At first glance, black holes would appear to be among the unlikeliest
of background radiation sources. In fact, experimental data on the
intensity of the EBL constrain black holes more strongly than any of
the dark-matter candidates we have discussed so far. Before explaining
how this comes about, we distinguish between "ordinary" black holes
(which form via the gravitational collapse of massive stars at the end
of their lives) and *primordial black holes* (PBHs) which may have
arisen from the collapse of overdense regions in the early Universe.
The existence of the former is very nearly an established fact,
while the latter remain hypothetical. However, it is PBHs which
are of interest to us as potential dark-matter candidates.

The reason for this is as follows. Ordinary black holes come from
baryonic progenitors (i.e. stars) and are hence classified with
the baryonic dark matter of the Universe. (They are of course not
"baryonic" in other respects, since among other things their baryon
number is not defined.) Ordinary black holes are therefore subject to
the nucleosynthesis bound (94), which limits them to
less than 5% of the critical density.
PBHs are *not* subject to this bound because they form during
the radiation-dominated era, before nucleosynthesis begins.
Nothing prevents them from making up most of the density in the Universe.
Moreover they constitute *cold* dark matter because their velocities
are low. (That is, they collectively obey a dust-like equation of state,
even though they might individually be better described as
"radiation-like" than baryonic.) PBHs were first proposed as
dark-matter candidates by Zeldovich and Novikov in 1966
[394]
and Hawking in 1971
[395].

Black holes contribute to the EBL via a process discovered by
Hawking in 1974 and often called *Hawking evaporation*
[396].
Photons cannot escape from inside the black hole, but they are produced
at or near the horizon by quantum fluctuations in the surrounding
curved spacetime. These give rise to a net flux of particles
which propagates outward from a black hole (of mass *M*) at a rate
proportional to *M*^{-2} (with the black-hole mass itself
dropping
at the same rate). For ordinary, stellar-mass black holes, this process
occurs so slowly that contributions to the EBL are insignificant, and the
designation "black" remains perfectly appropriate over the lifetime of
the Universe. PBHs, however, can in principle have masses far smaller
than those of a star, leading to correspondingly higher luminosities.
Those with
*M*
10^{15} g (the mass of a small asteroid) would
in fact evaporate quickly enough to shed all their mass over less than
~ 10 Gyr. They would already have expired in a blaze of
high-energy photons and other elementary particles as
*M* 0.

We will use the PBHs themselves as sources of radiation in what follows,
taking them to be distributed homogeneously throughout space.
The degree to which they actually cluster in the potential wells of
galaxies and galaxy clusters is not of concern here, since we are
concerned with their combined PBH contributions to the diffuse background.
Another subtlety must, however, be taken into account. Unlike the
dark-matter halos of previous sections, PBHs cover such a wide range
of masses (and luminosities) that we can no longer treat all sources
as identical. Instead we must define quantities like number density
and energy spectrum as functions of *PBH mass* as well as time,
and integrate our final results over both parameters.

The first step is to identify the distribution of PBH masses at the time when they formed. There is little prospect of probing the time before nucleosynthesis experimentally, so any theory of PBH formation is necessarily speculative to some degree. However, the scenario requiring the least extrapolation from known physics would be one in which PBHs arose via the gravitational collapse of small initial density fluctuations on a standard Robertson-Walker background, with an equation of state with the usual form (27), and with initial density fluctuations distributed like

(257) |

Here *M*_{i} is the initial mass of the PBH,
*M*_{f} is the mass lying
inside the particle horizon (or causally connected Universe) at PBH
formation time, and
is an unknown proportionality constant.

PBH formation under these conditions was originally investigated by Carr
[397],
who showed that the process is favoured over an extended
range of masses only if
*n* = 2/3. Proceeding on this assumption,
he found that the initial mass distribution of PBHs formed with masses
between *M*_{i} and *M*_{i} +
*dM*_{i} per unit comoving volume is

(258) |

where _{f}
is the mean density at formation time. The parameters
and
are formally
given by
2(2 - 1) /
and
exp[-
( -
1)^{2} /
2^{2}]
respectively, where
is the equation-of-state parameter as usual. However, in the interests
of lifting every possible restriction on conditions prevailing in the
early Universe, we follow Carr
[398]
in treating
and as free
parameters, not necessarily related to
and
. Insofar as the
early Universe was governed by the equation of state (27),
takes values
between 2 (dust-like or "soft") and 3 (stiff or "hard"), with
= 5/2
corresponding to the most natural situation (i.e.
= 4/3 as
for a non-interacting relativistic gas). We allow
to take
values as high as 4, corresponding to
"superhard" early conditions. The parameter
represents
the fraction of the Universe which goes into PBHs of mass
*M*_{f} at time *t*_{f}. It is a measure of
the initial inhomogeneity of the Universe.

