Spectra of primordial fluctuations in two-perfect-fluid regular bounces

We introduce analytic solutions for a class of two components bouncing models, where the bounce is triggered by a negative energy density perfect fluid. The equation of state of the two components are constant in time, but otherwise unrelated. By numerically integrating regular equations for scalar cosmological perturbations, we find that the (would be) growing mode of the Newtonian potential before the bounce never matches with the the growing mode in the expanding stage. For the particular case of a negative energy density component with a stiff equation of state we give a detailed analytic study, which is in complete agreement with the numerical results. We also perform analytic and numerical calculations for long wavelength tensor perturbations, obtaining that, in most cases of interest, the tensor spectral index is independent of the negative energy fluid and given by the spectral index of the growing mode in the contracting stage. We compare our results with previous investigations in the literature.Comment: 11 pages, 5 figure

Comments on "Growth of Covariant Perturbations in the Contracting Phase of a Bouncing Universe" by A. Kumar

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 other 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 realized 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 be checked 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. (Abstract abridged)Comment: 7 pages, revtex4-1, accepted for publication in PR

The accelerated expansion of the Universe as a quantum cosmological effect

We study the quantized Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) model minimally coupled to a free massless scalar field. In a previous paper, \cite{fab2}, solutions of this model were constructed as gaussian superpositions of negative and positive modes solutions of the Wheeler-DeWitt equation, and quantum bohmian trajectories were obtained in the framework of the Bohm-de Broglie (BdB) interpretation of quantum cosmology. In the present work, we analyze the quantum bohmian trajectories of a different class of gaussian packets. We are able to show that this new class generates bohmian trajectories which begin classical (with decelerated expansion), undergo an accelerated expansion in the middle of its evolution due to the presence of quantum cosmological effects in this period, and return to its classical decelerated expansion in the far future. We also show that the relation between luminosity distance and redshift in the quantum cosmological model can be made close to the corresponding relation coming from the classical model suplemented by a cosmological constant, for $z<1$. These results suggest the posibility of interpreting the present observations of high redshift supernovae as the manifestation of a quantum cosmological effect

The Wheeler-DeWitt Quantization Can Solve the Singularity Problem

We study the Wheeler-DeWitt quantum cosmology of a spatially flat Friedmann cosmological model with a massless free scalar field. We compare the consistent histories approach with the de Broglie-Bohm theory when applied to this simple model under two different quantization schemes: the Schr\"odinger-like quantization, which essentially takes the square-root of the resulting Klein-Gordon equation through the restriction to positive frequencies and their associated Newton-Wigner states, or the induced Klein-Gordon quantization, that allows both positive and negative frequencies together. We show that the consistent histories approach can give a precise answer to the question concerning the existence of a quantum bounce if and only if one takes the single frequency approach and within a single family of histories, namely, a family containing histories concerning properties of the quantum system at only two specific moments of time: the infinity past and the infinity future. In that case, as shown by Craig and Singh \cite{CS}, there is no quantum bounce. In any other situation, the question concerning the existence of a quantum bounce has no meaning in the consistent histories approach. On the contrary, we show that if one considers the de Broglie-Bohm theory, there are always states where quantum bounces occur in both quantization schemes. Hence the assertion that the Wheeler-DeWitt quantization does not solve the singularity problem in cosmology is not precise. To address this question, one must specify not only the quantum interpretation adopted but also the quantization scheme chosen.Comment: 13 pages, 1 figur