Starobinsky potential and power suppression in hybrid Loop Quantum Cosmology

Abstract

We study the effect on the primordial tensor power spectrum of varying the number of e-folds during slow-roll inflation in Loop Quantum Cosmology with a Starobinsky potential. Using the hybrid quantization approach, we derive the effective mass governing tensor mode evolution. The choice of vacuum state is crucial, especially since the preinflationary phase predicted by Loop Quantum Cosmology invalidates the choice of the Bunch-Davies state as the preferred vacuum. We adopt a choice which is optimally adapted to the dynamics, so that it provides a non-oscillating (NO) spectrum free of spurious contributions, and permits an asymptotic Hamiltonian diagonalization (AHD) of the perturbations. For this so-called NO-AHD vacuum, we compute the power spectrum using both analytic approximations and numerical integration. Our results confirm the accuracy of our approximations in a wide range of situations, including short- and long-lived inflationary scenarios. The primordial power spectrum exhibits a characteristic cutoff on a wavenumber scale determined primarily by the background dynamics around the bounce that replaces the big bang in Loop Quantum Cosmology.

Figures (3)

• Figure 1: Normalized PPS for short-lived inflation. We show the numerical computation of the oscillating PPS (continuous cyan line), the analytic approximation to this PPS (dot-dashed orange line), and the analytic approximation to the PPS of the NO-AHD vacuum (dashed blue line), obtained with a Bogoliubov transformation. We include an inset enlarging the region $1\leq k \leq 20$ (framed in the PPS) to see the details when power suppression appears. We have taken $\gamma=0.2375$ for the Immirzi parameter, $\phi_\text{B}=-1.45$ for the value of the inflaton at the bounce, positive inflaton time derivative at the bounce $\dot{\phi}_\text{B} >0$, and considered a Starobinsky inflaton potential with mass $m=2.51 \times 10^{-6}$.
• Figure 2: Normalized PPS for long-lived inflation. We show the numerical computation of the oscillating PPS (continuous cyan line), the analytic approximation to this PPS (dot-dashed orange line), and the analytic approximation to the PPS of the NO-AHD vacuum (dashed blue line), obtained with a Bogoliubov transformation. We include an inset enlarging the region $1\leq k \leq 20$ (framed in the PPS) to see the details when power suppression appears. We have taken $\gamma=0.2375$ for the Immirzi parameter, $\phi_\text{B}=0.97$ for the value of the inflaton at the bounce, positive inflaton time derivative at the bounce $\dot{\phi}_\text{B} >0$, and considered again a Starobinsky inflaton potential with mass $m=2.51 \times 10^{-6}$.
• Figure 3: Normalized analytic PPS for short- and long-lived inflation. The three PPS correspond to the analytic approximation to the PPS of the NO-AHD vacuum, including the final Bogoliubov transformation, for three different initial conditions on the inflaton and its derivative at the bounce. We show the analytic PPS for short-lived inflation with $\phi_\text{B}=-1.45$ and $\dot{\phi}_\text{B} >0$ (red solid line), and the analytic PPS for long-lived inflation with $\phi_\text{B}=0.97$ and $\dot{\phi}_\text{B} >0$ (green dot-dashed line). We include an inset enlarging the region $6\leq k \leq 20$ (framed in the PPS) to see the different tilts in detail. We have taken $\gamma=0.2375$ for the Immirzi parameter and considered a Starobinsky inflaton potential with mass $m=2.51 \times 10^{-6}$.