The fact that Eq. (258) has no exponential cutoff at
high mass is important because it allows us (in principle at least)
to obtain a substantial cosmological density of PBHs. Since
2
4, however, the power-law
distribution is
dominated by PBHs of *low* mass. This is the primary reason why
PBHs turn out to be so tightly constrained by data on background radiation.
It is the low-mass PBHs whose contributions to the EBL via Hawking
evaporation are the strongest.

Much subsequent effort has gone into the identification of alternative formation mechanisms which could give rise to a more favourable distribution of PBH masses (i.e. one peaked at sufficiently high mass to provide the requisite CDM density without the unwanted background radiation from the low-mass tail). For example, PBHs might arise from a post-inflationary spectrum of density fluctuations which is not perfectly scale-invariant but has a characteristic length scale of some kind [399]. The parameter in (258) would then depend on the inflationary potential (or analogous quantities). This kind of dependence has been discussed in the context of two-stage inflation [400], extended inflation [401], chaotic inflation [402], "plateau" inflation [403], hybrid inflation [404] and inflation with isocurvature fluctuations [405].

A narrow spectrum of masses might also be expected if PBHs formed
during a spontaneous phase transition rather than arising from
primordial fluctuations. The quark-hadron transition
[406],
grand unified symmetry-breaking transition
[407]
and Weinberg-Salam phase transition
[408]
have all been considered in this regard. The initial mass distribution in
each case would be peaked near the horizon mass *M*_{f} at
transition time. The quark-hadron transition has attracted particular
attention because PBH formation would be enhanced by a temporary
softening of the equation of state; and because *M*_{f} for
this case is coincidentally close to
*M*_{},
so that PBHs formed at this time might be responsible for
MACHO observations of microlensing in the halo
[409].
Cosmic string loops have also been explored as possible seeds for
PBHs with a peaked mass spectrum
[410,
411].
Considerable interest has recently been generated by the
discovery that PBHs could provide a physical realization of the
theoretical phenomenon known as critical collapse
[412].
If this is so, then initial PBH masses would no longer necessarily
be clustered near *M*_{f}.

While any of the above proposals can in principle concentrate the PBH
population within a narrow mass range, all of them face the same problem
of *fine-tuning* if they are to produce the desired present-day
density of PBHs. In the case of inflationary mechanisms it is the form
of the potential which must be adjusted. In others it is the bubble
nucleation rate, the string mass per unit length, or the fraction of
the Universe going into PBHs at formation time. Thus, while modifications
of the initial mass distribution may weaken the "standard" constraints
on PBH properties (which we derive below), they do not as yet have a
compelling physical basis. Similar comments apply to attempts to link
PBHs with specific observational phenomena. It has been suggested,
for instance, that PBHs with the right mass could be responsible for
certain classes of
-ray bursts
[413,
414,
415],
or for long-term quasar variability via microlensing
[416,
417].
Other possible connections have been drawn to diffuse
-ray emission
from the Galactic halo
[418,
419]
and the MACHO microlensing events
[420,
421].
All of these suggestions, while intriguing,
would beg the question: "Why this particular mass scale?"

In order to obtain the comoving number density of PBHs from their initial mass distribution, we use the fact that PBHs evaporate at a rate which is inversely proportional to the square of their masses:

(259) |

This applies to uncharged, non-rotating black holes, which is a reasonable
approximation in the case of PBHs since these objects discharge
quickly relative to their lifetimes
[422]
and also give up angular momentum by preferentially emitting particles with
spin
[423].
The parameter depends
in general on PBH mass *M* and its behaviour was worked out in
detail by Page in 1976
[424].
The most important PBHs are those with
4.5 × 10^{14} g
*M* 9.4 ×
10^{16} g.
Black holes in this range are light (and therefore "hot") enough to
emit massless particles (including photons) as well as ultra-relativistic
electrons and positrons. The corresponding value of
is

(260) |

For *M* > 9.4 × 10^{16} g, the value of
drops to
3.8 × 10^{25} g^{3} s^{-1} because the
larger black hole is "cooler" and no longer able emit electrons and
positrons. EBL contributions from PBHs of this mass are however of lesser
importance because of the shape of the mass distribution.

As the PBH mass drops below 4.5 × 10^{14} g, its energy
*kT* climbs past the rest energies of progressively heavier particles,
beginning with muons and pions. As each mass threshold is passed, the PBH
is able to emit more particles and the value of
increases further.
At temperatures above the quark-hadron transition (*kT*
200 MeV),
MacGibbon and Webber have shown that relativistic quark and gluon jets
are likely to be emitted rather than massive elementary particles
[425].
These jets subsequently fragment into stable particles,
and the photons produced in this way are actually more important
(at these energies) than the primary photon flux. The precise behaviour
of in this regime
depends to some extent on one's choice of particle physics. A plot of
(*M*) for the
standard model is found in the review by Halzen
[426],
who note that climbs to
7.8 × 10^{26} g^{3} s^{-1} at
*kT* = 100 GeV, and that its value would be at least three times higher
in supersymmetric extensions of the standard model
where there are many more particle states to be emitted.

