.1107 |  0.1097

##  V EHT Constraints and Parameter Estimation

The Event Horizon Telescope (EHT) observation results have revolutionized our
understanding of supermassive black holes, unveiling horizon-scale images of
M87* and Sgr A*. In 2019, the EHT Collaboration’s groundbreaking work
confirmed the existence of black holes with the unveiling of M87*’s image,
revealing an angular diameter of $\theta_{d}=42\pm 3~{}\mu as$ for its compact
emission region, defining the central flux depression as the black hole’s
shadow (EventHorizonTelescope:2019dse, ; EventHorizonTelescope:2019pgp, ;
EventHorizonTelescope:2019ggy, ). Subsequent observations in 2022 captured Sgr
A*, presenting a distinct ring with $\theta_{d}=48.7\pm 7~{}\mu as$ and
quantifying the Schwarzschild shadow deviation as
$\delta=-0.08^{+0.09}_{-0.09}$ (VLTI),$-0.04^{+0.09}_{-0.10}$ (Keck)
(EventHorizonTelescope:2022xqj, ). Sgr A* offers a unique opportunity due to
its probing of curvature orders higher than M87*, and the independent prior
estimates for the mass-to-distance ratio utilized for Sgr A*. The use of three
independent imaging algorithms ensures robustness, with the average diameter
of the Sgr A* shadow constrained to be within 46.9$\mu$as $\leq$
$\theta_{\text{sh}}$ $\leq$ 50 $\mu$as. The EHT result for Sgr A* also
calculated the emission ring angular diameter $\theta_{d}=(51.8\pm 2.3)~{}\mu
as$ (EventHorizonTelescope:2022xqj, ). These findings not only enable
exploration of strong-field gravity but also offer insights into theoretical
frameworks and put constraints on additional parameters within gravity
theories Afrin:2021wlj ; Zakharov:2021gbg ; EventHorizonTelescope:2021dqv ;
Ghosh:2020spb ; Kumar:2018ple ; KumarWalia:2022aop ; Ghosh:2022kit ;
Kumar:2023jgh ; Islam:2022wck . Leveraging the EHT data, we apply these
observations to assess the feasibility of string-inspired Euler-Heisenberg
black holes, placing constraints on deviation parameters $Q^{2}$ and
$\alpha-\beta$. By adopting the apparent radius of the photon sphere
$\theta_{\infty}$ as the angular measurement for the black hole shadow,
constraints are established at the 1-$\sigma$ confidence level. M87* and Sgr
A* are modelled as string-inspired Euler-Heisenberg black holes, underpinning
efforts to understand the implications of magnetically charged black holes in
cosmic phenomena. This synthesis of observational data and theoretical
frameworks advances our understanding of black hole physics and underscores
the interdisciplinary nature of contemporary astrophysical research.

#### Constraints from Sgr A*:

The shadow angular diameter ($\theta_{\text{sh}}$), is averaged at a range of
$(46.9,50)~{}\mu as$, with a 1-$\sigma$ interval of the Sgr A* black hole for
the EHT observation; hence it imposes constraints on the parameters of string-
inspired Euler-Heisenberg black holes, $Q^{2}$ and $\alpha-\beta$ as
illustrated in Fig. 10. Here $\theta_{\text{sh}}$ is depicted as a function of
parameters $Q^{2}$ and $(\alpha-\beta)$, the black and black dashed line
corresponds to $\theta_{\text{sh}}=50$ and $\theta_{\text{sh}}=46.9$
respectively. The constraints on the parameters range as $0.29278\leq
Q^{2}\leq 0.60778$, but there is no constraint on the parameter value
($\alpha-\beta$). So, within the above-defined parameter range, the string-
inspired Euler-Heisenberg black hole aligns with the Sgr A* black hole shadow
observations from the EHT.

Figure 10: Shadow angular diameter $\theta_{\text{sh}}(=2\theta_{\infty})$ as
a function of parameters $Q^{2}$ and $(\alpha-\beta)$, when Sgr A* is modelled
as string-inspired Euler-Heisenberg BH. The black and black dashed line
corresponds to $\theta_{\text{sh}}=50$ and $\theta_{\text{sh}}=46.9$
respectively. The region between these lines satisfies the Sgr A* shadow
1-$\sigma$ bound.

