# Shadows and optical appearance of black bounces illuminated by a thin
accretion disk

Merce Guerrero<EMAIL_ADDRESS>Departamento de Física Teórica and IPARCOS,
Universidad Complutense de Madrid, E-28040 Madrid, Spain  Gonzalo J. Olmo
<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia - CSIC. Universidad de Valencia, Burjassot-46100,
Valencia, Spain  Departamento de Física, Universidade Federal da Paraíba,
58051-900 João Pessoa, Paraíba, Brazil  Diego Rubiera-Garcia<EMAIL_ADDRESS>Departamento de Física Teórica and IPARCOS, Universidad Complutense de Madrid,
E-28040 Madrid, Spain  Diego Sáez-Chillón Gómez<EMAIL_ADDRESS>Department
of Theoretical Physics, Atomic and Optics, Campus Miguel Delibes,
University of Valladolid UVA, Paseo Belén, 7, 47011 - Valladolid, Spain

(August 27, 2024)

###### Abstract

We study the light rings and shadows of an uniparametric family of spherically
symmetric geometries interpolating between the Schwarzschild solution, a
regular black hole, and a traversable wormhole, and dubbed as black bounces,
all of them sharing the same critical impact parameter. We consider the ray-
tracing method in order to study the impact parameter regions corresponding to
the direct, lensed, and photon ring emission, finding a broadening of all
these regions for black bounce solutions as compared to the Schwarzschild one.
Using this, we determine the optical appearance of black bounces when
illuminated by three standard toy models of optically and geometrically thin
accretion disks viewed in face-on orientation.

##  I Introduction

The detection in 2019 by the Einstein Horizon Telescope (EHT) of the accretion
flow around the supermassive object at the center of the M87 galaxy
Akiyama:2019cqa  has triggered the beginning of a new era in the analysis of
electromagnetic phenomena around compact objects and on testing General
Relativity (GR) itself. The image released by the EHT shows a bright ring-
shaped lump of radiation surrounding a black central region of an estimated
$6.5$ billion solar masses. The canonical interpretation of this phenomenon
calls for the deviation of light rays in the gravitational field of an object
having a photon sphere (a critical unstable curve) when illuminated by an
accretion disk Falcke:1999pj . The inner region bounded by this critical curve
is commonly known as the black hole shadow Perlick:2021aok .

The implications of this discovery are far reaching. Not only does it allow us
to test the background geometry using gravitational light deflection
(lensing), but also constitute a test of the Kerr hypothesis on the nature of
astrophysical black holes as compared to its many competitors Cardoso:2019rvt
. This is so because gravitational lensing near critical curves involves
testing gravitational effects happening in a regime which is inaccessible to
weak-field limit tests Psaltis:2020lvx  and, as such, it allows plenty of room
for alternative compact objects to represent the shadow caster. On the other
hand, despite the fact that the main equations governing gravitational lensing
are known since a long time ago Bardeen  (for a in-depth analysis of such
equations in the strong-field regime see the paper by Bozza Bozza:2002zj ) and
many non-Kerr black hole shadows beyond GR have been studied in the literature
Johannsen:2010ru ; Atamurotov:2013sca ; Cunha:2015yba ; Abdujabbarov:2016hnw ;
Held:2019xde ; Kumar:2020owy ; Xavier:2020egv ; Wei:2020ght ; Herdeiro:2021lwl
; Devi:2021ctm ; Hou:2021okc , only very recently we have come to fully
appreciate the great richness of the physics of accretion disks around black
holes regarding its impact on the silhouettes of the latter Narayan:2019imo ;
Cunha:2019hzj ; Boero:2021afh .

A compact object having a critical curve and illuminated by an accretion disk
may yield a complex pattern of contributions to the total luminosity driven by
several light ray trajectories. Technically, the critical curve is defined as
the light ray received by the observer that, when traced backwards, would have
approached asymptotically a bound photon orbit. For a Schwarzschild black hole
this determines a critical impact parameter $b_{c}=3\sqrt{3}M\approx 5.197M$.
However, while the impact parameter region (number of orbits around the
critical curve) depends only on the background geometry, the optical
appearance of the object is not only a function of it, but also of the
geometry and physical properties of the illuminating accretion flow
Gralla:2019xty . For instance, for an optically and geometrically thin
accretion disk the total luminosity is largely dominated by the direct
emission (light rays deflected less than $90$ degrees) and, to lower extent,
by the lensing ring (light rays that intersect the equatorial plane just
twice), with the contribution of the critical curve to it being almost
negligible. This is why the authors of Gralla:2019xty  proposed to call the
inner region of this direct emission region (for Schwarzschild this is
$b\lesssim 6.17M$) the black hole shadow, instead of the one associated to the
inner region of the critical curve $b=b_{c}$. Whether this interplay between
gravitational theory and the modelling of the accretion disk allows to
discriminate GR black holes from alternative compact objects via
electromagnetic illumination, - likewise distinguishing horses from unicorns
via their respective shadow - is nowadays an exciting area of research
Glampedakis:2021oie ; Junior:2021atr ; Chael:2021rjo .

