# Stochastic inflation in phase space: is slow roll a stochastic attractor?

## Abstract

An appealing feature of inflationary cosmology is the presence of a phase-space 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 phase-space approach. A Hamiltonian formulation of stochastic inflation is presented, where it is shown that the coarse-graining 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 quantum-to-classical transition is also analysed and is shown to constrain the coarse-graining scale. In the case of free fields, we find that quantum diffusion is aligned in phase space with the slow-roll direction. This implies that the classical slow-roll 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 non-test fields or for test fields with non-linear self interactions however, quantum diffusion and the classical slow-roll flow are misaligned. We derive a condition onmore »

- Authors:

- Institut d'Astrophysique Spatiale, UMR8617, CNRS, Univ. Paris Sud, Université Paris-Saclay, Bt. 121, Orsay, F-91405 (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, E-mail: julien.grain@ias.u-psud.fr, E-mail: vincent.vennin@port.ac.uk.
```*Stochastic inflation in phase space: is slow roll a stochastic attractor?*. United States: N. p., 2017.
Web. doi:10.1088/1475-7516/2017/05/045.

```
Grain, Julien, & Vennin, Vincent, E-mail: julien.grain@ias.u-psud.fr, E-mail: vincent.vennin@port.ac.uk.
```*Stochastic inflation in phase space: is slow roll a stochastic attractor?*. United States. doi:10.1088/1475-7516/2017/05/045.

```
Grain, Julien, and Vennin, Vincent, E-mail: julien.grain@ias.u-psud.fr, E-mail: vincent.vennin@port.ac.uk. Mon .
"Stochastic inflation in phase space: is slow roll a stochastic attractor?". United States.
doi:10.1088/1475-7516/2017/05/045.
```

```
@article{osti_22676197,
```

title = {Stochastic inflation in phase space: is slow roll a stochastic attractor?},

author = {Grain, Julien and Vennin, Vincent, E-mail: julien.grain@ias.u-psud.fr, E-mail: vincent.vennin@port.ac.uk},

abstractNote = {An appealing feature of inflationary cosmology is the presence of a phase-space 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 phase-space approach. A Hamiltonian formulation of stochastic inflation is presented, where it is shown that the coarse-graining 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 quantum-to-classical transition is also analysed and is shown to constrain the coarse-graining scale. In the case of free fields, we find that quantum diffusion is aligned in phase space with the slow-roll direction. This implies that the classical slow-roll 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 non-test fields or for test fields with non-linear self interactions however, quantum diffusion and the classical slow-roll flow are misaligned. We derive a condition on the coarse-graining scale so that observational corrections from this misalignment are negligible at leading order in slow roll.},

doi = {10.1088/1475-7516/2017/05/045},

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}

}