#  The Dark Dimension, the Swampland, and the Dark Matter Fraction Composed of
Primordial Black Holes

Luis A. Anchordoqui Department of Physics and Astronomy,
Lehman College, City University of New York, NY 10468, USA  Department of
Physics,
Graduate Center, City University of New York, NY 10016, USA  Department of
Astrophysics,
American Museum of Natural History, NY 10024, USA  Ignatios Antoniadis
Laboratoire de Physique Théorique et Hautes Énergies - LPTHE,
Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France  SISSA, Via
Bonomea 265, 34136, Trieste, Italy  ICTP, Strada Costiere 11, 34151 Trieste,
Italy  Dieter Lüst Max–Planck–Institut für Physik,
Werner–Heisenberg–Institut, 80805 München, Germany  Arnold Sommerfeld Center
for Theoretical Physics,
Ludwig-Maximilians-Universität München, 80333 München, Germany

###### Abstract

Very recently, it was suggested that combining the Swampland program with the
smallness of the dark energy and confronting these ideas to experiment lead to
the prediction of the existence of a single extra-dimension (dubbed the dark
dimension) with characteristic length-scale in the micron range. We show that
the rate of Hawking radiation slows down for black holes perceiving the dark
dimension and discuss the impact of our findings in assessing the dark matter
fraction that could be composed of primordial black holes. We demonstrate that
for a species scale of ${\cal O}(10^{10}~{}{\rm GeV})$, an all-dark-matter
interpretation in terms of primordial black holes (PBHs) should be feasible
for masses in the range $10^{14}\lesssim M_{\text{BH}}/{\rm g}\lesssim
10^{21}$. This range is extended compared to that in the 4D theory by 3 orders
of magnitude in the low mass region. We also show that PBHs with $M_{\rm
BH}\sim 10^{12}~{}{\rm g}$ could potentially explain the well-known Galactic
511 keV gamma-ray line if they make up a tiny fraction of the total dark
matter density.

††preprint: MPP-2022-60††preprint: LMU-ASC 24/22

##  I Introduction

The swampland program Vafa:2005ui  seeks to demarcate the set of four-
dimensional effective field theories (EFTs) that can be coupled to quantum
gravity in a consistent way, e.g., the landscape of superstring theory vacua,
and discriminate these theories from those that cannot, to strengthen the
predictive power of quantum gravity in general, and superstring theory in
particular. This is accomplished by enumerating criteria that an EFT must
fulfill so as to be in the landscape, rather than be relegated to the
“swampland.” These criteria have evolved to some set of conjectures, which can
be used as new guiding principles to construct compelling UV-completions of
the Standard Model (SM). Moreover, the UV constraints on IR physics have led
to a shift in the way we approach cosmology model building. There are many
swampland conjectures in the market, indeed too many to be listed here and
readers are referred to comprehensive reviews Palti:2019pca ; vanBeest:2021lhn
.

It was argued some time ago that the cosmological hierarchy problem, i.e. the
smallness of dark energy in Planck units ($\Lambda\sim 10^{-122}M_{\rm
Pl}^{4}$), can be explained statistically Bousso:2000xa  or even anthropically
Weinberg:1987dv ; Susskind:2003kw ; Schellekens:2006xz  by the huge number of
vacua in the string landscape. However very recently, it was suggested
Montero:2022prj  that by a combination of the cosmological hierarchy problem
and the distance conjecture one naturally ends up in a peculiar corner of the
string landscape, namely with a single extra-dimension characterized by a
length-scale in the micron range.111Large extra dimension scenarios were
originally introduced to solve the electroweak hierarchy problem
Antoniadis:1990ew ; Arkani-Hamed:1998jmv ; Antoniadis:1998ig .

The distance conjecture predicts the appearance of infinite towers of states
that become light and imply a breakdown of the EFT in the infinite distance
limits in moduli space Ooguri:2006in . Stated in the form of the anti-de
Sitter (AdS) distance conjecture Lust:2019zwm , it suggests that there should
be an infinite tower of states, whose mass is related to the magnitude of the
cosmological constant. More precisely, the mass scale $m$ behaves as
$m\sim|\Lambda|^{\alpha}$, as the negative AdS vacuum energy $\Lambda\to 0$,
with $\alpha$ a positive constant of ${\cal O}(1)$. In Lust:2019zwm  also some
implications of the AdS distance conjecture for de Sitter space were
discussed, namely, when assuming this scaling behavior to hold in dS (or quasi
dS) space with a positive cosmological constant, approaching $\Lambda=0$ will
also lead to an unbounded number of massless modes.

