Note on Adiabatic Modes and Ward Identities In A Closed Universe
  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.
