Primordial black holes in the Dark Ages: Observational prospects for future 21cm surveys

Abstract

We consider the signatures of a population of primordial black holes (PBHs) in future observations of 21cm radiation from neutral hydrogen at high redshift. We focus on PBHs in the mass range $5 \times 10^{10} kg \lesssim M_{PBH} \lesssim 10^{14} kg$, which primarily influence the intergalactic medium (IGM) by heating from direct Hawking radiation. Our computation takes into account the black hole graybody factors and the detailed energy dependence of photon and e+/- absorption by the IGM. We find that for black holes with initial masses between $5 \times 10^{11} kg \lesssim M_{PBH} \lesssim 10^{14} kg$, the signal mimics that of a decaying dark matter species. For black holes in the range $5 \times 10^{10} kg \lesssim M_{PBH} \lesssim 5 \times 10^{11} kg$, the late stages of evaporation produce a characteristic feature in the 21cm brightness temperature that provides a unique signature of the black hole population. If no signal is observed, then 21cm observations will provide significantly better constraints on PBHs in the mass range $5 \times 10^{10} kg \lesssim M_{PBH} \lesssim 10^{12} kg$ than are currently available from the diffuse $\gamma$-ray background.

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arXiv:0805.1531v2 [astro-ph] 3 Sep 2008
DAMTP-2008-39
Primordial black holes in the Dark Ages:
Observational prospects for future 21cm surveys
Katherine J. Mack∗
Department of Astrophysical Sciences, Peyton Hall
Princeton University, Princeton, NJ 08544 USA
Daniel H. Wesley†
Centre for Theoretical Cosmology DAMTP, Cambridge University
Wilberforce Road, Cambridge, CB3 0WA United Kingdom
We consider the signatures of a population of primordial black holes (PBHs) in future observations
of 21cm radiation from neutral hydrogen at high redshift. We focus on PBHs in the mass range
5×1010kg ? MPBH ? 1014kg, which primarily influence the intergalactic medium (IGM) by heating
from direct Hawking radiation. Our computation takes into account the black hole graybody factors
and the detailed energy dependence of photon and e±absorption by the IGM. We find that for black
holes with initial masses between 5×1011kg ? MPBH ? 1014kg, the signal mimics that of a decaying
dark matter species. For black holes in the range 5 × 1010kg ? MPBH ? 5 × 1011kg, the late stages
of evaporation produce a characteristic feature in the 21cm brightness temperature that provides a
unique signature of the black hole population. If no signal is observed, then 21cm observations will
provide significantly better constraints on PBHs in the mass range 5 × 1010kg ? MPBH ? 1012kg
than are currently available from the diffuse γ-ray background.
I. INTRODUCTION
The 21cm hyperfine spin-flip transition of neutral hy-
drogen (HI) may allow observers to probe the cosmic
“Dark Ages” comprising the epoch between the last scat-
tering of the cosmic microwave background (CMB) at
z ∼ 1100, and the appearance of luminous sources at
z ∼ 30 [1, 2, 3]. Observations at wavelengths 21(1 + z)
cm could be used to slice the universe as a function of red-
shift z, and so produce a three-dimensional map of the HI
distribution when combined with angular measurements.
At moderate redshifts, this could allow reionization of the
universe to be studied in detail. Overall, the essentially
three-dimensional nature of 21cm data gives it the po-
tential to become one of the the richest data sets for cos-
mology [4]. It will not only enable the standard ΛCDM
cosmological model to be better tested and understood,
but will provide a new means to constrain – or detect –
more exotic possibilities. One such possibility, the pres-
ence of a population of primordial black holes (PBHs), is
ideally suited to study with the 21cm background, as we
describe in this work.
