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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 ^2 contours based on current data (Supernova Legacy Survey data set) are also constructed on top of the sectors, for a prior of ₌=0. 24. By demanding that the observed 2 ^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. 91H₀^2=-210^-20h^2 yrs^-2 at the 2 level which is complementary to solar system tests which constrain only the first derivative of G as | { G}G|<10^-14 yrs^-1 at 1.
Nesseris et al. (Wed,) studied this question.