Pre-inflationary dynamics of Starobinsky inflation and its generization
  in Loop Quantum Brans-Dicke Cosmology
  Recently, the nonperturbative quantization scheme of loop quantum gravity has
been extended to the Brans-Dicke theory and the corresponding loop quantum
Brans-Dicke cosmology has been derived, which provides an essential platform to
explore inflationary models in this framework. In this paper, we consider two
inflation models, the Starobinsky and $\alpha$-attractor inflation whose
cosmological predictions are in excellent agreement with Planck data, and study
systematically their pre-inflationary dynamics as well as the slow-roll
inflation. We show that for both models, the background evolution of a flat
Friedmann-Lema\^{i}tre-Robertson-Walker universe in general can be divided into
three different phases: the pre-inflationary quantum phase,
quantum-to-classical transition, and the slow-roll inflation. The
pre-inflationary dynamics are dominated by the quantum geometry effects of loop
quantum Brans-Dicke cosmology and the corresponding Universe could be either
initially expanding or contracting, depending on the initial velocity of
inflaton field. It is shown that the detailed evolution of pre-inflationary
quantum phase also depend on specific inflation models. After the
pre-inflationary quantum phase, the universe gradually evolves into the
slow-roll inflation with some of initial conditions for Starobinsky and
$\alpha$-attractor potentials. In addition, to be consistent with observational
data, we also find the restricted parameter space of initial conditions that
could produce at least $60$ $e$-folds during the slow-roll inflation.
