aainstitutetext: State Key Laboratory of Mountain Bridge and Tunnel
Engineering, Chongqing
Jiaotong University, Chongqing 400074, Chinabbinstitutetext: Department of
Mechanics, Chongqing Jiaotong University, Chongqing 400074,
Chinaccinstitutetext: Center for Gravitational Physics, Department of Space
Science, Beihang University, Beijing 100191, China.ddinstitutetext:
International Research Institute for Multidisciplinary Science, Beihang
University, Beijing 100191, China

# Influence of quintessence dark energy on the shadow of black hole

Xiao-Xiong Zeng c,d Hai-Qing Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the effects of quintessence dark energy on the shadows of black
hole, surrounded by various profiles of accretions. For the thin disk
accretion, the images of the black hole comprises the dark region and bright
region, including direct emission, lensing rings and photon rings. Although
their details depend on the form of the emission, generically, the direct
emission plays a major role for the observed brightness of the black hole,
while the lensing ring makes a small contribution and photon ring makes a
negligible contribution. The existence of cosmological horizon also plays an
important role in the shadows, since the observer in the domain of outer
communications is nearby the cosmological horizon. For the spherically
symmetric accretion, the static and infalling matters are considered. We find
that the positions of photon spheres are the same for both static and
infalling accretions. However, the observed specific intensity of the image
for the infalling accretion is darker than the static accretion, due to the
Doppler effect of the infalling movement.

##  1 Introduction

The Event Horizon Telescope (EHT) collaboration has recently released an image
of a supermassive black hole in M87 Akiyama:2019cqa ; Akiyama:2019brx ;
Akiyama:2019sww ; Akiyama:2019bqs ; Akiyama:2019fyp ; Akiyama:2019eap . The
image shows that there is a dark interior surrounding by a bright ring, which
are called the black hole shadow and photon ring, respectively. The shadow and
photon ring are resulted from the deflections of light, or gravitational
lensing by the black holes, a prediction from Einstein’s General Relativity
(GR) Wald:1984rg . Therefore, the shadow image of black hole in M87 from EHT
is another strong evidence for GR.

The deflection of light by an intense star or black holes was initiated in
Synge:1966okc ; Bardeen:1972fi . Later, it was extended to a black hole
surrounded by a thin accretion disk Luminet:1979nyg . According to these
papers, there exists a critical curve in the observed image of the black hole.
When traced backwards from the distant observer, the light ray from this
critical curve will asymptotically approach a bound photon orbit. This bound
orbits for Schwarzschild black hole is $r=3M$ ($M$ is the mass of the black
hole), and the critical curve has the radius of $b=\sqrt{27}M$ ($b$ is called
impact parameter). Thus, the shadow of the black hole represents the dark
interior of the critical curve. However, according to Gralla:2019xty , the
radius and the profile of the shadow would depend on the location and form of
the accretion surrounding the black hole. Besides the shadow, there also
exists the enhanced brightness, called photon ring, around the black hole.
This is because the light rays near the critical curve may orbit many times of
the black hole, thus picking up extra brightness. So far, the study of shadows
of black holes has become a subject of great interest Jaroszynski:1997bw ;
Falcke:1999pj ; Bambi:2013nla ; Shaikh:2018lcc ; Narayan:2019imo ;
Gralla:2019drh ; Allahyari:2019jqz ; Li:2020drn ; Banerjee:2019nnj ;
Vagnozzi:2019apd ; Vagnozzi:2020quf ; Safarzadeh:2019imq ; Davoudiasl:2019nlo
; Roy:2019esk ; Chen:2019fsq ; Cunha:2019hzj ; Konoplya:2020bxa ; Roy:2020dyy
; Islam:2020xmy ; Jin:2020emq ; Guo:2020zmf ; Wei:2020ght ; Zeng:2020dco ;
Caiyifu ; Khodadi:2020jij ; Cuadros-Melgar:2020kqn ; Konoplya:2019sns ;
Zhang:2019glo . In particular, the shadows of a high-redshift supermassive
black hole may serve as the standard rulers Vagnozzi:2020quf , which would
constrain the cosmological parameters. Moreover, the EHT data may impose
constraints in particle physics via the mechanism of superradiance
Davoudiasl:2019nlo ; Roy:2019esk . More references can be referred to
Shaikh:2018lcc .

Recent astronomical observations show that the Universe is accelerating
expansion Perlmutter ; Riess ; Garnavich , implying a state of negative
pressure. Quintessence dark energy is one of the candidates to interpret the
negative pressure. The state equation of quintessence dark energy is governed
by the pressure $p$ and the energy density $\rho$ with $p=w\rho$, where $w$ is
the state parameter. It is deemed that the range $-1<w<-1/3$ causes the
acceleration of the Universe. Black holes in quintessence dark energy models
were extensively studied Kiselev:2002dx . To investigate the shadows of black
hole in the quintessence dark energy model will bring new insights and impose
restrictions to the quintessence model. Previous work on shadows in dark
energy model can be found in Lacroix:2012nz ; Khan:2020ngg .

In this paper, we will investigate the shadow images of quintessence black
holes with different accretions. In particular, we study the geometrically
thin and optically thin disk accretion and spherically symmetric accretion.
Apparently different accretions will lead to different shadow images.
Specifically, for the thin disk accretion, the images of the black hole
comprises the shadows, lensing rings and photon rings. The laters are resulted
from the intersections of the light rays and the thin accretion for many
times. Although different emission profiles of the accretion is important to
the observed specific intensities, the direct image from accretion makes major
contributions to the brightness of the image. Besides, the lensing ring makes
small contributions and the photon ring makes negligible contributions to the
observed specific intensities. For the spherically symmetric accretions, the
shadows and photon rings are also spherically symmetric. The static accretion
and an infalling accretion will have different impacts to the observed
specific intensities, although the radii of the shadows are not changed.
Specifically, the infalling accretion will have extra Doppler effect, which
will lead to darker shadows than the static case. A key point in the
cosmological black hole is that there exists a domain of outer communication
between the event horizon and cosmological horizon Friedman ; Galloway .
Therefore, various quintessence state parameter $w$ will lead to different
distances between the event horizon and cosmological horizon. This will
certainly have impact on the distance between the observer (near cosmological
horizon) and the accretion (near the event horizon). Therefore, different
quintessence state parameters will lead to different forms of the observed
specific intensities. We expect that the shadow images of the quintessence
black hole will impose constraints of the quintessence dark energy model in
the Universe from the observation of EHT.

