# Astrophysical and electromagnetic emissivity properties of black holes
surrounded by a quintessence type exotic fluid in the Scalar-Vector-Tensor
Modified Gravity

Haidar Sheikhahmadi  Saheb Soroushfar  S. N. Sajadi  Tiberiu Harko

###### Abstract

The astrophysical consequences of the presence of a quintessence scalar field
on the evolution of the horizon and on the accretion disk surrounding a static
black hole, in the Scalar-Vector-Tensor version of Modified Gravity (MOG), are
investigated. The positions of the stable circular orbits of the massive test
particles, moving around the central object, are obtained from the extremum of
the effective potential. Detailed calculations are also presented to
investigate the light deflection, shadow and Shapiro effect for such a black
hole. The electromagnetic properties of the accretion disks that form around
such black holes are considered in detail. The energy flux and efficiency
parameter are estimated analytically and numerically. A comparison with the
disk properties in Schwarzschild geometry is also performed. The quantum
properties of the black hole are also considered, and the Hawking temperature
and the mass loss rate due to the Hawking radiation are considered. The
obtained results may lead to the possibility of direct astrophysical tests of
black hole type objects in modified gravity theories.

##  1 Introduction

According to the $\Lambda$ cold dark matter, $\Lambda$CDM, model, the material
content of the Universe mainly consists of two basic components, dark energy,
and dark matter, respectively [1, 2]. To explain the observed late-time cosmic
acceleration of the Universe [3, 4, 5], one can consider either a fundamental
cosmological constant $\Lambda$, which, for example, could be interpreted as
an intrinsic geometric property of the space-time, or a dark energy, an
ambiguous fluid component, of yet unknown physical origin, which spreads
throughout the whole Universe, and which would mimic a cosmological constant.
Currently, one of the main dark energy scenarios is based on the so-called
quintessence fields [6, 7, 8, 9]. In this model, the cosmological constant,
$\Lambda,$ is considered as a variable term, and it’s evolution and properties
can be explained by considering a scalar field with negative pressure,
representing the dark energy [10].

In cosmological studies the quintessence model faces some drawbacks for
providing a good concordance between theory and observations, especially in
local scales. These theoretical problems led to the introduction of some
extended scalar-tensor models, as chameleon, and Brans-Dicke chameleon models
[11, 12, 13, 14, 15, 16, 17]. Additionally, the explanation of the accelerated
expansion phase itself in a consistent quantum model of gravity is still a
challenge [18]. The origin of this drawback goes back to the so-called outer
horizon problem. In other word, this problem is related to the impossibility
of introducing an observable and physical S-matrix for describing the
asymptotic in and out states.

In the following we will not investigate the cosmological properties, with
their advantages and disadvantages, of the quintessence models. Instead of the
large scale cosmological studies we will consider the effects of the
quintessence fields on the behaviour of the compact objects, as, for instance,
the black holes.

Interestingly enough, in [18] it was shown that in the presence of the scalar
field and under some simple assumptions for the energy-momentum tensor, an
exact black hole solution for the Einstein equation can be obtained. One
interesting property of this solution is that when one considers the interval
$-1<\omega<-1/3$ for the equation of state parameter $\omega=p/\rho,$, then an
outer horizon is formed.

Before these investigations, in [19, 20] a solution based on a free
quintessence model was found, which leads to a bare singularity, with no hair
and no horizon, with the cosmic censorship conjecture not satisfied. Therefore
this solution does not represents a black hole solution. Recently, many
different classes of black holes models in the presence of quintessence fields
have been investigated. For the rotating charged black holes one can see[21].
For the black hole solutions originating from string theory we refer the
readers to [22]. The thermodynamics properties of the charged black holes in
the presence of quintessence have been investigated in [23]. In this work,
different conditions for the heat conductivity of the black holes’ three
horizons have been discussed in detail. Entropy calculations of the black
holes in the presence of quintessence can be found in [24]. For more details
on the black holes in the presence of quintessence fields see [25, 26, 27],
and references therein. It deserves to also note that there are some models of
black hole solutions that are able to remove the central singularity, namely,
the regular black holes, and the non-commutative black holes [33, 34, 35],
respectively.

Let us now briefly review the astrophysical and physical properties of the
accretion disks. Usually, in astrophysics the term accretion is associated to
the process of the attraction of matter, especially gaseous, by a central
massive object. When this ingoing process happens, gravitational energy, and
even momentum, are extracted from the central object. Obviously, the amount of
such energy should be proportional to the ratio $M/R$, where $M$ denotes the
mass, and $R$ denotes the radius of the central object, respectively.
Therefore it can be concluded that the more massive and compact the central
mass is, the larger amount of energy will flow in. Hence, from a
phenomenological point of view, the accretion into a black hole is one of the
important and interesting phenomena in the astrophysical context. Its
importance goes back to the fact that it indicates the transfer of
gravitational energy into radiation, and leads to an explanation of the
properties of the most luminous objects in the Universe, which recently have
been observed directly by the Event Horizon telescope, EHT [28]. Obviously,
the spherical symmetric accretion on a cental compact object is the simplest
case one can visualize. For the first time such a system was studied in [29],
and it is well-known as the Bondi accretion. Bondi did show that this solution
is unique, and consequently the flow becomes transonic crossing the sonic
radius. Observations have shown that accretion processes are present in many
astrophysical objects.

An important consequence of the accretion processes is that many compact
objects are surrounded by accretion disk. Astrophysical phenomena related to
accretion disks have been studied extensively. Here we give some simple
example of accretion disks. Usually, the protostellar clouds formed of a
molecular gas can form a protoplanetary disk around the host newly formed
star. As the second example, we can refer to the binary systems, in which mass
can be accreted to form an accretion disk due to the presence of strong
stellar winds.

The third possibility for the formation of an accretion disk is perhaps the
more interesting one, and recently it has been confirmed by direct
observations [28]. It is related to the super massive black holes in active
galactic nuclei, which can acquire mass by accreting matter from their
vicinity. It is important to emphasize that in almost all high luminosity
astrophysical objects there is a central mass surrounded by an accretion disk.
For a valuable discussion of the accretion disk properties of compact objects
we refer the reader to [30]. The study of the properties and of the physics of
the Newtonian accretion disks was initiated in the important and seminal paper
[31].

There are many different proposals advanced to explain the rotation curves of
galaxies, the dynamics of galaxy clusters, and other astrophysical and
cosmological phenomena. In one hand, as we have already mentioned above, the
dark matter model, for the first time hypothesized by Zwicky [32], has become
an essential part of the standard $\Lambda$CDM cosmological paradigm. On the
other hand, to explain the dark matter phenomenology, models based on modified
gravity have also been proposed. Fore example, some models, based on imposing
a modification in Newtonian version of gravity, the so called Modified
Newtonian dynamics, MOND [36] have been considered as alternatives to the
standard dark matter model.

Another proposal for a modified gravity theory is the scalar-vector-tensor
(S-V-T) modified gravity theory, or MOG [37]. This theoretical model
adequately explains Solar System observations [38], the rotation curves of
galaxies [39], the dynamics of galaxy clusters [40], and the Cosmic Microwave
Background predictions [41]. Recently, the physics of some black hole
solutions in the MOG version of gravity has been investigated, and it has been
shown that the well known exact solutions of general relativity can be
reproduced properly [42, 43]. For instance, although the Reissner-Nordtröm
solution in the standard version of gravity theory (general relativity)
corresponds to an electrically charged object, in MOG it can be considered as
describing electrically neutral objects. In the MOG theory, a gravitational
repulsive force, with an effective charge $Q=\sqrt{\alpha G_{N}}M$, can be
introduced. The parameter $\alpha=-1+G/G_{N}$ is known as the MOG parameter,
and it has to be determined from the observational constraints. The parameter
$G_{N}$ is Newton’s constant, and $G$ is the enhanced gravitational constant.
Moreover, by $M$ we have denoted the total mass of the central object, a black
hole, for example.

This physical solution for a MOG black hole motivated us to extend our
investigation to study the properties of the black holes in the presence of a
scalar field, that is, a quintessence field, and to consider the properties of
the compact objects in such an extension of the MOG theory. In the following
we will cal, this solution of the MOG theory as a QMOG black hole. More
exactly, we consider first the positions of the stable circular orbits of the
massive test particles, moving around the central MOG type black hole, which
are obtained from the extremum of the effective potential. Several important
astrophysical effects are investigated in detail, including the light
deflection, the shadow and the Shapiro effect, respectively. The
electromagnetic properties of the accretion disks are also analyzed in detail,
and the energy flux, the luminosity, and the efficiency parameter are obtained
by using both analytical and numerical methods. For each quantities we perform
a comparison with the disk properties in Schwarzschild geometry. The important
quantum properties of the MOG black holes are analyzed, and the Hawking
temperature, and the mass loss rate due to the Hawking radiation is obtained.
The results presented in this study may lead to the possibility of the
astrophysical tests of the black hole type objects in modified gravity
theories, and of the modified gravity theories themselves.

