A New Method to Measure the Post-reionization Ionizing Background from the Joint Distribution of Lyα and Lyβ 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 that 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 alpha forest, at z greater than or similar to 6 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 similar to 50 Mpc/h regions is less sensitive and suboptimal. In addition, the five times smaller oscillator strength of the Ly beta transition implies that the Ly beta forest is considerably more transparent at z greater than or similar to 6, even in the presence of contamination by foreground z similar to 5 Ly alpha forest absorption. In this work we present a novel statistical approach to analyze the joint distribution of transmission spikes in the cospatial z similar to 6 Ly alpha and Ly beta forests. Our method relies on approximate Bayesian computation (ABC), which circumvents the necessity of computing the intractable likelihood function describing the highly correlated Ly alpha and Ly beta 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 <= z <= 6.4 along this sightline with similar to 0.2 dex statistical uncertainties.