  We develop a possible cosmology for a Universe with 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. First, the radion becomes trapped at a
value much smaller than the size of the extra dimensions today. Second, the
Universe expands exponentially, but with a Planck mass smaller than its present
value. Because the Planck mass during inflation is small, density fluctuations
in agreement with observations can arise naturally. Third, 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 radion grows but the temperature
remains unchanged. Ultimately, the radion 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 of order the electroweak scale, and the
mature Universe evolves as in standard cosmological models. We show that the
present day energy density in radions can be smaller than the closure density
if the second inflationary epoch lasts about 8 e-foldings or more; the
present-day radion mass turns out to be small (less than or of the order of an
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.