As we will shortly see, however, EBL contributions from PBHs at these temperatures are suppressed by the fact that the latter have already evaporated. If we assume for the moment that PBH evolution is adequately described by (259) with = constant as given by (260), then integration gives

(261) |

The lifetime *t*_{pbh} of a PBH is found by setting
*M*(*t*_{pbh}) = 0,
giving *t*_{pbh} = *M*_{i}^{3} /
3.
Therefore the initial mass of a PBH which is just disappearing today
(*t*_{pbh} = *t*_{0}) is given by

(262) |

Taking *t*_{0} = 16 Gyr and using (260) for
, we find
that *M*_{*} = 4.7 × 10^{14} g. A
numerical analysis allowing for changes in the value of
over the full range of
PBH masses with 0.06
_{m,0}
1 and
0.4 *h*_{0}
1
leads to a somewhat larger result
[426]:

(263) |

PBHs with *M*
*M*_{*} are exploding at redshift *z*
0 and
consequently dominate the spectrum of EBL contributions. The parameter
*M*_{*} is therefore of central importance in what
follows.

We now obtain the comoving number density of PBHs with masses
between *M* and *M* + *dM* at any time *t*. This is
the same as the comoving number density of PBHs with *initial*
masses between *M*_{i} and
*M*_{i} + *dM*_{i} at formation time, so
*n*(*M*, *t*) *dM* = *n*(*M*_{i})
*dM*_{i}. Inverting Eq. (261) to get
*M*_{i} = (*M*^{3} +
3
*t*)^{1/3} and differentiating, we find from (258) that

(264) |

Here we have used (262) to replace
*M*_{*}^{3} with
3 *t*_{0}
and switched to dimensionless parameters
*M* /
*M*_{*} and
*t* /
*t*_{0}. The quantity
is formally given in terms
of the parameters at PBH formation time by
=
(
_{f} /
*M*_{f})(*M*_{f} /
*M*_{*})^{-1} and has the
dimensions of a number density. As we will see shortly, it corresponds
roughly to the comoving number density of PBHs of mass
*M*_{*}. Following Page and Hawking
[427],
we allow
to move up or down as required by observational constraints.
The theory to this point is thus specified by two free parameters: the
PBH normalization
and the equation-of-state
parameter .

To convert to the present mass density of PBHs with mass ratios
between and
+
*d*, we multiply
Eq. (264) by *M* = *M*_{*}
and put
= 1:

(265) |

The total mass in PBHs is then found by integrating over
from zero to infinity. Changing variables to
*x*
^{-3}, we obtain:

(266) |

where *a*
1/3( - 2) and
*b* 4/3.
The integral is solved to give

(267) |

where (*x*) is
the gamma function. Allowing
to take
values from 2 through 5/2 (the most natural situation)
and up to 4, we find that

(268) |

The total mass density of PBHs in the Universe is thus
_{pbh}
*M*_{*}. Eq. (267) can be recast
as a relation between the characteristic number density
and the PBH density parameter
_{pbh} =
_{pbh}
/ _{crit,0}:

(269) |

The quantities
and
_{pbh} are thus
interchangeable as free parameters. If we adopt the most natural value
for (=2.5)
together with an upper limit on
due to
Page and Hawking of
10^{4}
pc^{-3}
[427],
then Eqs. (24), (263), (268)
and (269) together imply that
_{pbh} is at
most of order ~ 10^{-8} *h*_{0}^{-2}. If
this upper limit holds (as we confirm
below, then there is little hope for PBHs to make up the dark matter.

Eq. (268) shows that one way to boost their importance would be
to assume a soft equation of state at formation time (i.e. a value of
close to 2
as for dust-like matter,
rather than 2.5 as for radiation). Physically this is related to the
fact that low-pressure matter offers little resistance to gravitational
collapse. Such a softening has been shown to occur during the quark-hadron
transition
[409],
leading to significant increases in
_{pbh} for
PBHs which form at that time (subject to the fine-tuning problem noted in
Sec. 9.1). For PBHs which arise from primordial density
fluctuations, however, such conditions are unlikely to hold throughout
the formation epoch. In the limit
2,
Eq. (258) breaks down in any case because it becomes
possible for PBHs to form on scales smaller than the horizon
[397].

**9.3. Spectral energy distribution**

Hawking
[428]
proved that an uncharged, non-rotating black hole
emits bosons (such as photons) in any given quantum state with energies
between *E* and *E* + *dE* at the rate

(270) |

Here *T* is the effective black-hole temperature, and
_{s} is the
absorption coefficient or probability that
the same particle would be absorbed by the black hole if incident upon
it in this state. The function
*d* is related to
the spectral energy distribution (SED) of the black hole by
*d* =
*F*(,
)
*d* / *E*,
since we have defined
*F*(,
) *d*
as the energy emitted
between wavelengths
and +
*d*. We
anticipate that *F* will depend explicitly on the PBH mass
as well as wavelength. The PBH SED thus satisfies

(271) |

The absorption coefficient
_{s} is a
function of
and *E* as well as
the quantum numbers *s*
(spin), (total angular
momentum) and *m* (axial angular momentum)
of the emitted particles. Its form was first calculated by Page
[424].
At high energies, and in the vicinity of the peak
of the emitted spectrum, a good approximation is given by
[429]

(272) |

This approximation breaks down at low energies, where it gives rise to
errors of order 50% for (*G* *M* *E* /
*c*^{3}) ~ 0.05
[430]
or (with *E* = 2
*c* /
and
*M* ~ *M*_{*}) for
~ 10^{-3} Å.
This is adequate for our purposes, as we will find that the strongest
constraints on PBHs come from those with masses *M* ~
*M*_{*} at wavelengths
~ 10^{-4} Å.

