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arxiv: 2606.00176 · v2 · pith:UFGZACREnew · submitted 2026-05-29 · 🌌 astro-ph.CO · gr-qc· hep-ph· hep-th· math-ph· math.MP

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

Swagat S. Mishra , Edmund J. Copeland , Anne M. Green This is my paper

Pith reviewed 2026-06-28 20:58 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-phhep-thmath-phmath.MP
keywords stochastic inflationprimordial black holeseigenvalue methodFokker-Planck equatione-fold distributioninflationary perturbationsprobability distribution function
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The pith

A new eigenvalue technique solves for the probability distribution of stochastic e-folds in inflation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Stochastic inflation tracks how quantum fluctuations during inflation create a distribution of possible numbers of e-folds, which can produce large perturbations that form primordial black holes. The distribution P(N) obeys an adjoint Fokker-Planck equation. The paper introduces a self-contained eigenvalue method to find the full set of eigenvalues and eigenfunctions that determine this distribution. The method first reproduces the known PDF for a flat potential with no drift, showing a power-law segment P(N) proportional to N to the minus 3/2 between the peak and an exponential tail. It then handles constant-drift cases, producing an analytic solution in the narrow-well limit and a piecewise spectrum in the broad-well limit that yields a qualitatively different PDF shape.

Core claim

We develop a new self-contained eigenvalue technique which can be used to determine P(N). For quantum diffusion along a flat potential we recover the PDF with an exponential tail and P(N) proportional to N to the minus 3/2 in the intermediate regime. In the narrow-well limit of constant-drift inflation the PDF is similar to the drift-free case with a mildly suppressed tail; in the broad-well limit the full spectrum requires piecewise construction and the PDF shows an enhanced peak with a strongly suppressed tail.

What carries the argument

Eigenvalue decomposition of the adjoint Fokker-Planck operator for P(N), with construction of the spectrum of eigenvalues and eigenfunctions to reconstruct the distribution.

If this is right

  • The narrow-well PDF remains close to the drift-free form except for mild suppression of the tail.
  • The broad-well PDF develops an enhanced peak and strongly suppressed tail once the piecewise spectrum is assembled.
  • The eigenvalue method supplies a closed procedure that avoids explicit use of characteristic functions.
  • The same spectral construction applies to other constant-drift regimes once the well boundaries are specified.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The technique can be tested on time-varying drift potentials by allowing the operator to change slowly between successive eigenvalue solves.
  • The intermediate power-law regime may translate into specific scaling of the tail probability for primordial black hole formation rates.
  • Piecewise spectrum construction for broad wells suggests a general strategy for potentials with multiple flat regions.
  • Comparison with Monte Carlo simulations of the underlying Langevin equation would directly validate the eigenvalue ordering and normalization.

Load-bearing premise

The probability distribution P(N) of the stochastic number of e-folds satisfies an adjoint Fokker-Planck equation.

What would settle it

Direct numerical integration of many stochastic inflation trajectories in the broad-well constant-drift model, followed by comparison of the resulting histogram of N values against the PDF obtained from the eigenvalue spectrum.

read the original 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, and a power-law behaviour, $P({\cal N}) \propto {\cal N}^{-3/2}$, in the intermediate regime between the peak and the tail of the PDF. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces a self-contained eigenvalue technique to solve the adjoint Fokker-Planck equation for the PDF P(𝒩) of the stochastic number of e-folds in stochastic inflation. It validates the approach on quantum diffusion along a flat potential (no classical drift), recovering the known PDF featuring an exponential tail and an intermediate power-law regime P(𝒩) ∝ 𝒩^{-3/2}. The method is then extended to constant-drift inflation, providing an analytic solution in the narrow-well limit (PDF similar to the drift-free case but with mildly suppressed tail) and a piecewise spectrum construction in the broad-well limit (enhanced peak and strongly suppressed tail).

Significance. If the eigenvalue method is correctly implemented and the recovered expressions match prior results, the work supplies a new, potentially efficient tool for computing P(𝒩) in stochastic inflation models relevant to large perturbations and primordial black hole formation. The explicit validation against the characteristic-function result for the flat case, together with the analytic narrow-well solution and the broad-well spectral construction, constitutes a concrete advance over purely numerical or characteristic-function approaches.

minor comments (2)
  1. [Introduction / §2] The abstract and introduction should explicitly state the boundary conditions imposed on the eigenfunctions at the end of inflation and at the absorbing barrier to allow readers to reproduce the spectrum construction without ambiguity.
  2. [§3] Notation for the adjoint operator and the inner product used to obtain the eigenvalues should be defined once in a dedicated subsection rather than introduced inline.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending acceptance of the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the standard adjoint Fokker-Planck equation for the first-passage time problem in stochastic inflation, which is an external premise not derived within the paper. The eigenvalue technique is presented as a new self-contained method applied first to the flat-potential case, where it recovers a previously known PDF (exponential tail plus 𝒩^{-3/2} regime) obtained independently via characteristic functions. The constant-drift applications (narrow-well analytic solution and broad-well piecewise spectrum) follow directly from the same framework without introducing fitted parameters renamed as predictions or self-citation chains that bear the central load. No step reduces by construction to its own inputs; the method is externally validated against known results and extends to new regimes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; no details on parameters or axioms beyond the stated Fokker-Planck premise.

axioms (1)
  • domain assumption The PDF, P(𝒩), of the stochastic number of e-folds, 𝒩, satisfies an adjoint Fokker-Planck Equation.
    Stated explicitly in the abstract as the basis for the eigenvalue technique.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Stochastic scalar-tensor inflation and beyond

    gr-qc 2026-06 unverdicted novelty 7.0

    Extends the stochastic inflation formalism to a wide class of scalar-tensor theories by mapping EFT of dark energy coefficients to stochastic equations of motion.

  2. Stochastic constant-roll inflation beyond the hilltop with the spectral method

    astro-ph.CO 2026-06 unverdicted novelty 6.0

    Spectral solution of the Fokker-Planck operator for hilltop constant-roll inflation shows rare crossing trajectories dominate the mean, so the median yields a coarse-grained ΔN distribution whose exponential tail flat...

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