This work is organised as follows. In Sec. 2, we write down the Reissner-
Nordström type black hole solution in the presence of a quintessence field in
the S-V-T theory. Moreover, the dynamical properties of the motion of the
massive particles in this geometry (effective potential, positions of the
event horizon, and the radii of the marginally stable orbits) are analyzed in
detail The astrophysical properties of the QMOG black holes, like the
deflection of light, the shadow of the black holes, and the Shapiro effects
are studied in Section 3. Furthermore, in Sec.(4), the flux, temperature and
the efficiency of the accretion disks that form around QMOG black hole are
obtained both analytically, and numerically. Finally, in Sec. 6 we discuss and
conclude our results.

##  2 Kiselev type S-V-T black hole in the presence of an exotic fluid

In the following Section we will first write down the metric of a static black
hole in the S-V-T (MOG) version of gravity in the presence of a quintessence
scalar field, aiming at investigating the implications of this modified
gravity theory on the behaviour and properties of the massive astrophysical
objects. Moreover, the horizon properties and the equations of motion of the
massive test particles are analyzed in detail. In the next step, the dynamical
properties of this compact, black hole type object, will be analyzed as well.

###  2.1 General properties of the metric, and the structure of the horizons

In this Section, we consider a static and spherically symmetric black hole,
surrounded by a quintessence scalar field, in the MOG theory. Generally, the
geometry of the static, spherically symmetric black hole is given by [18, 42,
43, 44, 45, 46]

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}).$ (2.1)

In the following we consider a Kiselev type black hole in MOG, with the metric
function given by

$f(r)=1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{\gamma G}{r^{3\epsilon+1}},$
(2.2)

where we have denoted $G=G_{N}(1+\alpha)$, and $M$ is the black hole mass,
$Q=\sqrt{\alpha G_{N}}M$ is the effective (non-electric) charge, while
$\alpha$ is the MOG parameter, $\gamma$ is the quintessence parameter, and
$\epsilon$ is the equation of state parameter of the quintessence field. The
allowed values for $\epsilon$ are in the range $-1<\epsilon<-1/3$. In the
following we will call the black hole described by the metric (2.2) as the
QMOG black hole.

In the case of $\epsilon=-1$, the metric (2.2) reduces to the general
relativistic black hole space-times, in the presence of a cosmological
constant, i.e., to the anti de-Sitter Schwarzschild geometry (for more details
see [42]). Moreover, in the limit of $\gamma=0$, the metric reduces to an
effective Reissner–Nordström type black hole, with a non-electric charge. For
$\gamma=0$, and $Q=0,$, it reduces to the well-known Schwarzschild black hole.
For the case $\epsilon=-2/3$, Eq. (2.1) describes an effective
Reissner–Nordström-like black hole, surrounded by quintessence field in the
S-V-T (MOG) version of gravity theories. In Fig. 1 the behaviour of the metric
coefficient is illustrated, for various numerical values of the metric
parameters.

Figure 1: The plots of $f(r)$ for $M=1,\epsilon=-2/3,G_{N}=1,\alpha=0.5$ and
$\gamma={\color[rgb]{0,0,1}0.1},{\color[rgb]{1,0,0}0.045},0.058$.

It deserves here to note that Moffat [42] obtained his MOG version of the
black holes by neglecting the energy momentum tensor of the matter sector,
i.e., $T_{\mu\nu}^{m}=0$. Following [18], and by introducing the matter
sector, as a quintessence scalar field, the metric (2.1) obtained accordingly.
To obtain the metric one must decompose the energy momentum tensor, in the
Einstein field equation $G_{\mu\nu}=-8\pi T_{\mu\nu}$, in two distinct
components, i.e.,

$T_{\mu\nu}=T_{\mu\nu}^{q}+T_{\mu\nu}^{V}\,,$ (2.3)

where $T_{\mu\nu}^{q}$ stands for the quintessence part [18], and
$T_{\mu\nu}^{V}$ describes the vector field [42]. For the derivation of the
metric one can find more details in [47, 48, 49] as well.

It is important here to note that the Kiselev black holes can be interpreted
as a solution of the Einstein gravitational field equations with the matter
energy-momentum tensor given by [50],

$T_{t}^{t}=T_{r}^{r}=\rho(r),$ (2.4)

and

$T_{\theta}^{\theta}=T_{\phi}^{\phi}=-\frac{1}{2}\left(1+3\varepsilon\right)\rho,$
(2.5)

where $\varepsilon$ is the parameter of the equation of state. By taking the
isotropic average over the angles of the components of the energy-momentum
tensor we obtain the barotropic equation of state $p=\varepsilon\rho$. Hence,
the interpretation of the Kiselev black hole as describing a black hole
solution in the presence of a quintessence field is problematic, and this
solution may be better interpreted as describing a black hole in the presence
of an exotic matter source.

###  2.2 Horizons of the QMOG black hole

The horizon of the spacetime given by the metric (2.1) can be determined by
setting the condition $g_{00}(r)=0$, i.e.,

$1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{G\gamma}{r^{3\epsilon+1}}=0.$
(2.6)

Consequently, we obtain for the position of the horizon the algebraic equation

$r^{2}-2GMr+GQ^{2}-G\gamma r^{1-3\epsilon}=0.$ (2.7)

For our purposes, and for mathematical convenience, we introduce the following
dimensionless parameters, defined as,

$\displaystyle
r_{\star}=\frac{\tilde{r}}{(1+\alpha)}=\frac{r}{(1+\alpha)G_{N}M},$ (2.8)
$\displaystyle
Q_{\star}=\frac{\tilde{Q}}{\sqrt{1+\alpha}}=\frac{Q}{\sqrt{G_{N}(1+\alpha)}M},$
$\displaystyle~{}\gamma_{\star}=(1+\alpha)^{2}{\tilde{\gamma}}=(1+\alpha)^{2}G_{N}^{2}\gamma
M\,.$

For $\epsilon=-2/3$, and by rearranging the parameters, Eq.(2.7),
interestingly, takes the following form,

$\gamma_{\star}{r}_{\star}^{3}-{r}_{\star}^{2}+2{r}_{\star}-{{Q}_{\star}^{2}}=0~{}.$
(2.9)

Another interesting case corresponds to $\epsilon=-\frac{1}{2}$. We have to
note that for this value of $\epsilon$, the dimensionless parameter
$\tilde{\gamma}$ takes the form $\tilde{\gamma}=G_{N}^{3}\gamma^{2}M$.
Accordingly, Eq.(2.7) can be rearranged as follows,

$\sqrt{\gamma_{\star}}{r}_{\star}^{\frac{5}{2}}-{r}_{\star}^{2}+2{r}_{\star}-{{Q}_{\star}^{2}}=0.$
(2.10)

In the following we would like to investigate the effects of the MOG parameter
$\alpha$ on the properties of the black holes. To this end, by taking into
account the definitions of the effective charge, and of the enhanced
gravitational constant, one can rewrite Eqs.(2.9) and (2.10), respectively, in
the alternative forms, namely,

$\tilde{\gamma}\tilde{r}^{3}-\frac{\tilde{r}^{2}}{(1+\alpha)}+2\tilde{r}-{\alpha}=0~{}\,,$
(2.11)

and

$\sqrt{\tilde{\gamma}(1+\alpha)}\tilde{r}^{\frac{5}{2}}-{\tilde{r}^{2}}+2(1+\alpha){{\tilde{r}}}-{\alpha(1+\alpha)}=0~{},$
(2.12)

respectively.

It is interesting to notice that there is another possibility of extending the
range of $\epsilon$ by also considering the case $\epsilon>0$. From the
asymptotically flat behaviour of such a configuration it can be concluded
that, obviously, it would not obey the properties required for a quintessence
field anymore. To make these things clearer, as an example, let’s consider the
value $\epsilon=2/3$. Then, the dimensionless equation (2.7) can be
reformulated as,

$\gamma_{\star}{r}_{\star}^{-3}-{{Q}_{\star}^{2}}{r}_{\star}^{-2}+2{r}_{\star}^{-1}-1=0~{},$
(2.13)

where, for the positive values of $\epsilon$, i.e., $\epsilon=2/3$, one has
$\bar{\gamma}=G_{N}^{2}\gamma M^{3}$. Now, from Eq.(2.8), and from the
definition of $Q$, one can rewrite Eq.(2.13) as follows,

$\bar{\gamma}(1+\alpha)^{3}\tilde{r}^{-3}-\frac{\alpha}{1+\alpha}{\tilde{r}^{-2}}+\frac{2}{1+\alpha}\tilde{r}^{-1}-\frac{1}{(1+\alpha)^{2}}=0~{}.$
(2.14)

We will return to this matter, and discuss it in detail, in the next Sections.