#### Constraints from M87*:

Fig. 11 depicts the angular diameter $\theta_{sh}$ as a function of parameters
$Q^{2}$ and $\alpha-\beta$, with the black solid line corresponding to
$\theta_{sh}=39~{}\mu$as for the string-inspired Euler-Heisenberg black holes
as M87*. When we investigated the string-inspired Euler-Heisenberg black holes
with the EHT results of M87* within the 1-$\sigma$ bound, it constrains the
parameters $Q^{2}$ and $\alpha-\beta$, viz., $0<Q^{2}\leq 0.08473$ but there
is no constraint on the value of the parameter ($\alpha-\beta$). Thus, based
on Fig. 11, a string-inspired Euler-Heisenberg black hole can be a candidate
for astrophysical black holes.

Figure 11: Shadow angular diameter $\theta_{\text{sh}}(=2\theta_{\infty})$ as
a function of parameters $Q^{2}$ and $(\alpha-\beta)$, when M87* is modelled
as string-inspired Euler-Heisenberg BH. The black line corresponds to
$\theta_{\text{sh}}=39$. The region within this line satisfies the M87* shadow
1-$\sigma$ bound.

###  V.1 Estimation of Black holes Parameters

By analyzing the angular shadow diameter of supermassive black holes like M87*
and Sgr A* using data from the Event Horizon Telescope (EHT), we have
estimated the parameters associated with the string-inspired Euler-Heisenberg
model Kumar:2022vfg ; Tsukamoto:2020bjm ; Islam:2022wck . This estimation of
black hole parameters helps us understand the implications of magnetic charge
and string coupling constants on these astrophysical objects.

In Table. 4 and 5, we estimate the parameters $Q^{2}$ and $\alpha-\beta$ when
Sgr A* is modelled as string-inspired Euler-Heisenberg BH at the value of
angular shadow diameter $\theta_{sh}=50~{}\mu as$ and $\theta_{sh}=46.9~{}\mu
as$. In Table. 6 we estimate the parameters $Q^{2}$ and $\alpha-\beta$ when
M87* is modelled as string-inspired Euler-Heisenberg BH at the value of
angular shadow diameter $\theta_{sh}=39~{}\mu as$. The analysis reveals that
substantial variations in the parameter $\alpha-\beta$ do not significantly
alter the angular shadow diameter ($\theta_{sh}$).

$\theta_{sh}=50\mu as$ | $\theta_{sh}=46.9\mu as$
---|---
$(\alpha-\beta)$ |  $Q^{2}$ |  $Q^{2}$
-1.75 |  0.28778 |  0.57651
-3.25 |  0.28373 |  0.55344
-4.75 |  0.27989 |  0.53304
-6.25 |  0.27623 |  0.51477
-7.75 |  0.27274 |  0.49824
-9.25 |  0.26941 |  0.48316
Table 4: Estimation of values of parameters $Q^{2}$ for different values of $(\alpha-\beta)$ corresponding to $\theta_{sh}=50$ $\mu$as and $\theta_{sh}=46.9$ $\mu$as for the string-inspired Euler-Heisenberg black holes as Sgr A*. $\theta_{sh}=50\mu as$ | $\theta_{sh}=46.9\mu as$
---|---
$Q^{2}$ |  $(\alpha-\beta)$ |  $Q^{2}$ |  $(\alpha-\beta)$
0.265 |  -11.34 |  0.505 |  -7.12
0.270 |  -8.977 |  0.520 |  -5.805
0.275 |  -6.77 |  0.535 |  -4.598
0.280 |  -4.706 |  0.550 |  -3.491
0.285 |  -2.77 |  0.565 |  -2.474
0.290 |  -0.96 |  0.580 |  -1.539
Table 5: Estimation of values of parameters $(\alpha-\beta)$ for different values of $Q^{2}$ corresponding to $\theta_{sh}=50$ $\mu$as and $\theta_{sh}=46.9$ $\mu$as for the string-inspired Euler-Heisenberg black holes as Sgr A*. $\theta_{sh}=39\mu as$ | $\theta_{sh}=39\mu as$
---|---
$(\alpha-\beta)$ |  $Q^{2}$ |  $Q^{2}$ |  $(\alpha-\beta)$
-1.75 |  0.08439 |  0.0825 |  -12.1
-3.25 |  0.08411 |  0.0829 |  -9.8
-4.75 |  0.08391 |  0.0833 |  -7.6
-6.25 |  0.08355 |  0.0837 |  -5.4
-7.75 |  0.08327 |  0.0841 |  -3.3
-9.25 |  0.0830 |  0.0845 |  -1.2
Table 6: Estimation of values of parameters $Q^{2}$ and $(\alpha-\beta)$ for
different values of $(\alpha-\beta)$ and $Q^{2}$ respectively corresponding to
$\theta_{sh}=39$ $\mu$as for the string-inspired Euler-Heisenberg black holes
as M87*.

