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Continuous Sensitivity Analysis for $δN$ Formalism

S. Mohammad Ahmadi

Abstract

The $δN$ formalism provides a powerful non-perturbative framework for following the evolution of primordial curvature perturbations on super-horizon scales. However, its standard implementation relies on the separate universe assumption, which neglects significant spatial gradient interactions. Recent work has addressed this limitation by incorporating gradient interactions directly into the background dynamics through an effective source term in the Klein--Gordon equation, thereby extending the applicability of the $δN$ framework beyond the separate universe approximation. Despite this conceptual progress, practical calculations within the $δN$ formalism remain technically challenging, as cosmological observables require evaluating the sensitivity of the total number of $e$-folds to initial conditions, a task that becomes even more involved once gradient contributions are included. In this work, we develop a systematic method to simplify these calculations by applying Continuous Sensitivity Analysis to the gradient-corrected $δN$ framework. In this approach, the required phase-space derivatives are obtained by solving a set of coupled first-order differential equations for the field Jacobian and Hessian, which significantly streamlines both analytical and numerical evaluations of $δN$ formalism. As an explicit demonstration, we apply the method to the Starobinsky model, which features a sharp transition into an ultra-slow-roll phase. Within this setup, we derive analytical expressions for the $k$-dependent power spectrum including full gradient corrections, and obtain an analytical estimate of the equilateral non-Gaussianity parameter $f_{\rm NL}^{\rm eq}$ that accurately captures the gradient-sourced contributions.

Continuous Sensitivity Analysis for $δN$ Formalism

Abstract

The formalism provides a powerful non-perturbative framework for following the evolution of primordial curvature perturbations on super-horizon scales. However, its standard implementation relies on the separate universe assumption, which neglects significant spatial gradient interactions. Recent work has addressed this limitation by incorporating gradient interactions directly into the background dynamics through an effective source term in the Klein--Gordon equation, thereby extending the applicability of the framework beyond the separate universe approximation. Despite this conceptual progress, practical calculations within the formalism remain technically challenging, as cosmological observables require evaluating the sensitivity of the total number of -folds to initial conditions, a task that becomes even more involved once gradient contributions are included. In this work, we develop a systematic method to simplify these calculations by applying Continuous Sensitivity Analysis to the gradient-corrected framework. In this approach, the required phase-space derivatives are obtained by solving a set of coupled first-order differential equations for the field Jacobian and Hessian, which significantly streamlines both analytical and numerical evaluations of formalism. As an explicit demonstration, we apply the method to the Starobinsky model, which features a sharp transition into an ultra-slow-roll phase. Within this setup, we derive analytical expressions for the -dependent power spectrum including full gradient corrections, and obtain an analytical estimate of the equilateral non-Gaussianity parameter that accurately captures the gradient-sourced contributions.

Paper Structure

This paper contains 20 sections, 161 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review of Gradient Expansion and $\delta N$ Formalism
  3. Gradient Expansion and Matching Methods
  4. $\delta N$ Formalism With Gradient Interactions
  5. Continuous Sensitivity Analysis in $\delta N$ Formalism
  6. Linear CSA Formulation
  7. Equivalence of CSA and Perturbation Theory
  8. Green's Function Solution for Sensitivity Equations
  9. Second-Order CSA Formulation
  10. Application to the Starobinsky Model
  11. Background Dynamics and Linear Perturbations
  12. First-Order Sensitivities
  13. Homogeneous Jacobian
  14. Leading Gradient Corrections to Jacobian
  15. Jacobian with Full Gradient Interactions
  16. ...and 5 more sections

Figures (3)

  • Figure 1: The power spectrum of the Starobinsky model evaluated using analytic solution of the MS solution \ref{['Power_MS']}, standard $\delta N$ formalism results \ref{['Standard_dN_1']} and \ref{['Standard_dN_2']}, $\delta N$ with leading gradient corrections \ref{['GC_J_1']} and \ref{['GC_J_2']}, with inclusion of $\mathcal{O}(k^4)$ corrections \ref{['k4_corrections']} and \ref{['k4_corrections2']}, and with full gradient interactions \ref{['J1a_full_gradients']} and \ref{['J1a_full_gradients2']}, as a function of wavenumber. The matching time is chosen at horizon crossing time, $\sigma = 1$. The numerical values of model parameters are chosen as $V_0 = 0.137 M_{\mathrm{Pl}}^4$, $A_+ = 4.56 \times 10^{-3} M_{\mathrm{Pl}}^3$, $A_- = 1.139 \times 10^{-3} A_+$, $\phi_T = 0$, and $\phi_{\rm in} = 0.4 M_{\mathrm{Pl}}$. The exact MS solution is denoted by the thick gray line. The standard $\delta N$ approximation, its $\mathcal{O}(k^2)$ corrections, and $\mathcal{O}(k^4)$ corrections are represented by the solid blue, dot-dashed green, and dashed red lines, respectively. The full gradient $\delta N$ result is shown using black bullets. The inset panel illustrates these same quantities over an extended range of wavenumbers.
  • Figure 2: The logarithm of the absolute value of the equilateral-type non-Gaussianity parameter, $\log |f_{\rm NL}^{\rm eq}|$, for the Starobinsky model. The numerical values of the model parameters are chosen to be identical to those used in Figure \ref{['powespectrum_starobinsky_model']}. These parameters are adopted from Ref. hazra2013bingo to facilitate a direct comparison between our results and those obtained using the standard in-in formalism. The gray line represents the exact numerical integration including full spatial gradients, while the solid blue line shows the purely homogeneous numerical limit. These are compared against two analytic approximations: a purely homogeneous expansion $(\Theta^{(0)}, J^{(0)})$ shown by the orange dashed line \ref{['fNL_staro']}; and a mixed approximation using the homogeneous Hessian and full gradients Jacobian $(\Theta^{(0)}, J^{\text{full}})$ shown by the green dashed line \ref{['fNL_mix']}. The vertical gray dashed line indicates the transition scale $k_T$, and the inset provides a magnified view of the gradient-corrected results close to the peak of the non-Gaussianity.
  • Figure 3: The left panel presents the first two Hubble flow parameters, while the right panel displays the power spectrum of curvature perturbations for the punctuated inflation model, obtained using a fully numerical approach. The power spectra are computed from the exact MS solution (solid black line) and from the full gradients $\delta N$ formalism (dashed blue line). The matching time is chosen at horizon crossing, $\sigma = 1$.