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During reionization, neutral hydrogen in the intergalactic medium (IGM)imprints a damping wing absorption feature on the spectrum of high-redshiftquasars. A detection of this signature provides compelling evidence for asignificantly neutral Universe, and enables measurements of the hydrogenneutral fraction $x_{\rm HI}(z)$ at that epoch. Obtaining reliable quantitativeconstraints from this technique, however, is challenging due to stochasticityinduced by the patchy inside-out topology of reionization, degeneracies withquasar lifetime, and the unknown unabsorbed quasar spectrum close to rest-frameLy$\alpha$. We combine a large-volume semi-numerical simulation of reionizationtopology with 1D radiative transfer through high-resolution hydrodynamicalsimulations of the high-redshift Universe to construct models of quasartransmission spectra during reionization. Our state-of-the-art approachcaptures the distribution of damping wing strengths in biased quasar halos thatshould have reionized earlier, as well as the erosion of neutral gas in thequasar environment caused by its own ionizing radiation. Combining thisdetailed model with our new technique for predicting the quasar continuum andits associated uncertainty, we introduce a Bayesian statistical method tojointly constrain the neutral fraction of the Universe and the quasar lifetimefrom individual quasar spectra. We apply this methodology to the spectra of thetwo highest redshift quasars known, ULAS J1120+0641 and ULAS J1342+0928, andmeasured volume-averaged neutral fractions $\langle x_{\rm HI}\rangle(z=7.09)=0.48^{+0.26}_{-0.26}$ and $\langle x_{\rm HI}\rangle(z=7.54)=0.60^{+0.20}_{-0.23}$ (posterior medians and 68% credibleintervals) when marginalized over quasar lifetimes of $10^3 \leq t_{\rm q} \leq10^8$ years.

Quantitative Constraints on the Reionization History from the IGM Damping Wing Signature in Two Quasars at z > 7