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We consider cosmic strings and magnetic monopoles in Bekenstein-type models and show that there is a class of models of this type for which the classical Nielsen-Olesen vortex and 't Hooft-Polyakov monopoles are still valid solutions. We show that Equivalence Principle constraints impose tight limits on the allowed variations of α induced by string networks on cosmological scales. We show that the results obtained using the spherical infall model for an infinite wavelength inhomogeneity are inconsistent with the results of a local linearised gravity study, and we argue in favour of the second approach. We also criticise the claim that the value of α inside collapsed regions could be significantly different from the background one by these findings.
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