Stochastic inflation with gradient interactions

TL;DR

This work addresses the breakdown of the stochastic-inflation, separate-universe approach during SR to USR transitions by incorporating gradient interactions. It develops a higher-dimensional Langevin framework in which gradient effects appear as a memory/coloured-noise contribution, but can be recast as a Markovian system with white noises on an enlarged set of variables. The authors validate the formalism in slow-roll, ultra-slow-roll, and Starobinsky-like transitions, showing that gradient corrections recover perturbation theory and reveal a pullback effect that damps the tails of first-passage-time distributions, with implications for primordial black hole formation. The results extend the stochastic-$\,\delta N$ program to transitions and backreaction in single-field inflation and open avenues for non-perturbative and multi-field explorations.

Abstract

Stochastic inflation rests on the separate-universe approximation, i.e. the ability to describe long-wavelength fluctuations in an inflating universe as homogeneous perturbations of its background dynamics. Although this approximation is valid in most cases, it has been recently pointed out that it breaks down during transition periods between attractor and non-attractor phases. Such transitions are ubiquitous in single-field models giving rise to enhanced perturbations at small scales, that are required to form primordial black holes. The current inability to apply the stochastic-inflation program in such models is therefore one of the main obstacles to investigating the role of backreaction in primordial-black-hole scenarios. In this work, we show how gradient interactions can be incorporated in stochastic inflation, via a set of Langevin equations of higher dimension. We apply our formalism to a few cases of interest, including one with a sharp transition. In all cases, in the classical limit we show that gradient corrections as predicted from cosmological perturbation theory are properly recovered. We uncover the existence of a "pullback" effect by which the tails of the first-passage-time distributions are dampened by gradient interactions. We finally discuss the role of backreaction in the presence of gradient interactions.

Figures (11)

• Figure 1: Second Hubble-flow parameter $\epsilon_2$, and $Z$ function normalised by the scale factor, in the piecewise Starobinsky model discussed in \ref{['subsec::Starobinsky_Linear_potential']}. The parameter values used in these figures are reported in \ref{['table:Starobinsky']}. The red dashed line corresponds to the transient-USR model \ref{['eq:Z approximation transient']}. Since $\eta<0$, time goes from right to left.
• Figure 2: Time evolution of a reference Hubble patch during inflation. As the expansion proceeds, this patch leads to several patches. The gradient interaction comes from the difference in the field value between the reference patch in orange and the neighbour patches in white. This difference is the result of the accumulation of distinct stochastic realisations of the noises along the two worldlines displayed in blue and red. Since these noises are perfectly correlated in the distant past, where they are evaluated within the same patch, gradient interactions can be interpreted as a local memory effect.
• Figure 3: First-passage-time distribution in the linear potential \ref{['eq:pot:lin']} for the parameters listed in \ref{['table:linear']}. The left panels correspond to $\sigma=0.5$ and the right panels to $\sigma=0.01$. Solid lines are obtained from $10^7$ realisations of the Langevin equations \ref{['eq:Langevin:standard']} without gradient interactions, i.e. without the $\xi_\Delta^{(i)}$ noises, while dashed curves include gradient interactions and are drawn from \ref{['eq:Langevin equations linear potential with gradient']}. The grey-shaded curves are Gaussian distributions following the prediction \ref{['eq:variance as integrated power spectrum']} from linear perturbation theory. The bottom panels zoom in on the tails, with a logarithmic scale on the vertical axis.
• Figure 4: Variance of the first-passage-time distribution, $\langle(\delta {\cal{N}})^2\rangle$, for different initial conditions along the slow-roll attractor, parametrised by its mean number of e-folds $\langle {\cal{N}}\rangle$. The colour code is the same as in \ref{['fig:PDF_linear']}, where perturbation theory, i.e. \ref{['eq:variance as integrated power spectrum']}, is displayed with black solid lines. $10^4$ realisations have been used for each point, whose error bars have been evaluated using Jackknife resampling.
• Figure 5: First-passage-time distribution in the USR model for the parameters listed in \ref{['table:pure_USR']}, for $\sigma=0.5$ (left panel) and $\sigma=0.01$ (right panel). Solid lines are obtained from $10^7$ realisations of the Langevin equations \ref{['eq:Langevin:standard']} without gradient interactions, while dashed curves include gradient corrections and are drawn from \ref{['eq:Langevin equations pure USR with gradient']}. The grey-shaded curves are Gaussian distributions following the prediction \ref{['eq:NvsdeltaN2_pure_USR']} from linear perturbation theory.