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Christopher M. Hirata developed the HyRec calculation of cosmological recombination with Yacine Ali-Haïmoud. Their method couples atomic level populations, the free electron fraction and radiative transfer, including two-photon transitions and diffusion near the Lyman-alpha line.^[ref]

The scientific need is predictive accuracy: the free electron population sets the Thomson scattering history that shapes Cosmic microwave background anisotropy. Equilibrium ionization alone cannot capture the entire decoupling process.

Assume nondegenerate hydrogen and helium in chemical equilibrium with a black-body radiation field at temperature $T$. Neglect doubly ionized helium, excited-state populations and the small ion/atom mass difference in translational factors. DefineThe Saha ionization equation and ground-state statistical weights givewhere $I_H$ and $I_{He}$ are the first ionization energies. The factor four comes from the ground-state degeneracy of neutral helium: the electronic spin singlet state has weight one, whereas the singly ionized helium ground state and free electron each have weight two. For hydrogen, the corresponding degeneracy ratio is one when nuclear spin is counted consistently.^[ref]

Both equations share the same free-electron abundance. Combining them with the free-electron fraction during hydrogen--helium recombination closes the algebraic system. As cosmological recombination falls out of chemical equilibrium, a kinetic evolution replaces these instantaneous Saha relations.

Yacine Ali-Haïmoud, with Christopher M. Hirata, produced the HyRec model of primordial hydrogen and helium recombination. It combines an effective multilevel-atom treatment with detailed photon transport while exploiting the sparse coupling structure to make repeated cosmological calculations practical.^[ref]

This connects atomic transition rates to cosmological inference: a changed recombination history changes the probability that a cosmic microwave background photon scatters before reaching the observer.