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Primordial black holes as possible candidates for
dark matter
Bradley Aldous (140172636)
(Dated: Friday 24th March, 2017, 16:50)
The capture of dark matter by pre-stellar cores is considered, subsequently the dark
matter will be trapped inside the compact remnant that the star becomes. If the dark
matter is made up of primordial black holes (PBHs) then these will rapidly destroy the
compact remnant and so constraints on the abundance of PBHs are implied by
observations of compact remnants. Observational constraints based on black hole
evaporation and gravitational lensing, as well as various dynamical constraints, are all
considered and applied to the three allowed PBH mass ranges in which they could
comprise the dark matter: 1016g - 1017g, 1020g - 1024g and 1M⊙ - 103M⊙. These mass
ranges are confronted with monochromatic and extended mass functions.
!ii
Contents
I. Introduction……………………………………………………………………………………………..1
II. Capture of dark matter during star formation by pre-stellar core……………………………………….1
A. Adiabatic contraction of dark matter………………………………………………………………..2
B. Constraints on PBHs………………………………………………………………………………..4
III. Capture of PBHs by neutron stars……………………………………………………………………….5
A. Energy loss of PBH…………………………………………………………………………………5
B. Capture rate…………………………………………………………………………………………8
C. Constraints on PBHs………………………………………………………………………………..9
IV. Summary of constraints on the contribution of PBHs to dark matter………………………………….11
A. Observational constraints from Hawking radiation……………………………………………….11
B. Observational constraints based on lensing by PBHs……………………………………………..13
C. Constraints based on interactions with astronomical objects……………………………………..14
V. Possible mass ranges for PBH dark matter…………………………………………………………….15
A. Submatomic-sized black holes…………………………………………………………………….15
B. Sublunar-mass black holes………………………………………………………………………..17
C. Intermediate-mass black holes…………………………………………………………………….17
VI. Conclusions……………………………………………………………………………………………19
VII. References……………………………………………………………………………………………..20
!1
I. INTRODUCTION
Primordial black holes (PBHs) are hypothetical black holes which formed from large density fluctuations due
to strong perturbations in gravitational field during the universe’s early expansion (Hawking 1971, Zabotin
and Nasel’skii 1980, Carr and Hawking 1974, Chapline 1975), and thus can have quite large mass, or low
mass in comparison to black holes formed today.
These black holes have been of great scientific interest for years (Zel’dovich and Novikov 1967) as the
number of PBHs that formed in the early universe have significant consequences to a number of effects. One
aspect of this scientific interest is related to dark matter and there is an exciting theory that PBHs could
constitute dark matter (Ivanov et al. 1994, Carr et al. 2016b). Observations of the cosmic microwave
background (CMB) implies that dark matter comprises 23% of the total amount of energy in the universe
compared to 4% which is the contribution from baryonic matter (Capela et al. 2013a). There is much
speculation regarding this theory as definitive evidence proving the existence of PBHs has yet to be found.
However the more traditional view of what dark matter could be is that it is a cold, neutral and long-lived
particle for which there is no candidate that has been found which exhibits these features in the framework of
the standard model.
The most favoured candidates for dark matter are exotic particles like the Weakly Interacting Massive
Particles (WIMPs) (Jungman et al. 1996, Preskill et al. 1983, Griest and Kamionkowski 2000) for which,
like PBHs, no evidence has been found despite years of accelerator experiments (Di Valentino et al. 2014).
This could just be due to the fact that, like PBHs, WIMPs are also extremely difficult to detect as they may
only interact via the weak and gravitational interactions.
In this paper I address the question: can PBHs be possible candidates for dark matter? In order to answer this
question I review the capture of PBHs during star formation in Sec. II and in Sec. III I review the capture of
PBHs by compact stars - more specifically neutron stars - both of these analyses yield constraints on the
fraction of dark matter that could be in PBHs (Capela et al. 2013a, Capela et al. 2013b). In Sec. IV I
summarise the constraints on the fraction of dark matter in PBHs, f(M), of non-evaporating black holes and
from this, in Sec. V, I review the mass ranges in which PBHs can still constitute the dark matter and the
constraints that apply to these mass ranges (Carr et al. 2016b).
II. CAPTURE OF DARK MATTER DURING STAR FORMATION BY PRE-STELLAR CORE
In their paper, Capela et al. (2013a) study the capture and adiabatic contraction of dark matter during star
formation. Stars form from overdense regions of molecular gas found within giant molecular clouds (GMCs)
which are extremely large (5 to 200 parsecs in diameter) and contain a huge amount of mass - roughly
! to !. These overdose regions within the GMC are supported by rotation and magnetic
pressures which prevent the clump of molecular gas from gravitationally collapsing. However, these
magnetic pressures are slowly lost due to a process called ambipolar diffusion, which is when positively
charged and negatively charged species diffuse due to their interaction through an electric field (Shu et al.
1987).
Due to the loss of the magnetic pressures, the core of the overdense region becomes increasingly unstable
until there is no force to support against the gravitational forces and so it collapses. The resulting object
which is in hydrostatic equilibrium is called a protostar, which is an early form of a star and hasn’t begun it’s
life cycle yet.
103M⊙
107M⊙
!2
A. Adiabatic contraction
In the following, Capela et al. (2013a) use adiabatic contraction as the main mechanism for the capture of
dark matter during star formation to find the density profile of dark matter in a prestellar core. As baryons
contract they lose energy and their time-dependent gravitational potential pulls the dark matter causing the
dark matter distribution to form a peak centred on that of the baryon’s. If the time scale of the baryonic
contraction is much larger than the free-fall time, given by
!,
with ! being the density of baryons within the cloud and G is just the gravitational constant. If this is the
case then the distribution of dark matter in the cloud is given by the conservation of the adiabatic invariant
!,
where E and T are the energy and orbital period of a dark matter particle respectively and p and q are the
coordinates of said dark matter particle in the phase-space - which is a space in which all possible states of a
system are represented, where each state corresponds to a specific point in the space (En.wikipedia.org,
2017). Adiabatic contraction tries to increase the phase-space density to it’s maximum value (Tremaine and
Gunn 1979)
!,
where ! is the dispersion of the initial velocity distribution of the dark matter and ! and ! refer to
the density and the mass of the dark matter respectively. By considering a spherical baryonic cloud with an
initial radius ! which contracts to a negligible size and by making the assumption that the dark matter
moves on circular orbits, the following relation for the dark matter density is found (Steigman et al. 1978).
