Gravitational clustering of galaxies: Derivation of two-point galaxy correlation function using statistical mechanics of cosmological many-body problem

Abstract

We derive the spatial pair correlation function in gravitational clustering for extended structure of galaxies (e.g. galaxies with halos) by using statistical mechanics of cosmological many-body problem. Our results indicate that in the limit of point masses (\(\epsilon=0\)) the two-point correlation function varies as inverse square of relative separation of two galaxies. The effect of softening parameter ‘\(\epsilon\)’ on the pair correlation function is also studied and results indicate that two-point correlation function is affected by the softening parameter when the distance between galaxies is small. However, for larger distance between galaxies, the two-point correlation function is not affected at all. The correlation length \(r_{0}\) derived by our method depends on the random dispersion velocities \(\langle v^{2}\rangle^{1/2} \) and mean number density \(\bar{n}\), which is in agreement with N-body simulations and observations. Further, our results are applicable to the clusters of galaxies for their correlation functions and we apply our results to obtain the correlation length \(r_{0}\) for such systems which again agrees with the data of N-body simulations and observations.