# Primordial Black Hole Constraints with Large Extra Dimensions

George Johnson

###### Abstract

We study how the constraints on the primordial black hole density arising from
the extragalactic photon background are modified in the scenario that there
exist extra large spatial dimensions. We find that though the overall
magnitude of the constraints is not substantially different, the mass ranges
to which they apply are, and for some choices of mass it is possible for the
black holes to constitute the entirety of the dark matter.

##  1 Introduction

There is still very little known about the nature of the dark matter in the
universe, besides that it is non-relativistic and interacts only weakly with
the Standard Model. Whilst it is typically assumed that the dark matter
consists of some as-yet undiscovered elementary particle, primordial black
holes (PBHs) are a compelling alternative, being naturally cold, dark, and
consistent with the framework of known physics. For this reason a great deal
of work has been done in understanding the astrophysical and cosmological
consequences of a large PBH background; for many choices of PBH mass, strict
constraints exist on the fraction of dark matter they could constitute. See
[1] for a review.

Many of these constraints, in particular those applying to smaller mass PBHs
(in the mass range $10^{10}$ g to $10^{17}$ g), are due to the effects of the
Hawking radiation these black holes emit. Whilst the existence of Hawking
radiation is under little doubt, being necessary for the consistency of a
thermodynamic description of black holes, Hawking radiation has never been
observed, and so the precise way black holes radiate is still under question.
For this reason, it is interesting to consider how modifying the nature of
Hawking evaporation modifies the constraints on the density of primordial
black holes in the universe.

In this work, we consider how the dominant constraints on the density of small
PBHs — those from the extragalactic gamma ray background (EGB) — differ in the
scenario that black holes can radiate into higher dimensions. The nature of
gravity on short scales is not well understood: whilst Coulomb’s law (or its
quantum field theoretic generalisation) is known to apply down to distances of
order $10^{-18}$ m, Newton’s law of gravity has only been tested on scales of
order several microns. Consequently, it is consistent for there to exist extra
large spatial dimensions, and being extraordinarily compact objects, even
black holes as heavy as $10^{17}$ g could be sensitive to these dimensions.

Most new physics, involving the introduction of extra degrees of freedom
available for black holes to radiate, would result in little reduction in
detectable evaporation products and at best modest weakening of existing
constraints. The motivation for studying extra-dimensional evaporation is the
critical fact that higher-dimensional black holes, if smaller than the size of
the extra dimensions, are significantly colder than their 4D counterparts, for
a given fixed mass. These PBHs are thus expected to produce fewer evaporation
products, and thus be subject to weaker constraints, than ordinary 4D black
holes.

This paper is divided into five sections. In Section 2 we review the latest
constraints on the PBH density across the entire range of masses. In Section 3
we discuss the behaviour of black holes in theories with large extra
dimensions, and explain how the evaporation rate is modified in such a way as
to substantially modify the nature of all constraints on low-mass PBHs. In
Section 4 we present the main result of our analysis, the modified constraints
on the PBH density arising from the extragalactic photon background, for
several choices of the number of extra dimensions. Finally, in Section 5, we
discuss the modifications we expect to occur to other constraints on the PBH
density.

We take $c=\hbar=k_{B}=1$ throughout.

##  2 Existing Primordial Black Hole Constraints

Constraints on the density of primordial black holes can be divided into two
classes: constraints from the gravitational effects of the black holes
themselves, and constraints from the particles they produce through Hawking
evaporation. Since the total rate of energy loss is less for larger black
holes, these evaporative constraints exist only for PBHs of mass less than
about $10^{17}$ g. Conversely, gravitational effects of PBHs are typically
negligible for black holes below this mass.

Figure 1: Constraints on the PBH density (at formation) as a fraction of the
dark matter density. Shaded regions are excluded. In the upper panel are those
constraints arising from the gravitational influence of the black holes,
adapted from [2]; from left to right the constraints arise from femtolensing
experiments (FL), capture by neutron stars (NS), microlensing experiments (HSC
and EROS), and effects on the CMB (PLANCK). The vertical grey lines indicate
the mass $M_{c}$ at which the black hole radius is equal to the size of the
extra dimensions, as discussed in Section 3. In the lower panel are those
constraints arising from the effects of the PBH evaporation products; the
constraints from BBN are taken from [3] and those from the EGB we have
reproduced in this work.