Once the tabletop is set, the community has launched to the search for shadows
cast by different compact objects when illuminated by different types of
accretion disks Zeng:2020dco ; Zeng:2020vsj ; Qin:2020xzu ; Lima:2020auu ;
He:2021htq ; Peng:2020wun ; Gan:2021pwu ; Li:2021riw ; 1865010 ;
Eichhorn:2021iwq ; Eichhorn:2021etc ; Shaikh:2021cvl  in order to compare them
with the GR (Kerr) expectations. Nonetheless, given the many ingredients
involved in the analysis of this problem - the underlying background geometry,
the assumptions on the symmetries of the problem, the geometrical, optical,
and emission aspects of the modeling of the accretions disk, etc -, it is
useful to consider some simplifying assumptions in order to investigate
prospective smoking guns of new Physics. In this sense, the assumption of
spherical symmetry, though seemingly too restrictive given the fact that real
astrophysical black holes do rotate, turns out to be a good approximation
since the size and shape of the shadow, as seen by an asymptotic observer,
depends very weekly on the spin of the black hole in combination with the
inclination with respect to the line of sight, with deviations from
circularity lying within $\sim 7\%$ for ultra-fast spinning black holes
Psaltis:2018xkc .

The main aim of this work is to study the optical appearances and shadows of
an uniparametric spherically symmetric family of extensions of the
Schwarzschild space-time recently introduced in Simpson:2018tsi  and dubbed as
black bounces, which have attracted quite some attention in the community
Huang:2019arj ; Churilova:2019cyt ; Lobo:2020kxn ; Nascimento:2020ime ;
Lobo:2020ffi ; Tsukamoto:2020bjm ; Zhou:2020zys ; Mazza:2021rgq ;
Shaikh:2021yux ; Cheng:2021hoc ; Islam:2021ful ; Fran:2021pyi ;
Bronnikov:2021liv ; Tsukamoto:2021caq ; Zeng:2021dlj . Despite its simple
mathematical structure, its interest lies in the following: i) it smoothly
interpolates between the Schwarzschild space-time, a family of regular black
hole solutions, and a family of traversable wormhole solutions; ii) it has the
same critical parameter as in the Schwarzschild solution; iii) it removes the
presence of space-time singularities, iv) it has not Cauchy horizons, thus
avoiding their associated instability issues Poisson:1989zz ; v) they can be
taken as parameterized deviations from the Schwarzschild solution in a theory-
agnostic way (for an example where solutions of this type arise as solutions
to modified gravity equations, see Olmo:2013gqa ; Olmo:2015bya ;
Bejarano:2017fgz ). Since black holes and traversable wormholes are
conceptually and operationally two different types of objects, the black
bounce geometry allows one to study the light rings and shadows cast by each
such object and compare them to that of the Schwarzschild solution. To this
end, in this paper we shall characterize the impact parameter regions for each
direct/lensed/photon ring trajectories using the ray-tracing method, and
moreover consider three standard toy models of geometrically and optically
thin accretion disks with different emission profiles in order to find the
corresponding optical appearances as compared to the Schwarzschild solution.

This paper is organized as follows: in Sec. II we describe the main aspects of
the black bounce geometries and discuss their geodesic motion equations and
associated effective potential. In Sec. III we use the ray-tracing method in
order to study the impact parameter region for the three types of emission
(direct/lensed/photon ring) for the different regions of interest of the black
bounce parameter. In Sec. IV we use the three toy models for the emission
profile of the accretion disk in order to study the observational appearance
of some samples of black bounces corresponding to the regular black hole and
traversable wormhole geometries. Finally in Sec. V we summarize our main
findings, discuss the limitations of our approach as well as future prospects.

##  II Black bounces

###  II.1 Geometry and horizons

Let us start by considering a static, spherically symmetric solution of the
form

$ds^{2}=-A(x)dt^{2}+B(x)dx^{2}+r^{2}(x)d\Omega^{2}\ ,$ (1)

where the radial coordinate $x$ spans the entire real line,
$x\in(-\infty,+\infty)$, while $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}$ is the line element on the two spheres. The areal radius is
measured by $S=4\pi r^{2}(x)$ and, in bouncing geometries such as in wormhole
ones, the radial function $r(x)$ is bounded by $r\geq r_{th}$ in a model-
dependent way VisserBook . One can note that the above line element can be
further simplified to just two free functions by introducing a new radial
coordinate $dy^{2}=B(x)dx^{2}$, though for the purposes of this paper we shall
keep it this form.

By black bounce (BB) we refer to the uniparametric family of solutions given
by the line element (1) with Simpson:2018tsi

$A(x)=B^{-1}(x)=1-\frac{2M}{r(x)}\hskip 2.84544pt;\hskip
2.84544ptr^{2}(x)=x^{2}+a^{2}\ ,$ (2)

where $a$ is the BB parameter, so in this geometry one has the wormhole throat
located at $r_{th}^{2}=a^{2}$. The most noticeable feature of such geometries
is the bounce (hence its name) in the radial function, in a simple
implementation of a wormhole geometry extending the Schwarzschild solution via
the replacement $x\to r(x)$, such that in the limit $a\to 0$ one has
$r^{2}(x)\approx x^{2}$. Whether the bounce is hidden behind an event horizon
or not can be found by looking at the location of the horizons,
$g^{xx}=A(x)=0$, which in the present case amounts to

$x_{h}^{\pm}=\pm\sqrt{4M^{2}-a^{2}}\ ,$ (3)

where the $\pm$ signs refer to the location of the horizon on both sides of
the throat. From these equations it can be easily seen that the bounce will be
hidden by an event horizon if $a<2M$, so in this case one finds a regular
black hole (BH) geometry111Indeed, the bounce allows for the extension of
geodesics beyond $x=0$ ($r=0$). For an extended discussion on geodesic
completeness restoration mechanisms, see e.g. Carballo-Rubio:2019fnb ., while
if $a>2M$ the bounce lies above the would-be horizon and the geometry
represents instead a traversable wormhole (WH) solution with its throat
located at $x_{th}=0$. Note that in terms of the radial function, the BH
solution has its horizon at $r_{h}=2M$, while the WH has its throat at
$r_{th}=a>2M$ instead, and no horizon is present. The case $a=2M$ was argued
in Simpson:2018tsi  to correspond to a non-traversable WH and, for the sake of
this paper, we shall use it as a limiting case in the transition BH/WH.