This paper is arranged as follows: In section 2 the black hole solutions
surrounded by quintessence matter and the light deflections by the
quintessence black hole are introduced; In section 3 the shadow images of the
black hole with thin disk accretion are given; In section 4, we show the
images of the black hole with spherically symmetric accretions; Finally, we
draw the conclusions and discussions in section 5.

##  2 Light deflection by a black hole in quintessence dark energy

The geometry of a black hole in a quintessence dark energy model can be
expressed as

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

where,

$f(r)=1-\frac{2M}{r}-\frac{a}{r^{3w+1}},$ (2)

in which, $M$ is the mass of the black hole, $a$ and $w$ are the normalization
factor and state parameter respectively. For the case of $w=-1$, the metric in
Eq.(2) reduces to the Schwarzschild-de-Sitter black hole with

$f(r)=1-\frac{2M}{r}-ar^{2},$ (3)

which has both the event horizon $r_{h}$ and cosmological horizon $r_{c}$. For
$w=-\frac{2}{3}$, we can obtain the event horizon and cosmological horizon as

$r_{h}=\frac{1-\sqrt{1-8Ma}}{2a},~{}~{}~{}r_{c}=\frac{1+\sqrt{1-8Ma}}{2a},$
(4)

where the restriction $8Ma<1$ has been imposed. For $w=-1/3$, the metric in
Eq.(2) reduces to the Schwarzschild black hole, which has only an event
horizon. In this paper, we are interested in the quintessence dark energy
model, in which the state parameter $w$ is restricted in the region
$-1<w<-\frac{1}{3}$, and there always exists an event horizon and a
cosmological horizon. The region between the two horizons is called the domain
of outer communication Friedman ; Galloway , since any two observers in this
region may communicate with each other without being hindered by a horizon.

The equation of state for the quintessence dark energy is

$p=w\rho,~{}~{}~{}\rho=-\frac{a}{2}\frac{3w}{r^{3(1+w)}}$ (5)

in which $p$ is the pressure while $\rho$ is the energy density. In order to
cause cosmic acceleration, the pressure of the quintessence dark energy $p$
should be negative, thus the energy density $\rho$ is positive if the
normalization factor $a$ is positive. In this paper, we will only focus on the
effect of state parameter on the shadows of a quintessence black hole, and the
we will set the normalization factor $a=0.05$ and black hole $M=1$ thereafter.

In order to investigate the light deflection in the quintessence black hole,
we need to find how the light ray moves in this spacetime. We will resort to
the following Euler-Lagrange equation

$\frac{d}{ds}\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)=\frac{\partial\mathcal{L}}{\partial
x^{\mu}},$ (6)

in which $s$ is the affine parameter of the light trajectory, the symbol
$\dot{}$ is the derivative with respect to $s$, thus $\dot{x}^{\mu}$ the four-
velocity of the light ray and $\mathcal{L}$ is the Lagrangian, taking the form
as

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\frac{1}{2}\left(-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}+r^{2}\left(\dot{\theta}^{2}+\sin^{2}{\theta}~{}\dot{\psi}^{2}\right)\right).$
(7)

We will also impose the initial conditions $\theta=\pi/2$, $\dot{\theta}=0$.
That is, the light rays always move in the equatorial plane. In addition,
since the metric coefficients do not depend explicitly on time $t$ and
azimuthal angle $\psi$, there are two conserved quantities correspondingly.
Combining Eqs.(2), (6) and (7), the time, azimuthal and radial components of
the four-velocity satisfy the following equations of motions

$\displaystyle\dot{t}=\frac{1}{b\left(1-\frac{2M}{r}-\frac{a}{r^{3w+1}}\right)},$
(8) $\displaystyle\dot{\psi}=\pm\frac{1}{r^{2}},$ (9)
$\displaystyle\dot{r}^{2}+\frac{1}{r^{2}}\left(1-\frac{2M}{r}-\frac{a}{r^{3w+1}}\right)=\frac{1}{b^{2}},$
(10)

where $+$ and $-$ in Eq.(9) correspond to the counterclockwise and clockwise
direction respectively for the motion of light rays. The parameter
$b={|L|}/{E}$ is the impact parameter, in which $E$ and $L$ are the conserved
quantities with respect to the time and azimuthal direction. Moreover, Eq.(10)
can be rewritten as

$\dot{r}^{2}+V(r)=\frac{1}{b^{2}},$ (11)

where

$V(r)=\frac{1}{r^{2}}\left(1-\frac{2M}{r}-\frac{a}{r^{3w+1}}\right),$ (12)

is an effective potential. At the photon sphere, the motions of the light ray
satisfy $\dot{r}=0$, and $\ddot{r}=0$, which also means

$\displaystyle V(r)=\frac{1}{b^{2}},~{}~{}~{}V^{{}^{\prime}}(r)=0,$ (13)

where the prime ′ denotes the derivative with respect to the radial coordinate
$r$. Based on Eq.(13), we can obtain the radius and impact parameter of the
photon sphere for different $w$. Generically, it is difficult to obtain the
analytic results of the radius and impact parameter. But for $w=-2/3$ the
metric function can be simplified, thus the radius and impact parameter of the
photon sphere can be expressed as

$\displaystyle r_{ph}=\frac{1-\sqrt{1-6aM}}{a},$ (14) $\displaystyle
b_{ph}=\frac{\sqrt{2}\sqrt{6M\sqrt{1-6aM}-\frac{\sqrt{1-6aM}}{a}+\frac{1}{a}-9M}}{\sqrt{8a^{2}M-a}}.$
(15)

For other $w$’s, the numerical results of radii and impact parameters of the
photon sphere, as well as the event horizons and cosmological horizons are
listed in the Table 1. From this table, we see that as one increases the
absolute value of $w$, the event horizon $r_{h}$ and impact parameter $b_{ph}$
increase as well, while the cosmological horizon $r_{c}$ decreases. The radius
of photon sphere however is non-monotonic, it increases at first and then
decreases around $w=-0.7$. In addition, from Table 1 we see that the locations
of the event horizon and cosmological horizon become closer as $|w|$
increases, which leads to a narrower domain of the outer communication.

