Generation of fluctuations during inflation: Comparison of stochastic and field-theoretic approaches

Abstract

We prove that the stochastic and standard field-theoretical approaches produce exactly the same results for the amount of light massive scalar field fluctuations generated during inflation in the leading order of the slow-roll approximation. This is true both in the case for which this field is a test one and inflation is driven by another field, and the case for which the field plays the role of inflaton itself. In the latter case, in order to calculate the mean square of the gauge-invariant inflaton fluctuations, the logarithm of the scale factor a has to be used as the time variable in the Fokker-Planck equation in the stochastic approach. The implications of particle production during inflation for the second stage of inflation and for the moduli problem are also discussed. The case of a massless self-interacting test scalar field in de Sitter background with a zero initial renormalized mean square is also considered in order to show how the stochastic approach can easily produce results corresponding to diagrams with an arbitrary number of scalar field loops in the field-theoretical approach (explicit results up to four loops included are presented)

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.…”

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.…”

Stochastic growth of quantum fluctuations during slow-roll inflation

“…Indeed, the standard slow-roll expression for the power spectrum of the adiabatic mode of primordial scalar (density) perturbations becomes infinite, i.e., meaningless, ifḢ becomes zero during inflation. Therefore, the study of quantum effects in a nearly de Sitter stage withḢ = 0, in particular, when the total change in H during inflation is not small compared with its value during the last e-folds of inflation [2,3,4,5,6] (see also the recent papers [7,8,9]), is not of just pure theoretical interest.…”

“…Following [5,6] we consider the stochastic growth of different test fields as massive minimally coupled scalar fields, massless non-minimally coupled scalar fields and moduli with an effective mass ∝ H 2 , in different inflationary models. On the other hand, it is known [3] that the mean square of gauge invariant Mukhanov variable [11] grows; it is therefore interesting to compare the amplification of the test fields above not only with the background inflaton dynamics, but also with the stochastic growth of this gauge invariant fluctuation.…”