###  2.1 Gravitational constraints

The constraints on the PBH density for masses above around $10^{17}$ g are
illustrated in the upper panel of Fig 1. Perhaps the most important of these
come from lensing experiments. For small black holes, the wave nature of
electromagnetic radiation is significant, and gravitational lensing around the
black hole can give rise to an interferometry pattern in the received
radiation (termed femtolensing) [4]111These results have been called into
question by [5].. For larger black holes this interference is not detectable,
though gravitational lensing can nevertheless result in the apparent
magnification of stars passing behind them (termed microlensing) [6, 7].

There also exist astrophysical effects associated with the collision and
subsequent capture of primordial black holes by compact objects such as
neutron stars and white dwarfs. In some mass ranges, such collisions would
result in the destruction of these objects; the number density of existing
neutron stars and white dwarfs hence places mild bounds on the density of
primordial black holes [8, 9].

For large black holes, of order $10^{35}$ g and larger, strict constraints
arise from the effects of these PBHs on the CMB. In particular, the accretion
of primordial gas around the black holes and subsequent injection of energy
into the background plasma would be detectable through its influence on the
angular distribution of temperature and polarization of the CMB. Data from
Planck strongly constrains this scenario [10].

###  2.2 Evaporative constraints

For masses below about $10^{17}$ g, all constraints on the PBH density are due
to the effects of black hole evaporation. The dominant constraints come from
the effects on big bang nucleosynthesis (BBN) and the size of the
extragalactic gamma ray background (EGB). In [3] a comprehensive analysis of
these constraints was performed. Injection of high-energy particles during BBN
can cause dissociation of heavier isotopes and induce extra interconversion of
protons and neutrons, the extent of which is strictly constrained by the known
abundances of the light elements. Constraints on black holes that have not
fully evaporated today arise from the extragalactic photon background.
Continual evaporation of PBHs over the course of the universe’s history would
have converted a considerable quantity of energy into photons (primarily gamma
rays), and this would be observable in the EGB. These constraints are
illustrated in the lower panel of Fig. 1.

The constraints from the extragalactic photon background are of most relevance
to this work, so we shall endeavour to explain the qualitative form of the
constraints. Assuming the black holes are formed very early in the universe,
there exists some mass $M_{0}$ such that they are disappearing today. For
black holes much larger than this, incomplete evaporation occurs, and thus not
all of the energy in the black holes is converted into Standard Model
particles. Since the lifetime of a black hole scales approximately like the
cube of its mass, a black hole need not be much heavier than $M_{0}$ before
its lifetime is far longer than the age of the universe and only a small
fraction of its energy is converted to photons. The constraints hence weaken
as $M$ is increased above $M_{0}$.

For masses smaller than $M_{0}$, the black holes have completely disappeared
by today. Though the total energy released by the black holes is the same for
all such $M$, the smaller the black holes, the earlier they disappeared, and
hence the greater the redshifting of the photons they produced. The energy
contributed to the photon background today hence decreases as $M$ is decreased
below $M_{0}$, and the constraints weaken. One needs to take a little more
care than this — smaller black holes emit predominantly higher-energy
radiation, and the constraints on the size of the photon background are
stricter at higher energy. However, sufficient redshifting of frequency occurs
that in fact the dominant constraints on small black holes come from the soft
end of the gamma ray background.

For black holes smaller than about $10^{13}$ g, complete evaporation has
occurred before photon decoupling. Such radiation is hence not visible in the
photon background, and so the constraints disappear completely for such black
holes.

We briefly mention that there are several other constraints of an evaporative
nature, arising from annihilation of electrons with positrons emitted by the
PBHs [11, 12], galactic gamma rays and antiprotons [13, 14], and effects on
CMB anisotropy [15]. Apart from over very narrow mass ranges, the constraints
from BBN and the EGB are the strictest of these.

