  The density spike of dark matter (DM) in the subparsec region near the
supermassive black hole at the Galactic Center can provide potentially
observable gamma-ray signals coming from DM annihilations. Taking Fermi-$LAT$
data for the gamma-ray flux from the point source 3FGL J1745.6-2859c (Sgr A*),
we calculate the resulting constraints on generic models of DM, allowing for
the possibility of a non-negligible velocity-dependent component of the
annihilation cross section. We consider a variety of selections for the
astrophysical parameters that describe the spike profile and find that the
gamma-ray flux is strongly dependent on these selections, particularly the
modelling of spike depletion effects due to gravitational interactions with
baryons, which affect the spike radius and steepness profile. We calculate
constraints on the DM parameter space for both attenuated and idealized spikes,
considering different choices for the steepness profiles in each case. We find
that for the most conservative selection of parameters, corresponding to a
depleted spike with an NFW cusp profile, the gamma-ray flux for a 100 GeV
thermal relic is lower than current observational constraints by several orders
of magnitude. For more optimistic choices of parameters corresponding to spikes
that have not been attenuated, bounds on DM masses can be obtained for a
variety of choices of steepness profiles. We then specialize to a class of
simplified models of fermionic DM that annihilate dominantly through the
$t-$channel exchange of two scalar mediators with arbitrary mixing angle
$\alpha$, and calculate the indirect detection constraints coming from the DM
spike. Along the way, we discuss constraints on the astrophysical parameters
describing the DM spike, taking the Galactic Center excess as a signal of DM
annihilation.
