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δN formalism

Precise understanding of nonlinear evolution of cosmological perturbations during inflation is necessary for the correct interpretation of measurements of non-Gaussian correlations in the cosmic microwave background and the large-scale structure of the universe. The "δN formalism" is a popular and powerful technique for computing non-linear evolution of cosmological perturbations on large scales. In particular, it enables us to compute the curvature perturbation, ζ, on large scales without actually solving perturbed field equations. However, people often wonder why this is the case. In order for this approach to be valid, the perturbed Hamiltonian constraint and matter-field equations on large scales must, with a suitable choice of coordinates, take on the same forms as the corresponding unperturbed equations. We find that this is possible when (1) the unperturbed metric is given by a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric; and (2) on large scales and with a suitable choice of coordinates, one can ignore the shift vector (g0i) as well as time-dependence of tensor perturbations to gij/a2(t) of the perturbed metric. While the first condition has