##  VI Conclusion

The study explores the influence of parameters $Q^{2}$ and ($\alpha-\beta$) on
gravitational lensing phenomena associated with string-inspired Euler-
Heisenberg black holes, comparing them with the Schwarzschild black hole of
General Relativity (GR). It reveals that the photon sphere radius $x_{ps}$
contracts with increasing magnitude of $Q^{2}$ and $\alpha-\beta$, indicating
smaller photon spheres in string-inspired Euler-Heisenberg black holes than
GR. Furthermore, the critical impact parameter $u_{ps}$ has a decreasing
behaviour for $Q^{2}$ and ($\alpha-\beta$), which is qualitatively similar to
the behaviour of the unstable photon orbit radius $x_{ps}$. Its value is
consistently lower than that of its GR equivalents. On the other hand, the
strong-gravitational lensing coefficients $\bar{a}$ tend to increase with
$Q^{2}$ as well as $(\alpha-\beta)$ but $\bar{b}$ tends to exhibit a gradual
decrease with increasing magnitude of $(\alpha-\beta)$ and $Q^{2}$. The
deflection angle $\alpha_{D}$, for a fixed impact parameter $u$, is smaller in
string-inspired Euler-Heisenberg black holes than in Schwarzschild black
holes, decreasing further with increasing $Q^{2}$.

Further investigations focus on strong lensing observables for supermassive
black holes, Sgr A* and M87*, under string-inspired Euler-Heisenberg black
hole spacetime. The study finds that string-inspired Euler-Heisenberg black
holes closely approximate GR predictions, with lensing observables such as
$\theta_{\infty}$, the separation $s$ between images, and the magnification
ratio $r_{\text{mag}}$ showing ranges consistent with GR expectations.
However, deviations from GR are observed, particularly in $\theta_{\infty}$,
which ranges between 23.81 - 26.28 $\mu$as for Sgr A* and 17.89 - 19.77
$\mu$as for M87*, with deviations as high as 2.452 $\mu$as and 1.89 $\mu$as,
respectively.

Finally, the study evaluates the time delay of relativistic images in string-
inspired Euler-Heisenberg black hole spacetime, finding them consistent with
GR predictions. For instance, the time delay for M87* is approximately $\sim
293$ minutes, while for Sgr A*, it’s around $\sim 0.19$ minutes. Except for
Sgr A*, these time delay differences are substantial enough for astronomical
observations, assuming sufficient angular resolution exists between the two
relativistic pictures. While string-inspired Euler-Heisenberg black holes
exhibit deviations from GR predictions for certain lensing observables, they
largely align with GR expectations, suggesting their potential as a valid
description for astrophysical black hole phenomena.

The constraints derived from EHT observations indicate that string-inspired
Euler-Heisenberg black holes may provide a viable description for the
spacetime of astrophysical black holes within limited parameter ranges.
Specifically, for M87*, $Q^{2}$ is constrained to $0.0<Q^{2}\leq 0.0847$,
while for Sgr A*, it falls within $0.292\leq Q^{2}\leq 0.607$, at the
$1\sigma$ confidence level. These constraints on $Q^{2}$ do not impose any
limitations on the parameter $\alpha-\beta$. The parameter estimation
conducted using the EHT data for the angular diameter shadow ($\theta_{sh}$)
of Sgr A* and M87* serves as a crucial validation step in affirming the
viability of string-inspired Euler-Heisenberg black holes as astrophysical
black hole candidates. While string-inspired Euler-Heisenberg black holes show
promise in modelling black hole spacetime, further exploration of the full
parameter space, particularly regarding $\alpha-\beta$, is warranted to
understand their viability in astrophysical contexts comprehensively.

##  VII Acknowledgments

A.V and S.G.G. would like to thank SERB-DST for project No. CRG/2021/005771.

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