Comment on “Growth of covariant perturbations in the contracting phase of a bouncing universe”

Abstract

A recent paper by Kumar (2012) (hereafter K12) claimed that in a contracting model, described by perturbations around a collapsing Friedmann model containing dust or radiation, the perturbations can grow in such a way that the linearity conditions would become invalid. This conclusion is not correct due to the following facts: first, it is claimed that the linearity conditions are not satisfied, but nowhere in K12 the amplitudes of the perturbations were in fact estimated. Therefore, without such estimates, the only possible conclusion from this work is the well-known fact that the perturbations indeed grow during contraction, which, per se, does not imply that the linearity conditions become invalid. Second, some evaluations of the linearity conditions are incorrect because third-order terms, instead of the appropriate second-order ones, are mistakenly compared with first-order terms, yielding artificially fast growing conditions. Finally, it is claimed that the results of K12 are in sharp contrast with the results of the paper by Vitenti and Pinto-Neto (2012) (hereafter VPN12), because the former was obtained in a gauge-invariant way. However, the author of K12 did not realize that the evolution of the perturbations were also calculated in a gauge-invariant way in VPN12, but some of the linearity conditions which are necessary to check cannot be expressed in terms of gauge-invariant quantities. In the present work, the incorrect or incomplete statements of K12 are clarified and completed, and it is shown that all other correct results of K12 were already present in VPN12, whose conclusions remain untouched, namely, that cosmological perturbations of quantum mechanical origin in a bouncing model can remain in the linear regime all along the contracting phase and at the bounce itself for a wide interval of energy scales of the bounce, ranging from the nucleosynthesis energy scale up to a few orders of magnitude below the Planck energy.