CERN-TH-2021-105

# Hawking radiation by spherically-symmetric static black holes for all spins:
II - Numerical emission rates, analytical limits and new constraints

Alexandre Arbey<EMAIL_ADDRESS>Univ Lyon, Univ Claude Bernard
Lyon 1,
CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France  Theoretical
Physics Department, CERN, CH-1211 Geneva 23, Switzerland  Institut
Universitaire de France (IUF), 103 boulevard Saint-Michel, 75005 Paris, France
Jérémy Auffinger<EMAIL_ADDRESS>Univ Lyon, Univ Claude Bernard
Lyon 1,
CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France  Marc Geiller
<EMAIL_ADDRESS>Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1,
CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France  Etera R. Livine
<EMAIL_ADDRESS>Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1,
CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France  Francesco
Sartini<EMAIL_ADDRESS>Univ Lyon, ENS de Lyon, Univ Claude
Bernard Lyon 1,
CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France

###### Abstract

In the companion paper [1] we have derived the short-ranged potentials for the
Teukolsky equations for massless spins $(0,1/2,1,2)$ in general spherically-
symmetric and static metrics. Here we apply these results to numerically
compute the Hawking radiation spectra of such particles emitted by black holes
(BHs) in three different ansatz: charged BHs, higher-dimensional BHs, and
polymerized BHs arising from models of quantum gravity. In order to ensure the
robustness of our numerical procedure, we show that it agrees with newly
derived analytic formulas for the cross-sections in the high and low energy
limits. We show how the short-ranged potentials and precise Hawking radiation
rates can be used inside the code BlackHawk to predict future primordial BH
evaporation signals for a very wide class of BH solutions, including the
promising regular BH solutions derived from loop quantum gravity. In
particular, we derive the first Hawking radiation constraints on polymerized
BHs from AMEGO. We prove that the mass window $10^{16}-10^{18}\,$g for all
dark matter into primordial BHs can be reopened with high values of the
polymerization parameter, which encodes the typical scale and strength of
quantum gravity corrections.

###### Contents

  1. 1 General equations
    1. 1.1 Short-ranged potentials
    2. 1.2 Greybody factors, low and high energy limits
      1. 1.2.1 High energy limit
      2. 1.2.2 Low energy limit
  2. 2 Examples of metrics
    1. 2.1 Charged black holes
    2. 2.2 Higher-dimensional black holes
    3. 2.3 Polymerized black holes
  3. 3 Results and discussion
    1. 3.1 Charged black holes
    2. 3.2 Higher-dimensional black holes
    3. 3.3 Polymerized black holes
    4. 3.4 Constraints from polymerized black holes
  4. A Tortoise coordinates
  5. B Detailed results for Hawking radiation

## Introduction

The most striking feature of black holes (BHs) might be that, in spite of
their name, they (supposedly) radiate particles and slowly evaporate, as first
discovered by Hawking [2]. Hawking radiation (HR) causes BHs to lose mass,
charge and angular momentum, up to a late evaporation stage where physics is
so far unsettled. This late stage is sometimes conjectured to lead either to
complete disappearance or to a stable remnant, but it is clear that a proper
understanding of the final state of a black hole after its evaporation
requires an understanding of the quantum nature of gravity.

BHs in the stellar mass range are observed e.g. at LIGO/VIRGO via the
gravitational waves they emit when binary BHs merge [3, 4], while shadows of
supermassive BHs at the center of galaxies can be probed by large array
interferometers [5]. Primordial BHs (PBHs) formed in the early universe could
be lighter that the Oppenheimer–Volkoff limit, down to the Planck mass. As
Hawking radiation (HR) gets more energetic when the BH mass decreases, one
could hope to observe these light PBHs thanks to the radiation they emit in
every direction. For now, there is currently no direct observational evidence
of HR, resulting in constraints on the abundance of PBHs in the universe, as
those which have not totally evaporated by now ($M_{\rm BH}\gtrsim 10^{14}\,$g
for a Kerr BH) account for some fraction of the dark matter (DM). We refer the
reader to [6] for a complete review on this topic.

Interestingly, there is still an open window in the asteroid mass range
$10^{17}\lesssim M_{\rm BH}\lesssim 10^{21}\,$g for PBHs to represent all of
DM, solving this long-standing issue in cosmology [7, 6], although the lower
part of this window may be closed by future gamma ray observatories such as
AMEGO [8]. This window is precisely constrained by HR predictions. However,
since different BH solutions to the Einstein equations provide different HR
signals, it is worth having a new look at the constraints on the PBH abundance
from various examples of spherically-symmetric static black holes.

In a companion paper [1] we have derived the short-ranged potentials of the
Teukolsky equations for a wide class of BH solutions, namely spherically-
symmetric and static BHs, setting the mathematical background for HR
computations. In the present paper, we use these potentials to compute the HR
of benchmark BH solutions: charged BHs, higher-dimensional BHs and polymerized
BHs. An abundant literature is dedicated to the case of charged BHs (e.g. [9,
10, 11, 12, 13, 14, 15]) and higher-dimensional BHs (e.g. [16, 17, 18, 19],
for a review see [20]), but much less work have focused on polymerized BHs
[21, 22, 23, 24, 25, 26] (see in particular [27]). We stress that our
derivation of the potentials for HR applies in particular to the class of
regular BHs, which are BH solutions showing no singularity at the coordinate
origin. Polymerized BHs, inspired by loop quantum gravity (LQG, see [28] for a
review), are an example of regular BHs [29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45]. Recent work on other regular BH solutions
include e.g. [46, 47, 48, 49, 50], some of which also discuss HR.

In this paper, we use the short-ranged potentials derived in the companion
paper [1] to compute numerically the HR rates for massless particles of spin
0, 1, 2 and $1/2$, thereby completing previous literature. We present in
details three examples as illustrations, namely charged BHs, higher-
dimensional BHs, and polymerized BHs. The important point we wish to emphasize
is that most of the previous studies on HR were performed with heavy use of
numerical methods due to the lack of a general derivation of the short-ranged
potentials. Here we take advantage of the analytical results derived in [1].
As a consistency check, we also compare our results at low and high energy
with analytical limits, some of which are derived for the first time here.
This gives strong support for the robustness of our numerical method based on
short-ranged potentials, which can then be used to predict HR radiation
signals for other BH metrics for which there is no analytical results yet.
During this study, we have also modified the public code BlackHawk [51],
written by a subset of the present authors, to produce the Hawking spectra of
BHs with the new metrics. This improvement will be part of a forthcoming
update of BlackHawk. As a novel application, we use the computed emission
rates for polymerized BHs to reassess the MeV-GeV photon constraint on PBHs
from AMEGO, and show that a sufficiently high value of the polymerization
parameter $\varepsilon$ reopens the mass window $M_{\rm PBH}\lesssim
10^{18}\,$g for all DM in the form of PBHs. This is the first constraint ever
set on polymerized BHs with Hawking radiation.

The paper is organized as follows. Section 1 briefly reviews the results of
the companion paper and gives the important analytical formulas necessary to
the study of Hawking radiation. In Section 2, we introduce the BH metrics on
which we focus, motivate their choice and physical relevance. Section 3
presents our main results, with the Hawking radiation numerical computations
and the new constraint on polymerized primordial black holes. Finally we
conclude and give perspectives for future work.

##  1 General equations

In this Section we summarize the results of the companion paper [1], which
establishes the mathematical framework necessary for the the study of Hawking
radiation emitted by spherically-symmetric static black holes. We introduce
the general metric ansatz and recall the main analytical formulas: short-
ranged potentials, Hawking radiation cross-section with low and high energy
limits.