###  II.2 Geodesic equations

A photon travels on a null geodesic, $g^{\mu\nu}k_{\mu}k_{\nu}=0$, with
$k^{\mu}=\dot{x}^{\mu}$ its wave number. In spherical symmetry there are two
conserved quantities, namely, the energy per unit mass,
$E=-g_{\mu\nu}t^{\mu}k^{\nu}=A\dot{t}$, and the angular momentum per unit
mass, $L=g_{\mu\nu}\phi^{\mu}k^{\nu}=r^{2}\dot{\phi}$ (dots indicating
derivatives with respect to the affine parameter). By spherical symmetry one
can assume the motion to take place in the plane $\theta=\pi/2$ without loss
of generality, and furthermore by introducing the impact parameter,
$b\equiv\tfrac{L}{E}=\tfrac{r^{2}\dot{\phi}}{A\dot{t}}$, one can cast the
geodesic equation for null trajectories as (a re-parametrization of the affine
parameter is introduced here to absorb a $L^{2}$ factor)

$\dot{x}^{2}=\frac{1}{b^{2}}-V(x)\ ,$ (4)

which is akin to the equation for a one-dimensional single particle moving in
an effective potential of the form

$V(x)=\frac{A(x)}{r^{2}(x)}\ ,$ (5)

which is depicted in Fig.1 for the BB solution in both the BH and WH cases as
compared to the Schwarzschild solution. Let us assume a photon approaching
from infinity with a given impact parameter $b$ such that at some value
$r_{0}$ the right-hand side of Eq.(4) vanishes. Such a photon will thus
approach to the closest distance $r_{0}$ before running away back to infinity.
The minimum impact factor for which that relation can be satisfied is given by
the critical value

$\displaystyle b_{c}^{2}=\frac{r^{2}(x_{ps})}{A(x_{ps})}\ ,$ (6)

which corresponds to the maximum of the effective potential (5), i.e.,
$V_{eff}(x=x_{ps})=\tfrac{1}{b_{c}^{2}},V_{eff}^{\prime}(x=x_{ps})=0,V_{eff}^{\prime\prime}(x=x_{ps})<0$.
At this point $r_{ps}$ it will turn an arbitrarily large number of times
around the compact object. However, this orbit is unstable since under a small
perturbation the photon will eventually fall into the inner region of the
object ($b<b_{c}$) or escape to asymptotic infinity $(b>b_{0}$, and therefore
this critical curve (following the notation of Gralla:2019xty ) defines the
unstable null circular orbit.

In the present BB case, the above conditions define the radius of this
critical curve (for which we shall also reserve the word “photon sphere”) as
Bronnikov:2021liv

$x_{ps}=\sqrt{9M^{2}-a^{2}}\to r_{ps}=3M$ (7)

A remarkable property of the BB family of solutions is that, when (7) is
introduced in (6), it yields the critical impact parameter
$b_{c}=3\sqrt{3}M\approx 5.19615M$ and, therefore, all BB solutions have the
same critical impact parameter as the Schwarzschild one. Note that the
condition (7) implies that such critical orbits will exist provided that
$a<3M$. Therefore the BB configurations relevant for shadows (i.e, having a
photon sphere) are naturally split into two families: those with $0<a<2M$
correspond to regular BHs while those with $2M<a<3M$ are traversable WHs, with
the $a=2M$ and $a=3M$ acting as limiting cases.

Figure 1: The effective potential $V(x)$ in (5) for the BB solutions with
$M=b^{2}=1$ as a function of $x$ for $a=0$ (dashed black, Schwarzschild
solution), $a=3/2$ (orange, BH case), $a=2$ (non-traversable WH, blue),
$a=5/2$ (traversable WH, red) and $a=3$ (gray, last photon orbit). Note that
only when $a>0$ are both sides of this figure physically connected, since in
the $a=0$ case, $r^{2}\approx x^{2}$ and because $r>0$ then the two regions
$x\in(-\infty,0),x\in(0,+\infty)$ are causally disconnected.

In order to study the optical appearance of a compact object as illuminated by
all the light rays passing close by, the geodesic equation (4) must be
suitably rewritten in terms of the variation of the azimuthal angle $\phi$
with respect to the radial coordinate, which in the present BB case reads

$\frac{d\phi}{dx}=\mp\frac{b}{r^{2}(x)\sqrt{1-\tfrac{b^{2}A(x)}{r^{2}(x)}}}\
,$ (8)

where the $\mp$ signs refer to ingoing/outgoing trajectories, respectively.
The few equations introduced in this section is all the setup we need in order
to start with the ray tracing of the BB solutions.

