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

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that superhorizon curvature perturbations remain conserved at one loop in single-field inflation with a transient non-slow-roll period, so enhanced small-scale modes do not alter the large-scale power spectrum.

desk verdict A careful, honest in-in derivation of one-loop conservation that is weaker than its abstract: the strict result depends on dropping an IR cutoff term the paper admits it cannot rigorously justify. read the letter →

arxiv 2502.08707 v3 pith:HJVHVM5K submitted 2025-02-12 astro-ph.CO gr-qchep-phhep-th

Keisuke Inomata This is my paper · ORCID
classification astro-ph.COgr-qchep-phhep-th
keywords curvatureperturbationconservationone-looppowerspectrumin-informalismbackreactionultraslow-rollprimordialblackholesrenormalizationseparateuniverse
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 paper tries to prove that in single-field inflation with a transient non-slow-roll period, the superhorizon curvature power spectrum is conserved at one-loop order: enhanced small-scale perturbations do not produce a scale-invariant correction to large-scale curvature perturbations. This matters because primordial-black-hole models often rely on such transient phases to boost small-scale power, and a one-loop correction leaking to CMB scales would constrain or invalidate them. The proof is done in the spatially flat gauge, where the curvature spectrum is the ratio of the inflaton fluctuation power spectrum to the squared background velocity, so the loop correction to the background velocity, the backreaction, must be computed together with the loop correction to the field fluctuations. Using the in-in formalism, the paper shows the two corrections cancel in the superhorizon limit, leaving the tree-level curvature power spectrum unchanged before and after the non-slow-roll phase, and it exhibits the counterterms that renormalize the UV divergences.

What carries the argument

The load-bearing object is the ratio identity $\lim_{q\to 0} (P_{\delta\phi}(q,\eta)-P_{\delta\phi,\mathrm{tr}}(q,\eta))/P_{\delta\phi,\mathrm{tr}}(q,\eta) = 2\left(\langle \delta\dot\phi(\eta)\rangle - \langle \delta\dot\phi\rangle_{\mathrm{IR}}\right)/\dot{\bar\phi}(\eta)$, which ties every one-loop contribution to the renormalized backreaction in the inflaton velocity. The in-in formalism supplies the time integrals for the one-vertex, two-vertex, and tadpole diagrams, and the retarded Green function of the linear perturbation equation converts the background-velocity backreaction into the same integrals. The counterterm $V_{c,(1)}$ fixes $V_{c,(2)}$ and cancels the quadratic and logarithmic UV divergences, while the logarithmic divergence in the two-vertex contribution cancels against the one-vertex counterterm contribution.

What would settle it

For a fixed SR-non-SR-SR model with finite $k_{\mathrm{IR}}$, keep the $\langle \delta\dot\phi\rangle_{\mathrm{IR}}$ term in Eq. (66) instead of dropping it and compute $\lim_{q\to 0} P_{\zeta,1\text{-loop}}(q)$; if the result is nonzero, the conservation claim fails. A direct alternative is to implement the field-dependent IR cutoff of Appendix C and check whether Eq. (C4) exactly cancels the IR term as claimed.

Watch

Extended reading notes

Core claim

The central discovery is that the superhorizon limit of the one-loop curvature power spectrum is conserved: $\lim_{q\to 0} \langle \zeta_q \zeta_{q'} \rangle = (2\pi)^3 \delta(q+q') (2\pi^2/q^3) P_{\zeta,\mathrm{tr}}(q)$ for $\eta<\eta_i$ and $\eta>\eta_e$. In the spatially flat gauge, $\zeta = -H\,\delta\phi/\langle \dot\phi\rangle$, so conservation requires that the one-loop correction to $\langle \delta\phi_q \delta\phi_{q'}\rangle$ be exactly matched by the one-loop backreaction to $\langle \dot\phi\rangle$. The paper derives this matching for the one-vertex, two-vertex, and tadpole contributions, after setting aside the homogeneous infrared-cutoff contribution $\langle \delta\dot\phi\rangle_{\mathrm{IR}}$, and shows that the same counterterm structure removes the UV divergences from both the fluctuation power spectrum and the background velocity.

