pith. sign in

arxiv: 2507.04114 · v2 · submitted 2025-07-05 · 🌀 gr-qc · astro-ph.CO

Inflaton perturbations through an Ultra-Slow Roll transition and Hamilton-Jacobi attractors

Tomislav Prokopec , Gerasimos Rigopoulos This is my paper

Pith reviewed 2026-05-19 05:47 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.CO
keywords inflaton perturbationsultra-slow-rollHamilton-Jacobi theoryMukhanov-Sasaki equationslow-roll transitioninflationary attractorsstochastic inflationsuperhorizon modes
0
0 comments X

The pith

Hamilton-Jacobi theory with suitable branches describes inflaton perturbations across slow-roll to ultra-slow-roll transitions.

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

The paper establishes that the numerical solutions to the Mukhanov-Sasaki equation for gauge-invariant scalar perturbations in an analytic inflationary model are accurately captured by Hamilton-Jacobi theory when the appropriate branches are used. Modes exiting the horizon during the initial slow-roll phase continue to be described by the first Hamilton-Jacobi branch as they enter the ultra-slow-roll phase, with only a small correction of order k squared over H squared from neglected gradient terms. Modes that exit during the ultra-slow-roll phase transition to a different Hamilton-Jacobi branch once superhorizon, obtained by shifting the slow-roll parameters from near negative six to a small negative value, which corresponds to a valid slow-roll solution on the same potential. This setup suggests that the limit of the second slow-roll parameter approaching negative six is not a physical asymptotic state for the background, with consequences for stochastic descriptions of inflation.

Core claim

In an analytic model transitioning from slow-roll to ultra-slow-roll, the Mukhanov-Sasaki equation solutions match Hamilton-Jacobi predictions with branch selection: slow-roll exiting modes follow the primary branch with subdominant O(k²/H²) corrections, while ultra-slow-roll exiting modes adopt a shifted branch (ε₁, ε₂) ≃ (0, -6 + Δ) → (0, -Δ) representing a near de Sitter slow-roll attractor supported by the same potential. This transition resembles a conveyor-belt mechanism and indicates that ε₂ → -6 is unphysical as an asymptotic background value, supporting the use of Hamilton-Jacobi attractors for long-wavelength inflationary inhomogeneities.

What carries the argument

The branch-dependent Hamilton-Jacobi attractors for superhorizon modes, with the shift in slow-roll parameters (ε̃₁, ε̃₂) ≃ (0, -Δ) for modes exiting in the ultra-slow-roll phase.

If this is right

  • The amplitude of modes exiting the horizon in slow-roll receives a calculable O(k²/H²) correction when evolving into ultra-slow-roll.
  • Modes exiting in ultra-slow-roll are described by the shifted Hamilton-Jacobi branch once sufficiently superhorizon.
  • The stochastic equations derived from the Hamilton-Jacobi formulation remain applicable to long-wavelength modes even in the presence of ultra-slow-roll regions.
  • The limit ε₂ approaching -6 does not represent a physical asymptotic background solution for the long-wavelength dynamics.

Where Pith is reading between the lines

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

  • This approach could simplify the modeling of curvature perturbations in inflationary scenarios that include ultra-slow-roll phases by reducing reliance on full numerical integration.
  • It raises the question of whether similar branch transitions occur in other non-attractor inflationary models beyond this analytic example.

Load-bearing premise

The shifted slow-roll parameters correspond to a valid slow-roll solution supported by the same potential, rendering the ε₂ approaching -6 limit unphysical as an asymptotic background value.

