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We consider a deformation of the $AdS_5\times S^5$ solution of IIBsupergravity obtained by taking the boundary value of the dilaton to be timedependent. The time dependence is taken to be slowly varying on the AdS scalethereby introducing a small parameter $\epsilon$. The boundary dilaton has aprofile which asymptotes to a constant in the far past and future and attains aminimum value at intermediate times. We construct the sugra solution to firstnon-trivial order in $\epsilon$, and find that it is smooth, horizon free, andasymptotically $AdS_5\times S^5$ in the far future. When the intermediatevalues of the dilaton becomes small enough the curvature becomes of order thestring scale and the sugra approximation breaks down. The resulting dynamics isanalysed in the dual SU(N) gauge theory on $S^3$ with a time dependent couplingconstant which varies slowly. When $N \epsilon \ll 1$, we find that a quantumadiabatic approximation is applicable, and use it to argue that at late timesthe geometry becomes smooth $AdS_5\times S^5$ again. When $N \epsilon \gg 1$,we formulate a classical adiabatic perturbation theory based on coherent stateswhich arises in the large $N$ limit. For large values of the 'tHooft couplingthis reproduces the supergravity results. For small 'tHooft coupling thecoherent state calculations become involved and we cannot reach a definiteconclusion. We argue that the final state should have a dual description whichis mostly smooth $AdS_5$ space with the possible presence of a small blackhole.

Slowly Varying Dilaton Cosmologies and their Field Theory Duals