Action approach to cosmological perturbations: the 2nd order metric in matter dominance

TL;DR

The paper develops a novel, action-based approach to nonlinear cosmological perturbations during matter domination by mapping a perfect barotropic fluid onto a derivatively coupled scalar with Lagrangian $P(X)$. Using ADM formalism in comoving gauge, it derives the exact second-order metric as a functional of the primordial inflaton-generated curvature perturbation $\zeta$, including scalar and tensor sectors and a gauge-transformation to Poisson gauge for comparison with prior results. A key finding is that $\zeta$ is not strictly conserved at second order in the matter-dominated era due to couplings with sub-Hubble modes, though $\zeta$ remains a useful initial condition variable on super-Hubble scales. The work also demonstrates the generation of gravitational waves from scalar perturbations at second order and provides a framework adaptable to other equations of state and multiple fluids, with potential implications for CMB and large-scale structure observables.

Abstract

We study nonlinear cosmological perturbations during the post-inflationary evolution, using the equivalence between a perfect barotropic fluid and a derivatively coupled scalar field with Lagrangian [-(\partial φ)^2]^[(1+w)/2w]. Since this Lagrangian is just a special case of k-inflation, this approach is analogous to the one employed in the study of non-Gaussianities from inflation. We use this method to derive the second order metric during matter dominance in the comoving gauge directly as function of the primordial inflationary perturbation ζ. Going to Poisson gauge, we recover the metric previously derived in the literature.