# Aspects of Quantum Gravity Phenomenology and Astrophysics

Arundhati Dasgupta and José Fajardo-Montenegro*
Physics and Astronomy, University of Lethbridge,
Lethbridge, Canada T1K 3M4.
*Departamento de Física, Universidad del Valle,
Cali 760032, Colombia

###### Abstract

With the discovery of gravitational waves, the search for the quantum of
gravity, the graviton, is imminent. We discuss the current status of the
bounds on graviton mass from experiments as well as the theoretical
understanding of these particles. We provide an overview of current
experiments in astrophysics such as the search for Hawking radiation in gamma-
ray observations and neutrino detectors, which will also shed light on the
existence of primordial black holes. Finally, the semiclassical corrections to
the image of the event horizon are discussed.

##  1 Introduction

The gravitational quantum is still elusive experimentally and somewhat
“elusive” theoretically [1, 2, 3]. In electrodynamics, the quantum of the
electromagnetic wave is known as the photon, and we work with the interactions
of photons to derive quantum electrodynamics (QED) phenomena. In the case of
gravity, gravitational waves have been discovered 100 years after their
prediction. The question is, are there “gravitons” or quanta of these waves?
Like QED, one can define the “Fock” space quantization for the linearized
Einstein equations and study free gravitons. However, introducing interactions
with gravitons to study scattering amplitudes leads to uncontrollable
infinities [3]. This is known as the “non-renormalizability” of perturbative
quantum gravity. General relativity might be nonperturbative in the quantum
regime, and the story of the quanta could be present in the geometry
measurements of area and volume [4]. These “nonperturbative” theoretical
explorations cannot be verified, as they are still in the realm of the
microscopic Planck length regime of $10^{-35}$ m. We investigate the
semiclassical fluctuations of the flat geometry using loop quantum gravity
(LQG) coherent states and discuss whether that can be interpreted as a
graviton quantum.

Further in the 1970s, the discovery of black hole thermodynamics and Hawking
radiation were studied as “semiclassical phenomena”, where gravity remained
classical and other fields were quantum. The isolated black hole was found to
have a temperature proportional to its surface gravity and entropy equal to
its horizon surface area. For a solar-mass black hole, which might have formed
using stellar collapse, this temperature is of the order of $10^{-8}$ K. If we
observe the current-day black holes, then they are immersed in the background
cosmic radiation, which has a temperature of $2.783$ K. As the heat flows from
higher to lower temperatures, the black holes would not radiate into the
surroundings, and as of now, there is no experimental evidence of Hawking
radiation. The study of black hole mergers using gravitational waves has
provided evidence for the area increase theorem [5]. How would one obtain a
verification of the temperature and radiative properties of black holes? The
existence of primordial black holes (PBH) of small mass, originating in
density fluctuations of the early universe, would allow for high-temperature
black holes and Hawking decays in the form of gamma-ray bursts. The search for
PBH has been a subject of experimental study [6]. We discuss this in some
detail, and the approximations which describe the theoretical derivation of
Hawking radiation are also discussed. The current experiments provide
stringent restrictions on the PBH contributions to photon and neutrino fluxes
observed on earth, as well as as fractions of dark matter [7, 8, 9, 10, 11].
Strangely, new observations from gravitational wave data suggest that there
are subsolar mass black holes. Recent work tries to find the origins of these,
either as PBH or from other processes without the Chandrasekhar limit in the
collapse process [12]. Whereas this is very interesting, this is not exactly
the realm of quantum gravity, though the research in this area might shed
light on semiclassical aspects.

However, astrophysical phenomena, such as the black hole merger event, the
collapse of a supernova to form a black hole, and neutron star mergers, are
strong gravitational events. The energies at which the events happen have
strongly coupled gravitational interactions. The quantum dynamics near these
events is interesting, and even though the effect is weak, one can try and
find indirect evidence in the observational data. Using LQG coherent states,
some of these can be studied semiclassically. We discuss these and also
comment on other observational results from the semiclassical gravity program
for astrophysical observations, including that for the image of the event
horizon [13, 14]. There are several collaborations in quantum gravity
phenomenology which, in particular, discuss Lorentz violations and quantum
anomalies. The appropriate discussions on these topics can be found in [15].
For a previous comprehensive review on quantum gravity phenomenology, see
[16]. One of the aims of this current review is to also provide a pedagogical
introduction to some aspects such as the search for primordial black holes,
which is a very active field currently.

This review has discussions on the (i) graviton, (ii) Hawking radiation, and
(iii) semiclassical corrections to strong gravity systems such as the event
horizon. The following section discusses the theory of the graviton and the
experimental bounds. Section 3 describes the phenomena of Hawking radiation,
as well as the experimental efforts to detect the emitted particles from PBH.
Section 4 describes the physics of the event horizon and quantum correction
predictions to the same. The final section concludes with the present status
of the field of research in the above and future avenues of quantum gravity
phenomenology.

##  2 Graviton

The electromagnetic (EM) wave is a solution to Maxwell’s equation and is
observed in nature. The visible spectrum is known as light, the infrared,
which we interpret as heat, and radio waves. The ultraviolet radiation is also
detectable and useful as are X-rays in many practical day-to-day events.
These, when quantized, give us the photon description of the EM wave, and
represent the source-free “free” EM fields. The actual production of EM
radiation is from accelerated charges, but as the waves propagate out in
space, they can be studied as “free” EM fields. In the case of gravity,
Einstein’s action is nonlinear, and the gravitational field has self-
interactions. To find the “free” plane wave which propagates on its own, we
take a linearized gravity, “weak fluctuations” over a flat background.
Nonperturbative waves, produced using strong gravitational interactions, have
been studied in [17]. As the linearized gravitational waves represent
classically “free” fields, one would expect that the Fock space quantization
of these would be obtained similarly to the photon quantum electrodynamics
description. However, herein lies the problem: the graviton theory is a
nonrenormalizable theory [3]. Is it because the graviton vacuum, which
represents the Minkowski spacetime is not a vacuum? Is flat space really a
vacuum state in a true theory of quantum gravity? Can we have a perturbation
over the flat-space system and describe a graviton as a quantum state in the
flat-space background? In the case of the EM theory, the EM field propagates
in a flat background that, however, serves as a noninteractive arena for the
EM fields to propagate. The photon is created and annihilated out of the QED
vacuum, which is a state with the photon quantum number as zero. In the
following, we discuss whether seeking a similar quantum field vacuum for the
graviton is relevant. We also discuss the question of which physics of the
systems we should experiment for the observation of the graviton.

###  2.1 The Linearized Theory of the Graviton

In the following, we discuss Einstein’s theory of the linearized metric. The
field equations for the Einstein action is “free” in its gauge-fixed form;
however, if we try to write the full Einstein Lagrangian for the gravitational
field, then there are interaction vertices to all orders for the graviton. The
quantum amplitudes including these interactions do not converge, and neither
can the theory be renormalized using standard techniques. To begin with, we
write the metric of spacetime $g_{\mu\nu}$ as a flat space $\eta_{\mu\nu}$ and
a weak fluctuation $h_{\mu\nu}$.

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}.$ (1)

It is assumed that $|h_{\mu\nu}|_{\rm max}\ll 1$ ($\mu,\nu,\alpha,\beta\ {\rm
etc}=0,\ldots,3$). Note that using standard convention, the metric is
dimensionless and the amplitude of the fluctuations are defined using the
absolute maximum value. From experiments [1], we are aware now that the
amplitude of the “gravitational wave” is of the order of $10^{-22}$ as
received on earth. One can write the Einstein Lagrangian density as a function
of this metric, its determinant $g$, and scalar curvature $R$,

${\cal
L}=\sqrt{g}~{}R=-\frac{1}{2}\sqrt{-1+h}\left[(h^{\mu\nu})(\eta^{\alpha\beta}\partial_{\alpha}\partial_{\mu}h_{\nu\beta}-\Box\
h_{\mu\nu})\right].$ (2)

In the above, we have kept the terms in the Lagrangian which are quadratic in
$h_{\mu\nu}$. The linear terms of the form
$\eta^{\mu\nu}\eta^{\lambda\rho}\partial_{\rho}\partial_{\mu}h_{\lambda\nu}$
are total derivatives and contribute only at the boundaries, which we ignore.
Further, $\Box\equiv\eta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$, and
$h$ is the trace of $h_{\mu\nu}$. The equation of motion from the above to a
linear order in “$h_{\mu\nu}$” is

$\eta^{\alpha\beta}\partial_{\alpha}\partial_{\mu}h_{\nu\beta}-\Box\
h_{\mu\nu}=0.$ (3)

This still has a gauge degree of freedom due to diffeomorphism invariance,
which can be fixed by putting the $\partial^{\alpha}h_{\alpha\beta}=0$
restriction on the linearized metric. The equation of motion reduces to a
“wave equation”

$\Box~{}h_{\mu\nu}=0.$ (4)

The solution for this is a transverse wave (due to Lorentz’s condition) and
has two polarizations as additional restrictions to fix the residual gauge
freedom keeping only two [18]. The two polarizations are taken as
$h_{+}=A_{+}\cos(\omega z-\omega t)$ and $h_{\times}=A_{\times}\cos(\omega
z-\omega t)$, if it is propagating in the z-direction [18], with angular
frequency $\omega$ and amplitude $A_{+},A_{\times}$. The question is: can
these waves, when quantized, give us “quanta” as it is possible for photon
quantization? In other words, can one define a Fock space representation for
the perturbative Hilbert space of Einstein’s gravity? The answer is
surprisingly difficult, as the Einstein action introduces self-interactions of
the gravitons to all orders, which cannot be renormalized using standard field
theory techniques. The gravitational propagator can be calculated, but the
quantum corrections cannot be made finite using regularization and
renormalization techniques. One can see the origin of self-interactions even
at this order in the Lagrangian in Equation (2) as the nonpolynomial “measure”
$\sqrt{-1+h}$ can give rise to the interaction terms upon expanding the square
root. A simple “degree of superficial divergence” counting of the
gravitational perturbative Feynman diagram gives the number as $D=2(k+1)$,
where $k$ is the number of independent momentum interactions [19]. This number
therefore increases with the number of loops in the scattering calculations
and cannot be absorbed by redefining the bare Lagrangian. For Yang–Mill’s (YM)
theory the same degree is given as $D=4-L_{e}$, where $L_{e}$ is the number of
external legs of the Feynman diagram. The YM theory is therefore
renormalizable, as the number of terms in the Lagrangian which need to be
renormalized is finite ($0<L_{e}<4$). One can use asymptotic techniques to
obtain a renormalizable effective Lagrangian for gravity, but we do not
discuss this in this review [20]. However, can there be a “free” graviton
theory where we can ignore all the interactions? Up to a certain length scale,
a “free graviton” quantization can be formulated, but the entire theory is
also complicated by the definition of the “gravitational vacuum”. In the
theory of gravitational physics, the metric is the basic degree of freedom,
and the graviton is a “perturbation” over the flat-space geometry. In a true
quantization of the theory, the flat spacetime geometry is also an emergent
“metric”. If the metric is an operator, then causality and therefore
quantization is not defined. The vacuum likely is the state with no metric or
the state that is such that

$\hat{g}_{\mu\nu}\ |0\rangle=0.$ (5)

There have been several attempts to obtain the perturbative quantum state
using a polymer state in the nonperturbative quantization framework of loop
quantum gravity. We report on those works briefly and then describe a
semiclassical description of a “gravitational wave” using LQG. It remains
though that the most complicated aspect of Einstein’s gravity is the fact that
the field which has to be quantized is the metric of the spacetime, the
causality of the system is complicated by the quantization, and macroscopic
configurations have to be emergent.

