Continuous Sensitivity Analysis for $\delta N$ Formalism

The $\delta N$ 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 $\delta N$ framework beyond the separate universe approximation. Despite this conceptual progress, practical calculations within the $\delta N$ 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 $\delta N$ 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 $\delta N$ 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.