(Nearly) Model-Independent Constraints on the Neutral Hydrogen Fraction in the Intergalactic Medium at $z\sim 5-7$ Using Dark Pixel Fractions in Ly$α$ and Ly$β$ Forests

Abstract: Cosmic reionization was the last major phase transition of hydrogen from neutral to highly ionized in the intergalactic medium (IGM). Current observations show that the IGM is significantly neutral at $z>7$, and largely ionized by $z\sim5.5$. However, most methods to measure the IGM neutral fraction are highly model-dependent, and are limited to when the volume-averaged neutral fraction of the IGM is either relatively low ($\bar{x}{\rm HI} \lesssim 10^{-3}$) or close to unity ($\bar{x}{\rm HI}\sim 1$). In particular, the neutral fraction evolution of the IGM at the critical redshift range of $z=6-7$ is poorly constrained. We present new constraints on $\bar{x}{\rm HI}$ at $z\sim5.1-6.8$, by analyzing deep optical spectra of $53$ quasars at $5.73<z<7.09$. We derive model-independent upper limits on the neutral hydrogen fraction based on the fraction of "dark" pixels identified in the Lyman $\alpha$ (Ly$\alpha$) and Lyman $\beta$ (Ly$\beta$) forests, without any assumptions on the IGM model or the intrinsic shape of the quasar continuum. They are the first model-independent constraints on the IGM neutral hydrogen fraction at $z\sim6.2-6.8$ using quasar absorption measurements. Our results give upper limits of $\bar{x}{\rm HI}(z=6.3) < 0.79\pm0.04$ (1$\sigma$), $\bar{x}{\rm HI} (z=6.5) < 0.87\pm0.03$ (1$\sigma$), and $\bar{x}{\rm HI} (z=6.7) < 0.94^{+0.06}_{-0.09}$ (1$\sigma$). The dark pixel fractions at $z>6.1$ are consistent with the redshift evolution of the neutral fraction of the IGM derived from the Planck 2018.