Abstract
Stochastic Inflation is an important framework for understanding the physics of de Sitter space and the phenomenology of inflation. In the leading approximation, this approach results in a Fokker-Planck equation that calculates the probability distribution for a light scalar field as a function of time. Despite its successes, the quantum field theoretic origins and the range of validity for this equation have remained elusive, and establishing a formalism to systematically incorporate higher order effects has been an area of active study. In this paper, we calculate the next-to-next-to-leading order (NNLO) corrections to Stochastic Inflation using Soft de Sitter Effective Theory (SdSET). In this effective description, Stochastic Inflation manifests as the renormalization group evolution of composite operators. The leading impact of non-Gaussian quantum fluctuations appears at NNLO, which is presented here for the first time; we derive the coefficient of this term from a two-loop anomalous dimension calculation within SdSET. We solve the resulting equation to determine the NNLO equilibrium distribution and the low-lying relaxation eigenvalues. In the process, we must match the UV theory onto SdSET at one-loop order, which provides a non-trivial confirmation that the separation into Wilson-coefficient corrections and contributions to initial conditions persists beyond tree level. Furthermore, these results illustrate how the naive factorization of time and momentum integrals in SdSET no longer holds in the presence of logarithmic divergences. It is these effects that ultimately give rise to the renormalization group flow that yields Stochastic Inflation.
Highlights
The study of quantum fields in de Sitter space provides insight into the foundations of inflationary cosmology
We argued that corrections to Stochastic Inflation are uniquely determined by the correlation functions of composite operators; computing those that are relevant to correcting Stochastic Inflation up to next-to-next-to-leading order (NNLO) is the topic of this section
Stochastic Inflation [21,22,23] informs much of the physical intuition for how we think about accelerating cosmologies, as we approach the eternally inflating regime that is dominated by quantum fluctuations [36,37,38,39]
Summary
The study of quantum fields in de Sitter (dS) space provides insight into the foundations of inflationary cosmology. The Fokker-Planck formalism obscures the connection to cosmological correlators, and it is not a prioi obvious how to incorporate higher-order corrections It would be ideal if we could understand how the success of Stochastic Inflation relates to other results regarding the IR behavior of fields in dS, such as the freeze-out of superhorizon metric fluctuations which has been shown to all orders in perturbation theory [44,45,46], or the loop generated anomalous scaling for the time-evolution of massive fields [47, 48]. Deriving the RG that yields Stochastic Inflation at NNLO requires performing this matching explicitly at one-loop order This provides a highly non-trivial check of the SdSET formalism, and these results can be utilized for a wide variety of correlator calculations.
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