Sterile Neutrino Dodelson–Widrow Relic Density Calculator

Introduction

Sterile neutrinos are hypothetical particles that extend the familiar neutrino sector of the Standard Model. Unlike the active neutrinos involved in weak interactions, a sterile neutrino would not couple directly to the weak force. Its main connection to ordinary matter would come through mixing with active neutrino states. That small mixing is enough to let the early universe produce a population of sterile neutrinos, and in some models that population can contribute to or even make up the dark matter. This calculator focuses on one of the best-known production channels: the Dodelson–Widrow mechanism, often abbreviated DW.

In the DW picture, sterile neutrinos are created non-resonantly from oscillations of active neutrinos in the hot primordial plasma. The production rate depends strongly on two user inputs: the sterile neutrino mass and the vacuum mixing angle, usually written as sin²(2θ). The output is the approximate relic density, written as Ω_sh², which is the standard cosmological way to express how much of today’s critical density is stored in that sterile neutrino component. A value near 0.12 is especially interesting because it is close to the observed total dark matter density inferred from cosmological data.

This page is meant as a quick estimate tool rather than a full numerical solver. The underlying expression is an analytic fit that captures the leading dependence on mass and mixing angle. That makes it useful for scanning parameter space, building intuition, and checking whether a chosen point is likely to underproduce dark matter, produce a substantial fraction of it, or overproduce it. It is especially helpful for students and researchers who want a fast back-of-the-envelope result before moving on to more detailed calculations.

How to Use

Using the calculator is straightforward. Enter the sterile neutrino mass in keV in the first field. This mass scale is typical for warm dark matter discussions, where particles are heavy enough to survive as dark matter but light enough to affect small-scale structure formation. In the second field, enter the mixing parameter sin²(2θ). This quantity is dimensionless and is usually very small, often written in scientific notation such as 3e-9 or 1e-10.

After you press the compute button, the calculator evaluates the DW relic-density fit and displays Ωₛh² in scientific notation. The output also includes a short qualitative note. If the result is above about 0.12, the page labels the point as overclosing, meaning that this simple DW estimate would predict too much dark matter compared with observations. Intermediate values are labeled substantial or minor, and very small values are labeled negligible. These labels are not experimental exclusions by themselves; they are only quick interpretations of the abundance level.

When entering values, keep the units and conventions in mind. The mass must be in keV, not eV, MeV, or GeV. The mixing input is not the angle θ itself, but the combination sin²(2θ). If you have an angle from another source, convert it before using the calculator. Also note that the form expects positive numbers. Zero or negative entries are rejected because they do not correspond to a physical DW production estimate.

Formula

The calculator uses a standard approximate fit for non-resonant sterile-neutrino production in the Dodelson–Widrow scenario. In words, the relic density grows linearly with the mixing parameter and quadratically with the sterile neutrino mass. That means increasing the mass has a particularly strong effect on the final abundance. 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 come from fitting more detailed kinetic calculations. They summarize the thermal history and production efficiency in a compact way. Because the mass appears squared, doubling the mass at fixed mixing increases the predicted relic density by a factor of four. By contrast, doubling the mixing angle parameter doubles the abundance.

This scaling is physically sensible. 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 formula is therefore a useful intuition builder even when one later turns to more sophisticated treatments that include momentum-dependent production, QCD effects, and modified cosmological histories.

Worked Example

Suppose you choose a sterile neutrino mass of 7 keV and a mixing parameter sin²(2θ) = 1 × 10⁻¹⁰. Plugging those values into the fit gives:

A result around 0.054 means this parameter choice would produce a noticeable but subdominant dark matter contribution in the simple DW estimate. It is below the full dark matter target of roughly 0.12, so by itself it would not account for all dark matter unless some additional production channel were present. On the other hand, it is large enough to be cosmologically relevant, so it would still be worth comparing with X-ray and structure-formation constraints.

As another comparison, if you kept the same mixing but increased the mass from 7 keV to 10 keV, the abundance would rise because of the mass-squared dependence. If instead you kept the mass fixed and increased the mixing by a factor of ten, the relic density would also increase by a factor of ten. These simple parameter changes show why the allowed region in the mass–mixing plane is often narrow once abundance and observational bounds are considered together.

Interpreting the 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 DW 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. Sterile neutrinos produced through DW are warm dark matter candidates, so they can suppress small-scale structure compared with cold dark matter. Their radiative decay can also produce X-ray photons, leading to strong observational 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

This calculator intentionally uses a simplified analytic fit. It does not solve the full Boltzmann or quantum kinetic equations for the sterile-neutrino distribution function. In detailed studies, the production history depends on the plasma temperature, the QCD equation of state, momentum-dependent matter effects, and the exact active flavor structure. Those ingredients can shift the predicted abundance relative to the compact formula used here.

The page also assumes the standard non-resonant Dodelson–Widrow scenario. It does not include resonant enhancement from a primordial lepton asymmetry, as in the Shi–Fuller mechanism. In resonant production, the same mass and mixing can yield a very different abundance and a cooler momentum distribution. Likewise, the calculator does not account for entropy dilution, low reheating temperatures, hidden-sector interactions, or decays of heavier particles into sterile neutrinos. Any of those effects can significantly alter the relic density.

Another important limitation is that the result says nothing by itself about observational consistency. X-ray searches, Lyman-α forest measurements, dwarf-galaxy structure, and other cosmological probes can exclude regions of parameter space even when the abundance looks attractive. The calculator should therefore be used as a first-pass estimate and teaching tool, not as a substitute for a full phenomenological analysis. It is best suited for understanding trends, checking orders of magnitude, and preparing for more complete model testing.

For quick reference, the table below shows a few sample parameter choices evaluated with the same formula used by the calculator. These examples illustrate the linear dependence on mixing and the 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 calculator like this is designed to provide.