To each tower we can associate two mass scales: $m$, which is the mass scale
of states in the tower, and $\hat{M}$, which is the scale local EFT
description breaks down. The latter is the so-called “species scale”
Dvali:2007hz ; Dvali:2007wp  that corresponds to the Planck scale of the
higher dimensional theory, $\hat{M}=m^{n/(n+2)}M_{\rm Pl}^{2/(n+2)}$, where
$n$ is the number of effective dimensions decompactifying. Requiring the
experimental bound on deviations from Newton’s gravitational inverse-square
law Lee:2020zjt  to be consistent with the theoretical bound from the
swampland conjectures leads to $\alpha=1/4$ and so the mass scale of the KK
modes in the tower is estimated to be $m\sim\lambda^{-1}\Lambda^{1/4}$.
Consistency with neutron star heating Hannestad:2003yd  yields $n=1$
Montero:2022prj , whereas consistency with the sharp cutoff observed in the
cosmic ray spectrum requires $\lambda\sim 10^{-3}$ Anchordoqui:2022ejw . On
the whole, swampland considerations combined with observational data lead to
the prediction of a single extra mesoscopic dimension of length
$R\sim\lambda\Lambda^{-1/4}\sim 1~{}\mu{\rm m}$, where
$\Lambda^{1/4}=2.31~{}{\rm meV}$. This extra dimension, nicknamed the dark
dimension, opens up at the scale $m$ of the tower, where physics must be
described by an EFT in higher dimensions up to the species scale $\hat{M}\sim
10^{10}~{}{\rm GeV}$.

In this paper we study some phenomenological aspects of black holes perceiving
the dark dimension and, in the spirit of Argyres:1998qn , we investigate the
impact of these higher dimensional objects in assessing the fraction of dark
matter that could be composed of primordial black holes, $f_{\rm PBH}$. The
layout of the paper is as follows. We begin in Sec. II with an overview of
existing limits on $f_{\rm PBH}$. This includes constraints from the isotropic
photon backgrounds, observations of the cosmic microwave background (CMB), and
measurements of the positron density at the Galactic bulge. In Sec. III we
first reexamine the rate of Hawking radiation of 5-dimensional black holes
with the new species scale in mind, and after that we confront our predictions
to experiment. In Sec. IV we reexamine within the 5D theory whether black
holes evaporating right now could be responsible for the excess of 511 keV
photons observed from the inner Galaxy by the SPI spectrometer on board the
INTEGRAL satellite Knodlseder:2005yq ; Jean:2005af . In Sec. V we discuss the
possibility for primordial black holes to grow via the accretion. The paper
wraps up with some conclusions presented in Sec. VI. Before proceeding, we
pause to note that a specific realization of the model proposed in
Montero:2022prj  should guarantee that the SM interacts with the extra
dimension only gravitationally, while the cosmological constant scale is fixed
by the size of the extra dimension. Although a priori this does not seem
obvious at all, we continue here on the assumption that such a realization can
indeed emerge from string theory.

##  II Primordial Black Holes as a Dark Matter Candidate

Figure 1: (Taken from Villanueva-Domingo:2021spv .) Compilation of contraints
on $f_{\rm PBH}$ as a function of the PBH mass $M_{\rm BH}$, assuming a
monochromatic mass function. The different probes considered are: the impact
of PBH evaporation (red) on the extragalactic $\gamma$-ray background
Carr:2009jm  and on the CMB spectrum Clark:2016nst ; non-observation of
microlensing events (blue) from the MACHO Macho:2000nvd , EROS EROS-2:2006ryy
, Kepler Griest:2013aaa , Icarus Oguri:2017ock , OGLE Niikura:2019kqi  and
Subaru-HSC Croon:2020ouk  collaborations; PBH accretion signatures on the CMB
(orange), assuming spherical accretion of PBHs within halos Serpico:2020ehh ;
dynamical constraints, such as disruption of stellar systems by the presence
of PBHs (green), on wide binaries Monroy-Rodriguez:2014ula  and on ultra-faint
dwarf galaxies Brandt:2016aco ; power spectrum from the Ly$\alpha$ forest
(cyan) Murgia:2019duy ; merger rates from gravitational waves (purple), either
from individual mergers Kavanagh:2018ggo ; LIGOScientific:2019kan  or from
searches of stochastic gravitational wave background Chen:2019irf .
Gravitational wave limits, denoted by dashed lines, are model dependent
Boehm:2020jwd . The dotted brown line corresponds to forecasts from the 21 cm
power spectrum with SKA sensitivities Mena:2019nhm  and from 21 cm forest
prospects Villanueva-Domingo:2021cgh .