##  III Ray tracing

Figure 2: The number of orbits, $n\equiv\tfrac{\phi}{2\pi}$, made by a light
ray on its trip from its emission source to the observer around the BB
solution for a BH with $a=3/2$ (blue) and a traversable WH with $a=5/2$ (red),
as compared to the Schwarzschild solution, $a=0$ (dashed black). $n$ diverges
at $b=b_{c}=3\sqrt{3}$ (in units of $M=1$), where it would perform an infinite
number of orbits around the BB object.

In the ray-tracing procedure, light rays arriving to the screen of the
observer at asymptotic infinity are traced back to the point of the sky they
originated from bearing in mind its deflection by the gravitational field of
the compact object, which in the BB case is determined by Eq.(8). The physical
scenario is that of a compact object being illuminated from behind by a planar
source which emits isotropically and with uniform brightness. In order to
understand the optical appearance of the BB solution by the ray-tracing
procedure, we first define the total number of orbits made by a single light
ray on its path from its source to the observer as the (normalized) change in
the azimuthal angle, that is, $n(b)\equiv\tfrac{\phi}{2\pi}$. This number of
orbits will obviously depend on how close the impact parameter is to the
critical one and, in addition, on the geometry of the different BB cases. Note
that within this setup, light rays in straight motion (i.e. not being
deflected at all by the BB solution) have $n=1/2$. In Fig.2 we depict the
number of orbits for two samples of the BB solutions, representative of the BH
and traversable WH families. As expected, the most salient feature of this
plot is the narrow spike in both cases at the critical impact parameter
$b_{c}=3\sqrt{3}M$, representing the location of the critical curve where a
light ray would have orbited the BB solution an arbitrarily large number of
times. For other values of the impact parameter one can see that the larger
the BB parameter $a$ is the more orbits for the corresponding BH/WH solutions
are found, an effect which is significantly enhanced in the inner region,
$b<b_{c}$, as we move towards the WH solutions.

The next step in our analysis is to integrate the geodesic equation (8) for a
bunch of light rays spanning the whole region of impact parameter values. For
impact parameters $b\geq b_{c}$ light rays will be deflected at some minimum
radius above the photon sphere one $r=r_{0}>r_{ps}$, and the corresponding
trajectories can be therefore classified according to the number of orbits
around the BB solution as follows:

  * •

Direct emission, for $n<3/4$, corresponding to trajectories that intersect the
equatorial plane (on its front side) just once.

  * •

Lensed emission, for $3/4<n<5/4$, corresponding to trajectories that intersect
the equatorial plane twice (on its front and back sides, respectively).

  * •

Photon ring emission, for $n>5/4$, corresponding to trajectories that
intersect the equatorial plane at least three times.

It should be noted that for $n>5/4$ one actually finds an infinite sequence of
concentric light rings converging to the critical curve $b=b_{c}$, which are
exponentially closer to each other and thinner as more number of orbits are
performed. Therefore, from now on we shall denote by photon ring just the
first of such orbits (whose range of impact parameters depends on the BB
parameter, see Table 1), and disregard all the others since their contribution
to the total luminosity will be negligible Gralla:2019xty .

Orbit/BB parameter | $a=0$ | $a=3/2$ | $a=2$ | $a=5/2$ | $a=3$
---|---|---|---|---|---
Direct | $b>6.17$ | $b>6.32$ | $b>6.46$ | $b>6.64$ | $b>6.88$
Lensed | $5.22<b<6.17$ | $5.25<b<6.32$ | $5.27<b<6.46$ | $5.33<b<6.64$ | $5.44<b<6.88$
Photon ring | $b_{c}<b<5.22$ | $b_{c}<b<5.25$ | $b_{c}<b<5.27$ | $b_{c}<b<5.33$ | $b_{c}<b<5.44$
Retro-photon ring | $5.19<b<b_{c}$ | $5.17<b<b_{c}$ | $5.04<b<b_{c}$ | $5.07<b<b_{c}$ | $5.13<b<b_{c}$
Retro-lensed | $5.02<b<5.18$ | $4.89<b<5.17$ | $4.42<b<5.04$ | $4.59<b<5.07$ | $4.83<b<5.13$
Retro-direct | $b<5.02$ | $b<4.89$ | $b<4.42$ | $b<4.59$ | $b<4.83$
Table 1: Range of impact parameters (in units of $M$, and taking only two
decimals in order not to overload the text) for different BB cases yielding
orbits being deflected above the BB solution photon sphere radius
(direct/lensed/photon ring) or emerging from within it (retro-photon
ring/retro-lensed/retro-direct). In the BH case ($a<2$) those-retro-orbits
will intersect the event horizon at $x_{h}=\sqrt{4-a^{2}}$, while in the WH
case ($a\geq 2$) they will travel all the way down to the wormhole throat
$x=0$ ($r_{th}=a>2$). In this plot $b_{c}=3\sqrt{3}\approx 5.19615$ is the
critical impact parameter for all the BB solutions.

For impact parameters $b<b_{c}$ the ray tracing yields trajectories that would
have emerged from the central region of the object inside the photon sphere
(note that having at least one intersection with the disk requires that
$n>1/4$). In the BH case, $a<2$, the backtrack of such trajectories intersects
with the event horizon, while in the WH case, $a\geq 2$, such trajectories
continue their path all the way down to the wormhole throat $x=0$ given the
absence of event horizon. For the sake of this paper we find it convenient to
also split such $b<b_{c}$ trajectories emerging out of the internal region of
the solutions into three cases depending on the number of orbits:

  * •

Retro-direct emission, for $n<3/4$.

