Abstract
We have recently proposed a novel self tuning mechanism to alleviate the famous cosmological constant problem, based on the general scalar tensor theory proposed by Horndeski. The self-tuning model ends up consisting of four geometric terms in the action, with each term containing a free potential function of the scalar field; the four together being labeled as the Fab-Four. In this paper we begin the important task of deriving the cosmology associated with the Fab-Four Lagrangian. Performing a phase plane analysis of the system we are able to obtain a number of fixed points for the system, with some remarkable new solutions emerging from the trade-off between the various potentials. As well as obtaining inflationary solutions we also find conventional radiation/matter-like solutions, but in regimes where the energy density is dominated by a cosmological constant, and where we do not have any explicit forms of radiation or matter. Stability conditions for matter solutions are obtained and we show how it is possible for there to exist an extended period of `matter domination' opening up the possibility that we can generate cosmological structures, and recover a consistent cosmology even in the presence of a large cosmological constant.