Researchers computed atomic reaction rates from first principles for a wide class of “atomic dark matter” models and used those results to update cosmological constraints based on the cosmic microwave background (CMB) and other data. Atomic dark matter is a simple idea: two new kinds of particles, a dark “electron” and a dark “proton,” carry opposite charge under a hidden U(1) gauge force and can bind into neutral atoms that interact with a bath of massless dark photons. The way these dark atoms form and radiate affects small-scale structure and leaves fingerprints in the CMB.

The team studied the minimal atomic dark matter model in which the dark electron and dark proton have masses m_eD and m_pD and interact with strength set by a dark fine‑structure constant α_D. They allowed the dark sector to have a different temperature than the visible CMB and to make up only a fraction f_D of the total dark matter. The hidden photon is taken to be massless and to have no direct mixing with ordinary light, so the dark sector communicates with our sector only through gravity.

To get reliable rates they solved the two‑body Schrödinger equation for the dark electron–dark proton system. From the resulting bound and scattering wavefunctions they computed radiative transition cross sections and recombination coefficients across mass ratios ranging from the hydrogen‑like limit (m_pD ≫ m_eD) to the positronium‑like case (m_eD = m_pD), and for couplings up to α_D ≤ 0.3. They also evaluated other processes important for cosmology, such as Thomson scattering and Bremsstrahlung, and compared their first‑principles numbers to the common practice of rescaling known atomic rates from ordinary hydrogen.

Their direct calculations show that the simple rescaling of Standard Model (ordinary hydrogen) rates is a good approximation in many cases. For the cosmologically relevant “case‑B” recombination channel—where electrons first go to an excited state before reaching the ground state—the rescaled rates differ from the first‑principles results by less than about ten percent across the explored parameter space. Larger differences appear for bound‑to‑ground (direct 1s) transitions at low temperatures, sometimes reaching order‑unity deviations, but those particular transitions do not strongly affect case‑B recombination.