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

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.