# Strong lensing as a probe of braneworld

Yi Zhang,11footnotetext: Corresponding author. Hong Liu  Dan Wen  Hongsheng
Zhang

###### Abstract

For the first time, we use the Event Horizon Telescope (EHT) data to constrain
the parameters of braneworld black holes which constrain
$\epsilon>-0.0655>-0.1218$ for the Anisotropic black hole and
$l^{2}=0.0745^{+0.2864+0.5156}_{-0.0745-0.0745}$ for the Garriga-Tanaka black
hole. Based on the fitted data, we calculate the photon deflection, the
angular separation and time delay between different relativistic images of the
the anisotropic black hole and the Garriga-Tanaka black hole. And furthermore,
we study the quasinormal modes (QNMs). The results shed light on existence of
extra dimension.

##  1 Introduction

Black hole is one of the most exciting predictions of Einstein’s general
relativity whose existence is confirmed directly by capturing images of the
black hole shadow from the Event horizon Telescope (EHT) [1, 2, 3, 4].
Theoretically, it is widely accepted that general relativity is an effective
infrared gravitational theory and should be modified in the ultraviolet regime
[5, 6, 7, 8]. To extend the theory to high energy regime, a natural way is to
introduce extra dimension, which may play a significant role in unification
and quantum gravity. Braneworld is one the most popular models for extra
dimension [9]. The braneworld paradigm views our universe as a slice of some
higher dimensional spacetime. Unlike the Kaluza-Klein picture of extra
dimensions, where we do not notice the extra dimensions because they are so
small and our physics is “averaged” over them. The braneworld picture can have
large, even non-compact but highly warped extra dimensions which are
unobservable at low energies since the gauge fields are confined to the brane.
This scenario provides a set-up in which we have standard four dimensional
physics confined to the brane, but gravity propagating in the bulk. Black
holes in braneworld have some key potential differences from the black hole of
general relativity [10].

Technically, by using Gauss-Codazzi approach [11], the classical five
dimensional braneworld black hole solution is reduced to the four dimensional
quantum radiating black hole. But the exact metric describing the spacetime
geometry around braneworld black holes is not yet known. Here, we concentrate
on the anisotropic and Garriga-Tanaka black holes. The anisotropic black hole
describes the properties near the horizon [12, 13, 14, 15]. Since this system
contains unknown bulk dependent term, assumptions have to be made either in
the form of metric or the Weyl term. This braneworld black hole is believed to
encodes quantum correction of black holes. Another solution of linearized
gravity in braneworld is given in [16, 17], which is called Garriga-Tanaka
black hole. It is shown that general relativity is recovered on the brane,
with higher order corrections due to presence of the extra dimension.

The highly bending, even looping of light rays around black holes in strong
fields a well-known and amazing predictions of general relativity [18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28] . It is significant and interesting to
investigate the braneworld effects through observations of EHT and thus
presents effectual constraint of braneworld model. In strong field, the light
deflection divergences at photo sphere. By an analytic approximation method,
Bozza proved that, when the angle between source and lens tends to zero, the
deflection angle diverges logarithmically. Bozza et al [27, 28] find an
interesting simplification for the lens equation in such regime, finding the
expression for observable quantities in the so-called strong deflection limit
regime. And theoretically, light bending by a compact body can exceed $2\pi$
and the light even can wind several loops before escaping, which develops
infinite discrete images on two sides of the body closely, called relativistic
images. A review can consult Ref. [29, 30]. Relativistic images, which is not
predicted by the classical weak gravitational lensing, provide a new way to
study the properties of spacetime in the strong gravitational field.
Differences in the deflection angle are immediately reflected on the
relativistic image. In recent years, more approaches have been developed, such
as time delay [31, 32, 33] and QNM [34, 35, 36]. The various strong deflection
lensing work can been seen in Refs. [37, 38, 39, 40, 47, 41, 42, 43, 44, 45,
46]. Based on the property that the divergence of deflection angle can be
integrated up to first order, one derives the gravitational lensing
observables. Using the M87* and Sgr A* black hole shadow data, one can
investigate the parameters of braneworld black holes. The $\chi^{2}$-test is
an effective method which extracts information from the observational data to
obtain the black hole parameter range [48, 49, 47].

We arrange the paper as below. In section 2, we introduce the two braneworld
black holes: The Anisotropic and Garriga-Tanaka ones. Then we apply the strong
field limit procedure [28] to the braneworld metrics in sections 3 and 4.
Then, in section 5, we calculate the observation effects, including the
positional separations $\theta_{\infty}$, brightness difference $s$ and
magnitude difference $r$. Besides the deflection angle, we will discuss quasi
normal mode and the time delay between images as well. Furthermore, we
calculate the time delay between different models. At last, we give a
conclusion in section 6.

