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

Abstract

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