Cosmological density fluctuations and gravity waves: A covariant approach to gauge-invariant non-linear perturbation theory

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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-Lemaitre-Robertson-Walker 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.