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REVIEW 3 major objections 4 minor 95 references

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

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper argues that dilatation invariance — symmetry under overall rescaling of space — forces the super-Hubble curvature perturbation into a form whose loop corrections on CMB scales are suppressed, for any inflaton potential and at any

desk verdict Solid, careful paper that cleanly derives super-Hubble loop suppression from dilatation invariance and reconciles the flat-gauge cubic/quartic cancellation; the abstract overreaches when it claims suppression for sub-Hubble hard modes at any loop order. read the letter →

arxiv 2608.03725 v1 pith:4Q6ZVWUS submitted 2026-08-04 astro-ph.CO gr-qchep-th

Danilo Artigas , Shi Pi , Takahiro Tanaka , Yuko Urakawa This is my paper
classification astro-ph.COgr-qchep-th PACS 98.80.Cq98.80.-k
keywords dilatationinvarianceloopcorrectionscurvatureperturbationultra-slow-rollinflationprimordialblackholesseparateuniverseformalismCMBpowerspectrumconsistencyrelations
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

Departures from slow-roll inflation, such as ultra-slow-roll phases used to amplify small-scale perturbations for primordial black holes, reopen a question: can non-linear interactions of those small-scale modes feed back through quantum loops into the large-scale perturbations we see in the cosmic microwave background? This paper argues they cannot. The load-bearing observation is that the super-Hubble action of single-field inflation has a residual symmetry under overall rescaling of spatial coordinates (dilatation invariance), which forces the non-linear curvature perturbation to take the form ζ = ζ* + g(ψ,ρ): the constant adiabatic mode enters additively, and every non-linear correction is built from the time-derivative part, whose amplitude on CMB scales is tiny. Consequently, one-loop and higher-loop corrections to the CMB power spectrum are suppressed for any inflaton potential, smooth or sharp, at any loop order, provided CMB modes left the horizon during slow roll. The paper shows where apparent counterexamples come from — truncating the action at cubic order breaks dilatation invariance and manufactures a spurious enhancement — and demonstrates that in the spatially-flat gauge the would-be divergent cubic term is cancelled exactly by the quartic interaction. The statement is complete for super-Hubble modes; extending it to sub-Hubble modes requires a locality condition on the initial quantum state that the paper states but does not compute.

What carries the argument

The central object is dilatation invariance: the residual symmetry of the super-Hubble action under an overall rescaling of spatial coordinates (x^i → x^i e^{-s}, ζ → ζ + s in the comoving gauge). It plays two roles. First, it restricts the non-linear solution to ζ = ζ* + g(ψ,ρ), eliminating dependence on the undifferentiated long-wavelength perturbation, so loop corrections on CMB scales are built only from the suppressed derivative mode. Second, its local version (slowly varying s(x)) defines the adiabatic mode, and the locality condition (Eq. 2.34) — short-wavelength operators respond to this mode purely through the inhomogeneous dilatation — extends the cancellation to loops containing s

What would settle it

Take a finite-time free-field vacuum (no short-long entanglement) as the initial state of a slow-roll–ultra-slow-roll–slow-roll model and compute the one-loop CMB-scale correction. The paper's logic predicts the locality condition (Eq. 2.34) fails and sub-Hubble loops no longer decouple, which should also show up as a violation of the consistency relation (Eq. 2.41) in the squeezed bispectrum. A lattice or stochastic simulation comparing this state with the Euclidean vacuum would quantify the difference.

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Extended reading notes

Core claim

The central claim: dilatation invariance — the symmetry of the long-wavelength action under an overall rescaling x^i → x^i e^{-s}, with ζ → ζ + s — forces the super-Hubble curvature perturbation to take the form ζ = ζ* + g(ψ,ρ): the constant adiabatic mode enters additively, and all non-linear corrections depend only on the derivative mode ψ,ρ, with no large coefficients. Since that derivative mode is tiny on CMB scales, loop corrections built from it are suppressed and the tree-level spectrum dominates, for any inflaton potential, smooth or sharp, and at any loop order, as long as CMB modes left the horizon during slow roll. Apparent large corrections trace to truncating the action at cubic

Load-bearing premise

The gradient-expansion argument is self-contained, but the claim that sub-Hubble modes cannot spoil CMB-scale loops rests on the locality condition (Eq. 2.34) — the initial state must let short-wavelength operators respond to the long-wavelength adiabatic mode purely through the inhomogeneous dilatation — which the paper states is 'not guaranteed a priori' and supports with no explicit calculation (Sec. 2.5, Sec. 4).

Editorial extensions

If this is right

  • For any single-field inflaton potential, smooth or sharp, and at any loop order, the CMB-scale power spectrum receives only suppressed corrections, so primordial-black-hole scenarios built on ultra-slow-roll phases are not invalidated by loop back-reaction.
  • The would-be divergent one-loop term from the cubic vertex in the spatially-flat gauge is cancelled exactly by the quartic vertex; keeping only the cubic action — or neglecting the boundary term that enforces the field redefinition — changes the variable and reproduces the spurious enhancement.
  • The suppression is the same physics as the squeezed-limit consistency relations: both follow from dilatation invariance plus the locality of the state, so the cancellation is not specific to particular potentials or transition shapes.
  • In a non-attractor phase the conjugate momentum of ζ acts as an independent soft variable not controlled by dilatation invariance; its contribution is suppressed only if the mode was projected onto the attractor before the non-attractor phase began.

