# Rings and images of Horndeski hairy black hole illuminated by various thin
accretions

Xi-Jing Wang<EMAIL_ADDRESS>Center for Gravitation and Cosmology,
College of Physical Science and Technology, Yangzhou University, Yangzhou,
225009, China  Xiao-Mei Kuang<EMAIL_ADDRESS>(corresponding author)
Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou, 225009, China  Yuan Meng
<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou, 225009, China  Bin Wang
<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou, 225009, China  Shanghai
Frontier Science Center for Gravitational Wave Detection, Shanghai Jiao Tong
University, Shanghai 200240, China  Jian-Pin Wu<EMAIL_ADDRESS>Center
for Gravitation and Cosmology, College of Physical Science and Technology,
Yangzhou University, Yangzhou, 225009, China

###### Abstract

In this paper, we analyze the light rays around a static hairy black hole in
Horndeski gravity with the use of ray-tracing procedure and investigate the
optical appearances of the Horndeski hairy black hole surrounded by various
thin accretions. Firstly, we consider that the Horndeski hairy black hole is
illuminated by the optically and geometrically thin accretion disk. We
carefully clarify the contributions from the direct, lensed ring and photon
ring intensities to the total observed intensity via the transfer function,
which is rarely discussed in this scenario. We find that the Horndeski hair
has significant influences on both shadow size and distributions of direct,
lensed ring and photon ring brightness in three standard emission profiles. As
a result, the rings and images of Horndeski hairy black hole and the
origination of their brightness differentiate from those of Schwarzschild
black hole (SBH) when viewed in face-on orientation. Then, when the Horndeski
hairy black hole is illuminated by thin spherical accretions, the hairy black
hole’s shadow surrounded by a bright ring is larger than that of SBH, but the
brightness of ring is fainter. Similar to that of SBH, the size of hairy black
hole shadow does not change as the radial moving of the spherical accretion,
and the brightness for the infalling accretion is fainter than that for the
static accretion due to the Doppler effect. Therefore, we argue that the black
hole image consisting of the shadow and accretion construction could, in
theory, reflect the observational differences between the Horndeski hairy
black hole and SBH.

###### Contents

  1. I Introduction
  2. II Trajectories of photons around Horndeski hairy black hole
  3. III Rings and images of hairy black hole illuminated by thin disk accretions
    1. III.1 Classification of rays: direct, lensed ring and photon ring emissions
    2. III.2 Observed intensities and optical appearances
  4. IV Rings and images of hairy black hole illuminated by thin spherical accretions
    1. IV.1 Static spherical accretions
    2. IV.2 Radially infalling spherical accretions
  5. V Conclusion and discussion
  6. A Inner-most stable circular orbit (ISCO)

##  I Introduction

It is generally accepted that our current understanding of gravity may be
incomplete, due to fundamental hints (e.g., the quantum theory of gravity) and
observational hints (e.g., the accelerated expansion of our Universe).
Moreover, most observations fall into the prediction of classical general
relativity (GR), but they allow the space for the modified theories of
gravity. In particular, the no-hair theorem, stating that black holes are only
described by mass, electric charge and spin, is one of the main
characteristics in classical GR, however, whether it can be evaded is still
pending. In fact, there are considerable affords to modify GR by means of
additional matter fields so that the theories admit black holes with hair.
Among them, the scalar-tensor gravity theory, which contains a scalar field
$\varphi$ as well as a metric tensor $g_{\mu\nu}$, are known as a simplest
nontrivial extension of GR Damour:1992we . One of the most famous four-
dimensional scalar-tensor theories is the Horndeski gravity proposed in 1974
Horndeski:1974wa , which contains higher derivatives of $\varphi$ and
$g_{\mu\nu}$ and is free of Ostrogradski instabilities because it posses at
most second-order differential field equations. Various observational
constraints or bounds on Horndeski gravity have been explored in
Bellini:2015xja ; Bhattacharya:2016naa ; Kreisch:2017uet ; Hou:2017cjy ;
SpurioMancini:2019rxy ; Allahyari:2020jkn . In fact, Horndeski gravity
attracts lots of attentions in cosmological and astrophysical communities
because it has significant consequences in describing the accelerated
expansion and other interesting features (please see Kobayashi:2019hrl  for a
review), as well as holographic applications Feng:2015oea ; Kuang:2016edj ;
Jiang:2017imk ; Baggioli:2017ojd ; Feng:2018sqm ; Wang:2019jyw ; Zhang:2022hxl
and references therein.

Hairy black holes in Horndeski gravity have also been widely constructed and
analyzed, including the radially dependent scalar field Rinaldi:2012vy ;
Cisterna:2014nua ; Feng:2015oea ; Sotiriou:2013qea ; Miao:2016aol ;
Kuang:2016edj ; Babichev:2016rlq ; Benkel:2016rlz ; Filios:2018xvy ;
Cisterna:2018hzf ; Giusti:2021sku  and time-dependent scalar field
Babichev:2013cya ; Babichev:2017lmw ; BenAchour:2018dap ; Takahashi:2019oxz ;
Minamitsuji:2019shy ; Arkani-Hamed:2003juy . However, the hairy solution with
scalar hair in linear time dependence was found to be unstable, and so this
type of hairy solution was ruled out in Horndeski gravity Khoury:2020aya .
Since Horndeski theory has diffeomorphism invariance and second order field
equations, which is similar to GR, so it is of importance to test the no-hair
theorem in Horndeski framework. In fact, in Hui:2012qt , the no-hair theorem
was demonstrated not be held when a Galileon field is coupled to gravity, but
the static spherical black hole only admits trivial Galileon profiles. Then,
the authors of Babichev:2017guv  further examined the no-hair theorem in
Horndeski theory and beyond. They demonstrated that the shift-symmetric
Horndeski theory and beyond allow for static and asymptotically flat black
holes with a nontrivial static scalar field, and the action they considered is
dubbed quartic Horndeski gravity

