**2.1 The Cosmological Principle**

Pietronero [13]
argues that the evidence from redshift
catalogs and deep galaxy counts is that the galaxy distribution
is best described as a
scale-invariant fractal with dimension *D* ~ 2. Others
disagree ([14],
[15]). I am
heavily influenced by another line of argument:
it is difficult to reconcile a fractal universe with the
isotropy observed in deep surveys (examples of which are
illustrated in Figs. 3.7 to 3.11 in
[11] and are
discussed in connection with the fractal universe in pp. 209 - 224 in
[11]).

Fig. 1 shows angular positions of particles in
three ranges of
distance from a particle in a fractal realization with dimension
*D* = 2 in
three dimensions. At *D* = 2 the expected number of neighbors
scales with distance *R* as *N* (< *R*)
*R*^{2}, and
I have scaled the fraction of particles plotted as *R*^{-2} to
get about the same number in each plot.
The fractal is constructed by placing a stick of length
*L*, placing on either end the centers of sticks of length
*L* / , where = 2^{1/D}, with
random orientation, and iterating to smaller and larger scales. The
particles are placed on the ends of the
shortest sticks in the clustering hierarchy. This construction
with *D* = 1.23 (and some adjustments to fit the galaxy three- and
four-point correlation functions) gives a good description of the
small-scale galaxy clustering
[16]. The fractal in
Fig. 1, with *D* = 2, the dimension
Pietronero proposes, does not look at all like deep sky maps of
galaxy distributions, which show an approach to isotropy with
increasing depth. This cannot happen in a scale-invariant
fractal: it has no characteristic length.

A characteristic clustering length for galaxies may be expressed in
terms of the dimensionless two-point correlation function
defined by the joint probability of finding galaxies centered in
the volume elements *dV*_{1} and *dV*_{2} at
separation *r*,

The galaxy two-point function is quite close to a power law,

where the clustering length is

and the Hubble parameter is

The rms fluctuation in galaxy counts in a randomly placed sphere
is *N/N* = 1 at sphere
radius *r* = 1.4*r*_{0} ~ 6*h*^{-1} Mpc, to
be compared to the Hubble distance (at which the recession
velocity approaches the velocity of light),
*cH*_{0}^{-1} = 3000*h*^{-1} Mpc.

The isotropy observed in deep sky maps is consistent with a
universe that is inhomogeneous
but spherically symmetric about our position. There are
tests, as discussed by Paczynski and Piran
[17]. For
example, we have a successful theory for the
origin of the light elements as remnants of the expansion and
cooling of the universe through *kT* ~ 1 MeV
[18].
If there were a strong radial matter density gradient out to the
Hubble length we could be using the wrong local entropy per
baryon, based on conditions at the Hubble length where
the CBR came from, yet the theory seems to be successful. But to
most people the compelling argument is that distant galaxies look
like equally good homes for observers like us: it would be
startling if we lived in one of the very few close to the center of symmetry.

Mandelbrot [19] points out that other fractal constructions could do better than the one in Fig. 1. His example does have more particles in the voids defined by the strongest concentrations in the sky, but it seems to me to share the distinctly clumpy character of Fig. 1. It would be interesting to see a statistical test. A common one expands the angular distribution in a given range of distances in spherical harmonics,

where is the surface mass density as a function of direction in the sky. The integral becomes a sum if the fractal is represented as a set of particles. A measure of the angular fluctuations is

where

In the approximation of the sum as an integral *e _{l}* is
the contribution to the variance of the angular distribution per
logarithmic interval of

I can think of two ways to define the dimension of a fractal that
produces a close to isotropic sky. First, each octant of a full
sky sample has half the diameter of the full sample, so
one might define *D* by the fractional departure of
the mean density within each octant from the mean in the full
sample,

Thus in Fig. 1, with *D* = 2, the
quadrupole anisotropy *e*_{2} is on
the order of unity. Second, one can see the idea that the mean
particle density varies with distance *r* from a particle as
*r*^{-(3-D)}. Then the small angle (large *l*) Limber
approximation to the angular correlation function *w* () is
[20]

(9)

To find *e _{l}* differentiate with respect to

The universe is not exactly homogeneous, but it seems to be
remarkably close to it on the scale of the Hubble length.
It would be interesting to know whether there is a fractal
construction that allows a significantly larger
value of 3 - *D* for given *e _{l}* than in this calculation.