Load-bearing premise

The proof drops the homogeneous infrared-cutoff contribution $\langle \delta\dot\phi\rangle_{\mathrm{IR}}$ after Eq. (66), and Appendix C concedes that the inhomogeneous, field-dependent cutoff that would fully cancel it cannot yet be justified.

Editorial extensions

If this is right

  • Small-scale enhancements from a transient non-slow-roll phase do not generate scale-invariant one-loop corrections to the large-scale curvature power spectrum, leaving CMB and LSS scales untouched at this order.
  • Primordial-black-hole single-field models are not constrained by one-loop backreaction on CMB scales, so higher-loop and non-perturbative small-scale effects remain the only open worries at this level.
  • Ignoring backreaction in the spatially flat gauge would break conservation unless counterterms are tuned; the explicit $V_{c,(1)}$ counterterm absorbs the UV divergences in both the power spectrum and the background velocity.
  • The conservation holds for $\eta<\eta_i$ and $\eta>\eta_e$, and the separate-universe argument extends it through the non-SR phase up to corrections suppressed by the choice of $k_{\mathrm{IR}}$.

Reading between the lines

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

  • The paper leaves open whether a physical regulator makes the homogeneous IR-cutoff contribution nonzero; if it does, conservation could fail at one loop.
  • The strict proof lives in the de Sitter limit $\epsilon\to 0$; a finite-$\epsilon$ extension could produce $\mathcal{O}(\epsilon)$ one-loop corrections not covered here.
  • The separate-universe picture is restored only by an inhomogeneous, field-dependent smoothing scale, suggesting that homogeneous cutoffs may spuriously break conservation in other gauge choices as well.
  • A natural next step is a two-loop version of the same backreaction-matching argument, where the cancellation pattern may or may not persist.
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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

2 major / 5 minor

Summary. The paper aims to prove that superhorizon-limit curvature perturbations are conserved at one-loop order in single-field inflation models with a transient non-slow-roll period, working in the spatially-flat gauge and using the in-in formalism. The author computes the one-loop power spectrum of δφ, including one-vertex, two-vertex, and tadpole contributions, and separately computes the one-loop backreaction of the perturbations on the background inflaton velocity ⟨δφ̇⟩. The central step is the matching in Eqs. (61)-(65), where the superhorizon limit of the one-loop power spectrum is shown to equal 2⟨δφ̇⟩/φ̇ plus an IR-cutoff term ⟨δφ̇⟩_IR. Eq. (66) then gives the ratio ⟨δφδφ⟩/⟨φ̇⟩^2 as the tree-level curvature power spectrum times [1 − 2⟨δφ̇⟩_IR/φ̇]; neglecting this IR term yields the conservation statement in Eq. (68). The paper also presents a UV renormalization procedure with counter terms, showing that the quadratic and logarithmic UV divergences in the one-vertex and two-vertex contributions cancel via the counter term in Eq. (76).

Significance. If the result holds, it is significant because it would refute claims that enhanced small-scale perturbations during a transient non-slow-roll period generate scale-invariant one-loop corrections to CMB/LSS-scale curvature perturbations, thereby supporting the separate-universe picture and relaxing constraints on PBH scenarios. The paper's strengths include a self-contained derivation of the loop-backreaction matching in the in-in formalism, an explicit and detailed treatment of UV renormalization with counter terms, and a clear comparison with earlier work. The derivation is internally consistent and the cancellation between the one-loop power spectrum and backreaction terms is demonstrated by explicit formulas. However, the conservation statement in Eq. (68) depends on dropping the IR cutoff term ⟨δφ̇⟩_IR, and the manuscript itself (Appendix C) states that the only known way to remove this term involves a field-dependent cutoff that cannot be justified. This gap currently leaves the central claim conditional rather than proven.

