The Newton constant and gravitational waves in some vector field adjusting mechanisms

As is well known, there exist some Lorentz breaking scenarios which explain the smallness of the cosmological constant in the present era. An important aspect to analyze is the propagation of gravitational waves and the screening or enhancement of the Newton constant $G_N$ in these models. The problem is that the Lorentz symmetry breaking terms may induce an unacceptable value of the Newton constant $G_N$ or introduce longitudinal modes in the gravitational waves. Furthermore there may spoil the standard dispersion relation $\omega=ck$. In [21] the authors have presented a model for which they suggest that the behavior of the gravitational constant is the correct one for asymptotic times. In the present work, an explicit checking is made and we finally agree with these claims. Furthermore, it is suggested that the gravitational waves are also well behaved for large times. In the process, some new models with the same behavior are obtained, thus enlarging the list of possible adjustment mechanisms.