###  2.2 Gravitons in Loop Quantum Gravity

It was shown in [21] that the SU(2) generators of the loop quantum gravity
(LQG) variables decouple into three independent gauge generators in the
linearized approximation. In LQG, the basic variables are obtained from the
ADM formulation of the canonical gravity. The spacetime is foliated by spatial
slices $\Sigma$ with a timelike normal vector along the fourth direction,
specified using the coordinate $t$. The induced three-metric on $\Sigma_{t}$
is given as $q_{ab}$, ($a,b=1,2,3$); the metric in the ADM formulation is
given as

$ds^{2}=-(N^{2}+N^{a}N_{a})dt^{2}+N^{a}dx_{a}dt+q_{ab}dx^{a}dx^{b},$ (6)

where $N^{2}$ is the lapse, $N^{a}$ is the shift, and $q_{ab}$ is the induced
metric of the time slices $\Sigma_{t}$. The second fundamental form of this
metric is $K_{ab}={\cal L}_{t}q_{ab}$ and is the extrinsic curvature tensor
which characterizes the embedding of the slice.

The LQG variables are defined using the soldering forms $e^{I}_{a}$ which
connect the tangent space $(I=1,2,3)$ of the three slices to the world volume.
The canonical variables are defined as

$e_{a}^{I}e_{bI}=q_{ab},\ \ E^{a}_{I}E^{bI}=q\ q^{ab},\ \ \
A_{a}^{I}=\Gamma_{a}^{I}-K_{ab}E^{bI},$ (7)

where $e_{a}^{I}$ is the triad, $E^{a}_{I}$ are densitized triads, and
$A_{a}^{I}$ have the properties of a connection due to their definition in
terms of the spin connection $\Gamma_{a}^{I}$ and the extrinsic curvature
tensor $K_{ab}$. The details of the variables can be found in [22]. There is
usually an Immirzi parameter in the definition of the gauge connection, and
this reflects an ambiguity in the system. We chose to set it to one, for the
purpose of this paper. The internal indices $I$ transform in the SU(2) group,
which is isomorphic to the group of rotations in the three-dimensional tangent
space [22]. The generators of the transformations in the internal directions
are the Gauss constraints

${\cal G}^{I}=\partial_{a}e^{aI}+\epsilon^{IJK}e_{J}^{a}A_{aK}.$ (8)

In the linearized approximation, $q=1$, $q^{ab}=\delta^{ab}+h^{ab}$ and
$A_{a}^{I}=0$, if one keeps the constraint up to a linear order in the fields,
the constraint algebra commutes, i.e.,

${\cal G}^{I}_{\rm Lin}=\partial_{a}(\delta
e^{aI})+\epsilon^{IJK}\delta_{J}^{a}\delta A_{aK},$ (9)

where due to the linearized metric, one has

$e^{aI}=\delta^{aI}+\delta e^{aI},\ \ \ A_{aK}=0+\delta A_{aK},$ (10)

and

$h^{ab}=\delta e^{aI}\delta^{b}_{I},\\\ $ (11)

$\\{\delta e^{I}_{a}(x),\delta
A_{Kb}(y)\\}=\kappa\delta^{3}(x-y)\delta^{I}_{K}\delta_{ab},$ (12)

where $\kappa$ is related to Newton’s constant $G$ [22, 23]. The $\delta
e^{I}_{a}$ and the $\delta A_{Kb}$ are the linearized dynamical fields, which
are quantized. In the limit $\kappa\rightarrow 0$,

$\left\\{{\cal G}^{I}_{\rm Lin},{\cal G}^{J}_{\rm Lin}\right\\}=0.$ (13)

Interestingly, if one keeps the next order in the constraint definition, the
algebra is not zero to a linear order as the Poisson bracket gives a linear
result in the fields.

${\cal G}^{I}_{\rm Lin}=\partial_{a}(\delta
e^{aI})+\epsilon^{IJK}\left(\delta_{J}^{a}+\delta e^{aJ}\right)\delta A_{aK},$
(14)

and

$\left\\{{\cal G}^{I}_{\rm Lin},{\cal G}^{J}_{\rm
Lin}\right\\}=\kappa\left(\delta A^{IJ}-\delta^{IJ}\delta
A^{b}_{b}\right)\delta^{3}(x-y).$ (15)

This term would go to zero in the limit $\kappa\rightarrow 0$. To avoid these
confusions about the algebra and also questions about the Minkowski “quantum
state” about which perturbation is being performed, we use the full SU(2)
degrees of freedom and imposed the linear metric only in the semiclassical
approximation. The details of the calculations appear in [24].

For the polymer quantization of linearized gravity using the $U(1)\times
U(1)\times U(1)$ Hilbert space, one can use the work of [26]. This approach is
based on the linearized algebra of LQG variables, as given in Equation (13).
The LQG phase space thus has a $U(1)\times U(1)\times U(1)$ symmetry in the
linearized approximation, instead of the full $SU(2)$. The Hilbert space
quantum states are of the form

$|\vec{\alpha},\\{q\\}\rangle=|\alpha_{1},q_{1}\rangle|\alpha_{2},q_{2}\rangle|\alpha_{3},q_{3}\rangle,$
(16)

where $|\alpha_{i},q_{i}\rangle$ are elements of a $U(1)$ Hilbert space.
$q_{i}$ label integers and $\alpha$ labels the discrete network. The flux
operator defined in terms of the triads is given as [26]

$X^{a}_{\vec{\alpha},\\{q\\}(r)}(\vec{x})=\sum_{I}q_{I}\int
ds_{I}(\vec{e}_{I}(s^{I}),\vec{x})\dot{e}^{a}_{I},$ (17)

where $s_{I}$ is a surface in three dimensions, which the discrete edge
$e_{I}$ of the graph $\alpha$ intersects once.

The Fock space quantum vacuum for the graviton is a transform of the state in
Equation (16). Whether this facilitates further study of the perturbation
theory of the graviton is yet to be investigated. The transform is given as

$\Phi_{0}:=\sum_{\alpha,{q}}c_{0\vec{\alpha},\\{q\\}}\langle\vec{\alpha},{q}|,$
(18)

where

$c_{0\vec{\alpha},\\{q\\}}=\exp\left(-\frac{\imath}{4}\int~{}d^{3}x\
G_{ab}^{\vec{\alpha},\\{q\\}(r)}(\vec{x})*X^{ab}_{\vec{\alpha},\\{q\\}(r)}(\vec{x})\right),$
(19)

where these are “smeared” operators in the LQG polymer space, and $r$ is a
measure of the Gaussian smearing ($X_{r}(\vec{x})=\int
d^{3}yX(\vec{y})\exp(-|\vec{x}-\vec{y}|^{2}/2r^{2})/((2\pi r^{2})^{3/2})$).

$X^{ab}_{\vec{\alpha},\\{q\\}(r)}=\sum_{i}X^{a}_{\alpha_{i},q_{i}}\delta^{b}_{i}.$
(20)

The $G_{ab}^{\vec{\alpha},\\{q\\}(r)}(\vec{x})$ is related to the flux of the
two “graviton” polarizations in the light cone. We refrain from getting into
the details of the above, but the reader is urged to follow the details of the
derivation in [26] and [27]. Whereas this approach to obtaining a “quantum” of
linearized gravity is technically rather involved and involves an additional
scale “$r$” apart from the usual discretization of quantum variables, it is
believed to give a polymer representation of the “graviton”.

The expectation values of the operators are preserved in the transform and
therefore, one loop corrections to the graviton propagator can be tested. A
derivation of a one-loop correction using a perturbation of reduced loop
quantum cosmology states exists in [28]. Another reference for the reduced
phase-space quantization of linearized gravitational waves is [29]. Moreover,
a more recent work uses the free graviton Lagrangian and obtains a “polymer
state” for the same. This approach obtains some corrections to the
gravitational wave propagator [30]. However, in none of the above papers the
emergence of the background Minkowski metric is discussed. The self-
interaction of gravitons is also not obtained to all orders, as predicted by
the Einstein Lagrangian. In the next section, we try to find some
phenomenological implications of the graviton’s existence in observational
data.

