Prepared for submission to JCAP
Primordial Black Holes as Dark
Matter: Converting Constraints
from Monochromatic to Extended
Mass Distributions
Nicola Bellomo
a,b
José Luis Bernal
a,b
Alvise Raccanelli
a
Licia
Verde
a,c
a
ICC, University of Barcelona, IEEC-UB, Martí i Franquès, 1, E08028 Barcelona, Spain
b
Dept. de Física Quàntica i Astrofísica, Universitat de Barcelona, Martí i Franquès 1, E08028
Barcelona, Spain
c
ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
E-mail: nicola.bellomo@icc.ub.edu, joseluis.bernal@icc.ub.edu, alvise@icc.ub.edu,
liciaverde@icc.ub.edu
Abstract. The model in which Primordial Black Holes (PBHs) constitute a non-negligible
fraction of the dark matter has (re)gained popularity after the first detections of binary black
hole mergers. Most of the observational constraints to date have been derived assuming a
single mass for all the PBHs, although some more recent works tried to generalize constraints
to the case of extended mass functions. Here we derive a general methodology to obtain con-
straints for any PBH Extended Mass Distribution (EMD) and any observables in the desired
mass range. Starting from those obtained for a monochromatic distribution, we convert them
into constraints for EMDs by using an equivalent, effective mass M
eq
that depends on the
specific observable. We highlight how limits of validity of the PBH modelling affect the EMD
parameter space. Finally, we present converted constraints on the total abundance of PBH
from microlensing, stellar distribution in ultra-faint dwarf galaxies and CMB accretion for
Lognormal and Power Law mass distributions, finding that EMD constraints are generally
stronger than monochromatic ones.
arXiv:1709.07467v2 [astro-ph.CO] 9 Jan 2018
Contents
1 Introduction 1
2 Equivalent Monochromatic Mass Distribution 2
3 Application to different observables 6
3.1 Microlensing 7
3.2 Ultra-Faint Dwarf Galaxies 9
3.3 Cosmic Microwave Background 11
4 Practical considerations and observational constraints 13
5 Conclusions 19
1 Introduction
The ΛCDM model has become the cosmological standard model thanks to its ability to
provide a good description to a wide range of observations, see e.g., [1]. However, it remains
a phenomenological model with no fundamental explanations on the nature of some of its key
ingredients, e.g., of dark matter, see e.g., [2]. Several possible dark matter candidates have
been proposed, ranging from yet undetected exotic particles like WIMPs [3] or axions [4], to
compact objects such as black holes [5], including the ones possibly forming at early times,
therefore called Primordial Black Holes (PBHs).
Since so far none of the numerous undergoing direct dark matter detection experiments
has given positive results (neither for WIMPs [6] nor for axions [7]), PBHs have started to
(re)gain interest after the first gravitational waves detection by the LIGO collaboration [8].
Those gravitational waves were generated by a merger of two black holes with masses around
30M
. Given the large mass of the progenitors, some authors [9, 10] have proposed that
it could be the first detection of PBHs, whose merger rate is indeed compatible with LIGO
observations. Since PBHs were first proposed as a candidate for dark matter, there have been
considerable efforts from both theoretical and observational sides to constrain such theory.
PBHs might produce a large variety of different effects because the theoretically allowed
mass range spans many order of magnitude. As a consequence, the set of constraints coming
from a variety of observables is broad too. Starting from the lower allowed mass, constraints
come from γ-rays derived from black holes evaporation [11], quantum gravity [12], γ-rays
femtolensing [13], white-dwarf explosions [14], neutron-star capture [15], microlensing of stars
[16–20] and quasars [21], stellar distribution in ultra-faint dwarf galaxies [22], X-ray and radio
emission [23], wide-binaries disruption [24], dynamical friction [25], quasars millilensing [26],
large-scale structure [27] and accretion effects [28–31]; given the strong interest in the model,
there have been recently suggestions for obtaining constraints from e.g. the cross-correlation
of gravitational waves with galaxy maps [32, 33], eccentricity of the binary orbits [34], fast
radio bursts lensing [35], the black hole mass function [36, 37] and merger rates [38].
These constraints have been obtained (mostly) for a PBH population with a Monochro-
matic Mass Distribution (MMD). This distribution has been always considered as stationary,
even if during its life any black hole changes its mass due to different processes, such as
– 1 –
Hawking evaporation [39], gravitational waves emission, accretion [29] and mergers events [9].
The magnitude of such effects has been analysed recently. In Ref. [40] the authors investigate
the importance of evaporation and Bondi accretion during the whole Universe history. They
found that PBHs with mass 10
−17
M
. M . 10
2
M
neither accrete or evaporate signifi-
cantly in a Hubble time (unless they are in a baryon-rich environment). On the other hand,
the mass lost in gravitational waves emission due to mergers has to be small, since the frac-
tion of dark matter converted into radiation after recombination cannot exceed the 1%
1
[41].
