The Void Probability functionand related statistics.ppt

The Void Probability functionand related statistics Sophie Maurogordato CNRS, Observatoire de la Cote d’Azur, France The Void probability function Count probability PN(V): probability of finding N galaxies in a randomly chosen volume of size V N= 0: Void Probability Function P0(V) Related to the hierarchy of n-point reduced correlation functions (White 1979) Why the VPF ? Statistical way to quantify the frequency of voids of a given size. Complementary information on high-order correlations that low-order correlations do not contain: strongly motivated by the existence of large-scale clustering patterns (walls, voids filaments). Straightforward calculated. But density dependent, denser samples have smaller voids: be careful when comparing samples with different densities. Scaling properties for correlation functions Observational evidence for low orders: n=3 (Groth Peebles, 1977, Fry Peebles 1978, Sharp et al 1984) n=4 (Fry Peebles 1978) Hierarchical models Generalisation for the reduced N-point correlation xN : a :tree shape L(a ) labellings of a given tree (Fry 1984, Schaeffer 1984, Balian and Schaeffer 1989) Scaling invariance expected for the correlation functions of matter In the linear- and mildly non linear regime: Evolution under gravitational instability of initial gaussian fluctuation; can be followed by perturbation theory predictions for SN’s (Peebles 1980, Jusckiewicz, Bouchet Colombi 1993, Bernardeau 1994, Bernardeau 2002) SN independant on W, L and z ! In the strongly non-linear regime: solution of the BBGKY equations Scaling of the VPF under the hierarchical ??ansatz?? The reduced VPF writes: VPF from galaxy surveys Zwicky catalog: Sharp 1981 CfA: Maurogordato Lachièze-Rey 1987 Pisces-Perseus: Fry et al. 1989 CfA2: Vogeley et al. 1991, Vogeley et al. 1994 SSRS: Maurogordato et al.1992, Lachièze-Rey et al. 1992 Huchra’s compilation: Einasto et al. 1991 Q