Free-Streaming Length of Dark Matter

• Free-streaming length of dark matter is the comoving scale over which particle velocities erase primordial density fluctuations, defining a cutoff for structure formation.
• It is calculated by integrating the velocity dispersion over cosmic time, and its value depends on dark matter production mechanisms and expansion history.
• Observational probes like strong lensing and Lyman-α forest analyses rigorously constrain free-streaming scales, influencing dark matter model viability.

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 $k \gg r_{fs}^{-1}$ (Amin et al., 26 Mar 2025).

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 (Amin et al., 26 Mar 2025), 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 (Long et al., 2024).
• 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: $$\lambda_{fs}^X = \int_{0}^1 \frac{da}{a^2 [H^2_{\Lambda{\rm CDM}}(a) + \rho_X(a)/3M_P^2]^{1/2}} \frac{q_*}{\sqrt{q_*^2 + m^2 a^2}}$$ (Long et al., 2024).

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 $T(k)$ (Gilman et al., 10 Nov 2025), 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

$$m_{hm} = 5 \times 10^8 M_\odot \left(\frac{m_{\rm therm}}{3~{\rm keV}}\right)^{-10/3}$$

(Gilman et al., 10 Nov 2025, Keeley et al., 2024, Hsueh et al., 2019). 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$/h$ (Gilman et al., 10 Nov 2025).

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 $k_J = 2\pi/\lambda_J$ (Amin et al., 26 Mar 2025, Piattella et al., 2013). 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 (Piattella et al., 2013).

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 (Long et al., 2024, Maccio' et al., 2012, Liu et al., 2024).
• 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 $T_{rel}(k)\sim \sin(k/k_{fs})/(k/k_{fs})$ (Liu et al., 2024, Ling et al., 2024).
• Non-thermal or decays: For decay/injection scenarios, e.g. inflaton decay to gravitinos (0705.0579), non-thermal production (Choi et al., 2023), or freeze-in (Huo, 2019), 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 (Haque et al., 2021).

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 $r_{fs}$ (Gilman et al., 10 Nov 2025, Keeley et al., 2024, Gilman et al., 2019, Hsueh et al., 2019, Gilman et al., 2017). 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 (Long et al., 2024, Garzilli et al., 2018).
• Milky Way satellites, subhalo counts: Subhalo mass functions and concentration-mass relations likewise probe $r_{fs}$, with Earth-mass scale sensitivity achieved in simulations (Ishiyama et al., 2019, Gilman et al., 2019).

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 (Maccio' et al., 2012).
• 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 (Huo, 2019).
• 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 (Liu et al., 2024).
• Nonlinear evolution, tidal stripping, and baryonic feedback can further modify observed subhalo populations, requiring careful modeling in forward-inference pipelines (Gilman et al., 10 Nov 2025, Keeley et al., 2024).

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 (Long et al., 2024)
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 $\omega_\phi$ (Haque et al., 2021)	Only $\lambda_{fs} < \lambda_{re}$ yields surviving microhalos
Substructure / Lensing	$m_{hm}$ from flux anomalies, $r_{fs} = \pi/k_{hm}$	$<$ 0.05 Mpc (Gilman et al., 10 Nov 2025)

References to Key Literature

• (Gilman et al., 10 Nov 2025, Keeley et al., 2024, Gilman et al., 2019, Hsueh et al., 2019, Gilman et al., 2017): Strong lensing constraints, semi-analytic and $N$-body approaches.
• (Amin et al., 26 Mar 2025, Long et al., 2024, Ling et al., 2024, Liu et al., 2024): BBGKY formalism, wave dark matter, analytical and numerical approaches to free-streaming.
• (Piattella et al., 2013, Maccio' et al., 2012): Kinetic and hydrodynamical treatments, Jeans scale comparisons.
• (Huo, 2019): Freeze-in models, early Brownian phases.
• (Haque et al., 2021, 0705.0579, Choi et al., 2023, Herring et al., 2019): Reheating, gravitational, and non-adiabatic production channels.
• (Garzilli et al., 2018): Lyman-α forest as diagnostic of $r_{fs}$, thermal and non-thermal effects.
• (Ishiyama et al., 2019): 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.