  Lyman-$\alpha$ (Ly$\alpha$) spectra provide insights into the small-scale
structure and kinematics of neutral hydrogen (HI) within galaxies as well as
the ionization state of the intergalactic medium (IGM). The former defines the
intrinsic spectrum of a galaxy, which is modified by the latter. These two
effects are degenerate. Using the IllustrisTNG100 simulation, we study the
impact of the IGM on Ly$\alpha$ spectral shapes between $z\sim 0$ and $5$. We
compute the distribution of the expected Ly$\alpha$ peaks and of the peak
asymmetry for different intrinsic spectra, redshifts and large-scale
environments. We find that the commonly used averaged transmission curves give
an incorrect perception of the observed spectral properties. We show that the
distributions of peak counts and asymmetry can lift the degeneracy between the
intrinsic spectrum and the IGM absorption. For example, we expect a significant
number of triple peaked Ly$\alpha$ spectra (up to 30% at $z\sim 3$) if the
galaxies' HI distribution become more porous at higher redshift as predicted by
cosmological simulations. We provide a public catalog of transmission curves to
be used in future simulations and observations to allow a more realistic IGM
treatment.

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics