On the conservation of second-order cosmological perturbations in a scalar field dominated universe

We discuss second-order cosmological perturbations on super-Hubble scales, in a scalar field dominated universe, such as during single field inflation. In this contest we show that the gauge-invariant curvature perturbations defined on the uniform density and comoving hypersurfaces coincide and that perturbations are adiabatic in the large scale limit. Since it has been recently shown that the uniform curvature perturbation is conserved on large scales if perturbations are adiabatic, we conclude that both the uniform and comoving curvature perturbations at second-order in a scalar field dominated universe are conserved. The study of second-order perturbation theory has recently become important [1, 2, 3, 4, 5] (see e.g., also [6, 7] for earlier work), especially because primordial non-Gaussianities generated by inflation are typically only of second-order level [2, 8, 9, 10, 11]. In the study of cosmological perturbations and non-Gaussianities it is very useful to establish results in terms of quantities that are conserved on large scales, i.e., on super-Hubble scales. At linear order, it is well known that the gauge invariant curvature perturbations defined on the uniform density and comoving hypersurfaces – the so called uniform and