Reading between the lines

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

  • A practical diagnostic follows: any loop calculation that finds a large CMB-scale correction must have broken dilatation invariance at an identifiable step — truncation, a dropped boundary term, or an initial state failing the locality condition. Re-examining published calculations for that step would locate the enhancement without redoing the full computation.
  • The locality condition is a property of the initial quantum state, so it suggests a concrete numerical experiment: initialize a lattice or stochastic simulation of a slow-roll–ultra-slow-roll–slow-roll model in a free-field vacuum at finite initial time; the paper's logic predicts un-suppressed short-mode back-reaction in that state, while the Euclidean (de Sitter-invariant) vacuum should decouple
  • The paper's own split between the adiabatic mode and the conjugate-momentum soft variable implies an unprotected corner: models where the growth begins before the mode settles onto the attractor — very early or prolonged non-attractor phases — lie outside the suppression argument and deserve separate treatment.
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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. This paper studies loop corrections to the CMB-scale curvature power spectrum in single-field inflation, allowing for non-slow-roll phases such as USR. In comoving gauge, using a long-wavelength action, the authors show that dilatation invariance forces the non-linear solution to take the form ζ = ζ* + g(ψ,ρ) (Eq. 2.21), so that one-loop corrections on CMB scales are suppressed by factors P_{ψψ,ρ}/P_{ψψ}, P_{ψ,ρψ,ρ}/P_{ψψ}, and k^3/p_c^3 (Eqs. 2.25-2.27). They argue that truncating the action at cubic order breaks dilatation invariance and produces spurious IR enhancements (Sec. 2.4). In the spatially flat gauge they compute ζ_n to third order with the Yang-Feldman formalism and show that singular contributions from cubic and quartic interactions cancel, yielding Eq. (3.34), which matches the non-linear gauge transformation Eq. (3.42). For sub-Hubble modes, Sec. 2.5 invokes a state-dependent locality condition (Eq. 2.34) to extend the cancellation beyond the gradient expansion.

Significance. If the central claim holds, the paper provides a unifying symmetry-based explanation for the suppression of loop corrections, reconciles comoving- and flat-gauge calculations, and diagnoses why cubic truncations overestimate loops. The explicit Yang-Feldman computation of the cubic/quartic cancellation (Sec. 3 and App. C) and the demonstration that Eq. (3.34) matches the gauge transformation (3.42) are valuable and largely self-contained. The main limitation is that the strongest wording of the abstract — 'any inflaton potential ... at any loop order', including sub-Hubble modes — rests on a locality condition that is asserted rather than derived.

major comments (3)
  1. [Sec. 2.5, Eq. (2.34); Sec. 4] The abstract claims suppression for 'any inflaton potential' and 'at any loop order' on CMB scales, which includes loops of sub-Hubble hard modes. The sub-Hubble extension, however, is explicitly conditional on the locality condition (2.34), which the authors state 'is not guaranteed a priori' and depends on initial short-long entanglement. Section 4 adds: 'We did not perform any detailed explicit calculations.' The argument following Eq. (2.43) is verbal. If Eq. (2.34) fails — e.g. for a free-field vacuum at finite initial time, as the authors themselves note — hard-mode contributions need not decouple. The abstract and conclusion should either restrict the claim to the super-Hubble gradient expansion or provide a derivation/test of Eq. (2.34) for the Euclidean/Bunch-Davies state.
  2. [Sec. 2.3, after Eq. (2.27)] The statement that 'the argument can easily be extended to any n-point correlation function and at any loop level' is an extrapolation from a one-loop estimate. Higher-loop diagrams introduce nested time integrals and products of g(ψ,ρ) factors; the observation that they bring only O(ψ_ρ^m) does not by itself exclude secular or resonant enhancements. Since 'any loop order' appears in the abstract, this step is load-bearing. Please provide a proof or a clearly stated weaker claim.
  3. [Sec. 2.5, Eq. (2.34) and Eqs. (2.42)-(2.43)] The locality condition (2.34) is equivalent to assuming that the response of short-wavelength operators to the Weinberg adiabatic mode is fully described by the inhomogeneous dilatation. The subsequent derivation of ⟨ψδζ⟩ suppression uses this assumption; without it, the conclusion does not follow. This is not a circularity in the super-Hubble derivation, but it is a genuine gap in the full-theory claim. A concrete check — e.g. computing the hard-mode contribution in a simple model where Eq. (2.34) is violated — would clarify the scope.
minor comments (4)
  1. [Eqs. (2.14) and (2.25)] The symbol q is overloaded: q(ρ) is the slow-roll combination in Eq. (2.14), while q is used as a loop momentum variable in Eq. (2.25) and surrounding text. Please rename one of them.
  2. [Sec. 2.4, text near Eq. (2.30)] Typo: 'purposese' should be 'purposes'.
  3. [Eq. (3.29)] The identity θ(ρ-ρ_J)δ(ρ-ρ_J) = (1/2)δ(ρ-ρ_J) is a distributional convention. It would be helpful to state explicitly how products of step and delta functions are defined in the sharp-transition limit.
  4. [App. B] The loop diagrams are described textually but not rendered as figures. The published version should include the actual diagrams for readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the super-Hubble suppression is solved from the action, and the flat-gauge cubic/quartic cancellation is explicit; the sub-Hubble extension is conditional on a stated, self-cited locality assumption, not a fitted prediction.

full rationale

The central super-Hubble result, Eq. (2.21), is obtained by solving the reduced action Eq. (2.12): the equation of motion (2.13) contains no undifferentiated ζ because of dilatation invariance, so the solution has ζ = ζ* + g(ψ,ρ). This is a derivation from the action, not an input. The one-loop suppression in Sec. 2.3 follows from an explicit Taylor expansion of ζ and the stated smallness of the non-adiabatic mode on CMB scales, Eqs. (2.25)-(2.27); no parameter is fitted to data and the conclusion is the stated SR condition, not a hidden definition. The spatially-flat gauge analysis is likewise self-contained: ζ_n is iterated to third order via the Yang-Feldman formalism, and the sharp-transition divergences cancel algebraically between ζ_n^(3,3pt) and ζ_n^(3,4pt), with the final expression (3.34) matching the gauge-transformation result (3.42). The only potentially load-bearing external input is the locality condition Eq. (2.34), attributed to the authors' own Refs. [76,77], which underpins the sub-Hubble extension and the abstract's 'any loop order' wording. However, the paper itself explicitly flags this condition as 'not guaranteed a priori' (Sec. 2.5) and states 'We did not perform any detailed explicit calculations' (Sec. 4). Thus the sub-Hubble claim is a conditional statement built on an acknowledged assumption, not a circular reduction of the prediction to its inputs. The self-citation is transparent and the main super-Hubble and flat-gauge derivations are independent of it.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a symmetry property of the truncated action, a smallness assumption for the non-adiabatic mode at CMB scales, and, for the extension to sub-Hubble loops, a locality condition on the initial state inherited from the authors' earlier work. No numbers are fitted to data; background SR parameters (H, ϵ_i) are inputs from the inflationary background.