$\displaystyle S=\int
d^{4}x\sqrt{-g}\big{[}Q_{2}+Q_{3}\Box\varphi+Q_{4}R+Q_{4,\chi}\left((\Box\varphi)^{2}-(\nabla^{\mu}\nabla^{\nu}\varphi)(\nabla_{\mu}\nabla_{\nu}\varphi)\right)+Q_{5}G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\varphi$
$\displaystyle-\frac{1}{6}Q_{5,\chi}\left((\Box\varphi)^{3}-3(\Box\varphi)(\nabla^{\mu}\nabla^{\nu}\varphi)(\nabla_{\mu}\nabla_{\nu}\varphi)+2(\nabla_{\mu}\nabla_{\nu}\varphi)(\nabla^{\nu}\nabla^{\gamma}\varphi)(\nabla_{\gamma}\nabla^{\mu}\varphi)\right)\big{]},$
(1)

where $\chi=-\partial^{\mu}\varphi\partial_{\mu}\varphi/2$ is the canonical
kinetic term, $Q_{i}(i=1,2,3,4,5)$ are arbitrary functions of $\chi$ and
$Q_{i,\chi}\equiv\partial Q_{i}/\partial\chi$, $R$ is the Ricci scalar and
$G_{\mu\nu}$ is the Einstein tensor. In particular, very recently a static
hairy black hole in a specific quartic Horndeski theory, saying $Q_{5}$ in the
above action vanishes, has been constructed in Bergliaffa:2021diw

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$
$\displaystyle\mathrm{with}~{}~{}~{}f(r)=1-\frac{2M}{r}+\frac{h}{r}\ln\left(\frac{r}{2M}\right).$
(2)

Here, $M$ and $h$ are the parameters related with the black hole mass and
scalar hair, respectively. The metric reduces to Schwarzschild case as $h\to
0$, and it is asymptotically flat. Its thermodynamical properties were
explored in Walia:2021emv  and the superradiant energy extraction was carried
on in Jha:2022tdl .

On the other hand, recently remarkable breakthroughs in the observation of
black holes have been made, which could promote us to investigate the physics
in strong gravity field regime. One of the most important achievements is that
the Event Horizon Telescope (EHT) collaboration has released images of the
supermassive black holes in M87* EventHorizonTelescope:2019dse ;
EventHorizonTelescope:2019uob ; EventHorizonTelescope:2019jan ;
EventHorizonTelescope:2019ths ; EventHorizonTelescope:2019pgp ;
EventHorizonTelescope:2019ggy , and further in SgrA* at the center of the
Milky Way system EventHorizonTelescope:2022wkp ; EventHorizonTelescope:2022apq
; EventHorizonTelescope:2022wok ; EventHorizonTelescope:2022exc ;
EventHorizonTelescope:2022urf ; EventHorizonTelescope:2022xqj . Those images
provide direct evidences of the existence of black hole, and also disclose
that there exists the central dark region usually known as black hole shadow
surrounded by a bright ring. Nowadays, the black hole shadow and the
observation data of EHT, which are believed to contain the information of
spacetime, are extensively used to further explore black hole physics, GR and
beyond, for example, parameter estimations of black holes Kumar:2018ple ;
Ghosh:2020spb ; Afrin:2021imp ; Ghosh:2022kit , extra dimension size
constraint Vagnozzi:2019apd ; Banerjee:2019nnj ; Tang:2022hsu  and testing
general relativity or alternative theories of gravity and fundamental physics
Mizuno:2018lxz ; Psaltis:2018xkc ; Stepanian:2021vvk ; Younsi:2021dxe ;
KumarWalia:2022aop ; Vagnozzi:2022moj ; Meng:2022kjs ; Kuang:2022ojj , and so
on.

The black hole images published by EHT reflect the optical features of black
holes, which result from the deflection of photon or the effect of
gravitational lensing by the black holes. In particular, the bright sector
around the dark region is radiated from the accretion matters surrounding the
real astrophysical black holes, which definitely affect the black hole shape
we observe due to the different distributions of accretion matters. Though it
is difficult to mimic the realistic accretion disk from theoretical study, the
first image of black hole with a thin accretion disk was calculated
analytically in Luminet:1979nyg , which shows that there are primary and
secondary images appeared outside black hole shadow. Then the author of
Bambi:2013nla  pointed out that it is relatively easy to distinguish a
Schwarzschild black hole from the static wormhole according to shadow images.
More recent investigations of a Schwarzschild black hole with thin and thick
accretion disks show that the lensed ring together with photon ring contribute
additional observed flux to the image Gralla:2019xty . As another kind of
accretion, the spherical accretion has been applied to analyze the image of a
Schwarzschild black hole Falcke:1999pj ; Narayan:2019imo . It is shown that
the shadow is a robust feature, and its size and shape are primarily
influenced by rather the spacetime geometry than the details of the accretion.
Currently the studies on photon ring and observational appearances of black
holes have attracted much more attentions Zeng:2020dco ; Zeng:2020vsj ;
Peng:2020wun ; Saurabh:2020zqg ; Qin:2020xzu ; Gan:2021pwu ; Okyay:2021nnh ;
Li:2021ypw ; Li:2021riw ; Guo:2021bhr ; Hu:2022lek ; Guo:2021bwr ; Wen:2022hkv
; Chakhchi:2022fls ; Hou:2022eev ; Kuang:2022xjp , which disclosed the images
of static black holes surrounded by various accretions beyond GR.

