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We present simple expressions for the relativistic first and second orderfractional density perturbations for FL cosmologies with dust, in fourdifferent gauges: the Poisson, uniform curvature, total matter and synchronousgauges. We include a cosmological constant and arbitrary spatial curvature inthe background. A distinctive feature of our approach is our description of thespatial dependence of the perturbations using a canonical set of quadraticdifferential expressions involving an arbitrary spatial function that arises asa conserved quantity. This enables us to unify, simplify and extend previousseemingly disparate results. We use the primordial matter and metricperturbations that emerge at the end of the inflationary epoch to determine theadditional arbitrary spatial function that arises when integrating the secondorder perturbation equations. This introduces a non-Gaussianity parameter intothe expressions for the second order density perturbation. In the special caseof zero spatial curvature we show that the time evolution simplifiessignificantly, and requires the use of only two non-elementary functions, theso-called growth supression factor at the linear level, and one new function atthe second order level. We expect that the results will be useful inapplications, for example, studying the effects of primordial non-Gaussianityon the large scale structure of the universe.

Second order density perturbations for dust cosmologies