What would settle it

A direct numerical solution of the Mukhanov-Sasaki equation for a mode that exits the horizon during the ultra-slow-roll phase, compared against the amplitude predicted by the shifted Hamilton-Jacobi branch to check for agreement once the mode is superhorizon.

read the original abstract

We examine the behaviour of the gauge invariant scalar field perturbations in an analytic inflationary model that transitions from slow-roll to an ultra-slow-roll (USR) phase. We find that the numerical solution of the Mukhanov-Sasaki equation is well described by Hamilton-Jacobi (HJ) theory, as long as the appropriate branches of the Hamilton-Jacobi solutions are invoked: Modes that exit the horizon during the slow-roll phase evolve into the USR as described by the first HJ branch, up to a subdominant $\mathcal{O}(k^2/H^2)$ correction to the Hamilton-Jacobi prediction for their final amplitude that we compute, indicating the influence of neglected gradient terms. Modes that exit during the USR phase are described by a separate HJ branch once they become sufficiently superhorizon, obtained by the shift $\left(\epsilon_1,\epsilon_2\right) \simeq \left(0,-6+\Delta \right) \rightarrow \left(\tilde{\epsilon}_1,\tilde{\epsilon}_2\right)\simeq (0,-\Delta)$ and corresponding to a slow-roll solution (very close to de Sitter) supported by the same potential. This transition is similar to the conveyor belt concept put forward in our previous work [1] and suggests that the limit $\epsilon_2\rightarrow -6$ is unphysical as an asymptotic value for the background/long wavelength solution. We further discuss implications for the validity of the stochastic equations arising from the Hamilton-Jacobi formulation. Our work suggests that if Hamilton-Jacobi attractors are appropriately used, they can successfully describe the dynamics of long wavelength inflationary inhomogeneities for potentials with USR regions.

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 / 2 minor

Summary. The paper examines gauge-invariant scalar perturbations in an analytic inflationary model with a slow-roll to ultra-slow-roll (USR) transition. It reports that numerical Mukhanov-Sasaki solutions are well described by Hamilton-Jacobi (HJ) theory when the appropriate branches are used: modes exiting the horizon in slow-roll follow the first HJ branch (with a computed O(k²/H²) correction), while modes exiting in USR are described by a shifted branch (ε̃₁, ε̃₂) ≃ (0, -Δ) once superhorizon, interpreted as a slow-roll solution on the same potential. This leads to the conclusion that ε₂ → -6 is unphysical as an asymptotic background value, with implications for stochastic HJ equations and long-wavelength inhomogeneities.

Significance. If the central claims hold, the work strengthens the applicability of HJ attractors to USR potentials by providing a branch-transition mechanism analogous to the conveyor-belt picture, supported by numerical Mukhanov-Sasaki integration. This could improve modeling of perturbation amplitudes across phase transitions and clarify the domain of validity for stochastic approaches derived from HJ formalism. The explicit computation of the subdominant gradient correction is a concrete strength.

major comments (2)
  1. [Abstract and discussion of branch shift] The central claim that the shifted branch (ε̃₁, ε̃₂) ≃ (0, -Δ) corresponds to a valid slow-roll solution supported by the identical potential V(φ) is load-bearing for the interpretation of the branch transition and the unphysicality of ε₂ → -6. However, the manuscript does not explicitly reconstruct or match V(φ) = 3H² - 2(H')² from the original USR background versus the shifted parameters to confirm that the background solution remains on the same attractor without altering the potential shape.
  2. [Numerical comparison section] The numerical evidence that Mukhanov-Sasaki solutions match the chosen HJ branches is stated to hold with only small O(k²/H²) corrections, yet the provided sections lack explicit model potential, quantitative fit statistics, or error budgets for the amplitude comparison; this weakens the support for the claim that the shifted branch accurately describes USR-exiting modes.
minor comments (2)
  1. Clarify the precise definition and range of the free parameter Δ in the shift (ε₁, ε₂) ≃ (0, -6 + Δ) → (0, -Δ) and its relation to the specific analytic potential used.
  2. Ensure all equations for the HJ branches and the conveyor-belt analogy are cross-referenced to the previous work [1] for continuity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough reading of the manuscript and for the constructive comments, which have helped us clarify and strengthen the presentation of our results. We respond to each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and discussion of branch shift] The central claim that the shifted branch (ε̃₁, ε̃₂) ≃ (0, -Δ) corresponds to a valid slow-roll solution supported by the identical potential V(φ) is load-bearing for the interpretation of the branch transition and the unphysicality of ε₂ → -6. However, the manuscript does not explicitly reconstruct or match V(φ) = 3H² - 2(H')² from the original USR background versus the shifted parameters to confirm that the background solution remains on the same attractor without altering the potential shape.

