Quantum Field Theory (QFT) is the universal language of modern physics, at the foundation of our formal understanding of particle physics, condensed matter and, in particular, cosmology. Cosmological observations are compatible with a period of slow-roll inflation preceding the classic hot Big Bang picture, in which the geometry of the universe was approximately that of a de Sitter spacetime. Developing the tools to study QFT on a de Sitter background is thus essential for understanding the details of the origin of structure in our universe.

In this thesis we present novel results that advance our understanding of QFT in de Sitter, from unitarity conditions on correlation functions, to constraints on renormalization group flows, and surprising properties of photons.

Chapter 1 is an introduction and a review of some aspects of QFT in de Sitter.

In chapter 2 we derive a K"allén-Lehmann-type decomposition for two-point functions of symmetric traceless tensors in de Sitter.

In chapter 3 we study renormalization group flows in two dimensional de Sitter spacetime and we prove the existence of two $c$-functions that interpolate between the UV and IR central charges of any quantum field theory. We connect one of these $c$-functions to a special representation of the de Sitter group that appears in the spectral decomposition of the stress tensor.

In chapter 4 we study some properties of photons in de Sitter. We show the surprising fact that states with the quantum numbers of a photon are created by generic massive operators acting on the vacuum. Moreover, we show that, among spin 1 excitations, photons do not dominate the IR of QFT in de Sitter. Finally, we prove a nonperturbative bound on the late-time scaling of the two-point functions of electromagnetic fields, which may be of interest in the context of primordial magnetogenesis.

In chapter 5 we conclude and discuss possible future directions of research.

Altogether this thesis argues that a representation-theoretic approach to QFT in de Sitter reveals powerful constraints on observables, as well as new surprising physical principles, which often go against our intuitions from QFT in flat space.