###  2.3 Equations of motion, and the effective potential

In this Section, to investigate the physical properties of the MOG black hole
in the presence of a quintessence field (or an exotic matter source), we are
going to study first the equations of motion of a massive test particle, and
to derive the effective potential as well. It is well known that for a point,
massive test particle, the Lagrangian $\mathcal{L}$ of the motion in the
spacetime given by Eq.(2.1), can be written as,

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}\,,$
(2.15)

Accordingly, in the equatorial plane, the conserved energy $E$ and the angular
momentum $L$ can be obtained according to,

$E=g_{tt}\frac{dt}{ds}=(1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{G\gamma}{r^{3\epsilon+1}})\frac{dt}{ds}\,,$
(2.16)

and

$L=g_{\varphi\varphi}\frac{d\varphi}{ds}=r^{2}\frac{d\varphi}{ds}\,,$ (2.17)

respectively. Plugging the above equations all together, the geodesic
equations are expressed in the form,

$\left(\frac{dr}{ds}\right)^{2}=E^{2}-\Bigg{(}1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{G\gamma}{r^{3\epsilon+1}}\Bigg{)}\left(1+\frac{L^{2}}{r^{2}}\right)\,,$
(2.18)
$\displaystyle\left(\frac{dr}{d\varphi}\right)^{2}=\frac{r^{4}}{L^{2}}\Bigg{[}E^{2}-\left(1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{G\gamma}{r^{3\epsilon+1}}\right)\left(1+\frac{L^{2}}{r^{2}}\right)\Bigg{]}\,,$
(2.19)
$\displaystyle\left(\frac{dr}{dt}\right)^{2}=\frac{1}{E^{2}}\Bigg{(}1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{G\gamma}{r^{3\epsilon+1}}\Bigg{)}^{2}\times$
(2.20)
$\displaystyle\Bigg{[}E^{2}-\left(1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{G\gamma}{r^{3\epsilon+1}}\right)(1+\frac{L^{2}}{r^{2}})\Bigg{]}\,.$

See [51, 52, 55, 54, 53, 56] for more details on the derivation of these
equations.

Eqs.(2.18)-(2.20) give a complete description of the dynamics of the massive
test particles moving around the MOG black hole. By considering Eq.(2.18), we
can define an effective potential of the motion as,

$V_{eff}=\Big{(}1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{G\gamma}{r^{3\epsilon+1}}\Big{)}\left(1+\frac{L^{2}}{r^{2}}\right).$
(2.21)

These definition allows us to compare our results with the standard form of
the equations of motion in other geometries, and black hole solutions.

Now we are prepared to discuss the properties of the accretion disks that form
around the central mass. We will begin this discussion in the next Subsection
by considering the properties of the event horizons, and of the stable
circular orbits for the QMOG black holes.

###  2.4 Event horizons, and stable circular orbits of the QMOG black hole

The accretion disks usually form through a simple astrophysical mechanism. In
an accretion disk, hot gas particles, carrying an electric charge, are moving
in stable circular orbits around the central compact object, which in the
present case we consider to be a black hole. For the case $\epsilon=-2/3$, the
specific energy $\tilde{E}$, the specific angular momentum $\tilde{L}$, and
the angular velocity $\tilde{\Omega}$, of the particles that move in a
circular orbit can be written as,

$\tilde{E}=\dfrac{\tilde{g}_{tt}}{\sqrt{\tilde{g}_{tt}-\tilde{g}_{\phi\phi}\tilde{\Omega}^{2}}}=\frac{\sqrt{2}\Bigg{[}\alpha(\alpha+1)-(\alpha+1)\tilde{\gamma}\tilde{r}^{3}+\tilde{r}^{2}-2(\alpha+1)\tilde{r}\Bigg{]}}{(\alpha+1)\tilde{r}^{2}\sqrt{\frac{2}{\alpha+1}-\frac{6\tilde{r}-4\alpha}{\tilde{r}^{2}}-\tilde{\gamma}\tilde{r}}},$
(2.22)
$\tilde{L}=\dfrac{\tilde{g}_{\phi\phi}{\tilde{\Omega}}}{\sqrt{\tilde{g}_{tt}-\tilde{g}_{\phi\phi}\tilde{\Omega}^{2}}}=\frac{\sqrt{{-2\alpha-\gamma\tilde{r}^{3}+2\tilde{r}}}}{\sqrt{\frac{2}{\alpha+1}-\frac{6\tilde{r}-4\alpha}{\tilde{r}^{2}}-\tilde{\gamma}\tilde{r}}},$
(2.23)

and,

$\tilde{\Omega}=\sqrt{\dfrac{\tilde{g}_{tt,\tilde{r}}}{\tilde{g}_{\phi\phi,\tilde{r}}}}=\frac{\sqrt{-2\alpha-\tilde{\gamma}\tilde{r}^{3}+2\tilde{r}}}{\sqrt{2}\tilde{r}^{2}},$
(2.24)

respectively.

Using Eqs.(2.8), (2.21) and (2.23), for the effective potential, and its
derivatives, we obtain,

$\tilde{V}_{eff}=\dfrac{1}{\tilde{r}^{2}}\Bigg{[}-\tilde{\gamma}\tilde{r}^{3}+\frac{\tilde{r}^{2}}{(1+\alpha)}-2\tilde{r}+{\alpha}\Bigg{]}\left(1+\frac{\tilde{L}^{2}}{\tilde{r}^{2}}\right),$
(2.25)
$\begin{array}[]{l}\dfrac{d\tilde{V}_{eff}}{d\tilde{r}}=-\frac{1}{{(1+\alpha){r^{5}}}}\bigg{[}-{\tilde{L}^{2}}\alpha\,\tilde{\gamma}\,{r^{3}}+\alpha\,\tilde{\gamma}{r^{5}}-{\tilde{L}^{2}}\tilde{\gamma}{r^{3}}+\tilde{\gamma}{r^{5}}+4{\tilde{L}^{2}}{\alpha^{2}}\\\
\quad\quad~{}~{}~{}~{}~{}~{}-6{L^{2}}\alpha
r+2{\tilde{L}^{2}}{r^{2}}+2{\alpha^{2}}{r^{2}}-2\alpha{r^{3}}+4{\tilde{L}^{2}}\alpha-6{\tilde{L}^{2}}r+2\alpha{r^{2}}-2{r^{3}}\bigg{]}\,,\end{array}$
(2.26)

and

$\displaystyle\begin{array}[]{l}\dfrac{d^{2}\tilde{V}_{eff}}{d\tilde{r}^{2}}=\frac{2}{{\big{(}1+\alpha\big{)}{{\tilde{r}}^{6}}}}\times\Bigg{[}-{{\tilde{L}}^{2}}\tilde{\gamma}{\tilde{r}^{3}}(\alpha+1)+10{{\tilde{L}}^{2}}{\alpha^{2}}-12{{\tilde{L}}^{2}}\alpha\tilde{r}+\\\
\quad\quad~{}~{}~{}~{}~{}~{}3{{\tilde{L}}^{2}}{{\tilde{r}}^{2}}+3{(\alpha\tilde{r})^{2}}-2\alpha{{\tilde{r}}^{3}}+10{{\tilde{L}}^{2}}\alpha-12{{\tilde{L}}^{2}}\tilde{r}+2\alpha{{\tilde{r}}^{2}}-2{{\tilde{r}}^{3}}\Bigg{]}\,.\end{array}\quad$
(2.29)

For more details on the derivation of the above equations one can see [68, 69,
71, 70, 73, 74]. It should be emphasised here that the equation
$d^{2}\tilde{V}_{eff}/d\tilde{r}^{2}=0$ is employed as a necessary condition
for the existence of innermost stable circular orbits. In addition to this,
the sign of $d^{2}\tilde{V}_{eff}/d\tilde{r}^{2}$, usually is a criterion to
show the stability of the orbits of the black hole.

By employing all these above mentioned conditions, the location of the event
horizons, and of the stable circular orbits, for a S-V-T (QMOG) black hole,
surrounded by a quintessence scalar field, or an exotic matter source, are
presented in Table 1, and Table 2, respectively.

