Loop Corrections to Cosmological Correlators

Abstract

Inflationary correlation functions are fundamental to probing the physics of the primordial universe. These correlations are the theoretical predictions, that can be mapped to present-day cosmological observations, such as the Cosmic Microwave Background (CMB) and Large-Scale Structure, to extract information about the inflationary era. While the tree-level structure of these correlators has been extensively analyzed, loop corrections in inflation remain technically challenging, and fully explicit computations are still lacking. These loop effects, although expected to be highly suppressed and practically too small to be observable, offer invaluable “theoretical data” that shed light on the behavior of inflationary dynamics at the 1-loop level. In this thesis, we work on a controlled set of 1-loop diagrams and explicitly compute them to investigate their analytic structure. One of the first loop computations in cosmology is that of the scalar power spectrum at 1-loop, performed by Weinberg [1]. This correction featured a worrisome and unphysical log(k/µ), where k is the comoving scale and µ is the physical renormalization scale. It was later pointed out in [2] that due to dimensional dependence of the modes in de-Sitter, the implementation of dimensional regularization is subtle, and once full d(= 3 + δ)-dimensional modes are used, the unphysical logarithms cancel. Motivated by this, we perform the first explicit computation of the scalar 3-point correlation function at 1-loop with interactions from the Effective Field Theory (EFT) of Inflation in [3], with the goal of exploring the structure of loop corrections beyond power spectrum. Interestingly, we found that these correlators indeed feature unphysical logarithms despite implementing dimensional regularization correctly, and the cancellation in the unrenormalized power spectrum was a happy coincidence. We show that for loops in higher point correlators these unphysical logarithms are removed only once the correlator is renormalized, a feature xiv which was previously unrecognized. It is important to understand how general physical principles, such as unitarity, locality and causality, manifest as analytic properties of cosmological observables in the complex plane of kinematics. In the flat space S-matrix program, a key ingredient to achieving the same was the knowledge of analytic properties of scattering amplitudes. However for cosmological correlators, the analytic structure beyond tree level was previously unexplored. To address this gap, we performed the first systematic study of the singularity structure of 1-loop (2-vertex loops) correlation functions of massless scalars in [4]. Interestingly, we find poles and branch cuts only along negative real axis in the complex plane of external energies. We also formulate diagrammatic rules to obtain singularity structure (higher order poles and logarithms) of any n-point function at 1-loop, based on the energy injection into specific subgraphs. Our analysis clarified the origin of total energy branch points in off-shell loop correlators as a feature of the time dependent cosmological background. Also, almost all results on loop correlations only apply to simple polynomial or derivative interactions. The effect of non-local interaction terms on the singularity structure of loop corrections was largely unexplored. This is an important gap, since realistic inflationary effective actions naturally contain non-local interactions, for example, when using the ADM formalism to enforce constraints. To address this gap, we investigate the effects of such interactions on the analytic structure of loops in [5], which poses considerable computational challenges due to complicated angular dependencies in the loop integrals. We find that the singularities arising still follow the diagrammatic rules, providing the first systematic investigation into non-local interactions in inflationary loops. We also establish for the first time the flat space limit of 1-loop correlators, where this correspondence was previously only known at tree level.