3 Variation of the Strength of Gravity

In GR the coupling strength of gravity is taken as a constant: G. However, not all theories require such a constraint. Within scalar-tensor theories the gravitational coupling can become a function of a dynamical scalar field. The variation of the gravitational constant then depends on the cosmological evolution of the scalar field. The exact form of the variation will depend on the specific cosmological scenario being proposed [71, 27, 46]. It has also been shown that dark energy is compatible with a wide range of models derived from the compactification of higher-dimensional theories if the gravitational constant varies with time [68].

A time variation of G will show up as an anomalous evolution of the orbital periods of astronomical bodies. This is easily seen from Kepler’s third law :

2 4π2r3 P = -----. (10 ) Gm
Taking the time derivative and rearranging we find:
G˙ ˙r- P˙ m˙ G = 3 r − 2P − m . (11 )
For solar system bodies, we can safely ignore the mass term, except for a small rate of mass loss by the Sun (for compact objects like neutron stars this term becomes important [53]). Both tidal-friction and a changing G influence the semimajor axis. However, one can separate the effects by taking into account their different proportional contributions to the orbital period.

This evolving range, linear with time, was used for analysis of the initial years of LLR data. However, a changing G affects both the monthly lunar orbit and the annual Earth-Moon orbit around the Sun. Solar perturbations on the lunar orbit are also large. Secular change in the annual orbital period from a varying G accumulates as an orbital longitude perturbation evolving quadratically with time [55]. This t2 effect on the phase of the solar perturbations provides a strong limit when measured over decades.

Recent analysis of LLR data by the JPL group sets a limit on G˙∕G = (6 ± 7 ) × 10− 13 year [69]. Similarly, Müller et al. find G˙∕G = (2 ± 7) × 10−13 year and G¨∕G = (4 ± 5 ) × 10− 15 year2 [35]. These limits translate to less than a 1% variation of G over the 13.7 billion year age of the universe. Because the dominant effect for a variation in G is quadratic in time, continued LLR measurements will significantly improve this limit. Additionally, a more optimal measurement schedule throughout the lunar month, now possible with APOLLO, will also put better constraints on a possible time variation of G [56].

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