  We evaluate the Lorentzian gravitational path integral in the presence of
non-vanishing torsion with the application of the Picard-Lefschetz theory for
minisuperspaces corresponding to a number of phenomenological bouncing
cosmological models as well as for the inflationary paradigm. It turns out that
the semi-classical wave function derived from the saddle points of the path
integral formalism coincides with the solutions of the Wheeler-DeWitt equation.
Intriguingly, our analysis showed that the relative probability, derived using
these semi-classical wave functions favors universes with smaller values of
torsion. Moreover, we find that in the inflationary case, non-zero values of a
certain parity-violating component of the torsion enhance the power in the
large physical length scales, which can have important observational
implications. On the other hand, in the case of bouncing models, the power
spectrum is characterized by an initial region of growth, an intermediate
oscillatory region, and then again a final region of growth. The shape of the
power spectrum in the initial and intermediate regions is sensitive to the
abundance of the bounce-enabling matter and torsion, along with the initial
wave function of the universe, while the final size modifies the behavior of
the power spectrum in the smaller length scales.
