  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 $\Omega=1$ and
a time independent comoving Jeans wavenumber, $k_J$. By working with a time
variable $\tau\equiv \ln(t^{2/3})$, 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 $\exp(-5\tau/4)\propto
t^{-5/6}$ to the limiting value $1/[1+(k/k_J)^2]$. For wavenumbers
$k>k_J/\sqrt{24}$, the decay is accompanied with oscillations of a period $
8\pi/\sqrt{24 (k/k_J)^2 -1}$ in $\tau$. In comparison, as $\tau $ increases in
a baryonic matter dominated Universe, the ratio approaches $1-(k/k_J)^2$ for
$k\le k_J$, and zero otherwise.

Astrophysics