Matter Bounce Loop Quantum Cosmology from $F(R)$ Gravity
  Using the reconstruction method, we investigate which $F(R)$ theories, with
or without the presence of matter fluids, can produce the matter bounce
scenario of holonomy corrected Loop Quantum Cosmology. We focus our study in
two limits of the cosmic time, the large cosmic time limit and the small cosmic
time limit. For the former, we found that, in the presence of non-interacting
and non-relativistic matter, the $F(R)$ gravity that reproduces the late time
limit of the matter bounce solution is actually the Einstein-Hilbert gravity
plus a power law term. In the early time limit, since it corresponds to large
spacetime curvatures, assuming that the Jordan frame is described by a general
metric that when it is conformally transformed to the Einstein frame, produces
an accelerating Friedmann-Robertson-Walker metric, we found explicitly the
scalar field dependence on time. After demonstrating that the solution in the
Einstein frame is indeed accelerating, we calculate the spectral index derived
from the Einstein frame scalar-tensor counterpart theory of the $F(R)$ theory
and compare it with the Planck experiment data. In order to implement the
resulting picture, we embed the $F(R)$ gravity explicitly in a Loop Quantum
Cosmology framework by introducing holonomy corrections to the $F(R)$ gravity.
In this way, the resulting inflation picture corresponding to the $F(R)$
gravity can be corrected in order it coincides to some extent with the current
experimental data.
