Compact Scalars at the Cosmological Collider

We study the dynamics of scalar fields with compact field spaces, or axions, in de Sitter space. We argue that the field space topology can qualitatively affect the physics of these fields beyond just which terms are allowed in their actions. We argue that the sharpest difference is for massless fields -- the free massless noncompact scalar field does not admit a two-point function that is both de Sitter-invariant and well-behaved at long distances, while the massless compact scalar does. As proof that this difference can be observable, we show that the long-distance behavior of a heavy scalar field, and thus its cosmological collider signal, can qualitatively change depending on whether it interacts with a light compact or noncompact scalar field. We find an interesting interplay between the circumference of the field space and the Hubble scale. When the field space is much larger than Hubble, the compact field behaves similarly to a light noncompact field and forces the heavy field to dilute much faster than any free field can. However, depending on how much smaller the field space is compared to Hubble, the compact field can cause the heavy scalar to decay either faster or slower than any free field and so we conclude that there can be qualitative and observable consequences of the field space's topology in inflationary correlation functions.