##  2 The Metric on the brane

Randall and Sundrum showed that a four dimensional Minkowskian braneworld can
be constructed although gravity was inherently five dimensional, while the
spacetime was strongly warped. In the Randall-Sundrum II model a single
membrane of positive tension imbedded in five dimensional AdS space,

$\displaystyle
ds^{2}=g_{ab}dx^{a}dx^{b}=d\tilde{z}^{2}+a^{2}(\tilde{z})\eta_{\mu\nu}dx^{\mu}dx^{\nu}.$
(2.1)

Here, $a(\tilde{z})=e^{-|\tilde{z}|l}$, where $l$ is the radius of the AdS,
and $\eta_{\mu\nu}$ is the Minkowski metric in four dimension. The Randall-
Sandrum II model offers a remarkable compactification, that is, on scales much
larger than $l$, four dimensional gravity is recovered on the brane. For some
five dimensional braneworld solutions, the difference in the observables is
found to be rather small from the four dimensional Schwarzschild case [53, 49,
54, 50, 51, 52].

Considering a static metric, there exists a five dimensional solution
analogous to the C-metric in four dimensions which has a timelike Killing
vector, and can therefore be “sliced” by the braneworld in such a way as to
create a static four dimensional black hole on the brane [12, 13, 14, 15].
Here, the spacetime is constructed so that there are four dimensional flat
slices stacked along the fifth $\tilde{z}$-dimension, which have a
$\tilde{z}$-dependent conformal pre-factor known as the warp factor. This warp
factor has a cusp at $\tilde{z}=0$, which indicates the presence of a domain
wall, or the braneworld, which represents an exact flat Minkowski universe
[12]. The Einstein equation for the simplest case (“vacuum” brane) can be
written as,

$\displaystyle G_{\mu\nu}={\cal E}_{\mu\nu}.$ (2.2)

where the ${\cal E}_{\mu\nu}$ is the Weyl term, consisting of projection of
the bulk Weyl tensor on the brane. In the AdS picture, the brane is not at the
AdS boundary, but at a finite distance, and the theory on the brane now
contains a conformal energy-momentum tensor, which appears as the Weyl term
${\cal E}_{\mu\nu}$. Using the symmetry of the physical set-up to put the Weyl
energy into the form [12, 13, 14, 15],

$\displaystyle{\cal E}_{\mu\nu}={\cal
U}(u_{\mu}u_{\nu}-\frac{1}{3}h_{\mu\nu})+\Pi(r_{\mu}r_{\nu}+\frac{1}{3}h_{\mu\nu}),$
(2.3)

where $u_{\mu}$ is a unit time vector, $r_{\mu}$ a unit radial vector, and
$h_{\mu\nu}$ is the metric perturbation. Then the field equations are

$\displaystyle G_{t}^{t}={\cal U},$ (2.4) $\displaystyle
G_{r}^{r}=-\frac{{\cal U}+2\Pi}{3},$ (2.5) $\displaystyle
G_{\theta}^{\theta}=-\frac{{\cal U}-\Pi}{3},$ (2.6)

where ${\cal U}$ is the Weyl energy and $\Pi$ is the anisotropic stress.

###  2.1 The Anisotropic metric

The simplest solution of Eq.(2.2) is based on the static spherically symmetric
metric on the brane which is

$\displaystyle ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)d\Omega_{II}^{2}$ (2.7)

where $C(r)=r^{2}$ and
$d\Omega_{II}^{2}=d\tilde{\theta}^{2}+\sin^{2}\tilde{\theta}d\tilde{\phi}^{2}$.
And, the horizon is the clear asymptotic regime which we could get some
information about the black hole. Then, when ${\cal U}=0$, by setting $G=1$,
there is a simple analytic solution which is near horizon in area gauge [12] ,

$\displaystyle
ds^{2}=-[(1+\epsilon)\sqrt{1-\frac{2M}{R}}-\epsilon]^{2}dt^{2}+(1-\frac{2M}{R})^{-1}dR^{2}+r^{2}d\Omega_{II}^{2},$
(2.8)

where $R=(r+r_{0})^{2}/r$, $M=2r_{0}$, $\epsilon=(-r_{1}+r_{0})/M$ and $r_{1}$
is the integral constant. When $\epsilon=0$, the metric reduces to
Schwarzchild metric. The anisotropic stress for this solution is
$\Pi=3M\epsilon/AC^{3}$. This metric describes the behaviors in the near
horizon regime. And as the metric ${\cal U}=0$, the anisotropic stress for
this solution is $\Pi=3M\epsilon/AC^{3}$. For convenience, we call such a
metric as the anisotropic metric which is based on the non-perturbative nature
of gravity. When $\epsilon=0$ which corresponds to ${\cal E}_{\mu\nu}=0$, it
is back to Schwarzchild . And, the area gives a familiar spatial part of the
metric, for $\epsilon>0$, $g_{tt}$ will be zero before $r=2M$, and the area
gauge holds outside the black hole. Then, this solution could not be treated
as black hole, as the ’horizon’ is singularity. Furthermore, the mathematical
constraint on the near horizon metric is $\epsilon>-1/2$ from the requirement
that $r_{1}$ is positive definite [12]. Therefore, we set the prior
$\epsilon<0$ in the $\chi^{2}$ test process. Here, we take $M$ as the measure
of distances, after defining $x=R/2M$, the anisotropic metric which shows near
horizon modification to general relativity is written as follows,