  * •

Retro-lensed emission, for $3/4<n<5/4$.

  * •

Retro-photon ring emission, for $n>5/4$.

These retro-orbits could therefore contribute to the luminosity in the
observer’s screen for impact parameters below the critical one, $b<b_{c}$, as
we shall see in Sec. IV.

Figure 3: Ray-tracing for BB solutions with $a=0$ (Schwarzschild case, left
figure), $a=3/2$ (BB BH case, middle figure) and $a=5/2$ (BB traversable WH
case, right figure), written all of them in terms of the radial function $r$.
The region inside the event horizon is represented by a black circle, while
the photon sphere radius is represented by the dashed thick black
circumference. The observer is located on the far right of the screen, and the
different regions of the image are associated to the number of crossings with
the equatorial plane outside the critical curve: direct emission (green),
lensed emission (orange), photon ring emission (red). In addition the retro-
trajectories (trajectories emerging out of the photon sphere) are also
depicted: retro-photon ring (blue), retro-lensed (purple) and retro-direct
(black). The dashed yellow circumference denotes the radius of the (spherical)
mouth in the WH case. The vertical black solid line represents the location of
the accretion disk in all cases.

In Table 1 we have displayed several impact parameters covering the ranges
between the critical cases $a=0$ (Schwarzschild), $a=2$ (transition BH/WH) and
$a=3$ (disappearance of the photon sphere), including two representative cases
of the BH/WH solutions, $a=3/2$, $a=5/2$, respectively, to be later used in
the illustration of the ray-tracing images as well as for the optical
appearance of the BB solutions when illuminated by accretion disks. There are
several aspects to be underlined in the modifications of the light rays’
impact parameters as compared to the Schwarzschild solution. First, for
$b>b_{c}$ one can note a broadening in the range of impact parameters
contributing to the direct/lensed/photon ring emissions as we increase the BB
parameter $a$. This therefore leads to wider lensing/photon rings, and
supposedly would contribute to enhance the corresponding luminosity in these
regions. Second, in the BH case the impact parameter region of the retro-
orbits ($b<b_{c}$) narrows for the retro-direct emission, but broadens for
both the retro-lensed and retro-photon ring until the WH branch is reached
($a=2$), where the tendency is reversed until the limit $a=3$ is attained.

Focusing for instance on the photon ring ones, one sees a sharp increase in
the width of the impact parameter region when moving from the BH to the WH
configurations ($a=2$), as allowed by the uncloaking of the wormhole mouth at
a radius $r_{th}=a>2M$. As the BB parameter $a$ is further increased the
contribution of the retro-direct emission increases, while those of the retro-
lensed and retro-photon ring decreases, but the total impact parameter region
(i.e. joint emission from both kinds of contributions) is slightly increased
for all the trajectories.

To illustrate this general discussion, let us take two representative samples
of the BH/WH configurations, namely, $a=3/2$ and $a=5/2$, and integrate the
geodesic equation (8) for a bunch of light rays spanning the relevant region
of impact parameters for these three plus three kinds of orbits. The
corresponding results are depicted in Fig.3 for values of the impact parameter
$b\in(0,10)$, alongside its comparison with the known results of the
Schwarzschild case ($a=0$), and we point out that the observer’s screen is
located at the far right side of this plot in all these cases. In these
figures one can clearly see the direct (green), lensed (orange) and photon
ring (red) trajectories outside the photon sphere ($r=3M$, dashed black), and
the impact parameter’s range they correspond to. In the BH case with $a=3/2$
(middle figure) the enhance in the impact parameter’s range as compared to the
Schwarzschild solution regarding these trajectories is barely visible, though
it is there, being much more noticeable in the WH case (right figure). In
addition, we have plotted the retro-photon ring (blue), retro-lensed (yellow),
and retro-direct (black), originated from inside the photon sphere, $b<b_{c}$.
In the BH case, $a=3/2$, these contributions would intersect the event horizon
at $r=2M$ (black circle), while in the WH case, $a=5/2$, such trajectories
corresponds to those originated from the throat (purple dashed circumference).
All these retro-trajectories reach the observer’s screen after circling the BB
solution and exiting the photon sphere.

Figure 4: The first three transfer functions $x_{m}(b)$ for a Schwarzschild
solution ($a=0$, left), a BB solution of BH type ($a=3/2$, middle) and a BB
solution of WH type ($a=5/2$, right). The colours depict direct (blue), lensed
(orange) and photon ring (green) emissions, and include the contributions from
the retro-orbits. The transfer functions get their minimum values in impact
parameter at the point in which no intersections with the accretion disk are
made ($n<1/4$), which are found at the horizon radius $x_{h}=\sqrt{4-a^{2}}$
for the Schwarzschild ($x_{h}=2,b\approx 2.852$) and BB ($x_{h}\approx
1.322,b\approx 2.572$) solutions, while in the BB WH case it corresponds to
the location of the wormhole throat ($x=0,b\approx 2.319$).

