# Constraints and Horizons for de Sitter with Extra Dimensions

## TL;DR

This paper derives a fundamental constraint on static extra-dimensional spacetimes to achieve 4D de Sitter expansion, showing it arises from the null Raychaudhuri equation and is independent of matter content, with implications for flux compactifications.

## Contribution

It establishes a null Raychaudhuri equation-based no-go theorem for de Sitter spacetimes with static extra dimensions, independent of matter and Einstein equations.

## Key findings

- Constraints require positive curvature, warping, or null energy violation.
- Known models like Freund-Rubin and Randall-Sundrum can evade the constraints.
- De Sitter spacetimes with static extra dimensions have apparent horizons and antitrapped regions.

## Abstract

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.