$\displaystyle
ds^{2}=-[(1+\epsilon)\sqrt{1-\frac{1}{x}}-\epsilon]^{2}dt^{2}+(1-\frac{1}{x})^{-1}dx^{2}+x^{2}d\Omega_{II}^{2}.$
(2.9)

###  2.2 The Garriga-Tanaka Metric

Furthermore, by considering perturbation $h_{\mu\nu}$ about the RS II
background, another important solution for gravity scenario is the Garriga-
Tanaka solution. A thick brane scenario that corresponds to a regularized
version of the Garriga-Tanaka solution describes black hole solution in RSII
model [11, 16], where the perturbation on the brane is written in terms of a
Green’s function which is dominated by the low-energy zero mode of the five
dimensional graviton. The RS II brane is a single 3 brane with positive
tension, embedded in a five dimensional bulk spacetime. When
$\Pi=5\cal{U}/$$4$, the Garriga-Tanaka metric is written as follows,

$\displaystyle
ds^{2}=-(1-\frac{2M}{r}-\frac{4Ml^{2}}{3r^{3}})dt^{2}+(1+\frac{2M}{r}+\frac{2Ml^{2}}{r^{3}})dr^{2}+r^{2}d\Omega_{II}^{2}.$
(2.10)

Here $l$ is the characteristic size of the extra dimension. The $r^{-3}$ term
is from the Kaluza-Klein excitations, which have non-vanishing momentum in the
fifth direction [52]. The coefficient in front of the correction
$l^{2}/r^{3}$, due to the Kaluza-Klein modes, is different in both cases,
because $d=5$ is in some sense four dimensional and contributes only to the
zero model. For the Garriga-Tanaka metric, the matter distribution with mass
$M$ is localized on the brane at the thin brane limit. For the first order
approximation Garriga-Tanaka metric implies the effects of Kaluza-Klein modes
in spherically symmetric source. This metric is valid on the brane $z=0$ in a
region far from the source $r>>M$. If we take $M$ as the measure of distances,
after defining $x=r/2M$, the Garriga-Tanaka metric is written as follows,

$\displaystyle
ds^{2}=-(1-\frac{1}{x}-\frac{2l^{2}}{3x^{3}})dt^{2}+(1+\frac{1}{x}+\frac{l^{2}}{x^{3}})dx^{2}+x^{2}d\Omega_{II}^{2}.$
(2.11)

##  3 The General introduction to strong lensing limit approach

In this section, for convenience to develop this paper we give a brief review
on the general formula of gravitational lensing in the strong field limit. Due
to the spherical symmetry, we only consider light rays moving on the
equatorial plane with $\tilde{\theta}=\frac{\pi}{2}$. The lens equation is
used to define the geometrical relations among the observer, the lens and the
source, the general lens equation reads,

$\displaystyle\tan\beta=\tan\theta-\frac{D_{LS}}{D_{OS}}[\tan\theta-\tan(\theta-\alpha)],$
(3.1)

where $\beta$ is the object position which provides the angle $\alpha$,where
$\beta$ is the angular separation between the source and the lens, $\theta$ is
the angular separation between the image and the lens, $D_{LS}$ and $D_{OS}$
are the projected distance of lens-source and observer-source along the
optical axis. Given a source position $\beta$, by solving this equation, the
value of $\beta$ denotes the position of the images observed by O.

We assume that both the observer and the source are far from the lens and the
spacetime of the lens is asymptotically flat. In this case, we are allowed to
expand $\tan\beta$ and $\tan\theta$ to the first order,

$\beta=\theta-\frac{D_{LS}}{D_{OS}}\Delta\alpha_{n}=\theta-\frac{D_{LS}}{D_{OS}}(\alpha(\theta)-2n\pi),$
(3.2)

where $\Delta\alpha_{n}=\alpha(\theta)-2n\pi$ is the extra angular deflection
angle after a photon with a deflection angle winding n loops. The deflection
angle $\alpha$ encodes the physical information about the deflector which can
be calculated through the integration of the geodesic of the light ray. Due to
the asymptotically approximated lens equation, the spacetime of the lens only
affects the deflection angle $\alpha(\theta)$ which will be calculated in the
strong lensing. We shall put our attention on situations where the source is
almost perfectly aligned with the lens.

Conserved quantities along the orbit are $E=A(r)\dot{t}$,
$L=C(r)\dot{\tilde{\phi}}$, where a dot denotes derivative with respect to the
affine parameter. Considering the conservation of energy and angular

$\displaystyle\frac{d\phi}{dx}=\frac{\sqrt{B}}{\sqrt{C}\sqrt{\frac{CA_{0}}{C_{0}A}-1}},$
(3.3)

we find

$\displaystyle
I(x_{0})=\int^{\infty}_{x_{0}}\frac{\sqrt{B}}{\sqrt{C}\sqrt{\frac{CA_{0}}{C_{0}A}-1}}dx,$
(3.4)

where $x_{0}$ is the closest distance of the photon to the black hole. The
deflection angle for the null geodesic of a photon in the black hole spacetime
can be found

$\displaystyle\alpha(x_{0})=-\pi+\int_{x_{0}}^{\infty}\frac{2\sqrt{B}dx}{\sqrt{C}\sqrt{\frac{CA_{0}}{C_{0}A}-1}}.$
(3.5)

When the light ray trajectory gets closer to the event horizon, the deflection
angle increases.

