Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

$\delta N$ formalism with gradient interactions

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A single effective source term extends the δN formalism to include spatial-gradient interactions, reproducing exact power spectra and equilateral non-Gaussianity in ultra-slow-roll inflation.

desk verdict The source-term reformulation is genuinely useful, but the f_NL result is not an independent test of the method — it feeds the exact MS solution into the source, and the 'fully nonlinear' language overreaches. read the letter →

arxiv 2602.00902 v3 pith:MJ2V2IN6 submitted 2026-01-31 gr-qc hep-th

S. Mohammad Ahmadi , Nahid Ahmadi This is my paper · ORCID
classification gr-qchep-th MSC 83F0583C25 PACS 98.80.Cq98.80.-k
keywords inflationδNformalismseparateuniverseapproximationgradientexpansionultra-slow-rollprimordialblackholesnon-Gaussianitycurvatureperturbations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The standard δN formalism computes curvature perturbations as the difference in expansion between perturbed and unperturbed homogeneous patches, and it assumes spatial gradients can be ignored. That assumption fails during ultra-slow-roll phases relevant to primordial black hole formation, where gradients keep the curvature perturbation evolving on super-Hubble scales. This paper tries to repair the gap by adding a single effective source term to the background Klein-Gordon equation, a term that acts as a proxy for the spatial Laplacian. The authors show that this sourced equation reproduces, order by order in k^2, the gradient expansion of the exact perturbation equation, and that when the source is supplied with the full linear solution the resulting power spectrum matches the exact result even for matching at horizon crossing. They then compute the equilateral non-Gaussianity parameter f_NL^eq for two potentials with transient non-slow-roll phases and find gradient-induced features that the standard δN misses but perturbative bispectrum calculations reproduce.

What carries the argument

The key object is the effective source term S_k added to the otherwise homogeneous Klein-Gordon equation. It stands in for the spatial Laplacian term that the separate-universe approximation discards, and it is built from the gauge-invariant curvature perturbation R_k, whose super-Hubble evolution is supplied by the gradient expansion of the Mukhanov-Sasaki equation (the exact linear equation for gauge-invariant scalar perturbations). Linearizing the sourced equation generates the same recursion as the k^2-expansion of the growing and decaying modes, which is the bridge that makes the δN formalism gradient-complete. The dictionary Q_k = (dφ/dt) R_k/H and the use of a time slicing with unifor

What would settle it

Compute f_NL^eq using the source term generated recursively from the gradient expansion of the curvature perturbation—rather than from the exact linear solution—and compare with a direct perturbative bispectrum calculation for the same two potentials. Agreement would confirm that the linear-order equivalence proven in the paper extends to the nonlinear quantity the method is designed to predict; disagreement would show that the reported f_NL depends on importing the exact solution as input.

Watch

Extended reading notes

Core claim

The central claim is that a modified background Klein-Gordon equation carrying the source S_k = -(k^2/(a^2 H)) (dφ/dt) R_k contains the same gradient information as the higher-order expansion of cosmological perturbation theory. Substituting the linear dictionary between the field fluctuation and the curvature perturbation, Q_k = (dφ/dt) R_k/H, turns the sourced equation into the recursion that builds the k^(2(i+1)) corrections to the growing and decaying super-Hubble modes; this is the formal equivalence to the gradient-expansion matching method. With that source evaluated from the full linear solution, the method reproduces the exact power spectrum for a smooth Gaussian-bump potential and

Load-bearing premise

The load-bearing premise is that all relevant spatial-gradient effects can be captured by a source term built from the linear curvature perturbation R_k; the paper's own bispectrum calculation feeds that source the exact linear solution, and the conclusions acknowledge that a fully self-consistent recursive evaluation, including backreaction and genuine nonlinearity, is not yet implemented. If the recursive version does not reproduce the same f_NL, the method is importing the

