Dynamical analysis of loop quantum R2 cosmology

The effective dynamics of loop quantum f(R) cosmology in the Jordan frame is considered by using the dynamical system method and numerical method. To make the analysis in detail, we focus on the R-2 model since it is simple and favored from observations. In classical theory, (phi = 1, (phi) over dot = 0) is the unique fixed point in both contracting and expanding states, and all solutions are either starting from the fixed point or evolving to the fixed point, while in loop theory, there exists a new fixed point (a saddle point) at (phi similar or equal to 2/3, (phi) over dot = 0) in the contracting state. We find that two of the critical solutions which start from the saddle point cause the solutions starting from the fixed point (phi = 1, (phi) over dot = 0) to bounce at small values of the scalar field in 0 < phi < 1. All other solutions, including the large field inflation solutions, have the history with phi < 0, which we think of as a problem of the effective theory of loop quantum f(R) theory. Another difference from loop quantum cosmology with the Einstein-Hilbert action is that there exist many solutions that do not have a bouncing behavior.