###  1.1 Short-ranged potentials

We are interested in spherically-symmetric and static metrics, which in
Boyer–Lindquist coordinates take the general form

${\rm d}s^{2}=-G(r){\rm d}t^{2}+\dfrac{1}{F(r)}{\rm d}r^{2}+H(r){\rm
d}\Omega^{2}\,,$ (1.1)

where ${\rm d}\Omega^{2}={\rm d}\theta^{2}+\sin(\theta){\rm
d}\varphi^{2}$.111From now on, we use natural units such that
$G=c=\hbar=k_{\rm B}=4\pi\varepsilon_{0}=1$. We restrict ourselves to black
hole solutions, by which here we mean metrics that are asymptotically flat

$F(r)\underset{r\rightarrow+\infty}{\longrightarrow}1\,,{\qquad\qquad}G(r)\underset{r\rightarrow+\infty}{\longrightarrow}1\,,{\qquad\qquad}H(r)\underset{r\rightarrow+\infty}{\sim}r^{2}\,,$
(1.2)

and present a horizon at some radius $r_{\rm H}$ which is a pole of $F$. For
such metrics the ADM mass of the BH is then [52]

$M=\underset{r\rightarrow+\infty}{\lim}\,\,\dfrac{1}{2}\left(\frac{r}{F}+\dfrac{H}{r}-\partial_{r}H\right).$
(1.3)

We have shown in the companion paper [1] that for these metrics the equations
of motion of spin 0, 1, 2 and $1/2$ massless particles can be transformed into
one-dimensional Schrödinger-like wave equations

$\partial_{*}^{2}Z+\Big{(}\omega^{2}-V\big{(}r(r^{*})\big{)}\Big{)}Z=0\,,$
(1.4)

where the tortoise coordinate is defined as

$\dfrac{{\rm d}r^{*}}{{\rm d}r}\equiv\dfrac{1}{\sqrt{FG}}\,,$ (1.5)

and we have introduced the notation $\partial_{*}\equiv\partial_{r^{*}}$.
These tortoise coordinates $r^{*}(r)$ are explicitly computed for the BHs of
interest in Appendix A. The potentials computed in [1] reduce to

$\displaystyle
V_{0}(r^{*})=\nu_{0}\frac{G}{H}+\frac{\partial_{*}^{2}\sqrt{H}}{\sqrt{H}}\,,$
(1.6a) $\displaystyle V_{1}(r^{*})=\nu_{1}\frac{G}{H}\,,$ (1.6b)
$\displaystyle
V_{2}(r^{*})=\nu_{2}\frac{G}{H}+\frac{(\partial_{*}H)^{2}}{2H^{2}}-\frac{\partial_{*}^{2}\sqrt{H}}{\sqrt{H}}\,,$
(1.6c) $\displaystyle
V_{1/2}(r^{*})=\nu_{1/2}\frac{G}{H}\pm\sqrt{\nu_{1/2}}\,\partial_{*}\left(\sqrt{\frac{G}{H}}\right),$
(1.6d)

where the intermediate $r$ dependency is not shown for conciseness. The spin-
dependent parameters $\nu_{i}$ are given by $\nu_{0}=l(l+1)=\nu_{1}$,
$\nu_{2}=l(l+1)-2$ and $\nu_{1/2}=l(l+1)+1/4$ where $l=s,s+1,\dots$ is the
total angular momentum of the wave and $m=-l,\dots,+l$ its projection. The
shape of the potentials and their discrepancies with the reference
Schwarzschild case as functions of the parameters of the various metric models
are detailed in the companion paper [1].

###  1.2 Greybody factors, low and high energy limits

With the short-ranged potentials at our disposal, it is numerically
straightforward to compute the greybody factors needed to estimate the Hawking
radiation rate. Hawking radiation [2] is a semi-classical phenomenon of quasi-
thermal emission of particles by BH horizons. The rate of emission of one
degree of freedom $i$ per unit time $t$ and energy $E$ is given by

$\dfrac{{\rm d}^{2}N_{i}}{{\rm d}t\,{\rm
d}E}=\sum_{l,m}\dfrac{1}{2\pi}\dfrac{\Gamma_{i}(E,M,x_{j})}{e^{E/T}-(-1)^{2s_{i}}}\,,$
(1.7)

where $s_{i}$ is the spin of the particle $i$ and $T$ is its Hawking
temperature given by

$T(M,x_{j})=\dfrac{\kappa}{2\pi}\,,$ (1.8)

where $\kappa$ is the surface gravity of the BH. This latter is obtained from
the formula

$\kappa^{2}\equiv\left.-\dfrac{1}{2}\nabla_{\mu}k_{\nu}\nabla^{\nu}k^{\mu}\right|_{\rm
hor}=\left.\dfrac{1}{4}\frac{FG^{\prime 2}}{G}\right|_{\rm hor}\,,$ (1.9)

where $k^{\mu}=(1,0,0,0)$ is the timelike Killing vector and “hor” denotes the
horizon $r=r_{\text{H}}$. The greybody factor is the probability that a
particle generated by thermal fluctuations at the horizon escapes to spatial
infinity. If we consider a wave function which is purely ingoing on the
horizon, i.e. with

$Z(r^{*})\underset{r^{*}\rightarrow-\infty}{\sim}A_{\rm hor}^{\rm
in}e^{-i\omega r^{*}}\,,$ (1.10)

and given at infinity by

$Z(r^{*})\underset{r^{*}\rightarrow+\infty}{\sim}A_{\rm\infty}^{\rm
in}e^{-i\omega r^{*}}+A_{\rm\infty}^{\rm out}e^{+i\omega r^{*}}\,,$ (1.11)

then the greybody factor is obtained as

$\Gamma_{i}(E,M,x_{j})=\left|\dfrac{A_{\rm hor}^{\rm in}}{A_{\rm\infty}^{\rm
in}}\right|^{2}\,.$ (1.12)

In general, the greybody factor and the temperature (or the surface gravity)
depend on the BH mass $M$ but also on the precise shape and parameters of the
metric, that is the set of $x_{j}$. As we have decomposed the wave in spin-
weighted spherical harmonics to obtain the radial potentials in [1], the
greybody factors $\Gamma_{i}$ also depend on the particle $i$ angular momentum
parameters $(l,m)$. The spherical symmetry reduces the sum on $m$ in Eq. (1.7)
to a factor $2l+1$. To compute these greybody factors, we have used the same
kind of Mathematica scripts that are given with the public code BlackHawk
[51]. Finally, we also define the cross-section $\sigma_{i}$ for a particle
$i$ by

$\sigma_{i}\equiv\dfrac{\pi}{E^{2}}\Gamma_{i}\,.$ (1.13)

We emphasize here that the new HR results obtained in this paper will be part
of a forthcoming update of BlackHawk, where the aforementioned scripts and
tabulated greybody factors will be publicly available.