major comments (2)
  1. [§V, Eq. (66)] The step from Eq. (66) to Eq. (68) drops the IR cutoff contribution ⟨δφ̇⟩_IR without a quantitative estimate. This term, defined in Eq. (57), is proportional to H V(3) P_{δφ,tr}(k_IR, η′) integrated with a^4 Im[u0 u0*]. For a scale-invariant tree-level spectrum, P_{δφ,tr}(k_IR) is of the same order as P_{δφ,tr}(q), and the paper provides no argument that the time integral suppresses this term relative to the kept one-loop contributions. Since Eq. (68) is the paper's central claim, the neglect of this term is load-bearing and needs to be justified by an explicit estimate or by reformulating the theorem to state the condition under which the term is negligible.
  2. [Appendix C] The manuscript acknowledges that the only known way to cancel the IR cutoff contribution is to adopt an inhomogeneous, field-dependent IR cutoff of the form k_IR[1 − Hδφ/φ̇], and then states in Appendix C that this choice "cannot be justified." This concession directly undermines the unconditional form of the conservation result stated in the abstract and in Eq. (68). The strict conservation theorem is therefore not proven for the homogeneous-cutoff calculation presented; it holds only if a particular local-smoothing prescription is imposed. The abstract and conclusion should be qualified accordingly, or the paper should provide a physical justification for the field-dependent cutoff within its framework.
minor comments (5)
  1. [§V, after Eq. (66)] The sentence "If we neglect the IR cutoff contribution..." introduces an unstated assumption into the derivation; the paper should clearly label this as an assumption in the statement of the theorem, so that readers can distinguish the proven result from the conditional result.
  2. [§V, paragraph after Eq. (68)] The claim that contributions of O(X P_{ζ,tr}(k1) P_{ζ,tr}(k2)) from smoothed modes are negligible "unless X is extremely large" is not quantified. The paper should either define a threshold or state explicitly that this condition is an assumption of the argument.
  3. [Eq. (58)] The notation ⟨δφ̇(η)⟩ in the denominator of Eq. (58) omits the spatial argument; since the backreaction is defined as a spatially homogeneous quantity in the main text, the notation should be made consistent, perhaps by writing ⟨δφ̇(x,η)⟩ or stating that the x-dependence drops out.
  4. [Figure 2] The Feynman diagram labels "Tadpole", "One vertex", and "Two vertices" are clear, but the one-vertex diagram includes both V(4) and Vc,(2) insertions; a sentence in the caption clarifying which vertex corresponds to which term would improve readability.
  5. [Appendix D, Eq. (D4)] The notation a1 in Eq. (D4) is used before it is defined in Eq. (D2), and the definition appears only in the surrounding text; consider defining a1 explicitly at first use.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the one-loop conservation result is derived in-in from the vertices and counter terms; the only notable caveat is the acknowledged IR-cutoff term, which is a rigor gap rather than a circular reduction.

full rationale

The central derivation is self-contained. The one-loop power spectrum is computed with the in-in formalism from the cubic and quartic vertices and the counter terms, and the key cancellation in Eq. (65) follows algebraically from Eqs. (61), (63), and (64) without assuming the final conservation. The proof uses standard external inputs (Bunch-Davies mode functions, Wronskian identities, the separate-universe/δN relation) and contains no fitted parameters or quantities that are renamed as predictions. The author cites his own previous work [58] for motivation, comparison, and some notation, and Appendix A explicitly reconciles the two formalisms, but the load-bearing steps are re-derived here rather than imported; the self-citations are not circular. The one substantive caveat is the IR-cutoff contribution: Eq. (66) contains the factor [1 - 2⟨δφdot⟩_IR/φdot], and Eq. (68) is obtained only after neglecting it. Appendix C concedes that the inhomogeneous, field-dependent cutoff that would fully remove this term 'cannot be justified.' This is an acknowledged limitation of the proof, not a circular step, because the dropped term is not an input that is later repackaged as the output. The result is therefore not equivalent to its assumptions by construction. Score 1 reflects minor self-citation and the flagged caveat, not circular reasoning.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The main assumptions are the de Sitter limit, the form of the potential, the Bunch-Davies vacuum, and the treatment of the infrared cutoff. No invented entities are introduced. The only regulator that survives in the final statement is the IR cutoff k_IR, whose contribution is neglected.

