We restrict to the weakly self-gravitating () and slow moving () localized material systems and follow . 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
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  and  lead respectively to the two constraints  allowed to set the constraint  , using the comparison of cesium and a strontium clocks derived that , allow one to set the three constraints as [34, 463] reanalyzed the data by  to conclude that . Combined with the constraint (218), it led to  also used the data of  to conclude [14, 210]. We refer to  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 . 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 N2H+ and N2D+ in the dark cloud L183.
A second analysis  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  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  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 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.
Living Rev. Relativity 14, (2011), 2
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