Second-order Perturbations of the Friedmann World Model

TL;DR

This work develops a gauge-ready, second-order cosmological perturbation framework around the Friedmann background with general curvature $K$ and cosmological constant $\Lambda$, accommodating a broad class of matter contents including imperfect fluids, scalar fields, EM fields, and generalized gravity theories. It presents complete perturbed equations in both ADM and covariant $1+3$ formalisms, together with a three-type (scalar, vector, tensor) decomposition that reveals second-order coupling among types and resolves gauge freedoms at each order. The authors derive closed-form, gauge-ready equations and demonstrate key results: a relativistic, second-order, pressureless fluid reproduces Newtonian behavior on the relevant scales, and large-scale conserved quantities (for pure scalar and tensor perturbations) persist beyond linear order, extending the known Salopek–Bond results. The framework supports multiple gauges and frame choices, enabling flexible, rigorous analyses of nonlinear structure formation, backreaction, and CMB anisotropies within a unified relativistic perturbation theory. It also connects to action-based approaches (Maldacena) and provides pathways for quasilinear, averaging, and Zel’dovich-type explorations in a fully relativistic setting.

Abstract

We consider instability of the Friedmann world model to the second-order in perturbations. We present the perturbed set of equations up to the second-order in the Friedmann background world model with general spatial curvature and the cosmological constant. We consider systems with the completely general imperfect fluids, the minimally coupled scalar fields, the electro-magnetic field, and the generalized gravity theories. We also present the case of null geodesic equations, and the one based on the relativistic Boltzmann equation. In due stage a decomposition is made for the scalar-, vector- and tensor-type perturbations which couple each other to the second-order. Gauge issue is resolved to each order. The basic equations are presented without imposing any gauge condition, thus in a gauge-ready form so that we can use the full advantage of having the gauge freedom in analysing the problems. As an application we show that to the second-order in perturbation the relativistic pressureless ideal fluid of the scalar-type reproduces exactly the known Newtonian result. As another application we rederive the large-scale conserved quantities (of the pure scalar- and tensor-perturbations) to the second order, first shown by Salopek and Bond, now from the exact equations. Several other applications are made as well.