####  1.2.1 High energy limit

The high energy limit is usually called the “geometrical optics”
approximation, because fields of all spins experience the BH as an optical
obstacle whose extension is given by the effective area $A_{\rm eff}(x_{j})$
enclosed by the last unstable circular orbit. This area depends on the set of
parameters $x_{j}$ of the BH metric. Let $b_{\rm c}$ be the critical impact
parameter for which the incoming massless fields would reach an unstable
circular orbit of radius $r_{\rm c}$. For a general BH metric of the form
(1.1), this is called the “photon sphere”, defined as the innermost unstable
circular orbit for a massless test particle in rotation around the black hole.
Photons follow null geodesics, meaning that for an affine parameter $\lambda$
we have

$0=g_{\mu\nu}\frac{{\rm d}x^{\mu}}{{\rm d}\lambda}\frac{{\rm d}x^{\nu}}{{\rm
d}\lambda}\,.$ (1.14)

Along every geodesic there are two conserved quantities, the energy $E$ and
the angular momentum $L$, associated respectively to the Killing vector fields
$\partial_{t}$ and $\partial_{\varphi}$. Using the metric ansatz (1.1) they
are given by

$E\equiv G\frac{{\rm d}t}{{\rm d}\lambda}\,,\qquad\qquad L\equiv H\frac{{\rm
d}\varphi}{{\rm d}\lambda}\,.$ (1.15)

Inserting these expressions into Eq. (1.14) and choosing a planar orbit at
$\theta=\pi/2$, we get

$\left(\frac{{\rm d}r}{{\rm
d}\lambda}\right)^{2}=\frac{1}{F}\left(\frac{E^{2}}{G}-\frac{L^{2}}{H}\right)\equiv-
V_{\rm{eff}}\,.$ (1.16)

The radial acceleration is then given by $\dfrac{{\rm d}^{2}r}{{\rm
d}\lambda^{2}}=-V_{\rm{eff}}^{\prime}$. Let us remark that we could obtain the
same result by directly calculating the radial geodesic equation. In order to
have an unstable circular orbit there must be a critical radius $r_{\rm c}$
such that $V_{\rm{eff}}^{\prime}(r_{\rm c})=0$ and
$V_{\rm{eff}}^{\prime\prime}(r_{\rm c})<0$. Since on this orbit the radial
velocity must be vanishing, we can also use Eq. (1.16) to constrain the energy
and the angular momentum to satisfy

$V_{\rm{eff}}(r_{\rm c})=0\,\Rightarrow\frac{L^{2}}{H(r_{\rm
c})}=\frac{E^{2}}{G(r_{\rm c})}\,.$ (1.17)

It is then straightforward to verify that the condition for the unstable orbit
reduces to

$\displaystyle V_{\rm{eff}}^{\prime}(r_{\rm c})=0\quad$
$\displaystyle\Rightarrow\quad\dfrac{G^{\prime}(r_{\rm c})}{H^{\prime}(r_{\rm
c})}-\dfrac{G(r_{\rm c})}{H(r_{\rm c})}=0\,,$ (1.18a) $\displaystyle
V_{\rm{eff}}^{\prime\prime}(r_{\rm c})<0\quad$ $\displaystyle\Rightarrow\quad
H(r_{\rm c})G^{\prime\prime}(r_{\rm c})-H^{\prime\prime}(r_{\rm c})G(r_{\rm
c})<0\,.$ (1.18b)

The critical impact parameter for a massless particle is then defined with
respect to the energy and the angular momentum as

$b_{\rm{c}}^{2}\equiv\frac{L^{2}}{E^{2}}=\frac{H(r_{\rm c})}{G(r_{\rm c})}\,.$
(1.19)

Finally, the effective area (classical scattering) of the BH is given in $4+n$
dimensions as

$\sigma_{\infty}=A_{\rm eff}(x_{j})=\dfrac{\pi^{(n+2)/2}b_{\rm
c}^{n+2}}{\Gamma\big{(}(n+4)/2\big{)}}\,,$ (1.20)

where $\Gamma$ is the Euler gamma function.

For the particular example of a so-called $tr$-symmetric metric, for which we
have $F(r)=G(r)\equiv h(r)$ and $H(r)=r^{2}$ in Eq. (1.1), the conditions
(1.18) reduce to [53, 54]

$h^{\prime}(r_{\rm c})-\dfrac{2}{r_{\rm c}}h(r_{\rm c})=0\,,\qquad
h^{\prime\prime}(r_{\rm c})-\dfrac{2}{r_{\rm c}^{2}}h(r_{\rm c})<0\,.$ (1.21)

Then, the impact parameter is given by

$b_{\rm c}=\dfrac{r_{\rm c}}{\sqrt{h(r_{\rm c})}}\,.$ (1.22)

In the Schwarzschild case, we obtain for all particle spins (see e.g. [55] and
references therein)

$\sigma_{\infty}=27\pi M^{2}\equiv\sigma_{\rm GO}\,,$ (1.23)

which is the usual geometric approximation cross-section.

####  1.2.2 Low energy limit

At low energy, massless fields of each spin behave differently. Several
methods have been used to obtain the low energy limits for the cross-section.
Classical scattering arguments apply to the low energy limit of the spin 0
field, as can be found in [21] which uses the partial wave decomposition and
the small angle approximation.

More generally, authors use a “matching” method which consists in reducing the
radial spin-dependent Teukolsky equation (see [1]) to a simplified version in
the “far field” region ($r\rightarrow+\infty$) and in the “horizon” region
($r\rightarrow r_{\rm H}$). Depending on the number of poles (number of
horizons) of the metric components $F$ and $G$, the differential equation
obtained is some form of a Heun equation. This dependency on the spin makes it
difficult to provide a general procedure to obtain the desired equation,
especially in the general case $F\neq G$ studied here. Then, solving in each
region independently, while applying the correct boundary conditions, and
matching the two solutions in the intermediate region gives an analytical
expression for the greybody factor. Arguments that justify the matching
procedure are given e.g. in [13]. This expression can be expanded in the limit
$\omega r_{\rm H}\rightarrow 0$ to obtain the analytical low energy limit for
the cross-section. One further simplification is that at low energy, only the
lowest momentum partial wave $l=s$ participates significantly to the result.
This is the method used e.g. in [13, 16, 17, 18, 56]. In the Schwarzschild
case (denoted by a superscript S), the low energy limits for the various spins
studied here are (see e.g. [55] and references therein)

$\begin{array}[]{ll}\sigma_{0}^{\rm S}=4r_{\rm
S}^{2}\,,&{\qquad\qquad}\sigma_{1}^{\rm S}=\dfrac{4}{3}r_{\rm
S}^{4}E^{2}\,,\\\ \sigma_{2}^{\rm S}=\dfrac{4}{45}r_{\rm
S}^{6}E^{4}\,,&{\qquad\qquad}\sigma_{1/2}^{\rm S}=\dfrac{1}{2}r_{\rm
S}^{2}\,.\end{array}$ (1.24)

##  2 Examples of metrics

To illustrate our results, we have selected three different metrics for which
we will compute the HR : $i)$ charged BHs, $ii)$ higher-dimensional BHs,
$iii)$ polymerized BHs arising from loop quantum gravity. In this Section we
briefly present the corresponding metrics and the parameters $x_{j}$ which
they contain.

###  2.1 Charged black holes

The no-hair theorem states that a BH (in general relativity coupled to
electromagnetism) is entirely characterized by its mass $M$, angular momentum
$J$ and electric charge $Q$. Astrophysical BHs are expected to have a sizeable
spin due either to their formation mechanism through the collapse of stars
(for stellar BHs) or due to long-term accretion of orbiting gas clouds (for
supermassive BHs at the center of galaxies). These are however not expected to
have a sizeable electric charge because their environment of formation is
typically electrically neutral. However, primordial BHs can have either zero
spin (if formed during radiation domination era) or high spin (if formed
during matter domination era) [57, 58, 59, 60]. The spin is bounded to
$a\equiv J/M<M$ to avoid the breaking of the horizon and the appearance of a
naked singularity. HR makes this spin decrease in time because the emission of
particles with an angular momentum aligned to the BH spin is preferred. The
paradigm is exactly the same for electric charge. The electric charge is
bounded to222The general relation for a Kerr–Newman (charged and rotating BH)
is $a^{2}+Q^{2}<M^{2}$. $Q<M$ and decreases in time because emission of
particles of the same charge as the BH is preferred [9, 10, 11, 12]. The
universe is supposed to be neutral at all times, but processes in the early
universe could produce electrically charged regions that collapse into
electrically charged PBHs (astrophysical BHs are generally assumed to be
neutral). The charge can be maintained until the present epoch if it was
initially very close to the extremal case $Q\lesssim M$, with the same kind of
evolution as described for the spin in [60]. Random fluctuations of the charge
would nevertheless remain until decay has reached the Planck mass [12, 15].

