Field Theory of the Correlation Function of Mass Density Fluctuations for Self-Gravitating Systems

The mass density distribution of Newtonian self-gravitating systems is studied analytically in field theoretical method. Modeling the system as a fluid in hydrostatical equilibrium, we apply Schwinger's functional derivative on the average of the field equation of mass density, and obtain the field equation of 2-point correlation function $\xi(r)$ of the mass density fluctuation, which includes the next order of nonlinearity beyond the Gaussian approximation. The 3-point correlation occurs hierarchically in the equation, and is cut off by the Groth-Peebles anzats, making it closed. We perform renormalization, and write the equation with three nonlinear coefficients. The equation tells that $\xi$ depends on the point mass $m$ and the Jeans wavelength scale $\lambda_{0}$, which are different for galaxies and clusters. Applying to large scale structure, it predicts that the profile of $\xi_{cc}$ of clusters is similar to $\xi_{gg}$ of galaxies but with a higher amplitude, and that the correlation length increases with the mean separation between clusters, i.e, a scaling behavior $r_0\simeq 0.4d$. The solution yields the galaxy correlation $\xi_{gg}(r) \simeq (r_0/r)^{1.7}$ valid only in a range $1<r<10 \,h^{-1}$Mpc. At larger scales the solution $\xi_{gg}$ deviates below the power law and goes to zero around $\sim 50 \, h^{-1}$Mpc, just as the observations show. We also derive the field equation of 3-point correlation function in Gaussian approximation and its analytical solution, for which the Groth-Peebles ansatz with $Q= 1$ holds.