# Parameter estimation of hairy Kerr black holes from its shadow and
constraints from M87*

Misba Afrina<EMAIL_ADDRESS>Rahul Kumara<EMAIL_ADDRESS>Sushant G.
Ghosh${}^{a,\;b}$<EMAIL_ADDRESS><EMAIL_ADDRESS>a Centre for
Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India  b
Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics
and Computer Science, University of KwaZulu-Natal, Private Bag 54001, Durban
4000, South Africa

###### Abstract

The recently obtained hairy Kerr black holes, due to additional sources or
surrounding fluid, like dark matter, conserving energy-momentum tensor, are
with a deviation $\alpha$, and primary hair $l_{0}$ apart from rotation
parameter $a$ and mass $M$. The Event Horizon Telescope (EHT) has released the
first image of supermassive black hole M87*, revealing that the observed image
is consistent with the shadow of a Kerr black hole and has opened up a new
window to probe the strong gravity regime. In the wake of the EHT
observations, the recent interest in black hole shadow suggests comparing the
black holes in general relativity (GR) and modified theories of gravity to
assess these models’ differences. Motivated by this, we consider the rotating
hairy Kerr black holes, which encompasses, in particular cases, Kerr
($\alpha=0$) black holes. Interestingly, for a set of parameters ($M$, $a$,
and $\alpha$), there exists an extremal value of the hair parameter $l_{E}$,
which corresponds to an extremal black hole with degenerate horizons. In
contrast, for $l_{0}>l_{E}$, it describes a nonextremal black hole with Cauchy
and event horizons and a naked singularity for $l_{0}<l_{E}$. We investigate
ergosphere and shadow of hairy Kerr black holes to infer that their size and
shape are affected due to the $l_{0}$ and have a richer chaotic structure. In
particular, the hairy Kerr black holes possess smaller size but more distorted
shadows when compared with Kerr black holes. We also estimate the parameters
$l_{0}$ and $a$ associated with hairy Kerr black holes using the shadow
observables. We show that the inferred circularity deviation $\Delta C\leq
0.1$ for the M87* black hole puts a constraint on hairy Kerr black hole
parameter $a$, whereas angular shadow diameter $\theta_{d}=42\pm 3\mu as$,
within $1\sigma$ region, for a given choice of $\alpha$, also places bounds on
these parameters.

##  I Introduction

Recently, the international Event Horizon Telescope (EHT) collaboration has
unveiled the first shadow image of a supermassive black hole M87* with angular
size gravitational radius $3.8\pm 0.4$ $\mu$as – an asymmetric bright emission
ring with a diameter of $42\pm 3$ $\mu$as, exhibiting a deviation from
circularity $\Delta C\leq 0.1$ and an axis ratio $\lesssim 4/3$, and it is
consistent with the shadow of a Kerr black hole of general relativity. The
black hole shadows have become a physical reality with the detection of the
M87* black hole’s horizon-scale image Akiyama:2019cqa ; Akiyama:2019abc ;
Akiyama:2019eap . The bright, sharp photon ring, a projection along the
photons’ null geodesics orbiting around the black hole, encompasses the shadow
and explicitly depends on the black hole parameters Johannsen_2010 . It has
been found that the quantitative features are not sufficient to distinguish
between black holes using different theories of gravity. Thus, using the M87*
black hole shadow, one can investigate the viability of black holes in
modified theories of gravity (MoG) in explaining the observational data and
constraints on the black hole parameters and the EHT observations thus became
an important tool to test strong gravitational fields.

The shadow of rotating black holes attracted a lot of attention in the last
decades, led by the seminal work by Synge Synge:1966  and Luminet Luminet:1979
, who gave the formula to calculate the angular radius of the photon capture
region around the Schwarzschild black hole. Later, Bardeen Bardeen1973
analysed the shadow of the Kerr black hole and showed that the spin would
cause distortion in the shape of shadows. Applications of shadow in
unravelling the useful information regarding the near horizon features of
gravity have aroused a flurry of activities in the analytical investigations,
and numerical simulation of shadows for black holes in GR Falcke:1999pj ; De ;
Shen:2005cw ; Yumoto:2012kz ; Atamurotov:2013sca ; Papnoi:2014aaa ;
Abdujabbarov:2015xqa ; Atamurotov:2016 ; Cunha:2018acu ; Kumar:2018ple , some
other configurations of black holes arising in modified theories of gravity
(MoG) Amarilla:2010 ; Amarilla:2011 ; Amarilla:2013 ; Amir:2017 ; Singh:2017 ;
Mizuno:2018 ; Ghosh:2020 ; 4DEGB_2020  and in higher dimensions Amir:2017slq ;
Eiroa:2018 ; Banerjee:2019nnj . One of the motivations of several works on
black hole shadow is that the trajectories of light near black holes are
related to the background geometry’s essential features and properties.
Moreover, by observing the size and deformation of its shadow, the spin, mass
parameter and possibly other global charges or hair-parameters of the black
holes can be calculated Hioki2009 ; Tsupko2017 ; CunhaPhysRevLett2019 ;
CunhaUniverse2020 ; Kumar:2018ple . Moreover, it is also useful in testing
theories of gravity Kramer1:2004hd .

Observations of the shadow of supermassive black holes M87* allow us to study
the properties of black holes and the nature of strong-field gravity. In
principle, one can use black hole shadow observations to place constraints on
the hairy Kerr black hole, which can also be a solution to modified gravity
theories that do not admit the Kerr solution. We here investigate, whether
black hole shadow observations can constrain hairy Kerr black holes
Ovalle:2021PRD , which has additional deviation parameters $\alpha$ and
primary hair $l_{0}$, from the Kerr black hole. Furthermore, we investigate
the horizons, ergoregions and shadow cast by the hairy Kerr black holes
Ovalle:2021PRD . We obtain the null geodesics equations of motions in the
first-order differential form using the Hamilton-Jacobi approach, and the
analytical expressions for the photon region are also determined. An
investigation of the deviation parameters’ effects shows that the hairy Kerr
black holes cast smaller and more distorted shadows than those for the Kerr
black holes. In turn, we use shadow observables to characterize the shadows’
size and shape and estimate the black hole parameters $a$ and $l_{0}$. The
M87* black hole shadow results, inferred from the recent Event Horizon
Telescope (EHT) collaborations observations, are further used to constrain the
parameters $a$ and $l_{0}$.

The hairy Kerr black hole that we are interested in, is obtained using the
gravitational decoupling approach Ovalle:2021PRD . It has a source satisfying
the strong energy condition (SEC) and gives an extended Kerr metric which was
termed as Kerr black holes with primary hair by Contreras et. al.
Ovalle:2021PRD . Hairy black holes are termed as stationary black hole
solution with new global charges, which are not associated with Gauss law
Herdeiro:2015waa , e.g., black holes with scalar hair Herdeiro:2014goa ;
yaun:2021 , or proca hair Herdeiro:2016tmi . See Ref. Herdeiro:2015waa  for a
recent review on black holes with hair due to global charge.