free parameters (1)
  • IR comoving cutoff k_IR = chosen such that q << k_IR and k_IR lies above the enhanced small-scale band
    The proof neglects the boundary term <delta dot-phi>_IR. Conservation in Eqs. (66) and (68) is stated only after this regulator is dropped, so the final statement depends on this choice.
assumptions (7)
  • domain assumption The bare potential is smooth and split as V_b = V + V_c with V_c of one-loop order or higher.
    Needed to expand in delta-phi around the background; introduced in Sec. II, Eq. (2).
  • domain assumption The de Sitter limit epsilon -> 0 is taken, with all metric perturbations neglected.
    Justifies dropping metric perturbations; Sec. II first paragraph. The result is derived only in this limit.
  • domain assumption V^(n>=3) is zero or exponentially suppressed outside the non-SR window, so interactions act only for eta_i < eta < eta_e.
    Makes the initial vacuum non-interacting and simplifies the loop integrals; Sec. II after Eq. (8).
  • domain assumption The initial state is the Bunch-Davies vacuum at eta_i and the i-epsilon prescription is unnecessary.
    Stated in Sec. II and footnote 5; relies on Hint being negligible before eta_i.
  • ad hoc to paper The IR cutoff contribution <delta dot-phi>_IR can be neglected by choosing k_IR much larger than the enhanced small-scale modes.
    Sec. V after Eq. (66); without this, Eqs. (66) and (68) show non-conservation. Appendix C admits the inhomogeneous cutoff that would cancel it cannot be justified.
  • domain assumption The separate universe / delta-N formula (Eq. 67) applies during the SR phases, with corrections suppressed by m^2/H^2.
    Used to convert the delta-phi correlation to the curvature power spectrum; Sec. V.
  • ad hoc to paper During the non-SR phase, contributions of O(X P_zeta,tr(k1) P_zeta,tr(k2)) from smoothed modes are negligible unless X is very large.
    Extends conservation through the non-SR period; Sec. V after Eq. (68). This is heuristic.

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Cite this review

Pith. "Pith review of Conservation of superhorizon curvature perturbations at one loop: Backreaction in the in-in formalism and Renormalization." pith.science (2026). https://pith.science/paper/HJVHVM5K

@misc{pith2026250208707,
  author       = {Pith},
  title        = {Pith review of: Conservation of superhorizon curvature perturbations at one loop: Backreaction in the in-in formalism and Renormalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJVHVM5K}},
  note         = {Machine review of arXiv:2502.08707}
}
read the original abstract

We show that the superhorizon-limit curvature perturbations are conserved at one-loop level in single-field inflation models with a transient non-slow-roll period. We take the spatially-flat gauge, where the backreaction plays a crucial role for the conservation of superhorizon curvature perturbations unless the counter terms are tuned. We calculate the backreaction with the in-in formalism. In addition, we explicitly show the renormalization of the UV divergences with the counter terms.

Figures

Figures reproduced from arXiv: 2502.08707 by the authors.