assumptions (5)
  • domain assumption Dilatation invariance of the long-wavelength action in the comoving gauge (footnote 1, Eq. 2.9)
    Derived from the reduced action (2.8) by neglecting gradients; this symmetry is the central input that forces ζ to appear only through derivatives.
  • domain assumption Gradient expansion: spatial-gradient terms are dropped; all relevant modes are treated as super-Hubble from the beginning (Sec. 2 opening paragraph)
    This is the separate-universe/δN framework; sub-Hubble modes are later argued to be harmless under an additional locality condition.
  • domain assumption CMB-scale modes are dominated by the constant adiabatic mode when they cross the horizon during SR, so P_{ψψ,ρ}(k)/P_{ψψ}(k) << 1 (Eq. 2.26)
    Needed to conclude that terms proportional to g(1) are negligible on CMB scales; stated and justified by SR decay of the non-adiabatic mode.
  • ad hoc to paper Locality condition for the quantum state of hard modes, Eq. (2.34): effects of the variation of the Weinberg adiabatic mode are described by the inhomogeneous dilatation δ_s
    Imported from the authors' earlier framework (Refs. 76-77); explicitly stated as not guaranteed a priori, and required for the sub-Hubble generalization of the suppression.
  • standard math Distributional identities for sharp transitions, Eq. (3.29): θ(ρ-ρJ)δ(ρ-ρJ)=1/2 δ(ρ-ρJ)
    Formal manipulation of distributions used to evaluate the sharp-transition limit; δ^2 divergences are subtracted and claimed to cancel between cubic and quartic pieces.

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Cite this review

Pith. "Pith review of How Dilatation Invariance Suppresses Loop Corrections to Curvature Perturbations on CMB Scales." pith.science (2026). https://pith.science/paper/4Q6ZVWUS

@misc{pith2026260803725,
  author       = {Pith},
  title        = {Pith review of: How Dilatation Invariance Suppresses Loop Corrections to Curvature Perturbations on CMB Scales},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4Q6ZVWUS}},
  note         = {Machine review of arXiv:2608.03725}
}
read the original abstract

Departures from standard slow-roll (SR) inflation have attracted increasing interest in recent years. In particular, scenarios that strongly enhance the power spectrum of the curvature perturbation are often proposed as a mechanism for producing primordial black holes, which could account for part or the totality of dark matter. An ongoing debate in these models is whether non-linear interactions of cosmological perturbations on small scales, namely loop corrections, can become sufficiently large to backreact on CMB scales. In this work, we adopt a non-linear framework, dropping spatial-gradient terms, to study loop corrections at super-Hubble scales. Our focus is to clarify the role of spatial-diffeomorphism invariance, especially dilatation invariance, which is the symmetry under overall rescaling of spatial coordinates, in demonstrating the suppression of loop corrections to the CMB power spectrum. This analysis is valid for any inflaton potential, both with smooth and sharp transitions, and at any loop order, as long as the CMB scales cross the horizon during an SR phase. We compare this result with the different explanations proposed in the literature using various gauge choices, and show our analysis is consistent both in the comoving and in the spatially-flat gauges. In particular, we show that in the spatially-flat gauge, the cubic interaction contains a term that diverges in the sharp transition limit. However, this contribution is exactly cancelled by the quartic interaction.

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Works this paper leans on

95 extracted references · 5 canonical work pages

  1. [1]

    Primordial Black Holes as Dark Matter

    Bernard Carr, Florian Kuhnel, and Marit Sandstad. Primordial Black Holes as Dark Matter. Phys. Rev. D, 94(8):083504, 2016.arXiv:1607.06077,doi:10.1103/PhysRevD.94.083504

    arXiv 2016

  2. [2]

    Primordial Black Holes as Dark Matter: Recent Developments

    Bernard Carr and Florian Kuhnel. Primordial Black Holes as Dark Matter: Recent Developments. Ann. Rev. Nucl. Part. Sci., 70:355–394, 2020.arXiv:2006.02838, doi:10.1146/annurev-nucl-050520-125911

    arXiv 2020

  3. [3]

    Green and Bradley J

    Anne M. Green and Bradley J. Kavanagh. Primordial Black Holes as a dark matter candidate. 7 2020.arXiv:2007.10722

    arXiv 2020

  4. [4]

    Primordial Black Holes

    Albert Escriv` a, Florian Kuhnel, and Yuichiro Tada. Primordial Black Holes. 11 2022. arXiv:2211.05767,doi:10.1016/B978-0-32-395636-9.00012-8

    arXiv 2022

  5. [5]

    Primordial Black Holes

    Christian Byrnes, Gabriele Franciolini, Tomohiro Harada, Paolo Pani, and Misao Sasaki, editors. Primordial Black Holes. Springer Series in Astrophysics and Cosmology. Springer, 2025.doi:10.1007/978-981-97-8887-3. – 24 –

    doi:10.1007/978-981-97-8887-3 2025

  6. [6]

    Starobinsky

    Alexei A. Starobinsky. Spectrum of adiabatic perturbations in the universe when there are singularities in the inflation potential. JETP Lett., 55:489–494, 1992

    1992

  7. [7]