This paper is organized as follows. In the Sec. II, we consider hairy black
holes, discuss the effect of $l_{0}$, for given choice of $\alpha$, on horizon
structure and ergoregions. Sec. III is devoted to the brief review of the null
geodesic equations in the hairy Kerr black hole spacetimes. The impact of the
hair parameter $l_{0}$ on the black hole shadow is the subject of Sec. IV. We
present the observables for shadow characterization and use them to estimate
the parameters associated with hairy Kerr black holes in Sec. V, and assuming
M87* as hairy Kerr black holes, we calculate the circularity deviation and
angular diameter observable over the entire parameter space for
$\theta_{0}=\pi/2$ in Sec. VI. Finally, we summarize our main findings in Sec.
VII. We use geometrized units $G=1$, $c=1$, unless units are specifically
defined.

##  II Hairy Kerr back holes

The Gravitational Decoupling (GD) approach is precisely designed to find
deformation of the known solution of GR due to the additional surrounding
sources like dark matter or dark energy Ovalle:2021PRD  (see also Ovalle:2017
; Ovalle:2019 ). The GD approach leads to deformed or hairy Schwarzchild black
holes Ovalle:2021PRD  given by

$ds^{2}=-\left[1-\frac{2M}{r}+\alpha
e^{{-r}/(M-\frac{l_{0}}{2})}\right]dt^{2}+\frac{1}{\left[1-\frac{2M}{r}+\alpha
e^{{-r}/(M-\frac{l_{0}}{2})}\right]}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}).$
(1)

Here, $M$ is the black hole mass, $\alpha$ is the deviation parameter and
$l_{0}\leq 2M$, which ensures asymptotic flatness, corresponds to primary
hair. The surrounding matter that leads to deformation of the Schwarzchild
black holes (1) is described by the conserved stress-energy tensor
$S_{\mu\nu}$ that satisfies the strong energy condition (SEC). The hairy black
hole (1) encompasses the Schwarzchild black hole in the absence of the
surrounding matter ($\alpha=0$). The rotating black hole solutions in modified
gravity theories are essential as they offer an arena to test these theories
through astrophysical observation. There is hardly any test for non-rotating
solutions since the black hole spin is crucial in any astrophysical process.
Hence, we consider the rotating counterpart of the spherically symmetric
solution. The stationary and axisymmetric counterpart of the spherically
symmetric solution (1) described by parameters $M$, $a$, $l_{0}$ and $\alpha$,
in the Boyer-Lindquist coordinates reads Ovalle:2021PRD

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left[\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\right]dt^{2}-2a\sin^{2}{\theta}\left[1-\frac{\Delta-a^{2}\sin^{2}{\theta}}{\Sigma}\right]dtd\phi$
(2) $\displaystyle+$
$\displaystyle\sin^{2}\theta\left[\Sigma+a^{2}\sin^{2}{\theta}\left(2-\frac{\Delta-a^{2}\sin^{2}{\theta}}{\Sigma}\right)\right]d\phi^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2},$

with $\Delta=r^{2}+a^{2}-2Mr+\alpha r^{2}e^{{-r}/(M-\frac{l_{0}}{2})}$ and
$\Sigma=r^{2}+a^{2}\cos^{2}\theta$, where $a$ denotes the angular momentum.
The $\alpha$ is a generic parameter that measures potential deviation of
metric 2 from the standard Kerr black holes and is related to $l_{0}$ via
$l_{0}=\alpha l$. The black hole 2 is termed as hairy Kerr black holes
Ovalle:2021PRD  which encompasses the Kerr black hole in the absence of
surrounding matter ($\alpha=0$). The solution 2 results from the surrounding
matter described by stress tensor $S_{\mu\nu}$ satisfying the strong energy
conditions. Thus metric (2) can be seen as a prototype non-Kerr black hole and
with an additional deviation parameter $\alpha$ and primary hair $l_{0}$
Ovalle:2021PRD . When written in Boyer-Lindquist coordinate, it is same as
Kerr black hole with $M$ replaced by
$m(r)=M-\frac{\alpha}{2}re^{-r/(M-\frac{l_{0}}{2})}$. In what follows, we
shall investigate how primary hair $l_{0}$ affects the Kerr horizons and
shadow.

The metric 2 is singular at spacetime points where $\Sigma\neq 0$ and
$g^{\alpha\beta}\partial_{\alpha}r\partial_{\beta}r=g^{rr}=\Delta=0$, which
corresponds to the event horizon. Thus zeros of

$r^{2}+a^{2}-2Mr+\alpha r^{2}e^{{-r}/(M-\frac{l_{0}}{2})}=0,$ (3)

give the black hole horizons. Equation (3), when $\alpha=0$ (Kerr black hole),
admits solutions

$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}.$ (4)

Here, $r_{+}$ denotes outer (event) horizon and $r_{-}$ the inner (Cauchy)
horizon. It is clear that the metric (2) describes non-extremal black hole for
$r_{+}>r_{-}$ and when $r_{+}=r_{-}$, one obtains an extremal black hole. To
ensure the event horizon’s existence, we require $M\geq a$. It defines a
region, parameter space, where metric represents a black hole and not a naked
singularity. The case $M<a$, contains a naked singularity.

Numerical analysis of Eq. (3) reveals that depending on the values of $M$,
$a$, $\alpha$ and $l_{0}$ there can exist a maximum of two distinct real
positive roots, or equal roots, or no-real positive roots. They respectively
correspond to the non-extremal black holes with Cauchy and event horizons,
extremal black holes, and no-black holes scenarios. The degenerate condition
of the horizons gives a bound on the black hole parameters $l_{0}$ and $a$ and
the ($l_{0}$-$a$) parameter space is shown in Fig. 1.

