  Understanding the nature of the Dark Matter (DM) is one of the current
challenges in modern astrophysics and cosmology. Knowing the properties of the
DM particle would shed light on physics beyond the Standard Model and even
provide us with details of the early Universe. In fact, the detection of such a
relic would bring us information from the pre-Big Bang Nucleosynthesis (BBN)
period, an epoch from which we have no data, and could even hint at
inflationary physics. In this work, we assume that the expansion rate of the
Universe after inflationary is governed by the kinetic energy of a scalar field
$\phi$, in the so-called "kination" model. We assume that the $\phi$ field
decays into both radiation and DM particles, which we take to be Weakly
Interacting Massive Particles (WIMPs). The present abundance of WIMPs is then
fixed during the kination period through either a thermal "freeze-out" or
"freeze-in" mechanism, or through a non-thermal process governed by the decay
of $\phi$. We explore the parameter space of this theory with the requirement
that the present WIMP abundance provides the correct DM relic budget. Requiring
that BBN occurs during the standard cosmological scenario sets a limit on the
temperature at which the kination period ends. Using this limit and assuming
the WIMP has a mass $m_\chi = 100\,$GeV, we obtain that the thermally-averaged
WIMP annihilation cross section has to satisfy the constraints $4 \times
10^{-16}{\rm \,GeV^{-2}} \lesssim \langle\sigma v\rangle \lesssim 2\times
10^{-5} {\rm \,GeV^{-2}}$ in order for having at least one of the production
mechanism to yield the observed amount of DM. This result shows how the
properties of the WIMP particle, if ever measured, can yield information on the
pre-BBN content of the Universe.
