Warm Dark Matter from the Dodelson–Widrow Mechanism
Sterile neutrinos are hypothetical fermions that interact with the Standard Model only through mixing with the active neutrino flavors. While they do not participate in weak interactions directly, their small mixing with the active states allows them to be produced in the early universe via oscillations. Among various production mechanisms, the Dodelson–Widrow (DW) scenario is a minimal and predictive framework in which sterile neutrinos are generated through non-resonant oscillations in the primordial plasma. In this mechanism the sterile neutrino abundance depends primarily on the mass of the sterile species and the vacuum mixing angle with active neutrinos. Our calculator implements an analytic fit to detailed kinetic calculations of this process, providing a quick estimate of the resulting relic density.
The DW mechanism hinges on the interplay between neutrino oscillations and thermal damping. In the early universe, active neutrinos are kept in equilibrium by weak interactions. As the universe expands and cools, the effective matter potential experienced by neutrinos diminishes, allowing oscillations into sterile states. These oscillations convert a fraction of the active neutrino population into sterile neutrinos. The production is most efficient when the temperature of the plasma is around 150 MeV, near the QCD phase transition, where changes in the number of relativistic degrees of freedom slightly modify the momentum distribution of the produced particles. Although the exact momentum spectrum is not purely thermal, it can be approximated by a redshifted Fermi–Dirac distribution with a slightly cooler effective temperature. The number density of sterile neutrinos generated through this non-resonant mixing scales with the square of the mixing angle and roughly linearly with the mass.
The resulting relic density is commonly expressed as . This fit, derived from the original Dodelson–Widrow analysis, captures the main dependence on mass and mixing while absorbing details of the thermal history into the numerical coefficients. When the resulting Ωsh² matches the measured dark matter density of approximately 0.12, the DW mechanism can account for all of the dark matter with just one new parameter, the sterile neutrino mass. However, X-ray observations, structure formation constraints, and cosmological data strongly restrict the allowed parameter space. For masses in the keV range, mixing angles large enough to yield the correct relic density often conflict with bounds from the non-observation of decay lines, since sterile neutrinos can radiatively decay into an active neutrino and a photon with energy E = ms/2.
Despite these constraints, the DW scenario remains a benchmark for studying warm dark matter and the cosmological implications of sterile neutrinos. Warm dark matter suppresses the formation of small-scale structures compared to cold dark matter, alleviating some tensions between simulations and observations of dwarf galaxies. The free-streaming length of a DW sterile neutrino is inversely proportional to its mass, making keV masses particularly interesting because they are heavy enough to avoid erasing too much small-scale structure while being light enough to leave imprints that could reconcile simulations with observations. By adjusting the mass and mixing angle, users can explore whether a given parameter set produces an acceptable relic density and what structural effects might be expected.
The sterile neutrino also features prominently in theoretical explorations of lepton-number generation and baryogenesis. If sterile neutrinos exist with masses in the GeV to TeV range, they could participate in leptogenesis processes that eventually create the matter–antimatter asymmetry. Although the DW mechanism focuses on keV-scale masses, the general framework of oscillation-driven production is conceptually linked to these higher-mass scenarios. In addition, sterile neutrinos may contribute to phenomena such as pulsar kicks, core-collapse supernova dynamics, and the cosmic neutrino background. Because they interact so weakly, detecting them directly is challenging, but their cosmological and astrophysical footprints offer indirect avenues for discovery.
The DW relic density estimate is not exact and carries uncertainties arising from QCD equation-of-state effects, lepton asymmetry, and potential new physics in the early universe. To incorporate these factors more accurately, one would need to solve the full Boltzmann equations for the sterile neutrino distribution function, a task best suited to numerical studies. Nevertheless, the analytic expression implemented here provides a robust back-of-the-envelope estimate. It allows researchers and students to quickly scan parameter space, identify regions where sterile neutrinos could plausibly account for dark matter, and assess how experimental or observational bounds impact those regions.
Below is a table illustrating how the relic density varies with mass and mixing angle in the DW mechanism. The examples highlight the strong quadratic dependence on the mass and the linear dependence on the mixing angle. For each combination, we compute Ωsh² using the same formula as the calculator, enabling quick comparisons across parameter sets.
| ms (keV) | sin²2θ | Ωsh² |
|---|---|---|
| 3 | 3×10−9 | 0.30 |
| 7 | 1×10−10 | 0.05 |
| 10 | 5×10−11 | 0.05 |
Parameter sets yielding Ωsh² far above the observed dark matter density are excluded, whereas those that underproduce dark matter might still be viable if additional production mechanisms exist. For instance, resonant production in the presence of a lepton asymmetry (the Shi–Fuller mechanism) can enhance the abundance without requiring large mixing angles. Similarly, sterile neutrinos could be part of a multi-component dark matter model, sharing the energy density with other particles. The calculator focuses on the DW channel to isolate the simplest case.
Beyond cosmology, laboratory and astrophysical searches for sterile neutrinos continue to progress. X-ray telescopes look for monoenergetic photons from radiative decays, while beta-decay experiments such as KATRIN probe deviations in the electron spectrum that might indicate heavy neutrino admixtures. These efforts constrain the same mass and mixing parameters that enter the relic density calculation, providing complementary tests. A confirmed signal in any channel could dramatically reshape our understanding of the neutrino sector and the composition of the universe’s dark matter.