From the horizon structure of the hairy Kerr black holes depicted in Fig. 2,
it turns out that, for given values of $\alpha$ and $l_{0}$, there exists
critical extremal value of $a$, $a_{E}$ and likewise there exists critical
extremal value of $l_{0}$, $l_{E}$ for given value of $\alpha$ and $a$, where
$\Delta=0$ has a double root. The hairy Kerr black hole exists when $a<a_{E}$
while for $a>a_{E}$ there is a naked singularity and the $a_{E}$ depends on
$l_{0}$, e.g., for $l_{0}=0.2M,0.5M$ respectively, $a_{E}=0.8350M,0.8638M$.
Also $l_{0}>l_{E}$ corresponds to hairy Kerr black holes with Cauchy and event
horizons whereas $l_{0}<l_{E}$ implies existence of a naked singularity. It
may be noted that, for $a=0.85M,0.90M$, $l_{E}=0.3562M,0.7778M$ respectively.
Indeed $a_{E}$ increases with the increase in $l_{0}$ likewise $l_{E}$
increases with increase in $a$. For fixed values of $M$, $a$ and $l_{0}$, the
hairy Kerr black holes have decreasing event horizon radius and increasing
Cauchy horizon radius with the increase in deviation parameter $\alpha$ from
the Kerr limit, $\alpha=0$ (cf. Fig. 2). Thus the hairy Kerr black holes have
smaller event horizon radii than the Kerr black hole. At the extremal value
$\alpha=\alpha_{E}$, the two horizons coincide and the radius of this extremal
event horizon ($r_{E}$) increases with the increase in rotational parameter
$a$ while the value of $\alpha_{E}$ decreases, for e.g., $\alpha_{E}=1.8283$,
$r_{E}=0.6009M$ for $a=0.7M$ and $\alpha_{E}=1.1652$, $r_{E}=0.7615M$ for
$a=0.8M$ (cf. Fig. 2).

Figure 1:  $l_{0}-a$ parameter space of the hairy Kerr black hole. The lines
separate black holes from naked singularity ($\Delta=0$ has no real root).

On the other hand, at the static limit surface (SLS), the asymptotic time-
translational Killing vector $\chi^{i}=(\frac{\partial}{\partial t})^{i}$,
becomes null, i.e.,

$\chi^{i}\chi_{i}=g_{tt}=r^{2}+a^{2}\cos^{2}{\theta}-2Mr+\alpha
r^{2}e^{{-r}/(M-\frac{l_{0}}{2})}=0.$ (5)

Thus, the location of SLS are the real positive roots ($r_{SLS}^{\pm}$) of Eq.
(5) which admits two possible. The larger of the two roots corresponds to the
outer surface of the SLS, denoted by $r_{SLS}^{+}$. The analysis of the zeros
of Eq. (5), for a given values of $a$, $l_{0}$ and $\theta$, disseminate a
critical parameter $\alpha_{SLS}^{E}$ such that Eq. (5) has no root if
$\alpha>\alpha_{SLS}^{E}$, a double root at $\alpha=\alpha_{SLS}^{E}$, and two
simple roots if $\alpha<\alpha_{SLS}^{E}$ .

|
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|
Figure 2: Horizons of the hairy Kerr black holes are shown with varying $a$
[top], $\alpha$ [middle] and $l_{0}$ [bottom] and compared with the Kerr black
holes ($\alpha=0$ or $l=l_{k}$).

The ergoregion bounded between $r_{+}<r<r_{SLS}^{+}$, where the timelike
killing vector $\chi^{i}$ becomes spacelike, and an observer necessarily
follows the worldline of $\chi^{i}$, has been shown in Fig. 3 for the hairy
Kerr black hole (2). The parameter $l_{0}$ has a diminishing effect on the
ergosphere’s size, as the ergoregion gets smaller with the increase in $l_{0}$
for fixed values of the parameters $\alpha$ and $a$. Thus, the hairy Kerr
black holes have smaller ergoregions than the Kerr black hole ($l=l_{k}$).
Moreover, in the non- Kerr limit ($l_{0}\neq l_{k}$), the increase in the
$\alpha$ increases the size of the ergoregion at fixed values of $l_{0}$ and
$a$. It is also worthwhile noting that the decrease in the parameter $l_{0}$
leads to disconnected event horizons for fast rotating hairy Kerr black holes
(cf. Fig. 3). The ergoregion acquired its name owing to the theoretical
possibility of extracting energy and mass from this region via the Penrose
process PENROSE1971 .

The frame dragging effect of the hairy Kerr black hole becomes evident from
the off diagonal elements of metric (2) viz., $g_{t\phi}$. Due to this effect,
a stationary observer outside the event horizon, moving with zero angular
momentum with respect an observer at spatial infinity can rotate with the
black hole with an angular momentum given by Pugliese:2018hju

$\tilde{\omega}=\frac{d\phi}{dt}=-\frac{g_{t\phi}}{g_{\phi\phi}}=\frac{2ar(M-\alpha\frac{r}{2}e^{{-r}/(M-\frac{l_{0}}{2})})}{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}{\theta}},$
(6)

which monotonically increases as the observer approaches the black hole and
ultimately at the event horizon, where the observer begins co-rotating with
the black hole, takes the maximum value:

$\Omega=\tilde{\omega}|_{r=r_{+}}=\frac{2ar_{+}(M-\alpha\frac{r_{+}}{2}e^{{-r_{+}}/(M-\frac{l_{0}}{2})})}{(r_{+}^{2}+a^{2})^{2}}.$
(7)

Here $\omega$ is the black hole angular velocity, which in the limit
$\alpha=0$ reads

$\Omega_{Kerr}=\frac{2Mar_{+}}{(r_{+}^{2}+a^{2})^{2}}$ (8)

and corresponds to the angular velocity of Kerr black hole Frolov:2014 . It is
clear that each point of the horizon has the same angular velocity (as
measured at the infinity) and in this sense the surface of the black hole is
rotating as a rigid body Frolov:2014

##  III Geodesics around Rotating Black Holes

The geodesic motion of photons around the black hole is necessary to study the
shadow formation, and for this purpose, we consider a motion of test particle
in stationary and axially symmetric spacetime. Since metric (2) doesn’t depend
on $t$ and $\phi$, hence these are cyclic coordinates and corresponding
killing vectors are given by $\chi_{(t)}^{\mu}=\delta_{t}^{\mu}$ and
$\chi_{(\phi)}^{\mu}=\delta_{\emptyset}^{\mu}$, whose existence further
implies that the corresponding momenta $p_{t}$ and $p_{\phi}$ are constants of
motion. The motion of the test particle, neglecting the back reaction, is
determined by the rest mass $m_{0}$, total energy $E$, angular momentum
$L_{z}$ and carter constant $\mathcal{Q}$ which associated with the second-
rank irreducible tensor field of hidden symmetry carter1 . We use the
Hamilton-Jacobi equation following the integral approach pioneered by Carter
carter1  to obtain the geodesic equations in the first-order differential
form, which for the metric (2) read Chandrasekhar1983

$\displaystyle\Sigma\frac{dt}{d\lambda}$ $\displaystyle=$
$\displaystyle\frac{r^{2}+a^{2}}{\Delta}(E(r^{2}+a^{2})-aL_{z})-a(aE\sin^{2}{\theta}-L_{z}),$
(9) $\displaystyle\Sigma\frac{d\phi}{d\lambda}$ $\displaystyle=$
$\displaystyle\frac{a}{\Delta}(E(r^{2}+a^{2})-aL_{z})-(aE-\frac{L_{z}}{\sin^{2}{\theta}})\
,$ (10) $\displaystyle\Sigma\frac{dr}{d\lambda}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\mathcal{R}(r)}\ ,$ (11)
$\displaystyle\Sigma\frac{d\theta}{d\lambda}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\Theta(\theta)}\ ,$ (12)

where $\lambda$ is the affine parameter along the geodesics and the effective
potentials $\mathcal{R}(r)$ and ${\Theta}(\theta)$ for radial and polar motion
are given by

