Second-order perturbations of the Friedmann world model

Abstract

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 perturbations͒ to second order, first shown by Salopek and Bond, now from the exact equations. Several other applications are shown as well.

Newtonian versus relativistic nonlinear cosmology

“…In fact, the magnetic part of Weyl tensor does not vanish even to the linear order in perturbations: this quantity valid to the second order is presented in eq. (96) of [6]. Our nonlinear perturbation approach is applicable to fully relativistic regimes including the super-horizon scales and the early universe.…”

“…(5), and take the same identifications made in eq. (6). To the second order the perturbed part of eq.…”

“…In all our relativistic (nonlinear) perturbation analyses we take the temporal comoving gauge (which together with the irrotational condition givesũ α = 0 for the fluid four-vectorũ a ) and the spatial γ = 0 gauge [6]. As these gauge conditions fix the gauge modes completely, all the remaining variables are equivalently gauge-invariant to all orders in perturbations; for technical details, see Section VI.C of [6] where an explicitly gaugeinvariant combination δµ v which is the same as the δµ in the temporal comoving gauge and the spatial γ = 0 gauge can be found in eq. (282).…”

Fully nonlinear and exact perturbations of the Friedmann world model: non-flat background

“…We have shown that the ϕ terms start to appear from the third order perturbation, thus in a flat background we have exact relativistic/Newtonian correspondences of the density and velocity perturbations to the second-order perturbation [5].…”

Cosmological perturbation theory and quantum gravity

“…According to [29], infinitesimal diffeomorphisms may be approximated by so-called knight diffeomorphisms, which are of the form exp L ξ with ξ = λξ 1 + 1 2 λ 2 ξ 2 +O(λ 3 ). Analogously we may expand a configuration Γ as Γ = Γ 0 +δΓ = Γ 0 +λδΓ 1 + 1 2 λ 2 δΓ 2 +O(λ 3 ), and determine the transformation behaviour of separate orders by considering exp L ξ Γ at fixed order in λ, see for example [4,6,19,21,23]. Assuming that ξ and δΓ vanish at spatial infinity, each order ξ i and δΓ i may be uniquely decomposed as in (2.4) and (2.3).…”