Warm Dark Matter Free-Streaming Horizon and Mass Scale Calculator

Introduction: why warm dark matter free-streaming matters

Warm dark matter free streaming is the physical reason small-scale structure gets erased before gravity can lock it in. This calculator turns that cosmology question into two concrete outputs: the comoving free-streaming horizon λ_fs and the associated cutoff mass M_fs. Together they show whether a candidate thermal relic would mainly affect dwarf-galaxy scales or leave the matter spectrum much closer to the cold dark matter case.

Because the form uses only the particle mass and the matter-density combination Ω_mh², the main task is choosing inputs that describe the same scenario. The result is most useful when the values are internally consistent, so the notes below focus on units, scaling, and interpretation rather than on generic calculator advice.

The sections below explain what warm dark matter scenario the tool is approximating, how to enter the inputs, how the horizon and mass scale respond when you change the mass, and which simplifying assumptions matter most when you interpret the estimate.

What problem does this warm dark matter calculator solve?

The question behind Warm Dark Matter Free-Streaming Horizon and Mass Scale Calculator is specific: given a candidate warm dark matter particle, how large is the region it can smooth out in the early universe, and what halo mass sits at the edge of that suppression? That makes it easier to compare lighter and heavier candidates on the same scale, even when the underlying particle model is only approximate.

Instead of treating free streaming as an abstract cosmology term, the calculator converts the mass and density inputs into two outputs you can compare directly. If you are screening models, that helps you see whether a proposed particle would erase structure on dwarf-galaxy scales or leave the small-scale universe much closer to the cold dark matter limit.

How to use this warm dark matter calculator

1. Enter WDM particle mass m x (keV) using the value and unit shown in the field.
2. Enter Ω m h² for the cosmology you want to test.
3. Click Compute to refresh the warm dark matter free-streaming estimate.
4. Check whether λ_fs and M_fs move in the direction expected for a lighter or heavier particle before comparing scenarios.

If you are testing more than one candidate mass, keep a short note of each warm dark matter scenario so you can compare the horizon and mass scale side by side.

Inputs: how to pick good warm dark matter values

The form needs only two numbers, but they should be chosen carefully because the free-streaming estimate is sensitive to the particle mass. Most mistakes come from mixing units or combining values from different references, so make sure the numbers you enter describe the same cosmological setup.

• Units: keep the particle mass in keV and enter Ω_mh² exactly as labeled, rather than borrowing a value in another convention and hoping it matches.
• Ranges: the calculator expects positive values, so use that as a prompt to double-check any zero, negative, or implausibly tiny input before you trust the result.
• Defaults: the prefilled 3 keV and 0.14 values are only a starting warm dark matter scenario; replace them with the mass and density you actually want to study.
• Consistency: if the mass and cosmological density come from different sources, confirm that they refer to the same thermal-relic-style setup before you compare outputs.

For this calculator, the particle mass usually has the strongest effect on the answer. A heavier warm dark matter particle shortens the free-streaming horizon and lowers the cutoff mass, while a lighter one does the opposite. Ω_mh² still matters, but it changes the result more gently than the mass input does.

Formulas: how warm dark matter inputs shape the result

This calculator does not use a generic weighted sum; it applies the warm dark matter scaling built into the result logic. In practical terms, the horizon shrinks as the particle gets heavier, and the suppression mass falls even faster because it depends on a steeper power of the particle mass than the horizon does.

That scaling is why it makes sense to vary the mass first when you are testing scenarios. If you only nudge Ω_mh², the outputs will move too, but usually not as dramatically as they do when the warm dark matter particle mass changes.

When the result looks surprising, the first things to re-check are the mass unit, the cosmology combination, and whether you intended to test a thermal-relic-like warm dark matter scenario. A large change in mass should produce a clear change in both outputs; if it does not, the issue is usually in the inputs rather than in the calculator itself.

Worked example: default warm dark matter inputs at 3 keV

Use the default values already filled into the form, m_x = 3 keV and Ω_mh² = 0.14, and the script returns λ_fs ≈ 0.0226 Mpc, or 22.6 kpc, together with M_fs ≈ 1.15×10⁶ M_☉. That is a good scale check for the form because the two outputs move in opposite ways when you vary the mass: a heavier thermal relic compresses the horizon and lowers the cutoff mass, while a lighter one pushes both upward.

If you want a quick self-check, change only the mass and leave Ω_mh² fixed. The horizon should move in the opposite direction from the mass, and the mass scale should move even more strongly. That direction test tells you far more about the warm dark matter scaling than a fake arithmetic sum ever could.

Sensitivity: how warm dark matter changes when mass shifts

The most important lever in this calculator is the warm dark matter particle mass. Raise the mass and both λ_fs and M_fs should decrease; lower it and both outputs should increase. Because the mass scale responds more steeply than the horizon, it usually separates scenarios more clearly.

Ω_mh² also shifts the estimate, but its effect is gentler. When you compare cases, change one warm dark matter input at a time so you can tell whether the mass or the cosmological density is responsible for the difference you see.

How to interpret the warm dark matter result

For warm dark matter, the result is best read as a scale estimate rather than a detailed prediction of halo counts. A larger free-streaming horizon means more early-time smoothing, and a larger free-streaming mass means the suppression reaches into more massive halos. If the output is far from your expectation, review the mass input first and the density input second.

