Improving photometric redshifts with Ly$\alpha$ tomography
  Forming a three dimensional view of the Universe is a long-standing goal of
astronomical observations, and one that becomes increasingly difficult at high
redshift. In this paper we discuss how tomography of the intergalactic medium
(IGM) at $z\simeq 2.5$ can be used to estimate the redshifts of massive
galaxies in a large volume of the Universe based on spectra of galaxies in
their background. Our method is based on the fact that hierarchical structure
formation leads to a strong dependence of the halo density on large-scale
environment. A map of the latter can thus be used to refine our knowledge of
the redshifts of halos and the galaxies and AGN which they host. We show that
tomographic maps of the IGM at a resolution of $2.5\,h^{-1}$Mpc can determine
the redshifts of more than 90 per cent of massive galaxies with redshift
uncertainty $\Delta z/(1+z)=0.01$. Higher resolution maps allow such redshift
estimation for lower mass galaxies and halos.
