Stochastic inflation in phase space: is slow roll a stochastic attractor?
Abstract
An appealing feature of inflationary cosmology is the presence of a phasespace attractor, ''slow roll'', which washes out the dependence on initial field velocities. We investigate the robustness of this property under backreaction from quantum fluctuations using the stochastic inflation formalism in the phasespace approach. A Hamiltonian formulation of stochastic inflation is presented, where it is shown that the coarsegraining procedure—where wavelengths smaller than the Hubble radius are integrated out—preserves the canonical structure of free fields. This means that different sets of canonical variables give rise to the same probability distribution which clarifies the literature with respect to this issue. The role played by the quantumtoclassical transition is also analysed and is shown to constrain the coarsegraining scale. In the case of free fields, we find that quantum diffusion is aligned in phase space with the slowroll direction. This implies that the classical slowroll attractor is immune to stochastic effects and thus generalises to a stochastic attractor regardless of initial conditions, with a relaxation time at least as short as in the classical system. For nontest fields or for test fields with nonlinear self interactions however, quantum diffusion and the classical slowroll flow are misaligned. We derive a condition onmore »
 Authors:
 Institut d'Astrophysique Spatiale, UMR8617, CNRS, Univ. Paris Sud, Université ParisSaclay, Bt. 121, Orsay, F91405 (France)
 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth, PO13FX (United Kingdom)
 Publication Date:
 OSTI Identifier:
 22676197
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2017; Journal Issue: 05; Other Information: Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ATTRACTORS; CORRECTIONS; COSMOLOGY; DIFFUSION; DISTRIBUTION; FLUCTUATIONS; HAMILTONIANS; INFLATIONARY UNIVERSE; INTERACTIONS; NONLINEAR PROBLEMS; PHASE SPACE; PROBABILITY; RELAXATION TIME; STOCHASTIC PROCESSES; VELOCITY; WAVELENGTHS
Citation Formats
Grain, Julien, and Vennin, Vincent, Email: julien.grain@ias.upsud.fr, Email: vincent.vennin@port.ac.uk. Stochastic inflation in phase space: is slow roll a stochastic attractor?. United States: N. p., 2017.
Web. doi:10.1088/14757516/2017/05/045.
Grain, Julien, & Vennin, Vincent, Email: julien.grain@ias.upsud.fr, Email: vincent.vennin@port.ac.uk. Stochastic inflation in phase space: is slow roll a stochastic attractor?. United States. doi:10.1088/14757516/2017/05/045.
Grain, Julien, and Vennin, Vincent, Email: julien.grain@ias.upsud.fr, Email: vincent.vennin@port.ac.uk. Mon .
"Stochastic inflation in phase space: is slow roll a stochastic attractor?". United States.
doi:10.1088/14757516/2017/05/045.
@article{osti_22676197,
title = {Stochastic inflation in phase space: is slow roll a stochastic attractor?},
author = {Grain, Julien and Vennin, Vincent, Email: julien.grain@ias.upsud.fr, Email: vincent.vennin@port.ac.uk},
abstractNote = {An appealing feature of inflationary cosmology is the presence of a phasespace attractor, ''slow roll'', which washes out the dependence on initial field velocities. We investigate the robustness of this property under backreaction from quantum fluctuations using the stochastic inflation formalism in the phasespace approach. A Hamiltonian formulation of stochastic inflation is presented, where it is shown that the coarsegraining procedure—where wavelengths smaller than the Hubble radius are integrated out—preserves the canonical structure of free fields. This means that different sets of canonical variables give rise to the same probability distribution which clarifies the literature with respect to this issue. The role played by the quantumtoclassical transition is also analysed and is shown to constrain the coarsegraining scale. In the case of free fields, we find that quantum diffusion is aligned in phase space with the slowroll direction. This implies that the classical slowroll attractor is immune to stochastic effects and thus generalises to a stochastic attractor regardless of initial conditions, with a relaxation time at least as short as in the classical system. For nontest fields or for test fields with nonlinear self interactions however, quantum diffusion and the classical slowroll flow are misaligned. We derive a condition on the coarsegraining scale so that observational corrections from this misalignment are negligible at leading order in slow roll.},
doi = {10.1088/14757516/2017/05/045},
journal = {Journal of Cosmology and Astroparticle Physics},
number = 05,
volume = 2017,
place = {United States},
year = {Mon May 01 00:00:00 EDT 2017},
month = {Mon May 01 00:00:00 EDT 2017}
}

Stochastic inflation describes the global structure of the inflationary universe by modeling the superHubble dynamics as a system of matter fields coupled to gravity where the subHubble field fluctuations induce a stochastic force into the equations of motion. The superHubble dynamics are ultralocal, allowing us to neglect spatial derivatives and treat each Hubble patch as a separate universe. This provides a natural framework in which to discuss probabilities on the space of solutions and initial conditions. In this paper we derive an evolution equation for this probability for an arbitrary class of matter systems, including DBI and kinflationary models, andmore »

Stochastic growth of quantum fluctuations during slowroll inflation
We compute the growth of the mean square of quantum fluctuations of test fields with small effective mass during a slowly changing, nearly de Sitter stage which takes place in different inflationary models. We consider a minimally coupled scalar with a small mass, a modulus with an effective mass {proportional_to}H{sup 2} (with H the Hubble parameter), and a massless nonminimally coupled scalar in the test field approximation and compare the growth of their relative mean square with the one of gaugeinvariant inflaton fluctuations. We find that in most of the single field inflationary models the mean square gaugeinvariant inflaton fluctuationmore » 
Spacedependent step features: Transient breakdown of slowroll, homogeneity, and isotropy during inflation
A step feature in the inflaton potential can model a transient breakdown of slowroll inflation. Here we generalize the step feature to include spacedependence, allowing it also to model a breakdown of homogeneity and isotropy. The spacedependent inflaton potential generates a classical curvature perturbation mode characterized by the wave number of the step inhomogeneity. For inhomogeneities small compared with the horizon at the step, spacedependence has a small effect on the curvature perturbation. Therefore, the smoothly oscillating quantum power spectrum predicted by the homogeneous step is robust with respect to subhorizon spacedependence. For inhomogeneities equal to or greater than themore » 
A parton picture of de Sitter space during slowroll inflation
It is wellknown that expectation values in de Sitter space are afflicted by infrared divergences. Long ago, Starobinsky proposed that infrared effects in de Sitter space could be accommodated by evolving the longwavelength part of the field according to the classical equations of motion plus a stochastic source term. I argue that  when quantummechanical loop corrections are taken into account  the separate universe picture of superhorizon evolution in de Sitter space is equivalent, in a certain leadinglogarithm approximation, to Starobinsky's stochastic approach. In particular the time evolution of a box of de Sitter space can be understood inmore »