Can past gamma-ray bursts explain both INTEGRAL and ATIC/PAMELA/Fermi anomalies simultaneously?

Gamma-ray bursts (GRBs) have been invoked to explain both the 511 keV emission from the galactic bulge and the high-energy positron excess inferred from the ATIC, PAMELA, and Fermi data. While independent explanations can be responsible for these phenomena, we explore the possibility of their common GRB-related origin by modeling the GRB distribution and estimating the rates. For an expected Milky Way long GRB rate, neither of the two signals is generic; the local excess requires a 2% coincidence, while the signal from the galactic center requires a 20% coincidence with respect to the timing of the latest GRB. The simultaneous explanation requires a 0.4% coincidence. Considering the large number of statistical "trials" created by multiple searches for new physics, the coincidences of a few per cent cannot be dismissed as unlikely. Alternatively, both phenomena can be explained by GRBs if the galactic rate is higher than expected. We also show that a similar result is difficult to obtain assuming a simplified short GRB distribution.