##  3 Black Holes in Large Extra Dimensions

The existence of extra compact spatial dimensions is an appealing explanation
for the observed weakness of gravity relative to the other fundamental forces
[16]. Informally, the force of gravity is weaker because it is ‘diluted’
amongst these extra dimensions. More formally, in the case that the geometry
of the extra dimensions is independent of ours, the gravitational action can
be written

$S=\frac{1}{16\pi}M_{*}^{2+n}\int\,\mathrm{d}^{4+n}x\sqrt{-g}\left(\mathcal{R}_{4}+\mathcal{R}_{n}\right)\,,$
(3.1)

where $M_{*}$ is the fundamental Planck scale, $\mathcal{R}$ is the Ricci
scalar, and $n$ is the number of extra dimensions. Neglecting the second term,
we can perform the integral over the extra spatial dimensions to generate an
effective 4D action. If $R$ denotes the size of the extra dimensions, we hence
find the relation

$M_{P}^{2}\sim M_{*}^{2+n}R^{n}\,.$ (3.2)

It is hence possible for the fundamental Planck scale $M_{*}$ to be far lower
than the 4D Planck scale, if the extra dimensions are sufficiently large. For
$M_{*}=10$ TeV, the above relation implies that $R\sim 10^{11}$ m for $n=1$ —
certainly such a large extra dimension is ruled out by gravitational
experiments on the solar system scale. For $n=2$ one finds $R\simeq
25\,\mu\mathrm{m}$, which is consistent with current short distance tests of
Newton’s law [17]. In this paper we will consider $2\leq n\leq 6$. In Section
4 we will describe qualitatively the nature of the constraints for $n$ larger
than this.

Naturally, experiments exist that test the Standard Model up to around
$1\,\mathrm{TeV}$ — far smaller distance scales than a micron. If there are
additional spatial dimensions, the particles of the Standard Model must not
‘feel’ these extra dimensions — they must be localised to a brane. The
mechanism by which this could occur is an important question, but we note here
that it is a natural scenario in the context of some string theories, in which
branes occur with gauge theories automatically localised to them.

###  3.1 The higher-dimensional Schwarzschild solution

What is the nature of black holes in this model? Black holes which are much
larger in size than these extra dimensions should be insensitive to them, and
behave as ordinary 4D black holes. On the other hand, black holes much smaller
than these extra dimensions should be insensitive to the finiteness of the
dimensions, and behave as $(4+n)$-dimensional objects. There will be some
intermediate regime in which the black hole is not adequately described by
either picture. However, the crossover between the two descriptions is
continuous, for one can show that there is some critical mass $M_{c}$ at which
the size of the extra dimensions, the 4D Schwarzschild radius of the black
hole, and the $(4+n)$-dimensional radius of the black hole all approximately
coincide. This critical mass is tabulated in Table 1. For a review of this
material, see [18].

$n$ |  $M_{c}$ / g
---|---
2 |  $1.62\times 10^{25}$
3 |  $1.52\times 10^{20}$
4 |  $4.65\times 10^{17}$
5 |  $1.44\times 10^{16}$
6 |  $1.45\times 10^{15}$
Table 1: The mass $M_{c}$ in grams of the black hole whose Schwarzschild
radius is equal to the size of the extra dimensions, for $M_{*}=10$ TeV.