###  2.3 Gravitons in Semiclassical Gravity

In this subsection, we derive the semiclassical phase space of the
gravitational wave metric and obtain a coherent state for the system using the
techniques of [22, 24]. To begin with, we find the triads for the metric and
the LQG gauge connection, which are the classical variables for the system.
The details can be found in [24]. The spatial metric for a standard
gravitational wave metric in the tt-gauge is (the lapse is one and shift is
zero in the ADM form of the four-metric)

$q_{ab}=\left(\begin{array}[]{ccc}1+h_{+}&h_{\times}&0\\\
h_{\times}&1-h_{+}&0\\\ 0&0&1\end{array}\right).$ (21)

In the process of obtaining the coherent state for the above metric, we
identify the classical phase space in terms of the LQG variables [25]. The
triads $e_{a}^{I}e_{bI}=q_{ab}$ are obtained as

${e}{{}^{I}_{a}}=\left(\begin{array}[]{ccc}\sqrt{\frac{1-(h_{+}^{2}+h_{\times}^{2})}{2(1-h_{\times})}}&\sqrt{\frac{1-(h_{+}^{2}+h_{\times}^{2})}{2(1-h_{\times})}}&0\\\
\frac{1-(h_{\times}-h_{+})}{\sqrt{2(1-h_{\times}})}&\frac{-1+(h_{\times}+h_{+})}{\sqrt{2(1-h_{\times}})}&0\\\
0&0&1\end{array}\right)=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}+\frac{h_{\times}}{2\sqrt{2}}&\frac{1}{\sqrt{2}}+\frac{h_{\times}}{2\sqrt{2}}&0\\\
\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}(h_{+}-\frac{h_{\times}}{2})&-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}(h_{+}+\frac{h_{\times}}{2})&0\\\
0&0&1\end{array}\right).$ (22)

Obviously, in our gauge choice, the triad is not diagonal at the zeroth order.
The extrinsic curvature of the metric is obtained using the definition
$K_{ab}=-\partial_{t}q_{ab}$, and the SU(2)-valued gauge connections defined
in Equation (7) are:

$\displaystyle{A}{{}^{{1}}_{x}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{2}}(\partial_{z}h_{\times}+\partial_{z}h_{+})={A}{{}^{{2}}_{y}}$
$\displaystyle{A}{{}^{{1}}_{y}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{2}}(\partial_{z}h_{\times}-\partial_{z}h_{+})=-{A}{{}^{2}_{x}}$
$\displaystyle{A}{{}^{{1}}_{z}}$ $\displaystyle=$
$\displaystyle{A}{{}^{{2}}_{z}}={A}{{}^{3}_{x}}={A}{{}^{{3}}_{y}}=0$
$\displaystyle{A}{{}^{{3}}_{z}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{z}h_{+}.$

We also computed the nonzero spin connections for this metric [25]. Next, we
take a discretization of the background geometry. This smearing of variables
is required to obtain smooth commutators of the quantum theory, instead of
distributional delta functions. For details, see [22], and the smearing of the
gauge connection on one-dimensional curves gives holonomies which involve
path-ordering.

$h_{e}(A)={\cal P}\exp\left(\int A\right).$ (23)

The discretization is not dictated by the theory but is motivated from the
flat geometry of the classical three-metric. We take a planar graph, which
form a cubic 3-d polyhedronal decomposition of the three-geometry, as shown in
Figure 1. Therefore, there are six links and/or six faces meeting at a given
vertex.

Figure 1: (a) Building block for the decomposition of the 3-geometry. (b)
Example of one of the smearing surfaces to calculate the momenta.

The holonomies and the momentum are calculated as smeared along the one-
dimensional edges of the graph, and the two-dimensional faces of the cube
which the links intersect precisely at one point. These calculations are done
using the techniques of [24]. The holonomies of the three independent links in
the $x$, $y$, and $z$ directions and the corresponding momenta are given up to
a linear order in the amplitudes $A_{+}$, $A_{\times}$,

$\displaystyle h_{e_{x}}$ $\displaystyle=$ $\displaystyle
1-i\frac{\epsilon}{2}A_{x}^{I}\sigma_{I}$ (24) $\displaystyle h_{e_{y}}$
$\displaystyle=$ $\displaystyle 1-i\frac{\epsilon}{2}A^{I}_{y}\sigma^{I}$ (25)
$\displaystyle h_{e_{z}}$ $\displaystyle=$ $\displaystyle
1+i\frac{A_{+}}{2}\sin\left(\omega\left(z_{0}-t_{0}+\frac{\epsilon}{2}\right)\right)\sin\left(\frac{\epsilon}{2}\right)\sigma_{3},$
(26)

where one has taken a vertex at $(x_{0},y_{0},z_{0})$ and the links are of
width $\epsilon$. $\sigma_{I}$ are the Pauli matrices. Next, one takes the
faces centred at the middle of the links, i.e., at $x_{0}+\epsilon/2$,
$y_{0}+\epsilon/2$, and $z_{0}+\epsilon/2$, and of area $\epsilon^{2}$. The
momenta are labelled by the edges which intersect the faces. The momenta are
defined as $P^{I}_{e}=\frac{1}{\kappa}\int_{S_{e}}*E^{I}$.

$\displaystyle P_{e_{x}}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\kappa}\left(\epsilon^{2}+\frac{\epsilon^{2}(A_{\times})}{2}\cos(\omega(z_{0}-t_{0}))\right)$
(27) $\displaystyle P_{e_{x}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\kappa}\left(\epsilon^{2}+\frac{\epsilon^{2}(2A_{+}-A_{\times})}{2}\cos(\omega(z_{0}-t_{0}))\right)$
(28) $\displaystyle P_{e_{y}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\kappa}\left(-\epsilon^{2}+\frac{\epsilon^{2}(2A_{+}+A_{\times})}{2}\cos(\omega(z_{0}-t_{0}))\right)$
(29) $\displaystyle P_{e_{y}}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\kappa}\left(\epsilon^{2}+\frac{\epsilon^{2}(A_{\times})}{2}\cos(\omega(z_{0}-t_{0}))\right)$
(30) $\displaystyle P_{e_{z}}^{3}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\epsilon^{2}.$ (31)

As the densitized triads are smeared over two-dimensional areas and acquire
dimensions, the momenta are defined with the dimensional constant $1/\kappa$,
$\kappa=8\pi G/c^{3}$ to make the variables dimensionless. In the quantum
version, this acquires the role of $1/\hbar\kappa=1/l_{p}^{2}$, where $l_{p}$
is the Planck length. The coherent states are defined as peaked at the
classical values of a complexified SL(2,C) element as specified by Hall [31],

$g_{e}=\exp(iT^{I}P_{e}^{I})h_{e},$

and a detailed coherent state can be written for the above classical phase
space, now described only using the discrete one-dimensional smeared
holonomies and corresponding momenta. Note these “coherent states”, as defined
in [22] for LQG, are representative semiclassical states and are not exactly
identifiable as “coherent states” as in completely solvable Hamiltonian
systems. However, these states have minimal uncertainty in the time slice they
are defined in. Next, we calculate the semiclassical corrections to the
geometry by using the results of [13]. The coherent states are given for one
such discrete element $e$ and the LQG smeared variables as,

$\psi^{t}(g_{e},h_{e})=\sum_{j}(2j+1)\exp(-\tilde{t}j(j+1)/2)\chi_{j}(g_{e}h_{e}^{-1}),$
(32)

where $\chi_{j}(h_{e})$ is the character of the $j$th irreducible
representation of SU(2). One can find the expectation value of the momentum
operator $\hat{P}_{e}^{I}$ in this state, and one obtains it to the first
order in the semiclassical parameter $\tilde{t}$ [13]

$\langle\psi^{t}|\hat{P}_{e}^{I}|\psi^{t}\rangle=P^{I}_{e}\left(1+\frac{\tilde{t}}{P_{e}}\left(\frac{1}{P_{e}}-\coth(P_{e})\right)\right)=P^{I}_{e}\left(1+\tilde{t}f(P_{e})\right),$
(33)

where $P_{e}=\sqrt{P_{e}^{I}P_{e}^{I}}$ and $f(p)=(1/p)(1/p-\coth(p))$.
Therefore, one can calculate the semiclassical corrections to the metric of
the classical gravitational wave, if one writes a coherent state for each
discrete element $e$ which comprises the entire Minkowski three-volume divided
into cubic cells as in the figure. The vertices of the cube which are shared
by three+three coherent states and these can have SU(2) intertwiners [32], but
the nature of the corrections remain the same. Note these coherent states are
not exactly similar to the coherent states for photons, which are Abelian.
These coherent states are non-Abelian in nature.

In fact, if we take the pure Minkowski space and use the coherent state as a
measure of the quantum fluctuation, what would we generate as the corrected
metric? All the $P^{I}_{e}$’s for the Minkowski metric can be obtained as
given above and, in the limit, $A_{+,\times}=0$ would represent the Minkowski
metric. In this particular gauge, the corrections generate semiclassical
fluctuations in the $\eta_{xx}$, $\eta_{yy}$, and $\eta_{zz}$ components but
not in the $\eta_{xy}$ directions.

Next, we discuss the fluctuations to the gravitational wave metric as
generated from the coherent state which peaks at the gravitational wave
metric. Obviously, the metric would fluctuate and generate semiclassical
corrections to the geometry at order $\tilde{t}$. We set the semiclassical
parameter (which has to be dimensionless) as a ratio of the Planck scale to
the gravitational wave, wavelength, or $\tilde{t}=l_{p}^{2}/\lambda^{2}$. We
take the wavelength as that is the length scale which characterizes the wave
system. A relevant-frequency gravitational wave, which might generate
detectable semiclassical fluctuations, has to be of very high frequency. Let
us say a $10^{35}$ Hz gravitational wave will have the semiclassical parameter
as $\tilde{t}\approx 10^{-16}$.

In the above, have we predicted a “quantum origin” of the gravitational wave
that would comprise the “graviton”? Obviously, the story is not about
particles in gravitational physics, or matter quanta, but the quantum of
geometry. The tiny area measurements in each basis state of the operator
$\hat{P}^{I}_{e}$ represent the “graviton”, the condensate of which is
represented by the coherent-state wave packet. It thus remains that from our
perspective, the Minkowski geometry is not the gravitational vacuum, but also
emergent from a semiclassical state. Therefore, one should not confuse the
quantum gravity vacuum state with the “matter vacua”.

We suggest two ways to search for quantum gravity bounds/origins in a
gravitational wave experiment:

  * (i)

As the coherent states are non-Abelian in nature, the expectation values of
operators have semiclassical corrections which originate due to self-
interactions. These can be detected for high-frequency gravitational waves.

  * (ii)

The search for individual “gravitons” or quanta of geometry would require much
more precise instruments, able to resolve the coarse-graining of geometry
itself.

The latter (ii) will require further investigations, in particular about what
the dynamical fundamental “quanta” of LQG is. One also has to find if there is
a gauge invariant observable which is measurable in experiments. Our questions
seem to seek answers by quantizing matter and the gravitational degrees of
freedom simultaneously. However, due to the hierarchy problem, it is preferred
that matter is quantized and the gravitational degrees of freedom are
semiclassical in the current epoch. In the combined Hilbert space of the
matter and gravitational degrees of freedom $H_{\rm matter}\otimes H_{\rm
grav}$, the combined matter–gravity state should be taken as

$|\Psi\rangle=|\psi_{\rm matter}\rangle\otimes|\psi^{\rm grav}_{\rm
semiclassical}\rangle.$ (34)

For previous work in adding matter interactions in LQG, refer to [23].

