  The big bang singularity could be understood as a breakdown of Einstein's
General Relativity at very high energies. Adopting this viewpoint, other
theories, that implement Einstein Cosmology at high energies, might solve the
problem of the primeval singularity. One of them is Loop Quantum Cosmology
(LQC) with a small cosmological constant that models a universe moving along an
ellipse, which prevents singularities like the big bang or the big rip, in the
phase space $(H,\rho)$, where $H$ is the Hubble parameter and $\rho$ the energy
density of the universe. Using LQC when one considers a model of universe
filled by radiation and matter where, due to the cosmological constant, there
are a de Sitter and an anti de Sitter solution. This means that one obtains a
bouncing non-singular universe which is in the contracting phase at early
times. After leaving this phase, i.e., after bouncing, it passes trough a
radiation and matter dominated phase and finally at late times it expands in an
accelerated way (current cosmic acceleration). This model does not suffer from
the horizon and flatness problems as in big bang cosmology, where a period of
inflation that increases the size of our universe in more than 60 e-folds is
needed in order to solve both problems. The model has two mechanisms to avoid
these problems: The evolution of the universe through a contracting phase and a
period of super-inflation ($\dot{H}> 0$).
