We have studied the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) model for the spatial and temporal variations of the fine structure constant, α, with the aid of full N-body simulations that explicitly and self-consistently solve for the scalar field driving the α-evolution. We focus on the scalar field (or equivalently α) inside the dark matter halos and find that the profile of the scalar field is essentially independent of the BSBM model parameter. This means that given the density profile of an isolated halo and the background value of the scalar field, we can accurately determine the scalar-field perturbation in that halo. We also derive an analytic expression for the scalar-field perturbation using the Navarro-Frenk-White halo profile and show that it agrees well with numerical results, at least for isolated halos; for non-isolated halos, this prediction differs from numerical result by a (nearly) constant offset, which depends on the environment of the halo.
Li, B., Mota, D., & Barrow, J. (2011). Varying alpha from N-body simulations. Astrophysical Journal, 728(2), Article 108. https://doi.org/10.1088/0004-637x/728/2/108