Stochastic instantons and the tail of the inflationary density perturbation

TL;DR

The paper develops a stochastic instanton framework to compute the rare tail of inflationary density perturbations within the stochastic δN formalism. By formulating transition probabilities in the Martin–Siggia–Rose path integral and connecting to Schwinger–Keldysh theory, it extends tail estimates beyond slow-roll and to multi-field setups, including time-dependent noise. The method reproduces known results in slow-roll linear potentials and yields new insights for USR, constant-roll, and decaying-noise scenarios, with a clear interpretation of the least-likely noise realization that drives extreme fluctuations. The approach offers a principled way to incorporate microscopic, non-Markovian effects via the influence functional and promises applica­tions to primordial black hole abundances and beyond.

Abstract

In the "stochastic $δN$ formalism", the statistics of the inflationary density perturbation are obtained from the first passage distribution of a stochastic process. We develop a general framework in which to evaluate the rare tail of this distribution, based on an instanton approximation to a path integral representation for the transition probability. We relate our formalism to the Schwinger-Keldysh path integral, by integrating out short wavelength degrees of freedom to produce an influence functional. This provides a principled way to extend the calculation beyond the slow-roll limit, and to models with multiple fields. We argue that our framework has a number of advantages in comparison with existing methods. In particular, it reliably captures the tail behaviour in cases where existing techniques do not apply, including cases where the noise amplitude has strong time dependence. We demonstrate the method by computing the tail probability in a number of scenarios, including a beyond-slow-roll analysis of a linear potential, ultra-slow-roll, and constant-roll inflation. We find close agreement with results already reported in the literature. Finally, we discuss a scenario with exponentially decaying noise amplitude. This is a model for the stochastic evolution of a fixed comoving volume of spacetime on superhorizon scales. In this case we show that the tail reverts to a Gaussian weight.

Figures (3)

• Figure 1: Background (noiseless) and stochastic trajectories for a field during a USR phase. Left: Case A ($\phi_{\text{end}} < \phi_{\infty} < \phi_0$). The target value $\phi_{\text{end}}$ is unreachable under noiseless evolution, and the stochastic trajectory approaches it monotonically. Right: Case B ($\phi_{\infty} < \phi_{\text{end}} < \phi_0$). A single noise realization $\mathsf{P}_1$ gives rise to two crossing times, $N_{\star}$ and $N_{\star\star}$.
• Figure 2: Left: Illustration of $\bm{\mathsf{P}}(\phi_{\text{end}}, N_{\star} \;|\; \phi_0)$ in an USR phase, as determined by Eq. \ref{['eq:SMSR_USR']}. The dotted region is classically inaccessible, whereas the diagonally hatched region corresponds to second crossings. $\phi_{\rm M}(N_{\star}) = (1+3N_{\star})\phi_0 e^{-3N_{\star}}$ and $\phi_0$ delimit this zone. Right: $\bm{\mathsf{Q}}(N_{\star}, \phi_{\text{end}} \;|\; \phi_0)$, as defined in Eq. \ref{['eq:Q_usr']}. Each curve has been rescaled so that its peak equals $1$. Dashed lines indicate a sign flip due to turnovers. Markers show the time the field would have reached $\phi_{\text{end}}$ under noiseless evolution. Note that these are not normalized distributions, and they are only meant to illustrate the exponential scaling behaviour captured by the instanton.
• Figure 3: $\bm{\mathsf{P}}(\phi_{\text{end}}, N_{\star} \;|\; \phi_0)$ in a CR phase, as given by Eq. \ref{['eq:Prob_CR']}. Dotted regions are classically inaccessible; diagonally hatched regions correspond to second crossings. $\phi_{\rm M}(N_{\star}) = \phi_0\, \mathrm{sech}(\sigma N_{\star})$ and $\phi_0$ delimit these zones. Left: $\epsilon_2 < 0$. The background field moves right to left, and the distribution peaks left of $\phi_0$, shifting toward zero as $N_{\star}$ grows. Right: $\epsilon_2 > 0$. The field moves left to right, with the peak right of $\phi_0$ and reduced impact from the turnover region.