Large-scale Fluctuations in the Cosmic Ionizing Background: the Impact of Beamed Source Emission

When modelling the ionisation of gas in the intergalactic medium after reionisation, it is standard practice to assume a uniform radiation background. This assumption is not always appropriate; models with radiative transfer show that large-scale ionisation rate fluctuations can have an observable impact on statistics of the Lyman-alpha forest. We extend such calculations to include beaming of sources, which has previously been neglected but which is expected to be important if quasars dominate the ionising photon budget. Beaming has two effects: first, the physical number density of ionising sources is enhanced relative to that directly observed; and second, the radiative transfer itself is altered. We calculate both effects in a hard-edged beaming model where each source has a random orientation, using an equilibrium Boltzmann hierarchy in terms of spherical harmonics. By studying the statistical properties of the resulting ionisation rate and HI density fields at redshift $z\sim 2.3$, we find that the two effects partially cancel each other; combined, they constitute a maximum $5\%$ correction to the power spectrum $P_{\mathrm{HI}}(k)$ at $k=0.04 \, h/\mathrm{Mpc}$. On very large scales ($k<0.01\, h/\mathrm{Mpc}$) the source density renormalisation dominates; it can reduce, by an order of magnitude, the contribution of ionising shot-noise to the intergalactic HI power spectrum. The effects of beaming should be considered when interpreting future observational datasets.