Figure 1
Figure 1. FIG. 1. The schematic picture of the inflaton potential we consider. The potential in the non-SR period can have any form, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The Feynman diagrams of the one-loop contributions in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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Reference graph

Works this paper leans on

76 extracted references · 1 canonical work pages · cited by 6 Pith papers

  1. [58]

    Inomata, Phys

    K. Inomata, Phys. Rev. Lett. 133, 141001 (2024), arXiv:2403.04682 [astro-ph.CO]

    arXiv 2024

  2. [1]

    J. M. Bardeen, Phys. Rev. D 22, 1882 (1980)

    1980

  3. [2]

    Kodama and M

    H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984)

    1984

  4. [3]

    Senatore and M

    L. Senatore and M. Zaldarriaga, JHEP 12, 008 (2010), arXiv:0912.2734 [hep-th]

    arXiv 2010

  5. [4]

    Senatore and M

    L. Senatore and M. Zaldarriaga, JHEP 01, 109 (2013), arXiv:1203.6354 [hep-th]

    arXiv 2013

  6. [5]

    G. L. Pimentel, L. Senatore, and M. Zaldarriaga, JHEP 07, 166 (2012), arXiv:1203.6651 [hep-th]

    arXiv 2012

  7. [6]

    D. H. Lyth, K. A. Malik, and M. Sasaki, JCAP 05, 004 (2005), arXiv:astro-ph/0411220

    arXiv 2005

  8. [7]

    G. F. Chapline, Nature 253, 251 (1975)

    1975

Show all 76 references
  1. [8]

    Ivanov, P

    P. Ivanov, P. Naselsky, and I. Novikov, Phys. Rev. D 50, 7173 (1994)

    1994

  2. [9]

    Yokoyama, Astron

    J. Yokoyama, Astron. Astrophys. 318, 673 (1997), arXiv:astro-ph/9509027

    1997 arXiv

  3. [10]

    Garcia-Bellido, A

    J. Garcia-Bellido, A. D. Linde, and D. Wands, Phys. Rev. D54, 6040 (1996), arXiv:astro-ph/9605094 [astro-ph]

    1996 arXiv

  4. [11]

    Afshordi, P

    N. Afshordi, P. McDonald, and D. N. Spergel, Astrophys. J. Lett. 594, L71 (2003), arXiv:astro-ph/0302035

    2003 arXiv

  5. [12]

    P. H. Frampton, M. Kawasaki, F. Takahashi, and T. T. Yanagida, JCAP 1004, 023 (2010), arXiv:1001.2308 [hep-ph]

    2010 arXiv

  6. [13]

    K. M. Belotsky, A. D. Dmitriev, E. A. Esipova, V. A. Gani, A. V. Grobov, M. Y. Khlopov, A. A. Kirillov, S. G. Rubin, and I. V. Svadkovsky, Mod. Phys. Lett. A 29, 1440005 (2014), arXiv:1410.0203 [astro-ph.CO]

    2014 arXiv

  7. [14]

    B. Carr, F. Kuhnel, and M. Sandstad, Phys. Rev. D94, 083504 (2016), arXiv:1607.06077 [astro-ph.CO]

    2016 arXiv

  8. [15]

    Inomata, M

    K. Inomata, M. Kawasaki, K. Mukaida, Y. Tada, and T. T. Yanagida, Phys. Rev. D96, 043504 (2017), arXiv:1701.02544 [astro-ph.CO]

    2017 arXiv

  9. [16]

    J. R. Espinosa, D. Racco, and A. Riotto, Phys. Rev. Lett. 120, 121301 (2018), arXiv:1710.11196 [hep-ph]

    2018 arXiv

  10. [17]

    S. Bird, I. Cholis, J. B. Mu˜ noz, Y. Ali-Ha ¨ ımoud, M. Kamionkowski, E. D. Kovetz, A. Raccanelli, and A. G. Riess, Phys. Rev. Lett. 116, 201301 (2016), arXiv:1603.00464 [astro-ph.CO]

    2016 arXiv

  11. [18]

    Clesse and J

    S. Clesse and J. Garc ´ ıa-Bellido, Phys. Dark Univ.10, 002 (2016), arXiv:1603.05234 [astro-ph.CO]

    2016 arXiv

  12. [19]

