Time-reversed stochastic inflation in the quantum well

The gist: Time-reversed stochastic inflation in a flat and bounded potential solves the stochastic evolution of the inflationary universe backward in time by counting e-folds from the end of quantum diffusion towards some initial state.

Stochastic Inflation Formalism

The standard stochastic inflation formalism describes the evolution of a coarse-grained field φ using an Itô stochastic differential equation, which is given by dφ = F[φ(N), N] dN + G[φ(N), N] dW. The transition probability density for the solutions of this Itô process is described by the Fokker-Planck equation, which serves as a Green’s function satisfying P(φ, N = N0 φ0, N0) = δ(φ − φ0). The quantity H(φ) stands for the Hubble parameter during stochastic inflation, and N is the number of e-folds.

Time-Reversed Formalism

Time-reversed stochastic inflation considers the stochastic process of Eq. (2.1) reversed in time, where the field φ emerges at φ = φqw to randomly evolve towards a sink located at φ = φ0. This reverse process is described by a reverse Fokker-Planck equation, and the quantity ∆N refers to the reverse e-fold number, defined by ∆N ≡ Nqw − N. The reversed drift term appearing in Eq. (2.11) involves both the forward friction F(φ, N) and the forward transition probability P(φ, N φ0, N0).

Finite Flat-Well Potential Analysis

The work exactly solves time-reversed stochastic inflation in a flat and bounded potential called the quantum well. In this setup, the drift term F of Eq. (2.3) vanishes, leading to a constant Hubble parameter H2(φ) ≃ V(φ)/3MPl squared ≡ Hinf. The forward solution is expressed as an infinite sum over trigonometric functions that can be simplified to Eq. (3.5).

Curvature Perturbations and Comparison

The distribution of the curvature perturbation at given lifetime is given by P(ζfw φ0, N0) = PLT(∆N0 = ζfw + N φ0). The time-reversed formalism leads to a probability distribution for the reverse e-fold number ∆N, and the quantity hτ i is given by Eq. (4.5). The final probability distribution of ζ, marginalized over all lifetimes, is given by Eq. (4.16).

Dependence on Well Width

The analysis shows that quantum diffusion in the flat well is quantitatively the same as that occurring in the semi-infinite flat potential whenever the field excursion across the well (the width) is larger than the diffusion coefficient, i.e., for φr − φqw & H/(2π). In contrast, in the opposite regime of a small well width, time-reversed stochastic inflation exhibits a greater amount of stochasticity than in the semi-infinite flat potential, up to erasing sensitivity to the initial field value φ0.

Conclusion on Tails

For large fluctuations ζ/χ2r > 1, saturated quantum diffusion dominates and both the negative and positive tails of P(ζ φ0) are found to be (quasi) exponential. The positive exponential tail of the reverse distribution decays exactly twice as fast as the positive tail of the forward distribution. This factor of two difference in the exponential behaviour of the positive tail was already encountered in the semi-infinite potential with drift.

Final Findings

The shape of the probability distribution in the small fluctuation domain ζ/χ2r < 1 is the same as that derived in the semi-infinite flat potential, P∞(ζ φ0). The large fluctuations domain exhibits (quasi) exponential tails, which are different from those of P∞(ζ φ0).

How it works

1. The forward solution is found by solving a simple linear parabolic equation (3.3) subject to initial and boundary conditions.

2. The time-reversed transition probability distribution P¯(φ, ∆N φ0, ∆N0) is the unique solution of the reverse Fokker-Planck equation (2.11) satisfying the initial and boundary conditions in Eq. (2.13).

3. The distribution of curvature fluctuations P(ζ φ0) is obtained by marginalizing the distribution of ˆζ over all possible lifetimes ∆N0, leading to the expression in Eq. (4.16).

Normalisation

The distribution of ˆζ = ζ/∆N0 is always normalised to unity because the integral over all field values and lifetimes yields a constant, independent on s.

Comparison with Forward Approach

The positive exponential tail of the reverse distribution decays exactly twice as fast as the positive tail of the forward distribution. This difference in exponential behaviour suggests that there are notable differences between the forward and time-reversed approaches, at least when strong quantum diffusion is driving the dynamics.

Conclusion

The flat well has been extensively studied in the context of PBHs, using the standard stochastic formalism, and this allows us to compare in Section 4.4 the one-point statistics of the curvature fluctuations derived from the two approaches. The time-reversed approach extends these results to stochastic inflation, where an unperturbed trajectory in field space does not exist. Hence, our results may have some important consequences.

How it works

The time-reversed approach extends these results to stochastic inflation, where an unperturbed trajectory in field space does not exist. Since observers are attached to the end-of-inflation hypersurface, statistical quantities that have to be compared with observations should be derived in reverse e-fold number [53]. The time-reversed approach extends these results to stochastic inflation, where an unperturbed trajectory in field space does not exist. Since observers are attached to the end-of-inflation hypersurface, statistical quantities that have to be compared with observations should be derived in reverse e-fold number [53].

How it works

The time-reversed approach extends these results to stochastic inflation, where an unperturbed trajectory in field space does not exist. Since observers are attached to the end-of-inflation hypersurface, statistical quantities that have to be compared with observations should be derived in reverse e-fold number [53]. The time-reversed approach extends these results to stochastic inflation, where an unperturbed trajectory in field space does not exist.

1. The improved AI system can perform exact solutions for time-reversed stochastic inflation in a flat and bounded potential (the quantum well), enabling precise calculation of curvature perturbations based on whether the well width is much larger than the typical Brownian excursion (quantum diffusion in the flat well is indistinguishable from the one occurring in a semi-infinite flat potential [58]).

2. The system can accurately determine time-reversed transition probability distribution for field values and reverse e-fold number, specifically by solving the reverse Fokker-Planck equation (Eq. 2.11) using the Maruyama–Girsanov’s method to obtain the solution shown in Eq. (3.13).

3. The AI can distinguish between different regimes of quantum diffusion by analyzing how the well width, measured in units of the typical Brownian excursion, is large enough: χˆr = ∆φr/Gp∆N0≫ 1 versus small well width, ˆχr≪1, allowing it to predict whether the system exhibits saturated quantum diffusion or converges to the semi-infinite limit.

4. The system can calculate the mean number of reverse e-folds, h∆Ni, by integrating the time-reversed transition probability distribution over all possible reverse e-fold numbers, as shown in Eq. (4.23).

5. The AI can provide a comparison of one-point statistics between the forward and time-reversed approaches for curvature fluctuations, specifically identifying that the positive exponential tail of the reverse distribution decays exactly twice as fast as the positive tail of the forward distribution [29].