Warm Dark Matter Free-Streaming Horizon and Mass Scale Calculator

Introduction to warm dark matter free streaming

Warm dark matter free streaming describes how particles with appreciable early-universe velocities move out of small overdense regions before gravity can preserve those perturbations. This calculator turns that physical idea into two characteristic outputs: the comoving free-streaming horizon λ_fs and the associated free-streaming mass M_fs. Together, they provide a quick indication of the length and mass scales on which a thermal warm dark matter candidate may suppress structure relative to cold dark matter.

The tool is intended for exploration and scale checking. It uses the particle mass in kiloelectronvolts and the physical matter-density combination Ω_mh². It does not calculate an entire transfer function or predict an exact number of galaxies. Instead, it helps students, researchers, and interested readers see how strongly the characteristic suppression scales depend on particle mass.

What the warm dark matter horizon and mass estimates mean

The warm dark matter free-streaming horizon is a comoving distance, meaning that it factors out the later expansion of the universe. A larger value means that particle motion can smooth primordial density fluctuations over a larger region. The corresponding free-streaming mass is a characteristic amount of matter associated with that smoothing scale. A larger M_fs therefore suggests that suppression may extend toward more massive structures, although it should not be interpreted as a perfectly sharp minimum halo mass.

Real warm dark matter transfer functions decline gradually rather than ending at one exact wavelength. Halo formation is also influenced by nonlinear growth, baryonic physics, the particle’s phase-space distribution, and the definition of the cutoff scale. The calculator’s two numbers are consequently best treated as intuitive reference scales for a thermal-relic-like model rather than exact observational boundaries.

How to use the warm dark matter calculator

To use this warm dark matter calculator, enter the candidate particle mass m_x in keV and the physical matter density Ω_mh². Select Compute free-streaming scales to update both outputs. If you are comparing candidates, change one input at a time. Holding Ω_mh² fixed while changing m_x makes the steep particle-mass dependence especially easy to see.

1. Use a positive thermal-relic-equivalent particle mass in keV.
2. Enter Ω_mh² as one combined dimensionless quantity, not Ω_m alone.
3. Compute the result and compare λ_fs in Mpc or kpc with M_fs in solar masses.
4. Copy the summary if you want to retain a scenario for comparison.

The default values, 3 keV and 0.14, are a convenient demonstration rather than a preferred or endorsed dark matter model. Values from a paper should be checked carefully because authors may quote a thermal-relic mass, a sterile-neutrino mass, or another model-dependent equivalent mass that cannot be substituted without a conversion.

Inputs for particle mass and physical matter density

The warm dark matter particle-mass input, m_x, represents the candidate mass under the calculator’s thermal-relic approximation. It is measured in keV. Raising this mass makes the candidate behave more like cold dark matter because the characteristic thermal motion and free-streaming reach become smaller. Since both fitting relations contain a negative power of m_x, even a moderate mass adjustment can produce a substantial change in the result.

The density input is Ω_mh², often called the physical matter density. Here Ω_m is the present-day total matter density as a fraction of the critical density, while h is the dimensionless Hubble parameter. Its definition is:

The form expects Ω_mh² as a combined value. For example, it expects approximately 0.14 rather than a separate Ω_m near 0.3. Entering Ω_m by itself would overstate the physical density and would make both reported scales inconsistent with the displayed assumptions.

Formulas for the warm dark matter free-streaming scales

The warm dark matter formulas are transparent scaling relations rather than a generic weighted calculation. For m_x expressed in keV and Ω_mh² entered as one quantity, the comoving horizon is approximated by:

In this relation, is the numerical mass in keV and the density ratio is dimensionless. Because the reciprocal mass is raised to the 4/3 power, the same relationship can be summarized as:

Doubling the particle mass therefore reduces λ_fs by a factor of 2^4/3, or about 2.52, when the density is unchanged. Density enters only through a cube root, so ordinary changes in Ω_mh² have a gentler effect than comparable fractional changes in particle mass.

The associated characteristic free-streaming mass is approximated by:

The particle-mass dependence of the characteristic mass can also be written explicitly as:

This m_x⁻⁴ dependence is steep. Doubling the mass lowers the characteristic mass scale by a factor of 16 at fixed density. This explains why M_fs often separates candidate scenarios more dramatically than λ_fs.

