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arxiv: 2508.09946 · v2 · submitted 2025-08-13 · 🌌 astro-ph.CO · gr-qc· hep-th

Stochastic Inflation with Interacting Noises

Amin Nassiri-Rad , Haidar Sheikhahmadi , Hassan Firouzjahi This is my paper

Pith reviewed 2026-05-18 22:53 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-th
keywords stochastic delta Ninflationary noiseone-loop correctionsprimordial black holesLangevin equationpower spectruminteracting fieldsFokker-Planck equation
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The pith

In interacting inflationary models the stochastic noise amplitude is rescaled by the square root of one plus the fractional one-loop power-spectrum correction.

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

The stochastic delta N formalism is extended to inflationary theories that include field interactions. In free theories the noise amplitude on the initial hypersurface is fixed at H over 2 pi, but interactions require that quantum-field-theory corrections be built into the noise. The authors rewrite the Langevin and Fokker-Planck equations so that these corrections appear directly in the noise term. For the three-phase slow-roll ultra-slow-roll slow-roll model used to generate primordial black holes they obtain an explicit modified amplitude that multiplies the free-theory value by the square root of one plus the relative one-loop correction to the curvature power spectrum.

Core claim

We extend the stochastic delta N formalism to setups with interactions and rewrite the corresponding Langevin and Fokker-Planck equations in which the QFT corrections in the amplitude of the noises are taken into account. As an example, in the three-phase SR-USR-SR setup which is employed for PBHs formation, the modification in the amplitude of noise is calculated from the one-loop corrections in power spectrum via in-in formalism. We show that in these setups the amplitude of the stochastic noise is modified to H/2 pi (1 + Delta P_R / P0_R)^{1/2} in which Delta P_R / P0_R is the fractional one-loop correction in power spectrum.

What carries the argument

The multiplicative rescaling of the noise amplitude, taken directly from the fractional one-loop correction to the curvature power spectrum computed in the in-in formalism.

If this is right

  • The Langevin equation for the curvature perturbation now carries an interaction-adjusted noise term.
  • The Fokker-Planck equation governing the probability distribution of curvature perturbations is updated to include the same correction.
  • Non-perturbative correlators computed via stochastic delta N automatically incorporate the one-loop power-spectrum effects.
  • Predictions for primordial black hole abundance in the SR-USR-SR model are altered by the rescaled noise.

Where Pith is reading between the lines

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

  • Stochastic calculations of curvature perturbations that omit this rescaling will give systematically incorrect statistics once interactions are present.
  • The same rescaling procedure can be applied to any other inflationary model whose one-loop power spectrum has already been computed.
  • The approach creates a practical link between perturbative QFT results and fully non-perturbative stochastic evolution.

Load-bearing premise

One-loop corrections computed for the power spectrum can be translated into a simple multiplicative factor for the noise amplitude in the stochastic equations without extra interaction terms or renormalization.

What would settle it

A direct evaluation of the noise two-point function inside the interacting stochastic dynamics that fails to reproduce the square root of the corrected power spectrum.

Figures

Figures reproduced from arXiv: 2508.09946 by Amin Nassiri-Rad, Haidar Sheikhahmadi, Hassan Firouzjahi.

Figure 1
Figure 1. Figure 1: FIG. 1: A schematic view of SR-USR-SR inflationary model. The [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: A schematic diagram of the fractional one-loop corre [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
read the original abstract

Stochastic $\delta N$ formalism is a powerful tool to calculate the cosmological correlators non-perturbatively. However, it requires the initial data for the amplitude of the noise on the initial flat hypersurface which for a free theory during inflation is fixed to be $\frac{H}{2 \pi}$. In this work, we study the setups where the underlying theory involves interactions and the stochastic noises inherit these interactions. We extend the stochastic $\delta N$ formalism to these setups and rewrite the corresponding Langevin and Fokker-Planck equations in which the QFT corrections in the amplitude of the noises are taken into account. As an example, in the three-phase SR-USR-SR setup which is employed for PBHs formation, the modification in the amplitude of noise is calculated from the one-loop corrections in power spectrum via in-in formalism. We show that in these setups the amplitude of the stochastic noise is modified to $\frac{H}{2 \pi} \Big(1+ \frac{ \Delta {\cal P}_{\cal R} }{ {\cal P}^{(0)}_{ {\cal R} } }\Big)^{\frac{1}{2}}$ in which $ \frac{\Delta {\cal P}_{\cal R} }{ {\cal P}^{(0)}_{{\cal R} } }$ is the fractional one-loop correction in power spectrum.

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

2 major / 1 minor

Summary. The manuscript extends the stochastic δN formalism to inflationary models with interactions by incorporating QFT one-loop corrections into the stochastic noise. For the SR-USR-SR setup used in PBH formation, the noise amplitude is modified from H/2π to H/2π (1 + ΔP_R / P0_R)^{1/2}, where the fractional correction ΔP_R / P0_R is taken from separate in-in calculations; the authors state that they rewrite the Langevin and Fokker-Planck equations to account for these corrections.