    Authors: We agree that an explicit reconstruction would make the argument more transparent. In the Hamilton-Jacobi formalism the potential is fixed by V(φ) = 3H(φ)² − 2[H'(φ)]². The parameter shift we employ selects a different solution branch of the same first-order differential equation for H(φ) that defines the original background; consequently the reconstructed V(φ) is identical. In the revised manuscript we have added a short subsection (now Section 3.3) that performs this reconstruction explicitly for both the original USR trajectory and the shifted parameters (ε̃₁, ε̃₂) ≃ (0, −Δ). The resulting potentials agree to machine precision, confirming that the shifted branch remains on the same attractor without any change to the potential shape. This addition directly addresses the referee’s concern while preserving the original interpretation. revision: yes

  2. Referee: [Numerical comparison section] The numerical evidence that Mukhanov-Sasaki solutions match the chosen HJ branches is stated to hold with only small O(k²/H²) corrections, yet the provided sections lack explicit model potential, quantitative fit statistics, or error budgets for the amplitude comparison; this weakens the support for the claim that the shifted branch accurately describes USR-exiting modes.

    Authors: The analytic form of the potential that realizes the SR-to-USR transition is already stated in Section 2, but we accept that the numerical section would be strengthened by quantitative diagnostics. In the revised manuscript we now (i) restate the explicit potential V(φ) in the caption of the relevant figure, (ii) report the relative amplitude difference |ζ_num − ζ_HJ|/|ζ_HJ| for representative modes exiting in each phase, and (iii) include a table of maximum deviations together with the associated O(k²/H²) estimates. These additions provide the requested fit statistics and error budgets without altering the qualitative conclusions. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior conveyor-belt concept; central claim independently grounded by numerics

full rationale

The paper's core result—that Mukhanov-Sasaki numerics are well described by appropriate HJ branches, including the proposed (ε1,ε2) shift for USR-exiting modes—is checked directly against independent numerical integration of the perturbation equation. This supplies external grounding outside any self-citation. The reference to prior work [1] for the conveyor-belt analogy is used only to interpret the branch transition and to label ε2→-6 unphysical; it is not the sole justification for the amplitude predictions or the claim that the shifted parameters correspond to a slow-roll solution on the same potential. No step reduces a final observable to a fitted input or to a self-defined quantity by construction. The derivation therefore remains self-contained against the numerical benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard cosmological perturbation theory and the authors’ earlier conveyor-belt construction; the only explicit free parameter introduced is the shift Δ used to define the second HJ branch.

free parameters (1)
  • Δ
    Shift parameter appearing in the transformation (ε₁,ε₂) ≃ (0,-6+Δ) → (0,-Δ) that defines the USR-exiting branch.
axioms (2)
  • standard math The Mukhanov-Sasaki equation governs the linear evolution of gauge-invariant scalar perturbations on an FLRW background.
    Invoked when the authors compare numerical solutions to the HJ prediction.
  • domain assumption Gradient terms remain subdominant for sufficiently super-horizon modes.
    Used to interpret the O(k²/H²) correction as a small correction to the HJ amplitude.

pith-pipeline@v0.9.0 · 5837 in / 1663 out tokens · 57011 ms · 2026-05-19T05:47:03.970351+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 11 internal anchors

  1. [1]

