  We compare the full-sky morphology of the 511 keV gamma ray excess measured
by the INTEGRAL/SPI experiment to predictions of models based on dark matter
(DM) scatterings that produce low-energy positrons: either MeV-scale DM that
annihilates directly into e+e- pairs, or heavy DM that inelastically scatters
into an excited state (XDM) followed by decay into e+e- and the ground state.By
direct comparison to the data, we find that such explanations are consistent
with dark matter halo profiles predicted by numerical many-body simulations for
a Milky Way-like galaxy. Our results favor an Einasto profile over the cuspier
NFW distribution and exclude decaying dark matter scenarios whose predicted
spatial distribution is too broad. We obtain a good fit to the shape of the
signal using six fewer degrees of freedom than previous empirical fits to the
511 keV data. We find that the ratio of flux at Earth from the galactic bulge
to that of the disk is between 1.9 and 2.4, taking into account that 73% of the
disk contribution may be attributed to the beta decay of radioactive 26Al.
