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.