Publisher's Note: Cosmological density fluctuations and gravity waves: A covariant approach to gauge-invariant nonlinear perturbation theory [Phys. Rev. D70, 103524 (2004)]

Abstract

We present a new approach to gauge-invariant cosmological perturbations at second order, which is also covariant. We examine two cases in particular for a dust Friedman-Lemaître-RobertsonWalker model of any curvature: we investigate gravity waves generated from clustering matter, that is, induced tensor modes from scalar modes; and we discuss the generation of density fluctuations induced by gravity waves -scalar modes from tensor perturbations. We derive a linear system of evolution equations for second-order gauge-invariant variables which characterise fully the induced modes of interest, with a source formed from variables quadratic in first-order quantities; these we transform into fully-fledged second-order gauge-invariant variables. Both the invariantly defined variables and the key evolution equations are considerably simpler than similar gauge-invariant results derived by other methods. By finding analytical solutions, we demonstrate that non-linear effects can significantly amplify or dampen modes present in standard linearised cosmological perturbation theory, thereby providing an important source of potential error in, and refinement of, the standard model. Moreover, these effects can dominate at late times, and on super-Hubble scales.

Gravitational waves generated by second order effects during inflation

“…Second-order perturbation theory has been investigated in various works [18,19,20,21,22,23,24] and a fully gauge-invariant approach to the problem was recently given in Ref. [25].…”

Cosmic electromagnetic fields due to perturbations in the gravitational field

“…This makes it possible to quantitatively evaluate the generated fields using CMB constraints. Progress in non-linear perturbation theory will allow us to investigate these non-linear effects in a manner that is free of spurious gauge modes [38,56].…”

“…where we have set the vorticity to zero (ω a = 0), see also [38]. Note that at first order in gravitational perturbations, the only source of vector modes is the vorticity ω a ; since, we neglect the effects of vorticity, ω a = 0, all the vector modes vanish.…”

“…Following [38] we make the following comments: (i) The magnetic fieldB a appearing in the equations 13 and 14 multiplied by the gravitational variables should not be the same as the B a appearing alone. The variable B a is a mixture of linear and non-linear quantities (the seed magnetic field and the induced field) while the terms involvingB a are a product of first order quantities.…”

Editorial note to: G. F. R. Ellis, Relativistic cosmology

“…Other significant programs strongly influenced by this paper are the development of gauge-invariant higher order perturbation theory by Clarkson and others [24], investigations of exact cosmological solutions, such as those by John Wainwright and his group exploring Bianchi universes, 3 the research on gravitational waves by Peter Hogan, George Ellis, and Emir O'Shea [31][32][33], the investigations of magnetic fields and gravitational waves in cosmology by Peter Dunsby and his collaborators [34][35][36], the work on the Newtonian limit in cosmology by Henk van Elst and George Ellis [37,38], and the work on integrability conditions for cosmologies which are special cases (e. g., when E = 0, H = 0 ("silent universes"), ∇ · E = 0, ∇ · H = 0, etc.) [39].…”