We present three-dimensional (3D) resonant radiative transfer simulations of the spatial and spectral diffusion of the Ly alpha radiation from a central source in the host galaxies of high column density absorption systems at z similar to 3. The radiative transfer simulations are based on a suite of cosmological galaxy formation simulations which reproduce a wide range of observed properties of damped Ly alpha absorption systems. The Lya emission is predicted to be spatially extended up to several arcsec, and the spectral width of the Ly alpha emission is broadened to several hundred (in some case more than thousand) km s(-1). The distribution and the dynamical state of the gas in the simulated galaxies are complex, the latter with significant contributions from rotation and both in-and out-flows. The emerging Ly alpha radiation extends to gas with column densities of N-HI similar to 10(18) cm(-2) and its spectral shape varies strongly with viewing angle. The strong dependence on the central HI column density and the HI velocity field suggests that the Ly alpha emission will also vary strongly with time on time-scales of a few dynamical times of the central region. Such variations with time should be especially pronounced at times where the host galaxy undergoes a major merger and/or starburst. Depending on the pre-dominance of in- or out-flow along a given sightline and the central column density, the spectra show prominent blue peaks, red peaks or double-peaked profiles. Both spatial distribution and spectral shape are very sensitive to details of the galactic wind implementation. Stronger galactic winds result in more spatially extended Ly alpha emission and - somewhat counterintuitively - a narrower spectral distribution.
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