Probing Large Scale Ionizing Background Fluctuation with Lyman $\alpha$ Forest and Galaxy Cross-correlation at z=2.4

The amplitude of the metagalactic ultraviolet background (UVB) at large-scales is impacted by two factors. First, it naturally attenuates at scales larger than mean-free-path of UVB photons due to the absorption by neutral intergalactic medium. Second, there are discrete and rare ionizing sources distributing in the Universe, emitting the UVB photons, and thus enhancing the local UVB amplitude. Therefore, for cosmological probe that is sensitive to the UVB amplitude and capable of detecting the large scale like Lyman-$\alpha$ forest spectrum, the fluctuation due to the clustering of ionizing sources becomes a significant factor for Lyman-$\alpha$ flux transmission and leave imprints on Lyman-$\alpha$ flux power spectrum at these large scales. In this work, we make use of a radiative transfer model that parametrizes the UVB source distribution by its bias $b_{\rm j}$ and shot noise $\overline{n}_{\rm j}$. We estimate the constraints on this model through the cross-correlation between Lyman-$\alpha$ forest survey and galaxy survey, using the DESI Lyman-$\alpha$ forest survey and the Roman Space Telescope emission line galaxy survey as an example. We show the detection sensitivity improvement for UVB parameters from disjoint to maximal overlap of DESI+Roman survey strategy. We also show that the degeneracy of two ionizing source parameters can be broken by increasing the overlapping survey area. Our results motivate survey strategies more dedicated to probe the UVB large-scale fluctuations.