This finding rejects the possibility of an intense merging period at z ≤ 1000. Although these
effects are small compared to current experimental precision and theoretical uncertainties in
the modelling of the processes involving PBHs, a comprehensive treatment must eventually
include a description of their evolution.
More importantly, a large variety of formation mechanisms directly produce Extended
Mass Distributions (EMDs) for PBHs. Such mechanisms generate PBHs as the result of,
among other precesses, collapse of large primordial inhomogeneities [42] arising from quantum
fluctuations produced by inflation [43], spectator fields [44] or phase transitions, like bubble
collisions [45] or collapse of cosmic string [46], necklaces [47] and domain walls [48].
As pointed out in Ref. [49], no EMD can be directly compared to MMD constraints.
Since re-computing the constraints for any specific EMD can be time-consuming, at least two
different techniques [49, 50] have been proposed so far to infer EMDs constraints from the
well-known MMD ones. In this paper we propose a new and improved way to compare EMDs
to MMD constraints, directly based on the physical processes when PBHs with different
masses are involved.
The paper is organized as follows. In Section 2 we present our method to convert between
monochromatic constraints and EMD ones and compare it with existing ones. In Section 3
first we introduce the EMDs we will analyse, then we provide some practical examples of
how our technique works for three different observables, namely microlensing (3.1), ultra-
faint dwarf galaxies (3.2) and the cosmic microwave background (3.3). In Section 4 we derive
constraints for EMDs and discuss the validity of the limits found in previous Sections. Finally,
we conclude in Section 5.
2 Equivalent Monochromatic Mass Distribution
Most of the constraints derived in previous works have been obtained under the simplifying
assumption that PBHs have a MMD, despite the fact that such distribution is unrealistic
from a physical point of view. Since EMDs have more robust theoretical motivations, it is
extremely important to derive accurate constraints for EMDs in order to establish if PBHs
could be a valid candidate for (at least a large fraction of) dark matter.
As pointed out for the first time in Ref. [49] and then in ref. [51], it is not straightforward
to interpret MMD constraints in terms of EMD. It is therefore important to derive constraints
precisely using directly the chosen EMD or to provide an approximated technique to convert
between MMD and EMD constraints, as done in [49, 50]. Advantages and shortcomings of
the presently available methods to convert between MMD and EMD constraints have been
discussed in Ref. [51]; in short they may bias (i.e., overestimate or underestimate depending
on the EMD) the inferred constraints.
1
A single merger event may surpass this limit, for example the LIGO event GW150914 has been estimated
to have converted about 5% of the mass in GW. Here however what matters is the overall integrated conversion.
– 2 –
As it is customary, hereafter f
PBH
denotes the fraction of dark matter in primordial
black holes, f
PBH
=
Ω
PBH
Ω
dm
. The fundamental quantity in our approach is the PBHs differential
fractional abundance
df
PBH
dM
≡ f
PBH
dΦ
PBH
dM
, (2.1)
defined in such a way that f
PBH
represents the normalisation and the distribution
dΦ
PBH
dM
describes the shape (i.e., the mass dependence) of the EMD and it is normalized to unity. By
definition this function is related to the differential PBH energy density or, equivalently, to
the differential PBH number density by
dρ
PBH
dM
=
dn
PBH
d log M
= f
PBH
ρ
dm
dΦ
PBH
dM
, (2.2)
since PBHs are a dynamically cold form of matter. Each EMD is specified by a different
number of parameters {ζ
j
} that define its shape and the mass range [M
min
, M
max
] where the
distribution is defined. Known theoretically-motivated models provide a variety of EMDs; in
what follows we consider two popular EMDs families, namely the Power Law (PL) and the
Lognormal (LN ) ones, which we will describe in Section 3 (for other examples of EMD, see
e.g., Ref. [52]).
We start from the same consideration done in Ref. [50], where it was noticed that PBHs
with different masses contributes independently to the most commonly considered observ-
ables. In order to account for a PBHs EMD, when calculating PBHs effects on astrophysical
observables we have to perform an integral of the form
Z
dM
df
PBH
dM
g(M, {p
j
}), (2.3)
where g(M, {p
j
}) is a function which encloses the details of the underlying physics and
depends on the PBH mass, M, and a set of astrophysical parameters, {p
j
}. Therefore,
g(M, {p
j
}) is different for each observable (some example of these functions are provided in
Section 3). Because of this integral over the mass distribution, there is an implicit degen-
eracy between different EMDs, which means that two distributions (indicated below by the
subscripts 1 and 2) such that
f
PBH,1
Z
dM
dΦ
1
dM
g(M, {p
j
}) = f
PBH,2
Z
dM
dΦ
2
dM
g(M, {p
j
}) (2.4)
will be observationally indistinguishable. As the constraints for MMDs have already been
computed in the literature, we set one of the two distributions in Equation 2.4 to be a
MMD and the other to be an arbitrary EMD i.e.,
df
PBH,1
dM
= f
MMD
PBH
δ(M − M
eq
) and
df
PBH,2
dM
=
f
EMD
PBH
dΦ
EMD
dM
, so that we can easily rewrite Equation 2.4 as
f
MMD
PBH
g(M
eq
, {p
j
}) = f
EMD
PBH
Z
dM
dΦ
EMD
dM
g(M, {p
j
}), (2.5)
where M
eq
will be called Equivalent Mass (EM). The equivalent mass is, by definition, the
effective mass associated with a monochromatic PBHs population such that the observable
effects produced by the latter are equivalent to the ones produced by the EMD under consid-
eration.