The aim of this paper is to disclose the effect of the Horndeski hair on the
image of static black hole by investigating the observational appearances of
Horndeski hairy black hole (I) illuminated by various accretions. In fact,
some observational investigations on this hairy black hole and its rotating
counterpart have been carried out, for example the strong gravitational
lensing Kumar:2021cyl , weak gravitational lensing Atamurotov:2022slw , shadow
constraint from EHT observations Afrin:2021wlj , which indicated that the non-
trivial scalar has significant influences on the optical phenomena of the
Horndeski black hole. Here, we will explore the light rays around the
Horndeski hairy black hole (I) using the ray-tracing method, and utilize this
to figure out the optical appearance of black hole surrounded by thin
accretion disk and thin spherical accretion flow, respectively. Especially, we
will peer the contributions from the direct, lensed ring and photon ring
brightness to the total brightness distribution of the optical appearance
under the thin disk accretion, and analyze the effect of the Horndeski hair in
this process by comparing to the Schwarzschild case. It is worthwhile to
emphasize that to our knowledge, all the existed literatures in this scenario
mainly concentrated on the distribution of the total brightness rather than
its sources of contribution, except the reference Wang:2020jek  in which the
authors traced back the sources of forming extra rings in the image of
gravastars. Before the main-body study, we should first clarify the range of
hairy parameter we will consider. Since the hairy parameter $h$ has the same
dimension as $M$ and $r$, we shall use the dimensionless quantity $h/M$. From
Eq.(I), it is not difficult to induce that for arbitrary $h$, $r=0$ is an
intrinsic singularity as the curvature scalar is singular and $f(r)=0$ always
admits a solution $r=2M$ which indicates a horizon at $r=2M$. In addition,
when $h/M>0$, $r=2M$ is the unique root of $f(r)=0$ , so it is the unique
horizon, i.e., the event horizon for the hairy black hole as for the
Schwarzschild black hole. While for $-2<h/M<0$, $f(r)=0$ has two roots: one is
$r_{+}=2M$ indicating the event horizon, and the other $r_{-}$ denotes the
Cauchy horizon which is smaller than the event horizon. $r_{-}$ increase as
$h$ decreases, and finally approaches $r_{+}$ as $h/M\to-2$, meaning the
extremal case. Thus, here we will focus on the hairy parameter in the range
$-2\leq h/M\leq 0$ as considered in Kumar:2021cyl ; Atamurotov:2022slw ;
Afrin:2021wlj  to explore the effect of Horndeski hair on the black hole’s
optical appearance.

This paper is organized as follows. In section II, we calculate the
trajectories of photons deflected by the Horndeski hairy black hole with the
help of Lagrangian formalism and ray tracing method. In section III, we
investigate the total observed intensity and the possible sources of its
contribution for three standard disk accretion profiles, and then compare the
optical appearances of the hairy black hole to that of Schwarzschild black
hole. In section IV, we consider the hairy black hole surrounded by the static
and radially infalling spherical accretions, respectively, and explore the
optical appearances. Finally, we conclude in section V.

##  II Trajectories of photons around Horndeski hairy black hole

In this section, we will analyze the light rays near the Horndeski hairy black
hole (I) from the null geodesics. The photons in the spacetime take the
Lagrangian

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\frac{1}{2}\left(-f(r)\dot{t}^{2}+\frac{1}{f(r)}\dot{r}^{2}+r^{2}\left(\dot{\theta}^{2}+\text{sin}^{2}\theta\dot{\phi}^{2}\right)\right),$
(3)

and their motions are determined by the Euler-Lagrange equation

$\frac{d}{d\lambda}\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)=\frac{\partial\mathcal{L}}{\partial
x^{\mu}},$ (4)

where $\lambda$ is the affine parameter and
$\dot{x}^{\mu}=d{x}^{\mu}/d\lambda$ represents the four-velocity of the
photon. Due to the spherical symmetry of the spacetime, we are safe to focus
on the photons moving on the equatorial plane ($\theta=\pi/2$) for
convenience, which leads to $\dot{\theta}=0$. Then, from the Lagrangian, we
can obtain two conserved quantities for the photons which are the energy $E$
and angular momentum $L_{z}$ respectively given by

$E\equiv-\frac{\partial\mathcal{L}}{\partial\dot{t}}=f(r)\dot{t},\qquad
L_{z}\equiv\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=r^{2}\dot{\phi}.$
(5)

Subsequently, by recalling $\mathcal{L}=0$ for photons and defining the impact
parameter $b\equiv L_{z}/E$, we can finally reduce (4) into three first-order
differential equations that control the photons’ motion in the spacetime

$\displaystyle\dot{t}=\frac{1}{bf(r)},$ (6)
$\displaystyle\dot{\phi}=\pm\frac{1}{r^{2}},$ (7)
$\displaystyle\dot{r}^{2}=\frac{1}{b^{2}}-V_{\text{eff}}(r),$ (8)

where the affine parameter $\lambda$ is redefined as $\lambda/L_{z}$ and the
effective potential takes the form

$V_{\text{eff}}(r)=\frac{f(r)}{r^{2}}.$ (9)

The sign $``+"$ and $``-"$ denote that the photon travels in the
counterclockwise and clockwise direction, respectively.

Before studying the complete trajectories of photon, we shall analyze the
circular orbit or photon sphere, which satisfies $\dot{r}=0$ and $\ddot{r}=0$.
According to Eq.(8), the conditions of photon sphere will be translated to

$V_{\text{eff}}(r)=\frac{1}{b^{2}},\qquad V_{\text{eff}}^{\prime}(r)=0,$ (10)

where the prime represents the derivative with respect to $r$. Based on the
above equation, one should numerically determine the radius of photon sphere,
$r_{ph}$, and the critical impact parameter $b_{ph}$, since analytical
expressions are difficult due to the logarithmic term in the metric. Their
explicit dependence on the hairy parameter are shown in Fig.1, and samples of
numerical results are listed in Table 1 . We find that though the radius of
event horizon is always $r=2M$, independent of the hairy parameter, both the
radius $r_{ph}$ and impact parameter $b_{ph}$ of the photon sphere for the
hairy black hole are larger than those of Schwarzschild black hole
($r_{ph}=3M$ and $b_{ph}=3\sqrt{3}M$), and they monotonously increase as $h$
decreases till the maximum in extremal case.

Figure 1: The radius of photon sphere $r_{ph}$ and critical impact parameter $b_{ph}$, as the function of the Horndeski hairy parameter $h/M$. $h/M$ | $0$ | $-0.5$ | $-1$ | $-1.5$ | $-2$
---|---|---|---|---|---
$r_{ph}/M$ | 3 | 3.07185 | 3.20941 | 3.52541 | 4.28807
$b_{ph}/M$ | 5.19615 | 5.81484 | 6.69978 | 8.05599 | 10.1676
Table 1: The numerical results of radius of photon sphere $r_{ph}$ and impact
parameter $b_{ph}$ of the photon sphere for different $h$ with $M=1$.