Using criterion (i) and the idea that matter quanta interact with
gravitational degrees of freedom at semiclassical length scales, one finds
that the semiclassical fluctuations of the metric are relevant. We therefore
calculate the metric corrections as predicted from the coherent states for LQG
constructed by Thiemann, Winkler, [22] and as observed in [13]. They emerge as

$\displaystyle g_{xx}$ $\displaystyle=$ $\displaystyle(1+h_{+})(1+2\tilde{t}\
f(P_{e_{x}}))$ (35) $\displaystyle g_{yy}$ $\displaystyle=$
$\displaystyle(1-h_{+})(1+2\tilde{t}\ f(P_{e_{y}}))$ (36) $\displaystyle
g_{xy}$ $\displaystyle=$ $\displaystyle h_{\times}(1+\tilde{t}\
f(P_{e_{x}})+\tilde{t}\ f(P_{e_{y}}))$ (37) $\displaystyle g_{zz}$
$\displaystyle=$ $\displaystyle 1+\tilde{t}\ f(P_{e_{z}}).$ (38)

The gauge invariant momenta are found to be:

$\displaystyle P_{e_{x}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{2}}{\kappa}\left(1+\frac{1}{2}h_{+}\right)$ (39)
$\displaystyle P_{e_{y}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{2}}{\kappa}\left(1-\frac{1}{2}h_{+}\right)$ (40)
$\displaystyle P_{e_{z}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{2}}{\kappa}.$ (41)

The continuum limit is obtained using $\lim_{\epsilon\to
0}P_{e}/\epsilon^{2}$. This gives the metric fluctuations at a location
$(x_{0},y_{0},z_{0})$ and one can solve the propagation of matter in this
corrected metric. As evident in the continuum limit, the corrections are
functions of the classical triads, and thus dependent only on the $z$
coordinate. Moreover, the corrections are relevant only at one instant
$t=t_{0}$ of the spacetime. For a 100 Hz frequency, the gravitational wave
will have a semiclassical correction of the order of $10^{-84}$, which is way
smaller than the gravitational wave amplitude. If one probes higher-frequency
gravitational waves, and therefore shorter wavelengths, the Planck scale
coarse-graining will start manifesting itself and the effects might be evident
in a gravitational wave detector. The Minkowski metric is also corrected
semiclassically, and one can probe these using quantum fields in these
geometries.

###  2.4 Summary

In this section, we gave a “semiclassical” state which could describe a
gravitational wave at one instant. It predicted fluctuations which could be
measurable for high-frequency waves $\geq 10^{30}$ Hz. These frequencies were
way above the ones observed in the LIGO detectors. From the current
observation of gravitational waves, there are bounds on the “graviton mass”.
From LIGO, the bound is $1.2\times 10^{-22}\ {\rm eV}$. This bound does not
shed light on the origins of the mass from the methodology. Theoretically, the
graviton mass can originate from quantum corrections to the Einstein theory,
as well as from matter interactions which preserve diffeomorphism invariance.
In this review, we do not discuss massive gravitons.

##  3 Search for Hawking Radiation and Primordial Black Holes

The discovery that quantum mechanics near black hole horizons results in
particle creation originates in the paper by SW Hawking [33]. In that paper, a
quantum field vacuum was time-evolved in the collapsing geometry of a star.
The quantum state evolved into a thermal state, with a temperature inversely
proportional to the mass of the black hole. In [33], it was shown that the
exact temperature of a solar-mass black hole was $10^{-8}$ K. However, it
would not radiate into the surrounding, which was at $2.78$ K. This led to the
search for black holes with mass $\sim 10^{14}\,{\rm g}$, and these could have
formed in the early universe. Due to the Chandrasekhar limit, astrophysical
black holes have a bounded mass if formed from stellar collapse. On the other
hand, early universe density fluctuations can lead to the formation of tiny
black holes, with horizon size fractions of a millimetre. These black holes
have intrinsic temperatures higher than the current CMB temperature of $2.78$
K. Even if the early universe had been hot, as the primordial universe cooled
down, these black holes would start radiating and evaporate eventually or form
Planck size remnants.

###  3.1 Formation of Primordial Black Holes (PBH)

The story of the collapse of matter to form black holes is well-studied in the
work of Choptuik [34]. Scalar data in an initial slice undergo collapse, and
the mass of the black hole formed has a scaling equation. This physics is true
for early universe cosmology. It is noted that the matter undergoing collapse
is taken as dust in most calculations and the Fermion/quark composition
(required for the Chandrasekhar limit) of the cosmic soup is mostly ignored.
For a comprehensive review of primordial black hole formation, one is referred
to [6]. Here, we briefly outline the methods used to study matter collapse in
the early universe. One of the main ingredient in the study of collapse in the
early universe is Jean’s instability. This instability characterizes density
fluctuations in a fluid. The formula for Jean’s instability is obtained by
equating the time for free fall (or the time taken for an object of radius $R$
to collapse under its own gravity) to the time taken by a sound wave to cross
the radius. It is therefore a critical radius for which a pressure wave in the
fluid gets trapped. Jean’s critical length can also be obtained by solving for
perturbations flowing in a fluid and the self-gravitational force generated by
the perturbation. In the following, we discuss Jean’s instability.

###  3.2 Jean’s Instability

In this section, we discuss the collapse in a fluid of density $\rho$. This
process also gives a rough description of the physics of a “density”
collapsing under “perturbations” or under its own weight. The time for “free
fall” of a mass in an elliptic orbit of eccentricity one, according to
Kepler’s laws (of planetary motion) is

$\tau^{2}=\frac{\pi^{2}}{2}\frac{R^{3}}{{GM}},$ (42)

where $M$ is the mass causing the orbit, and $R$ is the distance from the
focus of the ellipse. We use this to model self-collapse of a mass under its
own gravity. If the mass collapses, then only half of this time is taken.
Given that the total mass in a radius $R$ of a spherical distribution of
constant density $\rho$ is

$M=\frac{4\pi}{3}R^{3}\rho,$ (43)

approximating the mass using this formula, the time for free fall is given as
a function of density as

$\tau=\sqrt{\frac{3\pi}{32G\rho}}.$ (44)

If the speed of sound in the fluid is $c_{s}$, then the time for sound to flow
through a distance $R$ is

$\frac{R}{c_{s}}.$ (45)

This time would be the same as that a pressure wave flowing through the medium
would take. If the gravitational collapse time is greater than the pressure
wave time, the mass is unstable, and the critical length scale of the fluid
region is given as

$R_{\rm JL}=\left(\frac{3\pi}{32}\right)^{1/2}\frac{c_{s}}{\sqrt{G\rho}}.$
(46)

The same “collapse formula” can be derived using a spherical homogeneous mass
$M$, whose radius increases by a perturbation $\Delta R=-\alpha R$, where
$\alpha$ is a small perturbation. The change in pressure using the formula
$\delta p/\delta\rho=c_{s}^{2}$ can be related to the change in density due to
the compression, and this gives rise to a force and “acceleration” obtained as

$a_{p}=\frac{\delta p}{\rho_{0}R}=\frac{3\alpha c_{s}^{2}}{R}.$ (47)

In the above, we took $\delta\rho=3\alpha\rho_{0}$, where $\rho_{0}$ is the
original density. Simultaneously the shrinking of the radius gives rise to an
increase of the Newtonian acceleration

$a_{\rm g}=\frac{2GM\alpha}{R^{2}}.$ (48)

If the gravitational acceleration exceeds the “pressure acceleration”, the
mass is expected to collapse, which gives a critical length

$\frac{3\alpha
c_{s}^{2}}{R_{C}}=\frac{2GM\alpha}{R_{C}^{2}}=\frac{4\pi}{3}\rho_{0}R_{C}^{3}\frac{2G}{R_{C}^{2}}\rightarrow
R_{c}\propto\frac{c_{s}}{\sqrt{\rho_{0}G}}.$ (49)

Thus, the critical radius for the collapse in a fluid of density $\rho$ is
proportional to the speed of pressure waves $c_{s}$ in the medium. Here, one
of the important assumptions for the calculation of the speed of sound is the
assumption that for the early universe fluid, entropy is conserved. We next
discuss if a change in the description of the fluid of the early universe
might change this Jean’s length. The above discussion on Jean’s instability
can be found in many references, including [35, 36].

###  3.3 A Quantum Entropy Production Fluid and Jean’s Instability

In the above Newtonian derivation of gravitational collapse, the requirement
that the fluid be isentropic may not be true in the early universe. In fact,
entropy production causes the flow of the universe to be as in an “open
system”, where the big bang singularity is resolved [37]. We take a slight
detour and discuss the situation where there is entropy production in the
fluid as anticipated in [37]. In [37], it is conjectured that spacetime can
generate particles which add to the fluid, the energy momentum tensor of the
Einstein equation. This particle creation is a quantum process and might add
insight to the origins of today’s cosmological observations. In [37], it is
shown that in such open systems, cosmological singularity is not formed. In
this review, we briefly discuss whether the open system allows for PBH
formation. The conservation law for open thermodynamic systems is given as

$d(\rho V)+pdV-\frac{h}{n}d(nV)=0,$ (50)

where $n$ is the particle number and $h=\rho+p$ is the “enthalpy” of the
system. In most irreversible systems, as in systems with chemical reactions,
enthalpy is a measure of the energy of the system, and is a path-independent
quantity. The thermodynamics of these systems is controlled by the chemical
potential $\mu$, and the entropy per unit volume “$s$” is defined as

$\mu n=h-Ts,$ (51)

with $T$ being the temperature of the system.

