6.3 Implication for the universality of free fall

As we have seen in the previous sections, the tests of the universality of free fall is central in containing the model involving variations of the fundamental constants.

From Equation (14View Equation), the amplitude of the violation of the universality of free fall is given by ηAB, which takes the form

∑ ηAB = 1-- |fAi − fBi||∇αi|. gN i
In the case in which the variation of the constants arises from the same scalar field, the analysis of Section 6.1 implies that ∇ αi can be related to the gravitational potential by |∇ αi| = αisi(ϕ )αextgN so that
∑ ∑ ηAB = |fAi − fBi|si(ϕ )αiαext = |λAi − λBi|si(ϕ)αext. (226 ) i i
This can be expressed in terms of the sensitivity coefficient ki defined in Equation (214View Equation) as
∑ ηAB = |λAi − λBi|ki, (227 ) i
since |∇ α | = α k g i i i N. This shows that each experiment will yield a constraint on a linear combination of the coefficients ki so that one requires at least as many independent pairs of test bodies as the number of constants to be constrained.

While the couplings to mass number, lepton number and the electromagnetic binding energy have been considered [118] [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 λ ai [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 d and get constraints on ∗ D ˆm ≡ d g(d ˆm − dg) and De ≡ d∗gde where d ∗g ≡ dg + 0.093 (d ˆm − dg) + 0.00027de. For instance, the Be-Ti EötWash experiment and LRR experiment respectively imply

−11 −11 |D ˆm + 0.22De | < 5.1 × 10 , |D ˆm + 0.28De | < 24.6 × 10 .
This shows that while the Lunar experiment has a slightly better differential-acceleration sensitivity, the laboratory-based test is more sensitive to the dilaton coefficients because of a greater difference in the dilaton charges of the materials used, and of the fact that only one-third of the Earth mass is made of a different material.

The link between the time variation of fundamental constants and the violation of the universality of free fall have been discussed by Bekenstein [39] 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 (¨ϕ + 3H ϕ˙+ m2 ϕ + ...= 0), which implies that ϕ is damped as ϕ˙∝ a −3 if its mass is much smaller than the Hubble scale. Thus, in order to be varying during the last Hubble time, ϕ has to be very light with typical mass −33 m ∼ H0 ∼ 10 eV. As a consequence, ϕ 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, 166Jump To The Next Citation Point, 532Jump To The Next Citation Point] in which the dependence of αEM on the scalar field responsible for its variation is expanded as

( ) -ϕ-- -ϕ2- αEM = αEM (0) + λ M + 𝒪 M 2 . (228 ) 4 4
The cosmological observation from QSO spectra implies that λΔ ϕ∕M4 ∼ 10− 7 at best during the last Hubble time. Concentrating only on the electromagnetic binding energy contribution to the proton and of the neutron masses, it was concluded that a test body composed of n n neutrons and n p protons will be characterized by a sensitivity λ(νpBp + νnBn )∕mN where νn (resp. νp) is the ratio of neutrons (resp. protons) and where it has been assumed that mn ∼ mp ∼ mN. Assuming13 that νEna,prth ∼ 1 ∕2 and using that the compactness of the Moon-Earth system ∂ ln(mEarth∕mMoon )∕∂ ln αEM ∼ 10− 3, one gets η ∼ 10−3λ2 12. Dvali and Zaldarriaga [166] obtained the same result by considering that −2 −1 Δ νn,p ∼ 6 × 10 − 10. This implies that −5 λ < 10, which is compatible with the variation of αEM if −2 Δ ϕ ∕M4 > 10 during the last Hubble period. From the cosmology one can deduce that (Δ ϕ∕M4 )2 ∼ (ρϕ + Pϕ )∕ ρtotal. If ϕ dominates the matter content of the universe, ρtotal, then Δ ϕ ∼ M4 so that λ ∼ 10 −7 whereas if it is sub-dominant Δ ϕ ≪ M4 and λ ≫ 10−7. In conclusion 10− 7 < λ < 10−5. This makes explicit the tuning of the parameter λ. Indeed, an important underlying approximation is that the ϕ-dependence arises only from the electromagnetic self-energy. This analysis was extended in [143] who included explicitly the electron and related the violation of the universality of free fall to the variation of μ.

In a similar analysis [532], the scalar field is responsible for both a variation of αEM and for the acceleration of the universe. Assuming its equation of state is wh ⁄= 1, one can express its time variation (as long as it has a standard kinetic term) as

∘ ------------- ϕ˙= H 3Ωϕ(1 + wh ).
It follows that the expected violation of the universality of free fall is related to the time variation of αEM today by
( ) −2 ∂ lnαEM 2 (1 + &tidle;Q)Δ Z+ZN- η = − 1.75 × 10 ---∂z---- --(0)------(0)-, z=0 Ω ϕ (1 + wh )
where &tidle; Q is a parameter taking into account the influence of the mass ratios. Again, this shows that in the worse case in which the Oklo bound is saturated (so that − 6 ∂ lnαEM ∕ ∂z ∼ 10), one requires 1 + w (0h) ≳ 10−2 for η ≲ 10−13, hence providing a string bond between the dark energy equation of state and the violation of the universality of free fall. This was extended in [149] in terms of the phenomenological model of unification presented in Section 5.3.1. In the case of the string dilaton and runaway dilaton models, one reaches a similar conclusion [see Equation (203View Equation) in Section 5.4.1]. A similar result [348] was obtained in the case of pure scalar-tensor theory, relating the equation of state to the post-Newtonian parameters. In all these models, the link between the local constraints and the cosmological constraints arise from the fact that local experiments constrain the upper value of ˙ ϕ0, which quantify both the deviation of its equation of state from − 1 and the variation of the constants. It was conjectured that most realistic quintessence models suffer from such a problem [68].

One question concerns the most sensitive probes of the equivalence principle. This was investigated in [144] in which the coefficients λ Ai are estimated using the model (189View Equation). It was concluded that they are 2 – 3 orders of magnitude over cosmic clock bounds. However, [148] concluded that the most sensitive probe depends on the unification relation that exist between the different couplings of the standard model. [463] 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 z ∼ 1 and today. To finish, let us stress that these results may be changed significantly if a chameleon mechanism is at work.

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