Table 1: The values of radius $r_{ph}$ and impact parameter $b_{ph}$ of the photon sphere, as well as the event horizon $r_{h}$ and cosmological horizon $r_{c}$ for various $w$’s with $M=1$, $a=0.05$. | $w=-0.4$ | $w=-0.5$ | $w=-0.6$ | $w=-0.7$ | $w=-0.8$ | $w=-0.9$
---|---|---|---|---|---|---
$r_{ph}$ | 3.18038 | 3.21631 | 3.25038 | 3.27103 | 3.25476 | 3.16885
$b_{ph}$ | 5.72947 | 5.98805 | 6.42035 | 7.23434 | 9.22055 | 27.141
$r_{h}$ | 2.12343 | 2.15857 | 2.20806 | 2.28304 | 2.41487 | 2.8274
$r_{c}$ | 3199990 | 395.969 | 39.6448 | 13.1027 | 6.55098 | 3.65555

Figure 1:  The profile of the effective potential (blue lines) for $w=-0.5$
(left panel) and $w=-0.7$ (right panel) with $M=1$ and $a=0.05$. The dashed
lines indicate the radii of the photon sphere $r_{ph}$. Region 2 (green lines)
correspond to $V(r)=1/b_{ph}^{2}$, while Region 1 and Region 3 correspond to
$V(r)<1/b_{ph}^{2}$ and $V(r)>1/b_{ph}^{2}$ respectively.

From the Eq.(11), we see that the motion of the particle depends on the impact
parameter and the effective potential. In Figure 1, the effective potential is
plotted for different state parameters $w=-0.5$ and $w=-0.7$. We can see that
at the event horizon, the effective potential vanishes. Then it increases and
reaches a maximum at the photon sphere $r_{ph}$, and later decreases and
vanishes at the cosmological horizon. Let’s consider a light ray moves in the
radially inward direction. In Region 1, if the light ray starts its motion at
$r>r_{ph}$, the light ray will encounter the potential barrier and be
reflected back to the outward direction. If the photon starts the motion at
$r<r_{ph}$, the photons will fall into the singularity. In Region 2, namely
$b=b_{ph}$, as the light ray approaches the photon sphere, it will revolve
around the black hole infinitely many times since the angular velocity is non-
zero. Actually, this orbit is circular and unstable Gralla:2019xty . In Region
3, the light ray will continue moving in the inward direction since it does
not encounter the potential barrier. Finally it will enter the inside of the
black hole and fall into the singularity.

Since we have already obtained the effective potential, it is also convenient
to obtain the effective force on the photon, which is 111Here, we have divided
$dV/dr$ by 2 since the equation of motion has been written as Eq.(11).

$F=-\frac{1}{2}\frac{dV(r)}{dr}=-\frac{3M}{r^{4}}+\frac{1}{r^{3}}-\frac{1}{2}3a(w+1)r^{-3w-4},$
(16)

The first term $-{3M}/{r^{4}}$, called the Newtonian term, is attractive as it
is negative while the second term $1/r^{3}$ is repulsive. The contributions of
the quintessence dark energy comes from the third term
$-\frac{1}{2}3a(w+1)r^{-3w-4}$, which plays the role of an attractive force.

The trajectory of the light ray can be depicted on the basis of the equation
of motion. Combining Eqs.(9) and (10), we have

$\frac{dr}{d\psi}=\pm
r^{2}\sqrt{\frac{1}{b^{2}}-\frac{1}{r^{2}}\left(1-\frac{2M}{r}-\frac{a}{r^{3w+1}}\right)}.$
(17)

In order to integrate it out conveniently, we set $u=1/r$. Thus, Eq.(17)
becomes

$\frac{du}{d\psi}=\sqrt{\frac{1}{b^{2}}-u^{2}\left(-au^{3w+1}-2Mu+1\right)}\equiv\Phi(u).$
(18)

The geometry of the geodesics depends on the roots of the equation
$\Phi(u)=0$. In particular, for $b>b_{ph}$, light will be deflected at the
radial position $u_{i}$ which satisfies $\Phi(u_{i})=0$. Therefore, it is
important to find the radial position $u_{i}$ in order to obtain the
trajectory of the light ray. In addition, the location of the observer is also
important. Usually, the observer is located at infinite boundary for the
asymptotically flat spacetime. However, for the model of quintessence black
hole in our paper, the cosmological horizon is present. Physically, the
observer should be located in the domain of outer communication which is
between the event horizon and cosmological horizon, just as in de Sitter
spacetime. For convenience, the observer in our paper is set to be near the
cosmological horizon.

Figure 2:  The trajectory of the light ray for $w=-0.5$ (left) and $w=-0.7$
(right) with $M=1$, $a=0.05$. They are shown in the polar coordinates
$(r,\psi)$. The red lines, green lines and blue lines correspond to
$b=b_{ph}$, $b<b_{ph}$ and $b>b_{ph}$ respectively. The spacing of impact
parameter is $\Delta b=1/5$ for each light ray. Black hole is shown as the
solid black disk and the photon orbit as a dashed line.

By virtue of the Eq.(18), we can solve the trajectories of the light rays,
shown in Figure 2. All the light rays approach the black hole from the right
side. The green, red and blue lines correspond to $b<b_{ph}$, $b=b_{ph}$ and
$b>b_{ph}$ respectively. Specifically, for the case of $b<b_{ph}$, the light
rays (green lines) drop all the way into the black hole. For $b>b_{ph}$, the
light rays (blue lines) are deflected but never enter the black hole. In
particular, the light rays close to the black hole can even be reflected back
to the right side, i.e., the side they approach the black hole. For
$b=b_{ph}$, the light rays (red lines) revolve around the black hole. This
conclusion is consistent with those from the analysis of effective potential
in Figure 1. Indeed, region 1, 2, 3 in Figure 1 correspond to the blue, red
and green lines in Figure 2, respectively.

We need to emphasize that due to the existence of the cosmological horizon,
different $w$’s will result in different geometries of the light rays. We have
assumed that the observer is in the domain of the outer communications and
sitting close to the cosmological horizon. Therefore, for the case of $w=-0.5$
(left panel in Figure 2), the cosmological horizon is much far from the event
horizon (see Table 1 ). Therefore, the observed entering light rays are
approximately parallel, which can be seen from the left panel of Figure 2.
However, for the case of $w=-0.7$, the cosmological horizon is close to the
event horizon (see Table 1). Therefore, the observed entering light rays
behave significantly different from those of $w=-0.5$ (see the right panel of
Figure 2), i.e., the entering light rays for $w=-0.7$ are not parallel.

As the light is deflected, a shadow of the black hole will be formed. The
light is emitted by the accretion matters, the profiles of the accretion
matters thus are important for the shadows of the black holes. Next, we will
investigate the effect of the profiles of the accretions on the shadows of the
black holes. We will take the thin disc accretion and spherical accretion as
two examples.