Editorial extensions

If this is right

  • The matching time for δN calculations can be pushed to horizon crossing (σ = 1), removing the usual need to wait several e-folds for gradient and decaying modes to die out before switching to separate-universe evolution.
  • Equilateral non-Gaussianity sourced by gradients becomes computable from background dynamics plus linear perturbation equations, avoiding a full second-order calculation.
  • Standard δN predictions for f_NL in models with transient ultra-slow-roll should be revisited, because the standard approach fails to capture the dominant gradient-induced features.
  • The comparison with the spatial-curvature version of δN shows that a simple curvature assignment misses half the leading gradient correction; the source-term ladder gives a systematic way to add the missing pieces.
  • The framework sets up a direct comparison with stochastic-δN treatments of slow-roll-violating models, where noise and gradients are both important.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the paper's bispectrum test is not fully self-consistent, since the source term is fed the exact Mukhanov-Sasaki solution rather than the method's own recursive estimate; the sharper test is to run the recursion for f_NL and compare with in-in results.
  • Editorial extension: if the method holds, the usual ambiguity in choosing the δN matching scale may largely disappear, since the source term carries gradient information and allows matching at k = aH.
  • Editorial extension: the factor-of-two rescaling of spatial curvature (K to K/2) suggested by the appendix offers a cheap fix for existing curved separate-universe codes, although the appendix warns that higher-order contamination remains.
  • Editorial extension: because primordial black hole abundance is exponentially sensitive to non-Gaussian tails, the gradient-induced f_NL peaks found here could shift PBH abundance estimates by orders of magnitude in these models; the paper motivates but does not perform that calculation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes an extension of the δN formalism in which spatial-gradient effects are included by adding the source term S_k = -(k^2/a^2 H) dφ/dt \hat R_k to the background Klein–Gordon equation (Eq. 3.1). The authors argue that this source-term construction is formally equivalent, at linear order, to the higher-order gradient-expansion matching method, and that it permits tracking curvature perturbations from horizon crossing to the end of inflation in single-field models with transient ultra-slow-roll phases. They validate the power spectrum against the exact Mukhanov–Sasaki solution for a Gaussian-bump model and the Starobinsky linear-potential model, and compute the equilateral f_NL using Eq. (3.11), reporting agreement with existing in-in results and a failure of standard δN to capture these features. An appendix discusses an alternative spatial-curvature extension of δN and identifies a factor-of-two discrepancy.

Significance. If established, the proposed construction would be a useful and efficient bridge between the nonlinear separate-universe approach and full cosmological perturbation theory, with direct applications to PBH formation models where gradient effects are known to be important. The paper has genuine strengths: the formal correspondence between the linearized source-term equation and the higher-order gradient expansion is shown explicitly in §3.1; the power-spectrum comparisons in Fig. 1 include a step-by-step inclusion of adiabatic and non-adiabatic corrections rather than a single aggregate fit; and the numerical code is made publicly available. However, the central f_NL demonstration in §3.3 is not a validation of the method's own approximation scheme, because the source term is evaluated using the exact Mukhanov–Sasaki solution. Consequently, the headline claim that the gradient-corrected δN formalism captures essential non-Gaussian features missed by standard δN is not yet supported by the evidence presented.