It has long been suspected that black holes could emerge from the collapse of
large amplitude fluctuations in the very early universe Zeldovich:1967lct ;
Hawking:1971ei ; Carr:1974nx ; Carr:1975qj . Although the mass spectrum of
these primordial black holes (PBHs) is not set in stone, on cosmological
scales they would behave like a typical cold dark matter particle. Actually,
the idea that PBHs could be interesting dark matter candidates dates back at
least as far as 1975 Chapline:1975ojl , with punctuated revivals of activity
following the microlensing searches for massive compact halo objects (MACHOs)
in 1997 MACHO:1996qam , and the LIGO/Virgo detections of merging binary black
holes in 2016 LIGOScientific:2016aoc . The first microlensing searches
suggested that dark matter could be composed of MACHOs with mass $\sim
0.5M_{\odot}$, which is the expected mass scale for PBHs produced during the
quark-hadron phase transition Jedamzik:1998hc . However, more recent
observations exclude significant contributions of MACHOs to dark matter over
most of the plausible mass range MACHO:2000qbb ; Macho:2000nvd ;
EROS-2:2006ryy ; Griest:2013aaa ; Oguri:2017ock ; Niikura:2019kqi ;
Croon:2020ouk . The question of whether the LIGO/Virgo merger events
correspond to black holes of astrophysical or primordial origin is still under
debate Bird:2016dcv ; Sasaki:2016jop ; Hall:2020daa , and a mixed population
may also be compatible with observations DeLuca:2021wjr . However, data
suggest that the binary black hole merging rate is incompatible with an all-
dark-matter scenario and that PBHs could only contribute to less than 1% of
the total dark matter LIGOScientific:2019kan ; Wong:2020yig .

The mass distribution of PBHs is usually characterized by the mass function

$\psi(M_{\rm BH})=\frac{M_{\rm BH}}{\rho_{\rm CDM}}\ \frac{dn_{\rm
PBH}}{dM_{\rm BH}}\,,$ (1)

where $M_{\rm BH}$ is the black hole mass, $dn_{\rm PBH}$ is the number
density of PBHs within the mass range $(M_{\rm BH},M_{\rm BH}+dM_{\rm BH})$,
and $\rho_{\rm CDM}$ is the energy density of cold dark matter DeLuca:2020ioi
. Integrating $\psi(M_{\rm BH})$ gives the total fraction of dark matter in
PBHs,

$f_{\rm PBH}\equiv\frac{\rho_{\rm PBH}}{\rho_{\rm CDM}}=\int\psi(M_{\rm
BH})\,dM_{\rm BH}\,,$ (2)

where $\rho_{\rm PBH}=\int M_{\rm BH}\,dn_{\rm PBH}$ is the energy density of
PBHs. If all of the dark matter were made of PBH, we would have $f_{\rm
PBH}=1$. A compilation of upper limits on the dark matter fraction that can be
composed of PHBs is shown in Fig. 1.

The question we want to address herein is whether primordial black holes
perceiving the dark dimension could ameliorate the constraints on $f_{\rm
PBH}$ shown in Fig. 1. In terms of the size of the extra dimension $R$ and the
string length $l_{s}$, we can distinguish three definite regimes for the black
hole horizon $r_{s}$: (i) $r_{s}>R$, where the theory looks like 4D and
$r_{s}\simeq M_{BH}$; (ii) $l_{s}<r_{s}<R$, where the black hole perceives the
higher dimensional space; (iii) $r_{s}<l_{s}$, where the black hole turns into
a string state Horowitz:1996nw ; Horowitz:1997jc . In our analysis, only the
regime (ii) will be relevant, i.e. we will study black holes that are smaller
than the size of the extra dimension, but larger than the string size.

##  III Radiation Time-Scale of Five-Dimensional Black Holes

PBHs will Hawking evaporate, provided the semiclassical approximation is
valid. The average number Hawking:1975vcx ; Hawking:1974rv  and the
probability distribution of the number Parker:1975jm ; Wald:1975kc ;
Hawking:1976ra  of outgoing particles in each mode obey a thermal spectrum,
with temperature Anchordoqui:2001cg

$T_{\rm BH}=\frac{n+1}{4\,\pi\,r_{s}}$ (3)

and entropy

$S=\frac{4\,\pi\,M_{\text{BH}}\,r_{s}}{n+2}\,,$ (4)

where

$r_{s}(M_{\text{BH}})=\frac{1}{M_{{\rm
Pl},n}}\left[\frac{M_{\text{BH}}}{M_{{\rm
Pl},n}}\,\,\frac{2^{n}\pi^{(n-3)/2}\Gamma({n+3\over
2})}{n+2}\right]^{1/(1+n)}\,,$ (5)

is the radius of a $(4+n)$-dimensional Schwarzschild black hole,

$M_{{\rm Pl},n}=\left(\frac{m^{n}\ M_{\rm Pl}^{2}}{8\pi}\right)^{1/(n+2)}\,,$
(6)

and where $\Gamma(x)$ is the Gamma function Myers:1986un .

