  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.
