Comment on "Kinetic decoupling of WIMPs: Analytic expressions"
  Visinelli and Gondolo (2015, hereafter VG15) derived analytic expressions for
the evolution of the dark matter temperature in a generic cosmological model.
They then calculated the dark matter kinetic decoupling temperature
$T_{\mathrm{kd}}$ and compared their results to the Gelmini and Gondolo (2008,
hereafter GG08) calculation of $T_{\mathrm{kd}}$ in an early matter-dominated
era (EMDE), which occurs when the Universe is dominated by either a decaying
oscillating scalar field or a semistable massive particle before Big Bang
nucleosynthesis. VG15 found that dark matter decouples at a lower temperature
in an EMDE than it would in a radiation-dominated era, while GG08 found that
dark matter decouples at a higher temperature in an EMDE than it would in a
radiation-dominated era. VG15 attributed this discrepancy to the presence of a
matching constant that ensures that the dark matter temperature is continuous
during the transition from the EMDE to the subsequent radiation-dominated era
and concluded that the GG08 result is incorrect. We show that the disparity is
due to the fact that VG15 compared $T_\mathrm{kd}$ in an EMDE to the decoupling
temperature in a radiation-dominated universe that would result in the same
dark matter temperature at late times. Since decoupling during an EMDE leaves
the dark matter colder than it would be if it decoupled during radiation
domination, this temperature is much higher than $T_\mathrm{kd}$ in a standard
thermal history, which is indeed lower than $T_{\mathrm{kd}}$ in an EMDE, as
stated by GG08.
