Observational constraints on the slope of the radial acceleration
  relation at low accelerations
  The radial acceleration relation (RAR) locally relates the `observed'
acceleration inferred from the dynamics of a system to the acceleration implied
by its baryonic matter distribution. The relation as traced by galaxy rotation
curves is one-to-one with remarkably little scatter, implying that the dynamics
of a system can be predicted simply by measuring its density profile as traced
by e.g. stellar light or gas emission lines. Extending the relation to
accelerations below those usually probed by practically observable kinematic
tracers is challenging, especially once accounting for faintly emitting
baryons, such as the putative warm-hot intergalactic medium, becomes important.
We show that in the low-acceleration regime, the (inverted) RAR predicts an
unphysical, declining enclosed baryonic mass profile for systems with
`observed' acceleration profiles steeper than $g_{\rm obs}\propto r^{-1}$
(corresponding to density profiles steeper than isothermal - $\rho(r)\propto
r^{-2}$). If the RAR is tantamount to a natural law, such acceleration profiles
cannot exist. We apply this argument to test the compatibility of an
extrapolation of the rotation curve-derived RAR to low accelerations with data
from galaxy-galaxy weak lensing, dwarf spheroidal galaxy stellar kinematic, and
outer Milky~Way dynamical measurements, fully independent of the uncertainties
inherent in direct measurements of the baryonic matter distribution. In all
cases we find that the data weakly favour a break to a steeper low-acceleration
slope. Improvements in measurements and modelling of the outer Milky~Way, and
weak lensing, seem like the most promising path toward stronger constraints on
the low-acceleration behaviour of the RAR.