##  3 Shadows, photon rings and lensing rings with thin disk accretion

In this section, we consider an optically and geometrically thin disk
accretion around the black hole, viewed face-on. From Gralla:2019xty , we know
that an important feature of thin disk accretion is that there are photon
rings and lensing rings surround the black hole shadow. The lensing ring
consists of light rays that intersect the plane of the disk twice outside the
horizon, while the photon ring 222In our paper, the definition of photon ring
is different from photon sphere discussed in the later section. In references
such as Gralla:2019xty , the photon sphere is also called ‘photon ring’. We
distinguish them in our paper.  comprises light rays that intersect the plane
of the disk three or more times. Thus, the trajectory of the photon is
important to distinguish the photon ring and lensing ring.

###  3.1 Number of orbits of the deflected light trajectories

Figure 3:  Behavior of photons in the quintessence black hole as a function of
impact parameter $b$ with $M=1$, $a=0.05$, $w=-0.5$ (top row) and $w=-0.7$
(bottom row). (Left column) We show the fractional number of orbits,
$n=\psi/2\pi$, where $\psi$ is the total change in azimuthal angle outside the
horizon. The direct, lensing and photon ring correspond to $n<3/4$ (red),
$3/4<n<5/4$ (blue), and $n>5/4$ (green), respectively. (Right column) We show
a selection of associated photon trajectories in the polar coordinates
$(b,\psi)$. The spacings in impact parameter are 1/5, 1/100, 1/1000, for the
direct (red), lensing (blue), and photon ring (green) bands, respectively. The
black holes are shown solid black disks.

The trajectories of the light rays are shown on the right column of Figure 3
for $w=-0.5$ (top row) and $w=-0.7$ (bottom row), respectively. In these
figures, the red lines, blue lines and green lines are direct emission,
lensing rings and photon rings, respectively. This can be understood from the
definition in Gralla:2019xty  that these light rays intersect the disk plane
once, twice and more than twice. It should be noted that due to the domain of
outer communications, the light rays going towards the observer will behave
differently for different state parameters $w$. For the case of $w=-0.5$, the
observer locates near the cosmological horizon which is far away from the
event horizon (see Table 1). Thus, the light rays nearby the observer is
almost parallel, which is similar to the Schwarzschild black hole
Gralla:2019xty . However, for $w=-0.7$ the cosmological horizon is near the
event horizon (refer to Table 1). Therefore, the light rays going towards the
observer are not parallel, which is shown in the bottom row of right panel of
Figure 3.

One can also use the total number of orbits, defined by $n=\psi/2\pi$, to
distinguish the trajectories of light rays Gralla:2019xty . The total number
of orbits are shown in the left column of Figure 3 for $w=-0.5$ (top row) and
$w=-0.7$ (bottom row), respectively. We still use the red lines, blue lines
and green line to represent the direct emission, lensing rings and photon
rings, respectively. Therefore, from the definitions of these light rings we
can explicitly see that the direct emissions correspond to $n<3/4$, while
lensing rings correspond to $3/4<n<5/4$ and photon rings correspond to
$n>5/4$. For the case of $w=-0.5$ and $w=-0.7$, the parameter intervals of $b$
for the direct emission, photon rings and lensing rings are listed in
following Eq.(19) and Eq.(20).

$\displaystyle w=-0.5\begin{cases}\text{Direct
emission}:n<3/4,~{}~{}~{}b<5.71774~{}~{}\text{and}~{}~{}b>7.76682\\\
\text{Lensing
ring}:3/4<n<5/4,~{}~{}~{}5.71774<b<5.97314~{}~{}\text{and}~{}~{}6.04784<b<7.76682\\\
\text{Photon ring}:n>5/4,~{}~{}~{}5.97314<b<6.04784\end{cases}$ (19)
$\displaystyle w=-0.7\begin{cases}\text{Direct
emission}:n<3/4,~{}~{}~{}b<6.92848~{}~{}\text{and}~{}~{}b>8.36103\\\
\text{Lensing
ring}:3/4<n<5/4,~{}~{}~{}6.92848<b<7.2156~{}~{}\text{and}~{}~{}7.28972<b<8.36103\\\
\text{Photon ring}:n>5/4,~{}~{}~{}7.2156<b<7.28972\end{cases}$ (20)

###  3.2 Observed specific intensities and transfer functions

Next, we will investigate the observed specific intensity of the thin disk
accretion. We assume that the thin disk emits isotropically in the rest frame
of static worldlines. As in Gralla:2019xty , we take the disk to lie in the
equatorial plane of the black hole, with a static observer at the North pole.
We denote the emitted specific intensity and frequency as $I_{e}(r)$ and
$\nu_{e}$, the observed specific intensity and frequency as $I_{obs}(r)$ and
$\nu$. Since $I_{e}/\nu_{e}^{3}$ is conserved along a light ray from
Liouville’s theorem, the observed specific intensity thus can be written as

$I_{obs}(r)=f(r)^{3/2}I_{e}(r).$ (21)

The total specific intensity is obtained by integrating specific intensity
with different frequencies

$I=\int I_{obs}(r)d\nu=\int f(r)^{2}I_{e}(r)d\nu_{e}=f(r)^{2}I_{em}(r),$ (22)

where $I_{em}(r)=\int I_{e}(r)d\nu_{e}$ is the total emitted specific
intensity near the accretion.

If a light ray traced backwards from the observer intersects the disk, it will
pick up brightness from the disk emission. For $3/4<n<5/4$, the light ray will
bend around the black hole and hit the opposite side of the disk from the back
(see the blue lines in Figure 3). Thus, it will pick up additional brightness
from this second passage through the disk. For $n>5/4$, the light ray will
bend around the black hole more, and then hit the front side of the disk once
again (see the green lines in Figure 3). This leads to additional brightness
from the third passage through the disk. Therefore, the observed intensity is
a sum of the intensities from each intersection, that is

$I(r)=\sum\limits_{m}f(r)^{2}I_{em}\mid_{r=r_{m}(b)},$ (23)

where $r_{m(b)}$, called transfer function, is the radial position of the
$m^{th}$ intersection with the disk plane outside the horizon. For simplicity,
we have neglected absorption of the light for the thin accretion, which may
decrease the observed intensity from the additional passages.