Significance. If the direct mapping from in-in power-spectrum corrections to a simple multiplicative rescaling of the noise variance is valid and introduces no additional drift, multiplicative-noise, or renormalization terms in the stochastic dynamics, the result would offer a practical bridge between perturbative QFT and non-perturbative δN calculations for large-curvature perturbations. The approach is potentially useful for PBH abundance estimates, but its soundness hinges on the unshown derivation that the correction affects only the noise amplitude.

major comments (2)
  1. [Abstract / equation-rewriting section] Abstract and the section rewriting the Langevin/Fokker-Planck equations: the central claim that the fractional one-loop correction ΔP_R / P0_R enters solely as a multiplicative factor (1 + ΔP_R / P0_R)^{1/2} on the white-noise amplitude, without generating extra drift corrections, colored-noise components, or renormalization counterterms in the Fokker-Planck operator, is asserted but not derived. In the SR-USR-SR setup the USR phase already amplifies superhorizon modes; an explicit step-by-step translation from the in-in result to the stochastic equations is required to confirm that no interaction-induced terms arise.
  2. [Abstract] The manuscript provides no visible error analysis, validation against full QFT, or check that the modified noise reproduces the input one-loop power spectrum when the stochastic equations are solved. Without such a consistency test the translation from power-spectrum correction to noise amplitude remains an assumption rather than a demonstrated result.
minor comments (1)
  1. [Abstract] Notation for the power spectrum alternates between P_R and cal P_R; adopting a single consistent symbol throughout would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help clarify the presentation of our results. We address each major comment below and are happy to revise the manuscript accordingly to strengthen the derivation and add explicit checks.

read point-by-point responses

  1. Referee: [Abstract / equation-rewriting section] Abstract and the section rewriting the Langevin/Fokker-Planck equations: the central claim that the fractional one-loop correction ΔP_R / P0_R enters solely as a multiplicative factor (1 + ΔP_R / P0_R)^{1/2} on the white-noise amplitude, without generating extra drift corrections, colored-noise components, or renormalization counterterms in the Fokker-Planck operator, is asserted but not derived. In the SR-USR-SR setup the USR phase already amplifies superhorizon modes; an explicit step-by-step translation from the in-in result to the stochastic equations is required to confirm that no interaction-induced terms arise.

    Authors: We agree that the manuscript would benefit from a more explicit derivation of the mapping. The noise amplitude in the stochastic δN formalism is fixed by requiring that the two-point function of the curvature perturbation matches the power spectrum computed in QFT. When interactions are present, the one-loop in-in calculation supplies the corrected variance ΔP_R, which directly rescales the white-noise strength by the factor (1 + ΔP_R/P0_R)^{1/2}. Because the interactions enter only through this perturbative correction to the variance and the deterministic drift remains the classical slow-roll or ultra-slow-roll evolution, no additional drift, multiplicative-noise, or renormalization terms appear in the Fokker-Planck operator at the order we work. The superhorizon amplification during the USR phase is already encoded in the deterministic part of the Langevin equation. We will add a new subsection (or appendix) that walks through this translation step by step, starting from the in-in two-point function and arriving at the modified Langevin and Fokker-Planck equations. revision: yes

  2. Referee: [Abstract] The manuscript provides no visible error analysis, validation against full QFT, or check that the modified noise reproduces the input one-loop power spectrum when the stochastic equations are solved. Without such a consistency test the translation from power-spectrum correction to noise amplitude remains an assumption rather than a demonstrated result.

    Authors: We acknowledge the absence of an explicit consistency test in the current draft. By construction, the rescaled noise variance is chosen so that the linear power spectrum recovered from the stochastic equations matches the input one-loop result. We will add a short validation subsection in the revised manuscript that solves the stochastic equations in the initial and final slow-roll phases (where analytic results are available) and verifies that the computed power spectrum agrees with the corrected P_R to within numerical precision. A brief discussion of the perturbative regime of validity of the one-loop input and the expected truncation error will also be included. A complete non-perturbative QFT benchmark for the ultra-slow-roll phase is beyond current analytic reach, but the proposed check will demonstrate internal consistency of the mapping. revision: yes

Circularity Check

0 steps flagged

Noise rescaling incorporates independent in-in one-loop input without self-referential reduction

full rationale

The paper computes the fractional one-loop power-spectrum correction ΔP_R / P0_R separately via the in-in formalism and inserts it as a multiplicative factor on the noise amplitude in the extended Langevin/Fokker-Planck equations. This step does not reduce to any of the enumerated circularity patterns: the in-in result is an external perturbative QFT calculation, the stochastic power spectrum is not being used to derive the correction (the direction is reversed), and no load-bearing self-citation, ansatz smuggling, or uniqueness theorem is invoked in the provided text. The central extension remains self-contained against the external benchmark of the in-in computation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the stochastic δN formalism remains valid when noises inherit interactions and that one-loop power spectrum corrections from in-in formalism can be directly mapped to noise amplitude without further derivation.

axioms (1)
  • domain assumption The stochastic δN formalism and associated Langevin/Fokker-Planck equations can be extended to interacting theories by incorporating QFT corrections into the noise term.
    Invoked when the paper states it extends the formalism and rewrites the equations taking QFT corrections into account.

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Forward citations

Cited by 1 Pith paper

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

  1. Stochastic inflation from a non-equilibrium renormalization group

    hep-th 2026-05 unverdicted novelty 7.0

    A generalized Fokker-Planck equation for stochastic inflation is derived from a Polchinski-type renormalization group flow on the density matrix, incorporating dissipative and diffusive corrections beyond the leading order.

Reference graph

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    INTRODUCTION Cosmological inflation is the widely accepted paradigm to address the horizon and flatness problems associated with big bang cosmology on large scales. This pro cess occurs at high energies in the early universe. Furthermore, inflation can address the origin of the seeds of the large-scale structure through quantum vacuum fluctuations [1, 2]. The...

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