Possible time variations of G in scalar-tensor theories of gravity
  We estimate the possible variations of the gravitational constant G in the framework of a generalized (Bergmann-Wagoner-Nordtvedt) scalar-tensor theory of gravity on the basis of the field equations, without using their special solutions. Specific estimates are essentially related to the values of other cosmological parameters (the Hubble and acceleration parameters, the dark matter density etc.), but the values of G-dot/G compatible with modern observations do not exceed 10^{-12}. 
K.A. Bronnikov (Moscow), V.N. Melnikov (Moscow) and Mario Novello (Rio   de Janeiro)