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arxiv: 2511.00152 · v2 · submitted 2025-10-31 · ✦ hep-th · astro-ph.CO· gr-qc· hep-ph

Every Wrinkle Carries A Memory: An Integro-differential Bootstrap for Features in Cosmological Correlators

Sadra Jazayeri , Xi Tong , Yuhang Zhu This is my paper

Pith reviewed 2026-05-18 02:09 UTC · model grok-4.3

classification ✦ hep-th astro-ph.COgr-qchep-ph
keywords cosmological correlatorsbootstrapintegro-differential equationsaxion monodromycosmological colliderprimordial featuresscale-breakingmemory kernel
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0 comments X

The pith

Locality in the bulk implies integro-differential equations for cosmological correlators that carry a memory kernel of inflationary evolution.

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

This paper establishes that bulk locality for heavy fields with time-dependent masses and sound speeds produces integro-differential equations on the boundary for cosmological correlators. These equations include a built-in memory kernel in momentum-kinematic space that encodes the history of the universe during inflation. Specializing to sinusoidal masses, the combination of microcausality and analyticity yields an exact leading-order analytical solution. The resulting squeezed bispectrum exhibits a scale-breaking cosmological collider signal whose amplitude receives exponential enhancement from particle production triggered by the oscillations.

Core claim

Locality in the bulk implies a set of integro-differential equations for correlators on the boundary with a built-in memory kernel in momentum-kinematic space encapsulating the universe's evolution during inflation. Specialising to heavy fields with sinusoidal masses such as those found in axion monodromy scenarios, a synthesis of microcausality and analyticity allows an analytical solution at leading order in the amplitude of mass oscillations, with the squeezed bispectrum exhibiting an exponentially enhanced scale-breaking cosmological collider signal due to particle production.

What carries the argument

Integro-differential equations for boundary correlators that incorporate a memory kernel derived from bulk locality.

If this is right

  • The boundary equations admit an analytical leading-order solution for sinusoidal mass oscillations.
  • The squeezed bispectrum displays a scale-breaking cosmological collider signal with exponential enhancement over Boltzmann suppression.
  • Infrared divergences in the equations can be resummed as parametric resonances to extract non-perturbative information.
  • Numerical solution of the boundary equations directly maps the full solution space.

Where Pith is reading between the lines

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

  • The memory kernel offers a route to reconstruct aspects of bulk dynamics from measured boundary correlators.
  • The framework may extend to other time-dependent bulk parameters beyond mass, such as varying sound speeds.
  • The exponential enhancement could increase the visibility of primordial features in future CMB or large-scale structure surveys.

Load-bearing premise

The heavy fields have exactly sinusoidal time-dependent masses with small oscillation amplitudes so that a leading-order analytic treatment captures the dominant effects.

What would settle it

Detection of a squeezed bispectrum signal whose amplitude matches only the standard Boltzmann suppression without the predicted exponential boost for oscillating-mass models.

Figures

Figures reproduced from arXiv: 2511.00152 by Sadra Jazayeri, Xi Tong, Yuhang Zhu.

