We show the existence of a statistically significant, robust detection of a gamma-ray source in the Milky Way Galactic Center that is consistent with a spatially extended signal using about 4 years of Fermi-LAT data. The gamma-ray flux is consistent with annihilation of dark matter particles with a thermal annihilation cross-section if the spatial distribution of dark matter particles is similar to the predictions of dark matter only simulations. We find statistically significant detections of an extended source with gamma-ray spectrum that is consistent with dark matter particle masses of approximately 10 GeV to 1 TeV annihilating to b/b-bar quarks, and masses approximately 10 GeV to 30 GeV annihilating to tau+ tau- leptons. However, a part of the allowed region in this interpretation is in conflict with constraints from Fermi observations of the Milky Way satellites. The biggest improvement over the fit including just the point sources is obtained for a 30 GeV dark matter particle annihilating to b/b-bar quarks. The gamma-ray intensity and spectrum are also well fit with emission from a millisecond pulsar (MSP) population following a density profile like that of low-mass X-ray binaries observed in M31. The greatest goodness-of-fit of the extended emission is with spectra consistent with known astrophysical sources like MSPs in globular clusters or cosmic ray bremsstrahlung on molecular gas. Therefore, we conclude that the bulk of the emission is likely from an unresolved or spatially extended astrophysical source. However, the interesting possibility of all or part of the extended emission being from dark matter annihilation cannot be excluded at present.
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