Putting (272) into (271) and making the change of variable to wavelength
= *hc* /
*E*, we obtain the SED

(273) |

where *C* is a proportionality constant. This has the same form as the
blackbody spectrum, Eq. (78). We have made three simplifying
assumptions in arriving at this result. First, we have neglected the
black-hole charge and spin (as justified in Sec. 9.2).
Second, we have used an approximation for the absorption coefficient
_{s}. And
third, we have treated all the emitted photons as if
they are in the same quantum state,
whereas in fact the emission rate (270) applies separately to the
= *s* (= 1),
= *s* + 1 and
= *s* + 2 modes. There
are thus actually *three distinct* quasi-blackbody photon spectra
with different characteristic temperatures for any single PBH. However Page
[424]
has demonstrated that the
= *s* mode is overwhelmingly dominant, with the
= *s* + 1 and
= *s* + 2 modes
contributing less than 1% and 0.01% of the total photon flux
respectively. Eq. (273) is thus a reasonable approximation
for the SED of the PBH as a whole.

To fix the value of *C* we use the fact that the total flux of photons
(in all modes) radiated by a black hole of mass *M* is given by
[424]

(274) |

Inserting (273) and recalling that
*M* = *M*_{*}
, we find that

(275) |

The definite integral on the left-hand side of this equation can be
solved by switching variables to
= *c* /
:

(276) |

where (*n*)
and (*n*)
are the gamma function and Riemann
zeta function respectively. We then apply the fact that the temperature
*T* of an uncharged, non-rotating black hole is given by

(277) |

Putting (276) and (277) into (275) and rearranging terms leads to

(278) |

Using (3) = 2! = 2
and (3) =
1.202 along with (263) for *M*_{*}, we find

(279) |

We can also use the definitions (277) to define a useful new quantity:

(280) |

The size of this characteristic wavelength tells us that we will be
concerned primarily with the high-energy
-ray
portion of the spectrum. In terms of *C* and
_{pbh} the SED
(273) now reads

(281) |

While this contains no explicit time-dependence, the spectrum does of
course depend on time through the PBH mass ratio
.
To find the PBH luminosity we employ Eq. (57), integrating
the SED *F*(,
) over all
to obtain:

(282) |

This definite integral is also solved by means of a change of variable to frequency , with the result that

(283) |

Using Eqs. (263), (278) and (280) along with the values
(4) = 3! = 6 and
(4) =
^{4}/96,
we can put this into the form

(284) |

where

Compared to the luminosity of an ordinary star, the typical PBH
(of mass ratio
1) is not very
luminous. A PBH of
900 kg or so might theoretically be expected to reach the Sun's
luminosity; however, in practice it would already have exploded,
having long since reached an effective temperature high enough to
emit a wide range of massive particles as well as photons.
The low luminosity of black holes in general can be emphasized by
using the relation
*M* /
*M*_{*} to recast Eq. (284) in the form

(285) |

This expression is not strictly valid for PBHs of masses near
*M*_{},
having been derived for those with
*M* ~ *M*_{*} ~ 10^{15} g.
(Luminosity is *lower* for larger black holes, and one of
solar mass would be so much colder than the CMB that it would absorb
radiation faster than it could emit it.) So, Hawking evaporation or not,
most black holes are indeed very black.

To obtain the total bolometric intensity of PBHs out to a
look-back time *t*_{f}, we substitute the PBH number
density (264) and luminosity (282) into the integral (12) as usual.
Now however the number density *n*(*t*) is to be replaced by
*n*(,
)
*d*,
*L*() takes
the place of *L*(*t*), and we integrate over all PBH masses
as well as look-back times
:

(286) |

where

(287) |

Here
( + 2) / 3
and we have used (269) to replace
with
_{pbh}. In
principle, the integral over
should be cut off at a
finite lower limit
_{c}(),
equal to the mass of the lightest PBH which has not yet evaporated at
time . This arises because
the initial PBH mass distribution (258) requires a nonzero minimum
*M*_{min} in order
to avoid divergences at low mass. In practice, however, the cutoff
rapidly evolves toward zero from its its initial value of
_{c}(0) =
*M*_{min} / *M*_{*}. If
*M*_{min} is of the order of the Planck mass
as usually suggested
[431],
then _{c}() drops to zero
well before the end of the radiation-dominated era. Since we are
concerned with times later than this, we can safely set
_{c}() = 0.