The black hole, however, produces an effective potential barrier surrounding
the event horizon that backscatters part of the outgoing radiation, making
alterations to the Planckian spectrum. The black hole absorption cross
section, $\sigma_{s}$ (a.k.a. the greybody factor), depends on: (i) the spin
$s$ of particle being emitted, (ii) the particle’s energy $Q$, and (iii)
$M_{\text{BH}}$ Page:1976df . At high frequencies ($Qr_{s}\gg 1$) the greybody
factor for each kind of particle must approach the geometrical optics limit.
The integrated power emission is reasonably well approximated taking such a
high energy limit. In our calculations we adopt the geometric optics
approximation, where the black hole acts as a perfect absorber of a slightly
larger radius, with emitting area given by

$A_{4\subset
4+n}=4\pi\left(\frac{n+3}{2}\right)^{2/(n+1)}\frac{n+3}{n+1}\,r_{s}^{2}\,\,.$
(7)

Within this framework, we can conveniently write the greybody factor as a
dimensionless constant normalized to the black hole surface area seen by the
SM fields $\Gamma_{s}=\sigma_{s}/A_{4\subset 4+n},$ such that
$\Gamma_{s=0}=1$, $\Gamma_{s=1/2}\approx 2/3$, and $\Gamma_{s=1}\approx 1/4$
Han:2002yy .

All in all, a black hole emits particles with initial total energy between
$(Q,Q+dQ)$ at a rate

$\frac{d\dot{N}_{i}}{dQ}=\frac{\sigma_{s}}{8\,\pi^{2}}\,Q^{2}\left[\exp\left(\frac{Q}{T_{\rm
BH}}\right)-(-1)^{2s}\right]^{-1}$ (8)

per degree of particle freedom $i$. The change of variables $u=Q/T,$ brings
Eq. (8) into a more familiar form,

$\dot{N}_{i}=f\frac{\Gamma_{s}}{32\,\pi^{3}}\,\frac{(n+3)^{(n+3)/(n+1)}(n+1)}{2^{2/(n+1)}}\,T_{\rm
BH}\,\int\frac{u^{2}}{e^{u}-(-1)^{2s}}\,du.$ (9)

This expression can be easily integrated using

$\int_{0}^{\infty}\frac{z^{n-1}}{e^{z}-1}\,dz=\Gamma(n)\,\zeta(n)$ (10)

and

$\int_{0}^{\infty}\frac{z^{n-1}}{e^{z}+1}\,dz=\frac{1}{2^{n}}\,(2^{n}-2)\,\Gamma(n)\,\zeta(n)\,\,,$
(11)

yielding

$\dot{N_{i}}=f\frac{\Gamma_{s}}{32\,\pi^{3}}\,\frac{(n+3)^{(n+3)/(n+1)}\,(n+1)}{2^{2/(n+1)}}\,\Gamma(3)\,\zeta(3)\,T_{\rm
BH}\,,$ (12)

where $\zeta(x)$ is the Riemann zeta function and $f=1$ ($f=3/4$) for bosons
(fermions). Therefore, the black hole emission rate is found to be

$\dot{N}_{i}\approx 3.7\times
10^{21}\,\frac{(n+3)^{(n+3)/(n+1)}}{2^{2/(n+1)}\,(n+1)^{-1}}\,\left(\frac{T_{\rm
BH}}{{\rm GeV}}\right)\,\,{\rm s}^{-1}\,\,,$ (13) $\dot{N}_{i}\approx
1.8\times
10^{21}\,\frac{(n+3)^{(n+3)/(n+1)}}{2^{2/(n+1)}\,(n+1)^{-1}}\,\left(\frac{T_{\rm
BH}}{{\rm GeV}}\right)\,\,{\rm s}^{-1}\,\,,$ (14) $\dot{N}_{i}\approx
9.2\times
10^{20}\,\frac{(n+3)^{(n+3)/(n+1)}}{2^{2/(n+1)}\,(n+1)^{-1}}\,\left(\frac{T_{\rm
BH}}{{\rm GeV}}\right)\,\,{\rm s}^{-1}\,\,,$ (15)

for particles with $s=0,\,1/2,\,1,$ respectively Anchordoqui:2002cp .

