On the phantom barrier crossing and the bounds on the speed of sound in non-minimal derivative coupling theories

In this paper we investigate the so called "phantom barrier crossing" issue in a cosmological model based in the scalar-tensor theory with non-minimal derivative coupling to the Einstein's tensor. Special attention will be paid to the physical bounds on the squared sound speed. The numeric results are geometrically illustrated by means of a qualitative procedure of analysis that is based on the mapping of the orbits in the phase plane onto the surfaces that represent physical quantities in the extended phase space, that is: the phase plane complemented with an additional dimension relative to the given physical parameter. We find that the cosmological model based in the non-minimal derivative coupling theory -- this includes both the quintessence and the pure derivative coupling cases -- has serious causality problems related with superluminal propagation of the scalar and tensor perturbations. Even more disturbing is the finding that, despite that the underlying theory is free of the Ostrogradsky instability, the corresponding cosmological model is plagued by the Laplacian (classical) instability related with negative squared sound speed. This instability leads to an uncontrollable growth of the energy density of the perturbations that is inversely proportional to their wavelength. We show that independent of the self-interaction potential, for the positive coupling the tensor perturbations propagate superluminally, while for the negative coupling a Laplacian instability arises. This latter instability invalidates the possibility for the model to describe the primordial inflation.