##  4 The Bozza’s procedure

We follow the Bozza’s procedure[28] to discuss the strong lensing problem. It
has been proved that when a photon moves around a black hole, there exists an
innermost unstable orbit named photon sphere. First, we calculate $x_{m}$,
which is the largest root of the following equation,

$\frac{C^{\prime}(x)}{C(x)}=\frac{A^{\prime}(x)}{A(x)},$ (4.1)

where $A$, $C$, $A^{\prime}$ and $C^{\prime}$ must be positive for $x>x_{m}$.
And, this equation admits at least one positive solution. We shall call
$x_{m}$ the radius of the photon sphere. The deflection angle is divergent at
the photon sphere $x_{m}$.

We introduce the impact parameter $u=L/E$ which is the perpendicular distance
from the center of the mass of lens to the tangent of the null geodesics and
remains constant throughout the trajectory. By conservation of the angular
momentum, the closest distance is related to the impact parameter by

$\displaystyle u=\sqrt{\frac{C_{0}}{A_{0}}}=|\frac{L}{E}|.$ (4.2)

To expand the integral near the photon sphere is not only providing an
analytic re-presentation of the deflection angle but also showing behavior of
photons near the photon sphere. The minimum value of $u$ could be written as,

$\displaystyle u_{m}=\sqrt{\frac{C_{m}}{A_{m}}},$ (4.3)

where $A_{m}$ and $C_{m}$ are the values of $A$ and $C$ when $x=x_{m}$. The
track of a photon incoming from infinity with some impact parameter $u$ will
be curved while approaching the black hole. Detailedly, $\alpha$ becomes
higher than $2\pi$, resulting in a complete loop of the light ray around the
black hole.

Then, we define $y=A(x)$ and $z=(y-y_{0})/(1-y_{0})$ and rewrite the integral
$I(x_{0})$ as,

$\displaystyle I(x_{0})=\int_{0}^{1}R(z,x_{0})f(z,x_{0})dz,$ (4.4)

where

$\displaystyle
R(z,x_{0})=\frac{2\sqrt{By}}{CA^{\prime}}\left(1-y_{0}\right)\sqrt{C_{0}},$
(4.6) $\displaystyle
f(z,x_{0})=\frac{1}{\sqrt{y_{0}-[(1-y_{0})z+y_{0}]\frac{C_{0}}{C}}}.$ (4.7)

The function $R(z,x_{0})$ is regular for all values of its arguments, but the
function $f(z,x_{0})$ diverges as $z\rightarrow 0$. We divide the diverge part
into the regular one and the divergent one, then it could be

$\displaystyle I(x_{0})=I_{D}(x_{0})+I_{R}(x_{0}),$ (4.8)

where $I_{D}$ is the divergent part and $I_{R}$ is the regular part and
$x_{0}$ the closet distance,

$\displaystyle I_{D}(x_{0})=\int_{0}^{1}R(0,x_{m})f_{0}(z,x_{0})dz,$ (4.9)
$\displaystyle
I_{R}(x_{0})=\int_{0}^{1}g(z,x_{0})dz=\int_{0}^{1}(R(z,x_{m})f(z,x_{m})-R(0,x_{m})f_{0}(z,x_{m}))dz,$
(4.10)

where the latter gives the deflection angle to order $O(x_{0}-x_{m})$, and the
function $g(z,x_{m})$ is regular at $z=0$.

At last, we expand $u$ in defined Eq.(4.2), we have

$\displaystyle u-u_{m}=\hat{c}(x-x_{m})^{2}$ (4.11)

where
$\hat{c}=(C^{\prime\prime}_{m}y_{m}-C_{m}A^{\prime\prime}_{m})/(4\sqrt{y_{m}^{3}C_{m}})$.
Then, by using $u_{m}$ instead of $x_{m}$, the deflection angle could be
expressed as,

$\displaystyle\alpha(\theta)=-\bar{a}\ln\left(\frac{\theta
D_{OL}}{u_{m}}-1\right)+\bar{b},$ (4.12)

where

$\displaystyle\bar{a}=\frac{R(0,x_{m})}{2\sqrt{\beta_{m}}},$ (4.13)
$\displaystyle\bar{b}=b_{0}+b_{R}=-\pi+\bar{a}\ln\frac{2\beta_{m}}{y_{m}}+b_{R},$
(4.14)

where $b_{R}=I_{R}(x_{m})$ and
$\beta_{m}=\hat{\beta}\mid_{x_{0}=x_{m}}=2\hat{c}C_{m}^{3/2}y_{m}^{-1/2}(1-y_{m})^{2}$.
$\bar{a}$ and $b_{0}$ encode the divergent part, and $b_{R}$ represents the
regular part. Then, the two integrals in are expanded around the photo sphere
$x_{m}$ . As the value of $x_{m}$ is known, we could compute $\alpha$ by a
proper expansion in the parameters of the metric.

