About one year ago , it was speculated that decaying or annihilating Light Dark Matter ( LDM ) particles could explain the flux and extension of the 511 keV line emission in the galactic centre .
Here we present a thorough comparison between theoretical expectations of the galactic positron distribution within the LDM scenario and observational data from INTEGRAL/SPI .
Unlike previous analyses , there is now enough statistical evidence to put tight constraints on the shape of the dark matter halo of our galaxy , if the galactic positrons originate from dark matter .
For annihilating candidates , the best fit to the observed 511 keV emission is provided by a radial density profile with inner logarithmic slope \gamma = 1.03 \pm 0.04 .
In contrast , decaying dark matter requires a much steeper density profile , \gamma > 1.5 , rather disfavoured by both observations and numerical simulations .
Within the annihilating LDM scenario , a velocity-independent cross-section would be consistent with the observational data while a cross-section purely proportional to v ^ { 2 } can be rejected at a high confidence level .
Assuming the most simplistic model where the galactic positrons are produced as primaries , we show that the LDM candidate should be a scalar rather than a spin-1/2 particle and obtain a very stringent constraint on the value of the positron production cross-section to explain the 511 keV emission .
One consequence is that the value of the fine structure constant \alpha should differ from that recommended in the CODATA .
This is a very strong test for the LDM scenario and an additional motivation in favour of experiments measuring \alpha directly .
Our results finally indicate that an accurate measurement of the shape of the dark halo profile could have a tremendous impact on the determination of the origin of the 511 keV line and vice versa .