The ionizing background at the end of reionization

One of the most sought-after signatures of reionization is a rapid increase in the ionizing background (usually measured through the Ly alpha optical depth towards distant quasars). Conventional wisdom associates this with the 'overlap' phase when ionized bubbles merge, allowing each source to affect a much larger volume. We argue that this picture fails to describe the transition to the post-overlap Universe, where Lyman-limit systems (LLSs) absorb ionizing photons over moderate length-scales (less than or similar to 20-100 Mpc). Using an analytic model, we compute the probability distribution of the amplitude of the ionizing background throughout reionization, including both discrete ionized bubbles and LLSs (parametrized by an attenuation length, which we impose rather than attempt to model self-consistently). We show that the overlap does not by itself cause a rapid increase in the ionizing background or a rapid decrease in the mean Ly alpha transmission towards distant quasars. More detailed seminumeric models support these conclusions. We argue that the rapid changes should instead be interpreted as evolution in the attenuation length itself, which may or may not be directly related to overlap.