Abstract
In this paper, we first provide a brief review of the effective dynamics of two recently well-studied models of modified loop quantum cosmologies (mLQCs), which arise from different regularizations of the Hamiltonian constraint and show the robustness of a generic resolution of the big bang singularity, replaced by a quantum bounce due to non-perturbative Planck scale effects. As in loop quantum cosmology (LQC), in these modified models the slow-roll inflation happens generically. We consider the cosmological perturbations following the dressed and hybrid approaches and clarify some subtle issues regarding the ambiguity of the extension of the effective potential of the scalar perturbations across the quantum bounce, and the choice of initial conditions. Both of the modified regularizations yield primordial power spectra that are consistent with current observations for the Starobinsky potential within the framework of either the dressed or the hybrid approach. But differences in primordial power spectra are identified among the mLQCs and LQC. In addition, for mLQC-I, striking differences arise between the dressed and hybrid approaches in the infrared and oscillatory regimes. While the differences between the two modified models can be attributed to differences in the Planck scale physics, the permissible choices of the initial conditions and the differences between the two perturbation approaches have been reported for the first time. All these differences, due to either the different regularizations or the different perturbation approaches in principle can be observed in terms of non-Gaussianities.
Highlights
With the exact solution (3.7) as the initial conditions imposed at the moment t0 ( ≪ tB) in the contracting phase, it was found that the power spectrum of the cosmological scalar perturbations can be divided into three different regimes: 1) the ultraviolet (UV) (k > kmLQC−I); 2) intermedia√te; and 3) infrared (k < ki), where kmLQC−I ≡ aB RB/6 and ki ai Ri/6, and RB and Ri are the curvatures given at the bounce and the beginning of the slow-roll inflation, respectively [cf
The large relative difference of the power spectra between modified loop quantum cosmologies (mLQCs)-I and modified LQC-II (mLQC-II) happens in the infrared and oscillatory regimes, while in the UV regime it reduces to about 2%
In the past 2 decades, loop quantum cosmology (LQC) has been studied extensively, and several remarkable features have been found (Ashtekar and Singh, 2011; Ashtekar and Barrau, 2015; Bojowald, 2015; Agullo and Singh, 2017), including the generic resolution of the big bang singularity in the Planckian scale, the slow-roll inflation as an attractor in the post-bounce evolution of the Universe, and the scaleinvariant power spectra of the cosmological perturbations, which are consistent with the current cosmic microwave background (CMB) observations
Summary
Despite offering a solution to several fundamental and conceptual problems of the standard big bang cosmology, including the flatness, horizon, and exotic-relic problems, the cosmic inflation in the early Universe provides a mechanism to produce density perturbations and primordial gravitational waves (PGWs) (Starobinsky, 1980; Guth, 1981; Sato, 1981; Kodama and Sasaki, 1984; Mukhanov et al, 1992; Malik, 2001; Dodelson, 2003; Mukhanov, 2005; Weinberg, 2008; Malik and Wands, 2009; Senatore, 2017). We shall focus mainly on the states that are sharply peaked along the classical trajectories, so that the description of the “effective” dynamics of the Universe becomes available (Ashtekar and Singh, 2011; Ashtekar and Barrau, 2015; Bojowald, 2015; Agullo and Singh, 2017), and the questions raised recently in (Kaminski et al, 2020) are avoided This includes the studies of the “effective” dynamics of the homogeneous and isotropic mLQC-I and mLQC-II models, and their cosmological perturbations in the framework of the dressed metric and hybrid approaches. In mLQC-II and LQC, the BD vacuum can be imposed in the contracting phase as long as t0 is sufficiently early, so the Universe becomes so large that the spacetime curvature is very small, and particle creation is negligible With these in mind, the power spectra obtained in the three models, mLQC-I, mLQC-II and LQC, within the framework of the dressed metric approach were calculated and compared by imposing the initial conditions in the contracting phase.
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