$\displaystyle\mathcal{R}(r)$
$\displaystyle=E^{2}\left[\Big{(}(r^{2}+a^{2})-a\xi\Big{)}^{2}-\Delta\Big{(}{\eta}+(a-{\xi})^{2}\Big{)}\right],$
(13) $\displaystyle\Theta(\theta)$
$\displaystyle=E^{2}[\eta-\left(\frac{{\xi}^{2}}{\sin^{2}\theta}-a^{2}\right)\cos^{2}\theta]\
.$ (14)

The constant $\mathcal{K}$ is the separability constant related to the Carter
constant $\mathcal{Q}$ through $\mathcal{Q}=\mathcal{K}+(aE-L_{z})^{2}$
Chandrasekhar1983 . We introduce dimensionless quantities called impact
parameters Chandrasekhar1983

$\xi=\frac{L_{z}}{E}\;\;\text{,}\;\;\eta=\frac{\mathcal{K}}{E^{2}},$ (15)

|  |  |
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|  |  |
Figure 3: The cross-section of event horizon (outer red line), SLS (outer blue
dotted line) and ergoregion of hairy Kerr black holes. The black dotted curves
correspond to event horizon of Kerr black hole ($l_{0}=l_{k}$)

which are constant along the geodesics. The allowed region for photon motion
is $\mathcal{R}\geq 0$ and $\Theta(\theta)\geq 0$ and the sign of $\dot{r}$
and $\dot{\theta}$ can be independently chosen to be either positive or
negative. The change of sign occurs at turning points of motion, i.e. when
$\mathcal{R}=0$ or $\Theta=0$ Chandrasekhar1983 . Depending on the critical
parameters’ values, the photon may either get captured, scatter to infinity,
or form bound orbits. Here, we are interested in spherical lightlike geodesics
on a sphere of constant coordinate radius $r$ characterized by $\dot{r}=0$ and
$\ddot{r}=0$ called the spherical photon orbits. Mathematically, this
corresponds to the extremum of the radial potential at the unstable photon
orbit radius $r=r_{p}$ such that Frolov2011 ; Chandrasekhar1983

$\mathcal{R}=\mathcal{R}^{\prime}=0\,\,\text{and}\,\,\mathcal{R}^{\prime\prime}\leq
0.$ (16)

Solving Eq. (16) results into the critical values of impact parameters
($\xi_{crit},\eta_{crit}$) for the unstable orbits

$\displaystyle\xi_{crit}=$
$\displaystyle\frac{\left(a^{2}+r^{2}\right)\Delta^{\prime}(r)-4r\Delta(r)}{a\Delta^{\prime}(r)},$
$\displaystyle\eta_{crit}=$
$\displaystyle\frac{r^{2}\left(8\Delta(r)\left(2a^{2}+r\Delta^{\prime}(r)\right)-r^{2}\Delta^{\prime}(r)^{2}-16\Delta(r)^{2}\right)}{a^{2}\Delta^{\prime}(r)^{2}},$
(17)

where ′ stands for the derivative concerning the radial coordinate, Eq. (17),
in the limit $\alpha\to 0$, reduces to the critical impact parameter for the
Kerr black holes Hioki2009 . The unstable photon orbits have been particularly
studied with much interest for black holes and naked singularities Wilkins
(1972); Goldstein (1974); Jhonston1974 ; Izmailov1979 ; Izmailov1980 ;
Calvani1978 ; Teo2003  and in the case of the black hole, they define the
boundary between the capture or no-capture of a cross-section of light rays.
There can be two circular photon orbits in the equatorial plane of the black
hole viz in axially symmetric spacetimes viz. those move in the same direction
as the black hole’s rotation and those moving the opposite it which are
respectively called the prograde and retrograde photons. The prograde orbits
being smaller in radius than the retrograde ones BardeenAPJ1972 , which can be
attributed to the Lens-Thirring effect Thirring , whereby, due to the dragging
of the inertial frame by the black hole’s rotation, to an observer at
infinity, the prograde photons must have smaller orbits to account for the
excess angular momentum. In contrast, the retrograde ones, having lost some
angular momentum effectively, would have to rotate at larger radii Teo2003 .
The non-planar or 3-dimensional photon orbits only arise for
$\mathcal{\eta}_{crit}>0$ whereas, for $\mathcal{\eta}_{crit}=0$, the photon
orbits are planar and confined to the equatorial plane only. The radii of
prograde ($r_{p}^{-}$) and retrograde ($r_{p}^{+}$) orbits at the equatorial
plane are obtained as roots of $\eta_{crit}=0$. It turns out that these radii
for the Kerr black holes are Teo2003

$\displaystyle r_{p}^{-}\equiv
2M\left[1+\cos\left({\frac{2}{3}\arccos\left(-\frac{|a|}{M}\right)}\right)\right]\
,$ (18) $\displaystyle r_{p}^{+}\equiv
2M\left[1+\cos\left({\frac{2}{3}\arccos\left(\frac{|a|}{M}\right)}\right)\right]\
,$ (19)

and they fall in the range $M\leq r_{p}^{-}\leq 3M$ and $3M\leq r_{p}^{+}\leq
4M$ Kumar:2018ple . In the case of Schwarzschild black hole ($a=0$) the two
radii degenerate degenerate into a photon sphere of constant radius
$r_{p}^{-}=r_{p}^{+}=3M$ Kalb:2020 . While $r_{p}^{+}>r_{+}$, all spherical
photon orbits are confined to the region $r_{p}^{-}<r_{p}<r_{p}^{+}$. For the
$\eta_{crit}>0$ scenario, the non-planar ($\theta\neq\frac{\pi}{2}$ and
$\dot{\theta}\neq 0$) geodesics have an additional latitudinal motion governed
by the Carter constant which accounts for a hidden space-time symmetry.
Furthermore, these orbits oscillate symmetrically about the equatorial plane
while repeatedly crossing it.

