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$δN$ formalism with gradient interactions

S. Mohammad Ahmadi, Nahid Ahmadi

TL;DR

This work addresses the breakdown of the separate-universe approximation when gradient interactions are significant, as in transitions to ultra-slow-roll relevant for primordial black hole formation. It introduces a gradient-corrected $\\delta N$ framework by adding a $\\mathcal{S}_k$ term to the background Klein–Gordon equation, enabling nonlinear curvature evolution to be tracked from horizon exit. The authors demonstrate equivalence to higher-order gradient matching and show that including gradient corrections recovers linear perturbation theory results and key non-Gaussian features, notably the equilateral $f_{\\rm NL}^{\\rm eq}$. Tests on Gaussian-bump and Starobinsky-like potentials indicate substantial improvements over standard $\\delta N$ and provide a practical route to PBH-relevant predictions.

Abstract

The standard $δN$ formalism, a cornerstone for calculating nonlinear curvature perturbations on super-Hubble scales, relies on the separate universe assumption, in which spatial gradients are neglected. However, this approximation breaks down in scenarios critical for primordial black hole formation, such as transitions to an ultra-slow-roll phase, where gradient interactions induce significant non-conservation of the comoving curvature perturbation. In this Letter, we introduce a framework that incorporates gradient corrections into the $δN$ formalism at a desired order by adding an effective source term to the background Klein--Gordon equation. This approach allows for a nonlinear treatment of curvature perturbations at the end of inflation considering initial conditions at the time of horizon exit. By computing the equilateral non-Gaussianity parameter $f_{\mathrm{NL}}^{\mathrm{eq}}$, we demonstrate that our method captures essential features missed by the standard $δN$, offering a simple yet rigorous pathway to determine nonlinear evolution expected from cosmological perturbation theory.

$δN$ formalism with gradient interactions

TL;DR

This work addresses the breakdown of the separate-universe approximation when gradient interactions are significant, as in transitions to ultra-slow-roll relevant for primordial black hole formation. It introduces a gradient-corrected framework by adding a term to the background Klein–Gordon equation, enabling nonlinear curvature evolution to be tracked from horizon exit. The authors demonstrate equivalence to higher-order gradient matching and show that including gradient corrections recovers linear perturbation theory results and key non-Gaussian features, notably the equilateral . Tests on Gaussian-bump and Starobinsky-like potentials indicate substantial improvements over standard and provide a practical route to PBH-relevant predictions.

Abstract

The standard formalism, a cornerstone for calculating nonlinear curvature perturbations on super-Hubble scales, relies on the separate universe assumption, in which spatial gradients are neglected. However, this approximation breaks down in scenarios critical for primordial black hole formation, such as transitions to an ultra-slow-roll phase, where gradient interactions induce significant non-conservation of the comoving curvature perturbation. In this Letter, we introduce a framework that incorporates gradient corrections into the formalism at a desired order by adding an effective source term to the background Klein--Gordon equation. This approach allows for a nonlinear treatment of curvature perturbations at the end of inflation considering initial conditions at the time of horizon exit. By computing the equilateral non-Gaussianity parameter , we demonstrate that our method captures essential features missed by the standard , offering a simple yet rigorous pathway to determine nonlinear evolution expected from cosmological perturbation theory.
Paper Structure (8 sections, 24 equations, 2 figures)

This paper contains 8 sections, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Separate Universe Approach
  3. Homogeneous Matching
  4. The $\delta N$ Formalism
  5. Incorporating gradient terms into the $\delta N$ formalism
  6. Comparison with linear perturbation theory
  7. Non-Gaussianity
  8. Summary and discussion

Figures (2)

  • Figure 1: Comparison of the power spectra of the curvature perturbation, $\mathcal{P}_\mathcal{R} = \frac{k^3}{2 \pi^2} \left| \mathcal{R}_k \right|^2$ computed using the $\delta N$ formalism \ref{['expand_R_second']}, the higher-order matching method \ref{['higher_order_matching']}, and the exact MS equation \ref{['eq:MS_for_R']}, for the Gaussian bump (top panel) and Starobinsky (bottom panel) models. The gray solid line denotes the exact MS solution, while the colored lines represent the results obtained by step-by-step inclusion of $\mathcal{O}(k^2)$ corrections in the matching method. Colored markers indicate the corresponding $\delta N$ results. The black dot markers show the $\delta N$ results with full gradient interactions. The inset in each panel compares the MS solution with the $\delta N$ results including full gradients over a wider range of wavenumbers.
  • Figure 2: Equilateral-type non-Gaussianity parameter $f_{\rm NL}^{\rm eq}$ computed with gradient interactions (solid blue line) and using the standard $\delta N$ formalism (dashed green line) for the Gaussian bump (top panel) and Starobinsky (bottom panel) models. The orange dot--dashed line shows the gradient-corrected $f_{\rm NL}^{\rm eq}$ obtained by neglecting the contribution of $\dot{\delta\phi}_k$. In all panels, the matching time is chosen at horizon crossing, $\sigma = 1$. For ease of comparison, the parameter values of the Starobinsky model are chosen similar to Ref. hazra2013bingo.