The Matter Bounce Scenario in Loop Quantum Cosmology
  In the matter bounce scenario, a dust-dominated contracting space-time
generates scale-invariant perturbations that, assuming a nonsingular bouncing
cosmology, propagate to the expanding branch and set appropriate initial
conditions for the radiation-dominated era. Since this scenario depends on the
presence of a bounce, it seems appropriate to consider it in the context of
loop quantum cosmology where a bouncing universe naturally arises. For a
pressureless collapsing universe in loop quantum cosmology, the predicted power
spectrum of the scalar perturbations after the bounce is scale-invariant and
the tensor to scalar ratio is negligibly small. A slight red tilt can be given
to the scale-invariance of the scalar perturbations by a scalar field whose
equation of state is $P = - \epsilon \rho$, where $\epsilon$ is a small
positive number. Then, the power spectrum for tensor perturbations is also
almost scale-invariant with the same red tilt as the scalar perturbations, and
the tensor to scalar ratio is expected to be $r \approx 9 \times 10^{-4}$.
Finally, for the predicted amplitude of the scalar perturbations to agree with
observations, the critical density in loop quantum cosmology must be of the
order $\rho_c \sim 10^{-9} \rho_{\rm Pl}$.