    Leach and Andrew R

    Samuel M. Leach and Andrew R. Liddle. Inflationary perturbations near horizon crossing. Phys. Rev. D, 63:043508, 2001.arXiv:astro-ph/0010082, doi:10.1103/PhysRevD.63.043508

    arXiv 2001

  8. [8]

    Enhancement of superhorizon scale inflationary curvature perturbations

    Samuel M Leach, Misao Sasaki, David Wands, and Andrew R Liddle. Enhancement of superhorizon scale inflationary curvature perturbations. Phys. Rev. D, 64:023512, 2001. arXiv:astro-ph/0101406,doi:10.1103/PhysRevD.64.023512

    arXiv 2001

Show all 95 references
  1. [9]

    William H. Kinney. Horizon crossing and inflation with large eta. Phys. Rev. D, 72:023515, 2005.arXiv:gr-qc/0503017,doi:10.1103/PhysRevD.72.023515

    2005 arXiv

  2. [10]

    Curvature perturbation at the local extremum of the inflaton’s potential

    Shogo Inoue and Jun’ichi Yokoyama. Curvature perturbation at the local extremum of the inflaton’s potential. Phys. Lett. B, 524:15–20, 2002.arXiv:hep-ph/0104083, doi:10.1016/S0370-2693(01)01369-7

    2002 arXiv

  3. [11]

    Ultra slow-roll inflation demystified

    Konstantinos Dimopoulos. Ultra slow-roll inflation demystified. Phys. Lett. B, 775:262–265, 2017.arXiv:1707.05644,doi:10.1016/j.physletb.2017.10.066

    2017 arXiv

  4. [12]

    The attractive behaviour of ultra-slow-roll inflation

    Chris Pattison, Vincent Vennin, Hooshyar Assadullahi, and David Wands. The attractive behaviour of ultra-slow-roll inflation. JCAP, 08:048, 2018.arXiv:1806.09553, doi:10.1088/1475-7516/2018/08/048

    2018 arXiv

  5. [13]

    Primordial black hole formation in Starobinsky’s linear potential model

    Shi Pi and Jianing Wang. Primordial black hole formation in Starobinsky’s linear potential model. JCAP, 06:018, 2023.arXiv:2209.14183,doi:10.1088/1475-7516/2023/06/018

    2023 arXiv

  6. [14]

    Inflationary relics from an ultra-slow-roll plateau

    Albert Escriv` a, Jaume Garriga, and Shi Pi. Inflationary relics from an ultra-slow-roll plateau. JCAP, 03:018, 2026.arXiv:2512.04986,doi:10.1088/1475-7516/2026/03/018

    2026

  7. [15]

    Natura Non Facit Saltum: An Analytical Model of Smooth Slow-Roll to Ultra-Slow-Roll Transition

    Diego Cruces, Minxi He, Shi Pi, Jianing Wang, Masahide Yamaguchi, and Yuhang Zhu. Natura Non Facit Saltum: An Analytical Model of Smooth Slow-Roll to Ultra-Slow-Roll Transition. 3 2026.arXiv:2603.17465

    2026

  8. [16]

    Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Relation

    Jerome Martin, Hayato Motohashi, and Teruaki Suyama. Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Relation. Phys. Rev. D, 87(2):023514, 2013.arXiv:1211.0083, doi:10.1103/PhysRevD.87.023514

    2013 arXiv

  9. [17]

    ExtendedδN Formalism: Nonspatially Flat Separate-Universe Approach

    Danilo Artigas, Shi Pi, and Takahiro Tanaka. ExtendedδN Formalism: Nonspatially Flat Separate-Universe Approach. Phys. Rev. Lett., 134(22):221001, 2025.arXiv:2408.09964, doi:10.1103/PhysRevLett.134.221001

    2025 arXiv

  10. [18]

    Ultraslow-roll inflation on the lattice: Backreaction and nonlinear effects

    Angelo Caravano, Gabriele Franciolini, and S´ ebastien Renaux-Petel. Ultraslow-roll inflation on the lattice: Backreaction and nonlinear effects. Phys. Rev. D, 111(6):063518, 2025. arXiv:2410.23942,doi:10.1103/PhysRevD.111.063518

    2025 arXiv

  11. [19]

    Ultraslow-roll inflation on the lattice

    Angelo Caravano, Gabriele Franciolini, and S´ ebastien Renaux-Petel. Ultraslow-roll inflation on the lattice. II. Nonperturbative curvature perturbation. Phys. Rev. D, 112(8):083508, 2025. arXiv:2506.11795,doi:10.1103/39qd-gdfm

    2025

  12. [20]

    Constraining Primordial Black Hole Formation from Single-Field Inflation

    Jason Kristiano and Jun’ichi Yokoyama. Constraining Primordial Black Hole Formation from Single-Field Inflation. Phys. Rev. Lett., 132(22):221003, 2024.arXiv:2211.03395, doi:10.1103/PhysRevLett.132.221003

    2024 arXiv

  13. [21]

    Note on the bispectrum and one-loop corrections in single-field inflation with primordial black hole formation

    Jason Kristiano and Jun’ichi Yokoyama. Note on the bispectrum and one-loop corrections in single-field inflation with primordial black hole formation. Phys. Rev. D, 109(10):103541, 2024. arXiv:2303.00341,doi:10.1103/PhysRevD.109.103541

    2024 arXiv

  14. [22]

    Gangopadhyay, and M

    Sayantan Choudhury, Mayukh R. Gangopadhyay, and M. Sami. No-go for the formation of heavy mass Primordial Black Holes in Single Field Inflation. Eur. Phys. J. C, 84(9):884, 2024. arXiv:2301.10000,doi:10.1140/epjc/s10052-024-13218-2. – 25 –

    2024 arXiv

  15. [23]

    Sayantan Choudhury, Sudhakar Panda, and M. Sami. PBH formation in EFT of single field inflation with sharp transition. Phys. Lett. B, 845:138123, 2023.arXiv:2302.05655, doi:10.1016/j.physletb.2023.138123