##  IV Shadow of hairy Kerr black holes

The black hole shadow is the optical appearance of the black hole caused by
the strong gravitational lensing virbhadra_elis ; Bozza2001 ; Bozza2002 ;
Ghosh:2020spb ; Kumar:2020sag ; Islam:2020xmy ; Islam:2021dyk  of light near
these compact objects and appears as a two-dimensional dark zone for an
observer (for details see grenzebach2014 ; Cunha2018_1 ; Cunha2018_2 ;
Huang2018  and references therein). The shape and the size of the shadow
depend on the parameters associated with the geometry of black hole spacetime.
The shadow, in turn, has facilitated the estimation and measurement of various
black hole parameters like its mass, spin angular momentum as well as other
hairs Kumar:2018ple . Thus, it is a tool to test Einstein’s GR in the strong-
field regime Ayzenberg_2018 ; 4DEGB_2020 , and also the no hair theorem
Cunha2015_ScalarHair . The photon region around the black hole’s event horizon
is formed from the combination of all unstable and spherical photon orbits,
i.e. the separatrix between photon geodesics that escape to spatial infinity
and those that fall into the event horizon.

We assume uniformly distributed light sources at infinity and the photons
coming at all possible impact parameters and either getting scattered near the
vicinity of the black hole (2) or captured by it. We also assume the observer
at spatial infinity, at an inclination angle $\theta_{0}$ with the rotation
axis of the black hole. The celestial coordinates ($X$,$Y$) of the observer
are the apparent angular distances of the image measured from the line of
sight in directions perpendicular and parallel respectively, to the projected
axis of rotation black hole onto the celestial sphere Hioki2009 . By making a
stereographic projection of black hole shadow on the observer’s celestial sky
to the image plane, the shadow boundary can be described by the following
coordinates

$\displaystyle
X=\lim_{r_{o}\rightarrow\infty}\left(-r_{o}^{2}\sin{\theta_{o}}\frac{d\phi}{d{r}}\right),$
$\displaystyle
Y=\lim_{r_{o}\rightarrow\infty}\left(r_{o}^{2}\frac{d\theta}{dr}\right),$ (20)

where $r_{0}$ is the distance between the observer and the black hole. For an
asymptotic observer, Eq. (20) leads to

$\displaystyle X$ $\displaystyle=$ $\displaystyle-\xi_{crit}\csc\theta_{o},$
$\displaystyle Y$ $\displaystyle=$
$\displaystyle\pm\sqrt{\eta_{crit}+a^{2}\cos^{2}\theta_{o}-\xi_{crit}^{2}\cot^{2}\theta_{o}}\
.$ (21)

Further, if the observer in the equatorial plane ($\theta=\pi/2$), it
simplifies to

$\displaystyle X$ $\displaystyle=$ $\displaystyle-\xi_{crit},$ $\displaystyle
Y$ $\displaystyle=$ $\displaystyle\pm\sqrt{\eta_{crit}}\ .$ (22)

The contour of $X$ and $Y$ delineates the shadow for the hairy Kerr black
holes. Eq. (22), on using metric (1), yields

$\displaystyle X$ $\displaystyle=$
$\displaystyle-\frac{[a^{2}-3r_{p}^{2}]m(r_{p})+r_{p}[a^{2}+r_{p}^{2}][1+m^{\prime}(r_{p})]}{a[m(r_{p})+r_{p}[-1+m^{\prime}(r_{p})]]},$
$\displaystyle Y$ $\displaystyle=\pm$
$\displaystyle\frac{1}{a[m(r_{p})+r_{p}[-1+m^{\prime}(r_{p})]]}\Big{[}r_{p}^{3/2}\Big{[}-r_{p}^{3}(1+m^{\prime}(r_{p})^{2})$
(23)
$\displaystyle+m(r_{p})[4a^{2}+6r_{p}^{2}-9r_{p}m(r_{p})]-2r_{p}[2a^{2}+r_{p}^{2}$
$\displaystyle-3r_{p}m(r_{p})]m^{\prime}(r_{p})\Big{]}^{1/2}\Big{]},$

where for brevity, we have used
$m(r)=M-\frac{\alpha}{2}re^{-r/(M-\frac{l_{0}}{2})}$. and whereas for the Kerr
case, Eq. (23) simplifies to

$\displaystyle X$ $\displaystyle=$
$\displaystyle\frac{r_{p}^{2}(r_{p}-3M)+a^{2}(M+r_{p})}{a(r_{p}-M)},$
$\displaystyle Y$ $\displaystyle=$
$\displaystyle\pm\frac{r_{p}^{3/2}(4a^{2}M-r_{p}(r_{p}-3M)^{2})^{1/2}}{a(r_{p}-M)}$
(24)

which is exactly the same as obtained for the Kerr black hole Hioki2009 ,
which for the nonrotating black hole (1) takes the form

$X^{2}+Y^{2}=\frac{2r_{p}^{4}+[m(r_{p})+r_{p}m^{\prime}(r_{p})][-6r_{p}^{2}m(r_{p})+2r_{p}^{3}m^{\prime}(r_{p})]}{[m(r_{p})+r_{p}[-1+m^{\prime}(r_{p})]]^{2}}.$
(25)

The Eq. (25) ensures that the shadow of a nonrotating black hole (1) is
evidently a perfect circle and further Eq. (25) reverts to
$X^{2}+Y^{2}=27M^{2}$ for the Schwarzschild black hole.

|
---|---
Figure 4: Shadow geometry of non-rotating hairy black holes with varying
$l_{0}$ parameter (left) and Kerr black hole ($\alpha=0$) with varying spin
$a$ (right).

The shadow for the non-rotating case ($a=0$) and Kerr black hole ($\alpha=0$)
is depicted in Fig. 4. Whereas, a comparison of the hairy Kerr black hole
shadows with that of the Kerr black hole is shown in Fig. 5. The decrease in
$l_{0}$ from the Kerr limit $l_{0}=l_{k}$, causes a decrease in the shadow
size from that of the Kerr black hole (shown by the outermost solid circles in
Fig. 5). Alongside this, the parameter $\alpha$ is found to have a decremental
effect on the size of the hairy Kerr black hole shadow size, as for fixed
values of $a$ and $l_{0}$, the shadow size decreases with the increase in
$\alpha$.

The presence of parameters $\alpha$ and $l_{0}$ has a profound influence on
the approximate size and shape of the shadow (cf. Fig. 6). The shadow radius
$R_{s}$ of hairy Kerr black holes increases monotonically with $l_{0}$, but
interestingly $\alpha$ has a decremental effect, and the shadow radius of
hairy Kerr black holes are always smaller than that of the Kerr black hole
(solid black curve in Fig. 6), which is also evident from the shadow geometry
shown in Fig. 5. Contrastingly, the distortion parameter $\delta_{s}$ of hairy
Kerr black holes decreases with $l_{0}$ but increases with $\alpha$, i.e., the
shadows of hairy Kerr black holes are more distorted than the Kerr black hole
(solid black curve in Fig. 6), and this trend is again inferable from Fig. 5.

|
---|---
|
Figure 5: Shadow geometry of hairy Kerr black holes compared with Kerr black
hole (outermost solid curve corresponding to $l_{0}\sim l_{k}$).

