# Global Spacetime Structure of Compactified Inflationary Universe

## TL;DR

This paper analyzes the global structure of a compactified inflationary universe modeled as a torus de Sitter space, revealing issues like closed causal curves and quasi-regular singularities in its extensions.

## Contribution

It provides an explicit extension of torus de Sitter space and identifies inherent causal and singularity problems in such compactified inflationary models.

## Key findings

- Past incomplete null geodesics can be locally extended.
- Extensions introduce closed causal curves.
- Extended spacetime exhibits quasi-regular singularities.

## Abstract

We investigate the global spacetime structure of torus de Sitter universe, which is exact de Sitter space with torus identification based on the flat chart. We show that past incomplete null geodesics in torus de Sitter universe are locally extendible. Then we give an extension of torus de Sitter universe so that at least one of the past incomplete null geodesics in the original spacetime becomes complete. However we find that extended torus de Sitter universe has two ill behaviors. The first one is a closed causal curve. The second one is so called quasi regular singularity, which means that there is no global, consistent extension of spacetime where all curves become complete, nevertheless each curve is locally extensible.