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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 fine structure transition of neutral carbon, [CI], has been detected, in order to measure space-time variation of the fundamental constants. Comparison with the CO rotational lines measures gives the same combination of constants obtained from the comparison fine structure line of singly ionised carbon, [CII]. However, neutral carbon has the distinct advantage that it may be spatially coincident with the carbon monoxide, whereas [CII] could be located in the diffuse medium between molecular clouds, and so any comparison with CO could be dominated by intrinsic velocity differences. Using [CI], we obtain a mean variation of dF/F = (-3.6 +/- 8.5) x 10^-5, over z = 2.3 - 4.1, for the eight [CI] systems, which degrades to (-1.5+/- 11) x 10^-5, over z = 2.3 - 6.4 when the two [CII] 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, which describes a dipole and we see the same direction in dF/F. This trend is, however, due to a single source for which the [CI] spectrum is of poor quality. This also applies to one of the two [CII] spectra previously used to find a zero variation in alpha^2/mu. Quantifying this, we find an anti-correlation between |dF/F| and the quality of the carbon detection, as measured by the spectral resolution, indicating that the typical values of >50 km/s, used to obtain a detection, are too coarse to reliably measure changes in the constants. From the fluxes of the known z > 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 ... ABRIDGED