Our formalism is not adapted to treat the emission of particles that have
(electromagnetic) couplings with the background metric; therefore we did not
compute the Hawking rates of charged particles in this particular example.
With our notation for the general family (1.1), the metric of a
Reissner–Nordström BH is [1]

$\begin{matrix}F(r)=G(r)=1-\dfrac{r_{\rm
S}}{r}+\dfrac{r_{Q}^{2}}{r^{2}}=\dfrac{(r-r_{+})(r-r_{-})}{r^{2}}\,,\qquad
H(r)=r^{2}\,,\end{matrix}$ (2.1)

where $r_{\rm S}\equiv 2M$ is the Schwarzschild radius and $r_{Q}^{2}=Q^{2}$
in our system of units. The temperature is given by

$T_{Q}(M,Q)=\dfrac{\kappa_{Q}}{2\pi}=\dfrac{r_{+}-r_{-}}{4\pi r_{+}^{2}}\,,$
(2.2)

where the horizon radii are

$r_{\pm}(M,Q)\equiv M\left(1\pm\sqrt{1-\dfrac{Q^{2}}{M^{2}}}\right).$ (2.3)

In the limit $Q\rightarrow 0$ we recover the Schwarzschild case with
$r_{+}=r_{\rm S}$ and $r_{-}=0$. In the opposite limit $Q\rightarrow M$ we
obtain $r_{+}=r_{-}$ and $T_{Q}=0$, which means that there is no Hawking
emission; the so-called extremal BH is eternal. Overall, the temperature—and
thus the emission power—decreases as $Q$ increases, because $r_{-}\rightarrow
r_{+}$. The result of integration of Eq. (1.5) for a charged BH is given in
Appendix A.

###  2.2 Higher-dimensional black holes

Although general relativity is typically studied in four spacetime dimensions,
which coincides fairly well with astronomical observations, small extra
spatial dimensions are not ruled out by particle physics experiments, nor by
the propagation of gravitational waves (see [19] and references therein). As
in [19], we will consider here large extra dimensions, that is to say
dimensions with typical size larger than the Planck size $R\gg\ell_{\rm P}$,
and small BHs, that is to say BHs with horizon radius $r_{\rm H}\ll R$.
Hawking radiation on the bulk and in the brane of such BHs was previously
studied in e.g. [18]. On the 4-dimensional brane, the metric is

$F(r)=G(r)=h(r)\equiv 1-\left(\dfrac{r_{\rm H}}{r}\right)^{n+1},\quad
H(r)=r^{2}\,,$ (2.4)

where $n>0$ is the number of extra dimensions and the horizon radius is given
by

$r_{\rm
H}=\dfrac{1}{\sqrt{\pi}M_{*}}\left(\dfrac{M}{M_{*}}\right)^{1/(n+1)}\left(\dfrac{8\Gamma\big{(}(n+3)/2\big{)}}{n+2}\right)^{1/(n+1)},$
(2.5)

where $\Gamma$ is the Euler gamma function and the rescaled Planck mass is
$M_{\rm Pl}^{2}=M_{*}^{n+2}R^{n}$. The temperature is

$T_{n}=\dfrac{\kappa_{n}}{2\pi}=\dfrac{n+1}{4\pi r_{\rm H}}\,.$ (2.6)

We recover the Schwarzschild results for $n=0$. The result of integration of
Eq. (1.5) for a higher-dimensional BH is given in Appendix A.

###  2.3 Polymerized black holes

Polymerized BHs have emerged as an effective template for black holes in loop
quantum gravity333The term “polymerized” refers to the polymer-like
quantization scheme inherited from loop quantum gravity [28].. They are
studied by applying the techniques of loop quantum gravity and loop quantum
cosmology [61] to mini-superspace black hole spacetimes. Although many models
have been proposed in the literature (see e.g. the non-exhaustive list [29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]), depending on
the details of the regularization of the Hamiltonian, here we focus for
definiteness on the particular class of effective metrics derived in [62, 63].
These metrics are regular, i.e. do not admit a singularity at $r=0$, and
remain asymptotically flat. The resolution of the singularity arises from
effects of quantum geometry which become relevant at the Planck scale. Here we
treat these metrics as phenomenological ansatz for BHs taking into account
loop quantum gravity corrections at the semi-classical level.

Identifying possible signatures of quantum gravity has recently regained
interest due to the increasing precision in the detection of gravitational
waves. Indeed, the last phase of BH merging, known as the ringdown phase, is
extremely sensitive to the details of the metric structure of BHs. The
deformed BH resulting from the coalescence of two BHs settles down to a stable
(axially) symmetric shape by emitting gravitational waves of defined
frequencies known as quasi-normal modes. The determination of these modes for
a given metric and the comparison with the ringdown signal could discriminate
between different models of BHs [64].

The stability of polymerized BHs, the equations governing their geometry, and
their Hawking radiation were studied e.g. in [22, 23, 24, 25, 26, 27, 21] for
scalar waves and gravitational waves. The results for spin 1 are new to this
paper. In reference [27] one can find some results for the massless field of
spin $1/2$. Our results for this case however differ quantitatively from [27],
as we will see below. It is also noteworthy that no constraint has yet been
derived for these polymerized BHs using Hawking radiation. For the first time,
we give such a constraint on the primordial black holes abundance as a dark
matter component in Section 3.4.

We now turn to the precise description of the model. The polymerized BH metric
studied in [62, 63] has defining functions given by

$\displaystyle
G=\dfrac{(r-r_{+})(r-r_{-})(r+\sqrt{r_{+}r_{-}})^{2}}{r^{4}+a_{0}^{2}}\,,$
(2.7a) $\displaystyle
F=\dfrac{(r-r_{+})(r-r_{-})r^{4}}{(r+\sqrt{r_{+}r_{-}})^{2}(r^{4}+a_{0}^{2})}\,,$
(2.7b) $\displaystyle H=r^{2}+\dfrac{a_{0}^{2}}{r^{2}}\,.$ (2.7c)

Here $a_{0}$ and $\varepsilon$ are the two parameters encoding the quantum
gravity deformation from the Schwarzschild metric. The parameter $a_{0}$ is
the minimal area in loop quantum gravity, also referred to as the area gap. It
is typically of the Planck scale. The deformation parameter $\varepsilon\geq
0$ is an a priori independent parameter indicating the typical scale of the
geometry fluctuations in the Hamiltonian constraints of the theory as they get
renormalized from the Planck scale to astrophysical scales. Although one can
be tempted to keep it very small $\varepsilon\ll 1$, nothing a priori forbids
it from growing large and it can be interesting to consider possible high
values of this deformation parameter in order to understand the effects of
loop quantum gravity corrections.

The two roots of the metric components are $r_{+}=2m$ and
$r_{-}=2mP(\varepsilon)^{2}$. This identifies the first important effect of
the polymerization: the black hole, even without electric charge, acquires a
Cauchy horizon at $r=r_{-}$ on top of the event horizon at $r=r_{+}$. If the
deformation parameter $\varepsilon$ is sent to 0, the radius $r_{-}$ is also
sent to 0 (even if the area gap remains non-vanishing) and we recover a
Schwarzschild-like metric. On the other hand, as $\varepsilon$ grows large,
$r_{-}$ grows to $r_{+}$ (although always remaining smaller) and the
polymerized BH geometrically behaves as if it carried a non-vanishing charged
energy-momentum tensor although it of course does not create an
electromagnetic field. Finally, the mass parameter444We stress that the
effective mass quantity $m$ defined here has nothing to do with the angular
momentum projection $m$ used earlier; as we study spherically-symmetric BHs,
the angular momentum projection plays no particular role. $m$ is related to
the ADM mass $M$ by $M=m(1+P)^{2}$ where the polymerisation factor is

$P=\dfrac{\sqrt{1+\varepsilon^{2}}-1}{\sqrt{1+\varepsilon^{2}}+1}\,.$ (2.8)