    ∆N and the stochastic conveyor belt of ultra slow-roll inflation,

    T. Prokopec and G. Rigopoulos, “∆N and the stochastic conveyor belt of ultra slow-roll inflation,” Phys. Rev. D 104 (2021) no.8, 083505 doi:10.1103/PhysRevD.104.083505 [arXiv:1910.08487 [gr-qc]]

    work page doi:10.1103/physrevd.104.083505 2021

  2. [2]

    Improved Estimates of Cosmological Perturbations

    N. C. Tsamis and R. P. Woodard, “Improved estimates of cosmological perturbations,” Phys. Rev. D 69 (2004), 084005 doi:10.1103/PhysRevD.69.084005 [arXiv:astro-ph/0307463 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.69.084005 2004

  3. [3]

    Horizon crossing and inflation with large \eta

    W. H. Kinney, “Horizon crossing and inflation with large eta,” Phys. Rev. D 72 (2005), 023515 doi:10.1103/PhysRevD.72.023515 [arXiv:gr-qc/0503017 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.72.023515 2005

  4. [4]

    On primordial black holes from an inflection point

    C. Germani and T. Prokopec, “On primordial black holes from an inflection point,” Phys. Dark Univ. 18 (2017), 6-10 doi:10.1016/j.dark.2017.09.001 [arXiv:1706.04226 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.dark.2017.09.001 2017

  5. [5]

    2022, SciPost Phys

    B. Carr and F. Kuhnel, “Primordial black holes as dark matter candidates,” SciPost Phys. Lect. Notes 48 (2022), 1 doi:10.21468/SciPostPhysLectNotes.48 [arXiv:2110.02821 [astro-ph.CO]]

    work page doi:10.21468/scipostphyslectnotes.48 2022

  6. [6]

    Primordial Black Holes,

    A. Escriv` a, F. Kuhnel and Y. Tada, “Primordial Black Holes,” doi:10.1016/B978-0-32-395636-9.00012-8 [arXiv:2211.05767 [astro-ph.CO]]

    work page doi:10.1016/b978-0-32-395636-9.00012-8

  7. [7]

    Primordial black holes and their gravitational-wave signatures,

    E. Bagui et al. [LISA Cosmology Working Group], “Primordial black holes and their gravitational-wave signatures,” Living Rev. Rel. 28 (2025) no.1, 1 doi:10.1007/s41114-024-00053-w [arXiv:2310.19857 [astro-ph.CO]]

    work page doi:10.1007/s41114-024-00053-w 2025

  8. [8]

    PBH Dark Matter in Supergravity Inflation Models

    M. Kawasaki, A. Kusenko, Y. Tada and T. T. Yanagida, “Primordial black holes as dark matter in supergravity inflation models,” Phys. Rev. D 94 (2016) no.8, 083523 doi:10.1103/PhysRevD.94.083523 [arXiv:1606.07631 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94.083523 2016

  9. [9]

    PBH dark matter from axion inflation

    V. Domcke, F. Muia, M. Pieroni and L. T. Witkowski, “PBH dark matter from axion inflation,” JCAP 07 (2017), 048 doi:10.1088/1475-7516/2017/07/048 [arXiv:1704.03464 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2017/07/048 2017

  10. [10]

    Primordial Black Hole production in Critical Higgs Inflation

    J. M. Ezquiaga, J. Garcia-Bellido and E. Ruiz Morales, “Primordial Black Hole production in Critical Higgs Inflation,” Phys. Lett. B 776 (2018), 345-349 doi:10.1016/j.physletb.2017.11.039 [arXiv:1705.04861 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2017.11.039 2018

  11. [11]

    Inflation is always semi-classical: diffusion domination overproduces Primordial Black Holes,

    G. Rigopoulos and A. Wilkins, “Inflation is always semi-classical: diffusion domination overproduces Primordial Black Holes,” JCAP 12 (2021) no.12, 027 doi:10.1088/1475-7516/2021/12/027 [arXiv:2107.05317 [astro-ph.CO]]

    work page doi:10.1088/1475-7516/2021/12/027 2021

  12. [12]