The 21cm signal is sensitive to exotic physics through
its dependence on the thermal history of the intergalac-
tic medium (IGM). The IGM is visible in 21cm when
the spin temperature TSof the neutral hydrogen gas dif-
fers from the CMB temperature TCMB. The spin tem-
perature TS is itself determined by the competition be-
tween its coupling to TCMB through interactions with
∗Electronic address: mack@astro.princeton.edu
†Electronic address: D.H.Wesley@damtp.cam.ac.uk
CMB photons, its coupling to the gas kinetic temper-
ature TK through atomic collisions, and (especially at
lower redshifts) interactions with Ly-α photons produced
by luminous sources. Any process that heats the IGM
will influence TK and therefore affect TS. For redshifts
z ∼ 30−300, one expects TK∼ 10−103K ∼ 10−3−10−1
eV, and so little heating per hydrogen atom is necessary
to significantly change the thermal history of the IGM
and influence the 21 cm signal. By contrast, the CMB
is mainly sensitive to the Dark Age IGM thermal his-
tory through the effects of IGM heating on the ionized
fraction, which affects the optical depth to the last scat-
tering surface τLSS. The CMB is therefore less sensitive
to IGM heating, since the energy required to ionize HI is
much larger than the typical TKover the redshift range
of interest. Furthermore, changes in the CMB TT power
spectrum due to τLSS are degenerate with changes due
to the scalar spectral index ns(though measurements of
the TE power spectrum can break this degeneracy). The
projected sensitivity of 21cm emission to the IGM ther-
mal history has been exploited to show that the 21cm
signal can provide much more stringent constraints on a
population of long-lived, decaying particles than is avail-
able with CMB data [5].
Primordial black holes (PBHs) are an excellent exotic
physics target for 21cm observations because the mecha-
nisms by which they heat the IGM are under tight the-
oretical control. In this work, we compute the effects
of a population of PBHs with masses between 1010and
1015kg on 21cm observables, including the sky-averaged
brightness temperature and the fluctuation power spec-
trum. We also obtain predictions for the relic photon
density from the population of PBHs. We take into ac-
count the details of photon and e±emission by PBHs and

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the energy-dependent absorption properties of the IGM.
We find that future 21cm observations can provide better
constraints on PBHs than are currently available. The
techniques and computer code we develop can be easily
modified to study other exotic physics scenarios in the
Dark Ages, such as dark matter annihilation and decay,
if the precise energy spectrum of photons and e±pairs
produced by the exotic physics mechanism is known.
The effect of PBHs on the IGM is strongly dependent
on the PBH mass. Very massive PBHs would accrete
matter in accretion disks and thereby emit x-rays; this
scenario is explored in [6]. We focus our present discus-
sion on low-mass PBHs that would primarily affect the
IGM through the Hawking radiation [7, 8] of light par-
ticles: primarily photons and e±pairs, and heavier par-
ticles (and their decay products) for the smallest holes.
The total power emission by black holes through these
channels has been known for some time as a function of
the black hole mass [9]. Due to a coincidence between
the Hawking temperature of PBHs that evaporate dur-
ing the Dark Ages and a window of low optical thickness
of the IGM to photon absorption, it is vitally important
to know the spectrum of photons produced by the PBHs.
The radiated photons have energies such that the proba-
bility of absorption by the IGM, and thus their effective-
ness in heating it, is strongly dependent on the photon
wavelength. The emission rates as a function of photon
energy are given by the black hole “graybody factors,”
which have been calculated. The graybody factors arise
because black holes should not radiate with a blackbody
spectrum, even though they act as warm bodies with a
specific temperature. In the end, all that is needed to
completely characterize the PBH population is the total
density parameter ΩPBHand the mass of the individual
black holes MPBH. By comparison, while in principle
it is possible to calculate the spectrum of photons (and
other particles) produced in dark matter decay or anni-
hilation, in practice this dependence is subsumed in an
overall energy deposition rate per baryon ǫ. There is then
a model-dependent conversion between ǫ and quantities
such as the density parameter and lifetime of the relevant
particle species. For PBHs, the energy deposition rate is
uniquely determined once the mass of the black holes is
specified.
We obtain our predictions as follows. We numerically
integrate the equations governing IGM ionization and
temperature, including exotic sources of energy injection.
We use a modified version of the RECFAST [10] code at
high redshift for increased accuracy, and a simpler set of
IGM evolution equations at lower redshifts. The energy
injection itself is computed by using the graybody factors
for Hawking emission and the optical depths for a vari-
ety of IGM photon and e±absorption processes. Some
of these processes redistribute photons by energy, and
so we track the full photon population as a function of
energy through cosmic history. We use this information
to compute the IGM temperature and ionization history,
and use standard techniques to compute the 21cm bright-
ness temperature and the fluctuations. In the end, we
obtain the complete temperature and ionization history
of the IGM, the power spectrum and sky-averaged sig-
nal of 21cm brightness temperature fluctuations, and a
relic high-energy photon population. In order to obtain a
constraint on PBHs, we compare the power spectra from
several models to the power spectrum with no PBHs in-
cluded and use an estimate of the measurement error for
a realistic future experiment.
