Single-brane cosmological solutions with a stable compact extra dimension

We consider 5-dimensional cosmological solutions of a single brane. The correct cosmology on the brane, i.e., governed by the standard 4-dimensional Friedmann equation, and stable compactification of the extra dimension is guaranteed by the existence of a nonvanishing $T_5^5$ which is proportional to the 4-dimensional trace of the energy-momentum tensor. We show that this component of the energy-momentum tensor arises from the back reaction of the dilaton coupling to the brane. The same positive features are exhibited in solutions found in the presence of nonvanishing cosmological constants both on the brane $\left({\Lambda }_{\mathrm{br}}\right)$ and in the bulk $\left({\Lambda }_B\right).\mathrm{}$ Moreover, the restoration of the Friedmann equation, with the correct sign, takes place for both signs of ${\Lambda }_B$ so long as the sign of ${\Lambda }_{\mathrm{br}}$ is opposite ${\Lambda }_B$ in order to cancel the energy densities of the two cosmological constants. We further extend our single-brane thin-wall solution to allow a brane with finite thickness.