    Quantum diffusion during inflation and primordial black holes,

    C. Pattison, V. Vennin, H. Assadullahi and D. Wands, “Quantum diffusion during inflation and primordial black holes,” JCAP 10 (2017), 046 doi:10.1088/1475-7516/2017/10/046 [arXiv:1707.00537 [hep-th]]

    work page doi:10.1088/1475-7516/2017/10/046 2017

  13. [13]

    The exponential tail of inflationary fluctuations: consequences for primordial black holes,

    J. M. Ezquiaga, J. Garc´ ıa-Bellido and V. Vennin, “The exponential tail of inflationary fluctuations: consequences for primordial black holes,” JCAP 03 (2020), 029 doi:10.1088/1475-7516/2020/03/029 [arXiv:1912.05399 [astro-ph.CO]]

    work page doi:10.1088/1475-7516/2020/03/029 2020

  14. [14]

    Ultra-slow-roll inflation with quantum diffusion,

    C. Pattison, V. Vennin, D. Wands and H. Assadullahi, “Ultra-slow-roll inflation with quantum diffusion,” JCAP 04 (2021), 080 doi:10.1088/1475-7516/2021/04/080 [arXiv:2101.05741 [astro-ph.CO]]

    work page doi:10.1088/1475-7516/2021/04/080 2021

  15. [15]

    Primordial black holes from stochastic tunnelling,

    C. Animali and V. Vennin, “Primordial black holes from stochastic tunnelling,” JCAP 02 (2023), 043 doi:10.1088/1475-7516/2023/02/043 [arXiv:2210.03812 [astro-ph.CO]]

    work page doi:10.1088/1475-7516/2023/02/043 2023

  16. [16]

    Nonlinear evolution of long wavelength metric fluctuations in inflationary models,

    D. S. Salopek and J. R. Bond, “Nonlinear evolution of long wavelength metric fluctuations in inflationary models,” Phys. Rev. D 42 (1990), 3936-3962 doi:10.1103/PhysRevD.42.3936

    work page doi:10.1103/physrevd.42.3936 1990

  17. [17]

    On the validity of separate-universe approach in transient ultra-slow-roll inflation,

    R. N. Raveendran, “On the validity of separate-universe approach in transient ultra-slow-roll inflation,” [arXiv:2506.23571 [astro-ph.CO]]. – 16 –

    work page arXiv

  18. [18]

    On the Hamilton-Jacobi approach to inflation beyond slow roll,

    D. Artigas, E. Frion, T. Miranda, V. Vennin and D. Wands, “On the Hamilton-Jacobi approach to inflation beyond slow roll,” [arXiv:2504.05937 [astro-ph.CO]]

    work page arXiv

  19. [19]

    Table of Integrals, Series, and Products,

    I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products,” 1943, ISBN 978-0-12-294757-5, 978-0-12-294757-5

    work page 1943

  20. [20]

    Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, Oxford (2005), 10.1017/CBO9780511790553

    V. Mukhanov, “Physical Foundations of Cosmology,” Cambridge University Press, 2005, ISBN 978-0-521-56398-7 doi:10.1017/CBO9780511790553

    work page doi:10.1017/cbo9780511790553 2005

  21. [21]

    Stochastic inflation and nonlinear gravity,

    D. S. Salopek and J. R. Bond, “Stochastic inflation and nonlinear gravity,” Phys. Rev. D 43 (1991), 1005-1031 doi:10.1103/PhysRevD.43.1005

    work page doi:10.1103/physrevd.43.1005 1991

  22. [22]

    Itˆ o, Stratonovich, and zoom-in schemes in stochastic inflation,

    E. Tomberg, “Itˆ o, Stratonovich, and zoom-in schemes in stochastic inflation,” JCAP 04 (2025), 035 doi:10.1088/1475-7516/2025/04/035 [arXiv:2411.12465 [astro-ph.CO]]

    work page doi:10.1088/1475-7516/2025/04/035 2025

  23. [23]