The pressure for this fluid is given as

$p=\frac{n\dot{\rho}}{\dot{n}}-\dot{\rho}.$ (52)

If one assumes a fluid in the form of “radiation”, i.e., $\rho=aT^{4}$ and
$n=bT^{3}$, where $a$ and $b$ are dimensional constants [37], obviously, from
Equation (52), the equation of state is $p=\rho/3$. In such an open system, if
one obtains the propagation equation of a “pressure wave”, then the
conservation of mass and momentum equations are different. In previous work,
the speed of sound in such a fluid was taken as $c_{s}=\sqrt{1/3}$, which was
at constant entropy for the calculation of the Jean’s instability. However,
the speed of sound changes in a fluid with entropy production. We try to see
the origin of the speed of a pressure wave in a gravitating fluid, and it is
nonisentropic, with dynamics given by the equations above. To describe the
propagation of pressure waves in a system, one uses the following equations:
For the conservation of mass equation in the fluid, one has

$\frac{\partial\rho}{\partial t}+\vec{\nabla}\cdot(\rho\vec{v})=\dot{n_{i}},$
(53)

where we have the “convective” derivative of the density and any particle
production on the other side of the equation. The conservation of momentum
equation or Euler’s equation gives (we assume that the fluid is not viscous)

$\frac{\partial(\rho\vec{v})}{\partial
t}+\vec{v}\cdot\vec{\nabla}(\rho\vec{v})=-\vec{\nabla}{p}+\rho g.$ (54)

In the above, the Navier–Stokes equations have been reduced by setting the
viscosity to zero. On the right-hand side, there is a potential term which can
be a gravitational potential term. In all discussions for the speed of sound,
or the speed of pressure waves in the system, the velocity is taken to be
small, and the density and pressure undergo perturbations. We assume no
gravitational potential at this stage. If there is a linear perturbation in
the velocity, density, and pressure of the fluid, with the $\dot{n}$ remaining
the same, the perturbations lead to the following equations

$\frac{\partial\delta\rho}{\partial t}+\rho_{0}\vec{\nabla}\cdot\vec{\delta
v}=0,$ (55)

and

$\rho_{0}\frac{\partial\vec{\delta v}}{\partial t}=-\vec{\nabla}\delta p.$
(56)

If the system is isentropic, i.e., homogeneous, one can take a partial
derivative of Equation (55) and obtain

$\frac{\partial^{2}\delta\rho}{\partial
t^{2}}+\rho_{0}\vec{\nabla}\cdot\frac{\partial\vec{\delta v}}{\partial t}=0.$
(57)

In the above, using Equation (56), one obtains

$\frac{\partial^{2}\delta\rho}{\partial t^{2}}-\nabla^{2}\delta p=0.$ (58)

In the isentropic approximation

$\delta\rho=\left(\frac{\partial\rho_{0}}{\partial p_{0}}\right)_{s}\delta p,$
(59)

one plugs in the above and obtain

$\frac{\partial^{2}\delta\rho}{\partial
t^{2}}-c_{s}^{2}\nabla^{2}\delta\rho=0,$ (60)

and one obtains the speed of propagation of the density perturbations as

$\frac{1}{c_{s}}=\sqrt{\left(\frac{\partial\rho_{0}}{\partial
p_{0}}\right)_{s}}.$ (61)

In case the fluid has entropy changes, they induce a change in volume. One
therefore can obtain for nonisentropic fluids

$\delta\rho=\left(\frac{\partial\rho_{0}}{\partial p_{0}}\right)_{s}\delta
p+\left(\frac{\partial\rho_{0}}{\partial s_{0}}\right)_{p}\delta s.$ (62)

If we use the thermodynamic equation for entropy production as

$\delta s=\left(\frac{\partial s_{0}}{\partial\rho_{0}}\right)_{T}\delta p,$
(63)

then, in the formula for the “density perturbation” velocity, we have

$c=\sqrt{\frac{c^{2}_{s}c^{2}_{p}}{c_{s}^{2}+c_{p}^{2}}},$ (64)

where

$\frac{1}{c_{p}^{2}}=\left(\frac{\partial\rho_{0}}{\partial
s_{0}}\right)_{p}\left(\frac{\partial s_{0}}{\partial p_{0}}\right)_{T}.$ (65)

If we add the gravitational potential in Euler’s equation, then the wave
equation has an inhomogeneous term which has a “force driving term” obtained
from the gradient of a gravitational potential. If we take the potential to
originate from the density, we have $\nabla^{2}\phi_{1}=4\pi G\rho_{0}$, then

$\frac{\partial^{2}\delta\rho}{\partial t^{2}}-c^{2}\nabla^{2}\delta\rho=-4\pi
G\rho_{0}\delta\rho.$ (66)

We assume a plane wave solution for the density wave $\delta\rho\sim
e^{i(\omega t+\vec{k}\cdot\vec{x})}$, and we find

$\omega^{2}-c^{2}k^{2}=4\pi G\rho_{0},$ (67)

so a critical “pressure wave” is identified. For waves with wave numbers above
that, the system will see instability. The critical wave number is given as

$k^{2}=\frac{4\pi G\rho_{0}}{c^{2}}.$ (68)

Jean’s instability is thus identified as perturbations having a wavelength
greater than

$\lambda_{J}>\sqrt{\frac{\pi}{G\rho_{0}}}\,c.$ (69)

Unlike the previous estimate of the length scale where the gravitational
instability sets in, here, the speed of sound is not a mere $\sqrt{1/3}$ as
given in the formula for an isentropic radiation fluid but is obtained using
Equation (64). In a turbulent early universe, therefore, it is expected that
the fluid would be nonisentropic. In addition, the open universe will ensure
entropy production as spacetime generates particle species to add to the
fluid. As the speed differs, so will the threshold for the formation of PBH.
Note the origin of this change from an underlying quantum theory is implicit
in the velocity change of the pressure wave. Note our results for a
nonisentropic fluid is just one way to see how some of the formulas used for
PBH might change; for other origins of change in Jean’s instability formula in
cosmic fluids, see [38].

###  3.4 PBH Formation

How does one obtain the dynamics of formation of PBH in the early universe? It
is postulated that the FLRW universe metric could have perturbations induced
by the density fluctuations of the fluid. These can be modelled using a
spherical symmetry, and the conditions for the formation of “trapped surfaces”
or apparent horizons derived using the “Misner–Sharp” equations. These PBH can
then accrete and grow in size, and there can be PBH formed of masses which are
bigger than the solar masses of $10M_{\circ}$–$30M_{\circ}$. A great deal of
the current work on PBH discusses these and the fraction of PBH contributing
to dark matter halos $f_{PBH}$. For further reading on the PBH production and
the interest in them as contributors to dark matter and physical processes
such as microlensing, etc., refer to [8]. As the black hole formation follows
the same numerical flow as in the spherical collapse obtained by Choptuik, the
PBH’s mass has the following “scaling” formula

$M_{\rm PBH}=K\ M_{H}(t_{H})(\delta_{m}-\delta_{c})^{\gamma},$ (70)

where $\delta_{m}=(\rho-\rho_{b})/\rho_{b}$ is the fluctuation in the fluid
density over the Hubble density, at the radius where a compaction function is
maximum. $\delta_{c}$ is the fluctuation at the critical radius related to the
Jean’s instability in the fluid found earlier. $\delta_{c}$ represents the
threshold of black hole formation. This equation can only be trusted in the
regime $\delta_{m}-\delta_{c}\sim 10^{-2}$. $M_{H}(t_{H})$ is the Misner–Sharp
mass of the horizon, $K$ is a numerical constant. $\gamma$ is a universal
scaling exponent and varies depending on the fluctuation profile and the
equation of state of the fluid. This equation provides the basis for PBH
formation, though using classical equations. The compaction function $C(r,t)$
is defined as the excess of mass over the FLRW mass $M_{b}$ defined as
$M_{b}=4\pi\rho_{b}R^{3}/3$,

$C(r,t)=2\frac{M(r,t)-M_{b}(r,t)}{R(r,t)}.$ (71)

If one takes the perturbation of the FLRW metric to be modelled by a function
$\zeta(r,t)$, in the FLRW metric three-slice as
$a^{2}(t)e^{2\zeta(r,t)}r^{2}d\Omega$, one gets a formula for the compaction
function in terms of this parameterized fluctuation as

$C(r)=\frac{2}{3}\left(1-(1-r\zeta^{\prime}(r))^{2}\right).$ (72)

This facilitates the study of this function in terms of the curvature
fluctuations of the metric. The various calculations of the “peak” values of
this compaction function use different ensembles for the fluctuations and
accordingly, obtain different values. It is postulated that when the
compaction function exceeds a critical value, a collapse occurs, otherwise the
fluctuation dissipates away. The density contrast parameter is related to the
peak value of the compaction function as

$\delta_{m}=C(r_{m}).$ (73)

In this article, we refrain from discussing the various ways of finding PBH
compaction function but only show a way the change in threshold value
$\delta_{c}$ of PBH formation influences the collapse process. This critical
value is related to Jean’s instability in the cosmic fluid and as shown
previously, vary according to the approximations used. A dependence on the
formula for PBH on the nature of the fluid is discussed in [8]. As shown in
Equations (69) and (64), the threshold of the onset of the instability of a
fluid changes if quantum “particle creation” is allowed. In [37], the fluid
exchanges particles with the gravitational “quantum field”. In this open
universe, there is no initial singularity [37], and as we anticipate, the
formation of PBH would also differ. The masses would be different, and the
nature of the cosmological fluctuations of the gravitational metric would also
differ as per the “entropy production” of this open universe. A more detailed
calculation using quantum cosmology is required for the exact changes required
in the theoretical predictions of the PBH’s mass, and the PBH formation from
the cosmic soup.

The formation of PBH can vary from masses of the order of $10^{5}$ g -
$10^{50}$ g, and therefore, they can range from small black holes to larger-
than-solar-mass black holes. The lower limit is based on the Planck mass and
the upper limit is based on the cosmological mass. How can we verify the
existence of PBH? The existence of PBH can be verified using the observation
of particles received on earth, which might have originated from the PBH using
the Hawking radiation process. It is this process which we describe next. We
discuss PBH whose evaporation time $\propto M^{3}$ is about the age of the
universe. These PBH might have radiated away their mass in the form of photons
and neutrinos and would provide evidence for the phenomena of Hawking
radiation. The mass of these black holes is estimated as $<10^{14}{\rm g}$.

Curiously, there was an attempt to find quantum gravity effects on PBH
production using loop quantum cosmology (LQC) corrections to the scale factor
and the density [39]. The authors found that using the LQC-corrected early
universe cosmology, the production of PBH was increased theoretically compared
to estimates from other theoretical models as that of the Brans–Dicke gravity.

###  3.5 Evaporation of PBH

The mechanism of radiation from black holes can be studied using the power law
for the emission of particles. In the 1970s [33, 40, 41], one typically
calculated the power law using Hawking’s formula for the particle flux from
black holes. The total energy radiated per unit time from PBH of Hawking
temperature $T_{H}$ is given as

$\frac{dE}{dt}=\int d\omega\int d\Omega\sum_{lm}\frac{\Gamma_{\omega
slm}}{\exp(\omega/T_{H})\pm 1}$ (74)

where $\Gamma_{\omega slm}$ is the grey-body factor for the black hole
geometry and represents matter waves scattering off the gravitational
potential outside the black hole. $s,l,m$ represent the spin and angular
momentum quantum numbers of particles with frequency $\omega$. The sign in the
denominator is positive for bosons and negative for fermions. The Hawking
temperature for a nonrotating black hole is inversely proportional to the
mass. The grey-body factor is calculated using the solutions to the classical
equation of motion of the particles in the black hole background and is a
function of the spin, angular momentum, mass, and frequency of the emission.
The fraction of power radiated in different species can be calculated. The
total power radiated can be calculated numerically as

$P=2.011\times 10^{-4}\,\hbar c^{5}G^{-2}M^{-2},$ (75)

where $M$ is the mass of the black hole. Most of the above is radiated out in
the form of neutrinos (81.4%), 16.7% as photons and 1.9% as gravitons, as long
as the black holes have mass $M>10^{17}{\rm g}$ [40] After the black hole has
shrunk further, the temperature being higher, and the mass being denser, the
black hole radiates quarks in the form of muons and other particles such as
electrons and positrons. For this range of black holes, $10^{14}$ g
$<M<10^{17}$ g the power radiated was found to be

$P=3.6\times 10^{-4}\,\hbar c^{5}G^{-2}M^{-2},$ (76)

90$\%$ is equally divided in electrons, positrons, and neutrinos, 9$\%$ in
photons, and 1$\%$ in gravitons [40]. In this work, when computing the power
of Hawking particles, the numerical calculations of the grey-body factors were
used, and the above division into fractions were based on the spin of the
particles. The emission of massive particles would have a different
calculation, but for a detection on earth, the massless particles acquire
relevance.

