Critical Number of Fields in Stochastic Inflation

Abstract

Stochastic effects in generic scenarios of inflation with multiple fields are investigated. First passage time techniques are employed to calculate the statistical moments of the number of inflationary e-folds, which give rise to all correlation functions of primordial curvature perturbations through the stochastic δN formalism. The number of fields is a critical parameter. The probability of exploring arbitrarily large-field regions of the potential becomes non-vanishing when more than two fields are driving inflation. The mean number of e-folds can be infinite, depending on the number of fields; for plateau potentials, this occurs even with one field. In such cases, correlation functions of curvature perturbations are infinite. They can, however, be regularised if a reflecting (or absorbing) wall is added at large energy or field value. The results are found to be independent of the exact location of the wall and this procedure is, therefore, well-defined for a wide range of cutoffs, above or below the Planck scale. Finally, we show that, contrary to single-field setups, multi-field models can yield large stochastic corrections even at sub-Planckian energy, opening interesting prospects for probing quantum effects on cosmological fluctuations. IntroductionIn the inflationary paradigm [1-6], cosmological inhomogeneities result from the parametric amplification of the vacuum quantum fluctuations of the gravitational and matter fields during an early accelerated expansion [7][8][9][10][11][12]. The transition from quantum fluctuations to classical but stochastic density perturbations [13][14][15][16][17][18] plays an important role in this scenario. In particular, it implies that the open quantum system comprising the superHubble degrees of freedom can be described with a classical stochastic theory, the stochastic inflation formalism [10,[19][20][21][22][23][24][25][26][27]. This consists of an effective description of the long-wavelength parts of the quantum fields, which are "coarse grained" at a fixed physical scale, larger than the Hubble radius during the whole inflationary era. In this framework, the short-wavelength quantum fluctuations act as a classical noise on the dynamics of the super-Hubble scales, and at leading order in slow roll, the coarse-grained fields φ i follow Langevin equations

Quantum diffusion during inflation and primordial black holes

“…When stochastic inflation is employed to describe such a family of universes and to calculate the amount of expansion realised in them, this gives rise to the stochastic-δN formalism [51][52][53][54][55][56][57]. This approach is sketched in Fig.…”

“…Starting from φ = φ * at an initial time, the inflaton field evolves along the potential v(φ) under the Langevin equation (1.6), where hereafter we use the rescaled dimensionless potential 6) until it reaches φ end where inflation ends. A reflective wall is added at φ uv to prevent the field from exploring arbitrarily large values, which can be necessary to renormalise infinities appearing in the theory [57] (whose results are still independent of φ uv and of the exact nature of the wall, reflective or absorbing, provided φ uv lies in some range). The amount of expansion realised along a given trajectory is called N , which is a stochastic variable.…”

Stochastic inflation with quantum and thermal noise

“…The method of a description of inflation in terms of the fluctuations of e-folds (called δ N method) has been suggested long time ago [4,41,46], developed recently and applied to detailed estimates of inflation parameters [23][24][25][47][48][49][50][51]. In this paper we have extended this formalism to include a thermal noise.…”

“…In particular, the power spectrum of fluctuations could be calculated as discussed in [21,22] (see also Appendix B here). However, in [23] (following [11], see also [24,25]) an alternative method has been proposed for a calculation of the power spectrum of the quantum noise based on fluctuations of the e-folds. In this paper we extend the method to the calculation of the power spectrum of the system which contains both the quantum noise and the thermal noise.…”

“…There may be some problems with integrability in Eq. (36) with some potentials in an infinite interval [24,25] (if there is no thermal noise). Then, Q −V −1 and the integrability may fail if either V = 0 or V does not grow fast enough.…”