Cosmological scalar and tensor perturbations with a scalar field: quadratic-order effective energy-momentum tensor

Abstract: We introduce the scalar and tensor modes of the gravitational perturbation in the presence of a scalar field which describes inflation. We investigate the back-reaction of the perturbations to the background by studying the effective energy-momentum tensor (2EMT) which is the second order constructed by the quadratic terms of the linear perturbations. 2EMT is gauge dependent due to the scalar mode. We obtain 2EMT in the slow-roll stage of inflation, and get its cosmological expressions in three (longitudinal, spatially flat, and comoving) gauge conditions. We find that the pure scalar-mode part in 2EMT is stronger in the short-wavelength limit, while the parts involved with the tensor mode (the pure tensor-mode part and the scalar-tensor coupled part) are stronger in the long-wavelength limit.