Symmetries and Loops in Inflation

TL;DR

This work proves that in single-field inflation, the curvature perturbation $\zeta$ is conserved on superhorizon scales at all loop orders, as an operator statement constrained by locality and diffeomorphism invariance. The approach combines locality arguments, non-linear realizations of dilatation and SCT symmetries, and a renormalization framework for composite operators to show $\lim_{a\to\infty} \dot{\hat{\zeta}}_{\boldsymbol{k}} = 0$ up to $\mathcal{O}(k^2/a^2)$. The results illuminate the operator product expansion of $\zeta$ and its connection to Maldacena's single-field consistency relation, clarifying when the squeezed limit is fixed by symmetry. They also discuss potential violations in multi-field scenarios or with non-local dynamics, and contemplate extensions to tensor modes and eternal inflation, highlighting the broader utility of locality and symmetry in cosmological perturbation theory.

Abstract

In this paper, we prove that the superhorizon conservation of the curvature perturbation zeta in single-field inflation holds as an operator statement. This implies that all zeta-correlators are time independent at all orders in the loop expansion. Our result follows directly from locality and diffeomorphism invariance of the underlying theory. We also explore the relationship between the conservation of zeta, the single-field consistency relation and the renormalization of composite operators.

Figures (1)

• Figure 1: Composite operators ${\cal O}({\boldsymbol{x}},t)$ and ${\cal O}({\boldsymbol{y}},t)$ inserted at future infinity of de Sitter space. Locality requires that the operators are only influenced by sources that are inside their past light cones (shown in grey). The shaded regions near the operators indicate where an insertion of ${\cal H}_{\rm int}({\boldsymbol{x}}',t')$ yields an unsuppressed contribution to $\langle {\cal O}({\boldsymbol{x}},t) {\cal O}({\boldsymbol{y}},t) \rangle$. Divergences can therefore be removed by adding local counterterms.