Perturbations Of Quantum-mechanical Origin Research Articles

Consistent scalar and tensor perturbation power spectra in single fluid matter bounce with dark energy era

We investigate cosmological scenarios containing one canonical scalar field with an exponential potential in the context of bouncing models, where the bounce happens due to quantum cosmological effects. The only possible bouncing solutions in this scenario (discarding an infinitely fine tuned exception) must have one and only one dark energy phase, either occurring in the contracting era or in the expanding era. Hence,these bounce solutions are necessarily asymmetric. We calculate the spectral indexes and amplitudes of scalar and tensor perturbations numerically, considering the whole history of the model, including the bounce phase itself, without making any approximation or using any matching condition on the perturbations. As the background model is necessarily dust dominated in the far past, the usual adiabatic vacuum initial conditions can be easily imposed in this era. Hence, this is a cosmological model where the presence of dark energy behavior in the Universe does not turn problematic the usual vacuum initial conditions prescription for cosmological perturbation in bouncing models. Scalar and tensor perturbations end up being almost scale invariant, as expected. The background parameters can be adjusted, without fine tunings, to yield the observed amplitude for scalar perturbations, and also for the ratio between tensor and scalar amplitudes, $r = T/S \lesssim 0.1$. The amplification of scalar perturbations over tensor perturbations takes place only around the bounce, due to quantum effects, and it would not occur if General Relativity has remained valid throughout this phase. Hence, this is a bouncing model where a single field induces not only an expanding background dark energy phase, but also produces all observed features of cosmological perturbations of quantum mechanical origin at linear order.

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

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.

Cosmological Perturbations of Quantum Mechanical Origin: Are Nonvacuum Initial States Allowed?

We address the question of whether nonvacuum initial states for cosmologicalperturbations are allowed, or whether they are ruled out on the basis of presentexperimental and observational data. Our choice of a nonvacuum initial state isguided by the idea that the initial state could have a built-in characteristic scale.We find that a class of initial states can fit the data; however, the initial statesmust be close to the vacuum.