  The bulk of the extragalactic background between 10 keV and 10 GeV is likely
to be explained by the emission of Seyfert galaxies, type Ia supernovae, and
blazars. However, as revealed by the INTEGRAL satellite, the bulge of our
galaxy is an intense source of a 511 keV gamma-ray line, indicating the
production of a large number of positrons that annihilate. The origin of the
latter is debated, and they could be produced, in particular, by the (S- or
P-wave) annihilation of light Dark Matter particles into e+e-. In any case, the
cumulated effect of similar sources at all redshifts could lead to a new
background of hard X-ray and soft gamma-ray photons. On the basis of the
hierarchical model of galaxy formation, we compute analytically the SNIa
contribution to the background, and add it to Seyfert and blazars emission
models. We find that any extra contribution to this unresolved background at
511 keV should be lower than about 4 keV/cm^2/s/sr. We also estimate
analytically the extragalactic background due to Dark Matter annihilation,
increasing the accuracy of the earlier computations. Indeed, we take into
account the large positron escape fraction from low mass dark matter halos,
unable to confine a dense and magnetized interstellar medium. Our new
background estimate turns out to be one order of magnitude lower, so that the
hypothesis of a light Dark Matter candidate remains compatible with the
observed extragalactic background for a wider range of particle masses and
cross-sections.