!.
However, it is not realistic to assume that the dark matter will follow circular orbits and so calculations were
performed (Spolyar et al. 2008, Bambi et al. 2009) in the case of radial orbits to check if (4) is a good
approximation for non-circular cases and the results agreed. However, it is also not realistic to assume that
the dark matter will follow purely radial orbits and so the validity of the adiabatic approximation was tested
in numerical simulations (Jesseit et al. 2002) from which it was shown to be a good approximation even
when the collapse time is roughly the same as the free-fall time, dependent on the potential.
In order to resolve these issues Capela et al. (2013a) performed numerical simulations of two different
scenarios. They considered a point mass, with mass M = 0 at time t = 0, at the centre of a uniform spherical
cloud with a constant density profile. Dark matter particles with random initial velocities were then put in
one by one to make up for the fact they neglected the contribution to the gravitational potential from the dark
matter. The second scenario is the same as the first except that the dark matter particles were given initial
tff =1
G
ρ
0
ρ
0
p dq =ET
!∫
Qmax =3
2
π
⎛
⎝
⎜⎞
⎠
⎟
3
2
ρ
DM
mDM
4
υ
3
υ
ρ
DM
mDM
R
ρ
DM (r)=1
4
ρ
DM
R
r
⎛
⎝
⎜⎞
⎠
⎟
9
4
(1)
(2)
(3)
(4)
!3
velocities so that they would follow circular orbits. For the circular orbit case the same relation (4) was
reproduced. However, for the case where the particles had random initial velocities, Capela et al. found the
following relation for the dark matter density
!,
which is in agreement with Liouville’s theorem - which states that the density of system points in the vicinity
of a given system point travelling through a phase-space is constant with time, so is a better approximation
than the density relation for a circular orbit case and this is the relation Capela et al. use throughout the rest
of their paper (2013a).
Following on from this they then attempt to estimate the amount of dark matter that is bound gravitationally
to the baryonic cloud since this is the only dark matter that undergoes adiabatic contraction. In regions of star
formation, density of matter is dominated by baryons so this assumption is made for the following analysis.
Capela et al. consider the formation of a spherical baryonic cloud with baryonic density ! and radius !
and make the assumption that initially the dark matter has the following Maxwellian distribution in velocities
!,
with !, where m is the mass of the dark matter particle and ! is the dispersion.
Once a baryonic cloud has formed, some of these dark matter particles will become gravitationally bound to
the cloud and thus will feel a gravitational potential given by the following relation
!,
but only those particles with a kinetic energy lower than ! will become gravitationally bound. Capela et al.
obtained a relation giving the density of gravitationally bound dark matter by integrating (6) up to the limit
! to get the number density and then multiplying by the mass of the dark matter, ultimately getting
!,
and by substituting in the equation for !
!.
ρ
DM (r)=1
2
ρ
DM
R
r
⎛
⎝
⎜⎞
⎠
⎟
3
2
ρ
0
R0
dn =nDM
3
2
πυ
2
⎛
⎝
⎜⎞
⎠
⎟
3
2exp −3
υ
2
2
υ
2
⎧
⎨
⎩
⎫
⎬
⎭
d3
υ
nDM =
ρ
DM
m
υ
φ
~
φ
0=2
π
G
ρ
0R0
2
φ
0
υ
=2
φ
0
ρ
DM (bound )=
ρ
DM
4
π
3
3
φ
0
πυ
2
⎛
⎝
⎜⎞
⎠
⎟
3
2
φ
0
ρ
DM (bound )=
ρ
DM
4
π
3
6G
ρ
0R0
2
υ
2
⎛
⎝
⎜⎞
⎠
⎟
3
2
(5)
(6)
(7)
(8)
(9)
!4
C. Constraints on PBHs
Globular clusters (GCs) are spherical regions of high stellar density which orbit galactic cores as satellites.
They typically contain between 104 to 107 stars, with an average stellar density of 0.4 stars per cubic parsec,
and have an average diameter of 20-100 parsecs. They are among the oldest stellar systems in the universe
with an age of 8 to 13.5 billion years (Dotter et al. 2010). There a two different ways in which GCs are
thought to be formed - the “baryon-dominated” case and the “dark matter-dominated” case. In the former
case, GCs are thought to be made through mergers and other baryonic processes to later become part of the
Milky Way galaxy (Fall and Rees 1985, Ashman and Zepf 1992, Kravtsov and Gnedin 2005). In the latter
case GCs are said to be formed due to more dense regions of dark matter causing gravitational wells to pull
in baryon matter at the time of star formation (Bromm and Clarke 2002, Mashchenko and Sills 2005a). Both
of these mechanisms of GC formation could be working since observations of the metallicity distribution in
GCs suggests this. Capela et al. (2013a) base the following analysis on the assumption that some GCs are
formed via the dark matter dominated method even though there is uncertainty in whether dark matter exists
in GCs. However Mashchenko and Sills (2005b) have shown that, at the time of star formation, there could
have been dark matter present in GCs but would have been tidally disrupted by the host galaxy.
As of yet the conversation has been general and just about dark matter, unrelated to PBHs. By considering
PBHs captured by stars during their formation and subsequently being held inside of the resulting compact
remnant (white dwarf or neutron star), Capela et al. (2013a) have arrived at constraints on the fraction of
PBHs in the total amount of dark matter.
PBHs may be captured by stars during their formation, once inside the star they begin to gravitationally pull
on the matter that makes up the star and start to accrete mass as they sink to the centre of the star. This
process is very slow during the bulk of the star’s lifetime - the characteristic time for this process can be
greater than the age of the star. However once the star collapses into a compact remnant, either a white dwarf
or a neutron star - dependent on it’s initial mass, the accretion process becomes much more rapid and
consumes the compact remnant in a relatively short time scale. Thus, constraints on the number of PBHs
present in the universe can be obtained observationally from white dwarfs and neutron stars.
