Possible evidence for a large-scale enhancement in the Lyman-$\alpha$ forest power spectrum at redshift $\mathbf{\textit{z}\geq 4}$

Inhomogeneous reionization enhances the 1D Lyman-$\alpha$ forest power spectrum on large scales at redshifts $z\geq4$. This is due to coherent fluctuations in the ionized hydrogen fraction that arise from large-scale variations in the post-reionization gas temperature, which fade as the gas cools. It is therefore possible to use these relic fluctuations to constrain inhomogeneous reionization with the power spectrum at wavenumbers $\log_{10}(k/{\rm km^{-1}\,s})\lesssim -1.5$. We use the Sherwood-Relics suite of hybrid radiation hydrodynamical simulations to perform a first analysis of new Lyman-$\alpha$ forest power spectrum measurements at $4.0\leq z \leq 4.6$. These data extend to wavenumbers $\log_{10}(k/{\rm km^{-1}\,s})\simeq -3$, with a relative uncertainty of $10$--$20$ per cent in each wavenumber bin. Our analysis returns a $2.7\sigma$ preference for an enhancement in the Lyman-$\alpha$ forest power spectrum at large scales, in excess of that expected for a spatially uniform ultraviolet background. This large-scale enhancement could be a signature of inhomogeneous reionization, although the statistical precision of these data is not yet sufficient for obtaining a robust detection of the relic post-reionization fluctuations. We show that future power spectrum measurements with relative uncertainties of $\lesssim 2.5$ per cent should provide unambiguous evidence for an enhancement in the power spectrum on large scales.