The temperature is

$T_{\rm LQG}=\dfrac{\kappa_{\rm LQG}}{2\pi}=\dfrac{4m^{3}(1-P^{2})}{32\pi
m^{4}+2\pi
a_{0}^{2}}=\dfrac{r_{+}^{2}(r_{+}-r_{-})}{4\pi(r_{+}^{4}+a_{0}^{2})}\,.$ (2.9)

We can see that when $\varepsilon\ll 1$, the change in the temperature is
quite negligible compared to the Schwarzschild case. We thus expect rates of
emission for polymerized BHs close to the classical case. However, we see that
in the limit $\varepsilon\rightarrow+\infty$, the radii collapse
$r_{-}\rightarrow r_{+}$ and the temperature goes to $T_{\rm LQG}\rightarrow
0$, cancelling Hawking radiation, similarly to the charged BH with
$Q\rightarrow M$. We thus expect a close behaviour of the emission rates:
overall, the temperature—and thus the emission power—would decrease when
$\varepsilon$ increases. What we also remark is that unless the BH has a
horizon radius $r_{+}^{2}\gtrsim a_{0}$, the value of $a_{0}$ would have a
marginal effect on the temperature (and emission rates). The result of
integration of Eq. (1.5) for a polymerized BH is given in Appendix A.

##  3 Results and discussion

In this Section, we show the Hawking radiation of BHs with the different
metrics of Section 2. This is the first time Hawking radiation signals from
these metrics are compared, in particular in the spin 1 massless (photon) and
spin $1/2$ massless (Weyl neutrino) case of polymerized BHs emission. As
literature already exists for the case of charged Reissner–Nordström BHs [12]
and higher-dimensional Schwarzschild BHs [16, 17, 18, 19], we put the full
spectra of Hawking radiation for those in the Appendix B (see Figs. 5 and 6).

We confront our results to existing literature on analytical cross-sections
when available to conclude on the validity of our potentials and the precision
of the numerical computation. To compare our results to the limit at high
energies, we define the quantity

$\beta_{\infty}\equiv\dfrac{\sigma_{\infty}}{\sigma_{\rm GO}}=\dfrac{A_{\rm
eff}(x_{j})}{(27/4)\pi r_{\rm S}^{2}}\,,$ (3.1)

where $\sigma_{\rm GO}$ is given in Eq. (1.23). In the low energy limit, the
asymptotic expression for the cross-section depends on the spin $s$ of the
radiated field. We thus define the quantities

$\beta_{s}\equiv\dfrac{\sigma_{s,{\rm low}}}{\sigma_{s,{\rm S}}}\,,$ (3.2)

where the low energy limits in the Schwarzschild case are given in Eqs.
(1.24). We fit our numerical data at low and high energies to obtain the
coefficients $\beta_{\infty}$ and $\beta_{s}$. We also check that the exponent
of the energy dependency corresponds to the expected one up to $0.01\%$
precision (0 for spins 0 and $1/2$, 2 for spin 1 and 4 for spin 2). The
fitting of the high energy constant limit is complicated by the oscillatory
behaviour of the cross-section at high energies (see full spectra in the
Appendix B). As we reach the asymptotic value from below for spin 2, the
fitting procedure always slightly underestimates the value of
$\beta_{\infty}$, but the coherence with other spins results is clear. In all
3 examples presented here, we verify that when the extra-Schwarzschild
parameters go to 0 (charge, number of extra dimensions, polymerization
parameter) we recover the Schwarzschild results, that is $\beta_{\infty}=1$
and $\beta_{s}=1$.

###  3.1 Charged black holes

The full Hawking radiation spectra of charged BHs described by the metric
(2.1) for massless fields with spins 0, 1, 2 and $1/2$ is shown in Fig. 5 of
Appendix B. We checked that our results are consistent with existing
literature [12]. As expected, an increase in the charge $Q$ leads to a
decrease in the temperature and thus in the emitted power: the higher the BH
charge, the smaller the HR rate; with the emission peaking at a smaller
energy.

At high energy and for a Reissner–Nordström BH with charge $Q$, it was shown
in e.g. [14] that the asymptotic limit for the cross-section for all spins is

$\beta_{\infty}^{Q}=\dfrac{(3M+\sqrt{9M^{2}-8Q^{2}})^{4}}{54r_{\rm
S}^{2}(3M^{2}-2Q^{2}+M\sqrt{9M^{2}-8Q^{2}})}\,,$ (3.3)

which can also be obtained from Eq. (1.20). This quantity, compared to our
results, is shown in Fig. 1 (left panel). At low energies, the general study
of [13] applies and we can predict that the low frequency limits are for all
spins

$\displaystyle\beta_{0}^{Q}=r_{+}^{2}/r_{\rm S}^{2}\,,$ (3.4a)
$\displaystyle\beta_{1}^{Q}=r_{+}^{2}(r_{+}-r_{-})^{2}/r_{\rm S}^{4}\,,$
(3.4b) $\displaystyle\beta_{2}^{Q}=r_{+}^{3}(r_{+}-r_{-})^{3}/r_{\rm
S}^{6}\,,$ (3.4c) $\displaystyle\beta_{1/2}^{Q}=(r_{+}-r_{-})^{2}/r_{\rm
S}^{2}\,.$ (3.4d)

The comparison to our results is shown in Fig. 1 (right panel). From these
comparisons with theoretical limits at high and low energy, we can conclude
that our numerical computation of the HR from charged BHs is very satisfying.

Figure 1: Left: Comparison between the high energy asymptotic cross-section of
Eq. (3.3) for charged BHs (red solid line) and our computed results for all
spins (black markers). Right: Comparison between the low energy asymptotic
cross-section of Eqs. (3.4) (solid lines) and our computed results (markers)
for all spins. Be careful of the inverted log-scale $x$-axis.

###  3.2 Higher-dimensional black holes

The full HR spectra for massless fields of spins 0, 1, 2 and $1/2$ in the case
of higher-dimensional Schwarzschild BHs described by the metric (2.4) are
given in the Appendix B (see Fig. 6). The general trend is an increase in the
horizon temperature with an increasing number of large extra dimensions $n$
(see Eq. (2.6)), resulting in more energetic HR. One unusual feature is that
the oscillatory behaviour of the cross-section at high energies is damped. We
have checked for consistency with the spectra of [18, 19].

In [18] (and references therein) it is shown that for every dimension $n$, the
limiting value of the effective area of the horizon is $A_{\rm eff}(n)=4\pi
r_{\rm c}^{2}$ where

$r_{\rm
c}\equiv\left(\dfrac{n+3}{2}\right)^{1/(n+1)}\sqrt{\dfrac{n+3}{n+1}}r_{\rm
H}\,,$ (3.5)

which implies that the high energy cross-section satisfies

$\beta_{\infty}^{n}=\left(\dfrac{r_{\rm H}}{r_{\rm
S}}\right)^{2}\left(\dfrac{n+3}{2}\right)^{2/(n+1)}\left(\dfrac{n+3}{n+1}\right),$
(3.6)

which could also have been obtained thanks to Eq. (1.20). The agreement
between the theory and our results is shown in Fig. 2 (left panel). On the
other hand, at low energy we obtain cross-sections compatible with [16, 17,
18]

$\displaystyle\beta_{0}^{n}=r_{\rm H}^{2}/r_{\rm S}^{2}\,,$ (3.7a)
$\displaystyle\beta_{1}^{n}=\dfrac{4r_{\rm H}^{4}}{r_{\rm
S}^{4}}\left[\dfrac{\Gamma(1/(n+1))\Gamma(2/(n+1))}{(n+1)\Gamma(3/(n+1))}\right]^{2}\,,$
(3.7b) $\displaystyle\beta_{2}^{n}=\dfrac{16r_{\rm H}^{6}}{r_{\rm
S}^{6}}\left[\dfrac{\Gamma(1/(n+1))\Gamma(4/(n+1))}{(n+1)\Gamma(5/(n+1))}\right]^{2}\,,$
(3.7c) $\displaystyle\beta_{1/2}^{n}=2^{4-4/(n+1)}r_{\rm H}^{2}/r_{\rm
S}^{2}\,.$ (3.7d)

We note that the low energy asymptotic limit for spin 2 is not given
explicitly in [17]. However, their equations (37) and (38) are valid for spin
2, as shows a careful follow-up of all the steps from the Teukolsky master
equation. We also point out that in [65] only the analytical results for spin
2 emission in the bulk are given, while we focus here on brane emission of
massless gravitons. The results are shown in Fig. 2 (right panel). The
comparison of the theoretical asymptotic limits and our numerically computed
spectra once again shows very good agreement.