major comments (3)
  1. [§3.3, Eq. (3.1), Fig. 2] The f_NL computation is not an independent test of the gradient-source method. The text states: 'we instead numerically evaluate the source term with full gradients derived from the exact MS equation.' Substituting the exact R_k into Eq. (3.1) makes the linearized Klein–Gordon equation algebraically identical to the Mukhanov–Sasaki equation, so the resulting δφ_k and dδφ_k/dt are the exact linear solutions. Equation (3.11) then tests only the δN Taylor-map relation between those exact phase-space perturbations and the curvature perturbation, not the predictive content of the source-term construction. What is missing is a f_NL calculation using the method's own recursive, order-by-order source (the analogue of the step-by-step power-spectrum curves in Fig. 1), with convergence to the in-in result checked. As it stands, Fig. 2 cannot discriminate the gradient-corrected δN scheme from the s
  2. [Abstract and §4] The claim of a 'fully nonlinear treatment' is not supported by the implementation. The source term S_k in Eq. (3.1) is linear in R_k, and §4 explicitly concedes that the source is computed under first-order perturbation theory with backreaction neglected. The only nonlinearity retained is the δN derivative expansion of Eq. (2.19). The manuscript should either remove the 'fully nonlinear' language or restrict it to the δN sector, and should not describe the gradient interactions themselves as nonlinear beyond first order in the perturbation.
  3. [§3.2, Eqs. (3.2)–(3.3)] The formal equivalence established in Eqs. (3.2)–(3.3) is a linear-order statement: it shows that the i-th order source reproduces the (i+1)-th order term of the gradient expansion of the linear Mukhanov–Sasaki equation. It does not demonstrate equivalence at the level of the full nonlinear bispectrum, where the source would need to include quadratic (or higher) products of perturbations. The paper's central claim about f_NL therefore rests on this linear-order equivalence plus a separate, unvalidated assumption that the linear exact source is adequate for a three-point function. This limitation should be stated explicitly in §3.3 and the conclusion.
minor comments (4)
  1. [§2.3, Eq. (2.16)] The integration limits in Eq. (2.16) use an integral from 0 to η* and a reference time η_ref in Eq. (2.14). Please clarify the normalization and the meaning of the lower limit; the matching constants should be invariant under the choice of η_ref.
  2. [Fig. 1 and Fig. 2] The horizontal axis is labeled 'log(k)' without specifying units or normalization. Since the models have different parameter scales, the plots would be more informative if k were shown in units of the reference scale or in physical Mpc^-1, or if the normalization convention (large-scale power set to unity) were stated in the caption.
  3. [Footnote 1] The argument that ζ = R exactly even at k = aH relies on setting B = E = 0 in the separate-universe dictionary. The displayed perturbative relation contains the Bardeen potential Ψ, which is not generally zero, so the claim that the difference 'vanishes even at length scales k ~ aH' needs a more explicit derivation or a citation to a proof.
  4. [§2.4] The notation 'δN = B = 0' is introduced without defining the gauge variable B here; the same symbol is used for the shift component in Eq. (2.2). Please disambiguate the two uses.

Circularity Check

1 steps flagged · score 6.0 of 10

f_NL evidence is partially circular: the source term is fed the exact MS solution, so the full-gradient power spectrum and the f_NL results validate the δN mapping rather than the recursive gradient-source method.

  1. fitted input called prediction [§3.3 (Non-Gaussianity), with Eq. (3.1) and Fig. 2]
    "One could incorporate gradient corrections order by order into the source term and compute f_eq_NL using Eq. (3.11) at each step, analogous to the power spectrum analysis shown in Figure 1. However, to facilitate a direct and accurate comparison with previous works, we instead numerically evaluate the source term with full gradients derived from the exact MS equation."

    Eq. (3.1) defines the source as S_k = -(k^2 \dotφ/a^2H) \hat R_k, where \hat R_k is precisely the curvature perturbation whose evolution the method is supposed to predict. For the f_NL computation the paper does not use the recursive order-by-order \hat R_k from the gradient expansion; it uses the exact Mukhanov-Sasaki solution. Substituting Q = zR/a into the linearized Eq. (3.1) gives (z^2 R')' = -k^2 z^2 \hat R, which, with \hat R = R, is exactly the MS equation (2.11). Thus the full-gradient power spectrum in Fig. 1 is reproduced by construction, and the f_NL calculation in Fig. 2 inherits the exact linear solution as input. It therefore tests the δN map between phase-space perturbations and final curvature, not the gradient-source algorithm itself. The paper concedes in §4 that S_k is

full rationale

The paper contains a genuinely non-circular core: the order-by-order power-spectrum validation (colored lines and markers in Fig. 1) uses the source term built from the gradient expansion, not from the exact MS solution, and the formal identity Eqs. (3.2)-(3.3) showing equivalence to higher-order matching is a mathematical derivation, not a fit. However, the headline f_NL demonstration is not produced by that self-contained scheme. The paper explicitly replaces the recursive source with 'full gradients derived from the exact MS equation', i.e. the exact solution of the very equation the method aims to reproduce. At linear order this makes the full-gradient power-spectrum agreement tautological. The f_NL result is therefore not an independent prediction of the method: it is a δN computation driven by the exact linear solution, and the claimed agreement with in-in perturbation theory does not establish the gradient-source construction at second order. The §4 admission that the source is first-order and that backreaction is neglected further undercuts the 'fully nonlinear treatment' claim. These issues are localized to the f_NL evidence; the recursive matching validation and the formal equivalence remain independent content. Hence the circularity is partial, not total: a score of 6 is appropriate.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The main method introduces no new free parameters beyond the matching scale σ=1; the model potentials and coefficients are taken from prior literature (Refs [47,52]). The derivation relies on standard results of linear cosmological perturbation theory and on the uniform-expansion-gauge dictionary of Ref [42]. The most fragile assumption is the use of a first-order source term inside a 'nonlinear' δN expansion.