##  V Observables and black hole parameter estimation

The parameters associated with the hairy Kerr black holes are expected to be
constrained from EHT observations. The observed image of supermassive is per
the Kerr black hole of GR. However, the EHT observation did not mention black
holes in modified theories Akiyama:2019cqa ; Akiyama:2019abc ; Akiyama:2019eap
. We assume that the observer is in the equatorial plane, i.e., the
inclination angle $\theta_{0}=\pi/2$ for the parameter estimation.

#### Hioki and Maeda method:

Hioki and Maeda Hioki2009  characterized the distortion and shadow size by
proposing the two observables, viz. $\delta_{s}$ and $R_{s}$. They approximate
the shadow by a perfect reference circle, with $R_{s}$ being the radius of
this circle, and $\delta_{s}$, the deviation of the left edge of the shadow
from the circle boundary Hioki2009 . The shadow reference circle coincides at
the top ($X_{t}$, $Y_{t}$), bottom ($X_{b}$, $Y_{b}$) and right ($X_{r},0$)
edges with the shadow contour Ghosh:2020 .

|
---|---
|
Figure 6: Variation of the shadow observables $R_{s}$ (left) and $\delta_{s}$
(right) of the shadow of the hairy Kerr black holes. Table 1: Estimated values
of hairy Kerr black hole parameters $a$ and $l_{0}$ from known shadow
observables $R_{s}$ and $\delta_{s}$ for different values of $\alpha$ at
inclination angle $\theta=90$\mathrm{\SIUnitSymbolDegree}$$.

(a) $\alpha=0.5$ $R_{s}$ | $\delta_{s}$ | $a/M$ | $l_{0}/M$
---|---|---|---
5.15 |  0.25 |  0.6433 |  0.8255
5.10 |  0.15 |  0.4050 |  0.7147
5.075 |  0.100 |  0.3060 |  0.6214
5.05 |  0.035 |  0.2187 |  0.4028
5.00 |  0.01 |  0.0583 |  0.2220

(b) $\alpha=1.0$ $R_{s}$ | $\delta_{s}$ | $a/M$ | $l_{0}/M$
---|---|---|---
5.10 |  0.004 |  0.6651 |  0.1433
5.045 |  0.035 |  0.5248 |  0.3998
4.90 |  0.15 |  0.2610 |  0.6914
4.80 |  0.40 |  0.1123 |  0.8459
4.75 |  0.70 |  0.0432 |  0.9305

|
---|---
Figure 7: The contour maps for hairy Kerr black hole shadow observables
$R_{s}$ (red solid lines) and $\delta_{s}$ (blue dashed lines) at inclination
angle $\theta_{0}=90$\mathrm{\SIUnitSymbolDegree}$$.

where, ($X^{{}^{\prime}}_{l}$, 0) and ($X_{l}$, 0) are respectively the points
where the reference circle and leftmost edge of the contour of the shadow cut
the horizontal axis. To estimate the black hole parameters, we plot the
contours of $R_{s}$ and $\delta_{s}$ in the ($l_{0}-a$) parameter space of the
hairy Kerr black hole in Fig. 7. It is evident from the Fig. 7, the curves of
radius $R_{s}$ and distortion $\delta_{s}$ intersect each other and the
intersection points for a given $\alpha$ depends on both $a$ and $l_{0}$. An
intersection point of $R_{s}$ and $\delta_{s}$ for a given $\alpha$ and
inclination angle $\theta_{0}$, uniquely determines the parameters $a$ and
$l_{0}$ (cf. Table 1(b)). Thus, we have extended the Hioki and Maeda method
Hioki2009  to uniquely determine the parameters $a$ and $l_{0}$ of the hairy
Kerr black holes.

#### Kumar and Ghosh method:

The Hioki and Maeda method Hioki2009  of numerical estimation of black hole
parameters using shadow observables $R_{s}$ and $\delta_{s}$ was later
extended by Tsupko Tsupko2017  to analytical estimation. Further, Tsukamoto et
al. Tsukamoto:2014tja  proposed a method to distinguish the Kerr black hole
shadow from the other rotational black hole shadows arising in the MoG.
However, the observables $R_{s}$ and $\delta_{s}$ used in the above two
methods demand some symmetries in the shadow shape, and they may not work
precisely for the black hole shadows in modified gravity Schee:2008kz ;
Johannsen:2013 ; Tsukamoto:2014tja ; Abdujabbarov:2015xqa ; Younsi:2016azx ;
Tsupko:2017rdo . Also, the shadow shape may not be precisely circular due to
noisy data Abdujabbarov:2015xqa ; Kumar:2018ple . Motivated by this, Kumar and
Ghosh Kumar:2018ple  presented observables characterizing a haphazard (or not
circular shape) shadow to estimate the parameters associated with black holes.
They defined the Area ($A$) and oblateness ($D$) by

$A=2\int{Y(r_{p})dX(r_{p})}=2\int_{r_{p}^{-}}^{r_{p}^{+}}\left(Y(r_{p})\frac{dX(r_{p})}{dr_{p}}\right)dr_{p},$
(26)

and

$D=\frac{X_{r}-X_{l}}{Y_{t}-Y_{b}},$ (27)

Indeed, while $R_{s}$ designates approximately the actual size of the shadow,
$\delta_{s}$ connotes the shadow deformation from the circular reference
circle. These observables are defined as Hioki2009

$R_{s}=\frac{(X_{t}-X_{r})^{2}+Y_{t}^{2}}{2|X_{r}-X_{t}|},$ (28)

using the relations $X_{b}=X_{t}$ and $Y_{b}=-Y_{t}$, and

$\delta_{s}=\frac{|X_{l}-X^{{}^{\prime}}_{l}|}{R_{s}}\ ,$ (29)

where subscripts $r$, $l$, $t$ and $b$, respectively, stand for the right,
left, top and bottom of the shadow boundary. For an equatorial observer, the
distortion $D$ may vary between $\sqrt{3}/2\leq D<1$, with $D=1$ for the
Schwarzschild case ($a=0$) and $D=\sqrt{3}/2$ for extremal Kerr case ($a=M$)
Tsupko:2017rdo . We plot the observables $A$ and $D$ in Fig. 8, and it is
evident that all the three parameters $a$, $\alpha$ and $l_{0}$ have a
profound impact on the shadow area. Both the area $A$ and oblateness $D$
increases with $l_{0}$ for fixed values of $\alpha$ and $a$ and decreases with
$\alpha$ for fixed values of $l_{0}$ and $a$. Both $A$ and $D$ decrease with
$a$ for fixed values of $\alpha$ and $l_{0}$. Moreover, evidently from the
lower panels in Fig. 8, the area of the hairy Kerr Black holes is smaller than
that of the Kerr black hole and also is more distorted, which can also be
checked from Fig. 4.

