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

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,

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].

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 N_{2}H^{+} and
N_{2}D^{+} in the dark cloud L183.

A second analysis [324] using high resolution spectral observations of molecular core in lines of NH_{3},
HC_{3}N and N_{2}H^{+} 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 ^{13}CO probed the spatial variation of over the galaxy. It concluded
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.

Living Rev. Relativity 14, (2011), 2
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