  Following a dynamical encounter with Sgr A*, binaries in the Galactic Centre
(GC) can be tidally separated and one member star ejected as a hyper-velocity
star (HVS) with a velocity beyond the escape speed of the Milky Way. As GC-born
objects located in more observationally accessible regions of the sky, HVSs
offer insight into the stellar population in the inner parsecs of the Milky
Way. We perform a suite of simulations ejecting stars from the GC, exploring
how detectable HVS populations depend on assumptions concerning the GC stellar
population, focusing on HVSs which would appear in current and/or future data
releases from the \textit{Gaia} space mission with precise astrometry and
measured radial velocities. We show that predictions are sensitive to two
parameters in particular: the shape of the stellar initial mass function (IMF)
in the GC and the ejection rate of HVSs. The absence of confident HVS
candidates in \textit{Gaia} Data Release 2 excludes scenarios in which the HVS
ejection rate is $\gtrsim3\times10^{-2} \, \mathrm{yr^{-1}}$. Stricter
constraints will be placed on these parameters when more HVS candidates are
unearthed in future \textit{Gaia} data releases -- assuming recent
determinations of the GC IMF shape, one confident HVS \textit{at minimum} is
expected in \textit{Gaia} DR3 and DR4 as long as the HVS ejection rate is
greater than $\sim 10^{-3} \, \mathrm{yr^{-1}}$ and $\sim10^{-5} \,
\mathrm{yr^{-1}}$, respectively.