By calculating NBH, the number of black holes that would sink down to the centre of the star within it’s
lifetime and occupy a space with a radius smaller than rf - the radius of the future compact remnant the star
will become, one can calculate the maximum allowed fraction of PBHs in the total amount of dark matter
!.
This constraint is shown in Fig. 1.
As can be seen from this relation, when NBH < 1 this fraction becomes greater than 1 and so no valid
constraints can be made, so this relation is only valid for values of NBH that are smaller than 1. Once the star
has evolved and completed it’s life cycle to become a white dwarf or neutron star, the PBH trapped inside
will occupy a space with radius rc, which Capela et al. (2013a) call the “collection region”. The collection
region obviously has to be less than rf to allow for this to happen, otherwise the star would never collapse to
form a compact remnant, it would just collapse into the black hole as soon as the star had finished burning
it’s fuel. There is then a simple relation for calculating the number of black holes within the collection region
given by
ΩPBH
ΩDM
≤1
NBH
(10)
(11)
!5
!,
where mBH is the PBH mass inside the collection region and MDM(r) is the total mass of the dark matter
within the radius r. From Eq. (5) Capela et al. found the following relation
!,
where MDM(r) and M(r) and the dark matter and baryonic masses respectively. The constraints on the fraction
of PBHs in the total amount of dark matter from observations of white dwarfs and neutron stars in globular
clusters are shown in Fig. 1. However, it should be noted that this constraint is based on the assumption that
some GCs formed via the dark matter dominated mechanism and so would contain dark matter at the time of
their formation which is an issue still debated in the literature and brought up later in the paper.
III. CAPTURE OF PBHS BY NEUTRON STARS
Sec. II was focussed on the capture of PBHs by stars during their formation to then subsequently be trapped
within the final compact remnant whereas this section focusses on the capture of PBHs by these compact
remnants when already in their final form as discussed in the following analysis (Capela et al. 2013b).
A. Energy loss of PBH
When PBHs pass through stars they lose energy via two mechanisms. The first being dynamical friction
(Chandrasekhar 1949), which is the loss of momentum and kinetic energy of a travelling object due to
NBH =MDM (r
c)
mBH
MDM (r)=Mbound
M(r)r3
M*R*
3
⎛
⎝
⎜⎞
⎠
⎟
1
2
FIG. 1: Constraints on the fraction of PBHs in dark matter from capture during star formation,
figure is from Capela et al. (2013a). The region excluded by white dwarfs is shown coloured
purple and the region excluded by neutron stars is shown by the cyan coloured dotted lines.
(12)
!6
gravitational interactions with the surrounding matter. The second is similar to dynamical friction except
instead of gravitational interactions with surrounding matter causing the PBH to lose momentum, and thus
it’s energy, it’s the accretion of material from the star onto the PBH.
Dynamical friction can be thought of as a force, fdyn, acting in the opposite direction to the velocity of the
PBH, !, and is given by the following equation
.
This relation holds for PBH velocities which are greater than the velocities of the constituent particles of the
star that the PBH is passing through, where ! is simply just the density of the star and the ! factor is
called the Coulomb logarithm (Chandrasekhar 1949). As the PBH loses energy it becomes gravitationally
bound to the star. Once bound, the PBH continues to orbit and pass through the star until it loses all of it’s
energy and sinks to the centre of the star. In order for a PBH to be captured by a star the following relation
must be satisfied
!.
This states that the energy loss during the collision between the star and the PBH, Eloss, must be greater than
the kinetic energy of a black hole with asymptotic velocity and mass mBH.
Capela et al. (2013b) make the assumption that the flux of PBHs coming in is the same across the whole star,
this allows them to find the following relation for the average energy loss of PBHs passing through the star
!,
with R and M being the radius and the mass of the star respectively and the arrow brackets denote a density-
weighted average over the star’s volume. This average is shown on an arbitrary function f(r):
!.
The velocity of a PBH as it passes through the star is of the order of the escape velocity - much greater than
the PBH asymptotic velocity - and so can be taken as
!.
By making the assumption that ! is r-dependent, one can see, by simply substituting the above equation
for the PBH velocity into the energy loss equation (15), that you would arrive at a proportionality relation
given by
υ
fdyn =−4
π
G2mBH
2
ρ
ln Λv
υ
3
ρ
ln Λ
Eloss >mBH
υ
0
2
2
υ
0
Eloss =4G2mBH
2M
R2
ln Λ
υ
2
f(r)≡1
M4
π
r2dr
ρ
(r)f(r)
0
R
∫
υ
=2GM
R
ln Λ
(13)
(14)
(15)
(16)
(17)
(18)
!7
!,
where R is the radius of the star that the PBH is passing through. There are obviously some factors that
would arise from taking the density-weighted average over the star’s volume, however it can easily be seen
that the energy loss, Eloss, of a PBH passing through a star is inversely proportional to the star’s radius R.
Therefore a star with a smaller radius will cause a PBH passing through it to lose more energy during one
collision - neutron stars cause a much greater loss of energy than white dwarfs as they are smaller so Capela
et al. (2013b) only consider the case of the neutron star from now on.
However, there are two problems that appear to face calculating the energy loss in the neutron star case. The
first being that in a neutron star, accreting material makes a larger contribution to the energy loss of a PBH
during a collision and the second being that, due to a neutron star’s make up - degenerate neutron gas, the
equation for the dynamical friction force, fdyn, may not work. The first is solved by accounting for the
accretion of material in the Coulomb logarithm by adding in another term with a coefficient c(r) as shown by
!.
The second problem can also be solved by assuming the Coulomb logarithm is r-dependent and, by taking
the average value of !, can be incorporated into the equation for energy loss (15). Capela et al.
(2013b) have calculated this average value by using the neutron star density profile from Belvedere et al.
(2012)
!.
Through using this value and taking typical values for the parameters M and R, ! and
!, Capela et al. arrived at the following simplified relation for the energy loss of a PBH passing
through a neutron star
!.
By formulating an equation for the time it takes for a PBH to lose enough energy so that it is trapped inside
the neutron star one can obtain constraints on the masses of PBHs that could currently be trapped. Capela et
al. use an argument based on the time evolution of the apastron assuming that the orbit of the PBH is radial.
