Fermionic Casimir effect in toroidally compactified de Sitter spacetime

We investigate the fermionic condensate and the vacuum expectation values of the energy-momentum tensor for a massive spinor field in de Sitter spacetime with spatial topology $\mathrm{R}^{p}\times (\mathrm{S}^{1})^{q}$. Both cases of periodicity and antiperiodicity conditions along the compactified dimensions are considered. By using the Abel-Plana formula, the topological parts are explicitly extracted from the vacuum expectation values. In this way the renormalization is reduced to the renormalization procedure in uncompactified de Sitter spacetime. It is shown that in the uncompactified subspace the equation of state for the topological part of the energy-momentum tensor is of the cosmological constant type. Asymptotic behavior of the topological parts in the expectation values is investigated in the early and late stages of the cosmological expansion. In the limit when the comoving length of a compactified dimension is much smaller than the de Sitter curvature radius the topological part in the expectation value of the energy-momentum tensor coincides with the corresponding quantity for a massless field and is conformally related to the corresponding flat spacetime result. In this limit the topological part dominates the uncompactified de Sitter part. In the opposite limit, for a massive field the asymptotic behavior of the topological parts is damping oscillatory for both fermionic condensate and the energy-momentum tensor.