Stochastic Inflation in General Relativity

Abstract

We provide a formulation of Stochastic Inflation in full general relativity that goes beyond the slow-roll and separate universe approximations. We show how gauge invariant Langevin source terms can be obtained for the complete set of Einstein equations in their ADM formulation by providing a recipe for coarse-graining the spacetime in any small gauge. These stochastic source terms are defined in terms of the only dynamical scalar degree of freedom in single-field inflation and all depend simply on the first two time derivatives of the coarse-graining window function, on the gauge-invariant mode functions that satisfy the Mukhanov-Sasaki evolution equation, and on the slow-roll parameters. It is shown that this reasoning can also be applied to include gravitons as stochastic sources, thus enabling the study of all relevant degrees of freedom of general relativity for inflation. We validate the efficacy of these Langevin dynamics directly using an example in uniform field gauge, obtaining the stochastic e-fold number in the long wavelength limit without the need for a first-passage-time analysis. As well as investigating the most commonly used gauges in cosmological perturbation theory, we also derive stochastic source terms for the coarse-grained BSSN formulation of Einstein's equations, which enables a well-posed implementation for 3+1 numerical relativity simulations.

21 pages, 4 figures, reviewed, accepted by PRD and referee, to be published

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Cited by in corpus (16)

• The Open Effective Field Theory of Inflation
• Thermodynamics and decay of de Sitter vacuum
• Quantum diffusion and large primordial perturbations from inflation
• Cosmology using numerical relativity
• Ultra-Slow-Roll Inflation on the Lattice II: Nonperturbative Curvature Perturbation
• Stochastic inflation with gradient interactions
• Itô, Stratonovich, and zoom-in schemes in stochastic inflation
• Quantitative classicality in cosmological interactions during inflation
• On the Hamilton-Jacobi approach to inflation beyond slow roll
• Stochastic Inflation in Numerical Relativity
• Bunch-Davies initial conditions and non-perturbative inflationary dynamics in Numerical Relativity
• Eternal inflation near inflection points: a challenge to primordial black hole models
• Time-reversed Stochastic Inflation
• Deviations from Gaussian White Noise in Stochastic Inflation
• Nonlinear Lattice Framework for Inflation: Bridging stochastic inflation and the $\delta N$ formalism
• Effect of stochastic kicks on primordial black hole abundance and mass via the compaction function