{"paper":{"title":"Continuous Sensitivity Analysis for $\\delta N$ Formalism","arxiv_id":"2603.27366","primary_cat":"gr-qc","submitted_at":"2026-03-28T18:23:31+00:00","version_compiled":"v2"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. 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interactions.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.33,"method":0.0,"result":0.0,"background":0.51,"limitation":0.16,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.99,"id":"s2","own":false,"role":"background","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.0,"background":0.99,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.81,"id":"s3","own":false,"role":"gap","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.81,"method":0.0,"result":0.0,"background":0.16,"limitation":0.02,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":4,"p":0.96,"id":"s4","own":true,"role":"contribution","text":"In this work, we develop a systematic method to simplify these calculations by applying Continuous Sensitivity Analysis to the gradient-corrected $\\delta N$ framework.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.27,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.04,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.96,"interpretation":0.0}},{"i":5,"p":0.99,"id":"s5","own":true,"role":"method","text":"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 the $\\delta N$ formalism.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.78,"main_claim_p":0.29,"quantitative":false,"quantitative_p":0.25,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":6,"p":0.54,"id":"s6","own":true,"role":"method","text":"As an explicit demonstration, we apply the method to the Starobinsky model, which features a sharp transition into an ultra-slow-roll phase.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.54,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.46,"interpretation":0.0}},{"i":7,"p":0.98,"id":"s7","own":true,"role":"result","text":"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 main non-Gaussian 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