Research Note

Field equation of the correlation function of mass-density fluctuations for self-gravitating systems

Department of Astronomy, Key Laboratory for Researches in Galaxies and Cosmology, University of Science and Technology of China, Hefei, Anhui 230026, PR China
e-mail: yzh@ustc.edu.cn; cqpb@mail.ustc.edu.cn

Received: 28 November 2014
Accepted: 1 July 2015

Abstract

We study the mass-density distribution of Newtonian self-gravitating systems. Modeling the system as a fluid in hydrostatical equilibrium, we obtain from first principles the field equation and its solution of the correlation function ξ(r) of the mass-density fluctuation itself. We apply this to studies of the large-scale structure of the Universe within a small redshift range. The equation shows that ξ(r) depends on the point mass m and the Jeans wavelength scale λ₀, which are different for galaxies and clusters. It explains several long-standing prominent features of the observed clustering: that the profile of ξ_cc(r) of clusters is similar to ξ_gg(r) of galaxies, but with a higher amplitude and a longer correlation length, and that the correlation length increases with the mean separation between clusters as a universal scaling r₀ ≃ 0.4d. Our solution ξ(r) also shows that the observed power-law correlation function of galaxies ξ_gg(r) ≃ (r₀/r)^1.7 is only valid in a range 1 <r< 10h^-1 Mpc. At larger scales the solution ξ(r) breaks below the power law and goes to zero around ~50 h^-1 Mpc, just as observational data have demonstrated. With a set of fixed model parameters, the solutions ξ_gg(r) for galaxies, the corresponding power spectrum, and ξ_cc(r) for clusters, simultaneously agree with the observational data from the major surveys of galaxies and clusters.