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.