In $(4+n)$ dimensions the Schwarzschild metric is given by [19]

$\mathrm{d}s^{2}=-\left(1-\left(\frac{r_{h}}{r}\right)^{n+1}\right)\mathrm{d}t^{2}+\left(1-\left(\frac{r_{h}}{r}\right)^{n+1}\right)^{-1}\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\Omega_{n+2}^{2}\,,$
(3.3)

where the horizon radius is

$r_{h}=\frac{1}{M_{*}}\left(\frac{M}{M_{*}}\right)^{\frac{1}{n+1}}\left(\frac{8\,\,\Gamma((n+3)/2)}{(n+2)\pi^{(n+1)/2}}\right)^{\frac{1}{n+1}}\,.$
(3.4)

The Hawking temperature of such a black hole is given by

$T=M_{*}\left(\frac{M_{*}}{M}\right)^{\frac{1}{n+1}}\left(\frac{n+1}{4\sqrt{\pi}}\right)\left(\frac{n+2}{8\Gamma((n+3)/2)}\right)^{\frac{1}{n+1}}\,.$
(3.5)

The relation between the radius and temperature of the black hole is
particularly simple:

$T=\frac{n+1}{4\pi r_{h}}\,.$ (3.6)

When we restrict the metric Eq. (3.3) to the brane, we find a 4D black hole
solution whose geometry differs from the ordinary Schwarzschild solution. This
will affect the way the black hole gravitates. Consequently, some of the
aforementioned gravitational constraints will be modified in this scenario,
such as those from lensing experiments, since the bending of light around a
black hole is sensitive to the precise geometry. Similarly, capture of these
black holes by neutron stars and white dwarfs is sensitive to the potential
energy between the two objects, which depends fundamentally on the number of
extra dimensions.

The above notwithstanding, it transpires that for $M_{*}=10$ TeV, the critical
mass describing the transition from the 4D to the $(4+n)$-dimensional picture
occurs close to the mass at which existing constraints become dominantly
evaporative222For $n>2$ at least. For $n=2$, $M_{c}$ is appreciably larger
than this. . These critical masses are illustrated as vertical lines in Fig.
1. Black holes larger than this critical mass behave as ordinary 4D black
holes, and the existing gravitational constraints apply.

On the other hand, Eq. (3.5) shows that the temperature of a higher-
dimensional black hole can differ significantly from that of a 4D
Schwarzschild black hole of the same mass. This implies that the rate at which
it evaporates, and the energy of the particles it produces during this
evaporation, can differ considerably also. We thus expect the constraints on
low-mass PBHs to be substantially modified in this scenario.

###  3.2 Bulk and brane evaporation

A black hole smaller in size than the extra dimensions radiates gravitons into
the bulk and Standard Model particles onto the brane. The radiation of
Standard Model particles obeys the usual Stefan-Boltzmann law, with the
critical difference that the temperature of the radiation is no longer
inversely proportional to the mass of the black hole. As we discuss in the
appendix, for both modes of evaporation, the rate of loss of energy is
approximately equal, being given by

$-\frac{\mathrm{d}M}{\mathrm{d}t}\sim T^{2}\,.$ (3.7)

The differing dependence of the temperature on mass however, gives rise to a
black hole lifetime $\tau$ that depends critically on $n$:

$M_{*}\tau\sim\left(\frac{M}{M_{*}}\right)^{\frac{n+3}{n+1}}\,.$ (3.8)

By considering Eqs. (3.4), (3.5) and (3.8) in conjunction with Eq. (3.2), one
can show that black holes much smaller than the size of the extra dimensions
are larger, cooler, and live longer than 4D black holes of the same mass [20],
at least if one considers emission to involve only a single degree of
freedom333We consider the detail of the number of emitted degrees of freedom
in Section 4.. However, for black holes of order the size of the extra
dimensions, the numerical factors in Eq. (3.5) are not insignificant, and lead
to higher-dimensional black holes disappearing substantially faster.

Throughout the rest of this work we will take $M_{*}=10$ TeV. As mentioned at
the start of this section, this is approximately the bound for $n=2$ placed on
the size of the extra dimensions by measurements of the behaviour of Newton’s
law on short distances. However, we briefly mention here that there are
several other constraints on the size of $M_{*}$. Firstly, a weak-scale
fundamental Planck mass is subject to collider constraints. These are fairly
mild, and are consistent with $M_{*}=10$ TeV. There are also several
astrophysical and cosmological constraints due to the effects of the light
Kaluza-Klein modes of the graviton. These bounds are more strict, and
typically rule out $M_{*}=10$ TeV for $n=2$. However, we note that they are
subject to large systematic errors, and depend on the details of the decay of
the KK modes. See §106 of [21] for the latest constraints on $M_{*}$. In
Section 5 we describe the qualitative effects of choosing a larger value for
$M_{*}$.

