<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math>-attractor potentials in loop quantum cosmology

We perform in this work an analysis of the background dynamics for $\ensuremath{\alpha}$-attractor models in the context of loop quantum cosmology. Particular attention is given to the determination of the duration of the inflationary phase that is preceded by the quantum bounce in these models. From an analysis of the general predictions for these models, it is shown that we can be able to put constraints in the parameter $\ensuremath{\alpha}$ of the potentials and also on the quantum model itself, especially the Barbero-Immirzi parameter. In particular, the constraints on the tensor-to-scalar ratio and spectral tilt of the cosmological perturbations limit the $\ensuremath{\alpha}$ parameter of the potentials to values such that ${\ensuremath{\alpha}}_{n=0}\ensuremath{\lesssim}10$, ${\ensuremath{\alpha}}_{n=1}\ensuremath{\lesssim}17$ and ${\ensuremath{\alpha}}_{n=2}\ensuremath{\lesssim}67$, for the $\ensuremath{\alpha}$ attractors T, E, and $n=2$ models, respectively. Using the constraints on the minimal amount of $e$-folds of expansion from the quantum bounce up to the end of inflation leads to the upper bounds for the Barbero-Immirzi parameter for the $\ensuremath{\alpha}$-attractor models studied in this work: ${\ensuremath{\gamma}}_{n=0}\ensuremath{\lesssim}51.2$, ${\ensuremath{\gamma}}_{n=1}\ensuremath{\lesssim}63.4$ and ${\ensuremath{\gamma}}_{n=2}\ensuremath{\lesssim}64.2$, which are obtained when fixing the parameter $\ensuremath{\alpha}$ in the potential at the values saturating the upper bounds given above for each model.