At any given time, the rate of decrease in the black hole mass is just the
total power radiated

$\frac{\dot{M}_{\rm
BH}}{dQ}=-\sum_{i}c_{i}\,\frac{\sigma_{s}}{8\pi^{2}}\,\frac{Q^{3}}{e^{Q/T_{\rm
BH}}-(-1)^{2s}}\,\,,$ (16)

where $c_{i}$ is the number of internal degrees of freedom of particle species
$i.$ A straightforward calculation yields

$\dot{M}_{\rm
BH}=-\sum_{i}c_{i}\,\,\tilde{f}\,\,\frac{\Gamma_{s}}{32\,\pi^{3}}\,\frac{(n+3)^{(n+3)/(n+1)}(n+1)}{2^{2/(n+1)}}\,\Gamma(4)\,\zeta(4)\,\,T_{\rm
BH}^{2},$ (17)

where $\tilde{f}=1\,\,(\tilde{f}=7/8)$ for bosons (fermions). Herein, we
assume that the effective high energy theory contains approximately the same
number of modes as the SM (i.e. $c_{s=0}=1$, $c_{s=1/2}=90,$ and $c_{s=1}=27$)
and we neglect the effect of graviton emission.222A point worth noting at this
juncture is that at first sight it may appear that the KK modes must dominate
Hawking radiation because there are a large number – ${\cal O}(R/r_{s})^{2}$ –
light modes with masses below the $T_{\rm BH}$ scale. However, as noted in
Emparan:2000rs  it is incorrect to think of the individual KK modes of the
bulk graviton as massive spin two fields on the brane with standard (minimal)
gravitational couplings. Rather, since the KK modes are excitations in the
full transverse space, their overlap with the small (higher-dimensional) black
holes is suppressed by the geometric factor $(r_{s}/R)^{2}$ relative to the
brane fields. Thus, the geometric suppression precisely compensates for the
enormous number of modes, and the total contribution of all KK modes is only
the same order as that from a single brane field.

###  III.1 Black hole radiation rate for $\bm{n=0}$

The rate of Hawking radiation is estimated to be

$\left.\frac{dM_{\rm BH}}{dt}\right|_{n=0}\simeq-9\times 10^{73}~{}{\rm
GeV}^{4}\frac{1}{M_{\rm BH}^{2}}\,\,.$ (18)

Ignoring accretion and thresholds, i.e., assuming that the mass of the black
hole evolves according to Eq. (18) during the entire process of evaporation,
we can obtain an estimate for the lifetime of the black hole,

$\tau_{\rm BH}^{n=0}\simeq 1\times 10^{-74}~{}{\rm GeV}^{-4}\,\,\int M_{\rm
BH}^{2}\,\,dM_{\rm BH}\,\,.$ (19)

Using $\hslash=6.58\times 10^{-25}~{}{\rm GeV\,s},$ Eq. (19) can then be
rewritten as

$\displaystyle\tau_{\rm BH}^{n=0}\simeq 1.6\times 10^{-35}\,\,(M_{\rm BH}/{\rm
g})^{3}~{}{\rm yr}\,\,.$ (20)

Note that a black hole with $M_{\rm BH}\sim 5\times 10^{14}~{}{\rm g}$ will
have a lifetime of about 2 Gyr, comparable to the age of the Universe
MacGibbon:2007yq ; MacGibbon:2015mya . Therefore, PBHs with
$M_{\text{BH}}\lesssim 5\times 10^{14}~{}{\rm g}$ cannot form part of the
observed dark matter density. PBHs with masses small enough, but still alive
in the Universe, should emit strong photon and cosmic ray backgrounds which
could be observed Laha:2020ivk ; Iguaz:2021irx . Null results from detection
of these backgrounds exclude an all dark matter interpretation in terms of
PBHs for masses $M_{\text{BH}}\lesssim 10^{17}~{}{\rm g}$. As shown in Fig. 1,
the allowed mass range for the PHB dark matter interpretation is
$10^{17}<M_{\text{BH}}/{\rm g}\lesssim 10^{21}$.

###  III.2 Black hole radiation rate for $\bm{n=1}$

Figure 2: Compilation of constraints on $f_{\rm PBH}$ in the 5D theory as a
function of the PBH mass $M_{\rm BH}$, assuming a monochromatic mass function.
See Fig. 1 for details of the different probes considered.