We move on to study the trajectory of photon which can be depicted by solving
the equation of motion

$\frac{dr}{d\phi}=\pm r^{2}\sqrt{\frac{1}{b^{2}}-V_{\text{eff}}(r)},$ (11)

which is obtained from combining Eq.(7) and Eq.(8). The trajectory of photon
is somehow determined by effective potential $V_{\text{eff}}$ and the impact
parameter $b$ (due to the symmetry, we shall focus on the positive impact
parameter.). So we scan the hairy parameter space and find that the effective
potential always has one-single peak indicating the existence of single photon
sphere. In details, the effective potential firstly vanishes at the event
horizon, then as increasing $r$ it increases to a maximum at the photon
sphere, and finally decreases monotonously. This process is depicted in the
left panel of Fig.2 where we have fixed $h/M=-1$ as an example. Now let us
discuss the motion of photon in combination with the effective potential. In
Region 1 ($b>b_{ph}$), the photon will encounter the potential barrier and
then be scattered into infinity after passing through the turning point. In
Region 2 ($b=b_{ph}$), the photon asymptotically approaches the photon sphere
and then revolve around the black hole infinitely times. In Region 3
($b<b_{ph}$), the photon which does not encounter the potential barrier will
fall into the black hole. These analyses match well with the trajectories of
light rays we draw from solving (11). The distribution of light rays around
the hairy black hole are shown in the right panel of Fig.2, in which the solid
black disk stands for the hairy black hole and the black dashed circle denotes
the photon sphere, while the red, black and green curves correspond to
trajectory of photon emitted or ended at $b=b_{ph}$, $b<b_{ph}$ and
$b>b_{ph}$, respectively.

Figure 2: Left: The effective potential $V_{\text{eff}}$ as a function of
radius $r$ for $h=-1$ as an example with $M=1$. Right: The trajectories of
photons for $h=-1$ as an example with $M=1$ in the polar coordinates. The red,
black and green curves correspond to trajectories of photon emitted at
$b=b_{ph}$, $b<b_{ph}$ and $b>b_{ph}$, respectively. The black hole is shown
as the solid black disk and the photon sphere is denoted by the black dashed
circle.

Now, we are ready to study the optical appearance of the Horndeski hairy black
hole under various illumination conditions. We will consider the hairy black
hole illuminated by thin accretion disk and thin spherical accretions flow,
respectively.

##  III Rings and images of hairy black hole illuminated by thin disk
accretions

In this section, we shall explore the classification of light rays, and the
images of the Horndeski hairy black hole illuminated by optically and
geometrically thin accretion disks, which are located at rest on the
equatorial plane around the black hole, viewed face-on. In this case, the
light rays may intersect with the accretion disk for arbitrary times, which
should contribute differently to the total observed intensity. So we shall
first classify the light rays, then disclose the black hole images by studying
the total observed intensity for three standard accretion profiles.

###  III.1 Classification of rays: direct, lensed ring and photon ring
emissions

As illustrated in Gralla:2019xty , the photons are classified into three
classes in terms of the orbit numbers $n=\phi/(2\pi)$, where $\phi$ is the
total change of azimuthal angle that the photon orbits the black hole,
depending on the impact parameter via (11). Concretely, the first class is
defined as the direct emission with $n<3/4$, where light ray intersects the
accretion disk only once. The second class with $3/4<n<5/4$, where light ray
intersects the accretion disk twice, corresponds to the lensed ring emission.
The final class is photon ring emission where the light ray with $n>5/4$
crosses the accretion disk at least three times. For the above descriptions of
three classes of trajectories, one refers to Wielgus:2021peu ; Bisnovatyi-
Kogan:2022ujt ; Hu:2022lek  to see the schematic diagrams to help understand
well.

(a) $h=0$ (b) $h=-1$

(c) $h=-2$

Figure 3: The orbit numbers $n$ (top) and trajectories of photons (bottom) as the functions of impact parameter $b$ for different $h$ with $M=1$. The black, gold, and red curves correspond to the direct ($n<3/4$), lensed ring ($3/4<n<5/4$), and photon ring ($n>5/4$) emissions respectively. The black hole is shown as the solid black disk and the photon sphere is denoted by a black dashed curve. $h$ | Direct ($n<3/4$) | Lensed ring ($3/4<n<5/4$) | Photon ring ($n>5/4$)
---|---|---|---
$0$ |  $b<5.01514$ and $b>6.16757$ |  $5.01514<b<5.18781$ and $5.22794<b<6.16757$ | $5.18781<b<5.22794$
$-1$ |  $b<6.05358$ and $b>9.67298$ |  $6.05358<b<6.62643$ and $6.89312<b<9.67298$ | $6.62643<b<6.89312$
$-2$ |  $b<7.62893$ and $b>16.415$ |  $7.62893<b<9.54309$ and $10.7805<b<16.415$ | $9.54309<b<10.7805$
Table 2: The ranges of impact parameter $b$ correspond to the direct, lensed
ring, and photon ring emissions of the Horndeski black hole for different $h$
with $M=1$.

The orbit numbers with respect to impact parameter $b$ for different hairy
parameter $h$ are shown in the top panel of Fig.3. Similar to that in
Schwarzschild black hole, the ranges of impact parameter for both photon ring
(red curves) and lensed ring (gold curves) are narrow, and the direct emission
(black curves) consist of two separate parts; but the Horndeski hairy black
hole corresponds to wider ranges of impact parameter for both photon and
lensed rings, and larger photon sphere indicated by the position of the peak.
This phenomena can be simultaneously seen from the exact range of $b$ for the
direct, lensed ring, and photon ring listed in Table.2, where the borders in
each range are related with the intersections of each type of emission in the
top panel of Fig.3. Moreover, we also draw the photon trajectories in the
polar coordinates in the bottom panel of Fig.3. The Horndeski hairy black hole
is denoted by the solid black disk and the black dashed circle indicates its
photon sphere. Again, the red, gold, and black curves indicate the light rays
emitted from photon ring, lensed ring and direct emissions, respectively.

