Probing the Cosmic Reionization History with JWST: Gunn-Peterson and Ly$\alpha$ Damping Wing Absorption at $4.5 < z < 13$

We present a statistical analysis of Ly$\alpha$ absorption using 581 galaxies at $z=4.5$-13 observed with multiple JWST/NIRSpec spectroscopy programs, including JADES, UNCOVER, CEERS, and GO/DDT. We carefully construct composite spectra binned by redshift with homogeneous UV properties (UV magnitudes, UV slopes, and Ly$\alpha$ equivalent widths) and identify significant Ly$\alpha$ forest signals in galaxies at $z\sim5$-6, which diminish toward higher redshifts. We also find UV continuum breaks at rest-frame 1216\AA\ that soften beyond $z\gtrsim6$, confirming the effects of cosmic reionization through a self-consistent transition from Gunn-Peterson to Ly$\alpha$ damping wing absorption in galaxies. Fair comparisons of composite spectra with matched UV magnitudes and slopes across redshift reveal that UV-faint galaxies clearly show stronger Ly$\alpha$ absorption than UV-bright galaxies towards high redshift, providing insights into the topological evolution of reionization. We estimate Ly$\alpha$ transmission at the Gunn-Peterson trough and Ly$\alpha$ damping wing absorption by comparing the galaxy spectra to low-$z$ ($z\sim2$-5) galaxy templates that include galactic and circumgalactic absorption and Ly$\alpha$ emission. Using these measurements together with reionization simulations, we derive volume average neutral hydrogen fractions of $\langle x_{\rm HI} \rangle$ = ${0.00}^{+0.12}_{-0.00}$, ${0.25}^{+0.10}_{-0.20}$, ${0.65}^{+0.27}_{-0.35}$, ${1.00}^{+0.00}_{-0.20}$, and ${1.00}^{+0.00}_{-0.40}$ at $z\sim5$, 6, 7, 9, and 10, respectively. These values align with a reionization history characterized by a rapid transition around $z\sim7$-8, consistent with Ly$\alpha$ emitter observations. While the physical driver of this rapid reionization remains unclear, it may involve the emergence of hidden AGN populations and/or the onset of Lyman-continuum escape from galaxies.