Cosmology of the brane world

We develop a possible cosmology for a universe in which there are n additional spatial dimensions of variable scale and an associated scalar field, the radion, which is distinct from the field responsible for inflation, the inflaton. Based on a particular ansatz for the effective potential for the inflaton and radion (which may emerge in string theory), we show that the early expansion of the universe may proceed in three stages. During the earliest phase, the radion field becomes trapped at a value much smaller than the size of the extra dimensions today. Following this phase, the universe expands exponentially, but with a Planck mass smaller than its present value. Because the Planck mass during inflation is small, we find that density fluctuations, in agreement with observations, can arise naturally. When inflation ends, the universe reheats, and the radion becomes free to expand once more. During the third phase the universe is “radiation dominated” and tends toward a fixed-point evolutionary model in which the radius of the extra dimension grows, but the temperature remains unchanged. Ultimately, the radius of the extra dimensions becomes trapped once again at its present value, and a short period of exponential expansion, which we identify with the electroweak phase transition, ensues. Once this epoch is over, the universe reheats to a temperature $\lesssim m_{\mathrm{EW}},$ the electroweak scale, and the mature universe evolves according to standard cosmological models. We show that the present day energy density in radions can be smaller than the closure density of the universe if the second inflationary epoch lasts $\sim 8\hspace{0.5em}e$-foldings or more; the present-day radion mass turns out to be small ${(m}_{\mathrm{radion}}\lesssim$ eV, depending on parameters). We argue that although our model envisages considerable time evolution in the Planck mass, substantial spatial fluctuations in Newton’s constant are not produced.