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 approximate to 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 disfavor certain grand unification models.