Time-reversed Stochastic Inflation

TL;DR

This work tackles the challenge of deriving observables in stochastic inflation by reversing time and conditioning on the end of the quantum diffusion regime. Using a reverse Fokker-Planck equation and a backward $\mathcal{N}$-formalism, the authors obtain non-perturbative, exact results for a semi-infinite flat potential, revealing a normalisable curvature-fluctuation distribution with Levy-like heavy tails. The framework effectively regularises divergences inherent in forward stochastic inflation by conditioning on lifetimes, and it provides a practical, end-of-inflation–oriented measure for curvature perturbations $\zeta$ that can be extended to more general potentials and to primordial-black-hole scenarios. The approach offers new mathematical tools for connecting stochastic inflation to observable cosmological quantities while addressing eternal-inflation issues through lifetime partitioning.

Abstract

Cosmic inflation may exhibit stochastic periods during which quantum fluctuations dominate over the semi-classical evolution. Extracting observables in these regimes is a notoriously difficult program as quantum randomness makes them fully probabilistic. However, among all the possible quantum histories, the ones which are relevant for Cosmology are conditioned by the requirement that stochastic inflation ended. From an observational point of view, it would be more convenient to model stochastic periods as starting from the time at which they ended and evolving backwards in times. We present a time-reversed approach to stochastic inflation, based on a reverse Fokker-Planck equation, which allows us to derive non-perturbatively the probability distribution of the field values at a given time before the end of the quantum regime. As a motivated example, we solve the flat semi-infinite potential and derive a new and exact formula for the probability distribution of the quantum-generated curvature fluctuations. It is normalisable while exhibiting tails slowly decaying as a Levy distribution. Our reverse-time stochastic formalism could be applied to any inflationary potentials and quantum diffusion eras, including the ones that can lead to the formation of primordial black holes.

Figures (7)

• Figure 1: Normalised potential for the Starobinsky model in the Einstein frame as a function of the field values $\phi/M_{{\mathrm{Pl}}}$. In the semi-classical domain, for $\phi\in[\phi_\mathrm{end},\phi_\mathrm{qw}]$, the universe is inflating while the field rolls down the potential. This is the regime in which cosmological perturbations are generated, around $\phi=\phi_*$: quantum fluctuations are dealt with over a classical background. For $\phi>\phi_\mathrm{qw}$, quantum fluctuations become of larger amplitude than the classical field evolution, the semi-classical treatment breaks down: this is the regime of stochastic inflation. Because the quantum domain is at field values very far from $\phi_\mathrm{end}$, any structure formed in this region would be of length scales much larger than the Hubble radius today and, as such, can only affect background cosmological quantities.
• Figure 2: Sketch of a time-reversed stochastic inflation realisation when $\Delta N_{_0} \gg \Delta\phi_{_0}^2/G^2$. The field enters the quantum domain for $\Delta\phi=0$ ($\phi=\phi_\mathrm{qw}$), at the time $\Delta N=0$. In the white region, the repulsive drift $\bar{F}_{\infty}$ pushes the field away from $\Delta\phi=0$ whereas, in the blue domains, an elastic attractive force $\bar{F}_{{_0}}$ captures the field and drives it to $\Delta\phi_{_0}$. The sink at $\Delta\phi_{_0}$ is necessarily reached at $\Delta N=\Delta N_{_0}$, when the number of time-reversed $e$-folds has exhausted the lifetime of the forward process. In the opposite limit, $\Delta N_{_0} \ll \Delta\phi_{_0}^2/G^2$, the process is mostly driven by the attractive elastic force towards $\Delta\phi_{_0}$, white regions are almost non-existent (see also \ref{['fig:rvprob']}).
• Figure 3: Contour plots of the reverse probability distribution $\bar{P}\left(\phi,\Delta N|\phi_{_0},\Delta N_{_0}\right)$, for an arbitrary lifetime set at $\Delta N_{_0}=800$$e$-folds, with $\Delta\phi_{_0}=20G$ (upper panel) and $\Delta\phi_{_0}=60G$ (lower panel). For $\Delta\phi_{_0}\gg G \sqrt{\Delta N_{_0}}$, diffusion is reduced and the field flows from the quantum wall to the sink.
• Figure 4: Mean number of $e$-folds $\ev{\tau}=\ev{\Delta N}/\Delta N_{_0}$, in unit of the lifetime, as a function of the field $\hat{\chi} = \Delta\phi/(G\sqrt{\Delta N_{_0}})$, plotted for various values of $\hat{\chi}_{_0} = \Delta\phi_{_0}/(G\sqrt{\Delta N_{_0}})$. Generically, $\ev{\tau}$ increases for $\Delta\phi<\Delta\phi_{_0}$ and decreases to asymptote $1/2$ for $\Delta\phi > \Delta\phi_{_0}$. However, for small $\hat{\chi}_{_0}$, i.e. $\phi_{_0}$ small and/or very large lifetimes $\Delta N_{_0}$, one has $\ev{\tau}= 1/2$ in almost all the field domain.
• Figure 5: The normalised probability distribution of the rescaled curvature fluctuation $\hat{\zeta}=\zeta/\Delta N_{_0}$, at given lifetime, plotted for various values of $\hat{\chi}_{_0}=\Delta\phi_{_0}/(G\sqrt{\Delta N_{_0}})$. For $\hat{\chi}_{_0} \ll 1$, it converges to the rectangle function of \ref{['eq:Pzetatauhalf']}. In the opposite limit, $\hat{\chi}_{_0} \gg 1$, it develops an asymmetric structure, due to the effect of the quantum wall, while becoming strongly peaked around $\hat{\zeta}=0$. All dependence in the lifetime $\Delta N_{_0}$ are absorbed in the rescaled quantities, $\hat{\zeta}$ and $\hat{\chi}_{_0}$.