The PBH population is currently most tightly con-
strained for PBH masses near 1012kg, by the diffuse
γ-ray background as measured by EGRET [11, 12]. The
highest-energy EGRET photons cannot be of primordial
origin (z > 103) because the IGM is not transparent at
the corresponding energies. These photons are thought
to originate from outside the Galaxy, but their produc-
tion mechanism is as yet unknown. One possibility is
that these high-energy photons arise from processes at
redshifts lower than that of reionization, in which case it
is unlikely that 21cm observations will be a useful probe.
Another possibility is that these photons arise from some
energetic process occurring during the Dark Ages. In this
case the details of photon absorption and heating by the
IGM will be important. If these energetic processes heat
the IGM, then 21cm observations could provide a compli-
mentary measurement and a consistency check on exotic
physics mechanisms which purport to contribute to the
high-energy diffuse γ-ray background.
This paper is organized as follows.
physics of PBH formation and energy emission in Sec-
tion II. In Section III we describe our computation of
the energy injection rates from PBHs and how this in-
fluences the mean IGM evolution, the 21cm brightness
temperature, and the 21cm power spectrum. We give an
overview of our results in Section IV. We include pre-
dictions for the ionization history, 21cm observables, and
relic photon population for a selection of PBH models.
In Section V we present the constraints on PBH pop-
ulations that can be obtained with an ambitious 21cm
experiment. We discuss observational prospects in Sec-
tion VI and conclude in Section VII.
We review the
II.PRIMORDIAL BLACK HOLE EMISSION
A. Formation mechanisms
Several possible formation mechanisms for PBHs have
been discussed in the literature, and the resulting mass
spectra are highly model-dependent. We briefly mention
some possible PBH formation mechanisms here. For a
more in-depth review, see, e.g., [13].
As first proposed, PBHs result from fluctuations in the
high density primordial perturbations that collapse upon
horizon entry [14, 15]. This scenario is expected to pro-
duce an extended PBH mass spectrum [16]. As it gen-
erally requires a spectral index ns > 1, this scenario is
disfavored by the most recent observations which indi-

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cate ns< 1 (e.g., the WMAP five-year data set [17, 18]).
Other PBH formation scenarios invoke a phase transi-
tion or a period of a softening equation of state during
which fluctuations would be more likely to collapse (e.g.,
[19]). These models predict a very narrow mass spec-
trum, as PBH formation would occur only at the time
of the transition. Other formation scenarios invoke more
exotic mechanisms, such as the collapse of cosmic string
loops [20, 21, 22, 23, 24] and of domain walls [25, 26],
and predict a broader mass spectrum.
In this work, we assume all of the PBHs have the
same mass, amounting to a delta-function mass spec-
trum. This approximation is good for some PBH for-
mation scenarios and less so for others. It is the most
general in the sense that its results may be easily applied
to other models. Our results show that the constraints
on the PBH population depend strongly on mass, but
in a simple way. By assuming a single PBH mass, we
can determine the mass ranges most tightly constrained
by 21cm observations. We expect the constraints on a
more general PBH mass spectrum to be determined, to
a rough approximation, by the number density of PBHs
in the mass range to which 21cm observations are most
sensitive. Therefore assuming a single PBH mass gives a
good indication of the constraints on more general PBH
mass distributions.
B. Hawking radiation
The primary means by which black holes with masses
? 1014kg would have influenced the IGM is through
Hawking radiation. It has been known for some time that
a Schwarzschild (uncharged, non-rotating) black hole of
mass MPBHshould radiate [7, 8] as would a warm body
at the temperature TH, where
TH=
?c3
8πGMPBHkB
=
1.2 × 1013K
(MPBH/1010kg)
1.1 GeV/kB
(MPBH/1010kg).
=
(1)
The black hole should emit all massless and nearly-
massless particles (gravitons, photons, neutrinos), as well
as those massive particles whose mass is substantially be-
low kBTH. The energy emitted in neutrinos and gravi-
tons is essentially lost, for these particles interact very
weakly and do not affect the IGM. On the other hand,
the IGM is affected by the energy emitted in photons and
by the e±pairs emitted by smaller black holes.
