Eigenvalue formulation of Stochastic Inflation and application to large perturbation generating inflationary features

Abstract: Stochastic inflation is a powerful technique for calculating the probability distribution function (PDF) of large inflationary perturbations, which may collapse to form Primordial Black Holes. The PDF, $P({\cal N})$, of the stochastic number of e-folds, ${\cal N}$, satisfies an adjoint Fokker-Planck Equation. We develop a new self-contained eigenvalue technique which can be used to determine $P({\cal N})$. First we apply this method to the simple case of quantum diffusion along a flat potential without any classical drift. We recover the expression for the PDF that has previously been found using characteristic functions, with an exponential tail. We also identify an intermediate regime between the peak and the exponential tail of the PDF, which has not been emphasized in earlier studies, where it exhibits a power-law behaviour, $P({\cal N}) \propto {\cal N}^{-3/2}$. Finally we apply the method to constant drift inflation, in the narrow- and broad-well limits. In the narrow-well limit, there is an analytic solution and the PDF is similar to the drift-free case, with a mildly suppressed tail. In the broad-well limit, determining the full set of eigenvalues and eigenfunctions requires a piecewise construction of the spectrum, and the broad-well PDF is qualitatively different, with an enhanced peak and a strongly suppressed tail.

• The paper presents an eigenvalue-based spectral method to solve the Fokker-Planck equation, linking inflationary perturbations to primordial black hole formation.
• It derives analytic solutions for both drift-free and constant-drift regimes, revealing an intermediate N⁻³⁄² power-law behavior in the probability distribution function.
• Key implications include enhanced predictions of non-Gaussian tails, with potential applications to PBHs, ultra-compact minihalos, and extending to variable drift scenarios.

Eigenvalue Formulation of Stochastic Inflation: Spectral Method and Applications to Non-Gaussian Tails

Introduction and Motivation

The formation of Primordial Black Holes (PBHs) through the gravitational collapse of enhanced curvature perturbations during the early universe provides a unique probe into inflationary dynamics at small scales, far beyond Large Scale Structure and CMB probes. PBH abundance is exponentially sensitive to the non-Gaussian tails of the probability distribution function (PDF) describing these fluctuations. Accurate theoretical predictions for these non-Gaussian tails are therefore essential, especially for scenarios involving features in the inflationary potential—such as inflection points or localized wells—that can produce rare, large fluctuations.

Stochastic inflation, via a Fokker-Planck approach, offers a non-perturbative technique for computing the distribution of inflationary perturbations, taking into account both classical drift and quantum diffusion of the inflaton field. However, while characteristic function approaches have yielded partial analytic results for special cases, a comprehensive and self-contained spectral (eigenvalue) solution directly to the PDF has not previously been constructed. The paper "Eigenvalue formulation of Stochastic Inflation and application to large perturbation generating inflationary features" (2606.00176) addresses this gap by providing a rigorous eigenvalue-based framework for the Fokker-Planck equation in stochastic inflation, with systematic analysis of both drift-free and constant-drift stochastic dynamics—covering both narrow- and broad-well regimes.

Formalism: Spectral Decomposition of the Fokker-Planck Equation

The distribution for the first-passage number of e-folds, $P(\mathcal{N};\Phi)$, is governed by an adjoint Fokker-Planck operator acting on the phase-space of the coarse-grained inflaton field $\Phi$. The equation is: $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$ where $D_\phi$ is the classical drift and $\Sigma_{\phi\phi}$ is the stochastic diffusion coefficient.

The spectral method restates the solution as a sum over eigenmodes: $$P(\mathcal{N};\Phi) = \sum_n c_n \Psi_n(\Phi) \exp\left(-\Lambda_n \mathcal{N}\right)$$ with eigenfunctions $\Psi_n$ and quantized exponents $\Lambda_n$ determined by the boundary-value problem (reflecting and absorbing walls in field space).

Boundary conditions are:

• Absorbing at the exit boundary: $\Psi_n(\phi_A) = 0$
• Reflecting at the entry boundary: $\Psi_n'(\phi_R) = 0$

Completeness and orthogonality are established with a Sturm-Liouville weight $\Phi$0 appropriate to the drift, yielding a self-adjoint operator.

Figure 1: Schematic illustration of the inflaton potential with an intermediate diffusion-dominated feature of width $\Phi$1, highlighting the location of absorbing and reflecting boundaries.

The coefficients $\Phi$2 are fixed by matching to a sharply peaked initial condition at the absorbing boundary—physically, initial field configurations that immediately cross the boundary produce zero e-folds, mathematically leading to a derivative of the Dirac delta.

Drift-Free Diffusion: Analytical Solution and PDF Structure

For vanishing classical drift ($\Phi$3), the quantum diffusion equation reduces to that of a particle in a box. The solution is: $\Phi$4 with $\Phi$5 the dimensionless field coordinate, and $\Phi$6 the dimensionless diffusion width. The analytic PDF is: $\Phi$7 This exactly recovers previous characteristic-function-based results but frames the solution entirely in terms of spectral data.

