Second-order perturbations of the Friedmann world model

We consider the instability of the Friedmann world model to second order in perturbations. We present the perturbed set of equations up to second order in the Friedmann background world model with a general spatial curvature and cosmological constant. We consider systems with completely general imperfect fluids, minimally coupled scalar fields, an electromagnetic field, and generalized gravity theories. We also present the case of null geodesic equations, and one based on the relativistic Boltzmann equation. In due time, a decomposition is made for scalar-, vector-, and tensor-type perturbations which couple with each other to second order. A gauge issue is resolved to each order. The basic equations are presented without imposing any gauge condition, and thus in a gauge-ready form so that we can take full advantage of having gauge freedom in analyzing the problems. As an application we show that to 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 second order, first shown by Salopek and Bond, now from the exact equations. Several other applications are shown as well.