##  4 Modified Constraints from the Extragalactic Photon Background

In this section we present the constraints that exist on the density of
higher-dimensional PBHs that arise from the contribution they would make to
the extragalactic photon background. It transpires that the strongest
constraints come from the X-ray and gamma ray background, as in the 4D case.
In the $n=1$ case the black holes radiate primarily in the UV and soft X-ray
regions of the electromagnetic spectrum. There are only very poor measurements
of the extragalactic UV background, but this is of no consequence since $n=1$
is ruled out by gravitational experiments.

We assume for simplicity a monochromatic mass distribution — that is, that all
of the primordial black holes are formed at the same time with the same mass.
This is not particularly realistic, and it is known in the 4D case that
constraints tend to become more stringent with extended mass distributions, if
qualitatively similar [2]. A monochromatic distribution is nevertheless
sufficient to indicate the large modifications to the constraints that occur
in the extra-dimensional scenario. We also emphasise that the quantity
$\rho_{\mathrm{PBH}}$ we plot in Fig. 3 is the fraction of the dark matter
density the black holes constitute at formation, and likewise the mass $M$ is
their mass at formation. In order to constitute a fraction of the dark matter
today, the PBHs must have an initial mass larger than that mass $M_{0}$ which
would be evaporating now, tabulated in Table 2.

###  4.1 Methodology

To compute the spectrum of radiation emitted by a black hole, the public code
BlackHawk [22] was used. In its original form, this code computes the emission
rate of all Standard Model particles from a given black hole, accounting for
greybody factors and using PYTHIA to compute the subsequent decay of all
unstable particles. The code makes the simplifying assumption that a black
hole begins radiating a given particle only when its temperature exceeds the
particle’s mass, and thereafter begins radiating it as though it were
massless.

The code needs some modification to produce the emission rate in the large
extra dimensions scenario. Naturally the mass-radius and mass-temperature
relations are modified according to Eqs. (3.4) and (3.5). Furthermore, the
greybody factors differ in different dimensions. These greybody factors were
computed for all spins and for all dimensions $n$, using the numerical recipes
outlined in [23, 24]. Accuracy of the numerical results could be compared to
the results in [23, 25]. We found excellent agreement with the former,
although not with the latter. We note that the numerical results in [25] do
not agree with the expected low-energy analytic expressions (in particular,
all spin-2 greybody factors should go to zero at low energy), so we put the
discrepancy between our results down to an error in theirs. To produce the
high-energy and low-energy asymptotics of the greybody factors, the analytic
results from [26, 27] were used.

Given the spectrum of radiation from a black hole at each moment of its
lifetime, the density of background photons today (in units of energy per unit
volume per unit energy) is given by the formula

$n=\int_{t_{\mathrm{dec}}}^{t_{\mathrm{max}}}(1+z)\frac{\mathrm{d}^{2}N}{\mathrm{d}t\,\mathrm{d}E}((1+z)E)\,\mathrm{d}t\,,$
(4.1)

where the derivative in the integrand is the energy being emitted by the black
holes per unit time per unit volume per unit energy. The integral is taken
between the time of photon decoupling $t_{\mathrm{dec}}$ and
$t_{\mathrm{max}}=\min(t_{0},\tau)$, where $t_{0}$ is the time today and
$\tau$ the lifetime of the black hole. Those photons with energies between $E$
and $E+\mathrm{d}E$, if produced at an earlier time $t$, must have been
emitted with blueshifted energy $(1+z)E$ and belonged to a wider energy window
$(1+z)\mathrm{d}E$. This accounts for the two redshift factors in the
integrand. As appropriate for the matter-dominated era, we take
$1+z(t)=(t_{0}/t)^{2/3}$.

