[Paper Review] The nonlinear equation of correlation function of galaxies in the expanding universe and the solution in linear approximation

Yang Zhang, Bichu Li|arXiv (Cornell University)|Nov 3, 2021

Cosmology and Gravitation Theories80 references6 citations

TL;DR

This paper derives a nonlinear, hyperbolic integro-differential equation for the two-point correlation function ξ(r, t) of galaxies in an expanding universe with ΩΛ + Ωm = 1, using field theory and fluid dynamics. In linear approximation, it solves for ξ(r, z), revealing a growing main mountain (ξ ∝ r⁻¹) at small scales and periodic 100 Mpc bumps at large scales linked to the Jeans length λJ, with the bumps and wiggles in the power spectrum arising from initial conditions and acoustic oscillations, not just the sound horizon.

ABSTRACT

We present an analytic study of the density fluctuation of a Newtonian self-gravity fluid in the expanding universe with $\Omega_\Lambda+\Omega_m=1$, which extends our previous work in the static case. By use of field theory techniques, we obtain the nonlinear, hyperbolic equation of 2-pt correlation function $\xi$ of perturbation. Under the Zel'dolvich approximation the equation becomes an integro-differential equation and contains also the 3-pt and 4-pt correlation functions. By adopting the Groth-Peebles and Fry-Peebles ansatz, the equation becomes closed, contains a pressure term and a delta source term which were neglected in Davis and Peebles' milestone work. The equation has three parameters of fluid: the particle mass $m$ in the source, the overdensity $\gamma$, and the sound speed $c_s$. We solve only the linear equation and apply to the system of galaxies. We assume two models of $c_s$ and, take an initial power spectrum at a redshift $z=7$, which inherits the relevant imprint from the spectrum of baryon acoustic oscillations at the decoupling. The solution $\xi({\bf r}, z)$ is growing during expansion, and contains $100$Mpc periodic bumps at large scales, and a main mountain (a global maximum with $\xi \propto r^{-1}$) at small scales $r\lesssim 50$Mpc. The profile of $\xi$ agrees with the observed ones from galaxy and quasar surveys. The bump separation is given by the Jeans length $\lambda_J$ as the correlation scale, also modified by $\gamma$ and $c_s$. The main mountain is largely generated by the source $\propto m$ as the clustering scale. Since the outcome is affected by the initial condition and the parameters as well, it is hard to infer the imprint of baryon acoustic oscillations accurately. The difficulties with the sound horizon as a distance ruler are pointed out.

Motivation & Objective

• To derive a nonlinear field equation for the two-point correlation function ξ(r, t) of density perturbations in an expanding Newtonian self-gravitating fluid.
• To extend previous static solutions to the dynamic, expanding universe context with ΩΛ + Ωm = 1.
• To incorporate pressure and source terms neglected in Davis and Peebles' work, using Groth-Peebles and Fry-Peebles ansatz to close the equation.
• To solve the linearized version of the equation and apply it to galaxy systems, distinguishing local clustering from large-scale structure.
• To analyze the role of initial conditions, sound speed models, and parameters in shaping ξ and the power spectrum Pk.

Proposed method

• Derives a nonlinear, hyperbolic, integro-differential equation for ξ(r, t) using functional derivative methods applied to hydrodynamic equations of self-gravitating fluid.
• Applies the Zel'dovich approximation to reduce the equation to a form involving three- and four-point correlation functions.
• Closes the system using the Groth-Peebles and Fry-Peebles ansatz, introducing a pressure term and delta-function source term.
• Solves the linearized version of the equation under two models of sound speed cs, with initial power spectrum at z = 7 inherited from baryon acoustic oscillations (BAO) after Silk damping.
• Uses Green's function decomposition to separate the solution into homogeneous (initial condition-driven) and inhomogeneous (source-driven) parts.
• Performs Fourier transformation to obtain the power spectrum Pk and analyzes wiggles and peaks in relation to acoustic oscillations and λJ.

Experimental results

Research questions

• RQ1 How does the correlation function ξ(r, t) evolve in the expanding universe under nonlinear gravity and fluid dynamics?
• RQ2 What is the origin of the 100 Mpc periodic bumps in the observed galaxy correlation function?
• RQ3 How do initial conditions and sound speed models affect the growth and structure of ξ and Pk?
• RQ4 To what extent can the sound horizon be used as a standard ruler for cosmology, given the observed features in ξ and Pk?
• RQ5 What distinguishes the Jeans length λJ from the mass scale of galaxies or clusters in the context of large-scale structure?

Key findings

• The linear solution ξ(r, z) exhibits a main mountain with ξ ∝ r⁻¹ at small scales (r ≲ 50 Mpc), attributed to the inhomogeneous solution driven by the source term.
• Periodic bumps with separation ∼100 Mpc appear at large scales, identified as the Jeans length λJ, modified by overdensity γ and sound speed cs.
• The power spectrum Pk contains a main peak at k ∼ 2π/λJ corresponding to the 100 Mpc bumps, and multiwiggles at high k due to evolving acoustic oscillations.
• The wiggles in Pk develop during evolution even if the initial spectrum lacks them, indicating that the initial power spectrum imprints BAO features into the system at z = 7.
• The observed 100 Mpc bump separation is better explained by λJ than by the sound horizon, and the sound horizon is not a reliable distance ruler due to statistical washout and model dependence.
• The decomposition ξ = ξ₁ + ξ₂ shows that λJ governs large-scale structure (ξ₂), while galaxy mass m governs local clustering (ξ₁), clarifying that r₀ in the standard ξ = (r/r₀)⁻¹.⁷ is a phenomenological parameter, not a physical correlation length.