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We use cosmological hydrodynamical simulations to assess the feasibility ofconstraining the thermal history of the intergalactic medium duringreionisation with the Ly$\alpha$ forest at $z\simeq5$. The integrated thermalhistory has a measureable impact on the transmitted flux power spectrum thatcan be isolated from Doppler broadening at this redshift. We parameterise thisusing the cumulative energy per proton, $u_0$, deposited into a gas parcel atthe mean background density, a quantity that is tightly linked with the gasdensity power spectrum in the simulations. We construct mock observations ofthe line of sight Ly$\alpha$ forest power spectrum and use a Markov Chain MonteCarlo approach to recover $u_{0}$ at redshifts $5 \leq z \leq 12$. Astatistical uncertainty of $\sim 20$ per cent is expected (at 68 per centconfidence) at $z\simeq 5$ using high resolution spectra with a total redshiftpath 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 $\sim30$ per cent. This translates to distinguishing between reionisation scenarioswith similar instantaneous temperatures at $z\simeq 5$, but with an energydeposited per proton that differs by $2$-$3\, \rm eV$ over the redshiftinterval $5\leq z \leq 12$. 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.

Inferring the IGM thermal history during reionisation with the Lyman-$\alpha$ forest power spectrum at redshift $z \simeq 5$