Clustering Properties from Finite Galaxy Samples

Details Clustering Properties from Finite Galaxy Samples Clustering Properties from Finite Galaxy Samples

Feb 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994mnras.266..555p&link_type=abstract

Monthly Notices of the Royal Astronomical Society, Vol. 266, NO. 3/FEB1, P. 555, 1994

Mathematics

Logic

7

Scientific paper

The shape of the density correlation function g(r) = 1 + ξ(r) provides useful information on the properties of galaxy clustering. In particular, a power-law form of g(r) is proportional to r^-γ^ indicates the presence of a scaling regime with a correlation dimension D_2_ = 3 - γ, while a flattening of g(r) is a signature of homogeneity. A relevant issue, then, is whether the standard treatment of boundary effects may affect the behaviour of g(r) at large separations, for example by inducing a spurious flattening and/or a spurious scaling regime. To answer this question, we consider different types of fractal universe generated by the cosmological β-model and treat them in the same way as the observational data. The results of the analysis show that for a pure fractal universe, with no homogeneity at large separations, the function g(r) does not display any large-scale flattening, even when boundary effects are corrected by the standard normalization to an ensemble of random points distributed within the sample volume. Analogously, no spurious scaling regime is generated. This demonstrates that the flattening of g(r) detected in the analysis of the CfA and PP surveys does really indicate a transition to homogeneity; consequently, a pure fractal universe is not consistent with the available observational data. We then use a multi-scale β-model to reproduce the shape of g(r) observed for the PP survey; an intermediate scaling regime with D_2_ ~ 2 in the approximate range 4-25 h^-1^ Mpc is shown to be necessary to reproduce the observational results. As confirmed by recent N-body experiments, the small-scale regime has to be interpreted in terms of a truly fractal nature of the galaxy distribution, being associated with the outcome of `fully developed' non-linear gravitational clustering. Conversely, we suggest that the intermediate regime has a topological origin, being possibly produced by nearly flat structures resulting from the weakly non-linear gravitational evolution of significant large-scale fluctuations in the initial spectrum.

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