Critical Number of Fields in Stochastic Inflation

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 $\delta N$ formalism. The number of fields is a critical parameter. The probability of exploring arbitrarily large-field regions of the potential becomes nonvanishing 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 regularized 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, multifield models can yield large stochastic corrections even at sub-Planckian energy, opening interesting prospects for probing quantum effects on cosmological fluctuations.

Figure 1

Scalar power spectrum amplitude ${\mathcal{P}}_{\zeta }$ in $v=\alpha \sum _{i=1}^D{\varphi }_i^2$ potentials, as a function of the potential energy $v$ at which the scales for which ${\mathcal{P}}_{\zeta }$ is calculated exit the Hubble radius during inflation (left panel), and as a function of the upper reflecting wall location $v_{+}$ in the right panel, for a few values of the number of fields $D$.

Figure 2

Scalar power spectrum amplitude ${\mathcal{P}}_{\zeta }$ (left panel: full stochastic results, right panel: classical formula) for the single-field potential $v=v_{\text{end}}{e}^{\alpha \varphi }$ when the end of inflation is modulated by an extra field $\chi$ through ${\varphi }_{-}(\chi )=\mu \cos (\chi /{\chi }_0)$, for $\alpha =0.1$, ${\chi }_0=M_{\mathrm{Pl}}$, $\mu =0.5M_{\mathrm{Pl}}$, and $v_{\text{end}}=0.05$ (this last value does not lead to the right scalar power spectrum amplitude [54], but it is used for computational convenience). The black dashed lines are various level lines of ${\mathcal{P}}_{\zeta }$, and the white regions correspond to $\varphi <{\varphi }_{-}(\chi )$, which is outside the inflationary domain.