Figure 2: Left: Comparison between the high energy asymptotic cross-section of
Eq. (3.6) for higher-dimensional BHs (red solid line) and our computed results
for all spins (black markers). Right: Comparison between the low energy
asymptotic cross-section of Eqs. (3.7) for higher-dimensional BHs (solid
lines) and our computed results for all spins (markers).

###  3.3 Polymerized black holes

Figure 3: Left: Comparison between our high energy asymptotic limits for the
cross-section of high energy fields with polymerized BHs (black markers), in
the case $a_{0}=0$. The approximate formula (3.10) is shown as a red solid
line. Right: Comparison between the low energy asymptotic cross-section of
Eqs. (3.11) for polymerized BHs (blue line) and our computed results for spin
0 (blue crosses), as well as the computed low energy cross-sections for the
other spins (green, red and purple markers) for which literature is not
available.

The full HR spectra for massless fields of spins 0, 1, 2 and $1/2$ in the case
of a polymerized BH described by the metric (2.7) are given in the Appendix B
(see Figs. 7 and 8 for low and high values of $\varepsilon$ respectively).
Most of these results are new, as we now explain.

In the literature, the spin 0 case has historically been treated first,
theoretically in [23, 24, 25] and numerically in [21]. Then, spins 0 and $1/2$
have been studied in [27], where in the spin $1/2$ case the Teukolsky equation
is solved numerically without the intermediate step of deriving the short-
ranged potential. Concerning the massless spin 2 perturbations (and other
spins as well), they were only studied in the case of quasi-normal modes
(QNMs) [64, 66]. These studies invoke the same kind of potentials as in Eqs.
(1.6), but solve the Schrödinger-like wave equation with different boundary
conditions to find the quasi-normal frequencies. It was not possible to
compare our full spectra for low values of $\varepsilon$ for spins $0$ and
$1/2$ to those of [27] due to unspecified normalization factors in their
figures 2, 3 and 4. Moreover, is seems that we find results differing from
theirs for massless fields of spin $1/2$: they predict a distortion of the
spectra at high energies that we do not observe in the right panel of Fig. 7
corresponding to the fermionic field. The height of the first peak in the spin
$1/2$ cross-section seems to follow a different tendency in our results
compared to theirs when $\varepsilon$ increases. However we find, as they do,
that increasing the parameter $\varepsilon$ leads to a decreasing HR rate. A
linear scale has been used in the left panels of Fig. 7 to make this statement
more obvious, while a logarithmic scale was used for high values of
$\varepsilon$ in Fig. 8 as the HR is much more decreased. We also find that
the parameter $a_{0}$ plays no particular role in the rates of emission, at
least when varying between 0 and the often-used value
$a_{0}=\sqrt{3}\gamma/2\simeq 0.11$ where $\gamma\equiv\ln(2)/\sqrt{3}\pi$ is
the Barbero–Immirzi parameter, causing only a very slight decrease in the high
energy tail due to the different temperatures. The differences are at the
percentage level and not shown here. The effect of small values of
$\varepsilon$ on the Hawking radiation rates is small altogether.

The results are much more intriguing for high values of $\varepsilon$, and are
reported on Fig. 8. With values of $\varepsilon=\\{1,4,10\\}$ we obtain a
reduction in the temperature of a factor $\sim\\{1,1.6,3.2\\}$ (respectively).
The associated decrease in the emission rate is strongly spin-dependent, a
feature that could be explained with an analysis of the precise dependency of
the potentials in $\varepsilon$: the emission is more and more damped for high
values of $\varepsilon$ as the particle spin increases. This is a major result
which has important consequences on the Hawking radiation constraints
discussed in the next Section. A fundamental difference between
Reissner–Nordström (Kerr) and polymerized BHs is that the charge (angular
momentum) is radiated away during the BH evaporation. Thus, the behaviour of
the BH is expected to follow a Schwarzschild trajectory once these parameters
are back to small values. The polymerization factor, on the other hand, is a
constant inherited from the quantum nature of gravity. Thus, the specific
behaviour associated with a high value of $\varepsilon$ (namely a decrease of
the BH temperature compared to the Schwarzschild case, and smaller emission
rates) should last during the whole lifetime of the BH. These aspects are
discussed further in the next Section.

The high energy limit of the cross-section for polymerized BHs has not
appeared in the literature. Using conditions (1.18), we obtain the value for
the radius of the photon sphere in the limit $a_{0}\rightarrow 0$. This is

$r^{\rm LQG}_{\rm
c}=\dfrac{r_{+}}{6}\left(3P^{2}-4P+3+\dfrac{9+P\left(6+P(10+6P+9P^{2})\right)}{z}+z\right),$
(3.8)

where $z$ is

$\displaystyle z\equiv\Bigg{[}$ $\displaystyle
27+27P-63P^{2}-190P^{3}-63P^{4}+27P^{5}+27P^{6}$
$\displaystyle+3P(1+P)\sqrt{-675+3P(-238+P(49+P(636+P(49-P(238+225P)))))}\Bigg{]}^{1/3}.$
(3.9)

We therefore find that the high energy limit of the cross-section is given by

$\beta_{\infty}^{\rm LQG}=\left(\dfrac{r^{\rm LQG}_{\rm c}}{r_{\rm
S}}\right)^{2}\dfrac{1}{G\big{(}r_{\rm c}^{\rm LQG}\big{)}}\,.$ (3.10)

The comparison of this formula with our numerical results is shown in Fig. 3
(left panel), with excellent agreement even for high values of $\varepsilon$.
In the case where $a_{0}\neq 0$, and in particular when taking the fiducial
value $a_{0}=\sqrt{3}\gamma/2\simeq 0.11$ where
$\gamma\equiv\ln(2)/\sqrt{3}\pi$ is the Barbero–Immirzi parameter [27], we
cannot use the approximate formula (3.10). However, we have performed a
numerical estimation of the photon sphere radius $r_{\rm c}$ and compared the
resulting high energy cross-section to our computed results, showing great
agreement. The effect of taking $a_{0}\neq 0$ is small anyways. It was proven
by [21] that at low energy the scalar wave has a cross-section

$\beta_{0}^{\rm LQG}=\frac{4m^{2}(1+P^{2})}{r_{\rm
S}^{2}}\left(1+\dfrac{a_{0}^{2}}{16m^{4}}\right)=\dfrac{r_{+}(r_{+}+r_{-})}{r_{\rm
S}^{2}}\left(1+\dfrac{a_{0}^{2}}{r_{+}^{4}}\right).$ (3.11)

The agreement of this analytical result with our computed results is shown in
Fig. 3 (right panel). For the spin 0 field, the low $\varepsilon$ regime is
correctly reproduced, but a discrepancy between the analytical limit and the
numerical calculation is found at high values of $\varepsilon$. This
difference is of order unity, and concerns only the very low energy asymptotic
behaviour. It is not clear whether the derivation of the formula (3.11) is
valid at high values of $\varepsilon$. There is, to our knowledge, no
literature giving the asymptotic limits at low energy for the other spins. The
very good agreement between these theoretical limits and our numerically
computed results are convincing us that our potentials for this
non-$tr$-symmetric example are efficient. We have checked that the limit
$\varepsilon,a_{0}\rightarrow 0$ gives the Schwarzschild result for all spins.
We point out that the high-$\varepsilon$ damping of the HR rate is very clear
for spins $s>0$, while the spin 0 emission rate is reduced by much.

