Hysteresis and beats in loop quantum cosmology
  Differences in pressure during expansion and contraction stages in cosmic
evolution can result in a hysteresis-like phenomena in non-singular cyclic
models sourced with scalar fields. We discuss this phenomena for spatially
closed isotropic spacetime in loop quantum cosmology (LQC) for a quadratic and
a cosh-like potential, with and without a negative cosmological constant using
effective spacetime description of the underlying quantum geometry. Two
inequivalent loop quantizations of this spacetime, one based on holonomies of
the Ashtekar-Barbero connection using closed loops, and another based on the
connection operator, are discussed. Due to quantum geometric effects, both
models avoid classical singularities, but unlike the holonomy based
quantization, connection based quantization results in two quantum bounces. In
spite of differences in non-singular effective dynamics of both the models, the
phenomena of hysteresis is found to be robust for the $\phi^2$ potential.
Quasi-periodic beats exist for a cosh-like potential, irrespective of the
nature of classical recollapse whether by spatial curvature, or a negative
cosmological constant. An interplay of negative cosmological constant and
spatial curvature in presence of potentials results in rich features such as
islands of cluster of bounces separated by accelerated expansion, and a
universe which either undergoes a step like expansion with multiple turnarounds
or quasi-periodic beats depending on a "tuning" of the steepness parameter of
the potential.