In Table 1, we did investigate the role of the quintessence parameter
$\tilde{\gamma}$ on the black hole properties, when the MOG parameter $\alpha$
is fixed at $0$ and $0.25$. Then, in Table 2, we considered the effects of
changes in the parameter $\alpha$, when the parameter $\tilde{\gamma}$ takes
the values $0$ and $0.003$.

In addition to the points mentioned above, the location of the observer is
also important. Usually, the observer is considered to be at the infinite
boundary for the asymptotically flat spacetimes. However, for the model of
quintessence black hole, the cosmological horizon does matter. Physically, the
observer should be located in the domain of outer communication, which is
between the event horizon and cosmological horizon, similar to the de Sitter
case. For convenience, the observer in our paper is set to be near the
cosmological horizon.

To visualize the effects of the MOG parameter we present the Fig. 2, in which
the parameters $\tilde{\alpha}$ and $\tilde{\gamma}$ are plotted as a function
of the radius of the event horizon, for different values of $\tilde{Q}$ and
$\tilde{L}$. It is clear from Fig. 2 that as long as the values of $\tilde{Q}$
increase, the peak values of $\tilde{\gamma}$ become larger. Moreover,
considering Fig. 2, it can be seen that the increase in the values of
$\tilde{\gamma}$ leads to a decrease of the effective potential. The location
of the innermost stable circular orbits is represented, for different values
of the parameters of the black hole solution, in Fig. 3.

|  $\gamma$ |  $r_{-}$ |  $r_{+}$ |  $r_{c}$ |  $r_{ISCO}$ |  $r_{S}$ |  $r_{OSCO}$
---|---|---|---|---|---|---|---
$\alpha=0$ |  0 |  $\times$ |  2 |  $\times$ |  6 |  $\times$ |  $\times$
|  0 |  0.1320 |  2.368 |  $\times$ |  7.1106 |  $\times$ |  $\times$
$\alpha=0.25$ |  0.001 |  0.1319 |  2.3755 |  797.49 |  7.3186 |  24.29 |  2394.7
|  0.003 |  0.1319 |  2.3907 |  264.14 |  8.0783 |  12.542 |  794.6
|  0.004 |  0.1319 |  2.3984 |  197.469 |  493.59 |  $\times$ |  $\times$
Table 1: Location of the event horizons, and of the stable circular orbits in a S-V-T (MOG) black hole for $\alpha=0,~{}0.25$, when the quintessence parameter $\tilde{\gamma}$ takes different values, including $\tilde{\gamma}=0$ to $\tilde{\gamma}=0.004$. Here for the left side of the table one notice $r_{-}$ stands for Cauchy radius, $r_{+}$ refers event horizon and $r_{c}$ is cosmological horizon. Also considering the right panel $r_{ISCO}$ stands for ISCO radius, $r_{S}$ refers stable radii and $r_{OSCO}$ refers the most outer stable circular orbit.  |  $\alpha$ |  $r_{-}$ |  $r_{+}$ |  $r_{c}$ |  $r_{ISCO}$ |  $r_{S}$ |  $r_{OSCO}$
---|---|---|---|---|---|---|---
$\tilde{\gamma}$=0 |  0 |  $\times$ |  2 |  $\times$ |  6 |  $\times$ |  $\times$
|  0 |  $\times$ |  2.012 |  331.3 |  6.451 |  13.34 |  995.7
|  0.25 |  0.1319 |  2.3907 |  264.14 |  8.078 |  12.542 |  794.68
$\tilde{\gamma}$=0.003 |  0.5 |  0.1340 |  1.877 |  331.3 |  5.961 |  13.47 |  995.8
|  0.5 |  0.2751 |  2.7628 |  219.18 |  660.27 |  $\times$ |  $\times$
|  0.75 |  0.4269 |  3.1325 |  186.91 |  563.952 |  $\times$ |  $\times$
Table 2: The location of the event horizons, and of the stable circular orbits
in a S-V-T (MOG) black hole for $\tilde{\gamma}=0,~{}0.003$, and for different
values of the MOG parameter $\alpha$, indicated in the Table.

Figure 2: Right panel: the variation of the quintessence parameter
$\tilde{\gamma}$ as a function of the radial coordinate of the event horizon,
$\tilde{r}$, for different values of the MOG parameter $\alpha$. When
$\alpha=0$, the model reduces to the Schwarzschild black hole in the presence
of the quintessence field. Left panel: The variation of the effective
potential for the parameter $\alpha=0.25$, and for different values of
$\tilde{\gamma}$ and $\tilde{L}$. The red dots indicate the location of the
innermost stable circular orbits, ISCO.

As one can see from Fig. 2, the maximum value of the parameter
$\tilde{\gamma}$ corresponds to $\tilde{r}\simeq 8$. By increasing the value
of $\alpha$, the maximum value of the parameter $\tilde{\gamma}$ decreases,
and the effects of the maximum values of the scalar field do appear far from
the MOG black holes.

Figure 3: The innermost stable circular orbit (solid line) and the angular
momentum of the inflection point (dashed line) in terms of the coupling
constant of theory $\alpha$ ($\tilde{\gamma}=0$) (left panel) and
$\tilde{\gamma}$ ($\alpha=0$) (right panel).

##  3 Astrophysical properties of the QMOG black holes-deflection of light,
shadow and the Shapiro delay

In the present Section, for the sake of completeness, we will investigate in
detail some of the basic astrophysical tests for the QMOG black hole. In
particular, we will consider the deflection of light, the shadow of the black
hole, and the Shapiro time delay effect.

###  3.1 The deflection of light

The deflection angle of a photon as it moves from infinity to $r_{m}$ and off
to infinity for the metric (2.1) can be expressed as [76]-[78],

$\delta\varphi=\int_{r_{m}}^{\infty}{\dfrac{2dr}{\sqrt{\dfrac{r^{4}}{b^{2}}-f(r)r^{2}}}}-\bar{\pi}=I-\bar{\pi},$
(3.1)

where $b=\sqrt{r_{m}^{2}/f(r_{m})}$ is the impact parameter of the null ray,
and $r_{m}$ is the coordinate distance of the closest approach. Here
$\bar{\pi}$ is the change in the angle $\varphi$ for the straight line motion,
and is therefore subtracted out from the total deflection angle.

We now calculate the integral in Eq. (3.1). Writing the term in the
denominator of (3.1) as $f(r)r^{2}\left(r^{2}/b^{2}f(r)-1\right)$, for $M\ll
r$ one has

$\displaystyle\frac{f\left(r_{m}\right)}{f(r)}\frac{r^{2}}{r_{m}^{2}}-1=\left[\frac{1-\frac{2GM}{r_{m}}+\frac{GQ^{2}}{r_{m}^{2}}-\frac{G\gamma}{r_{m}^{3\epsilon+1}}}{1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\frac{G\gamma}{r^{3\epsilon+1}}}\right]\left(\frac{r^{2}}{r_{m}^{2}}\right)-1\,$
(3.2)
$\displaystyle=\left(\frac{r}{r_{m}}\right)^{2}\left[1+2GM\left(\frac{1}{r}-\frac{1}{r_{m}}\right)-GQ^{2}\left(\frac{1}{r^{2}}-\frac{1}{r_{m}^{2}}\right)+G\gamma\left(\frac{1}{r^{3\epsilon+1}}-\frac{1}{r_{m}^{3\epsilon+1}}\right)\right]-1\,$
$\displaystyle=\left(\frac{r^{2}}{r_{m}^{2}}-1\right)\left[1-\frac{2GMr}{r_{m}\left(r+r_{m}\right)}+\frac{GQ^{2}}{r_{m}^{2}}+\frac{G\gamma
r^{2}}{r^{2}-r_{m}^{2}}\left(\frac{1}{r^{3\epsilon+1}-r_{m}^{3\epsilon+1}}\right)\right]\,.$

Upon expanding in powers of $M/r$, $M/r_{m}$, and $G\gamma$, the integrand in
Eq. (3.1) then becomes,

$\displaystyle\int_{r_{m}}^{\infty}\dfrac{2dr}{\sqrt{\dfrac{r^{4}}{b^{2}}-f(r)r^{2}}}=\int\limits_{{r_{m}}}^{\infty}{\frac{1}{{\sqrt{\left({\frac{1}{{r_{m}^{2}}}-\frac{1}{{{r^{2}}}}}\right)}}}}\Bigg{\\{}1+\frac{{GM}}{r}\left[{1+\frac{{{r^{2}}}}{{{r_{m}}\left({r+{r_{m}}}\right)}}}\right]-\frac{{G{Q^{2}}}}{2}\left({\frac{1}{{{r^{2}}}}+\frac{1}{{r_{m}^{2}}}}\right)$
$\displaystyle+\frac{{G\gamma}}{2}\left[{\frac{1}{{{r^{3+1}}}}-\frac{{{r^{2}}}}{{2\left({{r^{2}}-r_{m}^{2}}\right)}}\left({\frac{1}{{{r^{3+1}}}}-\frac{1}{{r_{m}^{3+1}}}}\right)}\right]\Bigg{\\}}\frac{{dr}}{{{r^{2}}}}\,.$