###  3.4 Constraints from polymerized black holes

In this Section, we obtain for the first time Hawking radiation constraints on
polymerized primordial black holes using the results previously derived in
this paper. There are two parameters at stake, $a_{0}$ and $\varepsilon$.
First, $a_{0}$ is expected to be negligible for BHs with a radius
$r_{+}^{2}\gg a_{0}$, and to play a role only at the end of the BH
evaporation, when its radius reaches values close to the Planck length, out of
the Hawking radiation constraint range. The parameter $\varepsilon$, on the
other hand, has an effect which is proportional to its value, with small
values of $\varepsilon$ leading to very little changes in the Hawking
radiation emission, while larger values may have a dramatic impact (see
previous section). There are two major outcomes expected when considering the
evaporation constraints on polymerized PBHs:

  * •

a decrease of the Hawking temperature and emission rates at high
$\varepsilon$, which results in a longer lifetime, shifting the (time-
dependent) constraints towards smaller PBH masses;

  * •

this decrease will also lead to weaker (instantaneous) constraints.

Thus, the most striking result from this Section is that we expect the window
for light PBHs to represent all DM to be reopened in the case of high values
of $\varepsilon$, down to smaller PBH masses than in the Schwarzschild case.

In order to illustrate this proposal, we have chosen to compute the
prospective evaporation constraints from MeV to GeV photons as will be
measured by AMEGO, whose expected sensitivity can be found in [67]. There are
two reasons for this choice: $i$) this limit lies among the most stringent
ones in the disputed mass range where PBHs may represent all DM (see [8]),
$ii$) some of the authors of this paper are also authors of the public code
BlackHawk, which has been updated to compute precisely the secondary low
energy photon spectra and obtain robust constraints in the considered mass
range [8], as well as to compute the primary emission rates for photons in the
polymerized metric. We compute only the constraint from the AMEGO instrument
for PBHs evaporating in the galactic center, as it is the most stringent one.
The effects which we will describe below can also be applied to all the other
evaporation constraints. We follow exactly the setup chosen by [8]: a NFW
distribution of DM in the Milky Way, and observation in some small window of
angular width $5^{\circ}$, which gives $\Delta\Omega=2.39\times 10^{-2}\,$sr.
The expected emission is thus

$\dfrac{{\rm d}\Phi}{{\rm d}E}=\frac{1}{4\pi}\dfrac{f_{\rm PBH}}{M_{\rm
PBH}}\dfrac{{\rm d}^{2}N}{{\rm d}E\,{\rm d}t}\int_{\rm LOS}\rho_{\rm DM}\,{\rm
d}l\,,$ (3.12)

where the integral over the line of sight can be written as

$J\equiv\dfrac{1}{\Delta\Omega}\int_{\Delta\Omega}{\rm d}\Omega\int_{\rm
LOS}\rho_{\rm DM}\,{\rm d}l=1.597\times
10^{26}\,\text{MeV}\cdot\text{cm}^{-2}\cdot\text{sr}^{-1}\,.$ (3.13)

The numerical value comes from Refs. [8, 68]. The results are shown in Fig. 4
where we plot the constraints for $\varepsilon=\\{1,5,10\\}$ as well as the
fiducial constraint for the classical case and the constraint from [8] for
comparison. The constraints are obtained by maximizing $f_{\rm PBH}$ while
keeping $E^{2}{\rm d}\Phi/{\rm d}E$ below the AMEGO sensitivity (figure 5 of
[67]).

Figure 4: Constraints on the PBH fraction in DM from the measurement of MeV-
GeV photons in the galactic center by AMEGO. We show the constraints derived
for a classical BH in this work (solid black), to be compared to the same
limit from [8] (solid grey, denoted as [C21]). Then we show the constraints
computed for increasing values of $\varepsilon=\\{1,5,10\\}$ (dashed blue, dot
dashed green and dotted red, from right to left). The horizontal dashed line
denotes the limit $f_{\rm PBH}=1$.

We observe that the constraints we derive for the classical Schwarzschild case
have differences with the results of [8]. We were not able to explain their
origin, since we have carefully reproduced their low energy photon spectra and
use the same source for the AMEGO sensitivity. As expected, the constraints
for the classical Schwarzschild BH and the polymerized BH with $\varepsilon=1$
(small polymerization factor) are similar, as their Hawking radiation rates
are very close (see Fig. 7). Then, as we increase $\varepsilon$ to 5 and then
10, we observe that the constraints get weaker in the high mass range $M_{\rm
PBH}\gtrsim 10^{15}\,$g, allowing the DM fraction $f_{\rm PBH}$ of PBHs to be
1 for $M_{\rm PBH}\gtrsim 10^{17}\,$g (or $10^{16}\,$g for $\varepsilon=10$).
This is due to the fact that the main contribution to the photon spectra for
these PBHs comes from the directly emitted primary photons, whose emission
rate is strongly suppressed when $\varepsilon$ increases (see Fig. 8).
However, in the lower mass range $M_{\rm PBH}\lesssim 10^{15}\,$g, the
constraints remain of the same order of magnitude. In this energy range, the
constraints come from the secondary photons generated by neutral pion decay.
As pions are spin 0 particles, their emission rate decreases slowly as
$\varepsilon$ increases (see Fig. 8); the effect of the polymerization factor
becomes sizeable only for extreme values. In this constraint plot, we have
extended the mass range to masses $M_{\rm PBH}=10^{13}\,$g, which is 2 orders
of magnitude below the usual evaporation limit $M_{\rm PBH}\lesssim
10^{15}\,$g set by the lifetime of the PBHs, because we expect that the
decreased emission rates will result in an increased PBH lifetime, thus
allowing smaller PBHs to contribute to DM today. This effect will be
quantitatively studied in future work, as well as the modifications of the
other set of constraints (e.g. electrons [69, 70] and time-stacked extra-
galactic background [71]).

One last aspect, which we have not quantitatively explored, is the rate of
final bursts of PBHs and their observation by gamma ray instruments (see the
recent paper [72]). Since in the high $\varepsilon$ limit PBHs evaporating
today would have a lower initial mass than in the Schwarzschild case, their
number abundance should be larger if they represent some fixed fraction of DM.
Thus, the rate of nearby final bursts would be higher, leading to more
stringent constraints. However, the non-trivial modification of the final
light curves (energies, duration) makes it difficult to predict the
sensitivity of the gamma ray instruments to these polymerized PBHs burst.

## Conclusion

In this paper we have used the short-ranged potentials derived in the
companion paper [1] to compute numerically the Hawking radiation signals from
three spherically-symmetric and static BH solutions: charged BHs, higher-
dimensional BHs and polymerized BHs. We have checked the robustness of our
results by comparing the HR at low and high energy with analytical formulas,
some of which have been derived here for the first time. Focusing on the case
of polymerized BHs, whose peculiar metric form is the heart of our analytical
study, we conclude that the HR signals are not much different from the
Schwarzschild case in the limit $\varepsilon\ll 1$ of small polymerization
parameter. We have shown however that if the polymerization parameter takes
high values $\varepsilon\gtrsim 1$, then all the evaporation constraints would
need to be re-evaluated with two major effects: shifting the constraints
towards smaller PBH masses, and relaxing the constraints on the fraction of DM
which light PBHs can represent. The modification of the constraints depends in
a non-trivial way on the spin of the primary particles involved in the
secondary observed spectrum; the effect of the quantum deformation
$\varepsilon$ is very spin-dependent. The main consequence of this result is
that the mass range usually excluded by (future) evaporation limits
$10^{16}\,{\rm g}<M<10^{18}\,{\rm g}$ for all DM in the form of PBHs is
reopened. This is a striking result in PBH DM studies, and the first
constraint ever set on the fraction of polymerized primordial black holes as
dark matter using Hawking radiation signals. As a final remark, we have opened
a new window on the study of HR from other regular BH metrics, which is an
exciting prospect.