In a follow up work [41], the emission of gamma rays with energy of about 120
MeV was discussed, and a study of “gamma ray bursts” from evaporating PBH was
introduced. In there, a mass distribution was assumed for PBH, and this is an
ingredient in the current analysis of the data received on earth. The search
for Hawking radiation phenomena in the universe is thus a search for
primordial black holes and the particles emitted from them. There are several
searches for primordial black holes using gamma-ray bursts which might be the
evidence of these black holes evaporating. In the next, we describe some of
these searches in detail and provide a bibliography.

###  3.6 Archived Data

The Imaging Compton Telescope (COMPTEL) [42] was decommissioned in 2007, but
there remained the archived data to analyze gamma rays. The search from these
data has shown bounds for the primordial black holes (PBH) <$10^{17}{\rm g}$
[43].

###  3.7 Gamma-Ray Bursts

There are several satellite-based experiments, which are functional or at the
planning stage such as AMEGO and e-ASTROGRAM. AMEGO is an abbreviation for the
All-sky Medium Energy Gamma-ray Observatory experiment and comprises a silicon
tracker, a cesium iodide calorimeter, and a scintillator anti-coincidence
detector. All these will form the payload of a satellite. The detector will
operate in the MeV range and provide a wider field of view than the Fermi-LAT
detector. This detector is planned by NASA. e-Astrogram is a European Science
Commission gamma-ray detector, based on similar instrumentation as AMEGO [44].
The e-Astrogram project aims to observe the frequency range of 0.3 MeV to 3
GeV. It is also aiming to be more sensitive at a particular frequency than
previous instruments. These instruments will send data about the gamma-ray
bursts and other sources which will give a clue on the existence of primordial
black holes in the early universe.

###  3.8 HESS

The HESS is a gamma-ray observation experiment using an array of atmospheric
imaging Cerenkov telescopes with energy in the TeV range. The telescopes are
in Namibia. We report on the techniques of the HESS experiment in details here
as an example, but it is one of several developments for PBH observations
[45]. As the PBH which are smaller than $10^{17}$ g might have evaporated by
now, one searches for gamma-ray burst signals. The PBHs are expected to have
evaporated with an explosion of gamma rays, which have a high energy and last
only for a few seconds. Using statistics and the methods of [46] Feldman and
Cousins, one can estimate the “rate of” the PBH formation density
$\dot{\rho}_{PBH}$, with 95% and 99% confidence levels. Further, we discuss
this experiment’s data analysis [45] in details to illustrate the methodology
of the search of PBH. Let us say an unknown parameter $\mu$ is being assessed
using the measurements of a variable $x$. Usually, one uses Bayesian
statistics to estimate the “belief” in a system’s parameter being $\mu_{t}$.
This is given using the formula

$P(\mu_{t}|x_{0})={\cal L}(x_{0}|\mu)\frac{P(\mu_{t})}{P(x_{0})},$ (77)

where ${\cal L}(x_{0}|\mu_{t})$ is the “likelihood” of obtaining $x_{0}$ given
$\mu_{t}$. However, it is assumed that there is prior knowledge of the
probability $P(\mu_{t})$ of finding $\mu_{t}$ independent of what $x_{0}$ is,
which might not be the case. The probability $P(x_{0})$ can be absorbed in the
normalization of the conditional probability. In Bayesian methods, the belief
in finding $\mu_{t}$ given the measured values of $x$ is expressed as a
“confidence”. This is mathematically

$\int^{\mu_{2}}_{\mu_{1}}P(\mu_{t}|x_{0})d\mu_{t}=\alpha,$ (78)

where $\alpha$ is the degree of confidence for $\mu_{t}$ to be in the
confidence interval $[\mu_{1},\mu_{2}]$. In [46], a variation of this is
given, for estimating the value of a parameter $\mu$ given the measurements of
the variable $x$. If one takes the ratio of two likelihoods, then the “prior
knowledge” required in Bayesian statistics is not there.

$R=\frac{{\cal L}(x|\mu)}{{\cal L}(x|\mu_{\rm best})},$ (79)

where $\mu_{\rm best}$ is the value of the parameter which maximizes the
conditional probability. This ratio determines the acceptance region in the
$x$ variable, for a given value of $\mu$. A sum of the observation
probabilities in decreasing order of $R$, until the required confidence limit
is reached, provides a good estimate for the confidence intervals or upper
limits for a parameter.

In the HESS observations, gamma rays were detected using the Cerenkov
telescopes on earth. The number of photons detected could vary from one to
infinity in a given time interval $\Delta t$. A time interval of $\Delta t=10$
s was taken for the purpose. We assumed that the detection of photon
“clusters” of size $k$ followed a Poisson distribution

$P(k,N)=e^{-N}\frac{N}{k!},$ (80)

where $N(r,\alpha,\delta,\Delta t)$ is the number of $\gamma$ rays emitted
from PBH from a distance $r$ in the angular interval in the sky specified by
$\alpha,\delta$ in unit time $\Delta t$. Integrating this over all space,
i.e., $r,\alpha,\delta$, and over all runs of the experiment, the number of
significant clusters of photons detected were estimated to be

$n_{\rm sig}(k,\Delta t)=\dot{\rho}_{PBH}V_{\rm eff}(k,\Delta t),$ (81)

where

$V_{\rm eff}(k,\Delta t)=\sum_{i}T_{i}\int d\Omega_{i}\int
dr~{}r^{2}~{}P(k,N)=\sum_{i}T_{i}\Omega_{i}\frac{(r_{0}\sqrt{N_{0}})^{3}}{2}\frac{\Gamma(k-3/2)}{\Gamma(k+1)},$
(82)

where $N_{0}$ is the number of photons emitted from PBH at a distance of
$r_{0}$. $T_{i}$ is the run’s live time of the experiment, and $\Omega_{i}$ is
the solid angle of the observations. Based on the observed data, the
statistical analysis using the techniques of Feldman and Cousins was
implemented. The parameter being sought was $n_{\rm sig}$ given $n$ as the
observed variable. Note that these photon clusters, which might be from
evaporating PBH, were received along with the background photons, whose number
was taken as $\bar{n}$, or off photons.

$R=\prod_{n}\frac{{\cal L}(n|\bar{n}+n_{\rm sig})}{{\cal L}(n|\bar{n})}.$ (83)

Here, the maximal value of the likelihood function was taken as that of the
background $\bar{n}$.

The $\chi^{2}$ estimate of the above can be found as [46]:

${\rm LNR}=-2\ln(R)=2\sum_{n}n_{\rm sig}+n(\ln(n)-\ln(\bar{n}-n_{\rm sig})),$
(84)

where $n$ is the number of observed photon signals in the on position of the
telescopes and $\bar{n}$ is the number of mean observed signals in the off
data. This is an estimate of the background photons, obtained by averaging
over “scrambled” time intervals. In deriving the above, we used the Poisson
distribution.

This ${\rm LNR}$ had a maximum of $0.006$ in the preliminary data for $\Delta
t=10$ s and 6240 runs of four of the five telescopes [45]. This showed that
there was not much of the PBH excess data. However, if one sets ${\rm
LNR}=4,9$, one can obtain an upper-limit estimate for $\dot{\rho}_{PBH}$, with
95% and 99% confidence levels. The upper limit was found to be

$\displaystyle\dot{\rho}_{\rm PBH}$ $\displaystyle<$ $\displaystyle 2.5\times
10^{4}/pc^{3}yr\ \ (95\%),$ (85) $\displaystyle\dot{\rho}_{\rm PBH}$
$\displaystyle<$ $\displaystyle 5\times 10^{4}/pc^{3}yr\ \ (99\%),$ (86)

These data points were further updated with other experiments such as VERITAS,
MILAGRO, FERMI-LAT, and SWGO [47]. A comparative plot of the experimental
predictions of evaporating PBH or final bursts at the 99% confidence limit is
given in Figure 2. The data for this are quoted from [48] (2021). For some
recent updates in the field of constraints on PBH see [49].

Figure 2: The upper estimates of the number of final bursts at the 99%
confidence limit from some experiments [48].

For recent data on HESS, one can refer to the experiment’s website [50].

###  3.9 Neutrino Experiments

The Hawking radiation from PBH releases neutrinos. The flux of these as a
function of the PBH production and then a further analysis for “secondary
effects” producing neutrinos were analyzed. The data from several experiments
were taken and showed almost no or a very small estimation of the PBHs. Using
a recent work [51], we comment on the results. A neutrino spectrum rate was
defined using the Hawking emission spectrum as in Equation (74). Further,
there can be secondary neutrino production due to the decay of hadrons
produced initially:

$\frac{d^{2}N_{\nu}}{d\omega_{\nu}dt}=\int_{0}^{\infty}dM\,\frac{d{\cal
N}}{dM}\left(\frac{d^{2}N_{\nu}}{d\omega_{\nu}dt}_{\rm
prim}+\frac{d^{2}N_{\nu}}{d\omega_{\nu}dt}_{\rm sec}\right).$ (87)

where the black hole’s mass distribution could be taken as a Gaussian log-
normal profile,

$\frac{d{\cal N}}{dM}=\frac{1}{\sqrt{2\pi}\sigma
M}\exp\left(-\frac{\ln^{2}(M/M_{\rm PBH})}{2\sigma^{2}}\right),$ (88)

or simply a delta function profile centred at $M=M_{\rm PBH}$. In the above,
$M_{\rm PBH}$ is an average mass, and $\sigma$ is the standard deviation, as
the mass of the black hole is allowed to vary. A plot of the differential
neutrino flux from extragalactic sources and the milky way can be calculated
using publicly available software [51] and plotted. The differential flux of
the neutrino plotted as a function of the energy $\omega_{\nu}$ varied between
$10^{2}$ and $10^{-5}$, as the energy varied from 1 to 100 MeV for PBH of mass
$10^{13}$ g. The evaporated PBH were taken as a fraction of the cosmic
background which is $10^{-18}$ to obtain this result.

