Stringent null constraint on cosmological evolution of the proton-to-electron mass ratio

We present a strong constraint on variation of the proton-to-electron mass ratio, mu, over cosmological time scales using molecular hydrogen transitions in optical quasar spectra. Using high quality spectra of quasars Q0405-443, Q0347-383 and Q0528-250, variation in mu relative to the present day value is limited to (Delta mu)/mu=(2.6 +/- 3.0) x 10^(-6). We reduce systematic errors compared to previous works by substantially improving the spectral wavelength calibration method and by fitting absorption profiles to the forest of hydrogen Lyman alpha transitions surrounding each H_2 transition. Our results are consistent with no variation, and inconsistent with a previous approx 4 sigma detection of mu variation involving Q0405-443 and Q0347-383. If the results of this work and those suggesting that alpha may be varying are both correct, then this would tend to disfavour certain grand unification models.