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 Nakamuras 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.

A Concise Introduction to Perturbation Theory in Cosmology

We give a concise, self-contained introduction to perturbation theory in cosmology at linear and second order, striking a balance between mathematical rigour and usability. In particular we discuss gauge issues and the active and passive approach to calculating gauge transformations. We also construct gauge-invariant variables, including the second order tensor perturbation on uniform curvature hypersurfaces.