Abstract
We present a covariant formalism for studying non-linear perturbations of scalar fields. Inparticular, we consider the case of two scalar fields and introduce the notion of adiabaticand isocurvature covectors. We obtain differential equations governing the evolution ofthese two covectors, as well as the evolution equation for the covector associated with thecurvature perturbation. The form of these equations is very close to that of the analogousequations obtained in the linear theory, but our equations are fully non-linear andexact. As an application of our formalism, we expand these equations to secondorder in the perturbations. On large scales, we obtain a closed system of coupledscalar equations giving the evolution of the second-order adiabatic and entropyperturbations in terms of the first-order perturbations. These equations in generalcontain a non-local term which, however, decays rapidly in an expanding universe.
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