An isotropic photon background gives rise to an observed flux of

$I=\frac{c}{4\pi}n\,.$ (4.2)

The data for the gamma ray region of the electromagnetic spectrum are
collected by space-based telescopes, in particular EGRET and COMPTEL aboard
the Compton Gamma Ray Observatory and LAT aboard the Fermi Gamma Ray Space
Telescope [28, 29, 30]. More data exist for the intensity of the X-ray
background, for which we use a fit from [31]. In Fig. 2 we plot these data. In
determining the PBH constraints we make the conservative assumption that the
entirety of the photon background in the X-ray and gamma ray region of the
electromagnetic spectrum is due to black hole evaporation products.

Figure 2: The observed background photon flux in units of energy per square
centimetre per second per steradian per unit energy, as a function of energy.
The energy range plotted corresponds to the X-ray and gamma ray region of the
electromagnetic spectrum.

###  4.2 Results

Our results are plotted in Fig. 3. We see that independent of $n$, the shape
of the constraints is broadly similar. The explanation for this shape is as
outlined in Section 2.2: those black holes which are evaporating today
contribute most energy to the photon background, with the constraints
disappearing for black holes small enough to have evaporated before photon
decoupling. The primary qualitative difference is due to the fact that the
mass $M_{0}$ of those black holes evaporating today is dimension-dependent.
These masses are tabulated in Table 2. For $n\geq 4$ we note that the
constraints cut off sharply above a certain mass. This is the mass $M_{c}$
above which the 4D description of the black holes is valid, tabulated in Table
1. The total radiation rate is substantially reduced above this mass (as
explained in Section 3.2), which is why the constraints significantly soften.
Indeed, comparison with Fig. 1 shows that above this mass, the constraints
coincide with those in the 4D case. Since we have not treated the crossover
behaviour of the black hole solution precisely, the results are not reliable
at this mass.

From Eqs. (3.8) and (3.5) one can show that for a black hole which survives
until today, the typical temperature of the Hawking radiation is higher for
larger $n$. This explains why the constraints are slightly stronger for larger
$n$ (and indeed why they are noisier), the dominant constraints coming from
the higher-energy region of the electromagnetic spectrum (see Fig. 2). The
constraints also cover a wider mass range than the 4D constraints, on account
that the lifetime of the black holes Eq. (3.8) depends less strongly on mass
than in the 4D case.

$n$ |  $M_{0}$ / g
---|---
2 |  $2.44\times 10^{7}$
3 |  $5.33\times 10^{10}$
4 |  $1.83\times 10^{13}$
5 |  $2.53\times 10^{15}$
6 |  $1.45\times 10^{15}$
Table 2: The mass $M_{0}$ in grams of a black hole evaporating today, assuming
formation at the start of the universe.

Figure 3: Constraints on the PBH density (at formation) as a fraction of the
dark matter density for different choices of $n$, assuming a monochromatic
mass distribution. Shaded regions are excluded. In the upper panel are plotted
(left to right) the constraints for $n=2,3,4$ and in the lower panel the
constraints for $n=4,5,6$.

One might wonder whether radiation into gravitons would dominate photon
production for large $n$, and constraints might substantially weaken, on
account that the number of degrees of freedom in the graviton is quadratic in
$n$ ($g=(n+1)(n+4)/2$). Our results do not bear this hypothesis out. To
understand this, we note that the graviton greybody factor is suppressed at
low energies relative to those for the photon and neutrinos, and more so for
larger $n$. See Fig. 4 in the appendix.

For $n=7$, the mass $M_{c}$ is approximately equal to the mass of a 4D black
hole evaporating today ($M_{0}\simeq 7.09\times 10^{14}$ g). Black holes
smaller than this, for which the extra-dimensional picture is valid, have
disappeared by today and therefore could not constitute the dark matter. On
the other hand, for black holes larger than this the 4D description, and hence
the 4D constraints, apply. Since $M_{c}$ decreases for larger $n$, we find
that for all $n>6$ the constraints on the fraction of the dark matter the
black holes could constitute are unchanged from those in the 4D case.

