Constraints and horizons for de Sitter with extra dimensions

In order for spacetimes with static extra dimensions to have four-dimensional de Sitter expansion they must have at least positive curvature, warping sourced by the four-dimensional expansion, or violate the null energy condition everywhere in the extra dimensions. We show how this constraint arises from the null Raychaudhuri equation, and that it is independent of the matter content, the Einstein equations, and is true point by point in the extra dimensions (not integrated), setting it apart from other no-go theorems in the literature. We present two previously known examples-a Freund-Rubin compactification with bulk cosmological constant, and a Randall-Sundrum model where the mismatch between the brane and bulk cosmological constants sources nontrivial warping-which evade the constraint and discuss the implications for flux compactifications. We also show that any spacetime with static compact extra dimensions and four-dimensional de Sitter expansion has an apparent horizon and antitrapped region in the extra dimensions, which may have interesting implications for the dynamics of bulk fields in cosmology.