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 (14), the amplitude of the violation of the universality of free fall is given by
,
which takes the form
In
the case in which the variation of the constants arises from the same scalar field, the analysis of Section 6.1
implies that
can be related to the gravitational potential by
so that
This can be expressed in terms of the sensitivity coefficient
defined in Equation (214) as
since
. This shows that each experiment will yield a constraint on a linear combination of
the coefficients
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
[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
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
(
), 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 as
The cosmological observation from QSO spectra implies that
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
neutrons and
protons will be characterized by a sensitivity
where
(resp.
) is
the ratio of neutrons (resp. protons) and where it has been assumed that
.
Assuming
that
and using that the compactness of the Moon-Earth system
,
one gets
. Dvali and Zaldarriaga [166] 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 [143] who included explicitly the electron and related the violation of the universality of
free fall to the variation of
.
In a similar analysis [532], 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
It
follows that the expected violation of the universality of free fall is related to the time variation of
today by
where
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
), one requires
for
, 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 (203) 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
, which quantify both the deviation of its equation of state from
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
are estimated using the model (189). 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
and today. To finish, let us stress that these results may be changed significantly if a chameleon mechanism
is at work.