# Complexity in de Sitter Space

## TL;DR

This paper investigates holographic complexity in de Sitter space, comparing volume and action-based calculations, revealing differences in complexity growth and energy relationships in dual quantum field theories.

## Contribution

It provides a comparative analysis of complexity conjectures in de Sitter holography, highlighting discrepancies between volume and action approaches.

## Key findings

- Volume complexity results align with expectations but do not saturate growth bounds.
- Action complexity shows smaller energy states can have larger complexity than higher energy states.
- Differences between the two methods suggest nuanced behavior of complexity in de Sitter holography.

## Abstract

We consider the holographic complexity conjectures for de-Sitter invariant states in a quantum field theory on de Sitter space, dual to asymptotically anti-de Sitter geometries with de Sitter boundaries. The bulk holographic duals include solutions with or without a horizon. If we compute the complexity from the spatial volume, we find results consistent with general expectations, but the conjectured bound on the growth rate is not saturated. If we compute complexity from the action of the Wheeler-de Witt patch, we find qualitative differences from the volume calculation, with states of smaller energy having larger complexity than those of larger energy, even though the latter have bulk horizons.