  Ultracompact minihalos would be formed if there are larger density
perturbations ($0.0003 < \delta\rho/\rho < 0.3$) in the earlier epoch. The
density profile of them is steeper than the standard dark matter halos. If the
dark matter can annihilate or decay into the standard particles, e.g., photons,
these objects would be the potential astrophysical sources. In order to be
consistent with the observations, such as $Fermi$, the abundance of
ultracompact minihalos must be constrained. On the other hand, the formation of
these objects has very tight relation with the primordial curvature
perturbations on smaller scale, so the fraction of ultracompact minihalos is
very important for modern cosmology. In previous works, the studies are focused
on the dark matter annihilation for these objects. But if the dark matter is
not annihilated, the dark matter decay is another important possible case. On
the other hand, the abundance of ultracompact minihalos is related to many
other parameters, such as the mass of dark matter, the decay channels and the
density profile of dark matter halo. One of the important aspects of this work
is that we investigate the $\gamma$-ray signals from nearby ultracompact
minihalos due to dark matter decay and another important aspect is to study in
detail how the different decay channels and density profiles affect the
constraints on the abundance of ultracompact minihalos.
