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Many modern theories predict that the fundamental constants depend on time,position, or the local density of matter. We develop a spectroscopic method forpulsed beams of cold molecules, and use it to measure the frequencies ofmicrowave transitions in CH with accuracy down to 3 Hz. By comparing thesefrequencies with those measured from sources of CH in the Milky Way, we testthe hypothesis that fundamental constants may differ between the high and lowdensity environments of the Earth and the interstellar medium. For the finestructure constant we find \Delta\alpha/\alpha = (0.3 +/- 1.1)*10^{-7}, thestrongest limit to date on such a variation of \alpha. For theelectron-to-proton mass ratio we find \Delta\mu/\mu = (-0.7 +/- 2.2) * 10^{-7}.We suggest how dedicated astrophysical measurements can improve theseconstraints further and can also constrain temporal variation of the constants.

A search for varying fundamental constants using Hz-level frequency measurements of cold CH molecules