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arxiv: 2502.10287 · v2 · pith:2JSTA4TVnew · submitted 2025-02-14 · 🌌 astro-ph.CO · gr-qc· hep-ph· hep-th

Hamiltonians to all Orders in Perturbation Theory and Higher Loop Corrections in Single Field Inflation with PBHs Formation

Hassan Firouzjahi , Bahar Nikbakht This is my paper

Pith reviewed 2026-05-23 03:00 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-phhep-th
keywords single field inflationultra slow-rollprimordial black holesloop correctionseffective field theoryperturbation theoryGoldstone boson
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0 comments X

The pith

In single-field inflation with a transient ultra-slow-roll phase, the L-loop corrections to CMB-scale perturbations scale as (ΔN P_e L)^L and exit perturbative control at L=4 for typical PBH-forming parameters with ΔN ≈ 2.5.

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

The paper derives the full action and interaction Hamiltonians to all orders in perturbation theory for single-field inflation containing a brief ultra-slow-roll interval. It supplies a compact non-perturbative form for the interaction Hamiltonian in the Goldstone field π together with the all-order map from π to the curvature perturbation. These expressions make it possible to evaluate cosmological correlators at arbitrary loop order. As an application, the authors compute the L-loop corrections induced on long-wavelength modes by the peak in the power spectrum during the ultra-slow-roll phase and obtain the explicit scaling (ΔN P_e L)^L.

Core claim

The interaction Hamiltonian admits a compact non-perturbative expression in the decoupling limit, and the resulting L-loop corrections on long CMB scales in ultra-slow-roll models obey the scaling (ΔN P_e L)^L. For the conventional choice ΔN ≃ 2.5 used to produce primordial black holes, this scaling drives the corrections out of perturbative control already at L=4.

What carries the argument

Compact non-perturbative expression for the interaction Hamiltonian in terms of the Goldstone field π (in the decoupling limit), which organises all-order loop calculations.

If this is right

  • Cosmological correlators can be computed to any desired loop order using the all-order Hamiltonian.
  • In ultra-slow-roll models for primordial black holes the loop corrections grow rapidly with L and exceed unity by L=4 for standard parameters.
  • Perturbation theory breaks down at low loop order whenever the ultra-slow-roll phase is long enough to produce an appreciable peak P_e.

Where Pith is reading between the lines

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

  • Models that rely on a transient ultra-slow-roll phase to generate primordial black holes may require a non-perturbative resummation rather than a finite-order loop expansion.
  • The same all-order machinery could be applied to other transient features in the inflationary potential to test whether similar loop blow-ups occur.

Load-bearing premise

The peak value P_e of the power spectrum can be computed independently of the higher-loop corrections whose magnitude is being estimated.

What would settle it

An explicit four-loop calculation in a concrete ultra-slow-roll model with ΔN ≈ 2.5 that either confirms or contradicts the predicted size (ΔN P_e * 4)^4 relative to the tree-level result.

Figures

Figures reproduced from arXiv: 2502.10287 by Bahar Nikbakht, Hassan Firouzjahi.

Figure 1
Figure 1. Figure 1: The one-vertex, one-particle irreducible Feynman diagrams for various values of [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: The total resummed bulk loop corrections from Eq. (5.34). The curves from top to bottom correspond to h = −6, h = −9 and h = −12 respectively. Right: The comparison of the resummed bulk contributions for h = −6: the kinetic term Eq. (5.32) (dotted curve), the gradient term Eq. (5.33) (dashed curve) and the total contribution (5.34) (solid curve). 6 Loop Corrections from the Boundary of USR Here we ca… view at source ↗
Figure 3
Figure 3. Figure 3: Left: The comparison of the contributions from the bulk Eq. (5.29) (dashed curve) and the boundary Eq. (6.20) (solid curve) for ∆N = 2.3 and h = −6. We see that for large L the contribution of bulk falls off while that of boundary grows rapidly. Right: Lc as a function of ∆N for h = −6. The larger is ∆N, the smaller is Lc. correction come from the boundary term in which Rb(L) ≫ RB(L) for L ≫ 1. Third, as c… view at source ↗
Figure 4
Figure 4. Figure 4: The total loop corrections as given in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Some two-vertices Feynman diagrams at L-loop order. Left: (H4, H2L) vertices, middle: (H3, H2L+1) vertices, right: (HL+2, HL+2) vertices. tonians to calculate the L-loop corrections in power spectrum in models of inflation involving an intermediate phases of USR inflation. This analysis extends the previous results obtained for one￾loop case such as in [6, 16] and the two-loop analysis partially performed … view at source ↗
read the original abstract

