Proton-to-electron mass ratio

In physics, the proton-to-electron mass ratio, μ or β, is simply the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms). Because this is a ratio of like-dimensioned physical quantities, it is a dimensionless quantity, a function of the dimensionless physical constants, and has numerical value independent of the system of units, namely:
The number enclosed in parentheses is the measurement uncertainty on the last two digits. The value of "μ" is known to about 0.1 parts per billion.
It was also noticed that "μ" ≈ formula_1 
Discussion.
"μ" is an important fundamental physical constant because:
Variation of "μ" over time.
Astrophysicists have tried to find evidence that "μ" has changed over the history of the universe. (The same question has also been asked of the fine structure constant.) One interesting cause of such change would be change over time in the strength of the strong force.
Astronomical searches for time-varying "μ" have typically examined the Lyman series and Werner transitions of molecular hydrogen which, given a sufficiently large redshift, occur in the optical region and so can be observed with ground-based spectrographs.
If "μ" were to change, then the change in the wavelength "λi" of each rest frame wavelength can be parameterised as:
where Δ"μ"/"μ" is the proportional change in "μ" and "Ki" is a constant which must be calculated within a theoretical (or semi-empirical) framework.
Reinhold et al. (2006) reported a potential 4 standard deviation variation in "μ" by analysing the molecular hydrogen absorption spectra of quasars Q0405-443 and Q0347-373. They found that . King et al. (2008) reanalysed the spectral data of Reinhold et al. and collected new data on another quasar, Q0528-250. They estimated that , different from the estimates of Reinhold et al. (2006).
Murphy et al. (2008) used the inversion transition of ammonia to conclude that at redshift . Kanekar (2011) used deeper observations of the inversion transitions of ammonia in the same system at towards 0218+357 to obtain .
Bagdonaite et al. (2013) used methanol transitions in the spiral lens galaxy PKS 1830-211 to find at .
Kanekar et al. (2015) used near-simultaneous observations of multiple methanol transitions in the same lens, to find at . Using three methanol lines with similar frequencies to reduce systematic effects, Kanekar et al. (2015) obtained .
Note that any comparison between values of Δ"μ"/"μ" at substantially different redshifts will need a particular model to govern the evolution of Δ"μ"/"μ". That is, results consistent with zero change at lower redshifts do not rule out significant change at higher redshifts.