Redshift distortions of galaxy correlation functions

To examine how peculiar velocities can affect the 2-, 3-, and 4-point redshift correlation functions, we evaluate volume-average correlations for configurations that emphasize and minimize redshift distortions for four different volume-limited samples from each of the CfA, SSRS, and IRAS redshift catalogs. We find a characteristic distortion for the 2-point correlation, ${\text{xibar}}_2$: the slope $\gamma$ is flatter and the correlation length is larger in redshift space than in real space; that is, redshift distortions ``move’’ correlations from small to large scales. At the largest scales (up to $12 \Mpc$),theextrapowerintheredshiftdistributioniscompatiblewith$\Omega^{4/7}/b \approx 1$\mathrm{.}Weestimate$\Omega^{4/7}/b$tobe$0.53 \pm 0.15$,$1.10 \pm 0.16$and$0.84 \pm 0.45$ for the CfA, SSRS and IRAS catalogs. Higher order correlations ${\text{xibar}}_3$ and ${\text{xibar}}_4$ suffer similar redshift distortions, but in such a way that, within the accuracy of our analysis, the normalized amplitudes $S_3$ and $S_4$ are insensitive to this effect. The hierarchical amplitudes $S_3$ and $S_4$ are constant as a function of scale between 1–$12\text{Mpc}$ and have similar values in all samples and catalogues, $S_3\approx 2$ and $S_4\approx 6$, despite the fact that ${\text{xibar}}_2$, ${\text{xibar}}_3$, and ${\text{xibar}}_4$ differ from one sample to another by large factors (up to a factor of 4 in ${\text{xibar}}_2$, 8 for ${\text{xibar}}_3$, and 12 for ${\text{xibar}}_4$). The agreement between the independent estimations of $S_3$ and $S_4$