Stress-Energy Tensor of a Scalar Field on a Product Spacetime with a Time-Dependent Compact Dimension

We compute the vacuum expectation value of the stress-energy tensor of a scalar field on a product spacetime composed of an FLRW background times a compact dimension (${\mathcal{M}}^{1,d-1}\times {\mathcal{S}}^1$), where the size of the latter is allowed to vary with time. We modify the standard adiabatic regularization prescription to obtain analytic expressions for both $d=3$ and $d=4$. In the massless and conformally coupled limit, the leading order time-dependent results are consistent with known time-independent Casimir contributions. Furthermore, in this limit the higher-order time-dependent corrections, when the FLRW and compact-dimension scale factors coincide, match known results for ($1+d$)-dimensional FLRW spacetime.