Since the light ray will extract energy from the thin accretion disk each time
when passing through it, different types of light rays will contribute
differently to the observed light intensity. The above analysis indicates that
the Horndeski hair has a significant effect on the type of light rays in thin
accretion disk environment, which makes it interesting to further study the
observed intensities and see the hairy black hole’s observational appearances.

###  III.2 Observed intensities and optical appearances

Considering that the thin accretion disk emits isotropically in the rest frame
of static worldlines, the specific intensity received by the observer with
emission frequency $\nu_{e}$ is

$I_{o}(r,\nu_{o})=g^{3}I_{e}(r,\nu_{e}),$ (12)

where $g=\nu_{o}/\nu_{e}=\sqrt{f(r)}$ is the redshift factor, and
$I_{e}(r,\nu_{e})$ is the specific intensity of the accretion disk. The total
observed intensity $I_{obs}(r)$ can be obtained by integrating all observed
frequencies of $I_{o}(r,\nu_{o})$ written as

$I_{obs}(r)=\int I_{o}(r,\nu_{o})d\nu_{o}=\int
g^{4}I_{e}(r,\nu_{e})d\nu_{e}=f(r)^{2}I_{em}(r),$ (13)

where we denote $I_{em}(r)=\int I_{e}(r,\nu_{e})d\nu_{e}$ as the total emitted
intensity. We note that if the trajectory of photon followed backward from the
observer intersects the disk, the photon from accretion disk emission will
contribute the brightness to the observer Gralla:2019xty . Thus, ignoring the
absorption, the total observed intensities are determined by each
intersection, yielding

$I_{obs}=\sum_{m}f(r)^{2}I_{em}(r)|_{r=r_{m}(b)},$ (14)

where $r_{m}$ denotes for the coordinate of the $m-th$ intersecting position
between the light ray emitted with impact parameter $b$ and the accretion
disk. $r_{m}(b)$ is also known as the transfer function from the impact
parameter of photon to the $m-$th hitting positions on the disk, and its slope
$dr/db$ describes the demagnification factor at each $b$ Gralla:2019xty . We
plot the first three transfer functions $r_{m}(b)$ for different hairy
parameter in Fig.4, from which we can read off the following properties. (i)
For the first transfer function (black curves) which corresponds to the direct
image originating from direct, lensed and photon rings emission, the hairy
parameter slightly enhances the value of $b$ for the emergence of transfer
function, indicating the central dark region seen by the distant observer. In
addition, the slope is almost $1$, which means that this direct image can be
seen as the source profile after redshift. (ii) For the second transfer
function (gold curves) which can origin from lensed ring and photon ring
emission, the slope is large, and so this will contribute a bit into the total
photon flux. Therefore, a highly demagnified image will be observed. We see
that for the Horndeski hairy black hole, the demagnification factor is
suppressed by the hair, meaning that the lensed ring emission could be more
visible in the hairy black hole image. (iii) For the third transfer function
(red curves) which can only origin from photon ring emission, it is obvious
that this image is extremely demagnified, so it will contribute little to the
total flux and hardly visible. Again, the Horndeski hair slightly suppresses
the demagnification factor and makes it easier to be seen than in
Schwarzschild case.

(a) $h=0$ (b) $h=-1$

(c) $h=-2$

Figure 4: The first three transfer functions in Horndeski black hole for
different $h$ with $M=1$. They represent the radial coordinate of the first
(black), second (gold), and third (red) intersections with the emission disk.

(a) profile 1 (b) profile 2

(c) profile 3

Figure 5: The sketches of three accretion disk emission profiles plotted by
Eq.(15), Eq.(16), and Eq.(17) respectively.

To testify the prediction from the transfer function, we will consider some
specific emission profile of accretion disk, and then evaluate the total
observed intensity from each kind of emission via (14). We expect to see how
the direct, lensed and photon rings emissions contribute to the total
intensity and the image of black hole. Here we consider the following three
toy-model emission functions Wang:2022yvi ; Yang:2022btw . Firstly, we assume
that the emission of accretion disk is the power of second-order decay
function that starts from the innermost stable circular orbit $r_{isco}$

$\displaystyle\text{profile 1:}\quad I_{em}(r)$
$\displaystyle:=\begin{cases}I_{0}\left[\frac{1}{r-(r_{isco}-1)}\right]^{2},&\hskip
31.2982ptr>r_{isco}\\\ 0,&\hskip 31.2982ptr\leqslant r_{isco}\end{cases},$
(15)

where $I_{0}$ is the maximum intensity (the same below) and the $r_{isco}$ of
the Horndeski hairy black hole is calculated in the Appendix A. Secondly, the
emission function starts from the photon sphere $r_{ph}$ and exhibit a cubic
decay behaviour

$\displaystyle\text{profile 2:}\quad I_{em}(r)$
$\displaystyle:=\begin{cases}I_{0}\left[\frac{1}{r-(r_{ph}-1)}\right]^{3},&\hskip
32.72049ptr>r_{ph}\\\ 0,&\hskip 32.72049ptr\leqslant r_{\rm ph}\end{cases}.$
(16)

Thirdly, we consider a more moderate decay emission function starting from the
event horizon $r_{h}$

$\displaystyle\text{profile 3:}\quad I_{em}(r)$
$\displaystyle:=\begin{cases}I_{0}\frac{\frac{\pi}{2}-\arctan[r-(r_{isco}-1)]}{\frac{\pi}{2}-\arctan[r_{h}-(r_{isco}-1)]},&\quad
r>r_{h}\\\ 0,&\quad r\leqslant r_{h}\end{cases}.$ (17)

The sketches of three accretion disk emission profiles are explicitly drawn in
Fig.5.

Then we move on to study the optical appearance of the Horndeski hairy black
hole with thin accretion disks, and the results for the three accretion disks
are shown in Fig.6, Fig.7 and Fig.8, respectively. We specifically show the
different observed intensities originated from the first (black), second
(gold) and third (red) transfer function in Eq.(14) respectively, shown in the
first columns of the figures. Note that we regard the brightness contributed
by the first, second and third transfer function as the direct, lensed ring
and photon ring intensity respectively. The second columns correspond to the
total observed intensities, which is the sum of the contributions of direct,
lensed ring and photon ring intensity. Meanwhile, the third and fourth columns
translate these total observed intensities into the corresponding two-
dimensional images. Next, we will discuss the observational appearances of
hairy black hole with a thin accretion disk corresponding to three accretion
profiles, respectively.

