Title: On the phantom barrier crossing and the bounds on the speed of sound in non-minimal derivative coupling theories,
Abstract: 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.