Approaching a homogeneous galaxy distribution: results from the Stromlo-APM redshift survey

Recent results from a number of redshift surveys suggest that the Universe is well described by an inhomogeneous, fractal distribution on the largest scales probed. This distribution has been found to have fractal dimension, D, approximately equal to 2.1, in contrast to a homogeneous distribution in which the dimension should approach the value 3 as the scale is increased. In this paper we demonstrate that estimates of D, based on the conditional density of galaxies, are prone to bias from several sources. These biases generally result in a smaller measured fractal dimension than the true dime nsion of the sample. We illustrate this behaviour in application to the Str omlo-APM redshift survey, showing that this data set in fact provides evide nce for fractal dimension increasing with survey depth. On the largest scal e probed, r approximate to 60 h(-1) Mpc, we find evidence for a distributio n with dimension D = 2.76 +/- 0.10. A comparison between this sample and mo ck Stromlo-APM catalogues taken from N-body simulations (which assume a CDM cosmology) reveals a striking similarity in the behaviour of the fractal d imension. Thus we find no evidence for inhomogeneity in excess of that expe cted from conventional cosmological theory. We consider biases affecting fu ture large surveys and demonstrate, using mock SDSS catalogues, that this s urvey will be able to measure the fractal dimension on scales at which we e xpect to see full turn-over to homogeneity, in an accurate and unbiased way .