\(\delta \mathcal {N}\) formalism on the past light-cone

We apply the gradient expansion approximation to the light-cone gauge, obtaining a separate universe picture at non-linear order in perturbation theory within this framework. Thereafter, we use it to generalize the \(\delta \mathcal {N}\) $\delta \mathcal{N}$ formalism in terms of light-cone perturbations. As a consistency check, we demonstrate the conservation of the gauge invariant curvature perturbation on uniform density hypersurface \(\zeta \) $\zeta$ at the completely non-linear level. The approach studied provides a self-consistent framework to connect at non-linear level quantities from the primordial universe, such as \(\zeta \) $\zeta$ , written in terms of the light-cone parameters, to late time observables.