Cosmological scalar perturbations in Horndeski-like gravity

Scalar-tensor theories are promising dark energy models. A promising scalar-tensor theory, called Horndeski-like gravity, is coming from the application of the Horndeski gravity in string theory and cosmology that takes into account two dilaton fields. In this work we study the stability of the scalar sector of this theory and compare it with that coming from the previously studied tensor sector. With the first-order formalism we investigate the allowed background solutions. Focusing on the background solution with a single scalar field, the entropy coming from particle production $S_{in}$ and that of the apparent horizon $S$ will be studied, which translates into $\mathit{\text{entropy}}\mathit{\text{bounds}}$. These entropy bounds are compared with the stability of the scalar and tensor sector as well. The gravitational slip (minus one) to entropy ratio is also considered as a possible replacement for the usual shear viscosity to entropy ratio for black holes.