While black holes are expected to have a temperature,
they should not emit radiation with a blackbody spec-
trum.Instead, Hawking’s calculation shows that the
number of particles emitted with angular frequency ω
(measured at infinity), spin s, polarization p, and angu-
lar momentum quantum numbers lm is
?Nsplm(ω)? =
Γsplm(ω)
exp(?ω/kBTH) − (−1)2s, (2)
where the Γsplm(ω) are graybody factors. The Γsplm(ω)
parameterize the deviation of the black hole emission
spectrum from that of a blackbody.
Γsplm(ω) is the probability that a particle in an infalling
field mode described by splm is absorbed by the hole.
Since a black hole with temperature THmust be in ther-
mal equilibrium with a blackbody heat bath of the same
temperature, it follows that these absorption probabili-
ties must also determine the emission rate. In addition
to the parameters splm, the Γsplm(ω) also depend on the
mass of the particle species. Only particles with mass
m ≪ TH (in units where kB= c = 1) will be emitted at
an appreciable rate, and for these particles the Γsplm(ω)
are those of massless particles. In this work we only con-
sider the emission of particles with m ≪ TH and so use
the massless graybody factors.
It is essential to include the graybody factors when
studying the effect of black hole emission on the IGM.
They determine the total power emission in particles of
various spin, and therefore the evolution of the black
hole mass. The rate of IGM photon absorption is very
frequency-dependent, and so it is essential to know the
photon spectrum accurately. We use the power emis-
sion spectra computed in [9], obtained by integrating
the Press-Teukolsky equations for fields of different spin
[27, 28]. A general feature of these graybody factors is
that the emission of particles of high spin is suppressed,
and the peak emission moves to higher energies. For ex-
ample, the power emission per polarization is larger for
a spin-1/2 particle (such as a neutrino) than for a spin-1
particle (such as a photon). For black holes with masses
MPBH? 9.5×1013kg, which are too large (and thus too
cold) to emit ultrarelativistic e±pairs, most (∼ 81%) of
their energy emission is in neutrinos, which do not heat
the IGM. Black holes with MPBH? 9.5×1013kg, in con-
trast, emit ∼ 45% of their energy in e±pairs, compared
to only ∼ 9% in photons [9]. Black holes with masses
MPBH? 4.5×1011kg will emit relativistic µ±pairs, and
black holes with progressively smaller masses will emit
more and more massive particles. At sufficiently small
MPBHthese will contribute to the photon, neutrino and
electron emission by secondary decays and hadron jets.
By a coincidence between IGM physics and black hole
physics, the black holes that are evaporating near the
present epoch emit their photons into the window where
the IGM optical depth is strongly dependent on redshift.
Physically, each
III.IGM PHYSICS
The mean IGM evolution is determined by tracking the
kinetic temperature of the gas TK, the ionization fraction
xi, and the spin temperature TS. The evolution of TK
and xi with redshift is given by differential equations,
with initial conditions taken from standard cosmology
just before recombination, and with additional terms to
account for exotic energy injection mechanisms. The spin
temperature TSis then determined algebraically. Neutral

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hydrogen is visible in 21cm whenever the spin tempera-
ture TS differs from the CMB temperature, TCMB, and
due to the contrast, the signal will show up in either emis-
sion or absorption. The sky-averaged brightness temper-
ature Tbtherefore carries information about the mean gas
temperature and ionization state of the universe. How-
ever, because of the many bright foregrounds expected for
21cm experiments (discussed in more detail in §VIA), an
absolute all-sky signal will be difficult to detect. An eas-
ier target is a statistical detection of the power spectrum
of the 21cm brightness temperature perturbations. By
comparing the power spectrum of the 21cm signal with
and without a contribution from PBHs, we can retain
the ability to discriminate among models within the lim-
itations of realistic experiments. We discuss the power
spectrum calculation in §IIIB.
