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We present a new approach to gauge-invariant cosmological perturbations atsecond order, which is also covariant. We examine two cases in particular for adust Friedman-Lemaitre-Robertson-Walker model of any curvature: we investigategravity waves generated from clustering matter, that is, induced tensor modesfrom scalar modes; and we discuss the generation of density fluctuationsinduced by gravity waves -- scalar modes from tensor perturbations. We derive alinear system of evolution equations for second-order gauge-invariant variableswhich characterise fully the induced modes of interest, with a source formedfrom variables quadratic in first-order quantities; these we transform intofully-fledged second-order gauge-invariant variables. Both the invariantlydefined variables and the key evolution equations are considerably simpler thansimilar gauge-invariant results derived by other methods. By finding analyticalsolutions, we demonstrate that non-linear effects can significantly amplify ordampen 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 onsuper-Hubble scales.Comment: 11 pages, 1 figure, revtex4; v2 to appear in PR

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