##  5 The observables

In this section, we will show how to translate the parameters $u_{m}$,
$\bar{a}$ and $\bar{b}$ to the observables. Then, through the observations of
strong lens, we obtain the metric parameters , and then could probe the space
time structure. We consider the simplest condition where outermost image
$\theta_{1}$ is resolved as a single image, while all the remaining ones are
packed together at $\theta_{\infty}$. The strong field gravitational lensings
are helpful to distinguish black holes if we can separate the outermost
relativistic images and determine their angular separation, brightness
difference, time delay and QNM.

###  5.1  The parameter estimation on the positional separation
$\theta_{\infty}$

In theory, when the lens and the observer are nearly aligned and the black
hole has spherical symmetry, we can define the angular radius of shadow of
black hole as

$\displaystyle\theta_{\infty}=\frac{u_{m}}{D_{OL}}.$ (5.1)

Inversely, the impact parameter could be detected by the angular radius of
shadow of black hole as well $u_{m}=\theta_{\infty}D_{OL}$. In observation,
for M87*, the shadow angular diameter is $\theta_{\infty}=21\pm 1.5\mu as$,
the distance of the M87* from the Earth is $D_{OL}=16.8Mpc$, and the mass of
the M87* is $6.5\pm 0.90\times 10^{9}M_{\odot}$ . For Sgr. A* the shadow
angular radius is $\theta_{\infty}=24.35\pm 3.5\mu as$(EHT), the distance of
the $Sgr.A^{*}$ from the Earth is $D_{OL}=8277\pm 33pc$ and mass of the black
hole is $4.3\pm 0.013\times 10^{6}M_{\odot}$(VLTI). Then, to discuss the
observational constraint on $\theta_{\infty}$ by using the data from M87* and
Sgr A* of EHT, we make $\chi^{2}$ test which is defined as

$\displaystyle\chi^{2}=\frac{(\theta^{theory}_{\infty}-\theta^{observation}_{\infty})^{2}}{error^{2}}.$
(5.2)

We summarize our results in FIG.1. For the anisotropic black hole, the
observations show $0>\epsilon>-0.06551>-0.12181$ at $1\sigma$ and $2\sigma$
level of confidence and $\epsilon$ has no best-fitted value. For Garriga-
Tanaka black hole, we obtain $l^{2}=0.07451^{+0.2864+0.5156}_{-0.0745-0.0745}$
which have the $1\sigma$ and $2\sigma$ regimes of $l^{2}$ with the best-fitted
value. Interestingly, the $\chi^{2}$ curves of the two black holes seem to be
different sectors of an integrated curve.

As the constraints on anisotropic black hole fail to find the best fitted
value, we choose $\epsilon=-10^{-5}$ which has the lowest $\chi^{2}$ and could
represent the best fitted value to discuss. Based on the $1\sigma$ and
$2\sigma$ regimes and best fitted values, we list the related observations
(including angular separation, brightness difference, time delay and QNM) in
TABLEs 1 and 2. To do comparison, we also list these quantities of
Schwarzschild black hole as well which is the solution of Einstein equation
with vacuum, static and spherically and chargefree.

Figure 1: The left is $\chi^{2}$ for the parameter $\epsilon$ in the
anisotropic black hole. The right is $\chi^{2}$ for the parameter $l^{2}$ in
the Garriga-Tanaka black hole. The dashed lines denotes the $2\sigma$ value.
The dotted lines denotes the $2\sigma$ value. The solid line denote the best
fitted value.