We note that only for $n=2$ and $n=3$ are there wide mass windows in which
PBHs could constitute the entirety of the dark matter — $M\gtrsim 10^{12}$ g
for $n=2$ and $M\gtrsim 10^{15}$ g for $n=3$. For larger $n$ our results show
that the photon background places similar constraints on the PBH mass as in
the 4D case, requiring $M$ to be larger than about $10^{17}$ g, beyond which
other gravitational constraints need to be taken into consideration.

##  5 Discussion

The most important question to address next is the modification of the
constraints from BBN in the extra-dimensional scenario. Just as for the photon
background, we expect that the dominant constraints will arise from those
black holes which evaporate completely during BBN. Since such black holes will
be lighter than their 4D counterparts, we expect the constraints from BBN to
be shifted to lower mass, and more so for lower $n$. We leave a detailed study
of this to future work.

It is also necessary to understand how other evaporative and gravitational
constraints change with the introduction of extra dimensions. As mentioned in
Section 4.2, the typical temperature of the black holes increases as $n$ is
increased. For low $n$, the only evaporation products are neutrinos,
gravitons, and photons, and we hence expect the constraints from positron
annihilation or antiprotons to be non-existent. For large $n$, black holes
larger than $10^{17}g$ behave as 4D objects, and so we expect existing
gravitational constraints to apply. Lensing constraints in the low $n$ case,
and other evaporative constraints in the large $n$ case, would need to be
studied in greater detail.

The purpose of this work has been to demonstrate the significant qualitative
changes to the constraints on the PBH density that occur when extra dimensions
are present. In doing so we have made some simplifying assumptions — that the
black hole mass distribution is monochromatic, and that the black holes are
not rotating. In any number of dimensions, the Hawking temperature of a black
hole depends quite sensitively on its angular momentum; the constraints on a
population of spinning black holes could thus be appreciably different. To
understand how these two factors affect the constraints in the 4D case, see
[32].

A final natural question to ask is how the constraints would differ for a
different choice of $M_{*}$. In this work we have chosen the lowest value of
$M_{*}$ consistent with experiment. From Eq. (3.8) we see that a black hole
evaporating today would have larger mass for a larger choice of $M_{*}$. We
thus expect the constraints to be shifted to larger mass as $M_{*}$ is
increased, though with the same qualitative shape. For $M_{*}=10$ TeV we find
that the constraints for $n>6$ are just as in the 4D case, and we expect that
the larger we take $M_{*}$, the fewer choices of $n$ will give rise to novel
constraints.

## Acknowledgements

I am grateful to John March-Russell for many useful discussions, and for
initially suggesting the possibility that constraints on the PBH density could
differ depending on the number of dimensions the black holes exist in.

##  Appendix A Radiation in Higher Dimensions

In this appendix we state the formulae for computing the total emission rate

$\frac{\mathrm{d}^{2}N}{\mathrm{d}t\,\mathrm{d}\omega}\,,$ (A.1)

as appears in Eq. (4.1), for both bulk and brane emission. The most
straightforward expression, which applies in any number of dimensions, is

$\frac{\mathrm{d}^{2}N}{\mathrm{d}t\,\mathrm{d}\omega}=\frac{1}{2\pi}\sum_{\text{states}}\frac{|\mathcal{A}(\omega)|^{2}}{\exp(\omega/T)\mp
1}\,,$ (A.2)

where $|\mathcal{A}|^{2}$ is the absorption probability for a particle
incident upon the black hole. A formula that makes closer connection with the
usual Planck law is

$\frac{\mathrm{d}^{2}N}{\mathrm{d}t\,\mathrm{d}\omega}=\frac{g\,\Omega_{n+2}}{(2\pi)^{n+3}}\omega^{n+2}\frac{\sigma(\omega)}{\exp(\omega/T)\mp
1}\,,$ (A.3)

where $g$ is the number of degrees of freedom of the particle. Here the
quantity $(4+n)$ describes the number of dimensions the emitted particles feel
— even for a higher-dimensional black hole, we should take $n=0$ for brane
emission. These two expressions can be seen to be equivalent using the
relation between the greybody factor and the absorption amplitude:

