Understand
In order for spacetimes with static extra dimensions to have 4-dimensional de Sitter expansion they must have at least positive curvature, warping sourced by the 4-d 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 non-trivial warping -- which evade the constraint, and discuss the implications for flux compactifications.
- We also show that any spacetime with static compact extra dimensions and 4-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.