Measuring the temperature and profiles of Lyman-$\alpha$ absorbers.

The distribution of the absorption line broadening observed in the Ly$\alpha$ forest carries information about the temperature and widths of the filaments in the intergalactic medium (IGM). In this work, we present and test a new method for inferring the temperature of the IGM, the size of the absorbing filaments and the photo-ionization rate of hydrogen from the properties of absorption line broadening in the Ly$\alpha$ forest. We apply our method to mock spectra from the reference model of the EAGLE cosmological simulation, and we demonstrate that we are able to reconstruct the IGM properties. Our method explicitly takes into account the degeneracy between IGM temperature, the photo-ionization rate of hydrogen and the widths of the intergalactic filaments.