(a) $h=0$

(b) $h=-1$

(c) $h=-2$

Figure 6: Observational appearances of the profile 1 (15) of a thin disk for
different $h$ with $M=1$. First column: the different observed intensities
originated from the first (black), second (gold) and third (red) transfer
function in Eq.(14) respectively. Second column: the total observed
intensities $I_{obs}/I_{0}$ as a function of impact parameter $b$. Third
column: optical appearance: the distribution of observed intensities into two-
dimensional disks. Fourth column: the zoomed in sectors.

(a) $h=0$

(b) $h=-1$

(c) $h=-2$

Figure 7: Observational appearances of the profile 2 (16) of a thin disk for
different $h$ with $M=1$. First column: the different observed intensities
originated from the first (black), second (gold) and third (red) transfer
function in Eq.(14) respectively. Second column: the total observed
intensities $I_{obs}/I_{0}$ as a function of impact parameter $b$. Third
column: optical appearance: the distribution of observed intensities into two-
dimensional disks. Fourth column: the zoomed in sectors.

(a) $h=0$

(b) $h=-1$

(c) $h=-2$

Figure 8: Observational appearances of the profile 3 (17) of a thin disk for
different $h$ with $M=1$. First column: the different observed intensities
originated from the first (black), second (gold) and third (red) transfer
function in Eq.(14) respectively. Second column: the total observed
intensities $I_{obs}/I_{0}$ as a function of impact parameter $b$. Third
column: optical appearance: the distribution of observed intensities into two-
dimensional disks. Fourth column: the zoomed in sectors.

Firstly we analyze the observed intensities and black hole images caused by
the first emission profile of accretion disk (15) which are shown in Fig.6. In
leftmost figures, we find that the direct intensity moves towards the photon
ring intensity relatively. However, the relative positions of lensed and
photon ring intensities remain unchanged. In the movement of direct intensity,
it combines with different ring intensities which construct different optical
features. For example when $h=0$ (the Schwarzschild cases Gralla:2019xty ),
the total observed intensities show the three peaks without almost
superposition effect. But the photon ring intensity is the spike with a narrow
width at $b\sim 5.21M$, which contributes a totally negligible flux to the
observed feature and is only zoomed in in the rightmost plot. On the contrary,
the direct and lensed ring intensities make a prominent and small
contributions to the total brightness respectively that are clearly shown in
the image. For $h=-1$, we find that the maximum peak of the observed
intensities originated from the superposition of direct and lensed ring
intensities. Besides, these two intensities also contribute an extra tiny peak
at $b\sim 7.35M$ which is very close to the maximum peak and hard to see.
Similarly to the Schwarzschild case, the photon ring intensity also makes the
negligible contribution to the black hole image. However, for $h=-2$, the
direct and photon ring intensities make together a maximum peak with an extra
peak, where the latter can be seen clearly in the zoomed in sector. For the
lensed ring intensity combined with the direct intensity, it also can be seen
in the image due to its larger width.

Then we discuss the optical features of the Horndeski hairy black hole with
the second accretion disk profile (16) shown in Fig.7. One can see that unlike
the results of the first accretion profile, the positions of direct
intensities are always located inside the lensed and photon ring intensities.
However, the lensed ring intensity moves towards the photon ring intensity.
For $h=0$, the lensed ring intensity combined with photon ring intensity
construct together a maximum peak with an extra peak. But the extra peak is
very close to the maximum peak and difficult to be distinguished in the black
hole image. For the Horndeski hairy black hole, the extra peak gradually moves
away from the maximum peak and becomes visible even in the wider zoomed in
sector.

Finally, we analyze the results of Horndeski hairy black hole with the third
accretion profile (17) shown in Fig.8. We find that the three intensities
moving away from each other as the hairy parameter $h$ decreases, but the
photon ring intensity always makes the dominant contribute to the maximum
intensity. Moreover, the contribution of direct intensity to the maximum
intensity is larger than that of lensed ring intensity for $h=0$ and $h=-1$,
but the situation changes for $h=-2$ due to the intensities’ movement.
Meanwhile, comparing to the Schwarzschild case, the bright ring appeared
around the hairy black hole is more clear to see due to its larger width of
photon ring intensity.

In short, comparing to the Schwarzschild black hole under each thin disk
accretion, the Horndeski hair has significant influence on the brightness
distribution which can be arbitrary combinations among direct, lensed ring and
photon ring intensities, depending on each impact parameter. So, in Horndeski
hairy black hole’s optical appearance, the origination of the observed
intensity for each impact parameter, especially its maximum values which
correspond to rings, could be dependent of the Horndeski hair. Moreover, in
all three emission profiles, we find that the total observed intensity of the
Horndeski hairy black hole is always smaller than that of Schwarzschild black
hole.

##  IV Rings and images of hairy black hole illuminated by thin spherical
accretions

In this section, we will investigate the rings and images of Horndeski hairy
black hole surrounded by an optically and geometrically thin accretion flow
with spherically symmetric. Thus, the specific intensity $I(\nu_{o})$ observed
by a distant observer at $r=\infty$ (measured in erg s-1 cm-2 str-1 Hz-1 )
radiated by the accretion flow can be obtained by integrating the specific
emissivity along the photon path $\gamma$ Jaroszynski:1997bw ; Bambi:2013nla

$I(\nu_{o})=\int_{\gamma}g^{3}j_{e}(\nu_{e})dl_{prop},$ (18)

where $g=\nu_{o}/\nu_{e}$ is again the redshift factor, $\nu_{o}$ and
$\nu_{e}$ are observed photon frequency and emitted photon frequency
respectively. $j_{e}(\nu_{e})$ is the emissivity per unit volume in the rest
frame and we will set $j_{e}(\nu_{e})\propto\delta(\nu_{r}-\nu_{e})/r^{2}$ as
usual Bambi:2013nla , where $\nu_{r}$ is the emitter’s rest-frame frequency.
$dl_{prop}$ is the infinitesimal proper length. Integrating Eq.(18) over all
the observed frequencies, we get the total observed intensity given by

$I_{obs}=\int_{\nu_{o}}I(\nu_{o})d\nu_{o}=\int_{\nu_{e}}\int_{\gamma}g^{4}j_{e}(\nu_{e})dl_{prop}d\nu_{e}.$
(19)

Then, we will explore the influence of Horndeski hair on the black hole image
when the spherical accretion is static and radially free-infalling,
respectively.

