Measuring space-time variation of the fundamental constants with redshifted submillimetre transitions of neutral carbon

We compare the redshifts of neutral carbon and carbon monoxide in the redshifted sources in which the P-3(1) -> P-3(0) fine structure transition of neutral carbon, [C I], has been detected, in order to measure space-time variation of the fundamental constants. Comparison with the CO rotational lines measures F = alpha(2)/mu, where alpha is the fine structure constant and mu is the electron-proton mass ratio, which is the same combination of constants obtained from the comparison P-2(3/2) -> P-2(1/2) fine structure line of singly ionised carbon, [C II]. However, neutral carbon has the distinct advantage that it may be spatially coincident with the carbon monoxide, whereas [C II] could be located in the diffuse medium between molecular clouds, and so any comparison with CO could be dominated by intrinsic velocity differences. Using [C I], we obtain a mean variation of = (-3.6 +/- 8.5) x 10(-5), over z = 2.3-4.1, for the eight [C I] systems, which degrades to (-1.5 +/- 11) x 10(-5), over z = 2.3-6.4 when the two [C II] systems are included. That is, zero variation over look-back times of 10.8-12.8 Gyr. However, the latest optical results indicate a spatial variation in alpha, where Delta alpha/alpha describes a dipole and we see the same direction in Delta F/F. This trend is, however, due to a single source for which the [C I] spectrum is of poor quality. This also applies to one of the two [C II] spectra previously used to find a zero variation in alpha(2)/mu. Quantifying this, we find an anti-correlation between |Delta F/F| and the quality of the carbon detection, as measured by the spectral resolution, indicating that the typical values of greater than or similar to 50 km s (1), used to obtain a detection, are too coarse to reliably measure changes in the constants. From the fluxes of the known z greater than or similar to 1 CO systems, we predict that current instruments are incapable of the sensitivities required to measure changes in the constants through the comparison of CO and carbon lines. We therefore discuss in detail the use of ALMA for such an undertaking and find that, based upon the current CO detections only, the Full Array configuration is expected to detect similar to 100 galaxies in [C I] at better than 10 km s(-1) spectral resolution, while potentially resolving the individual molecular cloud complexes at redshifts of z greater than or similar to 3. This could provide greater than or similar to 1000 individual systems with which to obtain accurate measurements of space-time variation of the constants at look-back times in excess of 11 Gyr.