Non-linear power spectra in the synchronous gauge
  We study the non-linear corrections to the matter and velocity power spectra
in the synchronous gauge (SG). We consider the perturbations up to third order
in a zero-pressure fluid in flat cosmological background, which is relevant for
the non-linear growth of cosmic structure. As a result, we point out that the
SG is an inappropriate coordinate choice when handling the non-linear growth of
the large-scale structure. Although the equations in the SG happen to coincide
with those in the comoving gauge (CG) to linear order, they differ from second
order. In particular, the second order hydrodynamic equations in the the SG are
apparently in the Lagrangian form, whereas those in the CG are in the Eulerian
form. Thus, the non-linear power spectra naively presented in the original SG
show strange behavior quite different from the result of the Newtonian theory
even on sub-horizon scales. The power spectra in the SG show regularized
behaviors only after we introduce convective terms in the second order so that
the equations in two gauges coincide to the second order.