Importantly, the solution reveals an intermediate regime where $\Phi$8 before transitioning to the expected exponential tail—this intermediate power-law was not highlighted in earlier literature.

Figure 2: The exact rescaled PDF, $\Phi$9, for drift-free (pure diffusion) inflation, illustrating both the exponential tail and intermediate power-law regime.

Figure 3: The PDF $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$0 as a function of $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$1 for various $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$2 and $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$3, with the intermediate $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$4 scaling highlighted.

Constant-Drift Inflation: Spectral Structure in Narrow- and Broad-Well Regimes

With nonzero drift, the eigenvalue equation requires solution of a transcendental quantization condition, with physically distinct limits:

Narrow-Well Limit ($$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$5)

In this regime, the eigenvalue and PDF are analytic perturbations of the drift-free case: $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$6 The PDF closely resembles the pure diffusion result at small $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$7, deviating at large $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$8 due to mild suppression of the tail.

Figure 4: The rescaled PDF, $$\frac{\partial P}{\partial \mathcal{N}} = \hat{\mathcal{L}}_{\rm FP}^{\dagger} P = \left[D_\phi \frac{\partial}{\partial\Phi} + \frac{1}{2}\Sigma_{\phi\phi}\frac{\partial^2}{\partial\Phi^2}\right]P$$9, in the narrow-well constant-drift regime, showing the small deviation from free diffusion for various $D_\phi$0.

Broad-Well Limit ($D_\phi$1)

Here, classical drift dominates, leading to a sharply peaked PDF with a suppressed exponential tail. The eigenvalue spectrum is obtained by a piecewise analytic approximation, validated against full numerical solutions: $D_\phi$2 This suppresses the probability of large first-passage times (long $D_\phi$3), dramatically reducing the abundance of rare, large-amplitude events compared to the diffusion-dominated regime.

Figure 5: The expectation value of the stochastic number of e-folds, $D_\phi$4, compared to the classical value for constant-drift inflation; broad-well regime approaches the classical result.

Figure 6: The fractional error in the analytic (piecewise) estimate of the eigenvalue spectrum compared to the full numerical solution, validating the approximation for moderate-to-large $D_\phi$5.

Figure 7: Rescaled PDF, $D_\phi$6, for various field positions $D_\phi$7 and several choices of drift parameter $D_\phi$8, highlighting the strong suppression of tails in the broad-well regime.

Theoretical and Practical Implications

• Stochastic Enhancement and Suppression: In the diffusion-dominated (narrow-well, low-drift) limit, rare PBH-forming perturbations are exponentially more likely due to the long exponential tail in the PDF. Conversely, in the broad-well, drift-dominated regime, the probability of large perturbations is dramatically reduced.
• Intermediate Power-Law Regime: The identification of a $D_\phi$9 regime may affect predictions not just for PBHs but also for other rare-structure phenomena, such as ultra-compact minihalos, by altering the shape of the non-Gaussian PDF at intermediate e-fold numbers.
• Formal Rigour and Generalisability: The eigenvalue (spectral) method applies broadly, not just to $\Sigma_{\phi\phi}$0 (constant drift) inflation, but can be, with further development, adapted to cases of variable drift (e.g., constant-$\Sigma_{\phi\phi}$1, hilltop, or multiple-well scenarios). The formalism is immediately applicable to eternal inflation and the stochastic dynamics of light fields during inflation.
• Numerical and Analytical Tradeoffs: In regimes with rapidly changing drift, the PDF cannot be captured by simple analytic eigenvalue formulas, necessitating either numerical spectral solutions or appropriately matched piecewise approximations. The bias introduced by such approximations is quantitatively examined in the paper.

Future Directions

Further work will be required to generalize the spectral approach to scenarios with nontrivial field-dependent drift (e.g., constant or varying $\Sigma_{\phi\phi}$2), multi-field extensions, and more complex boundary conditions relevant for realistic inflationary features. Applications to spectator field dynamics and analyses of eternal inflation boundaries are natural extensions. The methodology provides a systematic, formally controlled framework for addressing a broad class of problems involving stochastic processes in early-universe cosmology.

Conclusion

This work provides a comprehensive spectral (eigenvalue) formulation for solving the Fokker-Planck equation in stochastic inflation, yielding detailed analytic and numerical characterizations of the full non-Gaussian PDF—including exponential tails and intermediate regimes—relevant to PBH production from inflationary features. The method resolves technical limitations of prior approaches by enabling self-contained computation of the PDF, is shown to recover and generalize earlier results, and quantitatively details strong suppression of rare-event rates in the drift-dominated regime. The spectral formalism developed is a robust tool for theoretical and phenomenological explorations of non-Gaussianity and rare event statistics in the early universe.

Figure 8: Illustration of the solutions of the eigenvalue quantization condition for the stochastic Fokker-Planck operator as a function of the drift parameter, indicating regime boundaries and the entailed change in stochastic dynamics.