Inflationary Fossils Beyond Perturbation Theory

In this work we provide the missing link between two approaches aimed at characterizing the effect of long perturbation modes in Inflation. We consider the Inflationary Fossils’ approach (arXiv:1203.0302 and related works) that characterizes the power-spectrum of the inflaton field in presence of other long and non dynamical fossil fields, and a technique, appeared in arXiv:2103.09244, that computes, beyond perturbation theory, the power-spectrum of a scalar field in presence of a large fluctuation of a second field. We clarify a few points on the applicability of the non-perturbative technique. We prove in six distinct cases, one involving a violation of the consistency conditions, that the non-perturbative approach, once expanded to first order in the coupling, matches the perturbative result following the Fossils’ approach. We believe that this non-perturbative technique extends to all orders the Fossils’ approach, resumming infinitely many diagrams of standard in-in perturbation theory.

💡 Research Summary

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This paper establishes a concrete bridge between two seemingly distinct frameworks used to describe the influence of long-wavelength perturbations on short-wavelength modes during inflation. The first framework, often called the “Inflationary Fossils” approach (originally introduced in arXiv:1203.0302 and subsequent works), treats a long, non‑dynamical mode χ as a classical background and adds to the short‑mode power spectrum a correction obtained from the squeezed limit of the three‑point function ⟨σ σ χ⟩. The second framework, introduced in arXiv:2103.09244, provides a non‑perturbative method for computing the power spectrum of a scalar field σ in the presence of a large fluctuation of a second field χ. The method works by integrating out the large χ fluctuation, treating it as a fixed background, and solving the resulting effective equation of motion for σ exactly (or to any desired order), thereby resumming an infinite set of in‑in diagrams.

The authors first consider three toy models in exact de Sitter space, each featuring a different cubic interaction between σ and χ: (i) a non‑derivative coupling λ χ σ², (ii) a time‑derivative coupling λ ∂₀σ ∂₀σ χ, and (iii) a spatial‑derivative coupling λ ∂ᵢσ ∂ᵢσ χ. For each model they impose a hierarchy of parameters: the coupling λ is weak (λ ≪ 1), the long mode momentum kχ is much smaller than the short mode momentum q, and the background amplitude of χ satisfies λ \barχ/H ∼ 1, i.e. the long mode is “large” compared to the typical Hubble fluctuation. Under these assumptions χ obeys the free de Sitter equation and can be replaced by its classical profile χ(η) ≈ \barχ(1 − ikχη)e^{ikχη}. Substituting this profile into the σ equation yields an effective action in which σ acquires either an effective mass (case i) or a modified sound speed (cases ii and iii). Solving the effective equation exactly gives closed‑form expressions for the σ power spectrum:

• Case (i): Pσ(k) ∝ k⁻³ (−kη)^{3−2ν} with ν = √{9/4 − λ \barχ/H}.
• Case (ii): Pσ(k) ∝ k⁻³ cₛ⁻¹ with cₛ² = 1 − λ \barχ/H.
• Case (iii): after a field redefinition σ → ϕ = σ√{1+λ \barχ/H}, the spectrum becomes Pσ(k) ∝ k⁻³ (1+λ \barχ/H)^{−3/2}.

Expanding each result to first order in λ reproduces a simple linear correction proportional to λ \barχ/H.

Next, the authors apply the Fossils prescription. They compute the bispectrum ⟨σ σ χ⟩ for each interaction using the standard in‑in formalism (the +/− contour). In the squeezed limit (q → 0) the bispectrum factorizes as ⟨σ σ χ⟩ ≈ α λ \barχ/H · Pσ(k) · Pχ(q), where α is a numerical coefficient that depends on the interaction (α = 2/3, 1/2, 3, etc.). Inserting this into the Fossils formula \