Second-order effective energy-momentum tensor of gravitational scalar
  perturbations with perfect fluid
  We investigate the second-order gravitational scalar perturbations for a
barotropic fluid. We derive the effective energy-momentum tensor described by
the quadratic terms of the gravitational and the matter perturbations. We show
that the second-order effective energy-momentum tensor is gauge dependent. We
impose three gauge conditions (longitudinal, spatially-flat, and comoving
gauges) for dust and radiation. The resulting energy-momentum tensor is
described only by a gauge invariant variable, but the functional form depends
on the gauge choice. In the matter-dominated epoch with dust-like fluid
background, the second-order effective energy density and pressure of the
perturbations evolve as 1/a^2 in all three gauge choices, like the curvature
density of the Universe, but they do not provide the correct equation of state.
The value of this parameter depends also on the gauge choice. In the
radiation-dominated epoch, the perturbations in the short-wave limit behave in
the same way as the radiation-like fluid in the longitudinal and the
spatially-flat gauges. However, they behave in a different way in the comoving
gauge. As a whole, we conclude that the second-order effective energy-momentum
tensor of the scalar perturbation is strictly gauge dependent.
