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 ${\mathrm{\backslash lambda}}_{\mathrm{fs}}=0.1\left(\frac{1}{\mathrm{m\_x}}\right){(}^4\left(\frac{{\mathrm{\backslash Omega}}_m}{0.15h^2}\right){(}^1\backslash text\{Mpc\}$ where $\mathrm{m\_x}$ is expressed in keV and ${\mathrm{\backslash Omega}}_m$ 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 $M_{\mathrm{fs}}\approx \mathrm{10^\{8\}}\left(\frac{1}{\mathrm{m\_x}}\right){\left(}^4\left(\frac{{\mathrm{\backslash Omega}}_m}{0.15h^2}\right)\backslash text\{M\}\_\{\backslash odot\}$. For a 3 keV particle and a fiducial cosmology with ${\mathrm{\backslash Omega}}_{m}h^2$ around 0.14, the calculator finds a free-streaming scale of tens of kiloparsecs and a mass of order $\mathrm{10^\{7\}}–\mathrm{10^\{8\}}$ solar masses, comparable to the smallest dwarf galaxies observed. Conversely, a heavier 7 keV particle, sometimes discussed in sterile neutrino scenarios, would have a much smaller free-streaming scale, 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 $\mathrm{m\_x}$, 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. Upcoming surveys with the James Webb Space Telescope and 21-cm experiments are expected to tighten constraints on warm dark matter properties by exploring these early epochs.

The table below provides example outputs for different particle masses, assuming ${\mathrm{\backslash Omega}}_{m}h^2$ = 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.