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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]).

Figure 1

Figure 1. Angular distributions of particles in a realization of a fractal with dimension D = 2 viewed from one of the particles in the realization. The fraction of particles plotted in each distance bin has been scaled so the expected number of particles plotted is the same in each bin.

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) propto R2, 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 / lambda, where lambda = 21/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 dV1 and dV2 at separation r,

Equation 1 (1)

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

Equation 2 (2)

where the clustering length is

Equation 3 (3)

and the Hubble parameter is

Equation 4 (4)

The rms fluctuation in galaxy counts in a randomly placed sphere is deltaN/N = 1 at sphere radius r = 1.4r0 ~ 6h-1 Mpc, to be compared to the Hubble distance (at which the recession velocity approaches the velocity of light), cH0-1 = 3000h-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,

Equation 5 (5)

where sigma is the surface mass density as a function of direction Omega 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

Equation 6 (6)


Equation 7 (7)

In the approximation of the sum as an integral el is the contribution to the variance of the angular distribution per logarithmic interval of l. It will be recalled that the zeros of the real and imaginary parts of Ylm are at separation theta = pi / l in the shorter direction, except where the zeros crowd together near the poles and Ylm is close to zero. Thus el is the variance of the fractional fluctuation in density across the sky on the angular scale theta ~ pi / l and in the chosen range of distances from the observer.

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,

Equation 8 (8)

Thus in Fig. 1, with D = 2, the quadrupole anisotropy e2 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 (theta) is [20]

Equation 9 (9)

To find el differentiate with respect to l. At D = 2 this gives el ~ 1: the surface density fluctuations are independent of scale. At 0 < 3 - D << 1, el ~ (3 - D) / l. The X-ray background fluctuates by about deltaf / f ~ 0.05 at theta = 5°, or l ~ 30. This is equivalent to D ~ 3 - l (deltaf / f)2 ~ 2.9 in the fractal model in Eq. (9).

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 el than in this calculation.

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