Conformal gravity rotation curves with a conformal Higgs halo

We discuss the effect of a conformally coupled Higgs field on Conformal Gravity (CG) predictions for the rotation curves of galaxies. The Mannheim-Kazanas (MK) metric is a valid vacuum solution of CG's 4-th order Poisson equation if and only if the Higgs field has a particular radial profile, S(r) = S₀ a/(r + a), decreasing from S₀ at r = 0 with radial scale length a. Since particle rest masses scale with S(r)/S₀, their world lines do not follow time-like geodesics of the MK metric g_μν, as previously assumed, but rather those of the Higgs-frame MK metric ḡ_μν = Ω² g_μν, with the conformal factor Ω(r) = S(r)/S₀. We show that the required stretching of the MK metric exactly cancels the linear potential that has been invoked to fit galaxy rotation curves without dark matter. We also formulate, for spherical structures with a Higgs halo S(r), the CG equations that must be solved for viable astrophysical tests of CG using galaxy and cluster dynamics and lensing.