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Over five years we have compared the hyperfine frequencies of 133Cs and 87Rb atoms in their electronic ground state using several laser cooled 133Cs and 87Rb atomic fountains with an accuracy of ∼10-15. These measurements set a stringent upper bound to a possible fractional time variation of the ratio between the two frequencies: d/dt In (νRb/νCs)=(0.2±7.0)×10-16 yr-1 (1σ uncertainty). The same limit applies to a possible variation of the quantity (μRb/μCs)α-0.44, which involves the ratio of nuclear magnetic moments and the fine structure constant. To improve this test, one needs more accurate cesium fountain clocks, for which the major limiting factor is the cold collision frequency shift. This effect can now be evaluated with great accuracy using a new method which we also present here. It is based on a transfer of population by adiabatic passage that allows to prepare cold atomic samples with a well defined ratio of atomic density and atom number. This method is used to perform a measurement of the cold collision frequency shift in a laser cooled cesium clock at the percent level. With improvements, the adiabatic passage would allow measurements of density-dependent phase shifts at the 10-3 level in high precision experiments. With this precision, reaching an accuracy of 10-16 is possible.