The transfer function describes the relation between the radial coordinate $r$
and the impact parameter $b$. The slope of the transfer function, $dr/db$, is
the demagnification factor. For different state parameters $w$, the transfer
functions with respect to impact parameter $b$ are shown in Figure 4. The
black line, orange line and red line represent the first ($m=1$), second
($m=2$) and third ($m=3$) transfer functions. The first transfer function
corresponds to the direct image of the disk, and its slope is about 0.85
(0.67) for $w=-0.5$ ($w=-0.7$). Therefore, the direct image profile is
actually the redshifted source profile. The second transfer function
corresponds to the lensing ring (including the photon ring). In this case, the
observer will see a highly demagnified image of the back side of the disk.
Finally, the third transfer function corresponds to the photon ring. In this
case, one will see an extremely demagnified image of the front side of the
disk since the slope is about infinite. It thus contribute negligibly to the
total brightness of the image.

Figure 4:  The first three transfer functions $r_{m}(b)$ for the thin disk in
the quintessence black hole with $w=-0.5$ (left panel) and $w=-0.7$ (right
panel). The black lines, orange lines and red lines stand for the radial
coordinate of the first, second, and third intersections with a face-on thin
disk outside the horizon.

###  3.3 Shadows, lensing rings and photon rings

Having obtained the transfer functions, we can further obtain the specific
intensity on the basis of Eq.(23) as the emitted specific intensity is given.
In this paper, we assume the following toy-model functions which may emit by
some thin matters. Firstly, we assume that $I_{em}(r)$ is a decay function of
the power of second order,

$\displaystyle I_{em}(r)=\begin{cases}\left(\frac{1}{r-5}\right)^{2},&r>6\\\
0,&r\leq 6.\end{cases}$ (24)

This profile is plotted in the top left panels in Figure 5 for $w=-0.5$ and
Figure 6 for $w=-0.7$. This emission is sharply peaked at $r=6M$, and then
drops to zero abruptly as $r<6M$. In this example, the region of emission is
well outside the photon sphere (see Table 1). Secondly, we assume that the
emission is a decay function of the power of third order as

$\displaystyle
I_{em}(r)=\begin{cases}\left(\frac{1}{r-(r_{ph}-1)}\right)^{3},&r>r_{ph}\\\
0,&r\leq r_{ph}\end{cases}$ (25)

But this time the peak of the mission is at the photon sphere $r_{ph}$. The
profiles of this emission can be seen from the middle left panels in Figure 5
for $w=-0.5$ and Figure 6 for $w=-0.7$. Finally, we assume a moderate decay of
emission as

$\displaystyle
I_{em}(r)=\begin{cases}\frac{\frac{\pi}{2}-\tan^{-1}(r-5)}{\frac{\pi}{2}-\tan^{-1}(-3)},&r>r_{h}\\\
0,&r\leq r_{h}\end{cases}$ (26)

This emission is outside of the horizon $r_{h}$, which is shown in the bottom
left panels in Figure 5 for $w=-0.5$ and Figure 6 for $w=-0.7$.

Figure 5:  Observational appearances of a geometrically and optically thin
disk with different profiles near black hole with $M=1$, $a=0.05$, $w=-0.5$,
viewed from a face-on orientation. The left column shows the profiles of
various emissions $I_{em}(r)$. The middle column exhibits the observed
emission $I(r)$ as a function of the impact parameter $b$. The right column
shows the 2-dim density plots of the observed emission $I(r)$. The details can
be found in the main texts.

For the case of $w=-0.5$, the observed specific intensities are shown in the
middle and right columns in Figure 5. The middle column is about the one
dimensional functions of $I(r)$ with respect to $b$, while the right column is
about the two dimensional density plots of $I(r)$, viewed faced-on. In the
first row the emission from the accretion is well outside the photon sphere
(left panel). The corresponding observed direct image decays similarly as
$b>7.5M$ due to the gravitational lensing (middle panel). However, the
observed lensing ring emission is confined to a thin region $6.4M<b<7.4M$,
since the back side image is highly demagnified for $r>6M$. The photon ring
emission is the spike at $b\sim 6M$, which is hardly to see in the right panel
of the density plots. One needs to zoom in the plot to see the photon ring.
Therefore, from the observer, the lensing ring makes a small contribution to
the total brightness while the photon ring makes negligible contributions.

In the second row of Figure 5, the emission starts from the photon sphere
$r_{ph}=3.216$. The most important difference from the first row is that the
observed lensing ring and photon ring emission are now superimposed on the
direct image emission, which decays from $b>4.25M$. The lensing ring has a
spike in the brightness in the range of $6M<b<6.5M$, while the photon ring has
an even narrower spike at $b\sim 6M$ which is hardly to be distinguished from
the lensing ring. In this case, the lensing ring still makes a very small
contribution to the total brightness and the photon ring makes a negligible
contribution.

Finally, in the third row of Figure 5, the emission extends to the event
horizon $r_{h}=2.158$ and the decay of the emission is much moderate compared
to the first and second row. The inner edge of the observed intensity at
$b\sim 3.2M$ is the lensed position of the event horizon. The increase of the
intensity outside of the central dark area is due to the gravitational
redshift. The very narrow spike at $b\sim 6M$ is the photon ring, while the
broader bump at $b\sim 6.2M$ is the lensing ring. In this case, the lensing
ring makes a prominent brightness to the observed intensity, but the photon
ring is still entirely negligible.

From Figure 5, we see that the dark central regions are different for
different profiles of the emissions. But the locations of the photon ring
always stay at $b\sim 6M$. In addition, for all the cases, we find that the
observed brightness mainly stems from the direct emission, while the lensing
ring provides only a small contribution, and the photon sphere provides a
negligible contribution.

Figure 6:  Observational appearances of a geometrically and optically thin
disk with different profiles near black hole with $M=1$, $a=0.05$, $w=-0.7$,
viewed from a face-on orientation. The left column shows the profiles of
various emissions $I_{em}(r)$. The middle column exhibits the observed
emission $I(r)$ as a function of the impact parameter $b$. The right column
shows the 2-dim density plots of the observed emission $I(r)$. The profiles
are similar to those of $w=-0.5$ in Figure 5.

For the case of $w=-0.7$, we plot the profiles of the emission and observed
intensities in Figure 6. They behave similarly to those of $w=-0.5$. The
differences are the quantities of the intensities, as well as the locations of
photon ring and lensing ring.

##  4 Shadows and photon spheres with spherical accretions

In this section, we are going to investigate the shadows and photon spheres
with the spherical accretions for various state parameters $w$.