A. Mean IGM evolution
The homogeneous IGM is described by the mean ki-
netic temperature TK, the spin temperature TS, and the
ionization fraction xi. The mean kinetic temperature of
the gas evolves according to [5, 29]
dTK
dt
= −2H(z)TK+xi(z)
η1tγ
(TCMB− TK) +χhǫ
kB
(3)
where H(z) is the Hubble parameter at redshift z, η1=
1 + fHe+ xi with fHe the helium fraction (defined by
nHe/[nH+ nHe]) [61], and
tγ=
3mec
8σTUCMB
(4)
with methe electron mass, σTthe Thomson cross section,
and UCMB the CMB energy density at redshift z. The
first term on the right-hand side of Equation (3) accounts
for the redshifting of kinetic energy with the expansion of
the universe. The second term includes the effect of IGM
heating by scattering of CMB photons from hydrogen
ions. The final term in (3) takes account of heating by
exotic energy injection into the IGM. The parameter ǫ
is an energy injection rate per baryon, and χh is the
fraction of the energy that goes into heating the IGM.
We describe in more detail below how ǫ is determined for
the PBH population.
The ionized fraction xiobeys [5, 29]
dxi
dt
= −α(TK)x2
inH+
χiǫ
Eion
(5)
where the first term on the right-hand side includes
hydrogen recombination through an effective coefficient
α(TK), which depends on the kinetic temperature of the
gas. Following [10] we use the Case B coefficient from Ta-
ble I of [30]. The second term in Equation (5) includes
the ionizations produced by an exotic energy injection
mechanism. As in Equation (3), ǫ is the energy injection
rate per baryon, χigives the fraction of energy going into
ionizations, and Eion= 13.6 eV is the ionization thresh-
old of hydrogen.
The spin temperature of the neutral hydrogen gas de-
pends on competition among collisional excitation [31],
heating by CMB photons, and interactions with Ly-α
photons [32, 33, 34]. The temperature itself is defined by
the ratio between the occupation numbers n1and n0of
the singlet and triplet hyperfine spin states, through
n1
n0
= 3exp
?
−TS
T∗
?
, (6)
where T∗= 0.068 K is the energy splitting between hy-
perfine states in temperature units. In equilibrium, the
spin temperature TSis given by
1 + xc+ xα
TS
=
1
TCMB
+xc
TK
+xα
Tc, (7)
where the parameters xc and xα are the collisional and
Wouthuysen-Field [32, 33] coupling parameters, respec-
tively, and Tc is the effective color temperature of the
radiation field [34, 35]. We omit the Wouthuysen-Field
coupling in this work, since we only expect it to be impor-
tant once the first luminous sources turn on at compar-
atively low redshift, at which point their heating effects
will easily swamp the contribution from PBHs. The col-
lisional coupling coefficient xcis
xc=4κ1−0(TK)nHT∗
3A10TCMB
, (8)
where A10= 2.87 × 10−15s−1is the spontaneous decay
rate of the hyperfine transition.
describes spin excitation through atomic collisions; its
value has been tabulated as a function of gas temperature
[36, 37]. For the range 1 K < TK < 300 K, we use the
values in column 4 of Table II in [37]. For TK> 300 K,
we use the fitting formulae suggested in [37] for the data
presented in [36].
Using Equations (3), (5) and (7) to evolve TK, xiand
TS, we calculate the differential 21cm brightness relative
to the background [1, 38]:
The parameter κ1−0
Tb=TS− TCMB
1 + z
?1 − e−τ?
?Ωbh
0.033
?1/2?TS− TCMB
(9)
≃ 28 mK
??Ωm
0.27
?−1/2
×
?1 + z
10TS
?
(10)
where we assume τ ≪ 1 in the second line.
B. 21cm power spectrum
Measurements of the power spectrum of fluctuations in
the 21cm signal will make it possible to extract statistical

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information about the structure in neutral hydrogen even
if detailed imaging or tomography remain beyond the
reach of experiments.
As evident in Equations (7) and (9) above, the bright-
ness of the 21cm signal depends on the density, temper-
ature and ionization state of the hydrogen gas. To trace
the shape of the power spectrum as a function of redshift,
we must track not only the density perturbations in the
baryonic matter, but also the fractional perturbations in
the ionization state and the temperature of the hydrogen
gas and the factors that influence the couplings among
them.
