Comparing different formulations of non-linear cosmological perturbation theory

TL;DR

This paper compares two metric-based formulations of non-linear cosmological perturbation theory, demonstrating their equivalence by analyzing gauge invariance and gauge choices in different gauges.

Contribution

It provides a detailed comparison and shows the equivalence of the MW2009 and Nakamura approaches in non-linear cosmological perturbation theory.

Findings

Nakamura's gauge invariant variables match Bardeen potentials in the longitudinal gauge.

The two approaches are equivalent in both longitudinal and uniform curvature gauges.

The relationship between gauge invariant variables in different gauges is consistent across methods.

Abstract

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.