Deviations from Gaussian White Noise in Stochastic Inflation

TL;DR

This work systematically analyzes how Gaussian white noise in stochastic inflation deviates when key standard assumptions are relaxed. It shows that an exact de Sitter background preserves whiteness, while deviations in the window function or the initial state produce colored noise with memory, and non-Bunch-Davies states induce non-Gaussian, non-stationary noise whose color is controlled by two-particle state amplitudes. By deriving explicit forms for the noise correlators and the instantaneous power spectrum in toy models (including a piecewise-linear window and a two-particle C2 state), the paper clarifies when stochastic inflation remains tractable and when memory or non-Gaussianity must be treated explicitly. These results have implications for the coarse-grained dynamics of IR modes and for constructing more accurate stochastic descriptions of inflationary perturbations.

Abstract

Stochastic inflation is widely used as a framework to study scalar field perturbations on an inflationary spacetime in a classical manner. In Starobinsky's seminal work and most of the subsequent literature, stochastic inflation is driven by a white noise. This is a consequence of a number of assumptions about the background metric, the window function, and the initial state. Given that noise is the central object in this approach, it is worthwhile to investigate how the noise is modified upon relaxing some of these assumptions. We show that while deviation from an exact de Sitter background maintains the white character of the noise (only with a time-dependent amplitude), deviation from the Heaviside window function or the Bunch-Davies initial state can produce colored noise. We calculate the power spectrum and the memory of the noise for a toy model with a piecewise linear window function. We also show that, in order to produce a colored noise, the deviation from the Bunch-Davies vacuum should essentially be a sum of two-particle states. The resulting noise is non-stationary and we find its instantaneous power spectrum in a concrete example. Furthermore, while deviations from de Sitter background and sharp cutoff do not affect Gaussianity, changing the initial state yields a non-Gaussian noise.

Figures (5)

• Figure 1: The power spectrum of $\xi_\phi$ in the quasi-dS toy model of eq. \ref{['a-toy-model']} (with $c=1$ and $\sigma=0.01$), calculated numerically (blue), by applying eq. \ref{['corr-Hksigma']} with $\bar{\epsilon}$ restored (orange), and by the analytic approximation of Green's method in appendix \ref{['app:Green']} (green). The horizontal axis has units of $H_0^{-1}$ and the vertical axis has units of $(H_0/2\pi)^2$.
• Figure 2: The window function \ref{['W']} used for our non-sharp cutoff. The width $2\delta$ the interval of $\kappa=k/k_\sigma$ over which the modes contribute to both $\phi_l$ and $\phi_s$.
• Figure 3: The noise statistics for a massless field in exact dS, using the window function \ref{['W']}. Left: The correlator \ref{['corr-window-dS']}. Right: The power spectrum \ref{['power-window-dS']}. In both cases the vertical axis has units of $(H/2\pi)^2$.
• Figure 4: The NLO part of the correlator (left) and the instantaneous power spectrum (right) for the noise of a massless field on dS in the state \ref{['Psi-C2']} with $C_2$ given by eq. \ref{['C2-exp']}. In both cases the vertical axis has units of $-8\sqrt2\varepsilon (H/2\pi)^2$.
• Figure 5: The plot of $|u_k|$ for the toy model \ref{['toy-model']} with $c=1$ and $k=0.01 H_0$, using Green's method (solid blue) and the numerical solution (dashed orange). The vertical and horizontal axes have units of $1/\sqrt{2k}$ and $1/H_0$, respectively.