We now assume the black hole can be treated as a flat $(4+n)$-dimensional
object. This assumption is valid for extra dimensions that are larger than the
Schwarzschild radius Argyres:1998qn . For $M_{{\rm Pl},n}\sim 10^{10}~{}{\rm
GeV}$, the rate of Hawking radiation is estimated to be

$\left.\frac{dM_{\rm BH}}{dt}\right|_{n=1}\simeq-4\times 10^{29}~{}{\rm
GeV}^{3}\frac{1}{M_{\rm BH}}\,\,,$ (21)

and so

$\tau_{\rm BH}^{n=1}\simeq 3\times 10^{-30}~{}{\rm GeV}^{-3}\,\,\int M_{\rm
BH}\,\,dM_{\rm BH}\,\,,$ (22)

which implies

$\displaystyle\tau_{\rm BH}^{n=1}\simeq 9\times 10^{-15}\,\,(M_{\rm BH}/{\rm
g})^{2}~{}{\rm yr}\,\,.$ (23)

For $n=1$, a black hole lives longer than a usual $n=0$ black hole of the same
mass. A black hole with $M_{\text{BH}}\sim 5\times 10^{11}~{}{\rm g}$ has a
lifetime approximately equal to the age of the Universe.

As shown in Fig. 1, PBHs in the 4D theory with $10^{15}\lesssim
M_{\text{BH}}/{\rm g}\lesssim 10^{17}$ are incompatible with an all-dark-
matter interpretation. However, black holes sensing the dark dimension slow
down the Hawking radiation. For a given mass, the black hole lives longer in
the 5D theory than in the 4D theory. This implies that black holes sensing the
extra mesoscopic dimension emit less particles and so the limits from
isotropic photon backgrounds Laha:2020ivk ; Iguaz:2021irx , CMB observations
Poulin:2016anj ; Poulter:2019ooo , and measurements of the positron density at
the Galactic bulge Laha:2019ssq ; DeRocco:2019fjq  can be relaxed. Via a
direct comparison of our calculations with the limits shown in Fig. 1 we can
conclude that an all-dark-matter interpretation in terms of PBHs in the 5D
theory should be feasible for $10^{14}\lesssim M_{\text{BH}}/{\rm g}\lesssim
10^{21}$, thus extending the allowed mass range by 3 orders of magnitude.

Three observations are in order: (i) The temperature of PBHs evaporating today
may not be enough to emit all the SM degrees of freedom. Herein we are
interested in the order of magnitude estimate and we take this to fall within
errors. (ii) The lack of femtolensing detection in the gamma-ray burst data
have been interpreted as evidence that PBHs in the mass range $5\times
10^{17}<M_{\rm BH}/{\rm g}<10^{20}$ cannot constitute a major fraction of dark
matter. This interpretation, however, has been disputed Carr:2020gox . (iii)
For $M_{\text{BH}}\sim 5\times 10^{11}~{}{\rm g}$, the Schwarzschild radius is
$r_{s}\sim 5\times 10^{-5}~{}\mu{\rm m}$, whereas for $M_{\text{BH}}\sim
10^{17}~{}{\rm g}$, we have $r_{s}\sim 2\times 10^{-2}~{}\mu{\rm m}$,
justifying our assumption that these black holes are 5-dimensional objects. It
is noteworthy that a black hole with $M_{\text{BH}}\sim 1\times 10^{21}~{}{\rm
g}$ has a horizon radius $r_{s}\sim 2~{}\mu{\rm m}$, saturating the range of
validity of our 5D description. In Fig. 2 we illustrate how the longer
lifetime of PBH perceiving the mesoscopic-scale extra-dimension modifies the
constraints on $f_{\rm PBH}$. Moreover, the Hawking temperature of the
lightest PBHs evaporating today, with $M_{\rm BH}\sim 10^{12}~{}{\rm g}$, is
roughly 1 MeV. For this mass scale, Hawking radiation could potentially
explain the well-known Galactic 511 keV gamma-ray line. It is this that we now
turn to study.

##  IV A new PBH window for explaining the Galactic 511 keV line

It has long been known that electron–positron annihilation proceeds at a
surprisingly high rate in the central region of the Galaxy Johnson:1972 . In
particular, the SPI spectrometer on the INTEGRAL satellite has detected an
intense 511 keV gamma ray line flux aligned with the Galactic center (bulge
component) Knodlseder:2005yq ; Jean:2005af , and also provided evidence for
the line in the disk or halo component Weidenspointner:2007rs .

A variety of potential astrophysical sources explaining this signal have been
proposed Prantzos:2010wi , including PHBs Frampton:2005fk ; Bambi:2008kx . The
source of positrons responsible for the 511-keV line must generate ${\cal
O}(10^{50})$ positrons per year Fuller:2018ttb . However, if these positrons
are injected at even mildly relativistic energies, higher-energy gamma rays
will also be produced. Diffuse Galactic gamma-ray data strongly constrain the
positron injection energy to be $\lesssim 3~{}{\rm MeV}$ Beacom:2005qv .
Another key constraint comes from the local $e^{+}e^{-}$ flux as measured by
Voyager 1 Boudaud:2018hqb .