In a binary system, the apastron is the furthest distance between the two orbiting objects. The time taken for
a PBH to complete half a full orbit is given by
!,
where rmax is the apastron. To get an equation of the time evolution of the apastron one can divide the energy
loss during a half orbit, which is given by Eq. (22), and expressing in terms of rmax to get the following
differential equation
Eloss ∝GmBH
2
R
ln Λ → ln Λ(r)=ln Λ+c(r)
υ
4
ln Λ/
υ
2
ln Λ
υ
2=14.7
MNS =1.4 M⊙
RNS =12km
Eloss
mBH
=6.3 ×10−12 mBH
1022 g
⎛
⎝
⎜⎞
⎠
⎟
ΔT=
π
r
max
3/2
GM
(19)
(20)
(21)
(22)
(23)
!8
!,
where ! is given by
!,
and ! is given by
!.
Capela et al. combine these terms to arrive at an equation relating the apastron to the energy loss time given
by
!,
where ! is just the initial value of !. By substituting all of the terms into this equation one arrived at the
more numerical relation
!.
It can be seen through a simple calculation using Eq. (27) that black holes of mass !
would lose their energy, and thus become trapped inside the neutron star, in a time scale that is greater than
the current age of the universe. So any PBHs with masses greater than this value could have been captured
by a neutron star, thus imposing the lower mass bound for which any constraints from this method apply.
B. Capture rate
The same assumption is made here as was made previously regarding dark matter bound to a baryonic cloud
Eq. (6). However, rather than the dark matter velocities following a Maxwellian distribution, it’s the PBHs
which follow
!,
with !, where ! is the density of PBHs in the star’s location
!
ξ
=−1
τξ
ξ
ξ
=r
max
R
τ
τ
=
π
R5/2
4GmBH GM
ln Λ
υ
2
−1
!8×106smBH
1022 g
⎛
⎝
⎜⎞
⎠
⎟
−1
tloss !2
τ ξ
0
ξ
0
ξ
tloss !4.1×104yr mBH
1022 g
⎛
⎝
⎜⎞
⎠
⎟
−3
2
MPBH <2.5 ×1018 g
dn =nBH
3
2
πυ
2
⎛
⎝
⎜⎞
⎠
⎟
3
2exp −3
υ
2
2
υ
2
⎧
⎨
⎩
⎫
⎬
⎭
d3
υ
nBH =
ρ
BH
mBH
ρ
BH
(24)
(25)
(26)
(27)
(28)
(29)
!9
!.
Work from Kouvaris (2008) has shown that the capture rate, F , follows a very similar relation given by the
following
!,
where F0 is the capture rate based on the assumption that the dark matter is entirely comprised of PBHs and
is given by the following relation
!,
where Rg is the Schwarzschild radius - which is the radius of a sphere around an object where, at any point
on the sphere, the escape velocity is equal to the speed of light.
Capela et al. (2013b) show two possibilities for the capture rate dependent on the energy loss of the PBH: the
first being small energy loss and the second being high energy loss. In the first case, !, the
exponential term gets expanded out, since the value inside the exponential will be some negative fraction -
thus the exponential will calculate to some value smaller than 1, and the relation simplifies to
!.
As can be seen by substituting Eq. (21) for the energy loss into this relation, all mBH terms cancel out so the
capture rate is independent of the PBH mass in this case. In the second case, !, this
exponential term can be neglected entirely, since the value inside the exponential will be some negative
number greater than 1 and so the exponential itself will work out to be some minute negligible fraction. This
leads to the following relation
!.
C. Constraints on PBHs
Through calculating the probability of a neutron star’s survival and by making the assumption that this
probability is not small, Capela et al. arrived at a constraint on the contribution of PBHs in dark matter. By
taking the age of a neutron star as tNS, the probability of it’s survival is given by exp(-tNSF) where F is given
by Eqs. (30) and (31). Assuming that this probability isn’t small leads to the following constraint
!,
ρ
BH =ΩPBH
ΩDM
ρ
DM
F=ΩPBH
ΩDM
F
0
F
0=6
πρ
DM
mBH
RgR
υ
(1 −Rg/R)1−exp −3Eloss
mBH
υ
2
⎛
⎝
⎜⎞
⎠
⎟
⎛
⎝
⎜⎞
⎠
⎟
Eloss << mBH
υ
2/ 3
F
0=3 6
πρ
DM
mBH
RgR
υ
3(1 −Rg/R)
Eloss
mBH
Eloss >> mBH
υ
2/ 3
F
0=6
πρ
DM
mBH
RgR
υ
(1 −Rg/R)
ΩPBH
ΩDM
≤1
tNS F
0
(30)
(31)
(32)
(33)
!10
where tNS is the age of the neutron star and F0 is the capture rate assuming all dark matter is in PBHs as it
appears in Eq. (31). This constraint is shown below in Fig. 2 along with numerous other constraints,
including the constraint from capture during star formation as shown in Sec. II.
However, F0 is dependent on the location of the neutron star, thus the location can alter the constraint by a
number of orders of magnitude. This constraint is at it’s most stringent when considering neutron stars within
GCs as F0 is high in these. This constraint is then also dependent on the assumption that dark matter is
contained within GCs as with the constraint derived in Sec. II.
FIG. 2: Constraints on the fraction of PBHs in dark matter from capture by
neutron stars in GCs (shown by the green shaded region). The red shaded regions
indicate constraints from other methods. Figure is from Capela et al. (2013b).
FIG. 3: Constraints on the fraction of PBHs in dark matter from neutron star
capture at three different dark matter densities of GC cores, taken from Capela
et al. (2013b). The three different cases are for densities of: 4 × 102 GeVcm-3
(the blue line shown), 2 × 103 GeVcm-3 (the black line shown) and 104
GeVcm-3 (the red line shown).
