Homogeneous on Large Scales ?

This morning's debate is focused on the question of the large-scale homo-geneity of the Universe. I shall present the aarmative position, that there is overwhelming evidence for large scale homogeneity on scales in excess of approximately 50h ?1 Mpc, with a fractal distribution of matter on smaller scales. My worthy opponent, Dr. Luciano Pietronero, will present the counter-argument, that the fractal distribution observed on smaller scales continues to the largest observed scales, and that there is no evidence for homogeneity on any scale. This is an important question, since the Friedmann-Robertson-Walker metric presumes large scale homogeneity and isotropy for the Universe. This is the simplest cosmological model, and it is a fair question to ask the degree to which it is supported by the observational evidence. We know that the galaxy distribution is far from homogeneous on small scales, and large-scale structure in the form of long laments and chains extends to lengths in excess of 100h ?1 Mpc 1]. Redshift surveys often show structures nearly as large as the entire survey size; how conndent can we be that homogeneity is a valid concept for the large-scale universe? 1 1.2 The Mean Density of Galaxies The question of large scale homogeneity revolves around the issue of whether the mean density of the Universe, n, is a well deened concept. In the standard cosmological model, n is perfectly well deened, but in the fractal model of the Universe, n has vanishing global value and has relevance only within a given survey volume. In the conventional picture of large scale structure, the rms value of the number density of galaxies as observed at radius r from the vantage point of any randomly chosen galaxy, n(r), is given from the usual deenition of the two-point galaxy correlation, (r), by n(r) rms = n (1 + (r)). The measured (r) is well characterized by a power law, (r) (r=r 0) ? , (r 0 5h ?1 Mpc, = 1:8). Alternatively, this can be expressed in terms of the power spectrum of density uctuations, P(k) / k ?1:2 , (M=M) rms / (k 3 P(k)) 1=2 = 1 at some well deened scale. In the conventional picture, there exists a distinct transition between \small" and \large" amplitude scales, depending on whether (r) < 1 or (r) > 1. On large scales, where (r) ! 0, one has simply n(r) M(r)=V (r) ! constant. On scales …