Eq. (286) can be used to put a rough upper limit on
_{pbh}
from the bolometric intensity of the background light
[432].
Let us assume that the Universe is flat, as suggested by most
observations (Sec. 4). Then its age
*t*_{0} can be obtained from Eq. (56) as

(288) |

Here _{0}
_{m}(0) where
_{m}(*z*)
is the dimensionless function

(289) |

Putting (288) into (287) and using Eqs. (18), (24), (263) and (284), we find:

(290) |

We are now ready to evaluate Eq. (286). To begin with we note that the integral over mass has an analytic solution:

(291) |

For the EdS case
(_{m,0} = 1),
_{0} = 1 and
Eq. (51) implies:

(292) |

Putting Eqs. (291) and (292) into (286), we find that

(293) |

The parameter _{f}
is obtained for the EdS case by inverting (292) to give
_{f} = (1 +
*z*_{f})^{-3/2}. The subscript "*f*"
("formation") is here a misnomer since we do not integrate back to
PBH formation time, which occurred in the early stages of the
radiation-dominated era. Rather we integrate out to the redshift
at which processes like pair production become significant enough to render
the Universe approximately opaque to the (primarily
-ray) photons
from PBH evaporation. Following Kribs
[430]
this is
*z*_{f} 700.

Using this value of *z*_{f} and substituting Eqs. (290) and
(291) into (293), we find that the bolometric intensity
of background radiation due to evaporating PBHs in an EdS Universe is

(294) |

This vanishes for
= 2 because
*k*_{}
in this limit, as discussed in Sec. 9.2. The case
= 4 (i.e.
= 2) is
evaluated with the help of L'Hôpital's rule, which gives
lim_{2}(1 - _{f}^{2-}) / (2 -
) =
- ln_{f}.

The values of *Q* in Eq. (294) are far higher than the
actual bolometric intensity of background radiation in an EdS universe,
2/5 *Q*_{*} = 1.0 × 10^{-4} erg
s^{-1} cm^{-2}
(Fig. 2.6). Moreover this background is already well accounted for by
known astrophysical sources. A firm upper bound on
_{pbh} (for
the most natural situation with
= 2.5) is
therefore

(295) |

For harder equations of state ( > 2.5) the PBH density would have to be even lower. PBHs in the simplest formation scenario are thus eliminated as important dark-matter candidates, even without reference to the cosmic -ray background.

For models containing dark energy as well as baryons and black holes,
the integrated background intensity goes up because the Universe is older,
and down because
*Q*
_{pbh}. The
latter effect is stronger, so that the above constraint on
_{pbh} will
be weaker in a model such as
CDM (with
_{m,0} =
0.3, _{,0} =
0.7). To determine the importance of this effect, we can re-evaluate the
integral (286) using the general formula (54) for
() in place
of (292). We will make the minimal assumption that PBHs
constitute the *only* CDM, so that
_{m,0} =
_{pbh} +
_{bar} with
_{bar} given
by (94) as usual. Eq. (56) shows that the parameter
_{f} is given for
arbitrary values of
_{m,0} by
_{f} =
_{m}(*z*_{f}) /
_{0} where
the function
_{m}(*z*)
is defined as before by (289).

Evaluation of Eq. (286) leads to the plot of bolometric
intensity *Q* versus
_{pbh} in
Fig. 40.
As before, *Q* is proportional to *h*_{0} because it
goes as both
_{pbh}
= _{pbh}
_{crit,0}
*h*_{0}^{2} and
*t*_{0}
*h*_{0}^{-1}. Since
*Q* 0 for
2 we have chosen a
minimum value of
= 2.2 as
representative of "soft" conditions.
Fig. 40 confirms that, regardless of
cosmological model, PBH
contributions to the background light are too high unless
_{pbh}
<< 1. The values in Eq. (294) are recovered at the right-hand
edge of the figure where
_{pbh}
approaches one, as expected. For all
other models, if we impose a conservative upper bound
*Q* < *Q*_{*} (as
indicated by the faint dotted line) then it follows that
_{pbh} <
(6.9 ± 4.2) × 10^{-5}*h*_{0}^{-1}
for = 2.5.
This is about 60% higher than the limit (295) for the EdS case.

Stronger limits on PBH density can be obtained from the
-ray
background, where these
objects contribute most strongly to the EBL and where we have good data
(as summarized in Sec. 8.6).
Spectral intensity is found as usual by substituting the comoving PBH
number density (264) and SED (281) into
Eq. (61). As in the bolometric case, we now have to
integrate over PBH mass
as well as time
= *t* /
*t*_{0}, so that

(296) |

Following the discussion in Sec. 9.4 we set
_{c}() = 0.
In light of our bolometric results it is unlikely that PBHs make up
a significant part of the dark matter, so we no longer tie the value of
_{m,0} to
_{pbh}.
Models with
_{m,0}
_{bar} must
therefore contain a second species of cold dark matter (other than PBHs) to
provide the required matter density. Putting (264) and
(281) into (296) and using (263), (269) and (278), we find that