(34)
!11
The content of dark matter within GCs is a topic of much debate and so casts some uncertainty on this
constraint. As stated earlier in Sec. II the GC metallicity distribution suggest that GCs were made through
two different mechanisms: metal-rich GCs were thought to be formed through baryonic processes like
mergers and metal-poor GCs were thought to be formed in dark matter halos by gravitational wells pulling in
baryonic matter around the time of star formation. Capela et al. focus on metal-poor GCs as these would
more likely have cores with high dark matter densities. An estimate of the density of dark matter close to the
core of a GC was made by Bertone and Fairbairn (2008) to be
!,
and this is the value adopted by Capela et al. (2013b) in Fig. 2. However, due to the uncertain dark matter
content in GCs, Capela et al. have also presented their constraint from capture of PBHs by neutron stars for
three different GC core dark matter densities: !, then ! as
adopted above, and !. It can be seen that the fraction of dark matter in PBHs is inversely
proportional to the core dark matter density in GCs by substituting (31) into (34) and so from looking at the
three densities listed above it can be seen that the first is the least stringent and the latter is the most stringent
case - this is shown in Fig. 3.
IV. SUMMARY OF CONSTRAINTS ON THE CONTRIBUTION OF PBHS TO DARK MATTER
Constraints on the fraction of dark matter contained in PBHs can be arrived at through numerous methods -
observational and lensing methods for lower mass black holes and other dynamical methods for higher mass
black holes. Constraints from Barrau et al. (2004) seem to exclude all masses apart from three mass ranges:
1016g - 1017g, 1020g - 1026g and . So in the following summary (Carr et al. 2016b) I have not
included the constraints that apply to masses outside this mass range or the constraints that apply to these
mass ranges but aren’t as stringent as others and so can be neglected as well as some constraints that have
been debated in the literature. In the following, ! is simply denoted as f(M).
A. Observational constraints from Hawking radiation
As stated in Sec. I black holes formed during the early universe could have very low mass up to quite large
masses as a result of strong perturbations in the gravitational field. By comparing the density of a black hole
of mass M with the cosmological density at time t after the big bang, PBHs would have a mass provided by
the following relation
!,
as stated by Carr et al. (2010).
About 40 years ago, S. W. Hawking (1975) made the prediction that, according to Heisenberg’s uncertainty
principle, a black hole which is rotating would emit particles. This means that a black hole would, over time,
lose energy, and thus lose it’s mass according to Einstein’s energy-mass relation, this is called black hole
evaporation. This evaporation corresponds to a temperature
!.
ρ
DM ~ 2 ×10 3GeVcm -3
ρ
DM =4×102GeVcm-3
ρ
DM =2×103GeVcm-3
ρ
DM =104GeVcm-3
10M⊙−105M⊙
ΩPBH /ΩDM
M~c3t
g
~ 1015 t
10−23 s
⎛
⎝
⎜⎞
⎠
⎟g
TBH =!c3
8
π
GMkB
~ 10−7M
M⊙
⎛
⎝
⎜⎞
⎠
⎟
−1
K
(35)
(36)
(37)
!12
If a black hole of mass M doesn't absorb any mass then it will evaporate away in a time scale that follows
!.
This is an important prediction with regards to PBHs as it allows us to calculate the mass of a black hole that
would’ve evaporated by today by setting τ as the age of the universe resulting in the minimum mass a PBH
has to be to still exist today, M* ≈ 5 × 1014g (Carr et al. 2010). PBHs with mass M > M* wouldn’t have
evaporated by today and are obvious dark matter candidates because they can be considered cold dark matter
as they would, at present, be dynamically cold.
The rate at which black holes emit particles increases as the mass of the black hole decreases. Black holes
formed today have a minimum mass of about and so will lose all of their mass in roughly 1069 years,
which is far greater than the current age of the universe (!years). Hawking radiation is especially
important when considering black holes which formed in the early universe since these tend to have much
lower masses than black holes formed today and so only PBHs are small enough for Hawking radiation to
play an important role.
When a PBH falls below the mass,!, it begins to emit jets of gluons and quarks. As illustrated by
Carr et al (2016b), for PBHs of mass ! this mass change can be ignored since the rate at which
mass is lost is so small. Now, we can take the instantaneous spectrum of photons emitted from a PBH,
, and multiply it by t0, the age of the universe, to get the time-integrated spectrum of photons
emitted from a PBH as shown by Carr et al. (2010) and investigated in more detail looking at the high-
energy photon tail by MacGibbon (1990).
!,
for (E < M-1) and (E > M-1) respectively.
The following constraint derivation is outlined by Carr et al. (2016b). One can obtain the number of
background photons per unit energy per unit volume from all the PBHs, !, by doing the following
integral
!,
which becomes
!,
for (E < M-1) and (E > M-1) respectively for a monochromatic mass function. It can be seen that is just
a factor of M smaller than !.
τ
(M) ~ G2M3
!c4~ 1064 M
M⊙
⎛
⎝
⎜⎞
⎠
⎟
3
yr
3M⊙
1.382 ×1010
Mq≈0.4 M*
M>2M*
dN
γ
/dE
dN
γ
dE ∝E3M3
E2M2e−EM
⎧
⎨
⎪
⎩
⎪
ε
(E)
ε
(E)=dM dn
dM
dN
γ
dE
m,E
( )
Mmin
Mmax
∫
ε
(E)∝f(M)×E3M2
E2Me−EM
⎧
⎨
⎪
⎩
⎪
ε
(E)
dN
γ
/dE
(38)
(39)
(40)
(41)
!13
The associated intensity of the photons is
!,
for (E < M-1) and (E > M-1) respectively. It can be seen in the above that I(E) is just a factor of E greater than
!. From looking at the ranges in which the two values apply it is easily deduced that the function peaks
at E ~ M-1. This peak corresponds to an intensity value of . The observed
extragalactic intensity is where . By substituting Imax and Iobs into the
following inequality
!,
we get the evaporation constraint
!,
for !.
B. Observational constraints based on lensing by PBHs
From looking at femtolensing of gamma ray bursts one can place constraints on low mass Massive
Astrophysical Compact Halo Objects (MACHOs), which are astronomical objects comprised of baryonic
matter that emit little radiation as they are not heavy enough to start fusion processes in their core. Because
of this the only way they can be detected are through their gravitational effects on light from stars, known as
gravitational lensing.
A couple of decades ago there was much excitement following these MACHO microlensing results from
Alcock et al. (2000) which stated that dark matter could be in compact objects of mass, !. This,
however, could only make up about 20% of the dark matter. White and red dwarfs have similar masses to
these objects but emit light and so can be detected by the Hubble space telescope which subsequently
disproved the theory that dark matter could be made up of these MACHOs.
