Exact solutions for a big bounce in loop quantum cosmology

Abstract

In this paper we study the flat ($k=0$) cosmological Friedmann-Robertson-Walker model with holonomy corrections of loop quantum gravity. The considered universe contains a massless scalar field and the cosmological constant $\ensuremath{\Lambda}$. We find analytical solutions for this model in different configurations and investigate its dynamical behavior in the whole phase space. Such an approach might be significant e.g. as a phenomenological reference for a further, fully quantum treatment. We show the explicit influence of $\ensuremath{\Lambda}$ on the qualitative and quantitative character of solutions. Even in the case of positive $\ensuremath{\Lambda}$ the oscillating solutions without the initial and final singularity appear as a generic case for some quantization schemes.