Figure 2:  The strong deflection lensing coefficients $\bar{a}$, $\bar{b}$, and deflection angle $\alpha$ vs $\epsilon$ and $l^{2}$. The blue dot denotes the Schwarzchild case. The $\epsilon$ parameter in the anisotropic black hole is in the constrained range $-0.12181<\epsilon<0$. The $l^{2}$ parameter in the Garriga-Tanaka black hole is $0<l^{2}<0.5901$.  Parameters | anisotropic BH | Schwarzschild BH | Garriga-Tanaka BH
---|---|---|---
$\epsilon=-0.1218$ | $\epsilon=-0.0655$ | $\epsilon=-10^{-5}$ | $l^{2}=0.0745$ | $l^{2}=0.3609$ | $l^{2}=0.5901$
$x_{m}$ | $1.397$ | $1.444$ | $1.500$ | $1.500$ | $1.552$ | $1.707$ | $1.803$
$u_{m}(R_{s})$ | $2.368$ | $2.474$ | $2.598$ | $2.598$ | $2.652$ | $2.822$ | $2.932$
$\bar{a}$ | $1.039$ | $1.020$ | $1.000$ | $1.000$ | $0.9779$ | $0.9285$ | $0.9064$
$\bar{b}$ | $-0.4319$ | $-0.4160$ | $-0.4004$ | $-0.4002$ | $-0.4112$ | $-0.4567$ | $-0.4870$
$\alpha$ | $6.500$ | $6.433$ | $6.364$ | $6.364$ | $6.223$ | $5.901$ | $5.753$
Table 1:  The values of parameters $x_{m}$, $u_{m}$, $\bar{a}$, $\bar{b}$ and $\alpha$ parameters. The $u_{m}$ is scaled by Schwarzschild radius $R_{s}=2GM_{\bullet}/c^{2}$.  Parameters | BH type | anisotropic BH | Schwarzschild BH | Garriga-Tanaka BH
---|---|---|---|---
$\epsilon=-0.1218$ | $\epsilon=-0.0655$ | $\epsilon=-10^{-5}$ | $l^{2}=0.0745$ | $l^{2}=0.3609$ | $l^{2}=0.5901$
$\theta_{\infty}~{}(\mu as)$ | M87* | $18.15$ | $18.96$ | $19.91$ | $19.91$ | $20.33$ | $21.63$ | $22.47$
Sgr $A^{*}$ | $24.19$ | $25.27$ | $26.54$ | $26.54$ | $27.10$ | $28.83$ | $29.95$
$s~{}(nas)$ | M87* | $28.34$ | $26.64$ | $24.91$ | $24.92$ | $21.63$ | $15.23$ | $12.82$
Sgr $A^{*}$ | $37.77$ | $35.51$ | $33.21$ | $33.22$ | $28.83$ | $20.30$ | $17.08$
$r~{}(mag)$ | $-$ | $6.565$ | $6.688$ | $6.822$ | $6.822$ | $6.976$ | $7.347$ | $7.526$
$\Omega_{m}$ | $-$ | $0.4223$ | $0.4042$ | $0.3849$ | $0.3849$ | $0.3770$ | $0.3543$ | $0.3411$
$\lambda$ | $-$ | $0.1265$ | $0.1223$ | $0.1176$ | $0.1176$ | $0.1166$ | $0.1125$ | $0.1096$
$\Delta T_{2,1}$ | $-$ | $15.14$ | $15.80$ | $16.57$ | $16.57$ | $16.90$ | $17.92$ | $18.59$
$\Delta T_{3,2}$ | $-$ | $14.89$ | $15.56$ | $16.33$ | $16.33$ | $16.67$ | $17.74$ | $18.43$
$\Delta T_{4,3}$ | $-$ | $14.88$ | $15.54$ | $16.32$ | $16.32$ | $16.67$ | $17.73$ | $18.42$
$\Delta T^{1}_{2,1}(10^{-3})$ | $-$ | $261.6$ | $255.2$ | $248.7$ | $248.7$ | $229.6$ | $190.0$ | $173.9$
$\Delta T^{1}_{3,2}(10^{-5})$ | $-$ | $1273$ | $1173$ | $1075$ | $1075$ | $924.3$ | $644.8$ | $543.3$
$\Delta T^{1}_{4,3}(10^{-7})$ | $-$ | $6191$ | $5392$ | $4644$ | $4645$ | $3720$ | $2188$ | $1697$
Table 2:  The values of observables $\theta_{\infty}$, $s$, $r$, the QNM and
the time delay parameters. The $\theta_{\infty}$ and $s$ are respectively in
the units of micro-arcsecond ($\mu$as) and nano-arcsecond (nas). The time
delay parameters $\Delta T_{n,m}$ and $\Delta T_{n,m}^{\ 1}$ are scaled by
$2GM_{\bullet}/c^{3}\approx 42.45s$.

Figure 3:  The strong deflection lensing coefficients $\bar{a}$, $\bar{b}$
(including $b_{R}$, $b_{0}$ as well) and deflection angle $\alpha$. The ranges
of parameters are the same as FIG.2.

###  5.2 Discussions of the deflection angle in theory

Firstly, we list the deflection angle related parameter in TABLE 1. Then, we
plot the shapes of the parameters $x_{m}$, $u_{m}$ and the observable
$\theta_{\infty}$ in FIG.2 . The shapes of them are similar, all of them are
increasing with respect to $\epsilon$ and $l^{2}$. In the case of Garriga-
Tanaka black hole, the $x_{m}$, $u_{m}$ and $\theta_{\infty}$ parameters are
always larger than the one of the Schwarzschild black hole with the same mass.
While in the anisotropic black hole, $x_{m}$, $u_{m}$ and $\theta_{\infty}$
are smaller than the ones in Schwarzchild black hole. The Schwarzchild black
hole connects the two braneworld black holes which denotes we could not
distinguish the two braneworld black holes from the Schwarzchild black hole.
As $u_{m}$ is determined by the non-linear relation (Eq.(4.11)) with $x_{m}$,
the slope of $u_{m}$ for the anisotropic black hole is smaller than that of
Garriga-Tanaka black hole.

