Warm Dark Matter: Models & Astrophysical Effects

• Warm dark matter is defined by keV-scale particles with significant thermal velocities that suppress the formation of small-scale cosmic structures.
• Production mechanisms such as thermal freeze-out, entropy dilution, and decay processes critically determine the relic abundance and observational signatures of WDM.
• Astrophysical consequences include altered halo profiles, delayed galaxy formation, and distinct X-ray and cosmic reionization signatures that help constrain WDM models.

A warm dark matter (WDM) component refers to a subpopulation of dark matter candidates characterized by non-negligible thermal velocities at decoupling, commonly with masses in the keV scale. Unlike cold dark matter (CDM), which clusters efficiently on all mass scales, WDM's free-streaming suppresses the growth of density perturbations below a characteristic scale, leading to modifications in the abundance and structure of small-scale cosmic objects. Recent research comprehensively examines the role, implications, and constraints on warm dark matter, both as a single-component and within mixed dark matter frameworks.

1. Physical Definition and Theoretical Models

The WDM paradigm is defined by dark matter candidates (usually fermions or scalars) with masses in the $\mathcal{O}(\mathrm{keV})$ range. These particles decouple while relativistic or semi-relativistic, resulting in a substantial free-streaming length and thus a suppression of structure formation below galaxy scales.

Original WDM models posited thermal relic particles, where the mass and effective temperature at decoupling are related by the number of degrees of freedom $g_{\mathrm{dec}}$:

$$n_X/n_\gamma \propto \frac{43/4}{g_{\mathrm{dec}}}.$$

Alternate production mechanisms, such as non-thermal resonant or decay production (e.g., Shi–Fuller sterile neutrinos), yield similar cutoffs in the matter power spectrum for higher mass particles—e.g., a 7 keV sterile neutrino can mimic a 2 keV thermal relic (Vega et al., 2013).

The "keVin" model (King et al., 2012) introduces two neutral fermions: a stable keVin (keV inert fermion, $\chi_1$) and an unstable GeVin (GeV inert fermion, $\chi_2$). The keVin's relic density is set by thermal overproduction and subsequent entropy dilution from late GeVin decay, parametrized by small couplings to the $Z$ (factors $\epsilon_1,\,\epsilon_2,\,\delta$) and characterized by the hierarchy:

$$\mathcal{L}_{ii} = g Z_\mu \epsilon_i^2 \left( \bar{\chi}_i \gamma^\mu \gamma^5 \chi_i \right), \qquad \mathcal{L}_{12} = g Z_\mu (\epsilon_1 \epsilon_2 \delta) \left( \bar{\chi}_1 \gamma^\mu \gamma^5 \chi_2 \right) + h.c.$$

The generic framework allows realization in E$_6$-inspired supersymmetric models, left–right symmetric sterile neutrino scenarios, and brane-world embeddings with large numbers of degrees of freedom at decoupling (Paduroiu et al., 2022).

2. Production Mechanisms and Cosmological Evolution

WDM candidates are produced via thermal freeze-out (relativistic or semi-relativistic) or nonthermal processes (e.g., particle decays). A common cosmological challenge for thermal WDM is an initial overproduction, requiring either entropy dilution or nonthermal production for phenomenological viability.

• Entropy Dilution: In models such as "keVins," the keVin relic abundance—naturally overabundant from relativistic freeze-out—is diluted via entropy injection from late-time GeVin decay. The dilution factor $\mathcal{S}$ is given as:

$$\mathcal{S} \approx 0.76 \frac{g_\chi}{2} \frac{g_*^{1/4} M_2}{g_* \sqrt{\Gamma_2 M_P}},$$

where $\Gamma_2$ is the GeVin decay width (King et al., 2012).