    Sasaki, T

    M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Phys. Rev. Lett.117, 061101 (2016), arXiv:1603.08338 [astro-ph.CO]

    2016 arXiv

  13. [20]

    Y. N. Eroshenko, J. Phys. Conf. Ser. 1051, 012010 (2018), arXiv:1604.04932 [astro-ph.CO]

    2018 arXiv

  14. [21]

    Sasaki, T

    M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Class. Quant. Grav. 35, 063001 (2018), arXiv:1801.05235 [astro- ph.CO]

    2018 arXiv

  15. [22]

    B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Rept. Prog. Phys. 84, 116902 (2021), arXiv:2002.12778 [astro-ph.CO]

    2021 arXiv

  16. [23]

    A. M. Green and B. J. Kavanagh, J. Phys. G 48, 043001 (2021), arXiv:2007.10722 [astro-ph.CO]

    2021 arXiv

  17. [24]

    Escriv` a, F

    A. Escriv` a, F. Kuhnel, and Y. Tada, (2022), 10.1016/B978-0-32-395636-9.00012-8, arXiv:2211.05767 [astro-ph.CO]

    2022 arXiv

  18. [25]

    Kristiano and J

    J. Kristiano and J. Yokoyama, Phys. Rev. Lett. 132, 221003 (2024), arXiv:2211.03395 [hep-th]

    2024 arXiv

  19. [26]

    Cheng, D.-S

    S.-L. Cheng, D.-S. Lee, and K.-W. Ng, Phys. Lett. B 827, 136956 (2022), arXiv:2106.09275 [astro-ph.CO]

    2022 arXiv

  20. [27]

    W. H. Kinney, Phys. Rev. D 56, 2002 (1997), arXiv:hep-ph/9702427

    1997 arXiv

  21. [28]

    Inoue and J

    S. Inoue and J. Yokoyama, Phys. Lett. B 524, 15 (2002), arXiv:hep-ph/0104083

    2002 arXiv

  22. [29]

    W. H. Kinney, Phys. Rev. D 72, 023515 (2005), arXiv:gr-qc/0503017

    2005 arXiv

  23. [30]

    Martin, H

    J. Martin, H. Motohashi, and T. Suyama, Phys. Rev. D 87, 023514 (2013), arXiv:1211.0083 [astro-ph.CO]

    2013 arXiv

  24. [31]

    Riotto, (2023), arXiv:2301.00599 [astro-ph.CO]

    A. Riotto, (2023), arXiv:2301.00599 [astro-ph.CO]

    2023 arXiv

  25. [32]

    Choudhury, M

    S. Choudhury, M. R. Gangopadhyay, and M. Sami, Eur. Phys. J. C 84, 884 (2024), arXiv:2301.10000 [astro-ph.CO]

    2024 arXiv

  26. [33]

    Kristiano and J

    J. Kristiano and J. Yokoyama, Phys. Rev. D 109, 103541 (2024), arXiv:2303.00341 [hep-th]

    2024 arXiv

  27. [34]

    Riotto, (2023), arXiv:2303.01727 [astro-ph.CO]

    A. Riotto, (2023), arXiv:2303.01727 [astro-ph.CO]

    2023 arXiv

  28. [35]

    Firouzjahi, JCAP 10, 006 (2023), arXiv:2303.12025 [astro-ph.CO]

    H. Firouzjahi, JCAP 10, 006 (2023), arXiv:2303.12025 [astro-ph.CO]

    2023 arXiv

  29. [36]

    Motohashi and Y

    H. Motohashi and Y. Tada, JCAP 08, 069 (2023), arXiv:2303.16035 [astro-ph.CO]

    2023 arXiv

  30. [37]

    Firouzjahi and A

    H. Firouzjahi and A. Riotto, JCAP 02, 021 (2024), arXiv:2304.07801 [astro-ph.CO]

    2024 arXiv

  31. [38]

