Update: In 2008, a new experiment with ammonia found no time-dependence in the last 6 billion years.Klaus Lange has pointed out that
describes a Dutch experiment performed primarily in the European Southern Observatory - hold your breath, this observatory is located in Chile. They measured the spectrum of the molecular hydrogen that depends on the proton-electron mass ratio "mu".
Note that this ratio is about 1836.15. Twenty years ago I played with the calculator and it turned out that this number can be written as
- 6.pi^5 = 1836.12.
This agreement has promoted me to the king of all crackpots: with only three characters, namely six, pi, five, I can match around 5 or 6 significant figures of the correct result.
Actually my calculator only had 8 significant figures (with no hidden figures) and I exactly matched 8 significant figures of 1836.1515 written in the mathematical tables of that time.
Later I learned that someone else has actually published this "discovery" fifty years ago, and the agreement got worse with better calculators and better measurements in particle physics.
More seriously, the Dutchmen now claim that the ratio was 1.00002 times higher twelve billion years ago. The New Scientist immediately speculates that this could prove extra dimensions or string theory. I, for one, have absolutely no idea where this statement comes from. I personally believe that these constants have been constant in the last 12 billion years - and moreover this opinion is completely and naturally compatible with string theory.
But of course, if some additional serious experimenters besides our European colleagues from Chile confirm the result, we will have to give an explanation. We will either have to add a new rolling scalar field analogous to quintessence that couples to the Standard Model nontrivially, or we will have to think about some dramatic non-local cosmological influences on local physics. Both approaches seem rather implausible to me and because I can't think of a third one, my guess is that the result will go away.
Alternatively it may be explained by some conventional physics that I am not able to identify right now - such as strong magnetic fields in the environment.