---
title: Free-Streaming Length of Dark Matter
url: https://www.emergentmind.com/topics/free-streaming-length-of-dark-matter
type: topic
---

# Free-Streaming Length of Dark Matter

The free-streaming length of dark matter quantifies the comoving scale below which particle velocities erase early universe density fluctuations, thereby suppressing small-scale structure formation. This concept is central to modeling the matter power spectrum, halo mass function, and the observable abundance of substructures across a wide range of dark matter scenarios, including traditional thermal relics, wave-like dark matter, macroscopic compact objects, and non-thermal production mechanisms.

## 1. Formal Definition and Physical Origins

The free-streaming length, often denoted $r_{fs}(a)$ or $\lambda_{fs}(a)$, is the comoving distance that a dark matter particle travels from its production (or kinetic decoupling) to a given cosmic epoch $a$:
\[
r_{fs}(a) = \int_{t_i}^t \frac{\langle v(t') \rangle}{a(t')} dt' = \int_{a_i}^a \frac{\langle v(a') \rangle}{a'^2 H(a')} da'
\]
Here, $\langle v \rangle$ is the physical velocity dispersion (linked to the momentum distribution), $a$ the scale factor, $H(a)$ the Hubble parameter, and $t_i$ (or $a_i$) marks the relevant production or decoupling epoch. In wave dark matter, the velocity follows directly from the comoving wavenumber of field modes, $v(q,a) = q / \sqrt{q^2 + m^2 a^2}$. For non-relativistic epochs and for sharply peaked momentum distributions, a leading-order approximation is $r_{fs}(a) \sim \sigma_v(a) (t - t_i)$ where $\sigma_v$ is the one-dimensional velocity dispersion.

This scale sets the threshold below which primordial density fluctuations are wiped out by streaming, with the power spectrum $P_\delta(k)$ exponentially suppressed as $\exp[-k^2 r_{fs}^2]$ for $k \gg r_{fs}^{-1}$ [2503.20881].

## 2. Analytical Expressions in Standard and Modified Cosmologies

In $\Lambda$CDM, splitting into relativistic and non-relativistic regimes and using the velocity dispersion at matter-radiation equality ($\sigma_{eq}$), the comoving free-streaming length takes the form:
\[
r_{fs}(a) = \frac{\sigma_{eq}}{\sqrt{2} k_{eq}} \ln \left(\frac{a/a_{eq}}{a_{nr}/a_{eq}} \frac{(1+\sqrt{1+a_{nr}/a_{eq}})}{(1+\sqrt{1+a/a_{eq}})} \right)^2
\]
where $k_{eq}=a_{eq} H_{eq}$, $a_{nr}$ marks transition to non-relativistic motion, and $\sigma_{eq} = q_*/(m a_{eq})$ for characteristic momentum $q_*$. In the matter-dominated era, the logarithmic dependence $r_{fs}(a) \sim \sigma_{eq}/k_{eq} \ln a$ emerges [2503.20881], reflecting slow growth.

Modified expansion histories alter $H(a)$, directly impacting $r_{fs}$. 
- Early matter domination reduces $\lambda_{fs}$ by up to $30\%$ for modes becoming non-relativistic in that epoch [2412.14322].
- Early/very early dark energy components yield sub-percent or up to $\sim50\%$ reductions, respectively.
The general prescription replaces $H(a)$ with the total rate including new components and numerically integrates:
\[
\lambda_{fs}^X = \int_{0}^1 \frac{da}{a^2 [H^2_{\Lambda{\rm CDM}}(a) + \rho_X(a)/3M_P^2]^{1/2}} \frac{q_*}{\sqrt{q_*^2 + m^2 a^2}}
\]
[2412.14322].

## 3. Connection to Structure Formation: Power Spectrum and Halo Mass Function

The free-streaming length sets the cutoff for linear and quasi-linear structure formation. In $N$-body simulations and transfer function modeling, the cutoff is parameterized by the half-mode wavenumber $k_{hm}=\alpha^{-1} (2^{1/\gamma}-1)^{1/\beta}$, where $T(k_{hm})=1/2$ for transfer function $T(k)$ [2511.07513], most commonly fitted as:
\[
T(k) = [1+(\alpha k)^\beta]^{-\gamma}
\]
with standard $\beta=2,\gamma=5$. The corresponding half-mode mass is $m_{hm}=(4\pi/3)\bar\rho (\pi/k_{hm})^3$.

