Nonlinear equation of correlation function of galaxies in an expanding universe and the solution in linear approximation

We present an analytic study of the density fluctuation a Newtonian self-gravity fluid in expanding universe with ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\Lambda}}}+{\mathrm{\ensuremath{\Omega}}}_{m}=1$, which extends our previous work static case. By use field theory techniques, we obtain nonlinear, hyperbolic equation two-point correlation function $\ensuremath{\xi}$ perturbation. Under Zel'dolvich approximation becomes integro-differential and contains also three-point four-point functions. adopting Groth-Peebles Fry-Peebles ansatz, closed, pressure term delta source were neglected Davis Peebles' milestone work. The has three parameters fluid; particle mass $m$ source, overdensity $\ensuremath{\gamma}$, sound speed ${c}_{s}$. solve only linear apply it to system galaxies. assume two models ${c}_{s}$, take initial power spectrum at redshift $z=7$, inherits relevant imprint from baryon acoustic oscillations decoupling. solution $\ensuremath{\xi}(\mathbf{r},z)$ is growing during expansion, 100 Mpc periodic bumps large scales, main mountain (a global maximum $\ensuremath{\xi}\ensuremath{\propto}{r}^{\ensuremath{-}1}$) small scales $r\ensuremath{\lesssim}50\text{ }\text{ }\mathrm{Mpc}$. profile agrees observed ones galaxy quasar surveys. bump separation given by Jeans length ${\ensuremath{\lambda}}_{J}$, modified $\ensuremath{\gamma}$ Using decomposition find that largely generated inhomogeneous come homogeneous spectrum. ${\ensuremath{\lambda}}_{J}$ identified as scale galaxies, distinguished clustering determined $m$. corresponding peak located around $k\ensuremath{\sim}\frac{2\ensuremath{\pi}}{{\ensuremath{\lambda}}_{J}}$ associated $\ensuremath{\xi}$, multiwiggles high $k>{k}_{J}$ are developing evolution even if no wiggles. Since outcome affected condition well, hard infer accurately. difficulties horizon distance ruler pointed out.