  Recent observations indicating the existence of a monochromatic gamma-ray
line with energy ~130 GeV in the Fermi-LAT data have attracted great interest
due to the possibility that the line feature stems from the annihilation of
dark matter particles. Many studies examining the robustness of the putative
line-signal have concentrated on its spectral attributes. Here, we study the
morphological features of the gamma-ray line photons, which can be used to
differentiate a putative dark matter signal from astrophysical backgrounds or
instrumental artifacts. Photons stemming from dark matter annihilation will
produce events tracing a specific morphology, with a statistical clustering
that can be calculated based on models of the dark matter density profile in
the inner Galaxy. We apply the DBSCAN clustering algorithm to Fermi gamma-ray
data, and show that we can rule out the possibility that 1 (2, 4) or fewer
point-like sources produce the observed morphology for the line photons at a
99% (95%, 90%) confidence level. Our study strongly disfavors the main
astrophysical background envisioned to produce a line feature at energies above
100 GeV: cold pulsar winds. It is highly unlikely that 4 or more such objects
have exactly the same monochromatic cosmic-ray energy needed to produce a
gamma-ray line, to within instrumental energy resolution. Furthermore, we show
that the larger photon statistics expected with Air Cherenkov Telescopes such
as H.E.S.S.-II will allow for extraordinarily stringent morphological tests of
the origin of the "line photons".