We calculate the action and the interaction Hamiltonians to all orders in perturbation theory in the model of single field inflation with a transient ultra slow-roll phase. Employing the formalism of EFT of inflation, we obtain a compact non-perturbative expression for the interaction Hamiltonian in terms of the Goldstone field $\pi$ in the decoupling limit. In addition, we also present a non-linear relation between $\pi$ and the curvature perturbations to all orders in perturbation theory. These are powerful results which enable us to calculate the cosmological correlators and loop corrections to any order in perturbation theory. As a non-trivial example, we calculate the $L$-loop corrections on long CMB scale perturbations in the USR models which are used for PBHs formation. We show that the loop corrections scale like $(\Delta N {\cal P}_e L) ^L$ in which ${\cal P}_e$ is the peak of the power spectrum and $\Delta N$ is the duration of the USR phase. This indicates that the loop corrections grow quickly out of perturbative control for large values of $L$. In the conventional USR setup for PBHs formation with $\Delta N \simeq 2.5$, this happens at $L=4$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper derives the action and interaction Hamiltonians to all orders in perturbation theory for single-field inflation featuring a transient ultra-slow-roll (USR) phase, using the EFT of inflation in the decoupling limit. It obtains a compact non-perturbative expression for the interaction Hamiltonian in terms of the Goldstone field π and a nonlinear relation between π and curvature perturbations. As an application, it computes the L-loop corrections to long CMB-scale perturbations in USR models for PBH formation and reports that these corrections scale as (ΔN P_e L)^L, implying loss of perturbative control at L=4 for conventional ΔN ≃ 2.5.

Significance. If the all-order Hamiltonian derivation and the reported scaling hold, the work supplies a systematic tool for computing arbitrary-order correlators in USR inflation and identifies a concrete threshold where perturbation theory fails for PBH-relevant parameters. The provision of non-perturbative expressions for the Hamiltonian and the π-to-ζ relation constitutes a technical advance that could be reused beyond the specific loop estimate.

major comments (2)
  1. [L-loop corrections section / abstract] The scaling (ΔN P_e L)^L for the L-loop correction (abstract and the section presenting the L-loop calculation) is obtained by treating the tree-level peak amplitude P_e as an external fixed input. The manuscript does not demonstrate that the same class of diagrams leaves the small-scale power spectrum peak uncorrected at O(1); if P_e itself receives large loop corrections, the numerical threshold L=4 becomes self-referential and cannot be read off directly.
  2. [Derivation of interaction Hamiltonian] The non-perturbative interaction Hamiltonian is asserted to follow from integrating out the USR phase, yet no explicit derivation or error estimate is supplied showing that the compact expression remains valid when the USR duration ΔN is finite (rather than strictly instantaneous). This step is load-bearing for the subsequent loop scaling.
minor comments (1)
  1. Notation: the symbol P_e is introduced without an explicit equation defining it as the tree-level peak; a numbered equation would clarify its relation to the power spectrum computed from the same Hamiltonian.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [L-loop corrections section / abstract] The scaling (ΔN P_e L)^L for the L-loop correction (abstract and the section presenting the L-loop calculation) is obtained by treating the tree-level peak amplitude P_e as an external fixed input. The manuscript does not demonstrate that the same class of diagrams leaves the small-scale power spectrum peak uncorrected at O(1); if P_e itself receives large loop corrections, the numerical threshold L=4 becomes self-referential and cannot be read off directly.