Figure 1
Figure 1. Figure 1: Right: The single-exchange diagram of our interest, characterised by four external conformally coupled fields φ and a heavy intermediate scalar σ, endowed with a time-dependent mass m(t). For constant masses, the diagram satisfies the ordinary differential equation (1.1). Left: For time-dependent masses, the integro-differential equation (1.2) takes over, relating the exchange diagram in one momentum confi… view at source ↗
Figure 2
Figure 2. Figure 2: The seed four-point function vs. the single-exchange diagrams for the bispectrum and the power spectrum of ζ, related by the action of weight-shifting operators, see Section 4.3 perturbations. We will achieve this in Section 4.3 through a set of weight-shifting operators and by taking appropriate soft limits to reduce from the four-point to the three- and the two-point kinematics, along the lines of [14, 1… view at source ↗
Figure 3
Figure 3. Figure 3: Four-point and three-point function with multiple mass insertions. f(k12, s) = 1 √ s X +∞ l=−∞ (−sη0) ilω fl(u). (3.27) Inserting the above mode expansions into the respective IDEs does not yield any major simplifi￾cation, other than producing a recursive set of IDEs for Fl and fl . Despite this recursive nature, the decomposition becomes especially convenient in perturbation theory, where—at a fixed order… view at source ↗
Figure 4
Figure 4. Figure 4: The plots depict two components of the Bogolyubov coefficient ∆β, defined by (4.21)– (4.22), as functions of µ. See the discussions around (4.31). We observe that the negative frequency component ∆β− (left) is enhanced within the mass window H ≪ µ ≲ ω, due to mass oscillations, while the positive frequency component ∆β+ (right) exponentially decays for large masses µ ≳ 1, regardless of the frequency ω. Mor… view at source ↗
Figure 5
Figure 5. Figure 5: The exchange diagram corresponding to F++(k12, k3, k3) at linear order in g 2 . The derivation of the exchange diagram parallels that of the three-point contact diagram f(k12, s). We start by decomposing F++ according to (3.17). Up to linear order in g 2 F++(k12, k34, s)|k4=0 = 1 s [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The modulus of the prefactor of the u 1/2+iµ x −iω 0 term in the resonant cosmological collider signal in (4.65), |C2(µ, −ω)|, for three sample values of the oscillation frequency ω, as a function of µ. Thanks to the mass oscillation induced particle production, the prefactor is amplified in the mass range m0 ≲ ω, exhibiting almost a flat behaviour apart from a sharp peak at µ = ω/2, which corresponds to t… view at source ↗
Figure 7
Figure 7. Figure 7: The s-channel bispectrum (4.85), normalised by two power spectrum P(k1) P(k3) and the coupling parameter r2, with an additional factor of (k1/k3) 3/2 for the better visualisation. The soft momentum k3 is fixed as we vary k1 and the scale-breaking phase is also at x0 = −k3η0 = 10. Left panel: the mass is fixed as µ = 5 while the oscillation frequency varies, ω = 13, 18, 23, with coupling g = 0.1. Right pane… view at source ↗
Figure 8
Figure 8. Figure 8: Schematic illustration of the two types of parametric resonances in our model. The black curve denotes the evolution of the effective frequency weff(t) of a massive particle with cosmic time, where it is dominated by the kinetic energy k/a(t) = −kη at early times (UV) and by the rest mass µ at late times (IR). Parametric resonances can be triggered when the effective frequency sweeps across the resonance b… view at source ↗
Figure 9
Figure 9. Figure 9: The time evolution of the modulus of the mode function (with comoving momentum s), from deep inside the horizon (|sη| ≫ 1) to the end of inflation (|sη| → 0), evaluated for the parameter values: (ω = 80.0, m0 = 12.0, g2 = 0.36) [left black curve] and (ω = 10.0, m0 = 4.0, g2 = 0.36) [right black curve]. The orange line in each case corresponds to g = 0 (with the same mass m0). The dashed lines mark the UV r… view at source ↗
Figure 10
Figure 10. Figure 10: Convergence tests of the numerical bootstrap solution with respect to the mesh points N (left panel) and the cutoff L = log(k12/s)max (right panel). The benchmark kinematics is chosen to be r∗ = log(k12/s) ∗ = 8 and the blue and yellow points correspond to g = 0.2 and g = 0.3, respectively. The dashed lines represent the prediction of the perturbative analytical solution. The other parameters are chosen a… view at source ↗
Figure 11
Figure 11. Figure 11: The dependence of the rescaled shape function on the momentum ratio r = log(k12/s). The left and right panels correspond to frequency choices ω = 2/3 and ω = 4/3, respectively. The grey, blue and yellow curves correspond to different sizes of the coupling g as specified in the right panel. Other model parameters are chosen as µ = 1, x0 = −sη0 = 2. The number of mesh point is N = 104 with a kinematic cutof… view at source ↗
Figure 12
Figure 12. Figure 12: The rescaled shape function in the resonant regime. The left and right panels correspond to frequencies at the primary resonance ω1 ≃ 2µ and the secondary resonance ω2 ≃ µ, respectively. The grey, blue and yellow curves correspond to different sizes of the coupling g as specified in the left panel. To maintain ω ≲ 1 for algorithm stability, we choose different masses for these two resonances, i.e. µ = 0.5… view at source ↗
Figure 13
Figure 13. Figure 13: 20 samples of numerical bootstrap with random boundary conditions near the primary (left) and secondary (right) resonances. The red solid lines are the prediction from boundary eigenfrequency analysis (see (5.17) and (5.20)) while the red dashed lines are the natural scaling in the absence of IR parametric resonances. Here the coupling constant is g = 0.5 and we choose µ = 0.5 and µ = 1 for the primary an… view at source ↗
read the original abstract