• Decay/Nonthermal Production: An alternative scenario involves heavy particle decay to lighter relics plus standard fermions (e.g., $\chi_2 \to \chi_1 + f + \bar{f}$). The relic abundance is directly related to the mother–daughter mass ratio as $$\Omega_{\mathrm{DM}} h^2 = (M_1/M_2) \Omega_{\chi_2} h^2$$ (Bari et al., 2013). The free-streaming length (and hence "warmth") depends on the lifetime and mass ratio, with representative scales $\lambda_{\mathrm{FS}} \sim 0.1\, {\rm Mpc}$ for $\tau \sim 10^4\,{\rm sec}$ and $M_1/M_2 \sim 10^{-2}$.
• Mixed Dark Matter (MWDM): Contemporary constraints and observational signatures increasingly favor mixed scenarios, where a fraction $f_{\rm WDM}$ is warm (either thermal relic, sterile neutrino, or boosted via thermalization) and the remainder cold. Small-scale observables, such as Milky Way satellite counts, tightly constrain $f_{\rm WDM}$ as a function of $m_{\rm WDM}$, typically allowing $f_{\rm WDM} \lesssim 0.45$ for $m_{\rm WDM} \lesssim 1.5$ keV (Tan et al., 27 Sep 2024).
• Boosted/CDM-WDM Interplay: In multi-component models, energy injection into a light DM component via decay or annihilation of a heavy component induces a "self-heating" effect, giving the light component WDM-like properties independent of its mass (i.e., without requirement for keV-scale particles) (Kim et al., 2023, Kim et al., 7 Oct 2024, Kamada et al., 2021).

3. Astrophysical and Cosmological Consequences

The principal dynamical consequence of a WDM or WDM-like component is the suppression of structure on scales below the free-streaming length, affecting:

• Matter Power Spectrum: Free-streaming causes a cutoff in $P(k)$ at high $k$, with $T_{\rm WDM}(k)$ typically parametrized as

$$T_{\rm WDM}(k) = [1 + (\alpha k)^{2\nu}]^{-5/\nu}, \quad \nu = 1.12,$$

with $\alpha$ set by $m_{\rm WDM}$ and cosmological parameters (Villanueva-Domingo et al., 2017).

• Subhalo and Dwarf Galaxy Counts: The suppression of low-mass halo formation is directly probed by observed satellite luminosity functions. Recent analyses use empirical suppression functions (\emph{e.g.}, $\beta(M, f_{\rm WDM})$) to connect theory to data and exclude models with $m_{\rm WDM} \lesssim 6.6$ keV for pure WDM ($f_{\rm WDM}=1$) and limit $f_{\rm WDM} \lesssim 0.45$ for $m_{\rm WDM} \lesssim 1.5$ keV, using observations from DES and Pan-STARRS1 (Tan et al., 27 Sep 2024).
• Halo Structure: WDM leads to lower halo concentrations, delayed halo assembly for $$M_{\mathrm{halo}} \lesssim 2 \times 10^9~h^{-1} M_\odot$$, shallower or cored inner profiles, and lower angular momenta in small halos (Bose et al., 2015).
• Cosmic Reionization: WDM delays the formation of low-mass galaxies, hence postponing and modifying the reionization history. Even when star formation efficiency is tuned to match UV luminosity functions, signatures such as the Gunn-Peterson optical depth distribution retain WDM-specific features (Villanueva-Domingo et al., 2017).
• Influence on Cosmological Tensions: Decaying warm components can affect the pre-recombination expansion rate, potentially ameliorating the Hubble tension by shifting the cosmic sound horizon and increasing $H_0$ as inferred from the CMB (Blinov et al., 2020).

4. Local and Global Distributions; Detection Implications

The mass fraction and spatial distribution of the WDM component exhibit distinctive features:

• Clustering and Local Abundance: Due to a longer free-streaming scale, the local fraction $f_{W,\odot}$ of WDM is suppressed relative to its global (cosmic) fraction $f_{\rm WDM}$. The empirical relation

$$\mathcal{A}(m, r) = [1 + \xi \cdot (\text{keV}/m_{\rm WDM})^2 \cdot (\text{kpc}/r)]^{-1}$$

with $\xi\simeq 0.008$ provides an accurate fit for the local WDM fraction as a function of $m_{\rm WDM}$ and galactocentric distance $r$ (Anderhalden et al., 2012).

• Direct and Indirect Detection: Analyses neglecting the suppression of the local WDM fraction can misestimate event rates or decay/annihilation signatures. For mixed models, this effect is significant for $m_{\rm WDM}\lesssim 1\;\mathrm{keV}$. Non-observation of predicted X-ray decay lines and precision measurements of the $Z$ invisible width further constrain sterile neutrino scenarios (Vega et al., 2013, King et al., 2012).
• Laboratory Constraints: For keV sterile neutrinos (the leading WDM candidate), experimental searches focus on kinks in beta decay (Tritium, Rhenium), electron capture spectra (Holmium), and X-ray line searches (e.g., the 3.56 keV line associated with 7.1 keV sterile neutrino decay (Vega et al., 2013, Babu et al., 2014)).