(297) |

Here the dimensional prefactor is a function of both
and
_{0}
and reads

(298) |

We have divided through by the photon energy
*hc* / _{0}
to put the results in units of CUs as usual. The range of uncertainty in
*I*_{}(_{0}) is smaller than that in *Q*, Eq. (290),
because *I*_{}(_{0}) depends only linearly on
*M*_{*} whereas *Q*
is proportional to *M*_{*}^{-3}.
(This in turn results from the fact that
*I*_{}
*C*
*M*_{*}^{ 2} whereas
*Q*
*L*_{pbh}
*M*_{*}^{-2}. One more factor of
*M*_{*}^{-1} comes from
_{pbh}
/ *M*_{*} in both cases.) Like *Q*,
*I*_{} depends linearly on *h*_{0} since
integrated intensity in either case is proportional to both
_{pbh}
_{crit,0}
*h*_{0}^{2} and
*t*_{0}
*h*_{0}^{-1}.

Numerical integration of Eq. (297) leads to the
plots shown in Fig. 41, where we have set
_{pbh} =
10^{-8}. Following Page and Hawking
[427]
we have chosen values of
_{m,0} =
0.06 in panel (a) and
_{m,0} = 1
in panel (b). (Results are
not strictly comparable in the former case, however, since we assume
that _{,0} =
1 - _{m,0}
rather than _{,0} = 0.) Our results are in
good agreement with the earlier ones except at the longest wavelengths
(lowest energies), where PBH evaporation is no longer well described
by a simple blackbody SED, and where the spectrum begins to be affected
by pair production on nuclei. As expected the spectra peak near
10^{-4} Å in the
-ray
region. Also plotted in
Fig. 41 are the data from SAS-2
([384];
heavy dashed line), COMPTEL
([382];
triangles) and EGRET
([385];
heavy solid line).

By adjusting the value of
_{pbh} up or
down from its value of 10^{-8}
in Fig. 41, we can match the theoretical PBH
spectra to those measured (e.g., by EGRET), thereby
obtaining the maximum value of
_{pbh}
consistent with observation. For
= 2.5 this
results in

(299) |

These limits are three orders of magnitude stronger than the one from bolometric intensity, again confirming that PBHs in the simplest formation scenario cannot be significant contributors to the dark matter. Using (269) this result can be translated into an upper limit on :

(300) |

These numbers are in good agreement with the original Page-Hawking bound
of < 1 ×
10^{4} pc^{-3}
[427],
which was obtained for *h*_{0} = 0.6.

Subsequent workers have refined the
-ray
background constraints on
_{pbh} and
in a number of
ways. MacGibbon and Webber
[425]
pointed out that PBHs whose effective temperatures have
climbed above the rest energy of hadrons probably give off more
photons by *indirect* processes than by direct emission.
This occurs because it is not bound states (i.e. hadrons) that are
most likely to be emitted, but their elementary constituents (quarks
and gluons in the form of relativistic jets). Accelerator experiments
and numerical simulations indicate that these jets subsequently
fragment into secondary particles whose decays (especially those of
the pions) produce a far greater flux of photons than that emitted
directly from the PBH. The net effect is to increase the PBH
luminosity, particularly in low-energy
-rays,
strengthening the constraint on
_{pbh} by
about an order of magnitude
[429].
The most recent upper limit obtained in this way using
EGRET data (assuming
_{m,0} = 1)
is _{pbh} <
(5.1 ± 1.3) × 10^{-9} *h*_{0}^{-2}
[433].

Complementary upper limits on PBH contributions to the dark matter have
come from *direct* searches for those evaporating within a few kpc
of the Earth. Such limits are subject to more uncertainty than ones
based on the EBL because they depend on assumptions about the degree
to which PBHs are clustered. If there is no clustering then
(295) can be converted into a stringent upper bound
on the local PBH evaporation rate,
<
10^{-7} pc^{-3} yr^{-1}.
This however relaxes to
10
pc^{-3} yr^{-1} if PBHs are strongly clustered
[426],
in which case limits from direct
searches could potentially become competitive with those based on the EBL.
Data taken at energies near 50 TeV with the CYGNUS
air-shower array has led to a bound of
< 8.5 ×
10^{5} pc^{-3} yr^{-1}
[434],
and a comparable limit of
< (3.0 ±
1.0) × 10^{6} pc^{-3} yr^{-1}
has been obtained at 400 GeV using an imaging atmospheric Cerenkov
technique developed by the Whipple collaboration
[435].
Very strong constraints have also been claimed based on balloon
observations of cosmic-ray antiprotons
[436].

Other ideas have been advanced which could weaken the bounds on PBHs as dark-matter candidates. It might be, for instance, that these objects leave behind stable relics rather than evaporating completely [437], a possibility that has recently been revived on the grounds that total evaporation would be inconsistent with a generalized gravitational version of the uncertainty principle [438]. This however raises a new problem (similar to the "gravitino problem" discussed in Sec. 8.5) because such relics would have been overproduced by quantum and thermal fluctuations in the early Universe. Inflation can be invoked to reduce their density, but must be finely tuned if the same relics are to make up an interesting fraction of the dark matter today [439].