Motivated by cosmological observations, we push the cosmological bootstrap program beyond the de Sitter invariance lamppost by considering correlators that explicitly break scale invariance, thereby exhibiting primordial features. For exchange processes involving heavy fields with time-dependent masses and sound speeds, we demonstrate that locality in the bulk implies a set of integro-differential equations for correlators on the boundary. These scale-breaking boundary equations come with a built-in memory kernel in momentum-kinematic space encapsulating the universe's evolution during inflation. Specialising to heavy fields with sinusoidal masses such as those found in axion monodromy scenarios, we show that a powerful synthesis of microcausality and analyticity allows an analytical solution of these equations at leading order in the amplitude of mass oscillations. Meanwhile, we also unveil non-perturbative information in the integro-differential equations by resumming apparent infrared divergences as parametric resonances. In addition, we provide a first-of-its-kind example of numerical bootstrap that directly maps out the solution space of such boundary equations. Finally, we compute the bispectrum and uncover, in the squeezed limit, a scale-breaking cosmological collider signal, whose amplitude can be exponentially enhanced (with respect to the Boltzmann suppression) due to particle production triggered by high-frequency mass oscillations.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript develops an integro-differential bootstrap for cosmological correlators that break scale invariance through time-dependent masses of heavy fields. Bulk locality and microcausality imply a set of boundary equations equipped with a memory kernel encoding the inflationary evolution. Specializing to sinusoidal mass profiles m²(t) = m₀² + A sin(ω t) with small A, microcausality plus analyticity yields a closed-form leading-order analytic solution; apparent infrared divergences are resummed as parametric resonances to extract non-perturbative information. A numerical bootstrap example is provided, and the squeezed bispectrum is shown to contain a scale-breaking cosmological collider signal whose amplitude can be exponentially enhanced relative to Boltzmann suppression by oscillation-triggered particle production.

Significance. If the central claims hold, the work meaningfully extends the cosmological bootstrap beyond de Sitter invariance by incorporating explicit scale-breaking features and memory effects directly into boundary equations. The combination of an analytical O(A) solution for sinusoidal masses, a resummation procedure for infrared divergences, and a first numerical bootstrap implementation supplies both practical tools and conceptual insight into particle production in axion-monodromy-type models. The prospect of exponentially enhanced collider signals in the bispectrum is observationally relevant for primordial non-Gaussianity searches.

major comments (2)
  1. [§5.1, Eq. (5.7)] §5.1, Eq. (5.7) and surrounding text: The leading-order analytic solution in the oscillation amplitude A is presented as capturing an exponentially enhanced squeezed bispectrum. However, the enhancement is attributed to particle production from high-frequency mass oscillations and to the resummation of apparent infrared divergences as parametric resonances. Parametric resonance growth is non-perturbative in A; the manuscript must explicitly demonstrate whether the O(A) truncation already produces the dominant exponential factor or whether the enhancement is obtained only after the separate resummation step described in §6.
  2. [§3.2, Eq. (3.12)] §3.2, Eq. (3.12): The integro-differential equation is derived from bulk locality with a memory kernel. While the constant-mass limit is recovered, the error control on the kernel truncation for time-dependent masses (especially when specializing to the sinusoidal case) is not quantified; this directly affects the reliability of both the analytic solution and the numerical bootstrap.
minor comments (2)
  1. The definition of the memory kernel K(k, t, t') should be written explicitly in the main text (rather than deferred to an appendix) to improve readability of the central equations.
  2. [Figure 4] Figure 4 (numerical bootstrap results): Adding a direct overlay of the analytic leading-order prediction would help readers assess the accuracy of the O(A) solution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised help clarify the separation between perturbative and non-perturbative effects as well as the control of approximations. We address each major comment below and have revised the manuscript to improve precision and add supporting estimates.

