Analytic Primordial Power Spectrum in the Dressed Metric Approach to Loop Quantum Cosmology and Thermodynamics of Spacetime

Abstract: We investigate the primordial power spectrum of cosmological tensor perturbations in the dressed metric approach to Loop Quantum Cosmology. We compute the background-dependent effective mass that affects their propagation using the effective description of Loop Quantum Cosmology and show that this mass can be approximated in different cosmological epochs by appropriate analytic functions. Moreover, in each of those epochs we can analytically solve the propagation of the perturbations, then obtaining the general solution globally by continuity requirements. On the other hand, since there are regimes far away from slow roll in the considered background evolution, the Bunch-Davies state does not provide a privileged choice of vacuum that would pick out a specific solution for the perturbations. Instead, we select the state of these perturbations by a recently proposed criterion that removes unwanted oscillations in the power spectrum. We compute the spectrum of this vacuum and compare it with other spectra obtained in the literature, especially with one corresponding to the hybrid approach to Loop Quantum Cosmology. Finally, we notice that the same type of background dynamics is found in a phenomenological approach to quantum gravity based on thermodynamics, allowing in this case a free value for the tantamount of the critical density. Extending the dressed metric proposal to this phenomenological model, one might expect a similar form for the associated primordial power spectrum.