The experimental bounds obtained from the Super-Kamiokande data showed that
for PBH which were already evaporated, the abundance ratio was about
$10^{-17}$ for $10^{13}$ g black holes and a confidence limit of 90$\%$. The
question is of course what the above bounds imply for quantum gravity
phenomenology? Whereas the PBH production cannot be ruled out completely,
using the above estimation methods, it remains that the mechanism of PBH
formation could be different, and the emission flux calculations greatly
modified by intervening cosmic flows and quantum effects. In this aspect, one
has to wait for future experiments such as JUNO, DARWIN, ARGO, and DUNE, and
perhaps quantum cosmology predictions of the PBH formation from a more
fundamental theory such as loop quantum gravity.

It is obvious from the above discussions that the detection of bursts of
photons and neutrinos on earth gives a very small window for the PBH to exist
which would be evaporating now, i.e., those having masses $10^{5}{\rm
g}$–$10^{14}{\rm g}$. However, as we know, there can still be the option that
there are PBH which have not evaporated away but have formed remnants. These
will still be candidate dark matter contributors. The fraction of PBH which
contribute to dark matter and have not been evaporated yet is also estimated
as $\sim 10^{-3}$ for masses of the order of $10^{16}{\rm g}$ as in [52].
There are other papers investigating this using various data sources such as
microlensing, accretion disk luminosity, radio signals, anisotropies of the
CMB, etc. We refer the reader to reviews in this field [8]; there are also
discussions of the PBH formation and evaporation using LQG corrected metrics,
though in reduced phase-space formulations [53]. In our opinion, whereas the
search is now much focused than earlier on what a gamma-ray burst or a
neutrino flux from PBH may be, the research is still nascent.

##  4 Event Horizon

In the initial days of the discovery of the black hole metric solution to
Einstein’s equation, the existence of the horizon was one of the most bizarre
predictions. The existence of trapped surfaces in general relativity was later
firmly established with the Ray–Chowdhury equations and Hawking–Penrose
singularity theorems. However, the debate continued on whether the event
horizon existed, as it was unobservable. With the discovery of compact objects
and the observation of X-rays from them, various models were tested for the
existence of the event horizon. As the conclusions were model-dependent, the
search continued, until the event horizon telescope project produced an
“assembled image” of the photon sphere surrounding a black hole [54, 14]. This
confirmed some of the predictions about the behaviour of geodesics near a
black hole’s horizon, but did it confirm the presence of an event horizon?
Perhaps not, but this is as “good as it gets”. The snapshot of the photon
sphere assimilated from eight infrared telescopes captured the electromagnetic
waves circulating a compact object. The question we are asking in this article
is: can we use the observations of geodesics around a black hole to measure
semiclassical physics? In a work using semiclassical states in loop quantum
gravity [13], it was shown that quantum fluctuations could cause instabilities
in black holes, and these could produce tangible detectable effects for
astrophysical black holes [13]. The main results of the paper were the
calculation of a nonpolynomial correction to the metric of the Schwarzschild
black hole. The semiclassically corrected metric was shown to be of the
following form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{r_{g}}{r}-\tilde{t}\
h_{tt}\right)dt^{2}+\tilde{t}\ h_{rt}\
dtdr+\left\\{\frac{1}{(1-r_{g}/r)}+\tilde{t}\ h_{rr}\right\\}dr^{2}+$ (89)
$\displaystyle+\left(r^{2}+\tilde{t}\
h_{\theta\theta}\right)d\theta^{2}+\left(r^{2}\sin^{2}\theta+\tilde{t}\
h_{\phi\phi}\right)d\phi^{2}.$

where $r_{g}$ is the Schwarzschild radius, and the location of the horizon is
at $r_{g}=2GM$, where $M$ is the mass of the black hole.
$h_{tt},h_{rt},h_{rr},h_{\theta\theta},$ and $h_{\phi\phi}$ are the
perturbations motivated from the corrections to the metric [13]. The
perturbations of the metric could be attributed to other quantum models of
gravity, but we used the one motivated from [13], and a shift was generated,
$h_{rt}$, breaking the “static” nature of the metric. The $\tilde{t}$ which
appears in this coherent state was obtained using the length scales of the
system and was thus a ratio of Planck’s area to the area of the horizon
$\tilde{t}=l_{p}^{2}/r_{g}^{2}$. Using this, we solved for the geodesics of
the black hole. The geodesics were taken as circular orbits and the radial
coordinate $r$ was solved as a function of the coordinate $\phi$. These orbits
described the trajectory of light rays which were incident on the black hole
geometry from a distance, and the impact parameter measured the perpendicular
distance of the light ray from the horizon. Using the invariant distance on
the Schwarzschild geometry, one can write the equation of motion for the
geodesic of a photon as a differential equation in the azimuth $\phi$, which
was taken as the affine parameter along the geodesic. The deviations in
geodesic computations for the rotating black hole from the nonrotating black
holes were small [55] but detectable. For rotating black holes, the cross
section of the photon scattering might not be circular [55], but the
difference was about 4%. However, quantum corrections might be different, and
one needs to formulate coherent states for rotating black holes separately.
The effect of the presence of “echoes” might still be true. The results stated
in this paper thus apply to nonrotating black holes strictly but pave the way
for realistic ones.

If we arrange the terms in a way they can be grouped into terms which are
zeroth order in $\tilde{t}$ and then first order in $\tilde{t}$ (in the
equatorial plane), one gets [14]:

$\frac{1}{r^{4}}\left(\frac{dr}{d\phi}\right)^{2}+\frac{1}{r^{2}}\left(1-\frac{r_{g}}{r}\right)\left(1+\tilde{t}\
\frac{h_{\phi\phi}}{r^{2}}-\tilde{t}\ h_{rr}\right)=\frac{1}{b^{2}}\left(1+2\
\tilde{t}\ \frac{h_{\phi\phi}}{r^{2}}-\tilde{t}\ h_{rr}+\tilde{t}\
\frac{h_{tt}}{1-r_{g}/r}\right).$ (90)

As one traces the trajectory through the entire path, the asymptotic angle of
“scattering” from the black hole geometry emerges as a function of the impact
parameter of the photon. The solution is obtained using a set of elliptic
integrals and one finds

$\exp(-\phi_{\infty})=\delta^{1+0.0203~{}\tilde{t}\
}\exp\left(+\frac{0.47~{}\tilde{t}^{1/2}\
}{(0.67\delta+0.225~{}\tilde{t})^{1/2}}+0.23~{}\tilde{t}+1.712~{}\frac{\tilde{t}}{\delta}\right),$
(91)

where $\delta=b-b_{c}$, and $\phi_{\infty}$ is the asymptotic angle the
geodesic makes as it re-emerges to the asymptotic region. The difference of
the photon geodesic impact parameter with the impact parameter of the critical
orbit $b_{c}=3\sqrt{3}M$ is expected to be zero as the photon can orbit an
infinite number of times round the horizon. One can see that in Equation (91),
the $\tilde{t}\rightarrow 0$ reduces to a linear term in $\delta$. Most
importantly, $\delta\rightarrow 0$ as
$\phi_{\infty}=\mu+2n\pi\rightarrow\infty$. $n$ counts the number of times the
geodesic encircles the black hole, and this goes to infinity for the critical
geodesic with the critical impact parameter. The photon circles the black hole
an infinite number of times, when the critical impact parameter is reached. If
we take the semiclassical corrections, then the plot of $w(\delta)$ (the RHS
of Equation (91) as a function of $\delta$ shows that the function does not
reach zero but bounces off (see Figures 3 and 4), and this we can associate
with the presence of a quantization.

Figure 3: Plot of the semiclassically corrected photon geodesic impact
parameter relation. The plot shows a bounce as the distance from the critical
radius approaches the semiclassical length scale of $\tilde{t}\sim 10^{-8}$
units. Figure 4: Plot of the semiclassically corrected photon geodesic impact
parameter relation. The plot shows a bounce as the distance from the critical
radius approaches the semiclassical length scale of $\tilde{t}\sim 10^{-66}$
units.

This observation is commensurate with the work in fuzzballs and ECHOS [56]. In
these models, the horizon is replaced by a “wall” at a particular distance
from the black hole. In our calculations with the LQG coherent states [13], we
found the explicit location of the “wall” as a function of the semiclassical
parameter $\tilde{t}$. We expect that our results can be eventually verified
from observational data from astrophysical black holes [56].

##  5 Conclusions

As it happens, the search for quantum gravity in experiments is still nascent.
However, we expect that in the early universe, the length scales were quantum,
and therefore the search for relics of quantum gravity is ongoing. There are a
number of papers in this Universe special issue in quantum gravity
phenomenology which discuss cosmology and the effect of quantum cosmology in
observational physics. In this review, the experiments we discussed only
provided bounds on the mass of the graviton, the PBH production. We discussed
the quantum effects which could be “directly” observable in recent experiments
including in gravitational wave detectors and event horizon telescope images.
We also reported on the numerous experiments which observe particles from
distant celestial events on earth. We showed theoretical calculations and
reported on bounds from experiments on Hawking emission from PBH. The
experimental bounds did not violate any theoretical predictions. The
observations provide directions for the experimental community to seek for
more precise measurements. The plot of the electric and magnetic polarizations
from the EHT [57] and the launching of LISA [58] are ongoing efforts in that
direction. The study of fast radio bursts (FRB) provided an effort towards
finding the quantum origins of astrophysical phenomena. The most promising
experiments on earth for the quantum effects of gravity remain the GW
detectors and the possibility that one would detect a “graviton” or its
semiclassical version in the near future.

Acknowledgments:This article is written for universe special issue in Quantum
Gravity Phenomenology. AD would like to thank the universe editorial team,
particularly Cici Xia for making this two volumes possible. AD is also
thankful to co-editor Alfredo Iorio for collaboration.

