Nine quasar absorption spectra at 21-cm and UV rest-wavelengths are used to estimate possible variations in x \equiv \alpha ^ { 2 } g _ { p } \mu , where \alpha is the fine structure constant , g _ { p } the proton g -factor and \mu \equiv m _ { e } / m _ { p } is the electron-to-proton mass ratio .
We find \langle \Delta x / x \rangle ^ { weighted } _ { total } = ( 0.63 \pm 0.99 ) \times 10 ^ { -5 } over a redshift range 0.23 \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \sim$ } \hss } \raise 2.0 pt \hbox { $ < % $ } } z _ { abs } \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \sim$ } \hss } \raise 2.0 % pt \hbox { $ < $ } } 2.35 which corresponds to look-back times of 2.7–10.5 billion years .
A linear fit against look-back time , tied to \Delta x / x = 0 at z = 0 , gives a best-fit rate of change of \dot { x } / x = ( -0.6 \pm 1.2 ) \times 10 ^ { -15 } { yr } ^ { -1 } .
We find no evidence for strong angular variations in x across the sky .
Our sample is much larger than most previous samples and demonstrates that intrinsic line-of-sight velocity differences between the 21-cm and UV absorption redshifts , which have a random sign and magnitude in each absorption system , limit our precision .
The data directly imply that the average magnitude of this difference is \Delta v _ { los } \sim 6 km s ^ { -1 } .

Combining our \Delta x / x measurement with absorption-line constraints on \alpha -variation yields strong limits on the variation of \mu .
Our most conservative estimate , obtained by assuming no variations in \alpha or g _ { p } is simply \Delta \mu / \mu = \langle \Delta x / x \rangle ^ { weighted } _ { total } .
If we use only the four high-redshift absorbers in our sample , we obtain \Delta \mu / \mu = ( 0.58 \pm 1.95 ) \times 10 ^ { -5 } , which agrees ( 2 \sigma ) with recent , more direct estimates from two absorption systems containing molecular hydrogen , also at high redshift , and which have hinted at a possible \mu -variation , \Delta \mu / \mu = ( -2.0 \pm 0.6 ) \times 10 ^ { -5 } .
Our method of constraining \Delta \mu / \mu is completely independent from the molecular hydrogen observations .
If we include the low-redshift systems , our \Delta \mu / \mu result differs significantly from the high-redshift molecular hydrogen results .
We detect a dipole variation in \mu across the sky , but given the sparse angular distribution of quasar sight-lines we find that this model is required by the data at only the 88 per cent confidence level .
Clearly , much larger samples of 21-cm and molecular hydrogen absorbers are required to adequately resolve the issue of the variation of \mu and x .