Adiabatic regularization method in inhomogeneous spacetime

It is well known that the adiabatic regularization method is applicable to evaluate numerically the renormalized stress-energy tensor for matter in homogeneous spacetime. In this paper we will extend this method to the more general spacetimes with a small inhomogeneity which is invariant under spatial reflection. The difficulty of mode-mixing behavior arising from the inhomogeneity of spacetime is overcome and a possible way of obtaining the WKB solution of the time-dependent mode function under the early-time approximation is presented. Our prescriptions are useful to study the back reaction of a quantized field in the early universe with a small inhomogeneity.