    2023 arXiv

  16. [24]

    Sayantan Choudhury, Sudhakar Panda, and M. Sami. Quantum loop effects on the power spectrum and constraints on primordial black holes. JCAP, 11:066, 2023.arXiv:2303.06066, doi:10.1088/1475-7516/2023/11/066

    2023 arXiv

  17. [25]

    Sayantan Choudhury, Sudhakar Panda, and M. Sami. Galileon inflation evades the no-go for PBH formation in the single-field framework. JCAP, 08:078, 2023.arXiv:2304.04065, doi:10.1088/1475-7516/2023/08/078

    2023 arXiv

  18. [26]

    Generating Large Primordial Fluctuations in Single-Field Inflation for Primordial Black Hole Formation

    Jason Kristiano and Jun’ichi Yokoyama. Generating Large Primordial Fluctuations in Single-Field Inflation for Primordial Black Hole Formation. 2025.arXiv:2405.12149,doi:10.1007/978-981-97-8887-3_3

    2025 arXiv

  19. [27]

    Comparing sharp and smooth transitions of the second slow-roll parameter in single-field inflation

    Jason Kristiano and Jun’ichi Yokoyama. Comparing sharp and smooth transitions of the second slow-roll parameter in single-field inflation. JCAP, 10:036, 2024.arXiv:2405.12145, doi:10.1088/1475-7516/2024/10/036

    2024 arXiv

  20. [28]

    One-loop corrections in power spectrum in single field inflation

    Hassan Firouzjahi. One-loop corrections in power spectrum in single field inflation. JCAP, 10:006, 2023.arXiv:2303.12025,doi:10.1088/1475-7516/2023/10/006

    2023 arXiv

  21. [29]

    Primordial Black Holes and loops in single-field inflation

    Hassan Firouzjahi and Antonio Riotto. Primordial Black Holes and loops in single-field inflation. JCAP, 02:021, 2024.arXiv:2304.07801,doi:10.1088/1475-7516/2024/02/021

    2024 arXiv

  22. [30]

    Primordial perturbations from ultra-slow-roll single-field inflation with quantum loop effects

    Shu-Lin Cheng, Da-Shin Lee, and Kin-Wang Ng. Primordial perturbations from ultra-slow-roll single-field inflation with quantum loop effects. JCAP, 03:008, 2024.arXiv:2305.16810, doi:10.1088/1475-7516/2024/03/008

    2024

  23. [31]

    Davies, Laura Iacconi, and David J

    Matthew W. Davies, Laura Iacconi, and David J. Mulryne. Numerical 1-loop correction from a potential yielding ultra-slow-roll dynamics. JCAP, 04:050, 2024.arXiv:2312.05694, doi:10.1088/1475-7516/2024/04/050

    2024 arXiv

  24. [32]

    Perturbativity in the presence of ultraslow-roll dynamics

    Gabriele Franciolini, Antonio Iovino, Junior., Marco Taoso, and Alfredo Urbano. Perturbativity in the presence of ultraslow-roll dynamics. Phys. Rev. D, 109(12):123550, 2024. arXiv:2305.03491,doi:10.1103/PhysRevD.109.123550

    2024 arXiv

  25. [33]

    Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics

    Jacopo Fumagalli. Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics. JHEP, 05:162, 2025.arXiv:2305.19263,doi:10.1007/JHEP05(2025)162

    2025 arXiv

  26. [34]

    Roles of boundary and equation-of-motion terms in cosmological correlation functions

    Ryodai Kawaguchi, Shinji Tsujikawa, and Yusuke Yamada. Roles of boundary and equation-of-motion terms in cosmological correlation functions. Phys. Lett. B, 856:138962, 2024.arXiv:2403.16022,doi:10.1016/j.physletb.2024.138962

    2024 arXiv

  27. [35]

    A note on the role of the boundary terms for the non-Gaussianity in general k-inflation

    Frederico Arroja and Takahiro Tanaka. A note on the role of the boundary terms for the non-Gaussianity in general k-inflation. JCAP, 05:005, 2011.arXiv:1103.1102, doi:10.1088/1475-7516/2011/05/005

    2011 arXiv

  28. [36]

    Suvashis Maity, H. V. Ragavendra, Shiv K. Sethi, and L. Sriramkumar. Loop contributions to the scalar power spectrum due to quartic order action in ultra slow roll inflation. JCAP, 05:046, 2024.arXiv:2307.13636,doi:10.1088/1475-7516/2024/05/046

    2024 arXiv

  29. [37]

    Loop corrections in the separate universe picture

    Laura Iacconi, David Mulryne, and David Seery. Loop corrections in the separate universe picture. JCAP, 06:062, 2024.arXiv:2312.12424,doi:10.1088/1475-7516/2024/06/062

    2024 arXiv

  30. [38]

    Superhorizon Curvature Perturbations Are Protected against One-Loop Corrections

    Keisuke Inomata. Superhorizon Curvature Perturbations Are Protected against One-Loop Corrections. Phys. Rev. Lett., 133(14):141001, 2024.arXiv:2403.04682, doi:10.1103/PhysRevLett.133.141001

    2024 arXiv

  31. [39]

    One-loop power spectrum in ultra slow-roll inflation and implications for primordial black hole dark matter

    Guillermo Ballesteros and Jes´ us Gamb ´ ın Egea. One-loop power spectrum in ultra slow-roll inflation and implications for primordial black hole dark matter. JCAP, 07:052, 2024. arXiv:2404.07196,doi:10.1088/1475-7516/2024/07/052. – 26 –

    2024 arXiv

  32. [40]

    Proving the absence of large one-loop corrections to the power spectrum of curvature perturbations in transient ultra-slow-roll inflation within the path-integral approach

    Ryodai Kawaguchi, Shinji Tsujikawa, and Yusuke Yamada. Proving the absence of large one-loop corrections to the power spectrum of curvature perturbations in transient ultra-slow-roll inflation within the path-integral approach. JHEP, 12:095, 2024. arXiv:2407.19742,doi:10.1007/...