Physically, $r_{fs}$ (or $\lambda_{hm} = \pi/k_{hm}$) defines the minimal scale for substructure formation. For thermal relics, empirical mappings give:
\[
\lambda_{fs} \simeq 0.049 (m_{\rm therm}/{\rm keV})^{-1.11} {\rm Mpc}/h
\]
and
\[
m_{hm} = 5 \times 10^8 M_\odot \left(\frac{m_{\rm therm}}{3~{\rm keV}}\right)^{-10/3}
\]
[2511.07513, 2405.01620, 1905.04182]. Numerical modeling of lensing and Lyman-α observables translate bounds on $m_{hm}$ to tightly constrained $r_{fs}$: e.g., $m_{\rm therm} \gtrsim 7.4$–$8.4$ keV corresponds to $\lambda_{fs} \lesssim 0.02$–$0.05$ Mpc$/h$ [2511.07513].

## 4. Comparison to Jeans Length and Other Scales

Free-streaming must be contrasted with the Jeans length $\lambda_J$, the scale where pressure from velocity dispersion balances gravitational collapse. In kinetic theory formalism, $\lambda_J = [\pi \sigma_v^2(a)/G \bar\rho(a)]^{1/2}$, with corresponding wavenumber $k_J = 2\pi/\lambda_J$ [2503.20881, 1306.3578]. The free-streaming length always exceeds the Jeans length by the logarithm of the expansion factor:
\[
\frac{r_{fs}(a)}{\lambda_J(a)} \sim \ln a \gg 1
\]
Hence, for structure suppression, $r_{fs}$ sets the dominant cutoff scale; $k_{fs}/k_J\sim 2$–$3$ at equality for viable particle masses [1306.3578].

## 5. Model Dependence: Production Mechanisms and Phase-Space Distributions

The value and impact of $r_{fs}$ depends sensitively on DM microphysics:
- **Thermal relics**: Fermi-Dirac (WDM) or Bose-Einstein (hot axions, neutrinos) distributions yield characteristic $r_{fs}$ based on late-time velocity and equilibrium moments [2412.14322, 1202.2858, 2406.12970].
- **Wave dark matter** (e.g., axions): Free-streaming arises from finite coherence scale $q_*\sim a_* m$, producing sharp cutoffs $k_{fs}=1/\lambda_{fs}(q_*)$ and transfer function suppression $T_{rel}(k)\sim \sin(k/k_{fs})/(k/k_{fs})$ [2406.12970, 2408.05591].
- **Non-thermal or decays**: For decay/injection scenarios, e.g. inflaton decay to gravitinos [0705.0579], non-thermal production [2304.07462], or freeze-in [1907.02454], the initial phase-space distribution yields an $r_{fs}$ that can be much smaller (for cold, low-momentum injection), or comparable (if kinetic energy is large compared to rest mass).
- **Gravitational production**: Highly non-thermal gravitationally produced DM during reheating often leads to $r_{fs}>\lambda_{re}$ unless particles become non-relativistic during reheating [2112.14668].

## 6. Observational Constraints and Impact

Observational probes sensitive to $r_{fs}$ include:
- **Strong gravitational lensing**: Statistical modeling of flux-ratio anomalies, image positions, and extended arcs in quadruple-image quasars provides tight bounds on $m_{hm}$ and hence $r_{fs}$ [2511.07513, 2405.01620, 1908.06983, 1905.04182, 1712.04945]. Current best limits from JWST and HST lensing require $r_{fs} \lesssim 0.05$–$0.12$ Mpc for thermal WDM (masses $\gtrsim 6$–$8.4$ keV).
- **Lyman-α forest**: The cutoff in the flux power spectrum at $z \gtrsim 5$ (k $\sim$ 1–4 Mpc$^{-1}$) enables constraints on $\lambda_{fs}$ and equivalent thermal masses, with current analyses consistent with lensing constraints [2412.14322, 1809.06585].
- **Milky Way satellites, subhalo counts**: Subhalo mass functions and concentration-mass relations likewise probe $r_{fs}$, with Earth-mass scale sensitivity achieved in simulations [1907.03642, 1908.06983].

