Correlation Functions in Stochastic Inflation

TL;DR

The paper develops a non-perturbative framework that combines stochastic inflation with the δN formalism to compute all scalar curvature correlation functions in single-field slow-roll inflation. It shows that classical results emerge as saddle-point limits and introduces a Planck-suppressed classicality criterion that marks when stochastic effects become important, particularly near flat regions of the potential. By employing first passage time analysis, it derives analytical expressions for the power spectrum, non-Gaussianity, and higher moments, along with their classical limits, and clarifies the role of the time variable by advocating the use of the number of e-folds N. The results indicate that stochastic corrections are typically small for CMB scales but can substantially modify the location of the observational window and non-linear phenomena such as primordial black hole formation in certain potentials. The framework lays groundwork for extensions to multi-field scenarios and tensor perturbations, offering a versatile tool for exploring quantum backreaction during inflation.

Abstract

Combining the stochastic and $δN$ formalisms, we derive non perturbative analytical expressions for all correlation functions of scalar perturbations in single-field, slow-roll inflation. The standard, classical formulas are recovered as saddle-point limits of the full results. This yields a classicality criterion that shows that stochastic effects are small only if the potential is sub-Planckian and not too flat. The saddle-point approximation also provides an expansion scheme for calculating stochastic corrections to observable quantities perturbatively in this regime. In the opposite regime, we show that a strong suppression in the power spectrum is generically obtained, and comment on the physical implications of this effect.

Figures (4)

• Figure 1: Sketch of the dynamics solved in section \ref{['sec:PSinstocha']}. The inflaton is initially located at $\phi_*$ and evolves along the potential $V(\phi)$ under the stochastic Langevin equation (\ref{['eq:Langevin']}), until it reaches one of the two ending values $\phi_1$ or $\phi_2$. The left panel is an example where inflation always terminates by slow-roll violation, while the right panel stands for a situation where one of the ending points, $\phi_2$, corresponds to where $V\sim M_{{\mathrm{Pl}}}^4$ above which inhomogeneities prevent inflation from occurring.
• Figure 2: Integration domain of Eq. (\ref{['eq:f:sol']}) when evaluated at $\varphi=\phi_2$, in the case $\phi_1<\phi_2$ (the opposite case proceeds the same way). The discrete parameter $x$ is integrated between $\phi_1$ and $\phi_2$, while $y$ varies between $x$ and $\bar{\phi}$. The resulting integration domain is displayed in green. When $x<\bar{\phi}$, one has $\mathrm{d} x\mathrm{d} y >0$ and one integrates a positive contribution to the mean number of $e$-folds. Conversely, when $x>\bar{\phi}$, one has $\mathrm{d} x\mathrm{d} y <0$ and one integrates a negative contribution. This is necessary in order for the overall integral to vanish. This is why $\bar{\phi}$ must lie between $\phi_1$ and $\phi_2$.
• Figure 3: Mean number of $e$-folds $\langle \mathcal{N}\rangle$ realized in the large field $V\propto\phi^2$ (left panel) and small field $V\propto 1-\phi^2/\mu^2$ (where $\mu=20M_{{\mathrm{Pl}}}$, right panel) potentials, as a function of $\phi_*$. The location $\phi_*^{50}$ refers to the value of $\phi_*$ for which the classical number of $e$-folds $N_\mathrm{cl}=50$ and $\phi_\mathrm{end}$ is where $\epsilon_1=1$. In both panels, the overall mass scale in the potential is set to the value that fits the observed amplitude of the power spectrum $\mathcal{P}_\zeta\sim 2\times 10^{-9}$, when the latter is calculated with the classical formula (\ref{['eq:deltaNform:Pzeta']}), $50$$e$-folds before the end of inflation. The green line corresponds to the analytical exact result (\ref{['eq:Nmean']}), and the red circles are provided by a numerical integration of the Langevin equation (\ref{['eq:Langevin']}) for a large number of realizations over which the mean value of $\mathcal{N}$ is computed. The orange dashed line corresponds to the classical limit (\ref{['eq:stocha:meanN:classtraj']}). The top axes display $v$ and the classicality criterion $\vert 2v-v^{\prime\prime}v^2/{v^\prime}^2\vert$. The yellow shaded area stands for $v > 1$, where inhomogeneities are expected to prevent inflation from occurring and our calculation cannot be trusted anymore.
• Figure 4: Scalar power spectrum $\mathcal{P}_\zeta$ for the large field $V\propto\phi^2$ (left panel) and small field $V\propto 1-\phi^2/\mu^2$ (where $\mu=20M_{{\mathrm{Pl}}}$, right panel) potentials, as a function of $\phi_*$. The conventions are the same as in Fig. \ref{['fig:Nmean']}. The green line corresponds to the analytical exact result (\ref{['eq:PS:fullstocha']}), and the orange dashed line to the classical limit (\ref{['eq:stocha:PS:cl']}).