A different suggestion due to Heckler [440, 441] has been that particles emitted from the black hole might interact strongly enough above a critical temperature to form a photosphere. This would make the PBH appear cooler as seen from a distance than its actual surface temperature, just as the solar photosphere makes the Sun appear cooler than its core. (In the case of the black hole, however, one has not only an electromagnetic photosphere but a QCD "gluosphere.") The reality of this effect is still under debate [433], but preliminary calculations indicate that it could reduce the intensity of PBH contributions to the -ray background by 60% at 100 MeV, and by as much as two orders of magnitude at 1 GeV [442].

Finally, as discussed already in Sec. 9.1, the limits obtained above can be weakened or evaded if PBH formation occurs in such a way as to produce fewer low-mass objects. The challenge faced in such proposals is to explain how a distribution of this kind comes about in a natural way. A common procedure is to turn the question around and use observational data on the present intensity of the -ray background as a probe of the original PBH formation mechanism. Such an approach has been applied, for example, to put constraints on the spectral index of density fluctuations in the context of PBHs which form via critical collapse [430] or inflation with a "blue" or tilted spectrum [443]. Thus, even should they turn out not to exist, primordial black holes provide a valuable window on conditions in the early Universe, where information is otherwise scarce.

In view of the fact that conventional black holes are disfavoured
as dark-matter candidates, it is worthwhile to consider alternatives.
One of the simplest of these is the extension of the black-hole concept
from the four-dimensional (4D) spacetime of general relativity to higher
dimensions. Higher-dimensional relativity, also known as Kaluza-Klein
gravity, has a long history and underlies modern attempts to unify
gravity with the standard model of particle physics
[444].
The extra dimensions have traditionally been assumed to be
compact, in order to explain their non-appearance in low-energy physics.
The past few years, however, have witnessed a surge of interest
in *non*-compactified theories of higher-dimensional gravity
[445,
446,
447].
In such theories the dimensionality
of spacetime can manifest itself at experimentally accessible energies.
We focus on the prototypical five-dimensional (5D) case, although the
extension to higher dimensions is straightforward in principle.

Black holes are described in 4D general relativity by the Schwarzschild metric, which reads (in isotropic coordinates)

(301) |

where
*d*^{2}
*d*^{2} +
sin^{2} *d*
^{2}.
This is a
description of the static, spherically-symmetric spacetime around a
pointlike object (such as a collapsed star or primordial density
fluctuation) with Schwarzschild mass *M*_{s}. As we have seen,
it is unlikely that such objects can make up the dark matter.

If the Universe has more than four dimensions, then the same object
must be modelled with a higher-dimensional analog of the Schwarzschild
metric. Various possibilities have been explored over the years, with
most attention focusing on a 5D solution first discussed in detail by
Gross and Perry
[448],
Sorkin
[449]
and Davidson and Owen
[450]
in the early 1980s. This is now generally known
as the *soliton metric* and reads:

302 |

Here *y* is the new coordinate and there are three metric parameters
(*a*, ,
) rather than just one
(*M*_{s}) as in Eq. (301).
Only two of these are independent, however, because a consistency
condition (which follows from the field equations) requires that
^{2}(^{2} -
+ 1) = 1. In the limit
where
0,
and
1, Eq. (302)
reduces to (301) on 4D hypersurfaces *y* = const. In this limit
we can also identify the parameter *a* as
*a* = 2*c*^{2} / *GM*_{s} where
*M*_{s} is the Schwarzschild mass.

We wish to understand the physical properties of this solution in
four dimensions. To accomplish this we do two things. First, we
assume that Einstein's field equations in their usual form hold in the
full *five*-dimensional spacetime. Second, we assume that the
Universe in five dimensions is *empty*, with no 5D matter fields
or cosmological constant. The field equations then simplify to

(303) |

Here _{AB} is the
5D Ricci tensor, defined in exactly the same way as the
4D one except that spacetime indices *A*, *B* run over 0-4 instead
of 0-3. Putting a 5D metric such as (302) into the vacuum
5D field equations (303), we recover the 4D field
equations (106) with a nonzero energy-momentum tensor
_{µ}.
*Matter and energy, in other words, are induced in 4D by pure
geometry in 5D.* It is by studying the properties of this
induced-matter energy-momentum tensor
(_{µ})
that we learn what the soliton looks like in four dimensions.

The details of the mechanism just outlined
[451]
and its application to solitons in particular
[452,
453]
have been well studied and we do not review this material here.
It is important to note, however, that the Kaluza-Klein soliton
differs from an ordinary black hole in several key respects. It
contains a singularity at its center, but this center is located at
*r* = 1/*a* rather than *r* = 0. (The point *r* = 0
is, in fact, not even part of the manifold, which ends at *r* = 1 /
*a*.) Its event horizon also shrinks to a point at *r* = 1 /
*a*. For these reasons the soliton is better
classified as a naked singularity than a black hole.