###  IV.1 Static spherical accretions

We firstly consider the static spherical accretions which are distributed
outside the event horizon of the hairy black hole (I). In this case, the
redshift factor is $g=f(r)^{1/2}$, and the proper length is

$dl_{prop}=\sqrt{\frac{1}{f(r)}dr^{2}+r^{2}d\phi^{2}}=\sqrt{\frac{1}{f(r)}+r^{2}\left(\frac{d\phi}{dr}\right)^{2}}dr.$
(20)

Therefore the total observed intensity observed by a distant observer is given
by

$I_{obs}=\int_{\gamma}\frac{f(r)^{2}}{r^{2}}\sqrt{\frac{1}{f(r)}+r^{2}\left(\frac{d\phi}{dr}\right)^{2}}dr.$
(21)

It is obvious that the observed intensity depends on the radial distance $r$
and its integral interval, which depends on the impact parameter $b$ as we
analyzed in the previous section.

Figure 9: The total observed intensity $I_{obs}$ with respect to impact
parameters $b$ for the hairy black hole surrounded by the static spherical
accretion flow.

Then we plot the observed intensity with respect to $b$ in Fig.9, which shows
that $I_{obs}$ for hairy black hole is always smaller than that for
Schwarzschild black hole. It is noted that with the decrease of parameter $h$,
the peak value of observed intensity at the $b=b_{ph}$ decreases which means
that the maximum brightness of Horndeski hairy black hole image is also lower
than that of the Schwarzschild case. As illustrated in Zeng:2020dco , the
intensity at $b=b_{ph}$ should be infinite because the photon travels around
the black hole infinite times which should contribute an arbitrarily large
intensity. However, the obtained intensity never goes to infinity due to the
numerical limitations.

(a) $h=0$

(b) $h=-1$

(c) $h=-2$

Figure 10: The black hole images with static spherical accretions for
different $h$ with $M=1$.

The distribution of observed intensity (21) in two-dimensional plane should
depict the image of hairy black hole illuminated by the static spherical
accretion for equatorial observers, as shown in Fig.10. The faint illuminating
region in the center is the black hole shadow and the surrounding represents
the spherical accretion flow. It is obvious that a bright photon ring can be
observed very close to the shadow. Moreover, comparing to the Schwarzschild
black hole, the Horndeski hairy black hole corresponds to fainter photon ring
but larger shadow which are consistent with the results reflected in Fig.9.

###  IV.2 Radially infalling spherical accretions

We then consider more realistic description that the black hole is surrounded
by the radially infalling freely spherical accretion because the most of
matters are dynamical in the Universe. Therefore, the redshift factor $g$
should be replaced by Bambi:2013nla

$g=\frac{k_{\mu}u_{o}^{\mu}}{k_{\nu}u_{e}^{\nu}}.$ (22)

Here, $k_{\mu}$ is the four-momentum of photon emitted from accretion matter
determined by $k_{\mu}=\partial\mathcal{L}/\partial\dot{x}^{\mu}$, which give
us

$k_{t}=-\frac{1}{b},\qquad
k_{r}=\pm\frac{1}{f(r)}\sqrt{\frac{1}{b^{2}}-\frac{f(r)}{r^{2}}},\qquad
k_{\theta}=0,\qquad k_{\phi}=\pm 1,$ (23)

where we have inserted the Eqs.(6-8). It is noted that $\pm$ in $k_{r}$
describe the photons moving radially inward and outward, while in $k_{\phi}$
they describe the photons moving counterclockwise and clockwise direction
respectively. $u_{o}^{\mu}$ and $u_{e}^{\nu}$ are four-velocity of distant
observer and accretion matters respectively. Considering an stationary distant
observer, we have $u_{o}^{\mu}=(1,0,0,0)$. $u_{e}^{\nu}$ is evaluated as

$u_{e}^{t}=\frac{1}{f(r)},\qquad u_{e}^{r}=-\sqrt{1-f(r)},\qquad
u_{e}^{\theta}=u_{e}^{\phi}=0.$ (24)

Subsequently, the redshift factor $g$ can be expressed as

$g=\frac{k_{t}}{k_{t}u_{e}^{t}+k_{r}u_{e}^{r}}=\frac{1}{\frac{1}{f(r)}\pm\sqrt{1-f(r)}\sqrt{\frac{1}{f(r)}\left(\frac{1}{f(r)}-\frac{b^{2}}{r^{2}}\right)}}.$
(25)

Moreover, the proper length can be defined as Bambi:2013nla

$dl_{prop}=k_{\mu}u_{e}^{\mu}d\lambda=\frac{k_{t}}{g|k_{r}|}dr.$ (26)

Thus, the total observed intensity (19) for the radially infalling accretion
flow is

$I_{obs}=\int_{\gamma}\frac{g^{3}k_{t}}{r^{2}|k_{r}|}dr,$ (27)

which is shown in the Fig.11(a). Similar to the static case, the observed
intensity increases firstly, then reaches a maximum at photon sphere
$b=b_{ph}$ and finally declines with the increase of $b$ for a fixed parameter
$h$. As $h$ decreases, the intensity will become weak for all the $b$. To
compare the difference of observed intensities between the static and
infalling spherical accretions, we collect them in Fig.11(b)-Fig.11(d). We can
see that for the same parameter $h$, the intensity of the black hole image in
the infalling spherical accretion is weaker than that in the static case,
which is due to the Doppler effect. Additionally, whether the black hole is
surrounded by the static or infalling spherical accretion, the shadow radius
is always the same as photon sphere, which implies that the shadow size is
independent of spherical accretion flow. Then, the image of hairy black hole
surrounded by the infalling accretion flow in the two-dimensional plane are
presented in Fig.12. It is obvious that comparing to that in the static case,
the central black hole shadow surrounded by a bright photon ring has a lower
brightness.