###  4.1 Shadows and photon spheres with a static spherical accretion

In this section, we intend to investigate the shadow and photon sphere of a
quintessence black hole with a static spherical accretion. We mainly focus on
the specific intensity observed by the observer
$\rm(ergs^{-1}cm^{-2}str^{-1}Hz^{-1})$, which can be expressed as
Jaroszynski:1997bw ; Bambi:2013nla

$I=\int_{\gamma}g^{3}j(\nu_{\rm e})dl_{\rm p},$ (27)

in which $g=\nu_{\rm o}/\nu_{\rm e}$ is the redshift factor, while $\nu_{\rm
e}$ is the radiated photon frequency and $\nu_{\rm o}$ is the observed photon
frequency, $j(\nu_{\rm e})$ is the emissivity per unit volume measured in the
static frame of the emitter, $dl_{\rm p}$ is the infinitesimal proper length,
and $\gamma$ stands for the trajectory of the light ray.

In the quintessence black hole (1), the redshift factor $g=f(r)^{1/2}$. In
addition, we assume that the radiation of light is monochromatic with fixed a
frequency $\nu_{s}$, thus the specific emissivity takes the form as

$j(\nu_{\rm e})\propto\frac{\delta(\nu_{e}-\nu_{s})}{r^{2}},$ (28)

where we have adopted the $1/r^{2}$ profile as in Bambi:2013nla . According to
Eq.(1), the proper length measured in the rest frame of the emitter is

$\displaystyle dl_{\rm p}$ $\displaystyle=$
$\displaystyle\sqrt{f(r)^{-1}dr^{2}+r^{2}d\psi^{2}}$ (29) $\displaystyle=$
$\displaystyle\sqrt{f(r)^{-1}+r^{2}\left(\frac{d\psi}{dr}\right)^{2}}~{}dr,$

in which $d\psi/dr$ is given by the inverse of Eq.(17). In this case, the
specific intensity observed by the static observer is

$I=\int_{\gamma}\frac{f(r)^{3/2}}{r^{2}}\sqrt{f(r)^{-1}+r^{2}\left(\frac{d\psi}{dr}\right)^{2}}~{}dr.$
(30)

Figure 7:  Profiles of the specific intensity $I(b)$ cast by a static
spherical accretion, viewed face-on by an observer near the cosmological
horizon. We set $M=1$, $a=0.05$, $w=-0.5$ (top row) and $w=-0.7$ (bottom row)
as two examples. The details can be seen in the main texts.

Next we will employ Eq.(30) to investigate the shadow of the quintessence
black hole with the static spherical accretion. Since the intensity depends on
the trajectory of the light ray, which is determined by the impact parameter
$b$, we will investigate how the intensity varies with respect to the impact
parameter $b$. The observed specific intensities are shown in Figure 7 with
$w=-0.5$ (top row) and $w=-0.7$ (bottom row). From Figure 7, we see that as
$b$ increases, the intensity ramps up first, then reaches a peak at $b_{ph}$
($b_{ph}=5.98M$ for $w=-0.5$ and $b_{ph}=7.23M$ for $w=-0.7$), and finally
drops rapidly to smaller values. This result is consistent with Figure 1 and
Figure 2. Because for $b<b_{ph}$, the intensity originated from the accretion
matter is absorbed mostly by the black hole, so the observed intensity is
little. For $b=b_{ph}$, the light ray revolves around the black hole several
times, thus the observed intensity is maximal. For $b>b_{ph}$, only the
refracted light contributes to the observed intensity. In particular, as $b$
becomes larger, the refracted light becomes less. The observed intensity thus
vanishes for large enough $b$.

From Figure 7, we can also see how the state parameter of dark energy will
affect the observed intensity. For a fixed $a$ and $M$, we find that the
larger the state parameter is, the stronger the intensity will be, i.e., the
intensity of $w=-0.5$ is stronger than that of $w=-0.7$. The 2-dimensional
plot of the shadows are shown in the right column of Figure 7, with the impact
parameter $b$ of the radius. We can see obviously that the shadow is
circularly symmetric for the quintessence black hole, and outside the black
hole, there is a bright ring, which is the photon sphere. The radius of the
photon sphere for different $w$’s have been listed in Table 1. Obviously, the
result in Figure 7 is consistent with that in Table 1. That is, the larger the
state parameter is, the smaller the radius of the photon sphere will be.

In addition, inside the photon sphere, we see that the specific intensity does
not vanish but has a small finite value. The reason is that part of the
radiation is escaped from the black hole. For $r<r_{ph}$, the solid angle of
the escaping radiation is $2\pi(1-\cos\theta)$, while for $r>r_{ph}$, the
solid angle is $2\pi(1+\cos\theta)$, in which $\theta$ is given by

$\sin\theta=\frac{r_{ph}^{3/2}}{r}\left(1-\frac{2M}{r}-\frac{a}{r^{3w+1}}\right)^{1/2}.$
(31)

In principle, one can calculate the escaped net luminosity by the following
relation

$L_{o}=\int_{r_{h}}^{r_{ph}}4\pi r^{2}j(\nu_{\rm
e})2\pi(1-\cos\theta)dr+\int_{r_{ph}}^{\text{observer}}4\pi r^{2}j(\nu_{\rm
e})2\pi(1+\cos\theta)dr.$ (32)

Obviously, for different state parameters, the net luminosity observed by the
static observer in the quintessence black hole is different, since solid angle
is dependent on the state parameter. This result is consistent with the result
in Figure 7.

###  4.2 Shadows and photon spheres with an infalling spherical accretion

In this section, the optically thin accretion is supposed to be infalling
matters. This infalling model is thought to be more realistic than the static
accretion model, since most of the accretion matters are dynamical in the
universe. We still employ Eq.(30) to investigate the shadow cast by the
infalling accretion. Different from the static accretion, the redshift factor
for the infalling accretion is related to the velocity of the accretion, that
is

$g=\frac{k_{\beta}u^{\beta}_{\rm o}}{k_{\gamma}u^{\gamma}_{\rm e}},$ (33)

in which $k^{\mu}=\dot{x_{\mu}}$ is the four-velocity of the photon,
$u^{\mu}_{\rm o}=(1,0,0,0)$ is the four-velocity of the static observer, and
$u^{\mu}_{\rm e}$ is the four-velocity of the accretion under consideration,
given by

$\displaystyle u^{t}_{\rm e}=\frac{1}{f(r)},~{}~{}u^{r}_{\rm
e}=-\sqrt{1-f(r)},~{}~{}u^{\theta}_{\rm e}=u^{\psi}_{\rm e}=0.$ (34)

The four-velocity of the photon has been obtained previously from Eq.(8) to
Eq.(10). From these equations, we know that $k_{t}=1/b$ is a constant, and
$k_{r}$ can be inferred from the equation $k_{\alpha}k^{\alpha}=0$, that is

$\frac{k_{r}}{k_{t}}=\pm\frac{1}{f(r)}\sqrt{1-\frac{b^{2}f(r)}{r^{2}}},$ (35)

in which the $+$/$-$ corresponds to the case that the photon gets close
to/away from the black hole. With Eq.(35), the redshift factor in Eq.(33) can
be simplified as

$g=\frac{1}{u^{t}_{e}+k_{r}/k_{e}u^{r}_{e}},$ (36)

which is different from the static accretion case.

