The spatial distribution of dark-matter-annihilation originated gamma-ray line signal

The GeV$-$TeV $\gamma-$ray line signal is the smoking gun signature of the dark matter annihilation or decay. The detection of such a signal is one of the main targets of some space-based telescopes, including Fermi-LAT and the upcoming CALET, DAMPE and Gamma-400. An important feature of the dark-matter-annihilation originated $\gamma-$ray line photons is their concentration at the center of the Galaxy. So far no reliable $\gamma-$ray line has been detected by Fermi-LAT and the upper limits on the cross section of annihilation into $\gamma-$rays have been reported. We use these upper limits to estimate the "maximal" number of $\gamma-$ray line photons detectable for Fermi-LAT, DAMPE and Gamma-400 and then investigate the spatial distribution of these photons. We show that usually the center of the distribution region will be offset from the Galactic centre (Sgr A$^{\star}$) due to the limited statistics. Such a result is almost independent of the dark matter distribution models and renders the reconstruction of the dark matter distribution with the $\gamma-$ray line signal very challenging for the foreseeable space-based detectors.