Slowly Varying Dilaton Cosmologies and their Field Theory Duals

We consider a deformation of the $AdS_5\times S^5$ solution of IIB supergravity obtained by taking the boundary value of the dilaton to be time dependent. The time dependence is taken to be slowly varying on the AdS scale thereby introducing a small parameter $\epsilon$. The boundary dilaton has a profile which asymptotes to a constant in the far past and future and attains a minimum value at intermediate times. We construct the sugra solution to first non-trivial order in $\epsilon$, and find that it is smooth, horizon free, and asymptotically $AdS_5\times S^5$ in the far future. When the intermediate values of the dilaton becomes small enough the curvature becomes of order the string scale and the sugra approximation breaks down. The resulting dynamics is analysed in the dual SU(N) gauge theory on $S^3$ with a time dependent coupling constant which varies slowly. When $N \epsilon \ll 1$, we find that a quantum adiabatic approximation is applicable, and use it to argue that at late times the geometry becomes smooth $AdS_5\times S^5$ again. When $N \epsilon \gg 1$, we formulate a classical adiabatic perturbation theory based on coherent states which arises in the large $N$ limit. For large values of the 'tHooft coupling this reproduces the supergravity results. For small 'tHooft coupling the coherent state calculations become involved and we cannot reach a definite conclusion. We argue that the final state should have a dual description which is mostly smooth $AdS_5$ space with the possible presence of a small black hole.