##  IV Shadows from geometrically and optically thin accretion disks

We consider an optically and geometrically thin disk222This model describes
accretion disks when the accretion rate is sub-Eddington with a very high
opacity Page:1974he , but leaves aside those with high mass accretion rates,
for instance around supermassive black holes, which may effectively turn the
accretion disk into an optically thin but geometrically thick one Riaz:2019bkv
. surrounding our BB solution on the equatorial plane (so that the observer is
located in the north pole), viewed face-on, and providing the main
contribution to the observed intensity. To this end, we shall consider three
toy models of accretion disks (therefore disregarding the complex modelling of
the plasma in realistic astrophysical scenarios, which requires full magneto-
hydrodynamic simulations) where the specific luminosities only depend on the
radial coordinate $x$, and where the disks are assumed to emit isotropically,
$I_{\nu}^{em}=I(x)$, with $\nu$ the emission frequency in the rest frame of
the emission:

  * •

Model I: the emission has a sharp peak at the innermost stable circular orbit
(ISCO) for time-like observers, while vanishing in the region internal to it
and falling off asymptotically to zero beyond of it. In the BB case, the ISCO
radius reads $x_{isco}=\sqrt{36M^{2}-a^{2}}$ (or $r_{isco}=6M$). Therefore we
model this emission profile as (taking $M=1$)

$I_{em}^{I}=\left\\{\begin{array}[]{c}\frac{1}{(x-(x_{isco}-1))^{2}}\hskip
2.84544pt\text{if}\hskip 2.84544ptx\geq x_{isco}\\\ 0\hskip
56.9055pt\text{if}\hskip 2.84544ptx<x_{isco}\end{array}\right.$ (9)

  * •

Model II: the emission has a sharp peak at the unstable circular orbit
location (7), having a qualitatively similar central and asymptotic behaviour
as Model I. This is described by

$I_{em}^{II}=\left\\{\begin{array}[]{c}\frac{1}{(x-(x_{pr}-1))^{3}}\hskip
2.84544pt\text{if}\hskip 2.84544ptx\geq x_{pr}\\\ 0\hskip
51.21504pt\text{if}\hskip 2.84544ptx<x_{pr}\end{array}\right.$ (10)

  * •

Model III: the emission starts at the event horizon and falls off more
smoothly to zero than in the previous two cases, being modelled as (in the WH
case we take instead the wormhole mouth, $x_{th}=0$)

$I_{em}^{III}=\left\\{\begin{array}[]{c}\frac{\pi/2-\arctan[x-5]}{\pi/2-\arctan[x_{hor}-5]}\hskip
2.84544pt\text{if}\hskip 2.84544ptx\geq x_{hor}\\\ 0\hskip
73.97733pt\text{if}\hskip 2.84544ptx<x_{hor}\end{array}\right.$ (11)

The observed intensity on the receiver’s screen is given by the
gravitationally red-shifted emitted density (disregarding effects associated
to absorption and reflection of light). Given the fact that $I_{\nu}/\nu^{3}$
is conserved along a photon trajectory, one finds that in the line element (1)
the observed intensity at a frequency $\nu^{\prime}$ scales with respect to
the emitted one as $I_{\nu^{\prime}}^{ob}=A^{3/2}(x)I(x)$. Therefore, the
total observed intensity will be the integration over the whole range of
received frequencies as $I^{ob}=\int I_{\nu^{\prime}}d\nu^{\prime}$ or, in
other words, $I^{ob}=A^{2}(x)I(x)$. In our ray-tracing setup developed in Sec.
III, whenever any light ray backtracked from the observer’s screen crosses the
accretion disk plane it will pick up additional brightness from it depending
on the number of orbits. Therefore, the total received luminosity will be the
sum of all the intensities from all these crossings with the accretion disk as

$I^{ob}(b)=\sum_{m}A^{2}I_{|_{x=x_{m}(b)}}\ ,$ (12)

where the so-called “transfer function” $x_{m}(b)$ contains the information
about the radius of the disk where a given light ray with impact parameter $b$
will have its $m^{th}$-intersection with the disk (in the coordinate $x$).
Moreover, its slope $dx/db$ defines the (de)magnification of the image for the
different types of emission (direct/lensed/photon ring). As it can be seen in
Fig. 4, the transfer function for the direct emission ($m=1$) has a constant
nearly unit slope, while the lensed ($m=2$) and photon ring ($m=3$) emissions
have quite a large slope, meaning that they are highly demagnified. Further
crossings with the disk will lead to exponentially demagnified images
Gralla:2019xty , so they can be safely ignored .

Model/$I^{ob}$ | BB parameter | $I_{Direct}$ | $I_{Lensed}$ | $I_{Photon}$
---|---|---|---|---
Model I | $a=0$ | $0.949$ | $0.0494$ | $0.00182$
$a=3/2$ | $0.942$ | $0.0549$ | $0.00264$
$a=5/2$ | $0.930$ | $0.0618$ | $0.00839$
Model II | $a=0$ | $0.901$ | $0.0940$ | $0.00412$
$a=3/2$ | $0.878$ | $0.1145$ | $0.00731$
$a=5/2$ | $0.798$ | $0.1760$ | $0.02589$
Model III | $a=0$ | $0.926$ | $0.0715$ | $0.00281$
$a=3/2$ | $0.908$ | $0.0871$ | $0.00506$
$a=5/2$ | $0.877$ | $0.1094$ | $0.01398$
Table 2: Contributions of the direct, lensed, and photon ring emissions
(including both $b\gtrless b_{c}$ trajectories) to the (normalized) total
emission as seen by the observer, for the Schwarzschild ($a=0$), BB BH
($a=3/2$) and WH ($a=5/2$) solutions.

