The nonlinear field equation of the three-point correlation function of galaxies

TL;DR

This paper derives an analytical nonlinear equation for the three-point correlation function of galaxies, incorporating second-order density fluctuations, and compares the non-Gaussian solution with observational data from SDSS.

Contribution

It presents a novel analytical derivation of the nonlinear 3-point correlation function equation using field theory and specific ansatzes, extending beyond Gaussian approximations.

Findings

Solution $ ilde{ ilde heta$ shows a shallow U-shape along $ heta$ and decreases along $r$.

The non-Gaussian reduced $Q$ deviates from 1, showing a deeper U-shape and weak $r$-dependence.

Results agree with SDSS observational data, indicating non-Gaussian features in galaxy correlations.

Abstract

Based on the field theory of density fluctuation under Newtonian gravity, we obtain analytically the nonlinear equation of 3-pt correlation function $\zeta$ of galaxies in a homogeneous, isotropic, static universe. The density fluctuations have been kept up to second order. By the Fry-Peebles ansatz and the Groth-Peebles ansatz, the equation of $\zeta$ becomes closed and differs from the Gaussian approximate equation. Using the boundary condition inferred from the data of SDSS, we obtain the solution $\zeta(r, u, \theta)$ at fixed $u=2$, which exhibits a shallow $U$-shape along the angle $\theta$ and, nevertheless, decreases monotonously along the radial $r$. We show its difference with the Gaussian solution. As a direct criterion of non-Gaussianity, the reduced $Q(r, u, \theta)$ deviates from the Gaussianity plane $Q=1$, exhibits a deeper $U$-shape along $\theta$ and varies weakly along $r$, agreeing with the observed data.