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

TL;DR

This paper derives a field equation for the correlation function of mass density fluctuations in self-gravitating systems, explaining observed large-scale structure features of the universe through a theoretical model.

Contribution

It introduces a first-principles field equation for the correlation function of mass density fluctuations in self-gravitating systems, linking theory with observed cosmic clustering.

Findings

The correlation function $\xi(r)$ depends on point mass and Jeans wavelength.

The model explains the similarity and differences in galaxy and cluster correlations.

Predicted correlation functions match observational data across scales.

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 $1<r<10 h^{-1}$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.