free parameters (1)
  • Matching scale σ = k/(aH) at horizon crossing = 1
    Matching between quantum and classical evolution is performed at horizon crossing (σ=1) rather than several e-folds later; the paper argues ζ=R at this scale, but this choice is central to the gradient-correction scheme.
assumptions (4)
  • domain assumption Uniform-expansion gauge provides a dictionary between cosmological perturbation theory and the separate-universe approach such that anisotropic modes decouple (Ref [42]).
    Invoked throughout to justify δN with a source term; if this dictionary fails at σ=1, the extension breaks down.
  • domain assumption In the gradient-expansion matching, higher-order corrections satisfy R^(i)_{k*} = R'^(i)_{k*} = 0 for i ≥ 1 at the matching time (just below Eq. 3.3).
    This is the standard prescription of the higher-order matching method (Ref [22]); it is used to prove equivalence between the source-term method and the gradient expansion.
  • domain assumption Single-clock inflation leaves only two independent integration constants, chosen as X = (φ_in, dφ/dt_in); R and R' are eliminated via the linear constraint R_k = H Q_k / (dφ/dt) (Sec. 3.1).
    Allows the δN expansion of Eq. (2.19) to be closed; if this constraint is violated in non-single-clock regimes, the method misses degrees of freedom.
  • ad hoc to paper The source term is computed using first-order perturbation theory for R_k, neglecting backreaction (stated in Sec. 4).
    This is a deliberate approximation; the paper acknowledges a recursive improvement is left for future work. It is load-bearing for the 'fully nonlinear' claim.

how reviews work

0 comments
Cite this review

Pith. "Pith review of $\delta N$ formalism with gradient interactions." pith.science (2026). https://pith.science/paper/MJ2V2IN6

@misc{pith2026260200902,
  author       = {Pith},
  title        = {Pith review of: $\delta N$ formalism with gradient interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MJ2V2IN6}},
  note         = {Machine review of arXiv:2602.00902}
}
abstract

The standard $\delta N$ formalism is a cornerstone technique for calculating curvature perturbations on super-Hubble scales. However, its validity relies heavily on the separate universe assumption, in which spatial gradients are neglected. This approximation is known to break down in scenarios that are critical for primordial black hole formation, such as transitions to an ultra-slow-roll phase, where gradient interactions induce a significant non-conservation of the comoving curvature perturbation. In this paper, we introduce a framework for incorporating gradient corrections into the $\delta N$ formalism by adding an effective source term to the background Klein-Gordon equation. While preserving the nonlinear separate-universe dynamics at zeroth order, this approach consistently incorporates the leading gradient-sensitive effects and thereby improves the treatment of the nonlinear evolution of curvature perturbations up to the end of inflation, given initial conditions specified at horizon exit. By computing the equilateral non-Gaussianity parameter $f_{\mathrm{NL}}^{\mathrm{eq}}$, we demonstrate that our method captures some essential physical features missed by the standard $\delta N$ approach, offering a simple pathway to determine the nonlinear evolution expected from cosmological perturbation theory.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. How Dilatation Invariance Suppresses Loop Corrections to Curvature Perturbations on CMB Scales

    astro-ph.CO 2026-08 conditional novelty 5.0 of 10

    Dilatation invariance forces the non-linear curvature perturbation on super-Hubble scales to depend only on the decaying/growing mode, suppressing loop corrections to the CMB power spectrum even with sharp transitions.

Reference graph

Works this paper leans on

54 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Starobinsky, A new type of isotropic cosmological models without singula rity, Physics Letters B 91 (1980) 99

    A.A. Starobinsky, A new type of isotropic cosmological models without singula rity, Physics Letters B 91 (1980) 99

    1980

  2. [2]

    Sato, First-order phase transition of a vacuum and the expansion of the universe , Monthly Notices of the Royal Astronomical Society 195 (1981) 467

    K. Sato, First-order phase transition of a vacuum and the expansion of the universe , Monthly Notices of the Royal Astronomical Society 195 (1981) 467

    1981

  3. [3]

    Guth, Inflationary universe: A possible solution to the horizon and flat ness problems, Physical Review D 23 (1981) 347

