Fully non-linear and exact perturbations of the Friedmann world model

In 1988 Bardeen suggested a pragmatic formulation of cosmological perturbation theory which is powerful in practice to employ various fundamental gauge conditions easily depending on the character of the problem. The perturbation equations are presented without fixing the temporal gauge condition and are arranged so that one can easily impose fundamental gauge conditions by simply setting one of the perturbation variables in the equations equal to zero. In this way one can use the gauge degrees of freedom as an advantage in handling problems. Except for the synchronous gauge condition, all the other fundamental gauge conditions completely fix the gauge mode, and consequently, each variable in such a gauge has a unique gauge-invariant counterpart, so that we can identify the variable as the gauge-invariant one. Here, we extend Bardeen's linear formulation to the fully non-linear order in perturbations, with the gauge advantage kept intact. The perturbation equations are exact (except for ignoring the transverse-tracefree part of the metric), and from these we can easily derive the higher order perturbation equations in a gauge-ready form. We consider scalar- and vector-type perturbations of an ideal fluid in a flat background; we also present the minimally coupled scalar field and the multiple components of ideal fluid cases. As applications, we present fully non-linear density and velocity perturbation equations in Einstein’s gravity in the zero-pressure medium, vorticity generation from pure scalar-type perturbation and fluid formulation of a minimally coupled scalar field, all in the comoving gauge. We also present the equation of gravitational waves generated from pure scalar- and vector-type perturbations.