Sterile Neutrino Dodelson–Widrow Relic Density Calculator

Dodelson–Widrow Sterile-Neutrino Introduction

The Dodelson–Widrow relic density calculator estimates how much present-day dark matter abundance a sterile neutrino can build through non-resonant production in the early universe. Because the mechanism depends on a small active–sterile mixing and a mass in the keV range, it is a fast way to test whether a point in parameter space could plausibly contribute to the cosmological dark matter budget.

In this calculator the two inputs are the sterile-neutrino mass m_s and the mixing parameter sin²(2θ). The mass sets how much energy each produced particle carries, while the mixing parameter controls how efficiently active neutrinos convert into the sterile state in the hot primordial plasma. The output is Ω_sh², the standard density parameter used to describe how much of today’s critical density is stored in that sterile-neutrino component.

The value should be read as a compact estimate rather than a full numerical production history. Even so, it is useful for checking whether a model point sits far below the observed dark-matter abundance, lands in the interesting middle range, or overshoots the target by too much. That makes the page handy when scanning sterile-neutrino models, comparing against astrophysical bounds, or deciding which parameter combinations deserve a more detailed treatment.

How to Use the Dodelson–Widrow Calculator

Using the Dodelson–Widrow calculator is straightforward: enter a sterile-neutrino mass in keV and a value for sin²(2θ). The first field should contain the mass scale you want to test, and the second should contain the dimensionless mixing strength in the convention commonly used for non-resonant sterile-neutrino production.

After you submit the form, the page evaluates the relic-density fit and prints Ω_sh² in scientific notation. The note that appears beside the number is only a quick interpretation of the abundance level. It tells you whether the result is extremely small, noticeable, or larger than the rough 0.12 benchmark associated with the observed dark-matter density, but it is not a full exclusion test by itself.

Keep the conventions in mind while entering values. The mass must be in keV, not eV or GeV, and the mixing input is sin²(2θ), not the angle θ itself. If you are starting from another parameterization, convert it before using the calculator so the result matches the same mixing convention as the fit. The calculator also expects positive numbers, because zero or negative entries do not represent a physical Dodelson–Widrow abundance estimate.

Dodelson–Widrow Formula for Ωₛh²

The Dodelson–Widrow relic-density fit implemented here captures the main scaling of non-resonant sterile-neutrino production. In this approximation the abundance grows linearly with the mixing input and quadratically with the sterile-neutrino mass, which is why the mass can have such a strong impact on the final result. The formula implemented in the script is:

Here, m_s is the sterile-neutrino mass and sin²(2θ) measures the active–sterile mixing strength. The normalization constants 0.3, 3 × 10⁻⁹, and 3 keV condense more detailed kinetic calculations into a form that is easy to scan by eye. Because the mass appears squared, doubling the mass at fixed mixing multiplies the predicted relic density by four. By contrast, doubling the mixing parameter doubles the abundance.

That scaling is physically sensible for Dodelson–Widrow production. A larger mixing angle makes oscillation-driven production more efficient, while a larger mass increases the energy density carried by each produced sterile neutrino. The result is therefore a practical intuition builder even when you later move on to momentum-dependent spectra, flavor-dependent effects, or nonstandard thermal histories.

Worked Example: 7 keV Dodelson–Widrow Point

Suppose you test the Dodelson–Widrow calculator with a sterile-neutrino mass of 7 keV and a mixing parameter sin²(2θ) = 1 × 10⁻¹⁰. Those values sit in the range often discussed for warm-dark-matter candidates, and they make the mass-squared dependence easy to see.

A value around 0.054 means the sterile neutrino would contribute a meaningful but subdominant fraction of the dark matter in this simple fit. It falls below the full target of about 0.12, so the chosen point would not by itself explain all dark matter unless some other production channel or additional component is also present.

If you keep the same mixing and raise the mass, the predicted abundance increases quickly because of the quadratic dependence. If you keep the mass fixed and increase the mixing, the abundance rises in direct proportion. That is why many sterile-neutrino scans look like narrow bands in the mass–mixing plane: the relic-density condition and observational limits squeeze the viable region from both sides.

Interpreting the Dodelson–Widrow Result

The output Ωₛh² should be read as the sterile-neutrino contribution to the present-day matter budget, not automatically as proof that the model point is viable. A value close to 0.12 suggests that the Dodelson–Widrow mechanism could, in principle, generate roughly the right abundance to explain all dark matter. A much smaller value means the sterile neutrino would be only one component of the dark sector unless another production mechanism boosts the abundance. A much larger value means the point would overproduce dark matter in this approximation and is therefore cosmologically disfavored.

Abundance is only one part of the story for Dodelson–Widrow sterile neutrinos. They are warm dark matter candidates, so they can suppress small-scale structure compared with cold dark matter, and their radiative decay can produce X-ray photons that place strong limits on the same mass and mixing parameters used here. For that reason, a point that gives the “right” relic density may still be ruled out by astrophysical data. Conversely, a point that underproduces dark matter may remain interesting in models with multiple production channels or multiple dark-matter components.

Limitations and Assumptions of the Dodelson–Widrow Estimate

The Dodelson–Widrow estimate used here is intentionally compact. It does not solve the full Boltzmann or quantum-kinetic equations for the sterile-neutrino phase-space distribution, so it should be treated as an analytic approximation rather than a precision cosmology code.

In a more complete treatment, the result can shift because of the plasma temperature evolution, the QCD equation of state, momentum-dependent matter effects, and which active flavor mixes most strongly with the sterile state. Those details can matter when you are close to an abundance boundary or trying to compare two nearby points, especially if you are mapping a thin allowed band in parameter space.

This page also assumes standard non-resonant Dodelson–Widrow production. It does not include lepton-asymmetry-driven resonant enhancement, entropy dilution, low reheating temperatures, hidden-sector interactions, or late decays of heavier particles into sterile neutrinos. Any of those ingredients can change both the size of Ω_sh² and the shape of the momentum distribution.

Finally, abundance alone does not tell you whether a sterile-neutrino point is allowed. X-ray searches, Lyman-α forest measurements, and dwarf-galaxy structure constraints can still exclude regions that look fine from the relic-density perspective. Use this calculator as a fast screening tool, a way to understand how mass and mixing compete, and a starting point for deeper phenomenology rather than a final verdict.

For quick reference, the table below shows a few sample parameter choices evaluated with the same Dodelson–Widrow fit used by the calculator. These examples illustrate the direct mixing dependence and the stronger quadratic dependence on mass.

In practice, users often explore several nearby points rather than relying on a single input pair. That approach makes the scaling behavior easier to see and helps identify whether a model point is robust or finely tuned. If a tiny change in mass or mixing moves the result from negligible to overclosing, then the parameter region is narrow and likely sensitive to the approximations built into the fit. That is exactly the kind of insight a compact Dodelson–Widrow calculator is designed to provide.