  It is important to understand the implications of current observational
constraints and potential signatures on the thermal history of dark matter. In
this paper, we build the connection between the present-day velocities and the
production mechanism of dark matter and find that the current observation on
structure formation can be imposed to constrain the decoupling temperatures and
the phase-space distribution of dark matter. We further explore the potential
of distinguishing different possible thermal histories of dark matter with
hypothetical future observational data. Using the freeze-in/-out scenarios as
templates, we find that future precision data may uniquely identify the allowed
parameter spaces for freeze-in and freeze-out, or even completely rule out one
of the scenarios. This method can be more generally applied to other scenarios.
