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We derive the evolution equation for the second order curvature perturbationusing standard techniques of cosmological perturbation theory. We do this fordifferent definitions of the gauge invariant curvature perturbation, arisingfrom different splits of the spatial metric, and compare the expressions. Theresults are valid at all scales and include all contributions from scalar,vector and tensor perturbations, as well as anisotropic stress, with all ourresults written purely in terms of gauge invariant quantities. Taking thelarge-scale approximation, we find that a conserved quantity exists only if, inaddition to the non-adiabatic pressure, the transverse traceless part of theanisotropic stress tensor is also negligible. We also find that the version ofthe gauge invariant curvature perturbation which is exactly conserved is theone defined with the determinant of the spatial part of the inverse metric.

Vector and tensor contributions to the curvature perturbation at second order