Analytic solutions for coupled linear perturbations

Analytic solutions for the evolution of cosmological linear density perturbations in the baryonic gas and collisionless dark matter are derived. The solutions are expressed in a closed form in terms of elementary functions, for arbitrary baryonic mass fraction. They are obtained assuming $\mathrm{\Omega }=1$ and a time independent comoving Jeans wavenumber, $k_J$. By working with a time variable $\tau \equiv \ln (t^{2\mathrm{/}3})$, the evolution of the perturbations is described by linear differential equations with constant coefficients. The new equations are then solved by means of Laplace transformation assuming that the gas and dark matter trace the same density field before a sudden heating epoch. In a dark matter dominated Universe, the ratio of baryonic to dark matter density perturbation decays with time roughly like $\exp(-5\tau/4)\propto t^{-5/6}$tothelimitingvalue$1/[1+(k/k_J)^2]$\mathrm{.}Forwavenumbers$k>k_J/\sqrt{24}$,thedecayisaccompaniedwithoscillationsofaperiod$ 8\pi/\sqrt{24 (k/k_J)^2 -1}$in$\tau$\mathrm{.}Incomparison,as$\tau $ increases in a baryonic matter dominated Universe, the ratio approaches $1-(k\mathrm{/}{k}_J{)}^2$ for $k\le k_J$, and zero otherwise.