The Kramers-Moyal Equation of the Cosmological Comoving Curvature Perturbation

Details The Kramers-Moyal Equation of the Cosmological Comoving Curvature Perturbation The Kramers-Moyal Equation of the Cosmological Comoving Curvature Perturbation

2011-03-30

arxiv.org/abs/1103.5876v1

Astronomy and Astrophysics

Astrophysics

Cosmology and Extragalactic Astrophysics

27 pages

Scientific paper

Fluctuations of the comoving curvature perturbation with wavelengths larger than the horizon length are governed by a Langevin equation whose stochastic noise arise from the quantum fluctuations that are assumed to become classical at horizon crossing. The infrared part of the curvature perturbation performs a random walk under the action of the stochastic noise and, at the same time, it suffers a classical force caused by its self-interaction. By a path-interal approach and, alternatively, by the standard procedure in random walk analysis of adiabatic elimination of fast variables, we derive the corresponding Kramers-Moyal equation which describes how the probability distribution of the comoving curvature perturbation at a given spatial point evolves in time and is a generalization of the Fokker-Planck equation. This approach offers an alternative way to study the late time behaviour of the correlators of the curvature perturbation from infrared effects.

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