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Constraining the variation of fundamental constants at z ~ 1.3 using 21-cm absorbers

We present high resolution optical spectra obtained with the Ultraviolet and Visual Echelle Spectrograph (UVES) at the Very Large Telescope (VLT) and 21-cm absorption spectra obtained with the Giant Metrewave Radio Telescope (GMRT) and the Green Bank Telescope (GBT) of five quasars along the line of sight of which 21-cm absorption systems at 1.17 < z < 1.56 have been detected previously. We also present milliarcsec scale radio images of these quasars obtained with the Very Large Baseline Array (VLBA). We use the data on four of these systems to constrain the time variation of x = g_p*alpha^2/mu where g_p is the proton gyromagnetic factor, alpha is the fine structure constant, and mu is the proton-to-electron mass ratio. We carefully evaluate the systematic uncertainties in redshift measurements using cross-correlation analysis and repeated Voigt profile fitting. In two cases we also confirm our results by analysing optical spectra obtained with the Keck telescope. We find the weighted and the simple means of Delta_x / x to be respectively -(0.1 +/- 1.3)x10^-6 and (0.0 +/- 1.5)x10^-6 at the mean redshift of <z> = 1.36 corresponding to a look back time of ~ 9 Gyr. This is the most stringent constraint ever obtained on Delta_x / x. If we only use the two systems towards quasars unresolved at milliarcsec scales, we get the simple mean of Delta_x / x = + (0.2 +/- 1.6)x10^-6. Assuming constancy of other constants we get Delta_alpha / alpha = (0.0 +/- 0.8)x10^-6 which is a factor of two better than the best constraints obtained so far using the Many Multiplet Method. On the other hand assuming alpha and g_p have not varied we derive Delta_mmu / mu = (0.0 +/- 1.5)x10^-6 which is again the best limit ever obtained on the variation of mu over this redshift range. [Abridged]