Work from Barnacka et al. (2012) gives a constraint on low mass MACHOs from femtolensing of gamma
ray bursts
!,
for !.
Following the MACHO microlensing experiment results (Alcock et al. 2000) stated above, the EROS
microlensing experiments (Tisserand et al. 2007) (EROS-I and EROS-II) which searched towards the Small
Magellanic Cloud (SMC) led to some more refined constraints on the PBH dark matter fraction
I(E)≡cE
ε
(E)
4
π
∝f(M)×E4M2
E3Me−EM
⎧
⎨
⎪
⎩
⎪
ε
(E)
Imax (M)∝f(M)M−2
Iobs ∝E−(1+
ε
)∝M1+
ε
0.1 <
ε
<0.4
Imax (M)≤Iobs (M)
f(M)≤2×10−8M
M*
⎛
⎝
⎜⎞
⎠
⎟
3+
ε
(5 ×1014 g<M<1018 g)
M=0.5M⊙
f(M)<0.1
(5 ×1016 g<M<1019 g)
(42)
(43)
(44)
(45)
!14
!,
for !, ! and ! respectively.
The Kepler space observatory was launched in 2009 designed to search for exoplanets of a similar size to
Earth in the Milky Way galaxy by monitoring the brightness of around 145,000 main sequence stars. Griest
et al. (2013) used the first two years of Kepler mission data in order to constrain the PBH dark matter in the
mass range, ! to !, arriving at the following limit
!,
for !.
C. Constraints based on interactions with astronomical objects
Another way to deduce different constraints on PBH dark matter is to focus on dynamical effects of PBHs -
collisions with astronomical objects. It is thought that PBHs can be captured by stars of any kind: white
dwarfs, neutron stars, main-sequence stars and red giants. When these stars capture a PBH, over time the
star’s mass will accrete onto the black hole until it is completely absorbed by the PBH. Obviously this
process would give off a huge amount of radiation. By considering PBHs being captured by neutron stars
(Capela et al. 2013b) and white dwarfs (Capela et al. 2014), Capela et al. found the constraint
!,
for !. In the paper by Carr et al. (2016b), this constraint also appears in a different
form elsewhere in this paper, shown by
!.
However, after notifying Carr, it was made clear that this was an error and (48) is the correct form as can be
seen by taking the limit M << 1023g, this constraint flattens out at around f = 0.06.
Binary star systems with stars which orbit each other separated by large distances of up to one light year are
known as wide binary star systems and are thought to be susceptible to tidal disruption from encounters with
MACHOs. From a sample of four wide halo binary systems, (Chanamé and Gould 2003), Quinn et al. (2009)
analysed the radial velocity measurements and found that three of the four wide halo binaries were genuine
however encountered problems with the fourth candidate, casting doubt on the previous constraint from Yoo
et al. (2004), that MACHOs of mass ! couldn’t provide the dark matter. The constraint then
became
f(M)<
1
0.1
0.04
⎧
⎨
⎪
⎩
⎪
6×10−8M⊙<M<30 M⊙
10−6M⊙<M<1M⊙
10−3M⊙<M<0.1M⊙
2×10−9M⊙
10−7M⊙
f(M)<0.3
(2 ×10−9M⊙<M<10−7M⊙)
f(M)<M
4.7 ×1024 g1−exp −M
2.9 ×1023 g
⎡
⎣
⎢⎤
⎦
⎥
⎛
⎝
⎜⎞
⎠
⎟
−1
(2.5 ×1018 g<M<1025 g)
f(M)<M
4.7 ×1018 g1−exp −M
2.9 ×1017 g
⎡
⎣
⎢⎤
⎦
⎥
⎛
⎝
⎜⎞
⎠
⎟
−1
M>43M⊙
(46)
(47)
(48)
(49)
!15
!,
for ! and ! respectively.
A dwarf galaxy is one which is comprised of up to several billion stars - far fewer than a standard galaxy like
the Milky Way galaxy which contains between 100-400 billion stars dependent on the method of estimation.
Ultra-faint dwarf galaxies are interesting with regards to the dark matter problem as they provide the best
constraints on annihilating WIMPs. An ultra-faint dwarf galaxy, Eridanus II, was recently found to have a
star cluster near the centre of it and, as shown by Brandt (2016), this provides strong constraints on
MACHOs of mass ! as the main component of dark matter
!,
for !.
V. POSSIBLE MASS RANGES FOR PBH DARK MATTER
A. Subatomic-sized black holes
f(M)<
500M⊙
M
0.4
⎧
⎨
⎪
⎩
⎪
(500M⊙<M<103M⊙)
(10 3M⊙<M<108M⊙)
M≥5M⊙
f(M)<3.7 M⊙
M1.1+0.1ln M⊙
M
⎡
⎣
⎢⎤
⎦
⎥
⎛
⎝
⎜⎞
⎠
⎟
−1
(M⊙<M<103M⊙)
FIG. 4: Constraints on the fraction of PBHs in dark matter in the subatomic-
sized black hole range, taken from Carr et al. (2016b). The red shaded region
shows the constraint from the absence of extragalactic γ-rays that should be
present from PBHs that lie within this mass range. The blue shaded region
shows the constraint from the femtolensing of γ-ray bursts (Barnacka et al.
2012).
(50)
(51)
!16
One of the allowed mass ranges in which PBHs could comprise dark matter is !.
PBHs that lie within this mass range have a radius of roughly 10-12cm - 10-10cm, thus are called subatomic-
sized black holes. The two constraints from Sec. II that apply to this mass range are: the extragalactic γ-ray
background constraint, given by (44), and the γ-ray burst (GRB) femtolensing constraint (Barnacka et al.
2012), given by (45).
The above figure from Carr et al. (2016b), Fig. 4, shows these two constraints with the dotted lines
corresponding to varying values of !. As can be seen in Fig. 4, there is no allowed mass whereby f = 1 as the
extragalactic γ-ray background constraint (44) hits this at roughly 1017g dependent on the value of and the
GRB femtolensing constraint (45) hits this at 1016.4g.