After making the substitution $\sin(\theta)=r_{m}/r$, the integral can be
expressed as follows,

$\displaystyle\int_{r_{m}}^{\infty}\dfrac{2dr}{\sqrt{\dfrac{r^{4}}{b^{2}}-f(r)r^{2}}}=\int_{0}^{\frac{\pi}{2}}d\theta\Bigg{\\{}1+\frac{GM}{r_{m}}\Bigg{(}\sin(\theta)+\frac{1}{1+\sin(\theta)}\Bigg{)}-\frac{GQ^{2}}{2r_{m}^{2}}\left(1+\sin^{2}(\theta)\right)$
(3.3)
$\displaystyle+\frac{G\gamma}{2r_{m}^{3\epsilon+1}}\left[\sin^{3\epsilon+1}(\theta)-\frac{1}{2\cos^{2}(\theta)}\big{\\{}\sin^{3\epsilon+1}(\theta)-1\big{\\}}\right]\Bigg{\\}}$
$\displaystyle=\frac{\pi}{2}+\frac{2GM}{r_{m}}-\frac{3\pi
GQ^{2}}{8r_{m}^{2}}+\frac{G\gamma}{4r_{m}^{3\epsilon+1}}\int_{0}^{\frac{\pi}{2}}\frac{\cos(2\theta)\sin^{3\epsilon+1}(\theta)+1}{\cos^{2}(\theta)}d\theta\,.$

Gathering together all the results above, the deflection angle is obtained in
the form,

$\Delta\varphi=\dfrac{4GM}{r_{m}}-\dfrac{3\pi
GQ^{2}}{4r_{m}^{2}}+\dfrac{G\gamma}{2r_{m}^{3\epsilon+1}}\int_{0}^{\frac{\pi}{2}}\dfrac{\cos(2\theta)\sin^{3\epsilon+1}(\theta)+1}{\cos^{2}(\theta)}d\theta\,,$
(3.4)

a relation that is valid for large $r$ ($r\rightarrow\infty$).

###  3.2 The shadow of the QMOG black hole

Now we are in the position to consider the shadow of QMOG classes of black
holes [58, 57, 79]. In fact, to obtain the shadow of the black hole we follow
up the null geodesics, on which the motion of the photons take place.

The angular radius of the shadow of the black hole, as seen by an observer
located at $r_{0}$, is defined as [80],

$\sin^{2}(\Gamma)=\dfrac{r_{ph}^{2}f(r_{0})}{r_{0}^{2}f(r_{ph})}\,,$ (3.5)

where for small values of $\Gamma$, $\sin(\Gamma)\approx\Gamma$, $r_{ph}$
denotes the radius of the photon sphere, and $\Gamma$ is the angle subtended
by the radius of the shadow, as seen by a typical observer located at $r_{0}$.

By using the expression of $f(r)$, as introduced in the metric (2.1), one
gets,

$\displaystyle\dfrac{r_{ph}^{2}f(r_{0})}{r_{0}^{2}f(r_{ph})}$
$\displaystyle=\dfrac{r_{ph}^{2}}{r_{0}^{2}}\left[\dfrac{1-\dfrac{2GM}{r_{0}}+\dfrac{GQ^{2}}{r_{0}^{2}}-\dfrac{G\gamma}{r_{0}^{3\epsilon+1}}}{1-\dfrac{2GM}{r_{ph}}+\dfrac{GQ^{2}}{r_{ph}^{2}}-\dfrac{G\gamma}{r_{ph}^{3\epsilon+1}}}\right]$
$\displaystyle=\dfrac{r_{ph}^{2}}{r_{0}^{2}}\left[\left(1-\dfrac{2GM}{r_{0}}+\dfrac{GQ^{2}}{r_{0}^{2}}-\dfrac{G\gamma}{r_{0}^{3\epsilon+1}}\right)\left(1+\dfrac{2GM}{r_{ph}}-\dfrac{GQ^{2}}{r_{ph}^{2}}+\dfrac{G\gamma}{r_{ph}^{3\epsilon+1}}\right)\right]$
$\displaystyle=\dfrac{r_{ph}^{2}}{r_{0}^{2}}\left[1+2GM\left(\dfrac{1}{r_{ph}}-\dfrac{1}{r_{0}}\right)-GQ^{2}\left(\dfrac{1}{r_{ph}^{2}}-\dfrac{1}{r_{0}^{2}}\right)+G\gamma\left(\dfrac{1}{r_{ph}^{3\epsilon+1}}-\dfrac{1}{r_{0}^{3\epsilon+1}}\right)\right],$

yielding in turn

$\sin(\Gamma)=\dfrac{r_{ph}}{r_{0}}+\dfrac{GM(r_{0}-r_{ph})}{r_{0}^{2}}-GQ^{2}\dfrac{(r_{0}^{2}-r_{ph}^{2})}{2r_{0}^{2}r_{ph}^{2}}+\dfrac{G\gamma}{2}\left(\dfrac{1}{r_{ph}^{3\epsilon+1}}-\dfrac{1}{r_{0}^{3\epsilon+1}}\right)\,,$
(3.6)

To get the shape of shadow, we obtain first the unstable circular orbits of
the null geodesics. They are determined by the equations,

$V_{eff}|_{r_{ph}}=V_{eff}^{\prime}|_{r_{ph}}=0\,.$ (3.7)

The impact parameters are now related as,

$\dfrac{f(r_{ph})}{r_{ph}^{2}}\left(\kappa+L^{2}\right)-E^{2}=0,\;\;\;\;\dfrac{r_{ph}f^{\prime}-2f(r_{ph})}{r_{ph}^{3}}(\kappa+L^{2})=0,$
(3.8)

where $\kappa$ is the Carter constant, with $r_{ph}$ obtained as a solution of
the constraint equation,

$r_{ph}f^{\prime}(r_{ph})-2f(r_{ph})=0.$ (3.9)

Defining the impact parameters $\xi$ and $\eta$, which are functions of the
energy $E$ and of the angular momentum $L$, and of the Carter constant
$\kappa$, as [81]

$\xi:=\dfrac{L}{E}\,\,\,\,\eta:=\frac{\kappa}{E^{2}}\,$ (3.10)

and using the conserved quantity $E$ one can plot the shadow of black holes
satisfactorily

Figure 4: The shadow of the QMOG black hole for
$M=1,\epsilon=-2/3,G_{N}=1,\alpha=0.5$ and
$\gamma={0.045},{\color[rgb]{1,0,0}0.04},0.035,0.03,0.025$ (left panel), and
for $\gamma=0.04$ and $\alpha=0.1,{\color[rgb]{1,0,0}0.2},0.3,0.4,0.5,0.6$
(Middle panel), and for $\alpha=0.1,\gamma=0.04$ and
$M=1,{\color[rgb]{1,0,0}1.5},2$ (right panel). Here the dashed line presents
the shadow of the pure Schwarzschild solution with $\alpha=\gamma=0$.

The shape of the shadow of the QMOG black hole is presented in Fig. 4. Since,
the black hole is static, the shadow is circular. As reported in [82], for the
$M87$ galaxy, the angular diameter of the shadow of the central black hole is
$\theta_{M87}=42\pm 3\mu as$ as, the distance of the $M87$ from the Earth is
$D=16.8Mpc$, and the mass of the $M87$ is $M_{M87}=6.5\pm 0.9\times
10^{9}M_{\odot}$. Similarly, for Sagittarius A*, the observational data are
provided in the recent EHT paper [83]. The angular diameter of the shadow is
$\theta_{Sgr.A*}=48.7\pm 7\mu as$ as (EHT), the distance of the Sgr. A* from
the Earth is $D=8277\pm 33pc$, and mass of the black hole is
$M_{Sgr.A*}=4.3\pm 0.013\times 10^{6}M_{\odot}$ [83].

Now, once we have the above data about the black holes, we can calculate the
diameter of the shadow size in units of mass, by using the following
expression,

$d_{sh}=\dfrac{D\theta}{M},$ (3.11)

Hence, the theoretical shadow diameter, however, can be obtained via
$d^{\theta}_{sh}=2R_{sh}$. Therefore, by using the above expression, we obtain
the diameter of the shadow image of $M87$ $d^{M87}_{sh}=(11\pm 1.5)M$, and for
Sgr. A* $d^{Sgr.A*}_{sh}=(9.5\pm 1.4)M$. The variation of the diameter of the
shadow image with coupling parameter $\alpha$ and $\gamma$ for $M87$ and for
Sgr. A is shown in Fig. 5. We observe that there is a lower bounds for
$\alpha$ and $\gamma$.

