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.