The effective two-dimensional phase space of cosmological scalar fields
  It has been shown by Remmen and Carroll that, for a model universe which contains only a kinetically canonical scalar field minimally coupled to gravity it is possible to choose 'special coordinates' to describe a two-dimensional effective phase space. The special, non-canonical, coordinates are $\phi$,$\dot{\phi}$ and the ability to describe an effective phase space with these coordinates empowers the common usage of $\phi-\dot{\phi}$ as the space to define inflationary initial conditions. This paper extends the result to the full Horndeski action. The existence of a two-dimensional effective phase space is shown for the general case. Subsets of the Horndeski action, relevant to cosmology are considered as particular examples to highlight important aspects of the procedure. 
David C. Edwards
gr-qc astro-ph.CO