Figure 5: The plots of diameter of the shadow of the black hole in terms of
$\alpha$ for
$M=1,\epsilon=-2/3,G_{N}=1,\gamma={\color[rgb]{1,0,0}0.001},0.002,0.003,0.004,0.005$
(left panel) and in terms of $\gamma$ for
$\alpha={\color[rgb]{1,0,0}0.001},0.005,0.01,0.05,0.1$ (right panel).

###  3.3 The Shapiro time delay

Finally, we consider the Shapiro time-delay to obtain a bound on the coupling
constant $\alpha$. The general expression for the time delay for a photon
moving in the metric of the QMOG black hole reads [59],

$t(r_{0},r)=\int_{r_{0}}^{r}{\dfrac{dr}{f(r)\sqrt{1-\dfrac{r_{0}^{2}}{r^{2}}\dfrac{f(r)}{f(r_{0})}}}}$
(3.12)

In order to evaluate the integral, we expand the metric in the asymptotic
regime. Similar manipulations as before yield for the integral,

$\displaystyle\int_{r_{0}}^{r}{\dfrac{dr}{f(r)\sqrt{1-\dfrac{r_{0}^{2}}{r^{2}}\dfrac{f(r)}{f(r_{0})}}}}=\frac{1}{\sqrt{1-\frac{r_{0}^{2}}{r^{2}}}}\Bigg{\\{}$
$\displaystyle
1+\frac{GM}{r}\left[1+\dfrac{r_{0}}{(r+r_{0})}\right]-\frac{3GQ^{2}}{2r^{2}}$
(3.13)
$\displaystyle+G\gamma\left[\frac{r_{0}^{2}}{2(r^{2}-r_{0}^{2})}\left(\frac{1}{r_{0}^{3\epsilon+1}}-\frac{1}{r^{3\epsilon+1}}\right)+\frac{1}{r^{3\epsilon+1}}\right]\Bigg{\\}}\,.$

Working in the asymptotic regime (to 2nd order in the continued fraction
expansion), approximately the for the time delay we obtain the expression,

$t(r,r_{0})=t_{SR}(r,r_{0})+\Delta t(r,r_{0})$ (3.14)

where $t_{SR}=\sqrt{r^{2}-r_{0}^{2}}$ is the special relativistic contribution
of the propagation of light in the flat spacetime.

Hence, the maximum round-trip excess time delay is given by,

$\Delta
t(r,r_{0})=2\left[t(r_{2},r_{0})+t(r_{1},r_{0})-\sqrt{r_{1}^{2}-r_{0}^{2}}-\sqrt{r_{2}^{2}-r_{0}^{2}}\right].$
(3.15)

Here $r_{0}$ is the distance of the closest approach of the radar wave to the
center of the Sun, $r_{1}$ is the distance along the line of sight from the
Earth to the point of closest approach to the Sun, and $r_{2}$ represents the
distance along the path from this point to the planet, where $r_{1,2}\gg
r_{0}$. In the case of $r_{1}=r_{2}=r$, this becomes,

$\displaystyle\Delta t(r,r_{0})$
$\displaystyle=4GM\ln\left(\dfrac{r+\sqrt{r^{2}-r_{0}^{2}}}{r_{0}}\right)$
(3.16) $\displaystyle+4GM\sqrt{\dfrac{r-r_{0}}{r+r_{0}}}-\dfrac{3\pi
GQ^{2}}{r_{0}}\left[1-\dfrac{2}{\pi}\tan^{-1}\left(\dfrac{r_{0}}{\sqrt{r^{2}-r_{0}^{2}}}\right)\right]$
$\displaystyle+2G\gamma\int_{r_{0}}^{r}\dfrac{r(r_{0}^{-3\epsilon+1}-3r_{0}^{2}r^{-3\epsilon-1}+2r^{-3\epsilon+1})}{(r^{2}-r_{0}^{2})^{\frac{3}{2}}}dr\,.$

Here it is supposed that $r_{1}=r_{2}\approx 10^{11}\;{\rm m}$ is the distance
from the Earth to the point of the closest approach to the Sun, and
$r_{0}\approx 10^{8}\;{\rm m}$ is the distance of closest approach of the
signal to the center of the Sun. After integration, and some manipulations,
for $Q=0$, and $\epsilon=-2/3$, one gets,

$\displaystyle\Delta
t(r,r_{0})=4GM\ln\left(\dfrac{r+\sqrt{r^{2}-r_{0}^{2}}}{r_{0}}\right)+4GM\sqrt{\dfrac{r-r_{0}}{r+r_{0}}}+2\gamma
G\,{\frac{\left({{r_{0}}}^{2}+r{r_{0}}+{r}^{2}\right)\sqrt{\left(r-{r_{0}}\right)}}{\sqrt{{{r}}+{{r_{0}}}}}}.$
(3.17)

The deviation of the Shapiro time delay from the prediction of the Einstein’s
general relativity is smaller than $1.2\times 10^{-5}\;{\rm s}$ [60, 61]. By
using the Solar System data in SI units, one can constrain the parameter of
the QMOG black hole model as $\gamma<10^{-17}$. To obtain this upper bound, we
have assumed that the parameter $\alpha$ is much less than unity in the Solar
System tests. This bound was predictable, since in the local regions the
effects of the quintessence scalar field is usually very small, as compared to
the cosmological scales.

##  4 Electromagnetic emissivity of the thin accretion disks around QMOG black
holes

The first systematic Newtonian theory of the thin accretion disks existing
around black holes dates back to the seminal work by Shakura and Sunyaev in
1973 [62]. In their paper, they showed that by considering the effects of the
friction between different layers of the Keplerian orbits around the black
hole, the motion around inward spirals resulted in the loss of angular
momentum. They concluded that due to such a mechanism, the kinetic energy of
the disk increases. Moreover, the release of the gravitational energy also
takes place. The warming up of the disk leads to the possibility of the
emission of a significant fraction of electromagnetic energy.

The generalization to the general relativistic case has been done by Novikov,
Thorne, and Page and Thorne, in 1973 and 1974, [63] and [64], respectively.
Therefore, more realistic astrophysical disk models can be constructed in this
framework. For their analysis they considered a stationary black hole, which
obeys axially or spherical symmetry, like, for example, the Schwarzschild
black hole. In the Novikov-Thorne model one considers that the central plane
of the disk coincides with the equatorial plane of the black hole. This
assumption significantly decreases the mathematical complexity of the
analysis, and leads to the result that all the diagonal and off-diagonal
components of the metric depend only on the radial coordinate $r$ [64].
Recently, in [65, 66] it was shown that for a Schwarzschild background metric,
for a specific type of accretion disk that contains force free magnetic field
[67], the form of the perturbed metric resulted in some modifications not only
in diagonal components of the background metric, but also in the appearance of
some off-diagonal components, with all of them showing a radial coordinate
dependence behaviour. One may consider these results as a confirmation for the
assumptions made by the authors of [64].

By taking into account all the points discussed above, we are now in a
position to study the electromagnetic emissivity properties, and the physical
properties for the QMOG black hole, and of its proposed disk. The flux $F$ of
the radiant energy over the disk can be written as [63, 68, 69, 71, 70, 72,
73, 74]

$F(\tilde{r})=\dfrac{-\dot{M}_{0}}{4\pi\sqrt{-g}}\dfrac{\Omega_{,\tilde{r}}}{(\tilde{E}-\Omega\tilde{L})^{2}}\int^{r}_{r_{\tiny
Isco}}(\tilde{E}-\Omega\tilde{L})\tilde{L}_{,\tilde{r}}d\tilde{r}\,,$ (4.1)

where $\dot{M}_{0}$ denotes the time averaged value of the accreted rest mass,
which has no dependency to the radial coordinate, and is taken as a constant
[64]. Substituting Eqs. (2.22), (2.23) and (2.24) into (4.1) one can obtain
for the radiation flux the expression

