Solvable time-dependent model and rolling closed string tachyon

We investigate the SL(2,R)/U(1) WZW model with level 0<k<2 as a solvable time-dependent background in string theory. This model is expected to be dual to the one describing a rolling closed string tachyon with a timelike linear dilaton. We examine its exact metric and minisuperspace wave functions. Two point functions and the one-loop vacuum amplitude are computed and their relation to the closed string emission is discussed. Comparing with the results from the minisuperspace approximation, we find a physical interpretation of our choice to continue the Euclidean model into the Lorentzian one. Three point functions are also examined.