Stochastic growth of quantum fluctuations during slow-roll inflation

Abstract

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 ∝H2 (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 gauge-invariant inflaton fluctuations. We find that in most of the single field inflationary models the mean square gauge-invariant inflaton fluctuation grows faster than any test field with a non-negative effective mass. Hybrid inflationary models can be an exception: the mean square of a test field can dominate over the gauge-invariant inflaton fluctuation one on suitably chosen parameters. We also compute the stochastic growth of quantum fluctuations of a second field, relaxing the assumption of its zero homogeneous value, in a generic inflationary model; as a main result, we obtain that the equation of motion of a gauge-invariant variable associated, order by order, with a generic quantum scalar fluctuation during inflation can be obtained only if we use the number of e-folds as the time variable in the corresponding Langevin and Fokker-Planck equations for the stochastic approach. We employ this approach to derive some bounds for the case of a model with two massive fields

Loop Corrections to Standard Model fields in inflation

“…This result can be found by simple estimation using Hartree-Fock approximation, or more rigorous stochastic approach [21][22][23][24][25][26], or through 1-loop resummation using dynamical renormalization group method [27,28] (see also [29][30][31][32] for related diagrammatic calculations). On the other hand, in the Euclidean version of dS, it was recognized that the 1-loop late-time divergence is from the improper treatment of scalar zero mode.…”

“…diagrams can be written as, 25) in which the partial derivatives act only on adjacent factors. Without evaluation it is clear that this expression is free from late-time divergence, because each τ -derivative of scalar mode is linear in τ variable, and thereby cancel one τ factor in the denominator.…”

One loop corrected conformally coupled scalar mode equations during inflation

“…Hence it would be necessary to use a nonperturbative resummation method such as Starobinsky's stochastic technique [48]. There are nonperturbative techniques for various interactions in the literature [5,11,13,[48][49][50][51][52][53][54][55][56][57]. However, we leave these considerations for future work.…”

The quantum Fokker-Planck equation of stochastic inflation

“…The Schrödinger equation then implies the following relations derived from the zeroth, quadratic, and quartic order terms in the fields, 13) and so on for yet higher orders of n. The infinite factor (2π) 3 δ 3 ( 0) that appears in the equation for the time dependence of the normalisation, and in several of the equations that will occur later, is the volume of a spatial hypersurface in de Sitter space. These volume factors always accompany contributions to the normalisation.…”