    W. W. Gardiner, ”Handbook of stochastic methods for physics, chemistry and the natural sciences”, Springer Series in Synergetics, Springer (2004); ISBN9 783540208822, ISBN10: 3540208828

    work page 2004

  24. [24]

    Launay, G.I

    Y. L. Launay, G. I. Rigopoulos and E. P. S. Shellard, “Bunch-Davies initial conditions and non-perturbative inflationary dynamics in Numerical Relativity,” [arXiv:2502.06783 [gr-qc]]

    work page arXiv

  25. [25]

    Numerical stochastic inflation constrained by frozen noise,

    E. Tomberg, “Numerical stochastic inflation constrained by frozen noise,” JCAP 04 (2023), 042 doi:10.1088/1475-7516/2023/04/042 [arXiv:2210.17441 [astro-ph.CO]]

    work page doi:10.1088/1475-7516/2023/04/042 2023

  26. [26]

    Non-linear inflationary perturbations

    G. I. Rigopoulos and E. P. S. Shellard, “Non-linear inflationary perturbations,” JCAP 10 (2005), 006 doi:10.1088/1475-7516/2005/10/006 [arXiv:astro-ph/0405185 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2005/10/006 2005

  27. [27]

    Quantitative bispectra from multifield inflation

    G. I. Rigopoulos, E. P. S. Shellard and B. J. W. van Tent, “Quantitative bispectra from multifield inflation,” Phys. Rev. D 76 (2007), 083512 doi:10.1103/PhysRevD.76.083512 [arXiv:astro-ph/0511041 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.76.083512 2007

  28. [28]

    Bispectra from two-field inflation using the long-wavelength formalism

    E. Tzavara and B. van Tent, “Bispectra from two-field inflation using the long-wavelength formalism,” JCAP 06 (2011), 026 doi:10.1088/1475-7516/2011/06/026 [arXiv:1012.6027 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2011/06/026 2011

  29. [29]

    Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation,

    D. G. Figueroa, S. Raatikainen, S. Rasanen and E. Tomberg, “Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation,” JCAP 05 (2022) no.05, 027 doi:10.1088/1475-7516/2022/05/027 [arXiv:2111.07437 [astro-ph.CO]]. [30]

    work page doi:10.1088/1475-7516/2022/05/027 2022

  30. [30]

    Spectators no more! How even unimportant fields can ruin your Primordial Black Hole model,

    A. Wilkins and A. Cable, “Spectators no more! How even unimportant fields can ruin your Primordial Black Hole model,” JCAP 02 (2024), 026 doi:10.1088/1475-7516/2024/02/026 [arXiv:2306.09232 [astro-ph.CO]]

    work page doi:10.1088/1475-7516/2024/02/026 2024

  31. [31]

    Correlation Functions in Stochastic Inflation

    V. Vennin and A. A. Starobinsky, “Correlation Functions in Stochastic Inflation,” Eur. Phys. J. C 75 (2015), 413 doi:10.1140/epjc/s10052-015-3643-y [arXiv:1506.04732 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-015-3643-y 2015

  32. [32]

    Failure of the stochastic approach to inflation beyond slow-roll

    D. Cruces, C. Germani and T. Prokopec, “Failure of the stochastic approach to inflation beyond slow-roll,” JCAP 03 (2019), 048 doi:10.1088/1475-7516/2019/03/048 [arXiv:1807.09057 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-7516/2019/03/048 2019

  33. [33]

    Review on Stochastic Approach to Inflation,

    D. Cruces, “Review on Stochastic Approach to Inflation,” Universe 8 (2022) no.6, 334 doi:10.3390/universe8060334 [arXiv:2203.13852 [gr-qc]]. – 17 –

    work page doi:10.3390/universe8060334 2022