## References

  * [1] Abbott, B. P.; Abbott, R.; Abbott, T. D.; Abernathy, M. R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R. X. ; et al. LIGO Scientific collaboration and Virgo Collaboration, Observation of Gravitational Waves from a Binary Black Hole Merger. _Phys. Rev. Lett._ 2016, 116, 061102.
  * [2] Shao, L.; Wex, N.; Zhou, S. New graviton mass bound from binary pulsars. _Phys. Rev._ 2020, D 102, 024069.
  * [3] Bern, Z. Perturbative Quantum Gravity and its Relation to Gauge Theory, Living Reviews in Relativity. _Living Rev. Relativ._ 2002, 5, 5. https://doi.org/10.12942/lrr-2002-5.
  * [4] Rovelli, C. _Loop Quantum Gravity_ : Cambridge University Press: Cambridge, UK, 2007\.
  * [5] Isi, M.; Farr, W. M.; Giesler, Matthew; Scheel, M. A.; and Teukolsky, S. Testing the Black-Hole Area Law with GW150914. _Phys. Rev. Lett._ 2021, 121, 011103.
  * [6] Escriva, A. PBH formation from spherically symmetric hydrodynamical perturbations: A review. _Universe_ 2022, 8, 66.
  * [7] Carr, B.; Kohri, K.; Sendouda, Y.; Yokoyama, J. Constraints on Primordial Black Holes. _Rep. Prog. Phys._ 2021, 84, 116902.
  * [8] Escriva, A.; Kuehnel, F.; Tada, Y. Primordial Black Holes. _arXiv_ 2022, arXiv:2211.05767v3.
  * [9] Laha, R. Primordial Black Holes as a Dark Matter Candidate Are Severely Constrained by the Galactic Center 511 keV $\gamma$-Ray Line. _Phys. Rev. Lett._ 2019, 123, 251101.
  * [10] Dasgupta, B.; Laha, R.; Ray, A. Neutrino and Positron Constraints on Spinning Primordial Black Hole Dark Matter. _Phys. Rev. Lett._ 2020, 125, 101101.
  * [11] Cappelutti, N.; Hasinger, G.; Natarajan, P. Exploring the High-redshift PBH-$\Lambda$CDM Universe: Early Black Hole Seeding, the First Stars and Cosmic Radiation Backgrounds. _Astro. J._ 2022, 926205 .
  * [12] Dasgupta, B.; Laha, R.; Ray, A. Low Mass Black Holes from Dark Core Collapse. _Phys. Rev. Lett._ 2021, 126, 141105.
  * [13] Dasgupta, A. Quantum Gravity Effects on Unstable Orbits in Schwarzschild Space-time. _Journ. Cosm. Astro. Phys._ 2010, _05_ , 011.
  * [14] Maharana, S.; Dasgupta, A. Semiclassical corrections to the photon orbits of a non-rotating black hole. _arXiv_ 2011, arXiv:2011.00676.
  * [15] Addazi, A.; Alvarez-Muniz, J; Alves Batista, R; Amelino-Camelia, G; Antonelli, V; Arzano, M; Asorey, M; Atteia, J. L.; Bahamonde, S; Bajardi, F; et al. Quantum gravity phenomenology at the dawn of the multi-messenger era—A review. _Prog. Part. Nucl. Phys._ 2022, 125, 103948.
  * [16] Amelino-Camelina, G. Are We at the Dawn of Quantum Gravity Phenomenology. _Lect. Notes. Phys._ 2000, 549, 1–49.
  * [17] Alcubierre, M.; Allen, G.; Brügmann, B.; Lanfermann, G.; Seidel, E.; Suen, W-M; Tobias, M.; Gravitational Collapse of Gravitational Waves in 3D Numerical Relativity. _Phys. Rev._ 2000, D 61, 041501.
  * [18] Bishop, N.T.; Rezzolla, L. Extraction of Gravitational Waves in Numerical Relativity. _Liv. Rev. Relat._ 2016, 19, 2.
  * [19] Dewitt, B.S. Quantum Theory of Gravity. III. Applications of the Covariant Theory. _Phys. Rev._ 1967, 162, 1239.
  * [20] Lauscher, O.; Reuter, M. Is Quantum Einstein Gravity Nonperturbatively Renormalizable? _Class. Quant. Grav._ 2002, 19, 483.
  * [21] Ashtekar, A.; Rovelli, C.; Smolin, L. Gravitons and loops. _Phys. Rev._ 1991, D 44, 1740.
  * [22] Thiemann, T.; Winkler, O. Gauge field theory coherent states (GCS): II. Peakedness properties. _Class. Quant. Grav._ 2001, 18, 2561–2636.
  * [23] Sahlmann, H. Coupling Matter to Loop Quantum Gravity Ph.D. Thesis, University of Potsdam, Potsdam, Germany, 2002.
  * [24] Dasgupta, A. Coherent states for black holes. _Journ. Cosm. Astro. Phys._ 2003, _08_ , 004.
  * [25] Fajardo-Montenegro, J.L.; Dasgupta, A. Semiclassical States for Gravitons work in progress
  * [26] Varadarajan, M. Gravitons from a loop representation of linearized gravity. _Phys. Rev._ 2002, 66 024017\.
  * [27] Varadarajan, M. The graviton vacuum as a distributional state in kinematic loop quantum gravity. _Class. Quant. Grav._ 2005, 22, 1207.
  * [28] Mortuza-Hossain, G. Large volume quantum correction in loop quantum cosmology: Graviton illusion? _arXiv_ 2005, arXiv:gr-qc/0504125.
  * [29] Hinterleitner, F.; Major, S. Toward loop quantization of plane gravitational waves. _Class. Quant. Grav._ 2012, 29, 065019.
  * [30] Garcia-Chung, A.; Mertens, J. B.; Rastgoo, S.; Tavakoli, Y.; and Moniz, P. V. Propagation of quantum gravity-modified gravitational waves on a classical FLRW spacetime. _Phys. Rev. D_ 2021, _103_ , 084053.
  * [31] Hall, B. The Segal-Bargmann "Coherent State” Transform for Compact Lie Groups. _Journ. Funct. Analysis_ 1994, 122, 103.
  * [32] Dasgupta, A. Semiclassical Loop Quantum Gravity and Black Hole Thermodynamics. _SIGMA_ 2013, 19, 013.
https://doi.org/10.3842/SIGMA.2013.013.

  * [33] Hawking, S.W. Particle creation by black holes. _Comm. Math. Phys._ 1975, 43, 199.
  * [34] Choptuik, M.W. Universality and scaling in gravitational collapse of a massless scalar field. _Phys. Rev. Lett._ 1993, 70, 12.
  * [35] Mcmillan, S. Available online: http://www.physics.drexel.edu/~ steve/Courses/Physics-431/jeans$\\_$instability.pdf (accessed on 31/01/2023.).
  * [36] Tomisaka, K. Lecture Notes. Available online: http://th.nao.ac.jp/MEMBER/tomisaka/Lecture$\\_$Notes/StarFormation/6/node36.html (accessed on 28/02/2023).
  * [37] Prigogine, I.; Geheniau, J.; Gunzig, E.; Nardone, P. Thermodynamics of cosmological matter creation. _Proc. Natl. Acad. Sci. USA_ 1988, 85, 7428.
  * [38] Kremer, G.M.; Richarte, M. G.; Teston, F. Jeans instability in a universe with dissipation. _Phys. Rev._ 2018, D 97, 023515.
  * [39] Dwivedee, D.; Nayak, B.; Jamil, M.; Singh, L. P.; Myrzakulov, R. Evolution of Primordial Black Holes in Loop Quantum Cosmology. _Journ. Astrop. Astron._ 2014, 35, 97.
  * [40] Page, D. Particle emission rates from a black hole: Massless particles from an uncharged, non-rotating hole. _Phys. Rev._ 1976, D 13, 198.
  * [41] Page, D.; Hawking, S.W. Gamma rays from primordial black holes. _Astrophys. Journ._ 1976, 206, 1–7.
  * [42] COMPTEL Data Source. Available online: http://cta.irap.omp.eu/ctools/users/tutorials/howto/comptel/index.html (accessed on 31/01/2023).
  * [43] Coogan, A.; Morrison, L. Profumo, Direct Detection of Hawking Radiation from Asteroid-Mass Primordial Black Holes. _Phys. Rev. Lett._ 2021, _126_ , 171101.
  * [44] AMEGO Collaboration. Available online: https://asd.gsfc.nasa.gov/amego/ (accessed on 31/01/2023).
  * [45] Tavernier, T.; Glicenstein, J.-F.; Brun, F. Search for Primordial Black Hole evaporations with H.E.S.S, _arXiv_ 2019, arXiv:1909.01620.
  * [46] Feldman, G.; Cousins, R.J. A Unified Approach to the Classical Statistical Analysis of Small Signals. _Phys. Rev._ 1998, D 57, 3873.
  * [47] Lopez-Coto, R.; Doro, M.; de Angelis, A.; Mariotti, M.; Harding, J. P. Prospects for the observation of Primordial Black Hole evaporation with the Southern Wide field of view Gamma-ray Observatory. _J. Cosmo. Astr. Physics._ 2021, _08_ , 040.
  * [48] Engel, K.; Peisker, A; Harding, P; Wood, J.;Martinez-Castellanos, I; Albert, A; Tollefson, K. on behalf of HAWC Collaboration, Setting Upper Limits on the Local Burst Rate Density of Primordial Black Holes Using HAWC, PoS ICRC2019 516 (2021). DOI: https://doi.org/10.22323/1.358.0516
  * [49] Korwar, M.; Profumo, S. Updated Constraints on Primordial Black Hole Evaporation. _arXiv_ 2023, arXiv:2302.04408.
  * [50] HESS Experiment. Available online: https://www.mpi-hd.mpg.de/hfm/HESS/ (accessed on 31/01/2023).
  * [51] Bernal, N.; Chu, X.; Garcia-Cely, C.; Hambye, T.; and Zaldivar, B. Production Regimes for Self-Interacting Dark Matter. _Journ. Cosmo. Astr. Phys._ 2022, 10, 68.
  * [52] Zhang, Z.H.; Yang, L. T.; Yue, Q.; Kang, K. J.; Li, Y. J.; An, H. P.; Greeshma C., Chang, J. P.; Chen, Y. H.; Cheng, J. P.; et al, Search for keV–MeV Light Dark Matter from Evaporating Primordial Black Holes at the CDEX-10 Experiment. _arXiv_ 2022, arXiv:2211.07477v1.
  * [53] Barrau, A.; Martineau, K.; Moulin, F. A Status Report on the Phenomenology of Black Holes in Loop Quantum Gravity: Evaporation, Tunneling to White Holes, Dark Matter and Gravitational Waves. _Universe_ 2018, 4, 102.
  * [54] Event Horizon Telescope Collaboration. Available online: http://eventhorizontelescope.org/ (accessed on 31/01/2023).
  * [55] Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. _Astr. Phys. J. Lett._ 2019, 875, https://doi.org.10.3847/2041-8213/ab0ec7.
  * [56] Cardoso, V.; Pani, P. Testing the nature of dark compact objects: a status report. _Living. Rev. Rel._ 2019, 22, 4.
  * [57] The Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon. _Astro. J. Lett._ 2021, 910, L12.
  * [58] The LISA Space Telescope. Available online: https://lisa.nasa.gov/ (accessed on 31/01/2023).