Note on Adiabatic Modes and Ward Identities In A Closed Universe

TL;DR

This paper investigates the existence of cosmological consistency relations in a closed universe, demonstrating how flat universe relations emerge as a limit and analyzing adiabatic modes in a toy model with spherical topology.

Contribution

It extends the concept of adiabatic modes and consistency relations to a closed universe setting, providing a new perspective on their behavior without spatial infinity.

Findings

Adiabatic modes and consistency relations are found in a closed universe model.

Flat FRW universe relations are recovered in the large radius limit.

Dilation and translation modes are linked through global dilation on the sphere.

Abstract

As statements regarding the soft limit of cosmological correlation functions, consistency relations are known to exist in any flat FRW universe. In this letter we explore the possibility of finding such relations in a spatially closed universe, where the soft limit $\textbf{q}\rightarrow 0$ does not exist in any rigorous sense. Despite the absence of spatial infinity of the spatial slices, we find the adiabatic modes and their associated consistency relations in a toy universe with background topology $R\times S^2$. Flat FRW universe adiabatic modes are recovered via taking the large radius limit $R\gg \mathcal{H}^{-1}$, for which we are living in a small local patch of Hubble size on the sphere. It is shown that both dilation and translation adiabatic modes in the local patch are recovered by a global dilation on the sphere, acting at different places.