    2024 arXiv

  33. [41]

    Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics

    Jacopo Fumagalli. Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics. Part 2. Quartic interactions and consistency relations. JHEP, 01:108, 2025. arXiv:2408.08296,doi:10.1007/JHEP01(2025)108

    2025 arXiv

  34. [42]

    Conservation of superhorizon curvature perturbations at one loop: Backreaction in the in-in formalism and renormalization

    Keisuke Inomata. Conservation of superhorizon curvature perturbations at one loop: Backreaction in the in-in formalism and renormalization. Phys. Rev. D, 111(10):103504, 2025. arXiv:2502.08707,doi:10.1103/PhysRevD.111.103504

    2025 arXiv

  35. [43]

    Role of the counterterms in the conservation of superhorizon curvature perturbations at one loop

    Keisuke Inomata. Role of the counterterms in the conservation of superhorizon curvature perturbations at one loop. Phys. Rev. D, 111(12):123517, 2025.arXiv:2502.12112, doi:10.1103/r8bh-s48f

    2025 arXiv

  36. [44]

    Cancellation of one-loop time dependence in superhorizon curvature perturbations from all scales

    Keisuke Inomata. Cancellation of one-loop time dependence in superhorizon curvature perturbations from all scales. 6 2026.arXiv:2606.28247

    2026 arXiv

  37. [45]

    The Primordial Black Hole Formation from Single-Field Inflation is Not Ruled Out

    Antonio Riotto. The Primordial Black Hole Formation from Single-Field Inflation is Not Ruled Out. 1 2023.arXiv:2301.00599

    2023 arXiv

  38. [46]

    A. Riotto. The Primordial Black Hole Formation from Single-Field Inflation is Still Not Ruled Out. 3 2023.arXiv:2303.01727

    2023 arXiv

  39. [47]

    Squeezed bispectrum and one-loop corrections in transient constant-roll inflation

    Hayato Motohashi and Yuichiro Tada. Squeezed bispectrum and one-loop corrections in transient constant-roll inflation. JCAP, 08:069, 2023.arXiv:2303.16035, doi:10.1088/1475-7516/2023/08/069

    2023 arXiv

  40. [48]

    Cancellation of quantum corrections on the soft curvature perturbations

    Yuichiro Tada, Takahiro Terada, and Junsei Tokuda. Cancellation of quantum corrections on the soft curvature perturbations. JHEP, 01:105, 2024.arXiv:2308.04732, doi:10.1007/JHEP01(2024)105

    2024 arXiv

  41. [49]

    Largeηapproach to single field inflation

    Gianmassimo Tasinato. Largeηapproach to single field inflation. Phys. Rev. D, 108(4):043526, 2023.arXiv:2305.11568,doi:10.1103/PhysRevD.108.043526

    2023 arXiv

  42. [50]

    Cancellation of loop corrections to soft scalar power spectrum

    Yohei Ema, Muzi Hong, Ryusuke Jinno, and Kyohei Mukaida. Cancellation of loop corrections to soft scalar power spectrum. 3 2026.arXiv:2603.01961

    2026

  43. [51]

    One-loop renormalization of the effective field theory of inflationary fluctuations from gravitational interactions

    Matteo Braglia and Lucas Pinol. One-loop renormalization of the effective field theory of inflationary fluctuations from gravitational interactions. Phys. Rev. D, 113(6):063513, 2026. arXiv:2504.07926,doi:10.1103/f6q6-5jxb

    2026

  44. [52]

    Freezing of the renormalized one-loop primordial scalar power spectrum

    Matteo Braglia and Lucas Pinol. Freezing of the renormalized one-loop primordial scalar power spectrum. Phys. Rev. D, 113(6):L061302, 2026.arXiv:2504.13136,doi:10.1103/5jkv-mj3w

    2026

  45. [53]

    Scale-Dependent Loop Corrections to the Inflationary Power Spectrum

    Matteo Braglia, Sebasti´ an C´ espedes, and Lucas Pinol. Scale-Dependent Loop Corrections to the Inflationary Power Spectrum. 3 2026.arXiv:2603.12216

    2026 arXiv

  46. [54]

    Non-Gaussian features of primordial fluctuations in single field inflationary models

    Juan Martin Maldacena. Non-Gaussian features of primordial fluctuations in single field inflationary models. JHEP, 05:013, 2003.arXiv:astro-ph/0210603, doi:10.1088/1126-6708/2003/05/013

    2003 arXiv

  47. [55]

    Starobinsky

    Alexei A. Starobinsky. Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations. Phys. Lett., 117B:175–178, 1982. doi:10.1016/0370-2693(82)90541-X

    1982 doi

  48. [56]

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

    1990 doi

  49. [57]

    Misao Sasaki and Ewan D. Stewart. A General analytic formula for the spectral index of the – 27 – density perturbations produced during inflation. Prog. Theor. Phys., 95:71–78, 1996. arXiv:astro-ph/9507001,doi:10.1143/PTP.95.71

    1996 arXiv

  50. [58]

    Superhorizon scale dynamics of multiscalar inflation

    Misao Sasaki and Takahiro Tanaka. Superhorizon scale dynamics of multiscalar inflation. Prog. Theor. Phys., 99:763–782, 1998.arXiv:gr-qc/9801017,doi:10.1143/PTP.99.763

    1998 arXiv

  51. [59]

    Malik, David H

    David Wands, Karim A. Malik, David H. Lyth, and Andrew R. Liddle. A New approach to the evolution of cosmological perturbations on large scales. Phys. Rev. D, 62:043527, 2000. arXiv:astro-ph/0003278,doi:10.1103/PhysRevD.62.043527

    arXiv

  52. [60]

    Lyth and David Wands

    David H. Lyth and David Wands. Conserved cosmological perturbations. Phys. Rev. D, 68:103515, 2003.arXiv:astro-ph/0306498,doi:10.1103/PhysRevD.68.103515

