A 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 by cosmological models that the loop quantisation in the two different frames will lead to inequivalent effective theories. Analytical solutions have been found in both frames for the effective loop quantum Brans-Dicke cosmology without potential in: (i) vacuum case, (ii) additional massless scalar field case. In Einstein frame the analytical solution for the Brans-Dicke potential $\propto\varphi^2$ has been found. In all of those solutions the bouncing evolution of the scale factor has been obtained around Planck regime. The differences between the loop quantisation of the two frames are reflected by: (i) the evolution of the scale factor around the bounce, (ii) the scale of the bounce in the physical Jordan frame.