The fractional perturbation to the brightness temper-
ature is defined as
δ21(x) ≡ [δTb(x) −¯
δTb]/¯
δTb. (11)
The Fourier transform δ21(k) can be written as a sum
of each contribution weighted by expansion coefficients
related to the various couplings [2, 5]:
δ21(k) = (β + µ2)δ + βHδH+ βαδα+ βTδT, (12)
where the δiare fluctuations in overdensity (δ), neutral
fraction (δH), Lyman-α coupling strength (δα) and tem-
perature (δT). The expansion coefficients are
β = 1 +
xc
xtot(1 + xtot)
xHH
c
− xeH
xtot(1 + xtot)
xα
xtot(1 + xtot)
Tγ
TK− Tγ
?
dlnTK
(13)
βH = 1 +
c
(14)
βα =
(15)
βT =
+
1
xtot(1 + xtot)×
dlnκeH
10
+ xHH
xeH
c
c
dlnκHH
dlnTK
10
?
,(16)
where the xc are collisional coupling coefficients [39],
for which xc = xeH
c
+ xHH
c
The parameters κeH
spin de-excitation from electron/hydrogen and hydro-
gen/hydrogen collisions, respectively [2].
The power spectrum of 21cm fluctuations contains con-
tributions proportional to the mean brightness temper-
ature of the 21cm line over the whole sky as well as
to the fluctuations in the spin temperature [40]. Tak-
ing a simplified estimate in which we consider only the
growing-mode perturbation on large scales, we can write
all perturbations as pure time-delay perturbations, which
means each separate perturbation, and hence the 21cm
power spectrum, is proportional to the matter power
spectrum Pδδ. As we discuss in more detail below, this
approximation leads to a small error when the brightness
temperature is near zero, but provides sufficient accuracy
for our purposes. Then, the 21cm power spectrum can
be written
and xtot = xc + xα.
are rate coefficients for
10 and κHH
10
P21(k,µ) =
¯
δT2
b(β′+ µ2)Pδδ(k), (17)
where µ = cos(θ) accounts for the angle between the
wavevector k and the line of sight, and we have
β′= β + βTgT−βH¯ xigi
(1 − ¯ xi)+ θuβα. (18)
Here, gi(z) ≡ δi/δ and gT(z) ≡ δT/δ are defined for
convenience to compare the fluctuations in the ioniza-
tion and temperature to the matter overdensity, and θu
encodes the uniformity of the energy deposition. It is de-
fined such that the energy deposition rate is proportional
to (1+θuδ) so that θu= 0 for uniform deposition. Recall
that we are neglecting the contribution of Wouthuysen-
Field coupling in this work, so we are effectively imposing
βα= 0.
In practice, the power spectrum is calculated by evolv-
ing giand gT with redshift along with the mean global
properties of the IGM. The evolution equations are [5]
dgT
dz
=
gT− 2/3
1 + z
+2
3
η1kB¯
gi
1 + z+α ¯ xi ¯
ǫ
Eion
+
¯ xi
η1tγ
gTTγ− gi(Tγ−¯
¯
TK(1 + z)H(z)
TKχh(1 − θu) + gT
(1 + z)H(z)
nH(1 + gi+ α′gT))
(1 + z)H(z)
χi
¯ xi
TK)
η2ǫ
(19)
dgi
dz
=
+η2
(1 − θu) + gi
(1 + z)H(z), (20)
where we have included bars over quantities that are
global averages, for clarity. The quantity α is the hy-
drogen recombination coefficient and α′≡ dlnα/dlnTK
[5].
C. Energy injection and the photon spectrum
The PBHs influence the IGM through the direct emis-
sion of photons and e±pairs. Given the black hole emis-
sion power in these channels we must take into account
the details of photon and e±absorption by the IGM in
order to give the power absorption per baryon ǫ which
appears in the evolution equations for TKand xi.
There are a number of processes that allow photons to
deposit energy in the IGM, all of which have frequency
dependencies, which themselves change with redshift and
with the ionized fraction. We follow the tabulation of op-
tical depths given in [41] for the following processes: pho-
toionization, Compton scattering, pair production from
atoms and ions, photon-photon scattering, and single and
double pair production from CMB photons. Individual
rates for each process are shown at a fixed redshift in
Figure 1, and the total photon-IGM interaction rate as a
function of redshift is shown in Figure 2. At all redshifts,
below roughly 100 eV, the optical depth is dominated by
photoionization, while at high energies it is dominated by
pair production from CMB photons. Between the two,
the absorption of photons depends strongly on energy.