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We present an analytic study of the density fluctuation of a Newtonianself-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 theorytechniques, we obtain the nonlinear, hyperbolic equation of 2-pt correlationfunction $\xi$ of perturbation. Under the Zel'dolvich approximation theequation becomes an integro-differential equation and contains also the 3-ptand 4-pt correlation functions. By adopting the Groth-Peebles and Fry-Peeblesansatz, the equation becomes closed, contains a pressure term and a deltasource term which were neglected in Davis and Peebles' milestone work. Theequation 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 linearequation 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 therelevant imprint from the spectrum of baryon acoustic oscillations at thedecoupling. The solution $\xi({\bf r}, z)$ is growing during expansion, andcontains $100$Mpc periodic bumps at large scales, and a main mountain (a globalmaximum with $\xi \propto r^{-1}$) at small scales $r\lesssim 50$Mpc. Theprofile 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 correlationscale, also modified by $\gamma$ and $c_s$. The main mountain is largelygenerated by the source $\propto m$ as the clustering scale. Since the outcomeis affected by the initial condition and the parameters as well, it is hard toinfer the imprint of baryon acoustic oscillations accurately. The difficultieswith the sound horizon as a distance ruler are pointed out.

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