The inhomogeneous ionizing background following reionization

We study the spatial fluctuations in the hydrogen-ionizing background in the epoch following reionization (z similar to 5-6). The rapid decrease with redshift in the photon mean free path (mfp), combined with the clustering of increasingly rare ionizing sources, can result in a very inhomogenous ionizing background during this epoch. We systematically investigate the probability density functions (PDFs) and power spectra of ionizing flux, by varying several parameters such as the mfp, minimum halo mass capable of hosting stars and halo duty cycle. In order to be versatile, we make use of analytic, seminumeric and numeric approaches. Our models show that the ionizing background indeed has sizable fluctuations during this epoch, with the PDFs being a factor of few wide at half of the maximum likelihood. The clustering of sources dominates the width of the PDFs, so analytic models must take large-scale clustering into account. The distributions also show marked asymmetries, with a high-value tail set by clustering on small scales, and a shorter low-value tail which is set by the mfp. The power spectrum of the ionizing background is much more sensitive to source properties than the PDF and can be well understood analytically with a framework similar to the halo model (usually used to describe dark matter clustering). Nevertheless, we find that Ly alpha forest spectra are extremely insensitive to the details of the ultraviolet background, despite marked differences in the PDFs and power spectra of our various ionizing backgrounds. Assuming a uniform ionizing background only underestimates the value of the mean ionization rate, <>, inferred from the Ly alpha forest by a few per cent. Instead, analysis of the Ly alpha forest is dominated by the uncertainties in the density field. Thus, our results justify the common assumption of a uniform ionizing background in Ly alpha forest analysis even during this epoch.