5. Quantum Effects and Core Formation

Below scales of $\sim 100$ pc, classical N-body simulations are insufficient for WDM. The quantum pressure, stemming from the Pauli exclusion principle for fermionic WDM, becomes significant. The Thomas-Fermi approach provides a self-consistent structure for galaxy cores:

$$\frac{d^2\mu}{dr^2} + \frac{2}{r}\frac{d\mu}{dr} = -4\pi G m \rho(r), \quad \rho(r) = \frac{m}{\pi^2 \hbar^3} \int_0^\infty dp\, p^2 f\left(\frac{p^2}{2m} - \mu(r)\right).$$

This framework produces core densities and velocity dispersions consistent with dwarf galaxy observations and is a key supporting argument for keV-mass sterile neutrino WDM (Vega et al., 2013).

The criterion for quantum regime is given by the ratio $\mathcal{R}=\lambda_{\mathrm{dB}}/d \gtrsim 1$, where $\lambda_{\mathrm{dB}}$ is the de Broglie wavelength and $d$ the mean interparticle distance, a condition satisfied in the cores of dwarf spheroidals.

6. Extensions: Multi-Component and Assisted Evolution

Recent work generalizes the WDM concept to dynamical scenarios:

• Self-Heating and Assisted Freeze-Out: In multi-component dark matter models, the lighter subdominant component (X$_1$) is produced both thermally and via conversion (annihilation, decay) from a heavier component (X$_0$). Continuous energy injection leads to self-heating, enhancing the effective temperature $T_1$ and rendering X$_1$ dynamically "warm" even for sub-GeV masses (Kim et al., 2023, Kamada et al., 2021, Kim et al., 7 Oct 2024). The pressure support is characterized by

$$c_{s,1}^2 = \frac{T_1}{m_1}\left(1-\frac{1}{3}\frac{d\ln T_1}{d \ln a}\right).$$

This thermal boost generates a small-scale cutoff and oscillatory features ("DMAO") in the linear matter power spectrum, mimics WDM-like suppression of halo formation, and produces cored density profiles in halos.

• Microscopic Consequences: Enhanced annihilation cross sections for sub-dominant components ($\langle\sigma v\rangle \propto 1/r_1^2$ or steeper) yield strong prospects for indirect detection even when fractional abundance is below 10%, with astrophysical and accelerator-based experiments jointly constraining the allowed parameter space (Kamada et al., 2021).
• Empirical Constraints: Satellite counts in the Milky Way, Lyman-$\alpha$ forest constraints, and halo mass functions provide robust exclusion limits for the warm fraction in MWDM scenarios (Tan et al., 27 Sep 2024, Bose et al., 2015, Anderhalden et al., 2012).

7. Theoretical and Observational Outlook

The warm dark matter component remains a compelling solution to several outstanding issues in cosmology and galaxy formation—most notably the missing satellites problem, the cusp–core discrepancy, and the overabundance of small halos in CDM-only simulations. However, the combination of astrophysical observations (especially ultra-faint Milky Way satellites), cosmic reionization histories, core/cusp and halo concentration measurements, and direct/indirect detection experiments currently limit the viable parameter space, particularly for pure WDM scenarios.

In models where WDM is a fraction of the total (MWDM), stringent upper bounds on the warm fraction as a function of free-streaming scale (related to $m_{\rm WDM}$ or sterile neutrino production mechanisms) will remain a central focus of future galaxy and halo surveys (Tan et al., 27 Sep 2024). Extensions involving two-component and self-heated dark matter suggest that the phenomenology of WDM-like structure suppression can arise from a broader range of microphysics—calling for nuanced modeling of velocity distributions, core formation, and small-scale power cutoffs (Kim et al., 7 Oct 2024, Kim et al., 2023).

Probing WDM thus remains at the intersection of particle physics, cosmological simulation, and small-scale astronomical measurement, with significant future advances expected from deeper surveys, continued X-ray line searches, improved N-body+quantum simulations, and direct detection campaigns targeting both canonical keV-scale particles and thermally boosted subpalatial-mass DM.