$\displaystyle\begin{array}[]{l}F(\tilde{r})=-\frac{{8(\alpha+1)\dot{M}_{0}\left({\alpha-\frac{{3\tilde{r}}}{4}+\frac{{\tilde{\gamma}{\tilde{r}^{3}}}}{8}}\right)}}{{\pi{\tilde{r}^{4}}\left({2\alpha-2\tilde{r}+\tilde{\gamma}{{\tilde{r}}^{3}}}\right){{\left({4{\alpha^{2}}+4\alpha-6(\alpha-1)\tilde{r}+2{{\tilde{r}}^{2}}+\tilde{\gamma}(-\alpha-1){{\tilde{r}}^{3}}}\right)}^{2}}}}\times\\\
\\\
\left[\begin{array}[]{l}{\alpha^{2}}(\alpha+1)-\frac{9}{4}\left({{\alpha^{2}}+\alpha}\right)\tilde{r}+\frac{3}{2}\left({\alpha+1}\right){{\tilde{r}}^{2}}+\frac{1}{8}\left({11{\alpha^{2}}\tilde{\gamma}+11\alpha\tilde{\gamma}-2}\right){{\tilde{r}}^{3}}-\\\
\frac{3}{2}(\alpha+1)\tilde{\gamma}{{\tilde{r}}^{4}}+\frac{{3}}{8}{\tilde{\gamma}{\tilde{r}}^{5}}-\frac{1}{8}(\alpha+1){{\tilde{\gamma}}^{2}}{{\tilde{r}}^{6}}\end{array}\right]\,.\end{array}$
(4.7)

Using the above expression for $F(\tilde{r})$, the variation of the energy
flux from the disk is presented in Fig. 6. In these plots different values for
the MOG and quintessence parameters are considered. Obviously, when both
$\alpha$ and $\tilde{\gamma}$ tends to zero, the configuration goes back to
the usual Schwarzschild solution. As an another interesting choice, one can
let $\tilde{\gamma}=0$, but $\alpha$ not, which we call it a Reissner-
Nordström like solution. From these plots, it is immediately seen that by
increasing the values of these parameters the peak of the flux, of the is
decreasing.

If one assumes that the radiation flux emitted by the disk’s surface is in
thermodynamical equilibrium, from the Stefan–Boltzmann law [73, 74],

$F(r)=\sigma T^{4}(r)\,,$ (4.8)

where $\sigma$ is the Stefan–Boltzmann constant, one can introduce the
temperature $T$ of the accretion disk, see Fig.(7).

Another important physical parameter of the accretion disks is the efficiency
parameter $\mathbf{\mathcal{\zeta}}$, which measures the amount of the
accreting mass transformed into radiation, in the presence of the central
compact object [72, 73, 74]. The efficiency is measured at infinity, and it is
defined as the ratio of two rates: the rate of energy of the photons emitted
from the disk surface, and the rate with which the mass-energy is transported
to the central body [72].

If all photons reach infinity, an estimate of the efficiency is given by the
specific energy of the particles in the disk measured at the marginally stable
orbit [72, 73, 74]

$\mathbf{\mathcal{\zeta}}=1-\tilde{E}(r_{{}_{Isco}})\,.$ (4.9)

The behaviour of the efficiency of the disk in the QMOG geometry is provided
in Table.(3)

Figure 6: The electromagnetic energy flux versus $\tilde{r}$, for different
values of the free parameters of the QMOG black hole model $\alpha$ and
$\tilde{\gamma}$. In the left panel the Schwarzschild case is also presented.
In the right panel a comparison with the Reissner-Nordstrom type black hole,
with $\gamma=0$, is depicted. In all cases we have considered
$\dot{M}_{0}=1.5\times 10^{17}kgs^{-1}$.

Figure 7: The behaviour of the disk temperature against $\tilde{r}$, for different values of $\alpha$ and $\tilde{\gamma}$. In the two panels the results based on the Schwarzschild and Reissner-Nordstrom type black holes are also presented. For the Stefan-Boltzmann constant we have adopted the value $\sigma=5.67\times 10^{-8}~{}Wm^{-2}K^{-4}$. |  $\gamma$ |  $r_{ISCO}$ |  $\zeta$
---|---|---|---
$\alpha=0$ |  0 |  6 |  $0.0571$
|  0 |  7.1106 |  $0.15927$
$\alpha=0.25$ |  0.001 |  7.3186 |  0.16694
|  0.003 |  8.0783 |  0.18313
|  0.004 |  493.59 |  $\times$
Table 3: By introducing the amount of $r_{Isco}$ provided in Tabel.(1), the
behaviour of efficiency parameter is investigated.

##  5 Hawking radiation of the QMOG black holes

In the present Section we would like to study the mass loss rate due to the
Hawking radiation, that is $\dot{M}$, with the aim of determining the lifetime
of the black hole [75]. It is an already known result that if the mass of a
black hole is sufficiently large, then the temperature will be low.
Accordingly, without affecting the generality, one can suppose that the mass
loss phenomenon that appears due to the Hawking radiation mechanism originates
from the emission of massless particles. In a similar approach used to
determine the relation between the flux parameter and the temperature, the
mass loss rate is given by Stefan-Boltzmann law as follows [84],

$\dot{M}=-\dfrac{\pi^{2}}{60}(\sigma_{g}+\sigma_{\gamma}+\dfrac{21}{8}\sigma_{\nu})T^{4}\,,$
(5.1)

where $\sigma_{i}$, are the thermally averaged cross sections of the black
hole for gravitons, $g$, photons, $\gamma$, and neutrinos, $\nu$. In the
following we define,

$\rho=\left(\sigma_{g}+\sigma_{\gamma}+\dfrac{21}{8}\sigma_{\nu}\right)\dfrac{1}{\sigma_{0}}\,,$
(5.2)

where $\sigma_{0}$ is the geometrical optics cross section of the black hole.
The cross sections for neutrinos, photons and gravitons can be estimated as,

$\sigma_{\nu}\sim 0.67\sigma_{0},\;\;\;\;\sigma_{\gamma}\sim
0.24\sigma_{0},\;\;\;\;\sigma_{g}\sim 0.03\sigma_{0}.$ (5.3)

To obtain the above values we have assumed $\rho\sim 2.02$, and $\sigma_{0}$
should be determined as follows. Since the emitted particles move along null
geodesics, from Eq. (2.18), it follows that their motion is governed by the
equation,

$\dot{r}^{2}=E^{2}+f(r)\left(\mu-\dfrac{L^{2}}{r^{2}}\right)\,,$ (5.4)

where, as previously discussed, $E$ and $L$ refer to the energy, and the
angular momentum, respectively.

For the emitted particles to reach the infinity, rather than to fall back into
the black hole horizon, one has to impose the following condition,

$\dfrac{1}{l^{2}}\equiv\dfrac{E^{2}}{L^{2}}\geq
f\left(\dfrac{1}{r^{2}}-\dfrac{\mu}{L^{2}}\right)\,.$ (5.5)

where by $\mu$ we have denoted the mass of the test particles. The geometrical
optical cross section of the black hole is given by,

$\sigma_{0}=\pi l_{c}^{2}=\dfrac{\pi
r_{c}^{2}}{f(r_{c})}\dfrac{1}{1-\dfrac{\mu r_{c}^{2}}{L^{2}}}.$ (5.6)

Using Eq. (5.5), one can obtain an equation for $r_{c}$ as follows,

$6GMr_{c}-4GQ^{2}+3G\epsilon\gamma r_{c}^{-3\epsilon+1}+3G\gamma
r_{c}^{-3\epsilon+1}-2r_{c}^{2}=0\,.$ (5.7)

For $Q=0$, and $\epsilon=-2/3$, we have,

$r_{c}=\dfrac{1-\sqrt{1-6G^{2}\gamma M}}{G\gamma}.$ (5.8)

For small $\gamma$ values, we obtain

$r_{c}\sim 3GM+\dfrac{9G^{3}M^{2}\gamma}{2}+\mathcal{O}(\gamma^{2})$ (5.9)

By substituting Eq. (5.9) into the equation (5.6) for massless particles,
i.e., for $\mu=0$, one gets,

$\sigma_{0}\sim 27\pi G^{2}M^{2}+243\pi
G^{4}M^{3}\gamma+\mathcal{O}(\gamma^{2})\,.$ (5.10)

Hence, the radius of the event horizon is obtained as,

$r_{+}=\dfrac{1-\sqrt{1-8G^{2}\gamma M}}{2G\gamma}\,.$ (5.11)

For $\gamma$ small, satisfying the condition $\gamma\ll 1$, we obtain

$r_{+}\sim
2GM+4G^{3}M^{2}\gamma+\mathcal{O}(\gamma^{2})\,\sim\dfrac{2}{3}r_{c}+\dfrac{G\gamma}{9}r_{c}^{2}+\mathcal{O}(\gamma^{2}).$
(5.12)

Considering the above expression of the event horizon, the temperature of the
black hole is given as follows,

$T=\dfrac{f^{\prime}(r)}{4\pi}=\dfrac{1}{8\pi
GM}-\dfrac{3G\gamma}{4\pi}+\mathcal{O}(\gamma^{2})\,,$ (5.13)

where ′ denotes differentiation with respect to $r$. The behaviour of $T$
against $r_{+}$ is shown in Fig.(8)

Figure 8: The behaviour of the black hole temperature against ${r_{+}}$, for
different values of $\alpha$ and $\tilde{\gamma}$. In the two panels the
results based on the Schwarzschild type black hole are also presented. Here
$G_{N}=1$ is assumed. For the left panel $\gamma=0.25$ and for the right panel
$\alpha=0.5$.

