Symmetries of the primordial Universe and cosmological correlation functions

Cosmological correlation functions are an exciting bridge between fundamental physics and observation. Using the Cosmic Microwave Background and large-scale structure of the Universe, we can infer information about the Universe at its earliest moments, including both its symmetries and the possible presence of heavy particles at energies far higher than we can hope to probe in collider experiments. I will discuss recent developments in our understanding of these connections, focusing on two areas: model-independent tests of the inflationary paradigm using cosmological soft theorems, and the signatures of massive spin-2 fields in correlation functions of the graviton. I will also briefly discuss the future, including the idea that systematic understanding of correlation functions in cosmological spacetimes can help us understand the emergence of spacetime itself.