The gamma-ray flux produced by Dark Matter (DM) annihilation is expected to be maximized in the inner regions of the Milky Way. Using the total fluxes (background+signal) measured by the Fermi Large Area Telescope (Fermi-LAT) in regions around the Galactic Center (GC), we derive constraints on the parameter space of generic DM candidates. These constraints are obtained with a conservative analysis, requiring simply that the DM-induced gamma-ray emission does not overshoot the total flux measured by the Fermi-LAT in an optimized region around the GC.

To quantify the uncertainty in the DM density profile we use several well motivated models whose parameters have been constrained from observational data of the Milky Way. In particular, using first NFW and Einasto DM profiles we find limits on the annihilation cross section of a generic DM particle that are comparable to the ones previously reported by the Fermi-LAT collaboration after a similar analysis of the Galactic halo.

When adiabatic contraction of the DM due to the infall of baryons to the GC is taken into account, the limits improve by almost three orders of magnitude, going below the thermal cross section for all annihilation channels (except $mumu$) when the DM mass is smaller than about 1 TeV. For the $mumu$ channel the thermal cross section can be constrained for DM masses below 200 GeV. Taking into account the effects of inverse Compton scattering in the modeling of the DM annihilation signal, we derive significantly stronger constraints for DM masses heavier than 100 GeV, subject to the assumed DM density profile, galactic cosmic-ray diffusion model, and strength of the inner galactic magnetic field.