In such a case, the fine-structure constant will behave as , the fluctuations being a stochastic variable. As we have seen earlier, enters the dynamics of recombination, which would then become patchy. This has several consequences for the CMB anisotropies. In particular, similarly to weak gravitational lensing, it will modify the mean power spectra (this is a negligible effect) and induce a curl component (B mode) to the polarization . Such spatial fluctuations also induce non-Gaussian temperature and polarization correlations in the CMB [466, 417]. Such correlations have not allowed to set observational constraints yet but they need to be included foe consistency, see e.g., the example of CMB computation in scalar-tensor theories . The effect on large the scale structure was also studied in [30, 363] and the Keck/HIRES QSO absorption spectra showed  that the correlation function of the fine-structure constant is consistent on scales ranging between 0.2 and 13 Gpc.
Recently, it has been claimed [50, 523] that the fine structure constant may have a dipolar variation that would explain consistently the data from the Southern and Northern hemispheres (see Section 3.4.3). Let assume a constant, say, depend on the local value of a dynamical scalar field . The value of at the observation point is compared to its value here and today,
This has lead to the idea  of the existence of a low energy domain wall produced in the spontaneous symmetry breaking involving a dilaton-like scalar field coupled to electromagnetism. Domains on either side of the wall exhibit slight differences in their respective values of . If such a wall is present within our Hubble volume, absorption spectra at large redshifts may or may not provide a variation in relative to the terrestrial value, depending on our relative position with respect to the wall.
Another possibility would be that the Copernican principle is not fully satisfied, such as in various void models. Then the background value of would depend, e.g., on and for a spherically symmetric spacetime (such as a Lemaître–Tolman–Bondi spacetime). This could give rise to a dipolar modulation of the constant if the observer (us) is not located at the center of the universe. Note, however, that such a cosmological dipole would also reflect itself, e.g., on CMB anisotropies. Similar possibilities are also offered within the chameleon mechanism where the value of the scalar field depends on the local matter density (see Section 5.4.2).
More speculative, is the effect that such fluctuations can have during preheating after inflation since the decay rate of the inflaton in particles may fluctuate on large scales [293, 294].
Living Rev. Relativity 14, (2011), 2
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