Cosmological expansion in the Randall-Sundrum brane world scenario

The cosmology of the Randall-Sundrum scenario for a positive tension brane in a 5D 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 $a\left(t\right):$ 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 paper, 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 (a fact already known), 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 degrees of freedom in the bulk. Thus, we find that there need not be any conflict between cosmology of the Randall-Sundrum scenario and the standard model of cosmology. We discuss how reheating at the end of inflation leads to the correct relationship between matter density and expansion rate, $H^2\to 8\pi G{\rho }_m/3,$ and the conditions that must be met for the expansion rate of the universe to be close to its standard model value around the epoch of cosmological nucleosynthesis.