Comparison between Jordan and Einstein frames of Brans-Dicke gravity a la loop quantum cosmology

It is well known that the Jordan and Einstein frames are equivalent to each other in classical Brans-Dicke theory, provided that one and the same metric is employed for the physical space-time. Nevertheless, it is shown in this paper that by cosmological models the loop quantization in the two different frames leads to inequivalent effective theories. Analytical solutions are found in both frames for the effective loop quantum Brans-Dicke cosmology without potential in (i) the vacuum case and (ii) the additional massless scalar field case. In the Einstein frame, the analytical solution for the Brans-Dicke potential alpha phi(2) is found. In all of those solutions, the bouncing evolution of the scale factor is obtained around the Planck regime. The differences between the loop quantization of the two frames are reflected by (i) the evolution of the scale factor around the bounce and (ii) the scale of the bounce in the physical Jordan frame.