The origin of the mass discrepancy-acceleration relation in $\Lambda$CDM

We examine the origin of the mass discrepancy--radial acceleration relation (MDAR) of disk galaxies. This is a tight empirical correlation between the disk centripetal acceleration and that expected from the baryonic component. The MDAR holds for most radii probed by disk kinematic tracers, regardless of galaxy mass or surface brightness. The relation has two characteristic accelerations; $a_0$, above which all galaxies are baryon-dominated; and $a_{\rm min}$, an effective minimum aceleration probed by kinematic tracers in isolated galaxies. We use a simple model to show that these trends arise naturally in $\Lambda$CDM. This is because: (i) disk galaxies in $\Lambda$CDM form at the centre of dark matter haloes spanning a relatively narrow range of virial mass; (ii) cold dark matter halo acceleration profiles are self-similar and have a broad maximum at the centre, reaching values bracketed precisely by $a_{\rm min}$ and $a_0$ in that mass range; and (iii) halo mass and galaxy size scale relatively tightly with the baryonic mass of a galaxy in any successful $\Lambda$CDM galaxy formation model. Explaining the MDAR in $\Lambda$CDM does not require modifications to the cuspy inner mass profiles of dark haloes, although these may help to understand the detailed rotation curves of some dwarf galaxies and the origin of extreme outliers from the main relation. The MDAR is just a reflection of the self-similar nature of cold dark matter haloes and of the physical scales introduced by the galaxy formation process.