In the 4D theory the black temperature scales as

$T_{{\rm BH},4D}\simeq 1.05\left(\frac{M_{\rm BH}}{10^{16}~{}{\rm
g}}\right)^{-1}~{}{\rm MeV}\,.$ (24)

Herein we are particularly interested in black holes with masses above the
evaporation limit. In this mass range, PBHs are Hawking evaporating today,
emitting particles with a characteristic spectrum centered around tens of MeV.
Note that most of these PBHs would be excluded by Galactic gamma-ray
observations.

In the 5D theory discussed herein, PBHs evaporating today are bigger, longer-
lived, and colder than in the 4D theory; the PBH temperature scales as

$T_{{\rm BH},5D}\sim\left(\frac{M_{\rm BH}}{10^{12}~{}{\rm
g}}\right)^{-1/2}~{}{\rm MeV}\,.$ (25)

In the most recent data analysis it was shown that in 4D theory primordial
black holes in mass range of $1<M_{\rm BH}/10^{16}~{}{\rm g}<4$ could
potentially produce the 511 keV gamma-ray signal if $10^{-4}<f_{\rm
PBH}<4\times 10^{-3}$ Keith:2021guq . This study takes into account the gamma-
ray fluxes measured by INTEGRAL in the 0.1-0.2 MeV, 0.2-0.6 MeV, 0.6-1.8 MeV
bands, and by COMPTEL in the 1-3 MeV, 3-10 MeV, 10-30 MeV bands, as well as
the local Voyager constraint on the flux of $e^{+}e^{-}$. Remarkably, the
regions of parameter space in which PBHs could accommodate the observed
511-keV excess require a PBH number density in the vicinity of the Solar
System of

$n^{\rm local}_{\rm PBH}=\frac{f_{\rm PBH}\ \rho^{\rm local}_{\rm DM}}{M_{\rm
BH}}\simeq 1.2\times 10^{-4}\,{\rm AU}^{-3}\times\bigg{(}\frac{f_{\rm
PBH}}{10^{-3}}\bigg{)}\,\bigg{(}\frac{M_{\rm BH}}{2\times 10^{16}\,{\rm
g}}\bigg{)}^{-1}\,,$ (26)

where $\rho^{\rm local}_{\rm DM}=0.4\,{\rm GeV}/{\rm cm}^{3}$ is the local
dark matter density Keith:2021guq . Since we have seen that black holes in the
5D theory with temperature of 1 MeV have masses roughly 4 orders of magnitude
smaller, to a first approximation a simple rescaling of the result of (26)
while demanding the same number density suggests that an interpretation of the
511-keV line would be, in principle, possible for $f_{\rm PBH}\sim 10^{-7}$.

As already noted in Keith:2021guq , for such particular number density, the
closest PBH would be located at a distance

$d\sim\left(\frac{3}{4\pi n^{\rm local}_{\rm
PBH}}\right)^{1/3}\sim\mathcal{O}(10\,{\rm AU})\,.$ (27)

This suggests that the Solar System could contain several hundred black holes
at any given moment and detectability of their Hawking evaporation could be at
reach of future gamma-ray telescopes Keith:2022sow . Moreover, e-ASTROGAM
e-ASTROGAM:2016bph  would not only be able to detect the Hawking radiation
from a PBH population responsible for the 511 keV excess, it would be able to
characterize the properties of such a population with remarkable precision
Keith:2022sow . A simultaneous study of the allowed $(M_{\rm BH},f_{\rm PBH})$
parameter space could then be used to disentangle a PBH evaporating in 4D from
one evaporating in 5D.

##  V Black Hole Growth by Accretion

In Sec. III we estimated the black hole lifetime assuming that black holes do
not accrete. In general, the net change of the black hole mass is given by

$\frac{dM_{\rm BH}}{dt}=\left.\frac{dM_{\rm BH}}{dt}\right|_{\rm
accr}+\left.\frac{dM_{\rm BH}}{dt}\right|_{\rm evap}\,\,,$ (28)

where $dM_{\rm BH}/dt|_{\rm evap}$ is given by Eq. (17) and

$\left.\frac{dM_{\rm BH}}{dt}\right|_{\rm
accr}\approx\pi\left(\frac{n+3}{2}\right)^{2/(n+1)}\ \frac{n+3}{n+1}\
r_{s}^{2}\ \varepsilon\,,$ (29)

where $\varepsilon$ is the energy density of the plasma in the vicinity of the
event horizon Chamblin:2003wg . Note, however, that any correction from the
$dM_{\rm BH}/dt|_{\rm accr}$ term will tend to enlarge the black hole
lifetime, and so the conclusions presented in Sec. III would still hold.