Figure 5: The observational appearance of the BB solutions within accretion
disk Model I (9) with $a=0$ (Schwarzschild, top), $a=3/2$ (BH case, middle)
and $a=5/2$ (WH case, bottom), viewed from a face-on orientation. From left to
right one finds the emitted profile, the observed one, and the optical
appearance (in celestial coordinates) for a given BB solution. In the emission
profiles we have made use of the radial coordinate $x$, related to the radial
function as $r^{2}=x^{2}+a^{2}$ [recall Eq.(2)], which for the Schwarzschild
case reads simply as $r^{2}\approx x^{2}$. The observed profiles and the
optical appearance are plotted as functions of the impact parameter. In these
plots $x_{isco}=\sqrt{36M^{2}-a^{2}}$ is the radius of the innermost stable
circular orbit for time-like observers, at which the emission of this Model I
starts.

Figure 6: The observational appearance of the BB solutions within accretion
disk Model I (9) with $a=0$ (Schwarzschild, top), $a=3/2$ (BH case, middle)
and $a=5/2$ (WH case, bottom), viewed from a face-on orientation, and with a
similar notation as in Fig. 5. In these plots $x_{ph}=\sqrt{9M^{2}-a^{2}}$ is
the photon sphere radius, at which the emission of this Model II starts.

Figure 7: The observational appearance of the BB solutions within accretion
disk Model III with $a=0$ (Schwarzschild, top), $a=3/2$ (BH case, middle) and
$a=5/2$ (WH case, bottom), viewed from a face-on orientation, and with a
similar notation as in Fig. 5. In these plots $x_{h}=\sqrt{4M^{2}-a^{2}}$ is
the horizon radius, since in Model III the emission goes all the way down to
it. In the WH case (bottom panel) we have slightly displaced the beginning of
this emission from $x=0$ (its throat) for numerical convergence reasons.

.

We can now proceed to study the optical appearance of the different families
of BB solutions for the three models of emission above. To this end we depict
in Figs. 5, 6 and 7 the emitted intensity (left), the observed intensity
(middle), and the optical appearance (right) for each model of emission in the
Schwarzschild case ($a=0$) and in the two samples of BB BH ($a=3/2$) and WH
($a=5/2$) solutions. As expected, the emission mode largely determines the
qualitative shape of the optical appearance of the BB object.

In Model I, due to gravitational lensing in the observed intensity we clearly
see the two isolated spikes representing the photon ring and lensing
emissions, together with the more gradual decrease of the direct emission at
larger impact parameter, neatly separated from each other. Therefore, the main
contribution to the total luminosity in the optical appearance is provided by
the direct emission yielding a wide ring, while inner to it we find the
lensing ring and in the innermost region the barely visible to naked eye is
the photon ring. In Model II, the direct, lensed, and photon ring emissions
are overlapped in the observed intensity in a wider range of impact
parameters. There are two peaks, one corresponding to the beginning of the
direct emission which falls off until a superposition of the lensing and
photon ring at almost coincident impact factor produces the large spike in
this figure. However, the photon ring emission sharply falls off, quickly
followed by the lensing one, until the direct emission dominates again. The
net result is that in the optical appearance the lensing and photon rings are
superimposed with the direct emission. The lensing ring contribution can be
appreciated in this figure, though the one of the photon ring is highly
diluted and barely visible. In Model III the direct observed region in impact
parameter extends all the way down to the event horizon, increasing from there
and getting again contributions at larger impact factors from the spike in the
light ring first and in the lensing ring shortly after, before smoothly
falling off to zero. The optical appearance in this case shows a narrow but
somewhat brighter extended ring, made up of the contributions of the direct,
lensed and photon ring emission, though as usual the latter can be safely
ignored. This description of the optical appearances in these three models is
completely consistent with the features obtained in similar images from the
original description of the Schwarzschild black hole introduced in
Gralla:2019xty .

Moving forward to discussing the modifications of the BB solutions as compared
to the Schwarzschild one ($a=0$), we first verify in Table 2 that the
contributions of both $I_{Lensed}$ and $I_{Photon}$ to the total luminosity as
compared to the direct emission $I_{Direct}$ (including trajectories both
above and below the critical impact parameter), though obviously emission-
model-dependent, are significantly increased as $a$ grows. Indeed, when moving
from $a=0$ (Schwarzschild) to $a=3/2$ (BB BH), the $I_{Lensed}$ contribution
slightly rises for all the three models of the accretion disk, while those
enhances are much more noticeable in the contributions from $I_{Photon}$,
though still pretty much negligible as compared to $I_{Direct}$. These
increases are much more severe when moving to $a=5/2$ (WH branch), where the
contribution of $I_{Lensed}$ can be twofold the original one (in Model II).
Moreover, the contribution of the $I_{Photon}$ can be up to a factor $\sim 5$
in Models II and III. This is due to the broadening of the impact factor
region for both the lensed and photon trajectories in the WH case, as
discussed in the ray-tracing of Sec. III, and that can be also seen in wider
regions for the peaks of the observed intensities in the middle panels of
Figs. 5, 6 and 7.

