Friction in Stochastic Inflation

Abstract

We solve time-reversed stochastic inflation in the semi-infinite flat potential with a constant drift term and derive an exact expression for the probability distribution of the curvature fluctuations. It exhibits exponential decaying tails which contrast to the Levy-like power law behaviour encountered without friction. Such a non-vanishing drift acts as a regulator for the conventional ``forward'' stochastic $δN$-formalism, which is otherwise ill-defined in the unbounded and flat potentials typical of plateau models of inflation. This setup therefore allows us to compare the curvature distribution derived from both approaches, reverse and forward in time. Up to similar exponential tails, we find quantitative differences. In particular, in the classical-like limit of very large drift, the tails become Gaussian but only in the time-reversed picture. As a toy model of eternal inflation, we finally discuss the case of negative drift in which inflation never ends for many field trajectories. The forward approach becomes pathological whereas the reverse formalism gives back a finite curvature distribution with always exponential tails. All these differences end up being related to the very definition of the background which is ambiguous when a classical trajectory does not exist.

Figures (7)

• Figure 1: Contour plots of the rescaled reverse probability distribution $G \sqrt{\Delta N_{_0}} \, \bar{P}\left(\phi,\Delta N|\phi_{_0},\Delta N_{_0}\right)$ as a function of $\hat{\chi} = \Delta\phi/(G\sqrt{\Delta N_{_0}})$ and $\tau=\Delta N/\Delta N_{_0}$ in the diffusion regime ($\hat{\chi}_{_0} \le 1$). The field emerges at $\hat{\chi}=0$ at time $\tau=0$, diffuses in the domain before being absorbed at $\hat{\chi}=\hat{\chi}_{_0}$ at $\tau=1$.
• Figure 2: Contour plots of the rescaled reverse probability distribution $G \sqrt{\Delta N_{_0}} \, \bar{P}\left(\phi,\Delta N|\phi_{_0},\Delta N_{_0}\right)$ as a function of $\hat{\chi} = \Delta\phi/(G\sqrt{\Delta N_{_0}})$ and $\tau=\Delta N/\Delta N_{_0}$ in the fluxing regime ($\hat{\chi}_{_0} > 1$). The field emerges at $\hat{\chi}=0$ at time $\tau=0$ and straightly jumps to $\hat{\chi}=\hat{\chi}_{_0}$ at $\tau=1$.
• Figure 3: Probability distribution of the rescaled curvature fluctuations $\hat{\zeta} = \zeta/\Delta N_{_0}$, at given lifetime $\Delta N_{_0}$, as a function of its sole parameter $\hat{\chi}_{_0}=\Delta\phi_{_0}/(G\sqrt{\Delta N_{_0}})$ (top panel). It asymptotes to the rectangle function of \ref{['eq:Pzetatauhalf']} in the large diffusion limit (small $\hat{\chi}_{_0}$), and to a Dirac-like distribution in the fluxing regime (large $\hat{\chi}_{_0}$). In between, it develops a complicated shape, some visible kicks coming from the field-domain selection functions appearing in \ref{['eq:PzetaphiGivenLT']}. The bottom panel is a zoom on the distribution in the fluxing regime, for $\hat{\chi}_{_0}=10^2$. The tails of the distribution are becoming Gaussian and decay as $e^{-2\chi_{_0}^2 \hat{\zeta}^2}/(2 |\hat{\zeta}|)$.
• Figure 4: Probability distribution of the curvature fluctuations $\zeta$ (solid lines) as a function of $\zeta/\chi_{_0}^2$ for various values of the rescaled drift coefficient $\hat{\alpha}$, defined in \ref{['eq:alphahatdef']}. The dashed lines are the analytical approximation of \ref{['eq:Pzetainf']}, obtained in the large diffusion limit. The tails are asymptotically exponentials. See \ref{['fig:pzeta']} for a zoom within the domain of small $\abs{\zeta}/\chi_{_0}^2$.
• Figure 5: Probability distribution of the curvature fluctuations $\zeta$ (solid lines) as a function of $\zeta/\chi_{_0}^2$ for various values of the rescaled drift $\hat{\alpha}$. Up to a change of amplitude, the shape of the distribution at small curvature is similar to the the one of the flat semi-infinite potential ($\hat{\alpha}=0$). It is skewed and the mode is at a positive small value which depends on $\hat{\alpha}$ (see \ref{['fig:mode']}). Despite the asymmetry, one has $\ev{\zeta}=0$. The dashed lines are the analytical approximation of \ref{['eq:Pzetainf']} providing a good fit of the tails (see \ref{['fig:logpzeta']}).