Cosmological Expansion in the Randall-Sundrum Brane World Scenario

The cosmology of the Randall-Sundrum scenario for a positive tension brane in a 5-D Universe with localized gravity has been studied previously. In the radiation-dominated Universe, it was suggested that there are two solutions for the cosmic scale factor a(t): the standard solution $a\sim t^{1/2}$, and a solution $a\sim t^{1/4}$, which is incompatible with standard big bang nucleosynthesis. In this note, we reconsider expansion of the Universe in this scenario. We derive and solve a first order, linear differential equation for $H^2$, the square of the expansion rate of the Universe, as a function of $a$. The differences between our equation for $H^2$ and the relationship found in standard cosmology are (i) there is a term proportional to density squared, which is small when the density is small compared to the brane tension, and (ii) there is a contribution which acts like a relativistic fluid. We show that this second contribution is due to gravitational wave degrees of freedom in the bulk (presumably a massless 5-D graviton mode which is trapped on the brane). Thus, we find that there need not be any conflict between cosmology of the Randall-Sundrum scenario and the standard model of cosmology. As an illustration, we discuss how reheating at the end of inflation matches smoothly onto the correct Big Bang behavior, $a(t)\sim t^{1/2}$.