Measuring the photo-ionization rate, neutral fraction and mean free path of HI ionizing photons at $4.9 \leq z \leq 6.0$ from a large sample of XShooter and ESI spectra

We measure the mean free path ($\lambda_{\rm mfp,HI}$), photo-ionization rate ($\langle \Gamma_{\rm HI} \rangle$) and neutral fraction ($\langle f_{\rm HI} \rangle$) of hydrogen in 12 redshift bins at $4.85<z<6.05$ from a large sample of moderate resolution XShooter and ESI QSO absorption spectra. The fluctuations in ionizing radiation field are modeled by post-processing simulations from the Sherwood suite using our new code "EXtended reionization based on the Code for Ionization and Temperature Evolution" (EX-CITE). EX-CITE uses efficient Octree summation for computing intergalactic medium attenuation and can generate large number of high resolution $\Gamma_{\rm HI}$ fluctuation models. Our simulation with EX-CITE shows remarkable agreement with simulations performed with the radiative transfer code Aton and can recover the simulated parameters within $1\sigma$ uncertainty. We measure the three parameters by forward-modeling the Ly$\alpha$ forest and comparing the effective optical depth ($\tau_{\rm eff, HI}$) distribution in simulations and observations. The final uncertainties in our measured parameters account for the uncertainties due to thermal parameters, modeling parameters, observational systematics and cosmic variance. Our best fit parameters show significant evolution with redshift such that $\lambda_{\rm mfp,HI}$ and $\langle f_{\rm HI} \rangle$ decreases and increases by a factor $\sim 6$ and $\sim 10^{4}$, respectively from $z \sim 5$ to $z \sim 6$. By comparing our $\lambda_{\rm mfp,HI}$, $\langle \Gamma_{\rm HI} \rangle$ and $\langle f_{\rm HI} \rangle$ evolution with that in state-of-the-art Aton radiative transfer simulations and the Thesan and CoDa-III simulations, we find that our best fit parameter evolution is consistent with a model in which reionization completes by $z \sim 5.2$.