    A.H. Guth, Inflationary universe: A possible solution to the horizon and flat ness problems, Physical Review D 23 (1981) 347

    1981

  4. [4]

    Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems , Physics Letters B 108 (1982) 389

    A.D. Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems , Physics Letters B 108 (1982) 389

    1982

  5. [5]

    Albrecht, P.J

    A. Albrecht, P.J. Steinhardt, M.S. Turner and F. Wilczek , Reheating an inflationary universe , Physical Review Letters 48 (1982) 1437. – 15 –

    1982

  6. [6]

    Lyth and A

    D.H. Lyth and A. Riotto, Particle physics models of inflation and the cosmological den sity perturbation, Physics Reports 314 (1999) 1

    1999

  7. [7]

    Zel’dovich and I

    Y.B. Zel’dovich and I. Novikov, The hypothesis of cores retarded during expansion and the hot cosmological model, Astronomicheskii Zhurnal 43 (1966) 758

    1966

  8. [8]

    Hawking, Gravitationally collapsed objects of very low mass , Monthly Notices of the Royal Astronomical Society 152 (1971) 75

    S. Hawking, Gravitationally collapsed objects of very low mass , Monthly Notices of the Royal Astronomical Society 152 (1971) 75

    1971

Show all 54 references
  1. [9]

    Carr and S.W

    B.J. Carr and S.W. Hawking, Black holes in the early universe , Monthly Notices of the Royal Astronomical Society 168 (1974) 399

    1974

  2. [10]

    Carr, The primordial black hole mass spectrum , Astrophysical Journal, vol

    B.J. Carr, The primordial black hole mass spectrum , Astrophysical Journal, vol. 201, Oct. 1, 1975, pt. 1, p. 1-19. Research supported by the Science Resear ch Council of England 201 (1975) 1

    1975

  3. [11]

    Chapline, Cosmological effects of primordial black holes , Nature 253 (1975) 251

    G.F. Chapline, Cosmological effects of primordial black holes , Nature 253 (1975) 251

    1975

  4. [12]

    Carr and M.J

    B. Carr and M.J. Rees, Can pregalactic objects generate galaxies? , Monthly Notices of the Royal Astronomical Society 206 (1984) 801

    1984

  5. [13]

    Starobinsky, Dynamics of phase transition in the new inflationary universe s cenario and generation of perturbations , Physics Letters B 117 (1982) 175

    A.A. Starobinsky, Dynamics of phase transition in the new inflationary universe s cenario and generation of perturbations , Physics Letters B 117 (1982) 175

    1982

  6. [14]

    Starobinski ˇ ı,Multicomponent de sitter (inflationary) stages and the gener ation of perturbations, Soviet Journal of Experimental and Theoretical Physics Lette rs 42 (1985) 152

    A. Starobinski ˇ ı,Multicomponent de sitter (inflationary) stages and the gener ation of perturbations, Soviet Journal of Experimental and Theoretical Physics Lette rs 42 (1985) 152

    1985

  7. [15]

    Sasaki and E.D

    M. Sasaki and E.D. Stewart, A general analytic formula for the spectral index of the densi ty perturbations produced during inflation , Progress of Theoretical Physics 95 (1996) 71

    1996

  8. [16]

    Sasaki and T

    M. Sasaki and T. Tanaka, Super-horizon scale dynamics of multi-scalar inflation , Progress of Theoretical Physics 99 (1998) 763

    1998

  9. [17]

    Lyth, K.A

    D.H. Lyth, K.A. Malik and M. Sasaki, A general proof of the conservation of the curvature perturbation, Journal of Cosmology and Astroparticle Physics 2005 (2005) 004

    2005

  10. [18]

    Wands, K.A

    D. Wands, K.A. Malik, D.H. Lyth and A.R. Liddle, New approach to the evolution of cosmological perturbations on large scales , Physical Review D 62 (2000) 043527

    2000

  11. [19]

    Lyth and D

    D.H. Lyth and D. Wands, Conserved cosmological perturbations, Physical Review D 68 (2003) 103515

    2003

  12. [20]

    Rigopoulos and E

    G. Rigopoulos and E. Shellard, Separate universe approach and the evolution of nonlinear superhorizon cosmological perturbations, Physical Review D 68 (2003) 123518

