A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation

TL;DR

This paper develops a covariant, multifield stochastic inflation framework in phase space, resolving discretisation ambiguities and ensuring invariance under field redefinitions, with implications for numerical simulations and conceptual understanding.

Contribution

It introduces a covariant formulation of stochastic inflation in phase space, addressing discretisation ambiguities and deriving equivalent covariant Langevin equations for multifield inflation.

Findings

Covariant Langevin equations with real noises are derived.

Discretisation ambiguity is resolved using Stratonovich scheme.

Phase-space Fokker-Planck equation is formulated for the probability density.

Abstract

Stochastic inflation is an effective theory describing the super-Hubble, coarse-grained, scalar fields driving inflation, by a set of Langevin equations. We previously highlighted the difficulty of deriving a theory of stochastic inflation that is invariant under field redefinitions, and the link with the ambiguity of discretisation schemes defining stochastic differential equations. In this paper, we solve the issue of these "inflationary stochastic anomalies" by using the Stratonovich discretisation satisfying general covariance, and identifying that the quantum nature of the fluctuating fields entails the existence of a preferred frame defining independent stochastic noises. Moreover, we derive physically equivalent It\^o-Langevin equations that are manifestly covariant and well suited for numerical computations. These equations are formulated in the general context of multifield inflation with curved field space, taking into account the coupling to gravity as well as the full phase space in the Hamiltonian language, but this resolution is also relevant in simpler single-field setups. We also develop a path-integral derivation of these equations, which solves conceptual issues of the heuristic approach made at the level of the classical equations of motion, and allows in principle to compute corrections to the stochastic formalism. Using the Schwinger-Keldysh formalism, we integrate out small-scale fluctuations, derive the influence action that describes their effects on the coarse-grained fields, and show how the resulting coarse-grained effective Hamiltonian action can be interpreted to derive Langevin equations with manifestly real noises. Although the corresponding dynamics is not rigorously Markovian, we show the covariant, phase-space Fokker-Planck equation for the Probability Density Function of fields and momenta when the Markovian approximation is relevant [...]