Quantum mechanics at high redshift -- Modelling Damped Lyman-$\alpha$ absorption systems

For around 100 years, hydrogen spectral modelling has been based on Voigt profile fitting. The semi-classical Voigt profile is based on a 2-level atom approximation. Whilst the Voigt profile is excellent for many circumstances, the accuracy is insufficient for very high column density damped Lyman-$\alpha$ absorption systems. We have adapted the quantum-mechanical Kramers-Heisenberg model to include thermal broadening, producing a new profile, the KHT profile. Interactions involving multiple discrete atomic levels and continuum terms, not accounted for in the Voigt model, generate asymmetries in the Lyman line wings. If not modelled, this can lead to significant systematics in parameter estimation when modelling real data. There are important ramifications in particular for measurements of the primordial deuterium abundance. However, the KHT model is complicated. We therefore present a simplified formulation based on Taylor series expansions and look-up tables, quantifying the impact of the approximations made. The KHT profile has been implemented within the widely-used VPFIT code.