    Franciolini, A

    G. Franciolini, A. Iovino, Junior., M. Taoso, and A. Urbano, Phys. Rev. D 109, 123550 (2024), arXiv:2305.03491 [astro- ph.CO]

    2024 arXiv

  32. [39]

    Tasinato, Phys

    G. Tasinato, Phys. Rev. D 108, 043526 (2023), arXiv:2305.11568 [hep-th]

    2023 arXiv

  33. [40]

    Cheng, D.-S

    S.-L. Cheng, D.-S. Lee, and K.-W. Ng, JCAP 03, 008 (2024), arXiv:2305.16810 [astro-ph.CO]

    2024

  34. [41]

    Fumagalli, (2023), arXiv:2305.19263 [astro-ph.CO]

    J. Fumagalli, (2023), arXiv:2305.19263 [astro-ph.CO]

    2023 arXiv

  35. [42]

    Maity, H

    S. Maity, H. V. Ragavendra, S. K. Sethi, and L. Sriramkumar, JCAP 05, 046 (2024), arXiv:2307.13636 [astro-ph.CO]

    2024 arXiv

  36. [43]

    Y. Tada, T. Terada, and J. Tokuda, JHEP 01, 105 (2024), arXiv:2308.04732 [hep-th]

    2024 arXiv

  37. [44]

    Firouzjahi, Phys

    H. Firouzjahi, Phys. Rev. D 109, 043514 (2024), arXiv:2311.04080 [astro-ph.CO]

    2024 arXiv

  38. [45]

    M. W. Davies, L. Iacconi, and D. J. Mulryne, JCAP 04, 050 (2024), arXiv:2312.05694 [astro-ph.CO]

    2024 arXiv

  39. [46]

    Iacconi, D

    L. Iacconi, D. Mulryne, and D. Seery, JCAP 06, 062 (2024), arXiv:2312.12424 [astro-ph.CO]

    2024 arXiv

  40. [47]

    Saburov and S

    S. Saburov and S. V. Ketov, Universe 10, 354 (2024), arXiv:2402.02934 [gr-qc]. 25

    2024 arXiv

  41. [48]

    Ballesteros and J

    G. Ballesteros and J. G. Egea, JCAP 07, 052 (2024), arXiv:2404.07196 [astro-ph.CO]

    2024 arXiv

  42. [49]

    Kristiano and J

    J. Kristiano and J. Yokoyama, JCAP 10, 036 (2024), arXiv:2405.12145 [astro-ph.CO]

    2024 arXiv

  43. [50]

    Kristiano and J

    J. Kristiano and J. Yokoyama, (2024), arXiv:2405.12149 [astro-ph.CO]

    2024 arXiv

  44. [51]

    Kawaguchi, S

    R. Kawaguchi, S. Tsujikawa, and Y. Yamada, JHEP 12, 095 (2024), arXiv:2407.19742 [hep-th]

    2024 arXiv

  45. [52]

    Fumagalli, JHEP 01, 108 (2025), arXiv:2408.08296 [astro-ph.CO]

    J. Fumagalli, JHEP 01, 108 (2025), arXiv:2408.08296 [astro-ph.CO]

    2025 arXiv

  46. [53]

    Sheikhahmadi and A

    H. Sheikhahmadi and A. Nassiri-Rad, (2024), arXiv:2411.18525 [astro-ph.CO]

    2024 arXiv

  47. [54]

    Frolovsky and S

    D. Frolovsky and S. V. Ketov, (2025), arXiv:2502.00628 [gr-qc]

    2025 arXiv

  48. [55]

    Sasaki and T

    M. Sasaki and T. Tanaka, Prog. Theor. Phys. 99, 763 (1998), arXiv:gr-qc/9801017

    1998 arXiv

  49. [56]

    Wands, K

    D. Wands, K. A. Malik, D. H. Lyth, and A. R. Liddle, Phys. Rev. D 62, 043527 (2000), arXiv:astro-ph/0003278

    2000 arXiv

  50. [57]