    2003 arXiv

  53. [61]

    G. I. Rigopoulos and E. P. S. Shellard. The separate universe approach and the evolution of nonlinear superhorizon cosmological perturbations. Phys. Rev. D, 68:123518, 2003. arXiv:astro-ph/0306620,doi:10.1103/PhysRevD.68.123518

    2003 arXiv

  54. [62]

    Lyth, Karim A

    David H. Lyth, Karim A. Malik, and Misao Sasaki. A General proof of the conservation of the curvature perturbation. JCAP, 05:004, 2005.arXiv:astro-ph/0411220, doi:10.1088/1475-7516/2005/05/004

    2005 arXiv

  55. [63]

    Lyth and Yeinzon Rodriguez

    David H. Lyth and Yeinzon Rodriguez. The Inflationary prediction for primordial non-Gaussianity. Phys. Rev. Lett., 95:121302, 2005.arXiv:astro-ph/0504045, doi:10.1103/PhysRevLett.95.121302

    2005 arXiv

  56. [64]

    Hamiltonian formalism for cosmological perturbations: the separate-universe approach

    Danilo Artigas, Julien Grain, and Vincent Vennin. Hamiltonian formalism for cosmological perturbations: the separate-universe approach. JCAP, 02(02):001, 2022.arXiv:2110.11720, doi:10.1088/1475-7516/2022/02/001

    2022 arXiv

  57. [65]

    BeyondδNformalism

    Atsushi Naruko, Yu-ichi Takamizu, and Misao Sasaki. BeyondδNformalism. PTEP, 2013:043E01, 2013.arXiv:1210.6525,doi:10.1093/ptep/ptt008

    2013 arXiv

  58. [66]

    Firouzjaee, and Hassan Firouzjahi.δN formalism in anisotropic inflation and large anisotropic bispectrum and trispectrum

    Ali Akbar Abolhasani, Razieh Emami, Javad T. Firouzjaee, and Hassan Firouzjahi.δN formalism in anisotropic inflation and large anisotropic bispectrum and trispectrum. JCAP, 08:016, 2013.arXiv:1302.6986,doi:10.1088/1475-7516/2013/08/016

    2013 arXiv

  59. [67]

    JCAP, 03:001, 2018

    Alireza Talebian-Ashkezari, Nahid Ahmadi, and Ali Akbar Abolhasani.δMformalism: a new approach to cosmological perturbation theory in anisotropic inflation. JCAP, 03:001, 2018. arXiv:1609.05893,doi:10.1088/1475-7516/2018/03/001

    2018 arXiv

  60. [68]

    JCAP, 05:047, 2018.arXiv:1803.03763, doi:10.1088/1475-7516/2018/05/047

    Alireza Talebian-Ashkezari and Nahid Ahmadi.δMformalism and anisotropic chaotic inflation power spectrum. JCAP, 05:047, 2018.arXiv:1803.03763, doi:10.1088/1475-7516/2018/05/047

    2018 arXiv

  61. [69]

    Anisotropic separate universe and Weinberg’s adiabatic mode

    Takahiro Tanaka and Yuko Urakawa. Anisotropic separate universe and Weinberg’s adiabatic mode. JCAP, 07:051, 2021.arXiv:2101.05707,doi:10.1088/1475-7516/2021/07/051

    2021 arXiv

  62. [70]

    Statistical Anisotropy of Primordial Gravitational Waves from GeneralizedδN Formalism

    Takahiro Tanaka and Yuko Urakawa. Statistical Anisotropy of Primordial Gravitational Waves from GeneralizedδN Formalism. Phys. Rev. Lett., 132(23):231003, 2024.arXiv:2309.08497, doi:10.1103/PhysRevLett.132.231003

    2024 arXiv

  63. [71]

    Mohammad Ahmadi and Nahid Ahmadi.δNformalism with gradient interactions

    S. Mohammad Ahmadi and Nahid Ahmadi.δNformalism with gradient interactions. 1 2026. arXiv:2602.00902

    2026 arXiv

  64. [72]

    Mohammad Ahmadi

    S. Mohammad Ahmadi. Continuous Sensitivity Analysis forδNFormalism. 3 2026. arXiv:2603.27366

    2026

  65. [73]

    Chen-Ning Yang and D. Feldman. The S Matrix in the Heisenberg Representation. Phys. Rev., 79:972–978, 1950.doi:10.1103/PhysRev.79.972

    1950 doi

  66. [74]

    Decoupling of large-scale, adiabatic inflationary perturbations from enhanced small-scale modes at one-loop

    Laura Iacconi, David Mulryne, and David Seery. Decoupling of large-scale, adiabatic inflationary perturbations from enhanced small-scale modes at one-loop. 1 2026. arXiv:2601.14229. – 28 –

    2026

  67. [75]

    Primordial black holes from inflation: on the decoupling between large and small scales

    Laura Iacconi. Primordial black holes from inflation: on the decoupling between large and small scales. In 60th Rencontres de Moriond on Cosmology: Moriond Cosmology 2026, 5 2026. arXiv:2605.04815

    2026 arXiv

  68. [76]

    Large gauge transformation, Soft theorem, and Infrared divergence in inflationary spacetime

    Takahiro Tanaka and Yuko Urakawa. Large gauge transformation, Soft theorem, and Infrared divergence in inflationary spacetime. JHEP, 10:127, 2017.arXiv:1707.05485, doi:10.1007/JHEP10(2017)127

    2017 arXiv

  69. [77]

    Locality in effective field theory for inflationary soft modes

    Takahiro Tanaka and Yuko Urakawa. Locality in effective field theory for inflationary soft modes. 5 2026.arXiv:2605.19331

    2026 arXiv

  70. [78]

    Adiabatic modes in cosmology

    Steven Weinberg. Adiabatic modes in cosmology. Phys. Rev., D67:123504, 2003. arXiv:astro-ph/0302326,doi:10.1103/PhysRevD.67.123504