(a) total intensity $I_{obs}$

(b) $h=0$

(c) $h=-1$

(d) $h=-2$

Figure 11: (a) The observed intensities for the infalling spherical accretion.
(b)-(d) The comparisons of the observed intensities between the static (red
curves) and radially infalling (blue curves) spherical accretions for
different $h$.

(a) $h=0$

(b) $h=-1$

(c) $h=-2$

Figure 12: The hairy black hole images with radially infalling spherical
accretions for different $h$ with $M=1$.

Thus, comparing to the Schwarzschild black hole, the existence of Horndeski
hair will enlarge the photon ring, which then corresponds to a larger black
hole shadow for a distant observer. Under the illumination of spherical
accretion flow, the deviations of shadow region, bright photon ring and the
brightness distribution due to the effect of Horndeski hair from the
Schwarzschild black hole are significant.

##  V Conclusion and discussion

Horndeski gravity has significant consequences in describing the accelerated
expansion and other interesting observations, so it attracts considerable
attentions in cosmological and astrophysical communities. Meanwhile, the black
hole solutions in Horndeski gravity may be a good platform to test the no-hair
theorem of classical black hole which also triggers great interest among
physicists, so their theoretical and observational properties have also been
widely studied. In this paper, we explored the optical appearance of a static
hairy black hole in Horndeski gravity, and analyzed the influence of Horndeski
hair on the photon rings and black hole image illuminated by various
accretions.

Firstly, we investigated the Horndeski hairy black hole image illuminated by
an optical and geometrically thin accretion disk, which located at rest on the
equatorial plane of hairy black hole. Comparing to Schwarzschild black hole,
the lensed ring and photon ring emission from the thin accretion disk around
hairy black hole always corresponds to wider range of impact parameter, which
can be easily found from Fig.3 and Table 2. In addition, the Horndeski hair
tends to suppress the demagnification factor in the second and third transfer
function, so we predicted that due to the Horndeski hair, the lensed ring and
photon ring could be more easily to be observed. Then by taking three toy
models of emission profile as examples, we carefully analyzed how the
brightness from the direct, lensed ring and photon ring intensity contribute
to the total observed intensities. Our novel analysis clearly show why the
rings in the images form and how is effected by the Horndeski hair. Our
results show that the Horndeski hair could significantly affect the brightness
distribution which can be arbitrary combinations among direct, lensed ring and
photon ring intensity, depending on each impact parameter, so the Horndeski
hairy black hole has a completely different optical appearance in contract to
the Schwarzschild black hole under illumination of all three emission profiles
(Figs.6-8). In addition, in all three emission profiles, we found that the
total observed intensity of the Horndeski hairy black hole is always smaller
than that of Schwarzschild black hole. It means that if we mimic the EHT
resolution of the black hole images by blurring them, the hairy black hole
will corresponds to a dimmer realistic observation. It is noted that here we
focused on the distance static observer viewing face-on in the north pole
direction, it is straightforward to extend our study into any observational
angles.

Then, we considered that the black hole is illuminated by a static and
radially free-falling spherical accretions flow, respectively. In both cases,
we can see that a bright photon ring closely surrounds a dark region indicated
the black hole shadow (Fig.10 and Fig.12); and comparing to the Schwarzschild
black hole, the shadow size for the hairy black hole is larger but the photon
ring is fainter. For fixed hairy parameter, the size of shadow keeps the same
in the two accretions cases, but its brightness for the infalling accretion is
fainter than the static accretion because of the Doppler effect.

In conclusion, according to the light rays distributions, we figured out the
optical appearances of Horndeski hairy black hole surrounded by various thin
disk or spherical accretion flow, which differentiate the Schwarzschild black
hole in GR. Our theoretical studies are ideal, especially, the accretion
profiles we consider are far away from the realistic astrophysical
environments, but this study provides potential way to distinguish Horndeski
hairy black hole from static black hole in GR. We expect that these
preliminary results could shed light on the future test of no-hair theorem
using black hole shadow.

###### Acknowledgements.

This work is partly supported by Natural Science Foundation of Jiangsu
Province under Grant No.BK20211601, Fok Ying Tung Education Foundation under
Grant No.171006, the Postgraduate Research & Practice Innovation Program of
Jiangsu Province under Grant No. KYCX22_3452 and KYCX21_3192, and Top Talent
Support Program from Yangzhou University.

##  Appendix A Inner-most stable circular orbit (ISCO)

For the motion of massive particle, it requires the Lagrangian
$\mathcal{L}=-1/2$. Similar to the calculations of null geodesic in section
II, one can also obtain the orbit equation
$(dr/d\phi)^{2}+\tilde{V}_{\text{eff}}(r)=0$ and introduce another effective
potential $\tilde{V}_{\text{eff}}$ given by

$\tilde{V}_{\text{eff}}=r^{4}\left(\frac{\tilde{E}^{2}}{\tilde{L}_{z}^{2}}-\frac{f(r)}{\tilde{L}_{z}^{2}}-\frac{f(r)}{r^{2}}\right),$
(28)

which is different from the photon’s. $\tilde{E}$ and $\tilde{L}_{z}$ are the
energy and angular momentum for the massive particle respectively. The ISCO is
determined by $\tilde{V}_{\text{eff}}=0$, $\tilde{V}_{\text{eff}}^{\prime}=0$
and $\tilde{V}_{\text{eff}}^{\prime\prime}=0$, where the prime represents the
derivative with respect to $r$. Finally the radius $r_{isco}$ of ISCO
satisfies the following relation

$r_{isco}=\frac{3f(r_{isco})f^{\prime}(r_{isco})}{2f^{\prime}(r_{isco})^{2}-f(r_{isco})f^{\prime\prime}(r_{isco})},$
(29)

which is numerically solved and its function of the hairy parameter is shown
in Fig.13.

Figure 13: The radius of innermost stable circular orbit $r_{isco}$ as the
function of the hairy parameter $h$.

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