226 Cosmic Structures
Turning to short wavelengths (scale about 10^12 kg or 10−^18 푀⊙), black holes provide
another constraint. Primordial perturbations on this scale must have been roughly
smaller than 1.0. Larger perturbations would have led to overproduction of black
holes, since large-amplitude perturbations inevitably cause overdense regions to col-
lapse before pressure forces can respond. From Equation (5.82) one sees that black
holes less massive than 10^12 kg will have already evaporated within 10Gyr, but those
more massive will remain and those of mass 10^12 kg will be evaporating and emit-
ting훾rays today. Large amplitude perturbations at and above 10^12 kg would imply
more black holes than is consistent with the mass density of the Universe and the훾
ray background. Combining the black hole limit on perturbations (up to around 1 for
푀≈ 1012 kg) with those from the CMB and galaxy formation also implies the spec-
trum must be close to the Harrison–Zel’dovich form.
The power spectra of theoretical models for density fluctuations can be compared
with the real distribution of galaxies and galaxy clusters. Suppose that the galaxy
number density in a volume element d푉 is푛G, then one can define the probability
of finding a galaxy in a random element as
d푃=푛Gd푉. (10.11)If the galaxies are distributed independently, for instance with a spatially homoge-
neous Poisson distribution, the joint probability of having one galaxy in each of two
random volume elements d푉 1 ,d푉 2 is
d푃 12 =푛^2 Gd푉 1 d푉 2. (10.12)There is then no correlation between the probabilities in the two elements. However,
if the galaxies are clustered on a characteristic length푟c, the probabilities in different
elements are no longer independent but correlated. The joint probability of having
two galaxies with a relative separation푟can then be written
d푃 12 =푛^2 G[ 1 +휉(푟∕푟c)]d푉 1 d푉 2 , (10.13)where휉(푟∕푟c)is thetwo-point correlation functionfor the galaxy distribution. This can
be compared with the autocorrelation function in Equation (10.8) of the theoretical
model. If we choose our own Galaxy at the origin of a spherically symmetric galaxy
distribution, we can simplify Equation (10.13) by setting푛Gd푉 1 =1. The right-hand
side then gives the average number of galaxies in a volume element d푉 2 at distance푟.
Analyses of galaxy clustering show [2] that, for distances
10 kpc≲hr≲10 Mpc, (10.14)
a good empirical form for the two-point correlation function is
휉(푟∕푟c)=(푟∕푟c)−훾, (10.15)with the parameter values푟c≈ 5. 0 ℎ−^1 Mpc,훾≈ 1 .7.
Irregularities in the metric can be expressed by the curvature radius푟푈defined in
Equation (5.54). If푟푈is less than the linear dimensions푑of the fluctuating region, it
will collapse as a black hole. Establishing the relation between the curvature of the
metric and the size of the associated mass fluctuation requires the full machinery of
general relativity, which is beyond our ambitions.