Analytical Equation of Three-point Correlation Function of Galaxies: to Third Order of Density Perturbation

TL;DR

This paper derives a third-order nonlinear equation for the three-point galaxy correlation function using density perturbation theory, and shows it aligns well with observational data, improving understanding of galaxy clustering.

Contribution

It presents the first derivation of a third-order correlation function equation for galaxies, incorporating nonlinear effects and renormalization, with solutions matching SDSS observations.

Findings

Third order correlation function $zeta$ is positive and U-shaped in angle.

The reduced $Q$ deviates from Gaussian predictions, showing a deeper U-shape.

The third order solution aligns closely with SDSS galaxy data, especially at large scales.

Abstract

Applying functional differentiation to the density field with Newtonian gravity, we obtain the static, nonlinear equation of the three-point correlation function $\zeta$ of galaxies, to the third order density perturbations. We make the equation closed and perform renormalization of the mass and the Jeans wavenumber. Using the boundary condition inferred from observations, we obtain the third order solution $\zeta(r, u, \theta)$ at fixed $u=2$, which is positive, exhibits a $U$-shape along the angle $\theta$, and decreases monotonously along the radial $r$ up to the range $r \leq 30\, h^{-1}$Mpc in our computation. The corresponding reduced $Q(r, u, \theta)$ deviates from 1 of the Gaussian case, has a deeper $U$-shape along $\theta$, and varies non-monotonously along $r$. The third order solution agrees with the SDSS data of galaxies, quite close to the previous second order solution, especially at large scales. This indicates that the equations of correlation functions with increasing orders of density perturbation provide a stable description of the nonlinear galaxy system.