## 7. Limitations, Nuances, and Systematic Issues

A single $r_{fs}$ does not always suffice to capture all nonlinear and dynamical effects:
- In mixed cold+warm scenarios, different warm fractions and particle masses can share $r_{fs}$ but differ strongly in halo concentrations and inner profiles [1202.2858].
- Production scenarios with strong early self-interactions (e.g., freeze-in with late Brownian decoupling) require both $r_{fs}$ and the decoupling epoch to characterize small-scale power [1907.02454].
- For specific modes (e.g., isocurvature patches in fuzzy dark matter), free-streaming erases coherent patches below $r_{fs}$, but incoherent wakes persist; only coherent contributions grow gravitationally [2406.12970].
- Nonlinear evolution, tidal stripping, and baryonic feedback can further modify observed subhalo populations, requiring careful modeling in forward-inference pipelines [2511.07513, 2405.01620].

**Summary Table: Free-Streaming Length Scaling and Constraints**

| DM Type / Scenario     |  Analytical $r_{fs}$ Expression    |  Scale (typical constraint)   |
|-----------------------|------------------------------------|------------------------------|
| Thermal relic (WDM)   | $r_{fs} \sim v(a) / H(a)$, $\lambda_{fs} \simeq 0.049\, (m_{\rm therm}/{\rm keV})^{-1.11}$ Mpc/$h$   |  $<$ 0.05 Mpc/$h$ ($m_{\rm therm}$ $>$ 8 keV) |
| Wave/axion DM         | $r_{fs} = \int v_q(a)/a^2 H(a)\,da$ or $k_{fs} = 1/\lambda_{fs}$  |  $\ll$ Mpc (for cold regime); $\sim$ 0.2–2 Mpc for warm axion [2412.14322] |
| Non-thermal decay     | $r_{fs} = \int v(a)/a^2 H(a)\,da$ (monoenergetic $p \sim M/2$ at injection) |  $<$ 0.1 Mpc for cold decay, $\sim$ few Mpc for relativistic decay |
| Gravitational reheating| $r_{fs}$ depends on initial $p \sim m_\phi$, expansion history, and $\omega_\phi$ [2112.14668] |  Only $\lambda_{fs} < \lambda_{re}$ yields surviving microhalos |
| Substructure / Lensing| $m_{hm}$ from flux anomalies, $r_{fs} = \pi/k_{hm}$ |  $<$ 0.05 Mpc [2511.07513] |

## References to Key Literature

- [2511.07513], [2405.01620], [1908.06983], [1905.04182], [1712.04945]: Strong lensing constraints, semi-analytic and $N$-body approaches.
- [2503.20881], [2412.14322], [2408.05591], [2406.12970]: BBGKY formalism, wave dark matter, analytical and numerical approaches to free-streaming.
- [1306.3578], [1202.2858]: Kinetic and hydrodynamical treatments, Jeans scale comparisons.
- [1907.02454]: Freeze-in models, early Brownian phases.
- [2112.14668], [0705.0579], [2304.07462], [1912.10859]: Reheating, gravitational, and non-adiabatic production channels.
- [1809.06585]: Lyman-α forest as diagnostic of $r_{fs}$, thermal and non-thermal effects.
- [1907.03642]: Microhalo abundance near $r_{fs}$, impact on indirect detection.

## Conclusion

The free-streaming length of dark matter is a fundamental scale set by the combination of particle velocity dispersion, production mechanisms, and cosmic expansion history. It governs the suppression of small-scale structure, appears naturally as an exponential cutoff in the power spectrum, and is constrained by multiple observational probes—most stringently by gravitational lensing and Lyman-α forest measurements. While $r_{fs}$ is the key controlling parameter for linear and quasi-linear suppression, detailed effects in nonlinear structure depend additionally on the phase-space properties, self-interactions, and environmental factors, necessitating a multidimensional modeling framework for precise cosmological inference.

Source: https://www.emergentmind.com/topics/free-streaming-length-of-dark-matter