Comparing two different formulations of metric cosmological perturbation theory

Abstract

We compare and contrast two different metric-based formulations of nonlinear cosmological perturbation theory: the MW2009 approach in Malik and Wands (2009 Phys. Rep. 475 1) following Bardeen and the recent approach of the paper KN2010 (Nakamura 2010 Adv. Astron. 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.