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 Lyα forest, at z≳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 ∼50 Mpc/h regions is less
sensitive and sub-optimal. Additionally, the five times smaller oscillator
strength of the Lyβ transition implies the Lyβ forest is
considerably more transparent at z≳6, even in the presence of
contamination by foreground z∼5 Lyα forest absorption. In this
work we present a novel statistical approach to analyze the joint distribution
of transmission spikes in the co-spatial z∼6 Lyα and Lyβ
forests. Our method relies on Approximate Bayesian Computation (ABC), which
circumvents the necessity of computing the intractable likelihood function
describing the highly correlated Lyα and Lyβ 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
∼0.2 dex statistical uncertainties.Comment: 15 pages, 18 figures, ApJ submitte
