The curvature perturbation in a box

TL;DR

This work argues that defining cosmological perturbations in a finite comoving box clarifies the stochastic properties of the curvature perturbation $\zeta$ and helps control infrared issues. It develops a framework to compute $\zeta$ correlators from the light-field perturbation $\delta\sigma$ via a $\delta N$ expansion, including quadratic and cubic truncations, and analyzes tree- and loop-level contributions with attention to box-size (minimal vs. super-large) effects. The paper shows that, under observational bounds on non-Gaussianity, loop corrections are typically negligible in a minimal box, while running with box size can absorb these contributions, though it raises concerns about cosmic variance and the relevance to our observable Universe. The discussion also contrasts inflaton and curvaton-type roles for $\sigma$, highlighting how stochastic approaches and renormalization-like running with the infrared cutoff shape the interpretation and robustness of predictions. Overall, the minimal-box approach provides robust, testable predictions for $\zeta$ while acknowledging substantial theoretical caveats associated with super-large-box treatments.

Abstract

The stochastic properties of cosmological perturbations are best defined through the Fourier expansion in a finite box. I discuss the reasons for that with reference the curvature perturbation, and explore some issues arising from it.