We plot the contour map of the observables $A$ and $D$ in the ($a$, $l_{0}$)
plane (Fig. 9). It is clear from the Fig. 9 that the intersection point gives
an exact value of parameters $a$ and $l_{0}$ for the chosen value of $A$ and
$D$. In the Table 2(b) we have shown the estimated values of parameters $a$
and $l_{0}$ of hairy Kerr black holes for given shadow observables $A$ and
$D$.

|
---|---
|
Figure 8: The shadow area $A$ and oblateness observables $D$ with varying
parameters for the hairy Kerr black holes. The black solid line in lower
panels correspond to Kerr black hole. Table 2: Estimated values of hairy Kerr
black hole parameters $a$ and $l_{0}$ from known shadow observables $A$ and
$D$ for different values of $\alpha$ at inclination angle
$\theta_{0}=90$\mathrm{\SIUnitSymbolDegree}$$.

(a) $\alpha=0.5$ $A/M^{2}$ | $D$ | $a/M$ | $l_{0}/M$
---|---|---|---
84 |  0.998 |  0.1758 |  0.8722
83 |  0.994 |  0.2961 |  0.7031
82 |  0.987 |  0.4211 |  0.6185
81 |  0.980 |  0.5061 |  0.5456
80 |  0.960 |  0.7089 |  0.8780

(b) $\alpha=1.0$ $A/M^{2}$ | $D$ | $a/M$ | $l_{0}/M$
---|---|---|---
84 |  0.998 |  0.1648 |  1.003
82 |  0.994 |  0.2657 |  0.7439
80 |  0.987 |  0.3666 |  0.6127
76 |  0.980 |  0.4203 |  0.3794
68 |  0.920 |  0.7230 |  0.1695

|
---|---
Figure 9: Contour plots of the observables $A/M^{2}$ and $D$ in the Hairy Kerr
black hole parameter space ($a,l_{0}$) with varying $\alpha$. Solid red curves
correspond to the constant $A$, and dashed blue curves are for the constant
oblateness parameter $D$.

###  V.1 Energy Emission

Observing the shadow can lead to witnessing exciting phenomena, and in the
present study, we turn to analyse the energy emission rate for hairy Kerr
black holes. For an observer at infinity, the black hole shadow corresponds to
a high energy absorption cross-section, which oscillates around a constant
limiting value $\sigma_{lim}$ for a spherically symmetric black hole. For a
black hole with a photon sphere, the $\sigma_{lim}$ is the same as the
geometrical cross-section of the photon sphere Wei:2013  taking the form

$\sigma_{lim}\approx\pi R_{s}^{2}.$ (30)

We intend to discuss the energy emission rate of hairy Kerr black holes which
can be computed for the hairy Kerr black holes (2) using the relation Wei:2013

$\frac{d^{2}E(\omega)}{d\omega
dt}=\frac{2\pi^{2}R_{s}^{2}}{e^{\omega/T}-1}\omega^{3},$ (31)

where $\omega$ is photon frequency and $T$ is the Hawking temperature at event
horizon $r_{+}$ of black hole (2) given by

$T=\frac{1}{2\pi\left(a^{2}+r_{+}^{2}\right)}\left(\frac{\alpha
e^{\frac{2r_{+}}{l_{0}-2M}}r_{+}(l_{0}-2M+r_{+})}{l_{0}-2M}+r_{+}-M\right).$
(32)

Interestingly, the Hawking temperature of hairy Kerr black holes depends on
the parameter $\alpha$ as well as on $l_{0}$. In Fig. 10, we plot the
behaviour of the energy emission rate of hairy Kerr black holes against the
photon frequency $\omega$ for various values of parameter $l_{0}$ for a given
$\alpha$ and $a$. The emission rate peak decreases with an increase in $l_{0}$
and shifts to a lower value of $\omega$ while an increase.

|
---|---
Figure 10: Evolution of the emission rate with the frequency $\omega$ for
different values of parameter $l_{0}$ and $a$ for the hairy Kerr black holes.

##  VI Constraints from EHT observation of M87*

The EHT collaboration using the VLBI technology unveiled the event horizon
scale image of the supermassive black hole M87* Akiyama:2019cqa ;
Akiyama:2019abc ; Akiyama:2019eap  This provides an exciting arena to
investigate gravity in the strong-field regime to determine in turn the exact
nature of black holes. The observed image of M87* is a nearly circular
crescent shape; the circularity deviation is $\Delta C\leq 0.10$ (10%) and the
angular size $\theta_{d}$ of the emission region in the observed image is
$42\pm 3\mu as$ Akiyama:2019cqa ; Akiyama:2019abc ; Akiyama:2019eap . The
observed image is constrained with the modelled image of Kerr black hole
Akiyama:2019cqa ; Akiyama:2019abc ; Akiyama:2019eap . It can be further used
to constrain the models of black holes in MoG.

The boundary of a black hole shadow can be defined by the polar coordinates
($R(\varphi),\varphi$), with origin at the shadow centre ($X_{c}$,$Y_{c}$)
such that $X_{c}=(X_{r}-X_{l})/2$ and $Y_{c}=0$. The average shadow radius
$\bar{R}$ 4DEGB_2020  is given by

$\bar{R}=\frac{1}{2\pi}\int_{0}^{2\pi}R(\varphi)d\varphi,$ (33)

where

$\bar{R}=\frac{1}{2\pi}\int_{0}^{2\pi}R(\varphi)d\varphi\ \text{and}\;\
\varphi\equiv\tan^{-1}\left(\frac{Y}{X-X_{C}}\right).$

The circularity deviation $\Delta C$ is defined as

$\Delta
C=2\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}\left(R(\varphi)-\bar{R}\right)^{2}d\varphi},$
(34)

The $\Delta C$ measures deviation from a perfect circle. The angular diameter
of the shadow $\theta_{d}$ 4DEGB_2020  is defined as

$\theta_{d}=2\frac{R_{a}}{d},\;\;\;\;R_{a}=\sqrt{A/\pi},$ (35)

Assuming M87* a hairy Kerr black hole, we can calculate $\Delta C$ for metric
(2) and use EHT observation results to put constraints on the hairy Kerr black
hole parameters. We shall use the deviation from circularity defined above in
Eq. (34) to perform a comparison between predicted hairy Kerr black hole
shadow and the EHT observations. Assuming Kerr black hole geometry of the
M87*, the EHT observations demonstrated that $\Delta C\leq 0.01$. Here we
shall use hairy Kerr black hole instead. The Eq. (34) implies that the $\Delta
C$ depends on the black hole metric parameters. Using mass of M87*,
$M=6.5\times 10^{9}M_{\odot}$ and the distance from the earth, $d=16.8Mpc$,
the circularity deviation $\Delta C$ is depicted in Fig. 11, with the black
curve denoting the EHT constraint $\Delta C=0.10$. Clearly, the Fig. 11
indicates that the hairy Kerr black hole shadow of M87* satisfies $\Delta
C\leq 0.10$ for some parameter space ($a/M$, $l_{0}/M$).