$\sigma(\omega)=\frac{2^{n}\pi^{(n+1)/2}(n+1)\Gamma((n+1)/2)}{\omega^{n+2}}\sum_{\ell}N_{\ell}|\mathcal{A}_{\ell}(\omega)|^{2}\,,$
(A.4)

where $N_{\ell}$ is the number of degrees of freedom per angular momentum mode
$\ell$, implicit in the sum over states in Eq. (A.2). In four dimensions we
have $N_{\ell}=2\ell+1$, though this is no longer true in higher dimensions,
and indeed $N_{\ell}$ depends on whether the degree of freedom being emitted
is a scalar, vector, or tensor perturbation. We remark that Eq. (2.8) in [23]
is not correct in general since it assumes the scalar form of $N_{\ell}$.

In general the greybody factor has only weak dependence on frequency. In
particular, for large energies the greybody factor asymptotes to a constant,
the absorption cross-section of the black hole. Note that for a perfect black
body, the greybody factor is independent of $\omega$ and equal to the cross-
sectional ‘area’ of the body, $\pi r^{2}$ for a 4D black hole. In this case we
can multiply Eq. (A.3) by energy and integrate to produce the total rate of
radiation. It is particularly simple:

$P=\frac{g(n+3)}{\pi}\,T^{n+4}\,r_{h}^{n+2}\begin{cases}\zeta(n+4)&\mathrm{bosons}\,,\\\
\eta(n+4)&\mathrm{fermions}\,,\end{cases}$ (A.5)

where $\zeta(z)$ is the Riemann zeta function and $\eta(z)$ the Dirichlet eta
function, both approximately unity for large $n$. To an order of magnitude, we
thus have

$P\sim T^{n+4}\,r_{h}^{n+2}\,.$ (A.6)

Since the horizon radius and Hawking temperature are inversely proportional,
as in Eq. (3.6), we arrive at the conclusion that the rate of loss of mass of
the black hole is always proportional to $T^{2}$, independent of the number of
dimensions, as claimed in Eq. (3.7). Studying Eq. (A.5) in more detail, we can
conclude that for a given $n$ the rates of bulk and brane emission are of the
same order of magnitude. However, note that the numerical factors in the
expression for the temperature Eq. (3.5) are not unimportant, and result in
orders-of-magnitude faster evaporation for $n=6$ as compared to $n=1$. For
this reason, constraints on the PBH density substantially weaken once the
radius of the black hole exceeds the size of the extra dimensions.

For black holes larger than the size of the extra dimensions, the spectrum
contains the 4D graviton and a tower of Kaluza-Klein modes, which the black
hole can also radiate in principle. However, the lightest of these has mass of
order $1/R$, which by Eq. (3.6) only begins to be radiated when $r_{h}\sim R$.
Such radiation can hence be neglected for large black holes, and evaporation
treated just as in the 4D case. The exact spectrum of Kaluza-Klein modes
depends on the geometry of the extra dimensions, but Weyl’s law guarantees
that as the black hole becomes much smaller than the size of the extra
dimensions, the spectrum tends towards that for infinite space.

We briefly mention that conventional Kaluza-Klein dimensional reduction does
give rise to other massless modes, arising from the symmetries of the extra
dimensions, such as the moduli fields which describe the dimensions’ size. For
both phenomenological and theoretical reasons such fields must by some
mechanism gain a mass: this evades the significant experimental implications
of additional massless particles, and stabilises the size and shape of the
extra dimensions. We have assumed in this work that this moduli problem is
solved in such a way as to raise the masses of these modes at least as high as
that of the lightest massive Kaluza-Klein mode. This ensures that for black
holes larger than the extra dimensions, ordinary 4D evaporation applies.

Figure 4: Sample greybody factors for $n=2$, for all types of particle. Note
the different scales on the $y$-axes, and the low-energy suppression of
graviton radiation.

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