In their analysis of this mass range Carr et al. (2016b) looked at the possibility of the dark matter being
explained by an extended mass function as a monochromatic mass function obviously can’t provide all the
dark matter due to the fact the limits don’t allow a mass where f = 1. The dotted lines of the extragalactic γ-
ray background constraint (44) corresponding to !, !, ! and ! intersect the GRB
femtolensing constraint (45) at f = 0.4 (at !), f = 0.45 (at !), and f = 0.55 (at
!) and f = 0.65 (at !) respectively.
Confronting these limits with an extended mass function gives rise to the following conclusions. For the f =
0.4 case, PBHs cannot provide all the dark matter with an extended mass function - it is stated that PBHs
cannot make up more than 80% of the dark matter in this case (Carr et al. 2016b). In the f = 0.45 case it is
also not possible to explain all the dark matter with an extended mass function. However, in the two latter
cases, f = 0.55 and f = 0.65, it becomes possible that the correct extended mass function could explain all the
dark matter.
Carr et al. (2016b) go further and conduct a more detailed analysis using two different models of inflation:
the axion-curvaton inflation model (Kasuya and Kawasaki 2009), which is an adaptation of the original
curvaton scenario (Lyth and Wands 2002, Lyth et al. 2003), and the running-mass inflation model (Stewart
1997a, Stewart 1997b). The inflationary potentials are
!,
and
!
for the running-mass and axion-curvaton models respectively. Carr et al. (2016b) chose these two models as
the parameters of these models can be tuned as to make a peak in the production of PBHs in these three
allowed mass ranges.
However, when confronting the subatomic-sized black hole mass range they found that neither model could
provide all of the dark matter. Although it has been argued (Carr et al. 2016a) that PBHs in this mass range
could provide all of the dark matter when the femtolensing constraint (45) is omitted.
B. Sublunar-mass black holes
The PBHs being considered in this subsection lie within the mass range ! and are
called sublunar-mass black holes as they have a lower mass than that of the moon (~ 7.35 × 1025g). The three
1016 g<MPBH <1017 g
ε
ε
ε
=0.1
ε
=0.2
ε
=0.3
ε
=0.4
M≈1017.2 g
M=1017 g
M≈1016.9 g
M≈1016.8 g
V
φ
( )
=V0+1
2
m
φ
2
φ
( )
φ
2
V
φ
( )
=1
2
λ
H2
φ
2
1020 g<MPBH <1024 g
(52)
(53)
!17
constraints from Sec. II that apply to this mass range are: the GRB femtolensing constraint (Barnacka et al.
2012), given by (45), the Kepler microlensing constraint (Griest et al. 2014), given by (47), and the neutron
star capture constraint (Capela et al. 2013b), given by (48).
Fig. 5 shows these and it can be seen that this mass range is very tightly constrained. However, the neutron
star capture constraint (Capela et al. 2013b) is based on the assumption that globular clusters possess the
dark matter densities exceeding several hundred GeVcm-3. It is uncertain that globular clusters contain dark
matter and so this constraint can be omitted as done by Carr et al. (2016b), which gives way to a very large
range of masses where f = 1, starting at roughly 1020g going up to ~ 4 × 1024g.
In their analysis of PBH capture by neutron stars, Capela et al. (2013b) focussed on three different cases of
the dark matter density : 4 × 102 GeVcm-3, 2 × 103 GeVcm-3 and 104 GeVcm-3. In Fig. 5, the constraint
(48) has three different lines corresponding to the three different cases of the dark matter density; the most
stringent one (dotted line) being the 104 GeVcm-3 case, and the least stringent (solid line) being the 4 × 102
GeVcm-3 case.
As can be seen from Fig. 5, the most stringent, 104 GeVcm-3, case does not allow PBHs to provide all the
dark matter with a monochromatic mass function at any mass - this is also true for the 2 × 103 GeVcm-3
(striped line) case. However for the least constrained case, 4 × 102 GeVcm-3, a monochromatic mass function
does allow for PBHs to provide all the dark matter within the mass range, 1024g - 4 × 1024g, where f = 1.
Through a more detailed analysis involving the two inflationary models stated above, Carr et al. (2016b)
conclude that, by tuning the parameters of the axion-curvaton model to allow a peak in PBH production in
this mass range, all the dark matter could be in PBHs as the extremely restrictive neutron star constraint is
satisfied for the two lower values of dark matter density. In the case of the running-mass inflationary model,
ρ
DM
FIG. 5: Constraints on the fraction of PBHs in dark matter in the sublunar-mass
black hole mass range, taken from Carr et al. (2016b). The blue-shaded region
shows the constraint from the femtolensing of γ-ray bursts (Barnacka et al.
2012). The green-shaded region shows the constraint from capture of PBHs by
neutron stars as derived in Sec. III (Capela et al. 2013b). The red-shaded region
shows the constraint from capture of PBHs by white dwarfs (Capela et al.
2013a). The turquoise-shaded region shows the constraints from microlensing by
the Kepler survey (Griest et al. 2014).
!18
all the dark matter could still be explained but only for the least restrictive neutron star constraint,
corresponding to the lowest dark matter density.
C. Intermediate-mass black holes
The highest allowed mass range for PBHs to comprise dark matter is ! and PBHs lying
within this mass range can be called intermediate-mass black holes (IMBH). This mass range is of particular
interest in light of the recent LIGO gravitational wave detections of black-hole mergers (Bird et al. 2016).
These black holes are limited by the following constraints: the EROS constraints (Tisserand et al. 2007), the
wide-binary constraints (Yoo et al. 2004, Chaname and Gould 2003), given by (50), and the Eridanus II
constraint (Brandt 2006), given by (51).
However, the wide-binary constraints can be neglected as these are less stringent than the weakest of the
Eridanus II constraints. These Eridanus II constraints can be rewritten (Carr et al. 2016b, Brandt 2016) as
!.
The Eridanus II cluster is thought to be at least 3Gyr old, however it has been stated that it can be up to
12Gyr old (Crnojević et al. 2016) whereby tighter constraints would be made (Brandt 2016).
