A New Method to Measure the Post-Reionization Ionizing Background from the Joint Distribution of Lyman-$α$ and Lyman-$β$ Forest Transmission
The amplitude of the ionizing background that pervades the intergalactic medium (IGM) at the end of the epoch of reionization provides a valuable constraint on the emissivity of the sources which reionized the Universe. While measurements of the ionizing background at lower redshifts rely on a simulation-calibrated mapping between the photoionization rate and the mean transmission of the Ly$α$ forest, at $z\gtrsim6$ the IGM becomes increasingly opaque, and transmission arises solely in narrow spikes separated by saturated Gunn-Peterson troughs. In this regime, the traditional approach of measuring the average transmission over large $\sim 50$ Mpc$/h$ regions is less sensitive and sub-optimal. Additionally, the five times smaller oscillator strength of the Ly$β$ transition implies the Ly$β$ forest is considerably more transparent at $z\gtrsim6$, even in the presence of contamination by foreground $z\sim 5$ Ly$α$ forest absorption. In this work we present a novel statistical approach to analyze the joint distribution of transmission spikes in the co-spatial $z\sim 6$ Ly$α$ and Ly$β$ forests. Our method relies on Approximate Bayesian Computation (ABC), which circumvents the necessity of computing the intractable likelihood function describing the highly correlated Ly$α$ and Ly$β$ transmission. We apply ABC to mock data generated from a large-volume hydrodynamical simulation combined with a state-of-the-art model of ionizing background fluctuations in the post-reionization IGM, and show that it is sensitive to higher IGM neutral hydrogen fractions than previous techniques. As a proof of concept, we apply this methodology to a real spectrum of a $z=6.54$ quasar and measure the ionizing background from $5.4\leq z \leq 6.4$ along this sightline with $\sim0.2$ dex statistical uncertainties.