Field Equation of Correlation Function of Mass Density Fluctuation for Self-Gravitating Systems

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 principle the field equation and its solution of correlation function $\xi (r)$ of the mass density fluctuation itself. We apply thid to studies of the large-scale structure of the Universe within a small redshift range. The equation tells that $\xi (r)$ depends on the point mass $m$ and the Jeans wavelength scale ${\lambda }_0$, which are different for galaxies and clusters. It explains several longstanding, prominent features of the observed clustering : that the profile of ${\xi }_{cc}(r)$ of clusters is similar to ${\xi }_{gg}(r)$ 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 $r_0\simeq 0.4d$. Our solution $\xi (r)$ also yields the observed power-law correlation function of galaxies ${\xi }_{gg}(r)\simeq (r_0/r{)}^{1.7}$ valid only in a range $1<r<10h^{-1}$Mpc. At larger scales the solution $\xi (r)$ breaks below the power law and goes to zero around $\sim 50h^{-1}$Mpc, just as the observational data have demonstrated. With a set of fixed model parameters, the solutions ${\xi }_{gg}(r)$ for galaxies, the corresponding power spectrum, and ${\xi }_{cc}(r)$ for clusters, simultaneously, agree with the observational data from the major surveys of galaxies, and of clusters.

arXiv comment11 pages, 4 figures,. arXiv admin note: substantial text overlap with arXiv:1408.5237; text overlap with arXiv:2107.09425

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