The Limits of Extended Quintessence
  We use a low redshift expansion of the cosmological equations of extended (scalar-tensor) quintessence to divide the observable Hubble history parameter space in four sectors: A forbidden sector I where the scalar field of the theory becomes imaginary (the kinetic term becomes negative), a forbidden sector II where the scalar field rolls up (instead of down) its potential, an allowed `freezing' quintessence sector III where the scalar field is currently decelerating down its potential towards freezing and an allowed `thawing' sector IV where the scalar field is currently accelerating down its potential. The dividing lines between the sectors depend sensitively on the time derivatives of the Newton's constant G over powers of the Hubble parameter. For minimally coupled quintessence which appears as a special case for a constant G our results are consistent with previous studies. Observable parameter \chi^2 contours based on current data (SNLS dataset) are also constructed on top of the sectors, for a prior of \Omega_m=0.24. By demanding that the observed 2\sigma \chi^2 parameter contours do not lie entirely in the forbidden sectors we derive stringent constraints on the current second time derivative of Newton's constant G. In particular we find {\ddot G}/G >-1.91 H_0^2=-2 10^{-20}h^2 yrs^{-2} at the 2\sigma level which is complementary to solar system tests which constrain only the first derivative of G as |{\dot G}/G|<10^{-14}yrs^{-1} at 1\sigma. 
S. Nesseris and L. Perivolaropoulos