$$\label{integral2}
\int^\infty\rho\sigma dt=\infty,$$ then the probability is unity. For constant $\sigma$, Eq.(\[integral2\]) has the limiting case $\rho\propto 1/t$ or $R\propto t^{1/3}$, where $R=R(t)$ denotes the scale factor in homogeneous isotropic cosmological solutions of Einstein’s field equations[@Peebles]. For steady-state cosmology both $\rho$ and $\sigma$ are constant, so Eq.(\[integral2\]) is satisfied.

The mean cross section for photon absorption by an electron is $$\sigma=\frac{A\rho}{T_i^{1/2}\omega^3}[1-\exp(-\omega/kT_i)],$$ where $T_i$ is the ion temperature, $\rho$ is the total heavy particle density and $A$ is a numerical factor. We now use that $\rho, T_i$ and $\omega$ are proportional to $1/R^3,
t^{2/3}/R^{8/3}$ and $1/R$, respectively, which yields $\sigma\propto R^{4/3}/t^{1/3}$. The density of electrons falls off as $1/R^3$, so complete absorption results in $$\int^{\infty}R^{-3}t^{-1/3}R^{4/3}dt=\infty$$ giving the limiting case $R\propto t^{2/5}$

Thus, the general conclusion is that all expanding matter conserving cosmological models do not permit complete (ideal) absorption along the future light cone. However, they lead to complete absorption along the past light cone due to the big-bang singularity and the increasing density of matter and radiation as the singularity is approached. For recontracting models which evolve to high density models in the future as the universe recollapses to a singularity, complete absorption can occur, leading to consistency with a 