From Equation (14), the amplitude of the violation of the universality of free fall is given by , which takes the form
While the couplings to mass number, lepton number and the electromagnetic binding energy have been considered  [see the example of Section 5.4.1] the coupling to quark masses remains a difficult issue. In particular, the whole difficulty lies in the determination of the coefficients [see Section 5.3.2]. In the formalism developed in [122, 121], see Section 5.4.1, one can relate the expected deviation from the universality of free fall to the 5 parameters and get constraints on and where . For instance, the Be-Ti EötWash experiment and LRR experiment respectively imply
The link between the time variation of fundamental constants and the violation of the universality of free fall have been discussed by Bekenstein  in the framework described in Section 5.4.2 and by Damour–Polyakov [135, 136] in the general framework described in Section 5.4.1. In all these models, the two effects are triggered by a scalar field. It evolves according to a Klein–Gordon equation (), which implies that is damped as if its mass is much smaller than the Hubble scale. Thus, in order to be varying during the last Hubble time, has to be very light with typical mass . As a consequence, has to be very weakly coupled to the standard model fields to avoid a violation of the universality of free fall.
This link was revisited in [96, 166, 532] in which the dependence of on the scalar field responsible for its variation is expanded as13 that and using that the compactness of the Moon-Earth system , one gets . Dvali and Zaldarriaga  obtained the same result by considering that . This implies that , which is compatible with the variation of if during the last Hubble period. From the cosmology one can deduce that . If dominates the matter content of the universe, , then so that whereas if it is sub-dominant and . In conclusion . This makes explicit the tuning of the parameter . Indeed, an important underlying approximation is that the -dependence arises only from the electromagnetic self-energy. This analysis was extended in  who included explicitly the electron and related the violation of the universality of free fall to the variation of .
In a similar analysis , the scalar field is responsible for both a variation of and for the acceleration of the universe. Assuming its equation of state is , one can express its time variation (as long as it has a standard kinetic term) as
One question concerns the most sensitive probes of the equivalence principle. This was investigated in  in which the coefficients are estimated using the model (189). It was concluded that they are 2 – 3 orders of magnitude over cosmic clock bounds. However,  concluded that the most sensitive probe depends on the unification relation that exist between the different couplings of the standard model.  concluded similarly that the universality of free fall is more constraining that the seasonal variations. The comparison with QSO spectra is more difficult since it involves the dynamics of the field between and today. To finish, let us stress that these results may be changed significantly if a chameleon mechanism is at work.
Living Rev. Relativity 14, (2011), 2
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