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]
