Inferring the IGM thermal history during reionisation with the
  Lyman-$\alpha$ forest power spectrum at redshift $z \simeq 5$
  We use cosmological hydrodynamical simulations to assess the feasibility of
constraining the thermal history of the intergalactic medium during
reionisation with the Ly$\alpha$ forest at $z\simeq5$. The integrated thermal
history has a measureable impact on the transmitted flux power spectrum that
can be isolated from Doppler broadening at this redshift. We parameterise this
using the cumulative energy per proton, $u_0$, deposited into a gas parcel at
the mean background density, a quantity that is tightly linked with the gas
density power spectrum in the simulations. We construct mock observations of
the line of sight Ly$\alpha$ forest power spectrum and use a Markov Chain Monte
Carlo approach to recover $u_{0}$ at redshifts $5 \leq z \leq 12$. A
statistical uncertainty of $\sim 20$ per cent is expected (at 68 per cent
confidence) at $z\simeq 5$ using high resolution spectra with a total redshift
path length of $\Delta z=4$ and a typical signal-to-noise ratio of $\rm S/N=15$
per pixel. Estimates for the expected systematic uncertainties are comparable,
such that existing data should enable a measurement of $u_0$ to within $\sim
30$ per cent. This translates to distinguishing between reionisation scenarios
with similar instantaneous temperatures at $z\simeq 5$, but with an energy
deposited per proton that differs by $2$-$3\, \rm eV$ over the redshift
interval $5\leq z \leq 12$. For an initial temperature of $T\sim 10^{4}\rm\,K$
following reionisation, this corresponds to the difference between early
($z_{\rm re}=12$) and late ($z_{\rm re}=7$) reionisation in our models.