Constraints for EMDs can be extracted from the previous equation through the following
procedure.
– 3 –
(A) Fix the ratio r
f
= f
EMD
PBH
/f
MMD
PBH
to a specific value. Here, since we want to reinterpret
f
MMD
PBH
as a f
EMD
PBH
and solve for M
eq
we set r
f
= 1, that is we assume that PBHs total
abundance in both scenarios is the same. In principle and for other applications one
may want to work with other values of r
f
or one may want to fix M
eq
and solve for
r
f
. For this reason in our equations we have left r
f
indicated explicitly, but in explicit
calculations it is set to unity.
(B) Given the (known, see e.g., Section 3) function g for the selected observable, solve for
M
eq
the equation
g(M
eq
, {p
j
}) = r
f
Z
dM
dΦ
EMD
dM
g(M, {p
j
}) (2.6)
to calculate the equivalent mass M
eq
(r
f
, {ζ
j
}) as a function of the parameters of the
EMD. As we will see below, in some case this can be done analytically (see e.g., Equation
3.20), but in other cases must be done numerically (see e.g., Equation 3.16). The
dependence of M
eq
on the EMD parameters describing its shape is helpful to understand
which observable effects are produced by a certain EMD.
(C) The allowed PBHs abundance (for the considered observable) is given by
f
EMD
PBH
({ζ
j
}) = r
f
f
MMD
PBH
(M
eq
(r
f
, {ζ
j
})) , (2.7)
where f
MMD
PBH
(M
eq
) is the largest allowed abundance for a MMD with M = M
eq
. If
we are interested in just one constraint in particular, then this formalism allows us to
immediately state if a given EMD is compatible or not with observations. If instead we
want to account for several constraints at once, we have to a find the set of Equivalent
Masses associated to each function g. Every mass calculated in this way has a maximum
allowed (MMD) PBHs fraction (e..g, as found in the literature); of these f
PBH
values,
the minimum one that satisfies all the constraints at once is the largest allowed PBH
abundance of that EMD. This is illustrated in Figure 1. Hereafter we refer to the
maximum allowed value of the PBH fraction as
ˆ
f
PBH
.
In Figure 1 we consider two specific EMDs, a PL (left) and a LN (right) and four
observational constraints obtained for MMD: microlensing, ultra-faint dwarf galaxies (UFDG)
and CMB. The adopted functions g for these observables will be described in Section 3. For
each observable and each EMD we show the corresponding M
eq
(dashed vertical lines) and
maximum allowed PBHs fraction
ˆ
f
PBH
. For the PL EMD the maximum allowed
ˆ
f
PBH
is
the lowest of the four i.e., the one obtained from EROS2 microlensing (for its corresponding
EM). On the other hand, for the chosen LN distribution the maximum allowed
ˆ
f
PBH
is that
provided by the UFDG for their EM.
An additional feature of this approach is that it allows one to understand which part of a
EMD (e.g., low-mass or high-mass tail) is more relevant for a given observational constraint.
Such information can be inferred from the value of the equivalent mass i.e., from the position
of the vertical dotted lines in Figure 1.
Our method extend existing ones [49, 50] in several ways. First of all it introduces
a clear physical connection between the effects of EMDs and those of a MMD thanks to
the introduction of the new concept of the Equivalent Mass. Thanks to this concept one
can predict the approximated strength of the constraint even without computing it, since
– 4 –
![Figure 2: Theoretical (ε(∆t) = 1) and experimental differential event rate for a MMD with Meq = 0.3M , two PL and a LN with Mmax = 100M and σ = 1.0. Mmin and µ have been obtained with Equation 3.12 imposing the same equivalent mass of the MMD. The stars source is the LMC, whose parameters can be found in [51, 58]. We assume fPBH = 1 for every distribution to calculate the EM. Notice the level of concordance of the expected number of events.](/figures/figure-2-theoretical-e-t-1-and-experimental-differential-1mpjhh7k.webp)
