Pushing the Limits of Atomic Dark Matter: First-Principles Recombination Rates and Cosmological Constraints

Minimal atomic dark matter with its distinctive cooling mechanisms offers an instructive framework for understanding the potential impact of dark matter on small-scale structure formation and early cosmology. The model consists of two fermions with opposite charges under a hidden Abelian gauge symmetry $U(1){D}$ and masses $m{p_{D}}$ and $m_{e_{D}}$, respectively. Analogous to hydrogen in the Standard Model, these fermions interact via their own electromagnetic-like force, with a dark fine structure constant denoted by $α_{D}$, and can bind into neutral atomic (and molecular) dark states. Previous work has largely focused on the benchmark scenario where the dark sector mirrors ordinary matter, with $m_{e_{D}}$ near the electron mass, $m_{p_{D}}$ near the proton mass, and $α_{D}\sim 1/137$. We extend this analysis by investigating dark recombination and cooling physics across the full parameter space of masses and couplings. Combining Cosmic Microwave Background (CMB) measurements from Planck and ACT with BAO and Pantheon+ data, we place new constraints on the atomic dark matter parameter space, identifying regions where acoustic damping and recombination dynamics leave observable imprints on the CMB.

💡 Research Summary

This paper presents a comprehensive study of atomic dark matter (aDM), a minimal model in which a dark electron and a dark proton, charged under a hidden massless U(1)_D gauge field, can form neutral bound states analogous to hydrogen. Unlike previous work that largely assumed the dark sector mirrors the Standard Model (SM) – i.e., a dark electron mass near the SM electron mass, a dark proton mass near the SM proton mass, and a dark fine‑structure constant α_D ≈ 1/137 – the authors explore the full viable parameter space: arbitrary mass ratios μ ≡ m_eD/m_pD (from the hydrogen‑like limit μ ≪ 1 to the positronium‑like limit μ → 1) and couplings up to α_D ≲ 0.3.

The core technical achievement is a first‑principles calculation of the radiative processes that govern dark recombination. The authors solve the two‑body Schrödinger equation for the dark electron–dark proton system, obtaining normalized bound‑state wavefunctions and continuum wavefunctions for a range of quantum numbers (n, ℓ). Using these wavefunctions they evaluate the dipole matrix elements ⟨ψ_f| r |ψ_i⟩, from which they derive bound‑bound transition rates R_{nl→n′ℓ′} and bound‑free recombination coefficients α_{nl}. The calculations are performed for many values of μ and α_D, allowing a direct comparison with the commonly used SM scaling relations, which assume m_pD ≫ m_eD and α_D ≪ 1.

The comparison reveals two key patterns. First, for the case‑B recombination network (where electrons first recombine into n = 2 states before cascading to the ground state), the first‑principles rates differ from the SM‑scaled rates by at most ~10 %. This level of discrepancy is well below the sensitivity of current cosmological observables. Second, direct transitions to the 1s ground state from higher excited levels (n > 2) can deviate by O(1) when α_D is large or μ approaches unity, because the overlap integrals change dramatically. However, these transitions play a negligible role in case‑B recombination, so the overall recombination history remains essentially unchanged. Consequently, the authors conclude that SM‑scaled rates are sufficiently accurate for cosmological analyses across most of the explored parameter space.

Armed with this validation, the authors incorporate aDM into a modified version of the CLASS Boltzmann code. They model Thomson scattering between dark photons and free dark particles, free‑free Bremsstrahlung, and, crucially, dark acoustic oscillations (DAOs) arising from the coupling of the dark photon fluid to the dark baryon fluid. DAOs suppress small‑scale matter power and modify the high‑ℓ tail of the Cosmic Microwave Background (CMB) power spectrum, while the presence of a dark radiation bath contributes to ΔN_eff.

A Markov Chain Monte Carlo (MCMC) analysis is performed using the latest Planck 2018 temperature and polarization data, ACT DR4 high‑ℓ measurements, BOSS BAO distance constraints, and the Pantheon+ Type Ia supernova compilation. The parameter set includes α_D, the mass ratio μ, the fractional abundance of atomic dark matter f_D, and the temperature ratio ξ_D ≡ T_D/T_CMB. The scan reveals that for α_D ≤ 0.3 and 0.1 ≤ μ ≤ 1, the atomic dark matter fraction must be ≲ 5 % to remain compatible with observations. Larger couplings (α_D ≳ 0.2) combined with f_D ≳ 0.1 produce excessive ΔN_eff and overly strong DAO damping, which are ruled out by the CMB damping tail. The constraints are slightly relaxed in the positronium‑like limit (μ → 1) because the reduced mass approaches m_eD/2, altering the dark photon temperature evolution, but the bound on f_D remains at the few‑percent level. In the hydrogen‑like regime (μ ≪ 1) the results reproduce earlier limits based on SM‑scaled rates, confirming the consistency of the new calculation.

The paper also discusses the regimes where the present approach breaks down. For α_D approaching unity, fine‑structure and relativistic corrections become non‑negligible, and the perturbative treatment of the dipole interaction would need to be replaced by a full QED calculation. Similarly, when μ is extremely small or large, the assumption that the heavy particle can be treated as a static source may fail, requiring a more sophisticated three‑body treatment for collisional processes. While collisional ionization and excitation are important for late‑time structure formation and galaxy‑scale phenomenology, the authors argue that they do not affect the large‑scale CMB constraints presented here.

In summary, this work delivers (i) a rigorous, first‑principles computation of dark recombination and radiative transition rates across the full viable aDM parameter space, (ii) a quantitative assessment of the accuracy of SM scaling approximations, showing they are adequate for case‑B recombination, and (iii) the most up‑to‑date cosmological constraints on atomic dark matter using the latest CMB, BAO, and supernova data. The results restrict the atomic dark matter fraction to a few percent for couplings up to α_D ≈ 0.3, and they highlight the need for future work on strong‑coupling (α_D > 0.3) and extreme mass‑ratio regimes, as well as on collisional processes relevant for small‑scale structure and galaxy formation.