### 6.1 Local scales

In order to determine the profile of the constant in the solar system, let us assume that their value is dictated by the value of a scalar field. As in Section 5.4.1, we can assume that at lowest order the profile of the scalar field will be obtained from the scalar-tensor theory, taking into account that all masses scale as where is the value of the field in the Einstein frame.

#### 6.1.1 Generalities

We restrict to the weakly self-gravitating () and slow moving () localized material systems and follow [124]. Using harmonic coordinates, defined with respect to the metric , the Einstein frame metric can be expanded as

so that Eqs. (162163) take the form
where is the flat d’Alembertian and where the scalar field has been assumed to be light so that one can neglect its potential. The source terms can be expressed in terms of the matter stress-energy tensor in the Einstein frame as
Restricting to the static case with a single massive point source, the only non-vanishing source terms are and so that the set of equations reduces to two Poisson equations
This set of equations can be solved by means of the retarded Green function. It follows that the Einstein frame gravitational potential is . The equation for can be solved iteratively, since at lowest order in it has solution
This can be used to determine the Jordan frame metric and the variation of the scalar field in function of the Jordan frame coordinates. It follows that at lowest order the Newton potential and the scalar field are given by
where we have neglected the corrections for the gravitational potential, which, given the solar system constraints on , is a good approximation.

Now let us consider any constant function of . Thus, its profile is given by so that

where is the sensitivity of the constant to a variation of the scalar field, . For laboratory in orbit on an elliptic trajectory,
where is the semi-major axis, the eccentricity and the true anomaly. It follows that
The first term represents the variation of the mean value of the constant on the orbit compared with its cosmological value. This shows that local terrestrial and solar system experiments do measure the effects of the cosmological variation of the constants [124, 461, 460, 462]. The second term is a seasonal modulation and it is usually parameterized [209] as
defining the parameters .

#### 6.1.2 Solar system scales

The parameters can be constrained from laboratory measurements on Earth. Since for the Earth orbit, the signal should have a peak-to-peak amplitude of on a period of 1 year. This shows that the order of magnitude of the constraints will be roughly of since atomic clocks reach an accuracy of the order of . The data of [214] and [37] lead respectively to the two constraints [209]

for and respectively. The atomic dysprosium experiment [100] allowed to set the constraint [193]
which, combined with the previous constraints, allows to conclude that
for and respectively. [61], using the comparison of cesium and a strontium clocks derived that
which, combined with measurement of H-maser [17], allow one to set the three constraints as
[34, 463] reanalyzed the data by [408] to conclude that . Combined with the constraint (218), it led to
[34] also used the data of [440] to conclude
All these constraints use the sensitivity coefficients computed in [14, 210]. We refer to [265] as an unexplained seasonal variation that demonstrated the difficulty to interpret phenomena.

Such bounds can be improved by comparing clocks on Earth and onboard of satellites [209, 444, 343] while the observation of atomic spectra near the Sun can lead to an accuracy of order unity [209]. A space mission with atomic clocks onboard and sent to the Sun could reach an accuracy of 10–8 [343, 547].

#### 6.1.3 Milky Way

An attempt [323, 358] to constrain from emission lines due to ammonia in interstellar clouds of the Milky Way led to the conclusion that , by considering different transitions in different environments. This is in contradiction with the local constraint (219). This may result from rest frequency uncertainties or it would require that a mechanism such as chameleon is at work (see Section 5.4.2) in order to be compatible with local constraints. The analysis was based on an ammonia spectra atlas of 193 dense protostellar and prestellar cores of low masses in the Perseus molecular cloud, comparison of N2H+ and N2D+ in the dark cloud L183.

A second analysis [324] using high resolution spectral observations of molecular core in lines of NH3, HC3N and N2H+ with 3 radio-telescopes showed that between the cloud environment and the local laboratory environment. However, an offset was measured that could be interpreted as a variation of of amplitude . A second analysis [322] map four molecular cores L1498, L1512, L1517, and L1400K selected from the previous sample in order to estimate systematic effects due to possible velocity gradients. The measured velocity offset, once expressed in terms of , gives .

A similar analysis [326] based on the fine-structure transitions in atomic carbon C i and low-laying rotational transitions in 13CO probed the spatial variation of over the galaxy. It concluded that

between high (terrestrial) and low (interstellar) densities of baryonic matter. Combined with the previous constraint on it would imply that . This was updated [319] to so that .

Since extragalactic gas clouds have densities similar to those in the interstellar medium, these bounds give an upper bound on a hypothetic chameleon effect, which are much below the constraints obtained on time variations from QSO absorption spectra.