Figure 8:  Profiles of the specific intensity $I(b)$ cast by an infalling
spherical accretion, viewed face-on by an observer near the cosmological
horizon. We set $M=1$, $a=0.05$, $w=-0.5$ (top row) and $w=-0.7$ (bottom row)
as two examples. The details can be seen in the main texts.

In addition, the proper distance is defined by

$dl_{\rm p}=k_{\gamma}u^{\gamma}_{\rm e}ds=\frac{k_{t}}{g|k_{r}|}dr,$ (37)

where $s$ is the affine parameter along the photon path $\gamma$. With regard
to the specific emissivity, we also assume that it is monochromatic, so that
Eq.(28) is still valid. Integrating Eq.(27) over the observed frequencies, we
obtain

$I\propto\int_{\gamma}\frac{g^{3}k_{t}dr}{r^{2}|k_{r}|}.$ (38)

Now we will use Eq.(38) to investigate the shadow of the quintessence black
hole numerically with an infalling accretion. Note that there is an absolute
sign for $k_{r}$ in the denominator. Therefore, when the photon changes its
motion direction, the sign before $k_{r}$ should also change. For different
state parameter $w$’s, the observed intensity with respect to $b$ are shown in
Figure 8, in which the top row is for $w=-0.5$ while the bottom row is for
$w=-0.7$. From Figure 8, we find that as $b$ increases the intensity will
increase as well to a peak $b=b_{ph}$, after the peak it drops to smaller
values. This behavior is similar to that in the static accretion in Figure 7.
We can also observe the effect of $w$ on the intensity, and find that the
intensity increases as the value of $w$ increases, i.e., the intensity for
$w=-0.5$ is stronger than $w=-0.7$. The two dimensional image of the intensity
is shown on the right column in Figure 8. We see that the radii of the shadows
and the locations of the photon spheres are the same as the static case. That
means the motion of the accretion does not affect the radii of the shadows and
the locations of the photon spheres. However, different from the static
accretion, the central region of the intensity for the infalling accretion is
darker, which can be accounted for by the Doppler effect. In particular, near
the event horizon of the black hole, this effect is more obvious.

##  5 Conclusions and discussions

We investigated the images of a black hole in the quintessence dark energy
model. In particular, we studied the quintessence black hole surrounded by a
geometrically and optically thin disk accretion, as well as a spherically
symmetric accretion. We found that for the observed specific intensities,
there existed a dark interior (shadow), lensing ring and photon ring for the
thin disk accretion. But their positions depended on the profiles of the
emission from the accretion near the black hole. Despite of this distinctions,
the direct image of the accretion made a major contribution to the brightness
of the black hole, while the lensing ring made a minor contribution and the
photon ring made a negligible contribution. The effects of the quintessence
state parameter was also important to the shadow images of the black hole,
since this parameter would have impact on the distances between the observer
and the event horizon.

We also studied the static and infalling spherically symmetric accretions, the
images of the black hole would have a dark interior and a photon sphere. The
infalling accretion would have a darker interior than the static case, which
was due to the Doppler effect of the infalling matter. Different quintessence
state parameter would change the positions of the photon spheres of the image.
We expect that this would bring insights and implications for the quintessence
dark energy model from the observations of the shadow images of the black
holes in future EHT experiments.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 11675140, 11705005, and 11875095).