Worked example: a 3 keV thermal relic

This worked warm dark matter example uses the default inputs m_x = 3 keV and Ω_mh² = 0.14. Substitution into the horizon relation gives:

The mass relation gives M_fs ≈ 1.15 × 10⁶ M_☉. These values mean that the approximation associates the candidate with smoothing over a few tens of comoving kiloparsecs and with a characteristic matter scale of roughly one million solar masses. Neither value says that every halo below that scale disappears; the transfer-function transition and subsequent nonlinear evolution are more complicated.

As a sensitivity test, keep Ω_mh² at 0.14 and raise the particle mass to 7 keV. The calculator then gives a horizon near 7.3 kpc and a mass scale near . The much smaller cutoff reflects the fact that a heavier candidate retains more small-scale structure. Conversely, reducing the mass to 1 keV increases the horizon to nearly 98 kpc and the characteristic mass to roughly 9.3 × 10⁷ solar masses.

Interpreting warm dark matter suppression in cosmology

Warm dark matter suppression should be interpreted as a transition across scales. Modes with wavelengths well above the characteristic free-streaming length behave more like their cold dark matter counterparts, while shorter-wavelength perturbations are increasingly damped. The transition is not a step. A complete model uses a scale-dependent transfer function, and observational constraints generally compare that function with probes such as the Lyman-α forest, satellite-galaxy populations, gravitational lensing, and the timing of early galaxy formation.

Warm dark matter can also influence reionization. If suppression delays the appearance of low-mass star-forming halos, ionizing sources may emerge later. A sufficiently heavy thermal relic produces a smaller suppression scale and approaches cold dark matter behavior. The numbers here can suggest which regime a candidate occupies, but they cannot determine whether a candidate is observationally allowed without a dedicated analysis.

By comparing structure observations with predictions for different values, astrophysicists can place limits on dark matter models. Such limits depend on assumptions about thermal histories, galaxy completeness, baryonic feedback, and particle production, so a quoted lower mass bound should always be read together with its model and data assumptions.

Reference values for several warm dark matter masses

These warm dark matter reference values use Ω_mh² = 0.14 and the same equations as the form. They offer a quick direction and magnitude check when experimenting with the calculator.

The qualitative trend is robust within this approximation: lighter thermal particles erase fluctuations over longer distances and larger characteristic masses. The precise relationship between a quoted particle mass and an observed cutoff, however, depends on the production mechanism and phase-space distribution. The tabulated numbers should therefore be used to check calculator operation and compare scaling, not as independent constraints on a particle candidate.

Assumptions behind the thermal-relic approximation

The warm dark matter scaling used here assumes that the entered mass can be treated as a thermal-relic-equivalent mass. That assumption packages a complicated early-universe momentum distribution into a single parameter. It also assumes that Ω_mh² is an appropriate density normalization for the fitting relations and that the requested output is a comoving characteristic scale rather than a present-day physical diameter.

The solar-mass output is a reference suppression mass, not the measured mass inside a particular observed object. Different studies may define a free-streaming mass, half-mode mass, filtering mass, or cutoff mass in different ways. Those quantities are related conceptually but are not automatically numerically identical. When comparing this result with a paper, check the paper’s transfer function, wavelength convention, radius definition, density convention, and particle-production model.

Limitations of this warm dark matter estimate

The principal limitation of this warm dark matter estimate is that it represents the physics with simplified thermal-relic scaling formulas. It does not integrate a momentum distribution through cosmic history, solve the Boltzmann equations, calculate a transfer function, or run a structure-formation simulation. The result is therefore an order-of-magnitude educational estimate rather than a precision constraint or a direct prediction of the smallest existing galaxy.

Non-thermal candidates require special care. Resonantly produced sterile neutrinos, particles created through decays, and mixed cold-plus-warm models may have momentum distributions that differ substantially from a thermal relic. Two models with the same particle mass can then possess different free-streaming behavior. A thermal-equivalent mass conversion may be needed, and that conversion can depend on the production parameters.

The estimate also omits detailed cosmological dependencies, baryonic feedback, nonlinear evolution, halo-definition choices, and observational selection effects. Do not use it alone to accept or exclude a particle model. For research use, consult the transfer function and numerical treatment adopted by the relevant study, verify all conventions, and propagate uncertainty in both cosmology and astrophysical modeling.