δ n formalism: A new formulation for the probability density of the curvature perturbation

$δN$ formalism is a useful method to calculate the curvature perturbation. Contrary to what it is typically done in the literature, we re-formulate the $δN$ formalism by using the e-folding number n counted forward in time. For a fixed initial time n̄₀, the probability density function (PDF) of the field perturbation δϕ₀ and its velocity δπ₀ are specified by the solutions of the perturbation equation on subhorizon scales. As δπ₀ is fully correlated with δϕ₀ after horizon exit, we find a novel $δn$ formalism to calculate the curvature perturbation as well as its PDF. It can naturally incorporate those trajectories for which inflation never ends, which are lost when counting N backward in time.