Use the Copy Result button to save the summary line after each run if you want to compare multiple candidates later. That is enough to reproduce a warm dark matter scenario later or compare it against a different candidate mass.

Understanding warm dark matter free-streaming

Warm dark matter (WDM) particles decouple from the primordial plasma while still semi-relativistic. Their residual thermal velocities allow them to stream out of overdense regions in the early universe, suppressing the formation of low-mass structures. The comoving distance a particle can travel before becoming non-relativistic defines the free-streaming horizon. Perturbations on scales smaller than this horizon are erased, leaving an imprint on the matter power spectrum that can be probed through observations of the Lyman-α forest, dwarf-galaxy counts, and weak gravitational lensing. This calculator uses the widely referenced fitting formulas from Bode, Ostriker, and Turok (2001) to estimate the free-streaming scale and the associated mass. These approximations capture the physics of free streaming for thermal relics with a Fermi-Dirac distribution and provide a quick tool for exploring how different WDM particle masses would modify structure formation.

The comoving free-streaming horizon is given by an approximate relation $${\lambda }_{\mathrm{fs}}=0.1\times {\frac{1}{m_x}}^{\frac{4}{3}}\times {\frac{{\Omega }_m}{0.15h^2}}^{\frac{1}{3}}\mathrm{Mpc}$$ where $m_x$ is expressed in keV and ${\Omega }_m$ is the present-day matter density parameter. Although this formula hides several approximations—such as assuming a thermal relic and ignoring detailed transfer-function features—it captures the scaling with particle mass and cosmological density. Lower mass WDM particles have longer free-streaming horizons because their higher velocities enable them to traverse larger distances before becoming non-relativistic.

The suppression of the matter power spectrum can also be characterized by a free-streaming mass scale, essentially the amount of matter contained within a sphere of radius half the free-streaming wavelength. This mass scale is roughly $$M_{\mathrm{fs}}\approx {10}^8\times {\frac{1}{m_x}}^4\times \frac{{\Omega }_m}{0.15h^2}{M}_{☉}$$. For a 3 keV particle and a fiducial cosmology with ${\Omega }_m{h}^2$ around 0.14, the calculator finds a free-streaming scale of about 22.6 kpc and a mass of about $1.15\times {10}^6$ solar masses. Conversely, a heavier 7 keV particle, sometimes discussed in sterile neutrino scenarios, would have a much smaller free-streaming scale of about 7.3 kpc and a cutoff mass of about $3.9\times {10}^4$ solar masses, allowing more small-scale structure to survive.

Interpreting these results requires appreciating how free streaming modifies cosmological observables. The transfer function describing the suppression of density perturbations is not a sharp cutoff but a gradual decline. Modes with wavelengths shorter than the free-streaming length are exponentially damped, while longer wavelengths retain the cold dark matter behavior. Observationally, this leads to a reduction in the number of low-mass halos, which can be probed by surveying faint dwarf galaxies in the Local Group or by studying absorption features in the Lyman-α forest of quasar spectra. By comparing the abundance of small-scale structures with predictions from varying $m_x$, astrophysicists can constrain the nature of dark matter.

Warm dark matter models also influence reionization and the formation of the first galaxies. If the free-streaming mass is too large, the first star-forming halos would appear later, potentially delaying the epoch of reionization. Conversely, a smaller free-streaming mass closer to the cold dark matter limit allows early structure formation to proceed unhindered. Future observations across galaxy surveys, quasar absorption spectra, and early-universe probes can tighten constraints on warm dark matter properties by exploring these epochs.

The table below provides example outputs for different particle masses, assuming ${\Omega }_m{h}^2$ = 0.14. It illustrates how rapidly the free-streaming horizon shrinks as the particle mass increases.

While the precise numbers depend on cosmological parameters and the particle’s production mechanism, the qualitative trend remains robust: lighter particles erase more structure. This suppression serves as the cornerstone for testing warm dark matter and distinguishing it from cold dark matter. Ultimately, a detection of a cutoff in the matter power spectrum would revolutionize our understanding of cosmic evolution and shed light on the fundamental properties of dark matter.

The methodology implemented in this calculator is intentionally transparent. Users can inspect the code to follow each computational step, from unit conversions through the application of the scaling relations. The goal is not merely to deliver numbers but to foster intuition about how cosmological parameters influence structure formation. By experimenting with different inputs, students and researchers can quickly gauge whether a proposed warm dark matter model conflicts with observed galaxy counts or other cosmological data. In an educational setting, the calculator can be used to demonstrate the interplay between particle physics and cosmology, highlighting how microscopic properties imprint on the macroscopic universe.

It is important to note the limitations inherent in such simplified formulas. The expressions from Bode et al. assume thermal relics and may not apply directly to non-thermal production mechanisms, such as resonant sterile neutrino production with lepton asymmetries. Moreover, the actual transfer function for warm dark matter depends on the detailed velocity distribution and on the epoch of decoupling. Nonetheless, the scaling relations provide a useful heuristic for exploring a broad class of models, and they align reasonably well with full numerical calculations across a range of parameter space.

Future refinements to this tool could incorporate more accurate transfer-function fits or allow users to specify alternative cosmological models, such as those with evolving dark energy or extra relativistic species. For now, the calculator focuses on the essentials, providing a streamlined interface for estimating the impact of warm dark matter on small-scale structure. By making these computations accessible in a browser, it aims to lower the barrier to entry for students and enthusiasts eager to delve into the frontier of dark matter research.