The nonlinear equation of correlation function of galaxies in the expanding universe and the solution in linear approximation

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.