In Figure 3, the parameters $\bar{a}$ and $\bar{b}$ play a prominent role in
measuring the angular difference from the outmost image and the adherent point
related to the sequence of subsequent images. Roughly speaking, a bigger
$u_{m}$ implies a smaller $\alpha$. Then the shape of $\bar{a}$ determines the
shape of $\alpha$ which represents the divergence part. And the first term of
$\alpha$ is more important than the other ones. Furthermore, corresponding to
Eqs.(4.13) and (4.14), the divergent parts $\bar{a}$ and $b_{0}$ have a
decreasing tendency, while the regular part $b_{R}$ contributes to the
increasing tendency. The parameter $\bar{b}$, which presents the main part of
the regular part, is one order lower than than $\alpha$, then its non-
monotonic value does not affect $\alpha$. But, the shape of $\alpha$ parameter
is not smooth as the $\bar{a}$.

###  5.3 The Observations

Besides $\theta_{\infty}$, there are other relations which could translate the
$\bar{a}$ and $\bar{b}$ parameter to the observables, e.g. $s$, $r$, the time
delays and the QNMs. We list all observables in TABLE 2 which shows the same
tendencies.

####  5.3.1 The $s$ and $r$ parameter

The observable $s$ is the angular separation between the outermost image
($n=1$) and the packed others $n=2,3,....\infty$, and $r$ is the magnitude
difference between the outermost image and the packed images,

$\displaystyle
s=\theta_{1}-\theta_{\infty}=\theta_{\infty}\exp{(\frac{\bar{b}}{\bar{a}}-\frac{2\pi}{\bar{a}})},$
(5.3) $\displaystyle
r=2.5\log_{10}(\frac{\mu_{1}}{\sum_{n=2}^{\infty}\mu_{n}})=2.5\log_{10}[\exp{(\frac{2\pi}{\bar{a}})}].$
(5.4)

We plot the strong lensing coefficients and the value of the observables $s$,
$r$ in FIG 4 as well. From FIG.4, the angular separation $s$ increases but
angular position ($\theta_{\infty}$) and flux magnitude ($\mu$) decrease with
respect to $\epsilon$ and $l^{2}$. The parameter $s$ is much smaller than
$\theta_{\infty}$($1/1000$). As Eq.(5.3) shows, then $\theta_{1}$ are close to
$\theta_{\infty}$. That means the $\theta_{\infty}$ parameter could present
the main effect of angular separation. The parameter $s$ in anisotropic black
hole is nearly linear, while in Garriga-Tanaka black hole is non-linear. The
$s$ and $r$ are hard to observe because the magnitude ratio is proportional to
the magnitude.

The $s$ is increasing with respect to $\epsilon$ and $l^{2}$, while the $r$ is
decreasing. If we try to distinguish them via observation data, the accuracy
of the measured separation between the first image and the surplus fringes
needs to be less than $1\mu as$, and the photometric uncertainty has to better
than $0.1$ mag.

Figure 4: The estimated observables $s$, $r$ as functions of $\epsilon$ (or
$l^{2}$) for Anistropic Black hole (or Garriga-Tanaka Black Hole). The ranges
of parameters are the same as FIG.2.

Figure 5:  The estimated observables $\Delta T_{n,m}$ and $\Delta T_{n,m}^{\
1}$ as functions of $\epsilon$ (or $l^{2}$) for Anistropic Black hole (or
Garriga-Tanaka Black Hole). The ranges of obervables are the same as in FIG.2.
$\Delta T_{n,m}$ and $\Delta T_{n,m}^{\ 1}$ are both represented in the unit
of $2GM_{\bullet}/c^{3}\approx 42.45$ s.

####  5.3.2  The time delay

The time delays between relativistic images are observations as well. Bozza
and Mancini obtain differential time delays among relativistic images due to
gravitational lensing by a general static spherically symmetry spacetime [31].
The time delay is the result of the different paths followed by the photons
while they cross around the black hole. Differences in the deflection angle
are immediately displayed on the relativistic images[31]. If the mass and
distance of the lens ( $D_{os}$) are known, then the any set of relativistic
images could probe the type of black hole. $\Delta T_{n,m}$ is the total time
delay between the $m$-loop image and the $n$-loop image

$\displaystyle\Delta T_{n,m}=\Delta T^{0}_{n,m}+\Delta T^{1}_{n,m},$ (5.5)

where leading term of time delay is

$\displaystyle\Delta T^{0}_{n,m}=2\pi(n-m)u_{m},$ (5.6)

while its much smaller correction is

$\displaystyle\Delta
T^{1}_{n,m}=2\sqrt{B_{m}/A_{m}}\sqrt{u_{m}/\hat{c}}\exp{(\bar{b}/(2\bar{a}))}[\exp{(-m\pi/\bar{a})}-\exp{(n\pi/\bar{a})}].$
(5.7)