    Authors: We agree that P_e enters the scaling as a tree-level input parameter. Our calculation computes the L-loop corrections specifically to the long-wavelength CMB-scale perturbations generated by the short-scale fluctuations during the USR phase. The same vertices could indeed produce loop corrections to the small-scale power spectrum peak. We will revise the abstract and the L-loop section to state explicitly that P_e denotes the tree-level peak amplitude and to note that a fully self-consistent treatment would require separate computation of loop corrections to the peak itself. Such corrections would renormalize the effective value of P_e but preserve the functional form of the scaling and the conclusion that perturbative control is lost at modest L for conventional parameters. This clarification will be added without altering the reported scaling for the long modes. revision: partial

  2. Referee: [Derivation of interaction Hamiltonian] The non-perturbative interaction Hamiltonian is asserted to follow from integrating out the USR phase, yet no explicit derivation or error estimate is supplied showing that the compact expression remains valid when the USR duration ΔN is finite (rather than strictly instantaneous). This step is load-bearing for the subsequent loop scaling.

    Authors: The compact expression is obtained within the EFT of inflation in the decoupling limit by expressing the action in terms of the Goldstone field π and incorporating the transient USR dynamics. We acknowledge that the manuscript presents the final result without a detailed derivation or error estimate for finite ΔN. We will add an appendix that derives the non-perturbative Hamiltonian step by step from the EFT action, including an error estimate showing that corrections arising from the finite duration of the USR phase are suppressed by the slow-roll parameters outside the USR interval and remain small for ΔN ≃ 2.5. This will confirm the validity of the expression used for the loop calculation. revision: yes

Circularity Check

0 steps flagged

No circularity; loop scaling derived from all-order Hamiltonian with P_e as external model input

full rationale

The paper derives a non-perturbative interaction Hamiltonian in the EFT of inflation and uses it to obtain the L-loop scaling (ΔN P_e L)^L for corrections to long CMB modes. P_e is the peak power spectrum amplitude on small scales set by the USR background evolution, treated as an independent input parameter of the model rather than a quantity computed from the loop diagrams under consideration. The derivation chain (all-order Hamiltonian → correlator computation → scaling) does not reduce to its own outputs by construction, contains no fitted parameters renamed as predictions, and invokes no self-citations for load-bearing steps. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the EFT framework in the decoupling limit and on treating the USR phase duration and power-spectrum peak as independent inputs.

free parameters (2)
  • ΔN = 2.5
    Duration of the transient ultra-slow-roll phase, chosen to produce the desired PBH abundance.
  • P_e
    Peak value of the curvature power spectrum during the USR phase, set by model parameters.
axioms (2)
  • domain assumption The effective field theory of inflation in the decoupling limit captures the relevant dynamics to all orders.
    Invoked to obtain the compact non-perturbative expression for the interaction Hamiltonian.
  • domain assumption The ultra-slow-roll phase is transient and can be consistently embedded in single-field inflation.
    Required for the model setup used to generate PBHs.

pith-pipeline@v0.9.0 · 5764 in / 1561 out tokens · 79121 ms · 2026-05-23T03:00:54.384298+00:00 · methodology

discussion (0)

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Forward citations

Cited by 2 Pith papers

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

  1. Stochastic Inflation with Interacting Noises

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    The stochastic noise amplitude is modified to (H/2π) * sqrt(1 + ΔP_R / P0_R) to account for one-loop corrections in interacting theories, demonstrated in a three-phase SR-USR-SR setup for PBH formation.

  2. Fixing the Renormalization of Inflationary Loops via Ward Identities

    gr-qc 2026-05 unverdicted novelty 5.0

    Ward identities from large gauge symmetry impose model-independent constraints on renormalizing inflationary loops and non-perturbatively govern the infrared power spectrum evolution.

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