Finally, using this definition for the temperature, the mass loss rate reads,

$\dfrac{dM}{dt}\sim\dfrac{27}{4096}\dfrac{\rho\varsigma}{\pi^{3}G^{2}M^{2}}-\dfrac{405}{4096}\dfrac{\rho\varsigma\gamma}{\pi^{3}M}+\mathcal{O}(\gamma^{2})\,.$
(5.14)

where $\varsigma=\pi^{2}/60.$ The behaviour of the mass loss rate for
different situations are illustrated in Fig.(9).

Figure 9: The behaviour of the mass loss against ${r_{+}}$, for different
values of $\alpha$ and $\tilde{\gamma}$. In the two panels the results based
on the Schwarzschild type black hole are also presented. Here $G_{N}=1$ is
assumed. For the left panel $\gamma=0.25$ and for the right panel
$\alpha=0.5$.

By integrating the above expression one obtains the lifetime of the black hole
as,

$t=\dfrac{4096}{81}\dfrac{G^{2}\pi^{3}M^{3}}{\varsigma\rho}+\dfrac{5120}{9}\dfrac{G^{4}\pi^{3}M^{4}\gamma}{\rho\varsigma}+\mathcal{O}(\gamma^{2})$
(5.15)

Here one should notice that, $G=G_{N}(1+\alpha)$. Beside this modification,
the second term appears due to the presence of the quintessence scalar field
surrounding the black hole. Here to realize the behaviour of the lifetime of
the black hole one can consider the plots appear in the Fig.(10)

Figure 10: The behaviour of the evaporation time against ${r_{+}}$, for
different values of $\alpha$ and $\tilde{\gamma}$. In the two panels the
results corresponding to the Schwarzschild type black hole are also presented.
Here $G_{N}=1$ is assumed. For the left panel $\gamma=0.25$, while for the
right panel $\alpha=0.5$.

##  6 Concluding remarks

In the present work we have investigated in detail some astrophysical effects
related to the possible presence of exotic forms of matter around black hole
type cosmic objects. We have adopted, as a particular black hole model, the
vacuum solution of the field equations in the MOG theory proposed, and
developed, in [37]. This theory is a scalar-vector-tensor type theory, and its
black-hole solutions has remarkable similarities with both the classic
Reissner-Nordström solution of general relativity, as well as with the Kiselev
black hole solutions [18], which initially interpreted as a black hole
solution in the presence of a quintessence field. However, this interpretation
is problematic [50], but the Kiselev solution, interpreted as a solution of
the field equations in the presence of an exotic fluid, still has many
interesting physical and astrophysical features that could lead to a better
understanding of the properties of black holes embedded in an exotic cosmic
environment. The solution of the field equations in the MOG gravity contains a
term similar to the one appearing in the Reissner-Nordström solution,
describing the properties of a charged black hole, of the form $Q^{2}/r^{2}$,
but with $Q$ unrelated to the electric charge, but proportional to the mass
and to the effective gravitational constant of the theory. Still, in order to
point out to the similarities to the charged general relativistic black holes
we call this class of vacuum solutions of the MOG theory as QMOG black holes.

The first important property of black holes is the position of their event
horizon. The radius of the event horizon essentially depends on the parameter
$\varepsilon$ of the equation of state of the exotic matter. In general, the
position of the event horizon can be obtained for the QMOG black hole as a
solution of a nonlinear algebraic equation. However, for some particular
values of the parameter of the equation of state $\varepsilon$, some
polynomial equations can be obtained.

The dynamical characteristics of the motion of massive test particles around a
QMOG black hole can be obtained from the properties of the effective
potential, which can be defined through the geodesic equations of motion. The
effective potential also allows the determination of another important
characteristic of the black hole geometry, the location of the innermost
stable circular orbits. For the QMOG black hole, the positions of the event
horizons and of the stable circular orbits depend on the solution parameters
$\alpha$ and $\gamma$, as summarized in Tables 1 and 2, respectively. The
effective potential is also strongly dependent on the solution parameters. for
example, as one can see from Fig. 2, the increase in the value of
$\tilde{\gamma}$, leads to a decrease of the magnitude of the effective
potential.

Three important effects that could help to discriminate between different
black hole types are the light deflection, the Shapiro delay and the shadow of
the black hole, respectively. We have investigated in detail these effects,
and we have explicitly obtained, by using some algebraic approximations, the
explicit expressions for the deflection angle, the angular radius and the
diameter of the shadow, as well as the time delay due to the presence of the
black hole. With the help of each of these observations, one can obtain some
constraints on the parameters of the black hole solution. As an interesting
result, we found an upper bound on the quintessence parameter,
$\gamma<10^{-17}$. To obtain this upper bound we have supposed that the
parameter $\alpha$ is much smaller than one in the Solar System tests. The
light deflection angle for a black hole in this configuration was also
obtained. Important information on the nature of the central compact object
can be found through the study of its shadow. We have studeied in detail the
shadow of the QMOG black hole, and obtained the angular radius, diameter, and
shape of the lack hole. These important information may provide the
theoretical basis for constraining the parameters of the modified gravity
model, and discriminate between the different types of black holes.

Many astrophysical objects interact gravitationally with the cosmic
environment, and grow through matter accretion. The Universe is filled with
interstellar matter, whose density was much higher in the early stages of
cosmological evolution. The presence of interstellar matter determines the
formation of accretion disks, located around compact objects. Accretion disks
are highly dynamical system, and in particular they are an important source of
energy, emitted mostly in the electromagnetic spectrum. Neutrino emissivity of
the disks is also possible. The emission of the electromagnetic radiation by
the disk is mainly a consequence of the presence and influence of the external
gravitational field of the central compact object. The gravitational field of
the massive star around which the disk is formed is determined by its nature -
black hole, neutron or quark star, naked singularity etc. Thus, the direct
observations of the electromagnetic emission spectra from astrophysical
accretion disks give us the chance of obtaining essential information on the
physical and astrophysical properties of the massive object around which the
accretion disk was formed due to the interaction with the cosmic environment.
Hence, modified gravity theories, can be tested and constrained by using the
properties of their accretion disks. Moreover, one can obtain a large amount
ofrelevant astrophysical information from the observation of the motion of
matter around the compact object.

In the present study we have performed a detailed analysis of the emissivity
properties of the accretion disks that form around the QMOG black holes. By
using the explicit form of the metric, the expression of the electromagnetic
flux can be obtained in an exact analytical form. The flux is dependent on the
parameters of the metric, and, for certain values of the parameters, it shows
a significant difference as compared to the general relativistic Schwarzschild
case. The differences do appear at the level of the maximum values of the
flux, and in the position of the maximum. For the QMOG balck holes, there is a
slight displacement of the maximum for higher $r$ values. A similar pattern
can be observed in the temperature distribution on the surface of the disk,
with the temperature maximum displaced towards the interior of the disk.
Similar differences do appear in the case of the efficiency parameter.

The quantum properties of the black holes are important for both theoretical,
and astrophysical point of view. Even they are not directly detectable at the
present moment, on a long run they can influence essentially the properties of
the black holes. We have obtained the corrections to the Hawking temperature
due to the extra terms present in the QMOG black hole solutions, and we have
investigated its dependence on the model parameters. The mass loss rate and
the evaporation time have also been obtained, and their dependencies on the
model parameters have been investigated. Interestingly, an increase in the
lifetime of the black hole is obtained for the QMOG black holes. This is due
to the presence of supplementary terms of the form
$\dfrac{5120}{9}\dfrac{G^{4}\pi^{3}M^{4}\gamma}{\rho\varsigma}$, which do
appear besides the usual general relativistic terms.

In the present study we have considered some basic astrophysical properties of
a particular modified gravity theory black hole. The obtained results may open
some new perspectives in the observational testing of this type of objects,
and in discriminating between different types of compact objects.
Investigating these corrections for other type of black holes, for example,
rotating black holes, can be the subject of future work.

## Acknowledgements

H.S. is grateful to H. Firouzjahi for the constructive discussions they had on
black holes and singularities. He also thanks J. Moffat for the discussions on
a primary version of the paper. The work of TH is supported by a grant of the
Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-
III-P4-ID-PCE-2020-2255 (PNCDI III).

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