Vacuum fluctuations for a charged scalar field in de Sitter spacetime with compact dimensions

Vacuum expectation value (VEV) of the field squared is investigated for a charged scalar field in dS spacetime with toroidally compactified spatial dimensions. Along compact dimensions general quasiperiodicity conditions are imposed and, in addition, the presence of a classical constant gauge field is assumed. The latter leads to Aharonov-Bohm type effect on the VEVs of the physical observables. The VEV of the field squared is an even periodic function of the magnetic fluxes enclosed by compact dimensions. For small values of the comoving length of a compact dimension, compared with the dS curvature scale, which corresponds to the early stages of the cosmological expansion, the contribution of the gravity in the VEV is small, and the field squared is related to the corresponding result in the Minkowski spacetime by a conformal relation. For large values of comoving length, which corresponds to the late stages of the cosmological evolution, depending on the field mass, two regimes are realized with the monotonic and oscillatory damping of the expectation value.