The mass discrepancy-acceleration relation: a universal maximum dark matter acceleration and implications for the ultra-light scalar field dark matter model

Recent analysis of the rotation curves of a large sample of galaxies with very diverse stellar properties reveal a relation between the radial acceleration purely due to the baryonic matter and the one inferred directly from the observed rotation curves. Assuming the dark matter (DM) exists, this acceleration relation is tantamount to an acceleration relation between DM and baryons. This leads us to a universal maximum acceleration for all halos. Using the latter in DM profiles that predict inner cores implies that the central surface density $\mu_{DM} = \rho_s r_s$ must be a universal constant, as suggested by previous studies in selected galaxies, revealing a strong correlation between the density $\rho_s$ and scale $r_s$ parameters in each profile. We then explore the consequences of the constancy of $\mu_{DM}$ in the context of the ultra-light scalar field dark matter model (SFDM). We find that for this model $\mu_{DM} = 648 \, M_\odot {\rm pc}^{-2}$, and that the so-called WaveDM soliton profile should be an universal feature of the DM halos. Comparing with data from the Milky Way and Andromeda satellites, we find that they are consistent with a boson mass of the scalar field particle of the order of $10^{-21} \, {\rm eV}/c^2$, which puts the SFDM model in agreement with recent cosmological constraints.