Regarding the optical appearances (right panels), there are some tiny changes
(for the BH case) but moderate ones (for the WH case) in the widths and
intensities of the different light rings for all the emission models, which
are barely visible in these plots for the BH case as compared to the
Schwarzschild solution, but much more noticeable in the WH one, as expected.
This is particularly true for Model III, where the extended impact parameter
region for both the lensing and photon rings (clearly visible in the
corresponding observed luminosities) manifest as an additional boost of
luminosity right in the middle of the direct emission, such that the
combination of the lensing and photon ring contributions are now clearly
visible. Large enhances of the contribution of the photon ring to the total
luminosity have also been observed in certain models of compact objects with
“flattened” regions in the effective potential Gan:2021pwu .

##  V Conclusion and prospects

In this work we have considered the optical appearance (light rings and
shadows) of an uniparametric family of (spherically symmetric) extensions of
the Schwarzschild solution when illuminated by a thin accretion disk. Such a
black bounce family smoothly interpolates between the Schwarzschild solution
and two classes of solutions: regular black holes and traversable wormholes,
and therefore it allows to compare the shadows cast by conceptually different
objects on an equal footing. Moreover, this model has the additional advantage
of having the last unstable orbit located exactly at the same critical impact
parameter as in the original Schwarzschild solution for all the different BB
configurations.

Using the ray-tracing procedure, we have classified the different light
trajectories according to the number of orbits performed around the BB
solution, splitting them into three main contributions according to the number
of intersections with the equatorial plane: direct (one), lensed (two) and
photon ring (three). Though in purity at the critical impact parameter the
light ray would turn an infinite number of times, the subsequent contributions
will be so demagnified that they can be safely ignored regarding their
contributions to the total luminosity of the object. We found that, as the BB
parameter increases, the impact parameter regions for these three
contributions to the total luminosity are moderately enhanced, particularly in
the WH case due to the retro-orbits flowing from the throat region thanks to
the absence of an event horizon.

Next we considered a scenario of optically and geometrically thin accretion
disks as the main source of illumination of the BB solutions, using three
standard toy models whose emitted intensity peaks at the innermost stable
circular orbit for time-like observers, at the last unstable circular orbit
for photons, and near the horizon (in the BH case), or the throat (in the WH
case). These three models are chosen on the grounds that they simulate
different physical scenarios and yield qualitative different observed
emissions and their respective optical appearances for a given solution. The
main modifications induced by the BB solutions as compared to the original
Schwarzschild solution are an increase in the contributions of the lensed and
photon ring emissions as compared to the direct one, which in the WH case are
noticeable enough to be perceived at naked eye in some of the optical
appearances plots.

The results found in this paper, though pointing to some differences in the
shape of light rings and shadows of the BB solutions as compared to the
Schwarzschild one, are in agreement with the running discussion on the
community regarding the difficulty for testing hints of new Physics given the
many elements involved in this analysis. In the model presented here, one
would need to generalize it to include rotation Fran:2021pyi  in order to
study the deviations in the circularity of the shadow when getting close to
extremality rotation ratios. Moreover, the description of the accretion disks
could be improved from the optically and geometrically thin modelling to a
geometrically thick one, and the face-on orientation should be upgraded to
consider modest inclinations of the disks and their effects in the optical
appearances of the BB solution which, together with the addition of rotation,
may significantly modify the total luminosity Beckwith:2004ae  and, therefore,
the optical appearance. Finally, the presence of wormhole structures yields
interesting new possibilities, such as shadows from objects without accretion
disks due to contribution of those disks on the other side of the wormhole and
flowing through the wormhole throat, or the generalization of our analysis to
reflection-asymmetric wormholes since they produce effective potentials with
two maxima (i.e. two unstable circular orbits), which can be capable to yield
additional light rings Guerrero:2021pxt ; Peng:2021osd .

To conclude, whether the combination of all the above elements could be able
to lead to further enhances in the brightness of the photon ring region, now
including also additional contributions of the bands with $m>3$, is yet to be
seen. Given the promises of the observational teams working on achieving
better resolution for the optical appearance of black hole candidates in order
to test GR to better precision, combined with some new ideas to test the
photon sphere using interferometer John  or via correlated intensity
fluctuations Hadar , this field is ripe for the existing zoo of non-canonical
compact objects to extract observational discriminators with respect to GR
predictions.

## Acknowledgements

MG is funded by the predoctoral contract 2018-T1/TIC-10431 and acknowledges
further support by the European Regional Development Fund under the Dora Plus
scholarship grants. DRG is funded by the _Atracción de Talento Investigador_
programme of the Comunidad de Madrid (Spain) No. 2018-T1/TIC-10431, and
acknowledges further support from the Ministerio de Ciencia, Innovación y
Universidades (Spain) project No.
PID2019-108485GB-I00/AEI/10.13039/501100011033, and the FCT projects No.
PTDC/FIS-PAR/31938/2017 and PTDC/FIS-OUT/29048/2017. DS-CG is funded by the
University of Valladolid (Spain), Ref. POSTDOC UVA20. This work is supported
by the Spanish project FIS2017-84440-C2-1-P (MINECO/FEDER, EU), the project
PROMETEO/2020/079 (Generalitat Valenciana), and the Edital 006/2018 PRONEX
(FAPESQ-PB/CNPQ, Brazil, Grant 0015/2019). This article is based upon work
from COST Action CA18108, supported by COST (European Cooperation in Science
and Technology). All images included in this paper where obtained with
<EMAIL_ADDRESS>
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