1M⊙<M<103M⊙
f(M)≤
0.5 1+0.046M⊙pc−3
ρ
⎛
⎝
⎜⎞
⎠
⎟10M⊙
M
⎛
⎝
⎜⎞
⎠
⎟
σ
10kms−1
⎛
⎝
⎜⎞
⎠
⎟
1+0.1ln 10M⊙
M
σ
10kms−1
⎛
⎝
⎜⎞
⎠
⎟
2
⎡
⎣
⎢⎤
⎦
⎥
⎛
⎝
⎜⎞
⎠
⎟
FIG. 6: Constraints on the fraction of PBHs in dark matter in the intermediate-mass
black hole mass range, taken from Carr et al. (2016b). The blue-shaded region shows
the constraint from the EROS and MACHO microlensing projects (Tisserand et al.
(2007). The red-shaded region shows the dynamical constraints from the ultra-faint
Eridanus II dwarf galaxy (Brandt 2006). The green-shaded region shows the
dynamical constraints from wide-binary star systems (Yoo et al. 2004, Chaname and
Gould 2003).
(54)
!19
There is a tiny mass range in which all of the dark matter can be explained by PBHs with a monochromatic
mass function at roughly 30M⊙, however this is only taking into account the least restrictive of the Eridanus
II constraints, as can be seen in Fig. 6. A monochromatic mass function, however, is unphysical and so Carr
et al. (2016b) confront these constraints with an extended mass function. The EROS microlensing constraint
and the most stringent Eridanus II constraint (!, !- the red solid curve) cross at f =
0.4 corresponding to a mass of roughly 10M⊙. Thus only 80% (40% below 10M⊙ and 40% above 10M⊙) of
the dark matter can be provided by PBHs in the intermediate mass range.
The next less stringent Eridanus II constraint (!, ! - the red dashed curve) crosses
with the EROS microlensing constraint at f = 0.5 for a mass of roughly 20M⊙ - just about allowing PBHs to
provide all the dark matter in this mass range, although this is unlikely because it is a very tight fit for a
tuned extended mass function. The Eridanus II constraint corresponding to ! and
! (the red dot-dashed line) crosses with the EROS constraint at f = 0.75 and so it is certainly
feasible that an extended mass function will allow all the dark matter to be explained by PBHs in the
intermediate mass range for this constraint as well as the least stringent constraint (!,
! - the red dotted line).
By confronting these constraints with the two models of inflation, as done previously, Carr et al. (2016b)
have shown that the axion-curvaton model as well as the running-mass model satisfy the constraints shown
in Fig. 6. However, Green (2016) has argued that this mass window is excluded from providing that all the
dark matter can be in PBHs from the updated version of Brandt (2016).
VI. CONCLUSIONS
In this paper I have considered the possibility that PBHs could be potential candidates for dark matter by
looking at constraints on the fraction of dark matter that could be PBHs. Dark matter, in the form of PBHs,
could have been captured by stars during their formation through adiabatic contraction to then consequently
be trapped inside the compact remnant left behind by the star. I have also reviewed the possibility that, rather
than being captured during the star’s formation to then be trapped in the compact remnant, a PBH could be
captured by a star once it’s already in it’s compact remnant state. PBHs, once trapped inside these compact
remnants, be it a neutron star or white dwarf, would act gravitationally on the surrounding material and begin
rapidly accreting this mass, destroying the compact remnant. Thus observations of these compact remnant
impose constraints on the abundance of PBHs.
The three mass ranges in which PBHs could constitute the dark matter have been considered: subatomic-
sized black holes, sublunar-mass black holes and intermediate-mass black holes. For all three there were
regions whereby an extended PBH mass function could explain the dark matter, however the constraints
applying to these mass ranges will undoubtedly become more stringent in the future, as to possibly exclude
these mass ranges entirely from explaining dark matter.
By looking at the relation for the density of radiation !, given by
!,
and the relation for the density of PBHs, given by
ρ
=0.1M⊙pc−3
σ
=5kms−1
ρ
=0.1M⊙pc−3
σ
=10kms−1
ρ
=0.01M⊙pc−3
σ
=5kms−1
ρ
=0.01M⊙pc−3
σ
=10kms−1
ρ
r
ρ
r=
ρ
ra0
( )
a0
a
⎛
⎝
⎜⎞
⎠
⎟
4
(55)
(56)
!20
!,
one can obtain the following equation for the fraction of the universe in PBHs
!.
However by forming a relation between the fraction of the universe in PBHs at the time of PBH formation
and at the time where the universe turned from radiation domination to matter domination we can make an
argument on the possibility of dark matter being in PBHs as is shown in the following:
!,
where f is the fraction of the universe in PBHs, ! is the density of the universe, ! is the density of PBHs
in the universe, ! is the scale factor at the time of PBH formation and ! is the scale factor at the time
where the universe turned from radiation domination to matter domination. The term with the scale factors
can be split into two fractions as shown by
!,
where ! and !, and so the relation becomes
!,
where zeq is the redshift at the time when the universe turned from radiation domination to matter domination
and zf is the redshift at the time of PBH formation. The fraction obtained from (60) is obviously dependent
on the values chosen for the redshifts, however this fraction would need to equal the fraction of the universe
made up of dark matter in order for PBHs to explain the dark matter.
In light of reviewing this material I believe it is unlikely that PBHs could make up the dark matter, however
Eq. (60) shows that the expansion of space acts like an amplifier and so, even though I find it unlikely that
PBHs can explain dark matter, PBHs are immensely powerful tools and unique probes of the early universe.
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ρ
PBH =f0
ρ
ra0
( )
a0
a
⎛
⎝
⎜⎞
⎠
⎟
3
ρ
PBH
ρ
r
=f0
a
a0
⎛
⎝
⎜⎞
⎠
⎟=f01+z
( )
f=
ρ
PBH (today)
ρ
(today)
=
ρ
PBH (formation)
ρ
(formation)
aeq
af
ρ
ρ
PBH
af
aeq
f=
ρ
PBH (formation)
ρ
(formation)
a(eq)
a(today)
a(today)
a(formation)
a(eq)
a(today)
=1
1+zeq
a(today)
a(formation)
=1+zf
f=ff
1
1+zeq
(1 +zf)
(57)
(58)
(59)
(60)
!21
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