    2003 arXiv

  71. [79]

    Dominance of gauge artifact in the consistency relation for the primordial bispectrum

    Takahiro Tanaka and Yuko Urakawa. Dominance of gauge artifact in the consistency relation for the primordial bispectrum. JCAP, 05:014, 2011.arXiv:1103.1251, doi:10.1088/1475-7516/2011/05/014

    2011 arXiv

  72. [80]

    The Observed Squeezed Limit of Cosmological Three-Point Functions

    Enrico Pajer, Fabian Schmidt, and Matias Zaldarriaga. The Observed Squeezed Limit of Cosmological Three-Point Functions. Phys. Rev. D, 88(8):083502, 2013.arXiv:1305.0824, doi:10.1103/PhysRevD.88.083502

    2013 arXiv

  73. [81]

    IR divergence does not affect the gauge-invariant curvature perturbation

    Yuko Urakawa and Takahiro Tanaka. IR divergence does not affect the gauge-invariant curvature perturbation. Phys. Rev. D, 82:121301, 2010.arXiv:1007.0468, doi:10.1103/PhysRevD.82.121301

    2010 arXiv

  74. [82]

    Natural selection of inflationary vacuum required by infra-red regularity and gauge-invariance

    Yuko Urakawa and Takahiro Tanaka. Natural selection of inflationary vacuum required by infra-red regularity and gauge-invariance. Prog. Theor. Phys., 125:1067–1089, 2011. arXiv:1009.2947,doi:10.1143/PTP.125.1067

    2011 arXiv

  75. [83]

    Loops in inflationary correlation functions

    Takahiro Tanaka and Yuko Urakawa. Loops in inflationary correlation functions. Class. Quant. Grav., 30:233001, 2013.arXiv:1306.4461,doi:10.1088/0264-9381/30/23/233001

    2013 arXiv

  76. [84]

    Strong restriction on inflationary vacua from the local gauge invariance II: Infrared regularity and absence of secular growth in the Euclidean vacuum

    Takahiro Tanaka and Yuko Urakawa. Strong restriction on inflationary vacua from the local gauge invariance II: Infrared regularity and absence of secular growth in the Euclidean vacuum. PTEP, 2013(6):063E02, 2013.arXiv:1301.3088,doi:10.1093/ptep/ptt037

    2013 arXiv

  77. [85]

    Strong restriction on inflationary vacua from the local gaugeinvariance III: Infrared regularity of graviton loops

    Takahiro Tanaka and Yuko Urakawa. Strong restriction on inflationary vacua from the local gaugeinvariance III: Infrared regularity of graviton loops. PTEP, 2014(7):073E01, 2014. arXiv:1402.2076,doi:10.1093/ptep/ptu071

    2014 arXiv

  78. [86]

    Primordial Black Holes from Inflation and Quantum Diffusion

    Matteo Biagetti, Gabriele Franciolini, Alex Kehagias, and Antonio Riotto. Primordial Black Holes from Inflation and Quantum Diffusion. JCAP, 07:032, 2018.arXiv:1804.07124, doi:10.1088/1475-7516/2018/07/032

    2018 arXiv

  79. [87]

    Primordial black hole formation with non-Gaussian curvature perturbations

    Vicente Atal, Jaume Garriga, and Airam Marcos-Caballero. Primordial black hole formation with non-Gaussian curvature perturbations. JCAP, 09:073, 2019.arXiv:1905.13202, doi:10.1088/1475-7516/2019/09/073

    2019 arXiv

  80. [88]

    Logarithmic Duality of the Curvature Perturbation

    Shi Pi and Misao Sasaki. Logarithmic Duality of the Curvature Perturbation. Phys. Rev. Lett., 131(1):011002, 2023.arXiv:2211.13932,doi:10.1103/PhysRevLett.131.011002

    2023 arXiv

  81. [89]

    Malik and David Wands

    Karim A. Malik and David Wands. Cosmological perturbations. Phys. Rept., 475:1–51, 2009. arXiv:0809.4944,doi:10.1016/j.physrep.2009.03.001

    2009 arXiv

  82. [90]

    Diego Cruces, Shi Pi, and Misao Sasaki.δn formalism: A new formulation for the probability density of the curvature perturbation. Phys. Rev. D, 114(2):023549, 2026.arXiv:2505.24590, doi:10.1103/85xd-bbgj

    2026 arXiv

  83. [91]

    Stochastic inflation beyond slow roll

    Chris Pattison, Vincent Vennin, Hooshyar Assadullahi, and David Wands. Stochastic inflation beyond slow roll. JCAP, 07:031, 2019.arXiv:1905.06300, doi:10.1088/1475-7516/2019/07/031. – 29 –

    2019 arXiv

  84. [92]

    Joseph H. P. Jackson, Hooshyar Assadullahi, Andrew D. Gow, Kazuya Koyama, Vincent Vennin, and David Wands. The separate-universe approach and sudden transitions during inflation. JCAP, 05:053, 2024.arXiv:2311.03281,doi:10.1088/1475-7516/2024/05/053

    2024 arXiv

  85. [93]

    Stochastic inflation with gradient interactions

    Vadim Briaud, Ryodai Kawaguchi, and Vincent Vennin. Stochastic inflation with gradient interactions. JCAP, 12:024, 2025.arXiv:2509.05124,doi:10.1088/1475-7516/2025/12/024

    2025

  86. [94]

    On Loops in Inflation

    Leonardo Senatore and Matias Zaldarriaga. On Loops in Inflation. JHEP, 12:008, 2010. arXiv:0912.2734,doi:10.1007/JHEP12(2010)008

    2010 arXiv

  87. [95]

    Inflationary background renormalization

    Jason Kristiano and Jun’ichi Yokoyama. Inflationary background renormalization. 4 2025. arXiv:2504.18514. – 30 –

    2025 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.