Studying the Lyman-$\alpha$ optical depth fluctuations at $z \sim 5.5$ using fast semi-numerical methods

We present a computationally efficient and fast semi-numerical technique for simulating the Lyman-$\alpha$ (Ly$\alpha$) absorption optical depth in presence of neutral hydrogen "islands" left over from reionization at redshifts $5 \lesssim z \lesssim 6$. The main inputs to the analysis are (i) a semi-numerical photon-conserving model of ionized regions during reionization (named SCRIPT) along with a prescription for simulating the shadowing by neutral islands and (ii) the fluctuating Gunn-Peterson approximation to model the Ly$\alpha$ absorption. Our model is then used for simulating the large-scale fluctuations in the effective optical depth as observed along sight lines towards high$-z$ quasars. Our model is fully described by five parameters. By setting two of them to default values and varying the other three, we obtain the constraints on reionization history at $5 \lesssim z \lesssim 6$ as allowed by the data. We confirm that reionization is \emph{not} complete before $z \sim 5.6$ at $\gtrsim 2\sigma$ confidence, with the exact confidence limits depending on how the non-detections of the flux in the data are treated. We also confirm that the completion of reionization can be as late as $z \sim 5.2$. With further improvements in the model and with more sight lines at $z \sim 6$, we can take advantage of the computational efficiency of our analysis to obtain more stringent constraints on the ionization fraction at the tail-end of reionization.