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We present a new investigation of the thermal history of the intergalacticmedium (IGM) during and after reionization using the Lyman-$\alpha$ forest fluxpower spectrum at $4.0\lesssim z\lesssim5.2$. Using a sample of 15high-resolution spectra, we measure the flux power down to the smallest scalesever probed at these redshifts ($-1\lesssim \log(k/$km$^{-1}$s)$\lesssim-0.7$). These scales are highly sensitive to both the instantaneous temperatureof the IGM and the total energy injected per unit mass during and afterreionization. We measure temperatures at the mean density of$T_{0}\sim7000$-8000 K, consistent with no significant temperature evolutionfor redshifts $4.2\lesssim z\lesssim5.0$. We also present the firstobservational constraints on the integrated IGM thermal history, finding thatthe total energy input per unit mass increases from $u_{0}\sim4.6$ ${\rm eV}$$m_{\rm p}^{-1}$ to 7.3 eV $m_{\rm p}^{-1}$ from $z\sim 6$ to $4.2$ assuming a$\Lambda$-CDM cosmology. We show how these results can be used simultaneouslyto obtain information on the timing and the sources of the reionizationprocess. Our first proof of concept using simplistic models of instantaneousreionization produces results comparable to and consistent with the recentPlanck constraints, favoring models with $z_{\rm rei}\sim 8.5^{+1.1}_{-0.8}$.

Revealing reionization with the thermal history of the intergalactic medium: new constraints from the Lyman-$\alpha$ flux power spectrum