Next we calculate diameter of the shadow, which apart from $a$, $l_{0}$,
$\alpha$, $\theta$, depends on mass $M$ and the distance $d$ of the black
hole. The mass of the black hole is $M\sim 6.5\times 10^{9}M_{\odot}$ using GR
Kerr model. The angular diameter $\theta_{d}=2R_{s}/d$, with $d=16.8Mpc$ for
M87* for the hairy Kerr black hole is calculated and depicted in Fig. 12 as
function of $a$ and $l_{0}$ for a given $\alpha$ (=0.5,1.0) taking inclination
angle $\theta_{0}=90$\mathrm{\SIUnitSymbolDegree}$$.

|
---|---
Figure 11: Circularity deviation observable $\Delta C$ for hairy Kerr black hole shadows as a function of parameters ($a/M$ and $l_{0}/M$). The black solid curve correspond to $\Delta C=0.01$ such that the region above the black line is excluded by the measured circularity of the M87* black hole reported by the EHT, $\Delta C\leq 0.01$ . The parameters of M87* used are $M=6.5\times 10^{9}M_{\odot}$, $\theta_{0}=90$\mathrm{\SIUnitSymbolDegree}$$ and $d=16.8Mpc$. The white region is forbidden for ($a/M$ and $l_{0}/M$). |
---|---
Figure 12: Angular diameter observable $\theta_{d}$ for hairy Kerr black hole
shadows as a function of parameters ($a/M$ and $l_{0}/M$). The black solid
curve correspond to $\theta_{d}=39\mu as$ within $1\sigma$ region of the
measured angular diameter of the M87* black hole reported by the EHT,
$\theta_{d}=42\pm 3\mu as$. The parameters of M87* used are $M=6.5\times
10^{9}M_{\odot}$, $\theta_{0}=90$\mathrm{\SIUnitSymbolDegree}$$ and
$d=16.8Mpc$. The white region is forbidden for ($a/M$ and $l_{0}/M$).

##  VII Conclusions

The no-hair theorem establishes the remarkable property of general-
relativistic black holes that precisely three parameters, viz. mass, spin, and
charge, uniquely determine their spacetimes and, hence, all of their
properties Israel:1967 ; Israel:1968 ; Carter:1971 ; Hawking:1972 . The
spacetimes of such black holes are described by the Kerr–Newman metric
kerrNewman , which reduces to the Kerr metric kerr  in electrically neutral
black holes or for an astrophysical black holes. Consequently, GR predicts
that the Kerr metric describes all astrophysical black holes. Nonetheless,
such black holes may nor exist in a perfect vacuum because the presence of
surrounding matter like dark matter, accretion disks could alter the Kerr
nature of the black hole. Non-Kerr spacetimes incorporate potential deviations
from the Kerr metric via introducing additional deviation parameters and
encompasses the Kerr metric if deviation vanishes. Also, there are theoretical
motivations (e.g. non-renormalizability and curvature singularities) to search
for modified theories to GR. Further, The EHT collaboration captured the image
of supermassive black hole M87* at 1.3 $mm$ wavelength with an angular
resolution of 20 $\mu\text{as}$ thereby opening a new window to test gravity
in the strong-field regime. The angular diameter of the shadow of M87* is
$\theta_{d}=42\pm 3\mu as$ exhibiting a deviation from circularity $\Delta
C\leq 10$ percent and the axis ratio $\lesssim 4/3$ consistent with the Kerr
black hole’s image as predicted by the GR, but the observation did not say
anything about most modified gravity theories or alternatives to the Kerr
black hole.

Motivated by this, we considered hairy Kerr black holes, which have additional
deviation parameters $\alpha$ and $l_{0}$ than the Kerr black hole, and the
parameter $\alpha$ produces deviation from Kerr geometry but with a richer
configuration. It is found that the hair parameter $l_{0}$, quantitatively has
an influence on the structure of the event horizon by reducing its radius
significantly with decreasing $l_{0}$ for a given $\alpha$ and the resulting
decrease in ergosphere area is thereby likely to have impact on energy
extraction, which has been investigated separately. The possibility of using
the available information of M87* shadow to constraint hairy Kerr black holes,
taking into account the surrounding dark matter, persuaded us to reconsider
the shadow cast by the black holes. We have discussed the photons geodesics
equations of motion, which are analytically solved in the first-order
differential form. Observables, namely area radius $R_{s}$ and oblateness
$\delta_{s}$, characterize the shadow size and shape and estimate the black
hole parameters $(a,l_{0})$. In addition, shadow observables, namely, area $A$
and oblateness $D$, are also used to explicitly determine the black hole
parameters $(a,l_{0}).$ Interestingly, the rotating hairy Kerr black holes,
with the parameter $l_{0}$, cast smaller and more distorted shadows, than the
Kerr black holes. Interestingly, both the shadow size decreases whereas the
distortion increases with the decreasing value of parameter $l_{0}$.

The EHT observational results are used to place constraints on the hair
parameter $l_{0}$ by taking the M87* black hole as the hairy Kerr black holes
and using the inferred shadow angular diameter and the circularity deviation
observables, we found that within a finite parameter space our results are in
agreement with the M87* shadow. Furthermore, our results put stringent
constraints on hairy Kerr black holes parameters to satisfy the M87* black
hole shadow observables deduced by the EHT. Thus, the M87* shadow observations
do not entirely rule out hairy Kerr black holes. The departure produced by the
hairy Kerr black holes parameters $l_{0}$ and $\alpha$ for the M87* black hole
shadow angular diameter is $\mathcal{O}(\mu$as). Hence, it is difficult to
distinguish the hairy black holes from the Kerr black hole, at least from the
present resolution of astronomical observations like that of the EHT, and we
may have to wait for the next-generation EHT (ngEHT).

The results presented here are the generalization of previous discussions, on
the Kerr black hole, to a more general setting, and demonstrate hairy Kerr
black hole properties are qualitatively different from the Kerr black hole.
The hairy Kerr black holes may allow studying the effect of surrounding dark
matter on the Kerr black holes. One can expect that ngEHT with highly improved
techniques would achieve the resolution required to observe the dark matter
influence on the shadow of the black hole M87*. The black hole shadow of M87*
due to EHT can also serve as a tool for distinguishing various models of
theories of gravity and thereby eventually shed light on the nature of
supermassive M87* and dark matter.

###### Acknowledgements.

S.G.G. would like to thank SERB-DST for the ASEAN project
IMRC/AISTDF/CRD/2018/000042. M. A. is supported by DST-INSPIRE Fellowship,
Department of Science and Technology, Govt. of India

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