Comparing different formulations of non-linear cosmological perturbation
  theory
  We compare and contrast two different metric based formulations of non-
linear cosmological perturbation theory: the MW2009 approach in [K. A. Malik
and D. Wands, Phys. Rept. 475 (2009), 1.] following Bardeen and the recent
approach of the paper KN2010 [K. Nakamura, Advances in Astronomy 2010 (2010),
576273]. We present each formulation separately. In the MW2009 approach, one
considers the gauge transformations of perturbative quantities, choosing a
gauge by requiring that certain quantities vanish, rendering all other
variables gauge invariant. In the KN2010 formalism, one decomposes the metric
tensor into a gauge variant and gauge invariant part from the outset. We
compare the two approaches in both the longitudinal and uniform curvature
gauges. In the longitudinal gauge, we find that Nakamura's gauge invariant
variables correspond exactly to those in the longitudinal gauge (i.e., for
scalar perturbations, to the Bardeen potentials), and in the uniform curvature
gauge we obtain the usual relationship between gauge invariant variables in the
flat and longitudinal gauge. Thus, we show that these two approaches are
equivalent.