## References

  * (1) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. 875, no. 1, L1 (2019)
  * (2) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. II. Array and Instrumentation,” Astrophys. J. 875, no. 1, L2 (2019)
  * (3) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration,” Astrophys. J. 875, no. 1, L3 (2019)
  * (4) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole,” Astrophys. J. 875, no. 1, L4 (2019)
  * (5) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring,” Astrophys. J. 875, no. 1, L5 (2019)
  * (6) K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole,” Astrophys. J. 875, no. 1, L6 (2019)
  * (7) R. M. Wald, “General Relativity,” The University of Chicago Press, (1984)
  * (8) J. L. Synge, “The Escape of Photons from Gravitationally Intense Stars,” Mon. Not. Roy. Astron. Soc. 131, no. 3, 463 (1966). doi:10.1093/mnras/131.3.463
  * (9) J. M. Bardeen, W. H. Press and S. A. Teukolsky, “Rotating black holes: Locally nonrotating frames, energy extraction, and scalar synchrotron radiation,” Astrophys. J. 178, 347 (1972).
  * (10) J.-P. Luminet, “Image of a spherical black hole with thin accretion disk,” Astron. Astrophys. 75, 228 (1979).
  * (11) S. E. Gralla, D. E. Holz and R. M. Wald, “Black Hole Shadows, Photon Rings, and Lensing Rings,” Phys. Rev. D 100, no. 2, 024018 (2019)
  * (12) M. Jaroszynski and A. Kurpiewski, “Optics near kerr black holes: spectra of advection dominated accretion flows,” Astron. Astrophys. 326, 419 (1997)
  * (13) H. Falcke, F. Melia and E. Agol, “Viewing the shadow of the black hole at the galactic center,” Astrophys. J. 528, L13 (2000)
  * (14) C. Bambi, “Can the supermassive objects at the centers of galaxies be traversable wormholes? The first test of strong gravity for mm/sub-mm very long baseline interferometry facilities,” Phys. Rev. D 87, 107501 (2013)
  * (15) R. Shaikh, P. Kocherlakota, R. Narayan and P. S. Joshi, “Shadows of spherically symmetric black holes and naked singularities,” Mon. Not. Roy. Astron. Soc. 482 (2019) no.1, 52-64 doi:10.1093/mnras/sty2624 [arXiv:1802.08060 [astro-ph.HE]].
  * (16) R. Narayan, M. D. Johnson and C. F. Gammie, “The Shadow of a Spherically Accreting Black Hole,” Astrophys. J. 885, no. 2, L33 (2019)
  * (17) S. E. Gralla and A. Lupsasca, “Lensing by Kerr Black Holes,” Phys. Rev. D 101, no. 4, 044031 (2020)
  * (18) A. Allahyari, M. Khodadi, S. Vagnozzi and D. F. Mota, “Magnetically charged black holes from non-linear electrodynamics and the Event Horizon Telescope,” JCAP 2002, 003 (2020)
  * (19) P. C. Li, M. Guo and B. Chen, “Shadow of a Spinning Black Hole in an Expanding Universe,” Phys. Rev. D 101, no. 8, 084041 (2020)
  * (20) I. Banerjee, S. Chakraborty and S. SenGupta, “Silhouette of M87*: A New Window to Peek into the World of Hidden Dimensions,” Phys. Rev. D 101, no. 4, 041301 (2020)
  * (21) S. Vagnozzi and L. Visinelli, “Hunting for extra dimensions in the shadow of M87*,” Phys. Rev. D 100, no. 2, 024020 (2019)
  * (22) S. Vagnozzi, C. Bambi and L. Visinelli, “Concerns regarding the use of black hole shadows as standard rulers,” Class. Quant. Grav. 37, no. 8, 087001 (2020)
  * (23) M. Safarzadeh, A. Loeb and M. Reid, “Constraining a black hole companion for M87* through imaging by the Event Horizon Telescope,” Mon. Not. Roy. Astron. Soc. 488, no. 1, L90 (2019)
  * (24) H. Davoudiasl and P. B. Denton, “Ultralight Boson Dark Matter and Event Horizon Telescope Observations of M87*,” Phys. Rev. Lett. 123, no. 2, 021102 (2019)
  * (25) R. Roy and U. A. Yajnik, “Evolution of black hole shadow in the presence of ultralight bosons,” Phys. Lett. B 803, 135284 (2020)
  * (26) Y. Chen, J. Shu, X. Xue, Q. Yuan and Y. Zhao, “Probing Axions with Event Horizon Telescope Polarimetric Measurements,” Phys. Rev. Lett. 124, no. 6, 061102 (2020)
  * (27) P. V. P. Cunha, N. A. Eiró, C. A. R. Herdeiro and J. P. S. Lemos, “Lensing and shadow of a black hole surrounded by a heavy accretion disk,” JCAP 2003, no. 03, 035 (2020)
  * (28) R. A. Konoplya and A. F. Zinhailo, “Quasinormal modes, stability and shadows of a black hole in the novel 4D Einstein-Gauss-Bonnet gravity,” arXiv:2003.01188 [gr-qc].
  * (29) R. Roy and S. Chakrabarti, “A study on black hole shadows in asymptotically de Sitter spacetimes,” arXiv:2003.14107 [gr-qc].
  * (30) S. U. Islam, R. Kumar and S. G. Ghosh, “Gravitational lensing by black holes in $4D$ Einstein-Gauss-Bonnet gravity,” arXiv:2004.01038 [gr-qc].
  * (31) X. H. Jin, Y. X. Gao and D. J. Liu, “Strong gravitational lensing of a 4-dimensional Einstein-Gauss-Bonnet black hole in homogeneous plasma,” arXiv:2004.02261 [gr-qc].
  * (32) M. Guo and P. C. Li, “The innermost stable circular orbit and shadow in the novel $4D$ Einstein-Gauss-Bonnet gravity,” arXiv:2003.02523 [gr-qc].
  * (33) S. W. Wei and Y. X. Liu, “Testing the nature of Gauss-Bonnet gravity by four-dimensional rotating black hole shadow,” arXiv:2003.07769 [gr-qc].
  * (34) X. X. Zeng, H. Q. Zhang and H. Zhang, “Shadows and photon spheres with spherical accretions in the four-dimensional Gauss-Bonnet black hole,” [arXiv:2004.12074 [gr-qc]].
  * (35) Sheng-Feng Yan, Chunlong Li, Lingqin Xue, Xin Ren, Yi-Fu Cai, Damien A. Easson, Ye-Fei Yuan, and Hongsheng Zhao, “Testing the equivalence principle via the shadow of black holes,” Phys. Rev. Research 2, 023164 (2020)
  * (36) M. Khodadi, A. Allahyari, S. Vagnozzi and D. F. Mota, “Black holes with scalar hair in light of the Event Horizon Telescope,” [arXiv:2005.05992 [gr-qc]].
  * (37) B. Cuadros-Melgar, R. D. B. Fontana and J. de Oliveira, “Analytical correspondence between shadow radius and black hole quasinormal frequencies,” [arXiv:2005.09761 [gr-qc]].
  * (38) R. A. Konoplya, “Shadow of a black hole surrounded by dark matter,” Phys. Lett. B 795, 1 (2019) doi:10.1016/j.physletb.2019.05.043 [arXiv:1905.00064 [gr-qc]].
  * (39) M. Zhang and M. Guo, “Can shadows reflect phase structures of black holes?,” [arXiv:1909.07033 [gr-qc]].
  * (40) S. Perlmutter et al. [Supernova Cosmology Project Collaboration], “Measurements of Omega and Lambda from 42 High-Redshift Supernovae", Astrophys. J. 517, 565 (1999) [arXiv:astro-ph/9812133];
  * (41) A. G. Riess et al. [Supernova Search Team Collaboration], “Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant", Astron. J. 116, 1009 (1998) [arXiv:astro-ph/9805201];
  * (42) P. M. Garnavich et al., “Supernova Limits on the Cosmic Equation of State", Astrophys. J. 509, 74 (1998) [arXiv:astro-ph/9806396].
  * (43) V. V. Kiselev, “Quintessence and black holes,” Class. Quant. Grav. 20 (2003), 1187-1198 [arXiv:gr-qc/0210040 [gr-qc]].
  * (44) T. Lacroix and J. Silk, “Constraining the distribution of dark matter at the Galactic Centre using the high-resolution Event Horizon Telescope,” Astron. Astrophys. 554 (2013), A36 doi:10.1051/0004-6361/201220753 [arXiv:1211.4861 [astro-ph.GA]].
  * (45) S. U. Khan and J. Ren, “Shadow cast by a rotating charged black hole in quintessential dark energy,” Phys. Dark Univ. 30 (2020), 100644 doi:10.1016/j.dark.2020.100644 [arXiv:2006.11289 [gr-qc]].
  * (46) J. L. Friedman, K. Schleich, and D. M. Witt, “Topological censorship", Phys. Rev. Lett. 71 (1993), no. 10, 1486-1489.
  * (47) G. J. Galloway, “On the topology of the domain of outer communication", Classical Quantum Gravity 12 (1995), no. 10, L99-L101.