Attracted to de Sitter II: cosmology of the shift-symmetric Horndeski models

Horndeski models with a de Sitter critical point for any kind of material content may provide a mechanism to alleviate the cosmological constant problem. Moreover, they could allow us to understand the current accelerated expansion of the universe as the result of the dynamical approach to the critical point when it is an attractor. We show that this critical point is indeed an attractor for the shift- symmetric subfamily of models with these characteristics. We study the cosmological scenario that results when considering radiation and matter content, and conclude that their background dynamics is compatible with the latest observational data.