Tensor power spectrum with holonomy corrections in LQC

Mielczarek, Jakub
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In this paper we consider tensor perturbations produced at a bounce phase in presence of the holonomy corrections. Here bounce phase and holonomy corrections originate from Loop Quantum Cosmology. We re-derive formulas for the corrections for the model with a scalar field content. Background dynamics with a free scalar field and multi-fluid potential are considered. Since the considerations are semi-classical effects of quantum fluctuations of the background dynamics are not taken into account. Quantum and classical back-reaction effects are also neglected. To find spectrum of the gravitational waves both analytical approximations as well as numerical investigations were performed. We have found analytical solutions on super-horizontal and sub-horizontal regimes and derived corresponding tensor power spectra. Also occupation number $n_{\bf k}$ and parameter $\Omega_{\text{gw}}$ were derived in sub-horizontal limit, leading to its extremely low present value. Final results are numerical power spectra of the gravitational waves produced in presence of quantum holonomy corrections. The obtained spectrum has two UV and IR branches where $\mathcal{P}_T \propto k^2$, however with the different prefactors. Spectrum connecting these regions is in the form of oscillations. We have found good agreement between numerical spectrum and this obtained from the analytical model. Obtained spectrum can be directly applied as initial conditions for the inflationary modes. We mention possible resulting observational features of the CMB in particular B-type polarization.
Comment: 13 pages, 12 figures. Numerical computations improved. Major revision
General Relativity and Quantum Cosmology, Astrophysics - Cosmology and Nongalactic Astrophysics, High Energy Physics - Theory