The mesoscopic-size dimension imposes generic constraints on the production of
PBHs. Namely, as in the context of large extra dimensions Arkani-Hamed:1998cxo
; Dienes:1998hx , the universe should remain 4D at the nucleosynthesis MeV
temperature, even if the compactification scale is much smaller (at meV). This
is attributed to the stabilization of the extra dimension which should happen
actually even before the reheating temperature. This of course assumes that
the inflation mechanism can be re-adapted in a higher dimensional theory and
implements also the stabilization. Hence, the 5D PBHs should be produced after
inflation but before reheating.

Since black holes and dark matter are diluted by cosmic expansion in the same
way, in the absence of accretion and decay the PBH mass function given in (1)
is a constant over time. Although a precise characterization of $\psi(M_{\rm
BH})$ is beyond the scope of this paper, we note that the bigger, longer-
lived, and colder 5D black holes are more prompted to growth through accretion
than those in the 4D theory. As noted in DeLuca:2021pls  PBHs may accrete and
increase their mass by several orders of magnitude. Moreover, accretion may
play a critical role in explaining the LIGO-VIRGO events in terms of PBHs
DeLuca:2020qqa . However, we stress once more that the 5D description breaks
down for $M_{\rm BH}\gtrsim 10^{-12}M_{\odot}$, and so while the 5D enhanced
accretion effects could influence $\psi(M_{\rm BH})$ in the golden window of
black-hole mass range ($10^{14}\lesssim M_{\rm BH}/{\rm g}\lesssim 10^{21}$)
where PBHs can account for all of the dark matter content of the Universe, it
will play a negligible role in the mass growth of $M_{\odot}$-scale black
holes.

##  VI Conclusions

We have studied some phenomenological aspects of black holes perceiving the
dark dimension and analyzed the impact of these higher dimensional objects in
assessing the fraction of dark matter that could be composed of PBHs. We have
shown that the rate of Hawking radiation slows down for 5-dimensional black
holes and thereby an all-dark-matter interpretation in terms of PBHs for
$10^{14}\lesssim M_{\text{BH}}/{\rm g}\lesssim 10^{21}$ should be possible. We
have also shown that an explanation of the Galactic 511-keV line could be
possible for $M_{\rm BH}\sim 10^{12}~{}{\rm g}$ if $f_{\rm PBH}\sim 10^{-7}$.
Of course, for a PBH distribution that peaks at $M_{\rm BH}\sim 10^{15}~{}{\rm
g}$, one can, in principle, obtain a simultaneous all-dark-matter
interpretation, with an explanation of the Galactic 511 keV gamma-ray signal.
These results are strongly dependent on the choice of $M_{{\rm Pl},n}$ and to
a lesser degree on the behavior of $\sigma_{s}$ with $Q$.

It is interesting to note that a rotating black hole would first shed its spin
radiating particles, dominantly in the equatorial plane. Roughly 25% of its
mass is lost in the so-called “spin down phase” Giddings:2002kt . Spin down
leaves a Schwarzschild black hole which continues to Hawking radiate in the
“Schwarzschild phase.” Since the radiation temperature in the Schwarzschild
phase is larger than the one in the spin-down phase Page:1976ki , we conclude
that existing limits on spinning PBHs Laha:2020vhg  would also be relaxed. All
in all, our results are also valid for PBHs produced with angular momentum.

It is also interesting to note that the black hole decay rate could be slowed
down due to quantum effects, when compared to the semiclassical Hawking
radiation adopted in our calculations Dvali:2020wft . These quantum
corrections would become important if the black hole half-time is comparable
to the age of the Universe.

In summary, within the 5D model proposed in Montero:2022prj , with a species
scale at $10^{10}~{}{\rm GeV}$, PBHs sensing the extra dimension would have a
larger horizon radius, which scales as $M_{{\rm Pl},n}^{-(2+n)/(1+n)}$ as
shown in (5). From (3) we see that a larger horizon radius in turn implies a
smaller black hole temperature $T_{\rm BH}$. Now, one can check in (17) that
the Hawking radiation $\propto T_{\rm BH}^{2}$, and therefore the PBHs in the
5D theory live longer, automatically relaxing existing bounds on $f_{\rm
PBH}$. In conclusion, the PBHs as dark matter candidates could provide a
complete picture for the swampland and its cosmology.

## Acknowledgments

We thank Gia Dvali and Ranjan Laha for discussion, and Sergio Palomares-Ruiz
for permission to reproduce Fig. 1. The work of L.A.A. is supported by the
U.S. National Science Foundation (NSF Grant PHY-2112527). The work of D.L. is
supported by the Origins Excellence Cluster and by the German-Israel-Project
(DIP) on Holography and the Swampland.

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