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 (Supernova Legacy Survey data set) 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 G/G >-1.91H(0)(2)=-2x10(-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 vertical bar G/G vertical bar < 10(-14) yrs(-1) at 1 sigma.