    D. H. Lyth and D. Wands, Phys. Rev. D 68, 103515 (2003), arXiv:astro-ph/0306498

    2003 arXiv

  51. [59]

    Baumann and D

    D. Baumann and D. Green, JCAP 09, 014 (2011), arXiv:1102.5343 [hep-th]

    2011 arXiv

  52. [60]

    Pajer, G

    E. Pajer, G. L. Pimentel, and J. V. S. Van Wijck, JCAP 06, 009 (2017), arXiv:1609.06993 [hep-th]

    2017 arXiv

  53. [61]

    Inomata, M

    K. Inomata, M. Braglia, X. Chen, and S. Renaux-Petel, JCAP 04, 011 (2023), [Erratum: JCAP 09, E01 (2023)], arXiv:2211.02586 [astro-ph.CO]

    2023 arXiv

  54. [62]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. D 72, 043514 (2005), arXiv:hep-th/0506236

    2005 arXiv

  55. [63]

    Arroja and T

    F. Arroja and T. Tanaka, JCAP 05, 005 (2011), arXiv:1103.1102 [astro-ph.CO]

    2011 arXiv

  56. [64]

    Kawaguchi, S

    R. Kawaguchi, S. Tsujikawa, and Y. Yamada, Phys. Lett. B 856, 138962 (2024), arXiv:2403.16022 [hep-th]

    2024 arXiv

  57. [65]

    S. B. Giddings and M. S. Sloth, JCAP 01, 023 (2011), arXiv:1005.1056 [hep-th]

    2011 arXiv

  58. [66]

    C. T. Byrnes, M. Gerstenlauer, A. Hebecker, S. Nurmi, and G. Tasinato, JCAP 08, 006 (2010), arXiv:1005.3307 [hep-th]

    2010 arXiv

  59. [67]

    Inomata, E

    K. Inomata, E. McDonough, and W. Hu, Phys. Rev. D 104, 123553 (2021), arXiv:2104.03972 [astro-ph.CO]

    2021 arXiv

  60. [68]

    Inomata, E

    K. Inomata, E. McDonough, and W. Hu, JCAP 02, 031 (2022), arXiv:2110.14641 [astro-ph.CO]

    2022 arXiv

  61. [69]

    Iacconi and D

    L. Iacconi and D. J. Mulryne, JCAP 09, 033 (2023), arXiv:2304.14260 [astro-ph.CO]

    2023 arXiv

  62. [70]

    Fumagalli, S

    J. Fumagalli, S. Bhattacharya, M. Peloso, S. Renaux-Petel, and L. T. Witkowski, JCAP 04, 029 (2024), arXiv:2307.08358 [astro-ph.CO]

    2024 arXiv

  63. [71]

    Caravano, K

    A. Caravano, K. Inomata, and S. Renaux-Petel, Phys. Rev. Lett. 133, 151001 (2024), arXiv:2403.12811 [astro-ph.CO]

    2024 arXiv

  64. [72]

    Caravano, G

    A. Caravano, G. Franciolini, and S. Renaux-Petel, (2024), arXiv:2410.23942 [astro-ph.CO]

    2024 arXiv

  65. [73]

    Tanaka and Y

    T. Tanaka and Y. Urakawa, Class. Quant. Grav. 30, 233001 (2013), arXiv:1306.4461 [hep-th]

    2013 arXiv

  66. [74]

    J. M. Maldacena, JHEP 05, 013 (2003), arXiv:astro-ph/0210603

    2003 arXiv

  67. [75]

    Hinterbichler, L

    K. Hinterbichler, L. Hui, and J. Khoury, JCAP 08, 017 (2012), arXiv:1203.6351 [hep-th]

    2012 arXiv

  68. [76]

    M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory (Addison-Wesley, Reading, USA, 1995)

    1995

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