Constraints on changes in the proton-electron mass ratio using methanol
  lines
  We report Karl G. Jansky Very Large Array (VLA) absorption spectroscopy in
four methanol (CH$_3$OH) lines in the $z = 0.88582$ gravitational lens towards
PKS1830-211. Three of the four lines have very different sensitivity
coefficients $K_\mu$ to changes in the proton-electron mass ratio $\mu$; a
comparison between the line redshifts thus allows us to test for temporal
evolution in $\mu$. We obtain a stringent statistical constraint on changes in
$\mu$ by comparing the redshifted 12.179 GHz and 60.531 GHz lines, $[\Delta
mu/\mu] \leq 1.1 \times 10^{-7}$ ($2\sigma$) over $0 < z \leq 0.88582$, a
factor of $\approx 2.5$ more sensitive than the best earlier results. However,
the higher signal-to-noise ratio (by a factor of $\approx 2$) of the VLA
spectrum in the 12.179 GHz transition also indicates that this line has a
different shape from that of the other three CH$_3$OH lines (at $> 4\sigma$
significance). The sensitivity of the above result, and that of all earlier
CH$_3$OH studies, is thus likely to be limited by unknown systematic errors,
probably arising due to the frequency-dependent structure of PKS1830-211. A
robust result is obtained by combining the three lines at similar frequencies,
48.372, 48.377 and 60.531 GHz, whose line profiles are found to be in good
agreement. This yields the $2\sigma$ constraint $[\Delta \mu/\mu] \lesssim 4
\times 10^{-7}$, the most stringent current constraint on changes in $\mu$. We
thus find no evidence for changes in the proton-electron mass ratio over a
lookback time of $\approx 7.5$ Gyrs.
