Solving Stochastic Inflation for Arbitrary Potentials

TL;DR

The paper develops a perturbative, noise-based solution of the Langevin equation for stochastic inflation including backreaction, enabling analytic access to the probability distribution of the inflaton for arbitrary potentials and including volume effects. By expanding the inflaton field as φ = φ_cl + δφ1 + δφ2 + ..., it derives a Gaussian posterior for the coarse-grained field with mean φ_cl + ⟨δφ2⟩ and variance ⟨δφ1^2⟩, and provides a volume-weighted extension that shifts the mean by a computable term 3 I^T J. The method is applied to large-field, small-field, hybrid, and running-mass inflation, yielding new insights: stochastic effects can be substantial in new inflation, negligible in vacuum-dominated hybrid inflation, and can blur distinctions among running-mass variants, with self-reproduction regimes likely in the RM case. The framework offers a general, analytically tractable approach to quantify quantum backreaction across inflationary models and sets the stage for detailed accuracy assessments in follow-up work.

Abstract

A perturbative method for solving the Langevin equation of inflationary cosmology in presence of backreaction is presented. In the Gaussian approximation, the method permits an explicit calculation of the probability distribution of the inflaton field for an arbitrary potential, with or without the volume effects taken into account. The perturbative method is then applied to various concrete models namely large field, small field, hybrid and running mass inflation. New results on the stochastic behavior of the inflaton field in those models are obtained. In particular, it is confirmed that the stochastic effects can be important in new inflation while it is demonstrated they are negligible in (vacuum dominated) hybrid inflation. The case of stochastic running mass inflation is discussed in some details and it is argued that quantum effects blur the distinction between the four classical versions of this model. It is also shown that the self-reproducing regime is likely to be important in this case.

Figures (3)

• Figure 1: Characteristic mass scale $M$ obtained from the CMB normalization as a function of the spectral index for the new inflationary scenario. The exact, numerically calculated, curve (solid blue line) blows up at $1-n_{_{\rm S}}=6/(1+4N_*)$ while the approximated one (red dashed line) vanishes at the origin. A different approximation, usually found in the literature (see for instance Refs. KMJMbraz), is also shown for comparison (green dotted-dashed line). The difference between these two lines is approximatively a factor of $2$ in $M$ and is significant since this leads to a factor of $16$ in the normalization factor $M^4$ of the potential and, thus, in the variance and the mean value of the fluctuations.
• Figure 2: Evolution of the single-point (solid blue line) and volume-weighted (dashed red line) probability distributions, $P_\mathrm{c}(\varphi)$ and $P_\mathrm{v}(\varphi)$, for the large field model $V\propto\varphi^2$ (LF) and the small field model $V\propto 1-(\varphi/\mu)^2$ (SF). The initial values are $\varphi_\mathrm{in}=7.6\times 10^5m_{_\mathrm{Pl}}$ (corresponding to $V_\mathrm{in}=m_{_\mathrm{Pl}}^4/2$) for LF and $\varphi_\mathrm{in}/\mu=10^{-5}$ for SF. The initial shape of the two probability density functions is always chosen to be $\delta(\varphi-\varphi_\mathrm{in})$. The vertical dotted black lines represent the location of the classical field. Three successive snapshots of the distributions (numbered $1$, $2$ and $3$) are shown on the left panels while the evolution of $\left\langle \varphi \right\rangle_\mathrm{c}$ and $\left\langle \varphi \right\rangle_\mathrm{v}$ is displayed on the right panels. The classical field $\varphi_\mathrm{cl}$ evolves from the right to the left in LF and from the left to the right in SF. In both cases, $P_\mathrm{c}(\varphi)$ rolls down the potential, spreads significantly around its mean value and penetrates into a classically forbidden region ($\varphi<0$ for LF and $\varphi>\mu$ for SF). The quantity $\left\langle \varphi \right\rangle_\mathrm{c}$ stays "behind" the classical value in LF but is "ahead" in SF. On the other hand, $P_\mathrm{v}(\varphi)$ (not shown in the LF left panel) reverses its motion and climbs towards the trans-Planckian region (in LF) or towards the maximum of the potential at $\varphi=0$ in SF.
• Figure 3: Evolution of the single-point (solid blue line) and volume-weighted (dashed red line) probability distributions for the two different running mass inflation models RM1 and RM2. The initial probability density function is $\delta(\varphi-\varphi_\mathrm{in})$ for the two models. The initial values are chosen to be $\varphi_\mathrm{in}/\varphi_0=1-1.5\times 10^{-5}$ for RM1 ($\varphi<\varphi_0$, inflation proceeding from the right to the left) and $\varphi_\mathrm{in}/ \varphi_0=1+10^{-3}$ for RM2 ($\varphi>\varphi_0$, inflation proceeding from the left to the right). The vertical lines (dotted black lines) represent the location of the classical field. On the two left panels, three snapshots of $P_\mathrm{c}(\varphi)$ and $P_\mathrm{v}(\varphi)$ (numbered from $1$ to $3$) are shown at three different times (respectively corresponding for RM1 to $\varphi /\varphi _0=0.9, 0.5$ and $0.01$) together with the corresponding values of the classical field. In the right panels, the evolution of $\left\langle \varphi \right\rangle_\mathrm{c}$ and $\left\langle \varphi \right\rangle_\mathrm{v}$ is followed until the end of inflation. The physical interpretation of these results is discussed in the text.