View FImP Miracle (by Scale Invariance) $\acute{\rm a}$ $\rm la$
  Self-interaction
  Combining feebly interacting massive particle (FIMP) dark matter (DM) with
scale invariance (SI) leads to extremely light FIMP (thus the FImP) with FImP
miracle, i.e., the mass and relic generations of FImP DM share the same
dynamics. In this paper we show that due to the lightness of FImP, it,
especially for a scalar FImP, can easily accommodate large DM self-interaction.
For a fermionic FImP, such as the sterile neutrino, self-interaction
additionally requires a mediator which is another FImP, a scalar boson with
mass either much lighter or heavier than the FImP DM. DM self-interaction opens
a new window to observe FImP (miracle), which does not leave traces in the
conventional DM searches. As an example, FImP can account for the offsets
between the centroid of DM halo and stars of galaxies recently observed in the
galaxy cluster Abel 3827.
