δNformalism

Abstract

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 ``$\ensuremath{\delta}N$ formalism'' is a popular and powerful technique for computing nonlinear evolution of cosmological perturbations on large scales. In particular, it enables us to compute the curvature perturbation $\ensuremath{\zeta}$ 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-Lema\^{\i}tre-Robertson-Walker metric; and (2) on large scales and with a suitable choice of coordinates, one can ignore the shift vector (${g}_{0i}$) as well as time dependence of tensor perturbations to ${g}_{ij}/{a}^{2}(t)$ of the perturbed metric. While the first condition has to be assumed a priori, the second condition can be met when (3) the anisotropic stress becomes negligible on large scales. However, in order to explicitly show that the second condition follows from the third condition, one has to use gravitational field equations, and thus this statement may depend on the details of the theory of gravitation. Finally, as the $\ensuremath{\delta}N$ formalism uses only the Hamiltonian constraint and matter-field equations, it does not a priori respect the momentum constraint. We show that the error in the momentum constraint only yields a decaying mode solution for $\ensuremath{\zeta}$, and the error vanishes when the slow-roll conditions are satisfied.