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We present a gauge-invariant formalism to study the evolution of thecurvature and entropy perturbations in the case in which spatial and timevariations of the inflaton decay rate into ordinary matter are present. Duringthe reheating stage after inflation curvature perturbations can vary with timeon super-horizon scales sourced by a a gauge-invariant inflaton decay rateperturbation. We show that the latter is a function not only of the spatialvariations of the decay rate generated during inflation, as envisaged in arecently proposed scenario, but also of the time variation of the inflatondecay rate during reheating. If only the second source is present, the finalcurvature perturbation at the end of the reheating stage is proportional to thecurvature perturbation at the beginning of reheating with a coefficient ofproportionality which can be either smaller or larger than unity depending uponthe underlying physics governing the time variation of the inflaton decay rate.As a consequence, we show that the standard consistency relation between theamplitude of curvature perturbations, the amplitude of tensor perturbations andthe tensor spectral index of one-single field models of inflation is violatedand there is the possibility that the tensor-to-curvature amplitude ratio islarger than in the standard case.Comment: LaTeX file, 21 pages. A few typos correcte

Large-scale curvature perturbations with spatial and time variations of the inflaton decay rate