##  Appendix A Tortoise coordinates

In order to solve the Schrödinger wave equation (1.4) with the potentials
(1.6), we need to translate from the radial coordinate to the tortoise
coordinate. Here we give the results of the analytical integration of Eq.
(1.5) for the three metrics considered in Section 2.

##### Charged black holes.

For the metric (2.1), integration of Eq. (1.5) gives

$r^{*}(r)=r+\dfrac{r_{+}^{2}}{r_{+}-r_{-}}\ln\left(\dfrac{r}{r_{+}}-1\right)-\dfrac{r_{-}^{2}}{r_{+}-r_{-}}\ln\left(\dfrac{r}{r_{-}}-1\right).$
(A.1)

with $r_{\pm}$ as in Eq. (2.3).

##### Higher-dimensional black holes.

For the metric (2.4), integration of Eq. (1.5) gives for $n=1$

$x_{1}^{*}(x)=x+\dfrac{1}{2}\ln\left(\dfrac{x-1}{x+1}\right),$ (A.2)

for $n=2$

$x_{2}^{*}(x)=x+\dfrac{1}{3}\ln(x-1)+\dfrac{1}{\sqrt{3}}\arctan\left(\dfrac{\sqrt{3}}{1+2x}\right)-\dfrac{1}{6}\ln\left(x^{2}+x+1\right),$
(A.3)

for $n=3$

$x_{3}^{*}(x)=x+\dfrac{1}{4}\ln\left(\dfrac{x-1}{x+1}\right)+\dfrac{1}{2}\arctan\dfrac{1}{x}\,,$
(A.4)

for $n=4$

$\displaystyle x_{4}^{*}(x)=x$
$\displaystyle+\dfrac{1}{5}\ln(x-1)+\dfrac{\sqrt{\sqrt{5}}}{5}\left[\sqrt{\varphi_{+}}\arctan\left(\dfrac{\sqrt{\sqrt{5}}\sqrt{\varphi_{+}}}{2x+\varphi_{-}}\right)+\sqrt{\varphi_{-}}\arctan\left(\dfrac{\sqrt{\sqrt{5}}\sqrt{-\varphi_{-}}}{2x+\varphi_{+}}\right)\right]$
$\displaystyle-\dfrac{1}{10}\left[\varphi_{-}\ln(x^{2}+\varphi_{-}x+1)+\varphi_{+}\ln(x^{2}+\varphi_{+}x+1)\right],$
(A.5)

where $\varphi_{\pm}\equiv(1\pm\sqrt{5})/2$, for $n=5$

$x_{5}^{*}(x)=x+\dfrac{1}{6}\ln\left(\dfrac{x-1}{x+1}\right)+\dfrac{1}{2\sqrt{3}}\left[\arctan\left(\dfrac{\sqrt{3}}{2x+1}\right)+\arctan\left(\dfrac{\sqrt{3}}{2x-1}\right)\right]+\dfrac{1}{12}\ln\left(\dfrac{x^{2}-x+1}{x^{2}+x+1}\right),$
(A.6)

and for $n=6$

$\displaystyle x_{6}^{*}(x)=x+\dfrac{1}{7}\ln\left(x-1\right)$
$\displaystyle+\dfrac{2}{7}\Bigg{[}\cos\left(\pi/14\right)\arctan\left(\dfrac{\cos(\pi/14)}{x+\sin(\pi/14)}\right)+\cos\left(3\pi/14\right)\arctan\left(\dfrac{\cos(3\pi/14)}{x-\sin(3\pi/14)}\right)+\sin\left(\pi/7\right)\arctan\left(\dfrac{\sin(\pi/7)}{x+\cos(\pi/7)}\right)\Bigg{]}$
$\displaystyle+\dfrac{1}{7}\Bigg{[}\sin\left(3\pi/14\right)\ln(x^{2}-2\sin(3\pi/14)x+1)-\sin\left(\pi/14\right)\ln(x^{2}+2\sin(\pi/14)x+1)-\cos\left(\pi/7\right)\ln(x^{2}+2\cos(\pi/7)x+1)\Bigg{]}\,.$
(A.7)

In these relations, we have defined $x^{*}\equiv r^{*}/r_{\rm H}$ and $x\equiv
r/r_{\rm H}$.

##### Polymerized black holes.

For the metric (2.7), integration of Eq. (1.5) gives

$r^{*}(r)=r-\dfrac{a_{0}^{2}}{r_{+}r_{-}r}+\dfrac{a_{0}^{2}(r_{+}+r_{-})}{r_{+}^{2}r_{-}^{2}}\ln\left(\dfrac{r}{r_{+}+r_{-}}\right)+\dfrac{a_{0}^{2}+r_{+}^{4}}{r_{+}^{2}(r_{+}-r_{-})}\ln\left(\dfrac{r}{r_{+}}-1\right)+\dfrac{a_{0}^{2}+r_{-}^{4}}{r_{-}^{2}(r_{-}-r_{+})}\ln\left(\dfrac{r}{r_{-}}-1\right)\,.$
(A.8)

##  Appendix B Detailed results for Hawking radiation

The various plots mentioned in the core of the article are gathered below.
They feature the different spins $(0,1,2,1/2)$ from top to bottom.

Figure 5: Hawking radiation of massless particles of spin $(0,1,2,1/2)$ (top
to bottom) from charged BHs with $Q=\\{0.010,\,0.758,\,0.999\\}M$ (solid blue,
dashed green and dot-dashed red respectively). The Schwarzschild BH with $Q=0$
is in dotted black. The $Q=0.010$ curves are indistinguishable from the
Schwarzschild ones. The vertical lines on the left panels represent the
temperature of the BH. Be careful of the different $x$ and $y$ axes values.
Figure 6: Hawking radiation of massless particles of spin $(0,1,2,1/2)$ (top
to bottom) from higher-dimensional BHs with $n=\\{2,\,4,\,6\\}$ (solid blue,
dashed green and dot-dashed red respectively) and $M^{*}=1$. The Schwarzschild
BH with $n=0$ is in dotted black. The vertical lines on the left panels
represent the temperature of the BH. Be careful of the different $x$ and $y$
axes values. Figure 7: Hawking radiation of massless particles of spin
$(0,1,2,1/2)$ (top to bottom) from polymerized BHs with
$\varepsilon=\\{10^{-1},\,10^{-0.6},\,10^{-0.1}\\}$ (solid blue, dashed green
and dot-dashed red respectively) and $a_{0}=\sqrt{3}\gamma/2$ where
$\gamma\equiv\ln(2)/\sqrt{3}\pi$ is the Barbero–Immirzi parameter [27]. The
Schwarzschild BH is in dotted black. The vertical lines on the left panels
represent the temperature of the BH, which are indistinguishable within our
choice of parameters. Be careful of the different $x$ and $y$ axes values and
scales. Figure 8: Hawking radiation of massless particles of spin
$(0,1,2,1/2)$ (top to bottom) from polymerized BHs with
$\varepsilon=\\{1,\,4,\,10\\}$ (solid blue, dashed green and dot-dashed red
respectively) and $a_{0}=0$. The Schwarzschild BH is in dotted black. The
vertical lines on the left panels represent the temperature of the BH. Be
careful of the different $x$ and $y$ axes scales.

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