    2003

  13. [21]

    Pattison, V

    C. Pattison, V. Vennin, H. Assadullahi and D. Wands, Stochastic inflation beyond slow roll , Journal of Cosmology and Astroparticle Physics 2019 (2019) 031

    2019

  14. [22]

    Jackson, H

    J.H. Jackson, H. Assadullahi, A.D. Gow, K. Koyama, V. Ve nnin and D. Wands, The separate-universe approach and sudden transitions during in flation, Journal of Cosmology and Astroparticle Physics 2024 (2024) 053

    2024

  15. [23]

    Briaud, R

    V. Briaud, R. Kawaguchi and V. Vennin, Stochastic inflation with gradient interactions , Journal of Cosmology and Astroparticle Physics 2025 (2025) 024

    2025

  16. [24]

    Artigas, S

    D. Artigas, S. Pi and T. Tanaka, ExtendedδN formalism: Nonspatially flat separate-universe approach, Physical Review Letters 134 (2025) 221001

    2025

  17. [25]

    Byrnes, E.J

    C.T. Byrnes, E.J. Copeland and A.M. Green, Primordial black holes as a tool for constraining non-gaussianity, Physical Review D—Particles, Fields, Gravitation, and Cosm ology 86 (2012) 043512

    2012

  18. [26]

    Young and C.T

    S. Young and C.T. Byrnes, Primordial black holes in non-gaussian regimes , Journal of Cosmology and Astroparticle Physics 2013 (2013) 052. – 16 –

    2013

  19. [27]

    Passaglia, W

    S. Passaglia, W. Hu and H. Motohashi, Primordial black holes and local non-gaussianity in canonical inflation, Physical Review D 99 (2019) 043536

    2019

  20. [28]

    Atal and C

    V. Atal and C. Germani, The role of non-gaussianities in primordial black hole format ion, Physics of the Dark Universe 24 (2019) 100275

    2019

  21. [29]

    Biagetti, V

    M. Biagetti, V. De Luca, G. Franciolini, A. Kehagias and A. Riotto, The formation probability of primordial black holes , Physics Letters B 820 (2021) 136602

    2021

  22. [30]

    Taoso and A

    M. Taoso and A. Urbano, Non-gaussianities for primordial black hole formation , Journal of Cosmology and Astroparticle Physics 2021 (2021) 016

    2021

  23. [31]

    Young, Peaks and primordial black holes: the effect of non-gaussiani ty, Journal of Cosmology and Astroparticle Physics 2022 (2022) 037

    S. Young, Peaks and primordial black holes: the effect of non-gaussiani ty, Journal of Cosmology and Astroparticle Physics 2022 (2022) 037

    2022

  24. [32]

    Ferrante, G

    G. Ferrante, G. Franciolini, A.J. Iovino and A. Urbano, Primordial non-gaussianity up to all orders: Theoretical aspects and implications for primordial black hole models , arXiv preprint arXiv:2211.01728 (2022)

    2022 arXiv

  25. [33]

    A.D. Gow, H. Assadullahi, J.H. Jackson, K. Koyama, V. Ve nnin and D. Wands, Non-perturbative non-gaussianity and primordial black holes , Europhysics Letters 142 (2023) 49001

    2023

  26. [34]

    Komatsu and D.N

    E. Komatsu and D.N. Spergel, Acoustic signatures in the primary microwave background bispectrum, Physical Review D 63 (2001) 063002

    2001

  27. [35]

    Maldacena, Non-gaussian features of primordial fluctuations in single fi eld inflationary models, Journal of High Energy Physics 2003 (2003) 013

    J. Maldacena, Non-gaussian features of primordial fluctuations in single fi eld inflationary models, Journal of High Energy Physics 2003 (2003) 013

    2003

  28. [36]

    Bartolo, E

    N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Non-gaussianity from inflation: Theory and observations , Physics Reports 402 (2004) 103

    2004

  29. [37]

    Liguori, E

    M. Liguori, E. Sefusatti, J.R. Fergusson and E. Shellar d, Primordial non-gaussianity and bispectrum measurements in the cosmic microwave background and large-scale structure , Advances in Astronomy 2010 (2010) 980523

    2010

  30. [38]

    Chen, Primordial non-gaussianities from inflation models , Advances in Astronomy 2010 (2010) 638979