The unit of time delay is $2GM_{\bullet}/c^{3}\sim 42.45$ s. The dominant term
in the time delay is not a new independent factor of the black hole. But the
second term is. We consider three cases ( ($n=2,m=1$), ($n=3,m=2$),
($n=4,m=3$)) satisfying $n-m=1$ in which we consider two nearby loops and set
same $T^{0}_{n,m}$. As shown in FIG. 5, the time delay in the two nearby loops
is from $\Delta T^{1}_{n,m}$ which decreases to $0$ fast as $n$ increases. The
same tendency occurs in the leading term $T_{n,m}$ when $n-m=1$, which is
consistent with small $s$. The phenomenon shows there is no significant
difference between the outmost image and the stacked images. Our present
observational facilities do not reach the required resolution yet. For
galactic black hole, the required resolution is of the order of $0.01$ micro-
arcsecs. As the unit is $2GM_{\bullet}/c^{3}\sim 42.45$, the detection of
$\Delta T_{2,1}$ needs to have an accuracy better than the level of $\sim
2\times 10^{-2}$ . After multiplying the units, it corresponds to the level of
about $1s$ for Sgr A*.

####  5.3.3 The quasi normal modes (QNMs)

The strong lensing is useful to explain the characteristic modes of black hole
as well. In Ref. [34, 35, 36], the quasi normal modes (QNMs) and the strong
lensing are found to connect each other. The QNM describes the decay rates of
perturbations around a black hole. It is expected to detect these
perturbations in the further observations. At eikonal limit, the real and
imaginary parts of the QNMs of any spherically symmetric, asymptotically flat
spacetime are given by (multiples of) the frequency ($\Omega_{m}$) and
instability timescale of the unstable circular null geodesics.

$\displaystyle\omega_{QNM}=\Omega_{m}l-i(n+1/2)|\lambda|,$ (5.8)

where $l$ and $n$ are constants and

$\displaystyle\lambda=\frac{1}{u_{m}\bar{a}},$ (5.9)
$\displaystyle\Omega_{m}=1/u_{m}.$ (5.10)

The real part of the complex QNM frequencies is determined by the angular
velocity at the unstable null geodesic, the imaginary part the QNM is related
to the instability timescale of the orbit which is called the Lyapunov
exponent. The Lyapunov exponent is in turn reflected in the associated QNMs in
the geometrical optical approximation. We list the values of QNMs in FIG.6.
The tendencies of $\lambda$ and $\Omega_{m}$ are similar. For the Schwarzchild
model, it is the dot when $l^{2}=0$, while for the two branneworld black
holes, it decreases with respect to $\epsilon$ and $l^{2}$. Our constrained
Lyapunov exponents for M87* and SgrA* are positive. A positive Lyapunov
exponent indicates a divergence between nearby trajectories, i.e., a high
sensitivity to initial conditions.

Figure 6:  The estimated observables $\lambda$ and $\Omega_{m}$ as functions
of $\epsilon$ (or $l^{2}$) for Anistropic Black hole (or Garriga-Tanaka Black
Hole). The ranges of parameters are the same as in FIG.2.

###  5.4 A short summary

To derive the deflection angle, we need three parameter ($u_{m}$, $\bar{a}$
and $\bar{b}$). The observations $\theta_{\infty}$, $T_{n,m}$ and $\Omega_{m}$
are all related to $u_{m}$. The parameters $\lambda$ and $r$ are related to
$\bar{a}$. The parameters $s$ and $\Delta T^{1}_{n,m}$ are related to
$\bar{b}$. The Higher order effects, such as $s$, $r$, $\Delta T^{1}_{2,1}$
will distinguish the two black holes. We also show the total effect of
$\alpha$. The non-linear relation between $u$ and $x_{m}$ make the non-
smoothness between the the braneworld models.

##  6 Conclusion

In this work, based on the EHT data ($\theta_{\infty}$), we first use the
$\chi^{2}$ test to estimate the range of parameters of braneworld black holes.
Then, the $1\sigma$ and $2\sigma$ regimes of the model parameters are
$\epsilon>-0.0655>-0.1218$ for the anisotropic black hole, and
$l^{2}=0.0745^{+0.2864+0.5156}_{-0.0745-0.0745}$ for the Garriga-Tanaka black
hole. The braneworld model are consistent with the observation which shows the
braneworld black holes possess richer structure than ordinary black holes. And
following the fitted data, we calculate the photon deflection angle, the
angular separation, time delay values and QNM values between different
relativistic images of the anisotropic black hole and the Garriga-Tanaka black
hole. The anisotropic black hole and the Garriga-Tanaka black hole could not
be distinguished by using observation data. And, this result is useful to
probe extra dimension.

## Acknowledgments

YZ is supported by National Natural Science Foundation of China under Grant
No.12275037 and 12275106. DW is supported by the NSFC under Grants No.
12205032, CQ RLSBJ under grant cx2021044, and the Talent Introduction Program
of Chongqing University of Posts and Telecommunications under grant No.
E012A2020248. HZ is supported by National Natural Science Foundation of China
Grants Nos.12275106 and 12235019.

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