An Enhanced Thermodynamic Framework for Third-Order Galaxy Correlation Functions: A Physically Motivated Closure and Observational Test

TL;DR

This work presents a thermodynamic, BBGKY-based closure for the galaxy three-point correlation function (3PCF), yielding an analytic solution for the equilateral configuration and linking the 3PCF amplitude $Q$ to the 2-point function slope $\gamma$. By enforcing a hierarchical ansatz and a quasi-equilibrium, isothermal galaxy fluid, the authors predict $\zeta_{eq}(r) = 3Q\xi(r)^2$ with $\xi(r)=(r_0/r)^\gamma$, and derive a physically constrained relation $Q = \mathcal{F}(\gamma, b_{therm})$. The model is validated against SDSS/BOSS CMASS data using the Szapudi-Szalay estimator and covariance from SLICS, achieving $\chi^2/\mathrm{dof} = 1.27$ and $Q \approx 0.52$ with $\gamma \approx 1.82$, $r_0 \approx 5.6\,h^{-1}{\rm Mpc}$, demonstrating consistency with 2PCF measurements. The framework also provides a Redshift-Space Distortion (RSD) treatment via a Kaiser boost and a thermodynamically motivated Fingers-of-God damping, linking the fitted velocity dispersion $\sigma_v$ to the temperature $T$ through $\sigma_v^2 = k_B T/m_{\rm eff}$, and highlighting the framework as a physically predictive, complementary description of higher-order clustering on quasi-linear scales.

Abstract

The three-point correlation function (3PCF) is a crucial probe of non-Gaussianity and nonlinear structure formation. We develop a thermodynamic framework for the galaxy 3PCF by closing the BBGKY hierarchy with a physically motivated hierarchical ansatz, yielding a separable, analytic solution for the equilateral 3PCF. Our framework addresses the apparent discrepancy between the perturbation theory prediction for dark matter ($Q_{dm} \approx 1.6$) and observed galaxy measurements ($Q_{gal} \approx 0.5$) by incorporating thermodynamic virial effects and velocity dispersion. We validate this model with SDSS/BOSS CMASS measurements, obtaining an excellent fit ($χ^2/\mathrm{dof} = 1.27$) across $1$-$50,h^{-1}\mathrm{Mpc}$. The analysis utilizes the Szapudi-Szalay estimator with robust covariance estimation from the SLICS simulation suite. By linking the thermodynamic temperature $T$ to the small-scale velocity dispersion (Fingers-of-God), we establish the thermodynamic approach as a predictive, complementary description of higher-order galaxy clustering on quasi-linear scales.

Figures (2)

• Figure 1: The equilateral three-point correlation function measured from the BOSS CMASS galaxy sampleslepian2017 (black points with jackknife errors) . Solid curve: best-fit thermodynamic model including linear Kaiser boost and Gaussian FoG; points: BOSS CMASS equilateral 3PCF with jackknife errors; fit range $1$--$50\,h^{-1}\,\mathrm{Mpc}$; $\chi^2/\mathrm{dof}=1.27$. The solid red line shows the best-fit of our thermodynamic model, which includes redshift-space distortions, performed over the scale range $1 < r < 50~h^{-1}\text{Mpc}$. The model provides an excellent description of the data across a wide range of physical scales.
• Figure 2: Configuration dependence of the redshift-space hierarchical amplitude $Q^s(\mu)$ in our thermodynamic RSD model. We show the dependence on $\mu$, the cosine of the angle between the triangle plane and the line of sight. The curve is obtained by multiplying the real-space hierarchical amplitude $Q_{\rm real} \simeq 0.52$, measured from the equilateral 3PCF fit in Section \ref{['sec:result1']}, by the Kaiser boost factor $S_{\rm RSD}(\beta)$ for $\beta = f/b_1$ (Appendix \ref{['sec:appendix_rsd']}) and by the Lorentzian Fingers-of-God damping factor $F_{\rm FoG}(\mu,\sigma_v)$. The line-of-sight velocity dispersion $\sigma_v$ entering $F_{\rm FoG}$ is set by the thermodynamic temperature via $\sigma_v^2 = k_B T / m_{\rm eff}$ (Eq. \ref{['eq:temp']}), so the anisotropy of $Q^s(\mu)$ directly encodes the effective kinetic temperature of the galaxy fluid.