On open superstring partition function in inhomogeneous rolling tachyon background

We consider open superstring partition function Z on the disc in time-dependent tachyon background T = f(xi)e μx0 where the profile function f depends on spatial coordinates. We compute Z to second order in derivatives of f and compare the result with some previously suggested effective actions depending only on first derivatives of the tachyon field. We also compute the target-space stress-energy tensor in this background and demonstrate its conservation for the linear profile f = f0 + qix i corresponding to a marginal perturbation. We comment on the role of the rolling tachyon with the linear spatial profile in the decay of an unstable D-brane. Also at Lebedev Physics Institute, Moscow. 1