Hello!
As a theorist who occasionally thinks about 'varying constants' I am interested in the latest observational position. This short paper is not about new observations, but it does highlight a significant aspect : how the observational errors are estimated.
Relevant previous works are astroph/0310318, astroph/0401094, astroph/0511765.
The method for the astrophysical observations is simply to compare the redshifts of many different lines in a given absorber system, where the ratios between line wavelengths depend theoretically on alpha or some other 'constant'. For each system a fit is made for the redshift and the value of the 'constant' under consideration.
The tricky part for error estimation is getting from the measured spectra to a set of line velocities (aka redshifts or wavelengths) with errors. I quote:
'Most metalline QSO absorption profiles display a complicated velocity structure and oneusually focuses on the propeties of individual velocity components, each of which is typically modelled by a Voigt profile. However, it is important to realize that Delta alpha / alpha and its uncertainty are integrated quantities determined by the entire absorption profile.'
What is actually measured is a set of pixels, each of which has an flux and an associated 1sigma error, and a wavelength. The velocity uncertainty for a given component can be found from a formula taken from Bouchy, Pepe, Queloz, Astron. Astrophys. 374, 733 (2001), which depends on the individual pixel errors and the derivative of flux wrt wavelength.
Here the authors simply calculate this velocity uncertainty for the lines used in previous results for alpha, and derive a minimum possible uncertainty for the derived value of alpha in each absorber. (Other contributions to uncertainty come from different lines having different optical depths, for example.)
The last page has a plot of this minimum possible error versus the quoted error. Many points lie well *below* the line of minimum possible error  meaning that the quoted error was too small.
Funny thing is that all the Webb group points (which suggest a nonzero variation) lie above the line  while the more recent results which claim no variation are the ones with underestimated errors. The authors make a strong conclusion that the analysis of the recent negative results was not well understood.
Any reaction to this method of estimating the minimum possible error?
[astroph/0611080] Revisiting VLT/UVES constraints on a varying finestructure constant
Authors:  M. T. Murphy (1), J. K. Webb (2), V. V. Flambaum (2) ((1) Institute of Astronomy, U. Cambridge; (2) U. New South Wales) 
Abstract:  Current analyses of VLT/UVES quasar spectra disagree with the Keck/HIRES evidence for a varying finestructure constant, alpha. To investigate this we introduce a simple method for calculating the minimum possible uncertainty on Delta(alpha)/alpha for a given quasar absorber. For many absorbers in Chand et al. (2004) and for the singleabsorber constraint of Levshakov et al. (2006) the quoted uncertainties are smaller than the minimum allowed by the UVES data. Failure of this basic consistency test prevents reliable comparison of the UVES and HIRES results. 
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