    X. Chen, Primordial non-gaussianities from inflation models , Advances in Astronomy 2010 (2010) 638979

    2010

  31. [39]

    Wang, Inflation, cosmic perturbations and non-gaussianities , Communications in Theoretical Physics 62 (2014) 109

    Y. Wang, Inflation, cosmic perturbations and non-gaussianities , Communications in Theoretical Physics 62 (2014) 109

    2014

  32. [40]

    Y.-F. Cai, X. Chen, M.H. Namjoo, M. Sasaki, D.-G. Wang an d Z. Wang, Revisiting non-gaussianity from non-attractor inflation models , Journal of Cosmology and Astroparticle Physics 2018 (2018) 012

    2018

  33. [41]

    Namjoo and B

    M.H. Namjoo and B. Nikbakht, Geometry of non-gaussianity in transient non-attractor inflation, arXiv preprint arXiv:2512.11020 (2025)

    2025

  34. [42]

    Artigas, J

    D. Artigas, J. Grain and V. Vennin, Hamiltonian formalism for cosmological perturbations: the separate-universe approach, Journal of Cosmology and Astroparticle Physics 2022 (2022) 001

    2022

  35. [43]

    Yokoyama, T

    S. Yokoyama, T. Suyama and T. Tanaka, Primordial non-gaussianity in multiscalar inflation , Physical Review D—Particles, Fields, Gravitation, and Cosm ology 77 (2008) 083511

    2008

  36. [44]

    Wands, Local non-gaussianity from inflation , Classical and Quantum Gravity 27 (2010) 124002

    D. Wands, Local non-gaussianity from inflation , Classical and Quantum Gravity 27 (2010) 124002

    2010

  37. [45]

    Hooshangi, M.H

    S. Hooshangi, M.H. Namjoo and M. Noorbala, Rare events are nonperturbative: Primordial black holes from heavy-tailed distributions , Physics Letters B 834 (2022) 137400

    2022

  38. [46]

    Hooshangi, M.H

    S. Hooshangi, M.H. Namjoo and M. Noorbala, Tail diversity from inflation , Journal of Cosmology and Astroparticle Physics 2023 (2023) 023. – 17 –

    2023

  39. [47]

    Mishra and V

    S.S. Mishra and V. Sahni, Primordial black holes from a tiny bump/dip in the inflaton potential, Journal of Cosmology and Astroparticle Physics 2020 (2020) 007

    2020

  40. [48]

    Starobinskij, Spectrum of adiabatic perturbations in the universe when there a re singularities in the inflationary potential

    A. Starobinskij, Spectrum of adiabatic perturbations in the universe when there a re singularities in the inflationary potential. , Soviet Journal of Experimental and Theoretical Physics Lette rs 55 (1992) 489

    1992

  41. [49]

    Martin and L

    J. Martin and L. Sriramkumar, The scalar bi-spectrum in the starobinsky model: The equilateral case, Journal of Cosmology and Astroparticle Physics 2012 (2012) 008

    2012

  42. [50]

    Martin, L

    J. Martin, L. Sriramkumar and D.K. Hazra, Sharp inflaton potentials and bi-spectra: Effects of smoothening the discontinuity , Journal of Cosmology and Astroparticle Physics 2014 (2014) 039

    2014

  43. [51]

    Ahmadi, M

    N. Ahmadi, M. Noorbala, N. Feyzabadi, F. Eghbalpoor and Z. Ahmadi, Quantum diffusion in sharp transition to non-slow-roll phase , Journal of Cosmology and Astroparticle Physics 2022 (2022) 078

    2022

  44. [52]

    Cole, A.D

    P.S. Cole, A.D. Gow, C.T. Byrnes and S.P. Patil, Primordial black holes from single-field inflation: a fine-tuning audit , Journal of Cosmology and Astroparticle Physics 2023 (2023) 031

    2023

  45. [53]

    The Python codes used to plot all figures in this work are p ublicly available at https://github.com/mohammadahmadi-physics/dN-formal ism-with-gradient-interactions

  46. [54]

    Hazra, L

    D.K. Hazra, L. Sriramkumar and J. Martin, Bingo: A code for the efficient computation of the scalar bi-spectrum, Journal of Cosmology and Astroparticle Physics 2013 (2013) 026. – 18 –

    2013

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.