Solitons in the induced-matter picture are further distinguished from conventional black holes by the fact that they have an extended matter distribution rather than having all their mass compressed into the singularity. It is this feature which proves to be of most use to us in putting constraints on solitons as dark-matter candidates [454]. The time-time component of the induced-matter energy-momentum tensor gives us the density of the solitonic fluid as a function of radial distance:

(304) |

From the other elements of
_{µ} one finds that
pressure can be written *p*_{s} = 1/3
_{s}
*c*^{ 2}, so that the soliton has
a radiation-like equation of state. In this respect the soliton
more closely resembles a primordial black hole (which forms during the
radiation-dominated era) than one which arises as the endpoint of stellar
collapse. The elements of
_{µ} can also be
used to calculate the gravitational mass of the fluid inside *r*:

(305) |

At large distances *r* >> 1 / *a* from the center the
soliton's density (304) and gravitational mass (305) go over to

(306) |

The second of these expressions shows that the asymptotic value of
*M*_{g} is in general not the same as *M*_{s}
[*M*_{g}() =
*M*_{s}
for *r* >> 1 / *a*], but reduces to it in the limit
1.
Viewed in four dimensions, the soliton resembles a hole in the geometry
surrounded by a spherically-symmetric ball of ultra-relativistic matter
whose density falls off at large distances as 1 /
*r*^{4}. If the Universe
does have more than four dimensions, then objects like this should be
common, being generic to 5D Kaluza-Klein gravity in exactly the same
way that black holes are to 4D general relativity.

We therefore assess their impact on the background radiation, assuming that the fluid making up the soliton is in fact composed of photons (although one might also consider ultra-relativistic particles such as neutrinos in principle). We do not have spectral information on these so we proceed bolometrically. Putting the second of Eqs. (306) into the first gives

(307) |

Numbers can be attached to the quantities
, *r* and
*M*_{g} as follows.
The first () is
technically a free parameter. However, a natural
choice from the physical point of view is
~ 1. For this case the
consistency relation implies
~ 1 also,
guaranteeing that
the asymptotic gravitational mass of the soliton is close to its
Schwarzschild one. To obtain a value for *r*, let us assume that
solitons are distributed homogeneously through space with average
separation *d* and mean density
_{s} =
_{s}
_{crit,0}
= *M*_{s} / *d*^{3}. Since
_{s} drops
as *r*^{-4} whereas the number of solitons at a distance
*r* climbs only as *r*^{3}, the local density of
solitons is largely determined by the nearest
one. We can therefore replace *r* by
*d* = (*M*_{s} /
_{s}
_{crit,0})^{1/3}.
The last unknown in (307) is the soliton mass *M*_{g} (=
*M*_{s}
if = 1). The fact that
_{s}
*r*^{-4}
is reminiscent of the density profile of the Galactic dark-matter halo,
Eq. (174). Theoretical work on the classical tests
of 5D general relativity
[455]
and limits on violations of the equivalence principle
[456]
also suggests that solitons are
likely to be associated with dark matter on galactic or larger scales.
Let us therefore express *M*_{s} in units of the mass of
the Galaxy, which from (175) is
*M*_{gal}
2 × 10^{12}
*M*_{}.
Eq. (307) then gives the local energy density of solitonic fluid as

(308) |

To get a characteristic value, we take
*M*_{s} = *M*_{gal} and adopt our
usual values *h*_{0} = 0.75 and
_{s} =
_{cdm} =
0.3. Let us moreover
compare our result to the average energy density of the CMB, which
dominates the spectrum of background radiation
(Fig. 1).
The latter is found from Eq. (141) as
_{cmb}
*c*^{ 2} =
_{}
_{crit,0}
*c*^{ 2} = 4 × 10^{-13} erg
cm^{-3}. We therefore obtain

(309) |

This is of the same order of magnitude as the limit set on anomalous contributions to the CMB by COBE and other experiments. Thus the dark matter could consist of solitons, if they are not more massive than galaxies. Similar arguments can be made on the basis of tidal effects and gravitational lensing [454]. To go further and put more detailed constraints on these candidates from background radiation or other considerations will require a deeper investigation of their microphysical properties.

Let us summarize our results for this section. We have noted that standard (stellar) black holes cannot provide the dark matter insofar as their contributions to the density of the Universe are effectively baryonic. Primordial black holes evade this constraint, but we have reconfirmed the classic results of Page, Hawking and others: the collective density of such objects must be negligible, for otherwise their presence would have been obvious in the -ray background. In fact, we have shown that their bolometric intensity alone is sufficient to rule them out as important dark-matter candidates. These constraints may be relaxed if primordial black holes can form in such a way that they are distributed with larger masses, but it is not clear that such a distribution can be shown to arise in a natural way. As an alternative, we have considered black hole-like objects in higher-dimensional gravity. Bolometric arguments do not rule these out, but there are a number of theoretical issues to be worked out before a more definitive assessment of their potential can be made.