read point-by-point responses

  1. Referee: [§5.1, Eq. (5.7)] §5.1, Eq. (5.7) and surrounding text: The leading-order analytic solution in the oscillation amplitude A is presented as capturing an exponentially enhanced squeezed bispectrum. However, the enhancement is attributed to particle production from high-frequency mass oscillations and to the resummation of apparent infrared divergences as parametric resonances. Parametric resonance growth is non-perturbative in A; the manuscript must explicitly demonstrate whether the O(A) truncation already produces the dominant exponential factor or whether the enhancement is obtained only after the separate resummation step described in §6.

    Authors: We thank the referee for highlighting this distinction. The O(A) analytic solution obtained in §5.1 via microcausality and analyticity yields the leading perturbative correction to the correlators, including oscillatory features induced by the sinusoidal mass. The exponential enhancement of the squeezed bispectrum, however, originates from the non-perturbative resummation of the apparent infrared divergences interpreted as parametric resonances in §6. We have revised the text around Eq. (5.7), the bispectrum discussion, and the abstract to state explicitly that the dominant exponential factor arises after the resummation step, while the O(A) result illustrates the perturbative onset of the effect. This removes any ambiguity between the two regimes. revision: yes

  2. Referee: [§3.2, Eq. (3.12)] §3.2, Eq. (3.12): The integro-differential equation is derived from bulk locality with a memory kernel. While the constant-mass limit is recovered, the error control on the kernel truncation for time-dependent masses (especially when specializing to the sinusoidal case) is not quantified; this directly affects the reliability of both the analytic solution and the numerical bootstrap.

    Authors: We agree that an explicit error estimate strengthens the reliability assessment. In the revised manuscript we have added a paragraph in §3.2 that quantifies the truncation error for time-dependent masses. For the sinusoidal profile with small amplitude A the leading error is O(A²) and is controlled by the convergence of the perturbative expansion in the kernel; we provide a concrete bound derived from the analytic properties of the memory integral. For the numerical bootstrap we have included additional convergence tests with respect to truncation order, confirming that the reported solutions remain stable within the quoted precision. revision: yes

Circularity Check

0 steps flagged

Derivation chain from bulk locality to boundary equations and leading-order solutions is self-contained

full rationale

The paper starts from the assumption of bulk locality to derive integro-differential equations with a memory kernel for boundary correlators, then explicitly specializes to sinusoidal mass profiles and applies microcausality plus analyticity to obtain an O(A) analytic solution. The resummation of apparent IR divergences as parametric resonances is framed as extracting non-perturbative content already present in those derived equations rather than introducing new fitted inputs or self-referential definitions. No load-bearing step reduces a claimed prediction or first-principles result to an input by construction, and no self-citation chain is invoked to justify uniqueness or the central ansatz.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard assumptions of locality, microcausality and analyticity in the bulk, plus the specific choice of sinusoidal mass profile; no new particles or forces are postulated, but the memory kernel is an emergent structure whose independence from fitting is not fully demonstrated in the abstract.

free parameters (1)
  • amplitude of mass oscillations
    Treated perturbatively at leading order; its smallness is required for the analytic solution to hold.
axioms (2)
  • domain assumption Bulk locality implies integro-differential boundary equations with memory kernel
    Invoked when moving from de Sitter-invariant bootstrap to scale-breaking cases.
  • domain assumption Microcausality and analyticity permit closed-form solution at leading order
    Used to specialise to sinusoidal masses.

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