Probing Light Dark Matter with Cosmic Gravitational Focusing (2509.21213v1)

Abstract: We investigate the possibility of using the cosmic gravitational focusing (CGF) to probe the minor light dark matter (DM) component whose mass is in the range of $(0.1 \sim 100)$\,eV. Being a purely gravitational effect, the CGF offers a mode-independent probe that is complementary to the existing ways such as Lyman-$\alpha$ and $\Delta N_{\rm eff}$. Such effect finally leads to a dipole density distribution that would affect the galaxy formation and hence can be reconstructed with galaxy surveys such as DESI. Both the free-streaming and clustering limits have been studied with analytical formulas while the region in between is bridged with interpolation. We show the projected sensitivity at DESI with the typical phase space distribution of a freeze-in DM scenario as illustration.

• The paper presents a theoretical framework using cosmic gravitational focusing to detect subdominant light dark matter in the 0.1–100 eV range.
• It employs analytic and numerical methods, including galaxy cross-correlations from DESI BGS, to reconstruct the DM-induced dipole density from relative velocities.
• Projected sensitivity improves constraints by up to two orders of magnitude compared to Lyman-α and CMB bounds, demonstrating robust probe potential.

Probing Light Dark Matter via Cosmic Gravitational Focusing

Introduction

This work presents a comprehensive theoretical and phenomenological paper of probing subdominant light dark matter (DM) components in the mass range $0.1$–$100$ eV using cosmic gravitational focusing (CGF). The CGF effect, a purely gravitational phenomenon, induces a dipole density distribution in the minor DM component due to its relative bulk velocity with respect to the dominant cold dark matter (CDM). This dipole can be reconstructed from galaxy cross-correlations in large-scale structure (LSS) surveys, providing a mode-independent and complementary probe to existing constraints from Lyman-$\alpha$ forest, CMB $\Delta N_{\rm eff}$, and 21 cm cosmology.

Theoretical Framework

The CGF effect arises when a light DM species $X$ with non-negligible velocity dispersion passes through CDM halos, resulting in a focused overdensity downstream. The density perturbation $\delta(\bm{x})$ exhibits a dipole symmetry, and its Fourier transform $\tilde{\delta}(\bm{k})$ acquires an imaginary component, parameterized by a phase $\tilde{\phi}_X$. The total matter overdensity is then expressed as $(1 + i \tilde{\phi}_X)\tilde{\delta}_m$, where the real part is dominated by CDM and the imaginary part encodes the CGF effect from $X$.

For a thermal relic, the phase is given by:

$$\tilde{\phi}_X \equiv \frac{G a^2}{|\bm{k}|^2} (\bm{v}_{Xc} \cdot \hat{\bm{k}}) \left( m_X^4 f_0 + 3 m_X^2 T_A^2 f_1 + 2 T_A^4 f_2 \right)$$

where $G$ is Newton's constant, $a$ the scale factor, $T_A$ a spectral parameter, and $f_n$ are moments of the phase space distribution. For non-relativistic $X$, the $m_X^4$ term dominates, leading to strong mass dependence.

In realistic scenarios, light DM is produced via freeze-in mechanisms, yielding a non-thermal phase space distribution:

$$f_X(\bm{p}) \approx C_X \frac{e^{-|\bm{p}|/T_A(a)}}{\sqrt{|\bm{p}|/T_A(a)}}$$

with $T_A(a) = T_{A0}/a$ and normalization $C_X$ fixed by the present energy density. The resulting mass scaling for the CGF phase is modified to $$\tilde{\phi}_X \propto |\bm{v}_{Xc}|^{1/2} m_X^{5/2}$$, and accounting for $|\bm{v}_{Xc}| \propto 1/m_X$, the net scaling is $\tilde{\phi}_X \propto m_X^2$.

Free-Streaming and Clustering Limits

The CGF effect exhibits distinct behavior in the free-streaming and clustering regimes, separated by the free-streaming scale $k_{\rm fs}^{-1}$. For $|\bm{k}|^{-1} < k_{\rm fs}^{-1}$ (free-streaming), the phase is dominated by the freeze-in solution. For $|\bm{k}|^{-1} \gg k_{\rm fs}^{-1}$ (clustering), the phase is determined by the linear response of the DM fluid, with the mass dependence entering only through the relative velocity, which scales as $1/m_X^2$.

An interpolation function $g(|\bm{k}|)$ bridges these regimes, allowing for a unified treatment across the full mass range.

Galaxy Cross-Correlation and Sensitivity Forecasts

The CGF-induced dipole can be reconstructed from the imaginary part of the cross-correlation between different galaxy types:

$$\mathcal{S} \equiv \mathrm{Im} \langle \tilde{\delta}_{g\alpha} \tilde{\delta}_{g\beta} \rangle$$

where $\tilde{\delta}_{g\alpha}$ is the overdensity of galaxy type $\alpha$, with bias $b_\alpha$ for CDM and $b_X$ for $X$. The signal-to-noise ratio (SNR) is computed by integrating over survey volume, wavenumber, and bias differences, incorporating the ensemble averages of $\tilde{\phi}_X$ and its time derivative.

Projected sensitivities using DESI BGS and faint galaxy catalogs are shown in the following figure.

Figure 1: The projected CGF sensitivity (red solid) on the light DM energy fraction $F_X \equiv \Omega_X / \Omega_{\rm DM}$ as a function of $m_X$ from DESI BGS observations, compared to Lyman-$\alpha$ (green dashed), CMB $\Delta N_{\rm eff}$ (blue dash-dotted), and 21 cm (black dotted) constraints.

For $m_X < 1$ eV, the sensitivity improves with $m_X^2$ scaling, while for $m_X > 10$ eV, the constraint weakens as $1/m_X^2$. The strongest constraint is achieved in the intermediate regime $1$–$10$ eV, with $F_X < 10^{-3}$, surpassing existing Lyman-$\alpha$ and CMB bounds by two orders of magnitude.

Numerical Implementation and Scaling Behavior

The velocity dispersion and its mass scaling are computed using the CLASS Boltzmann code, with the filter scale set to $R = 5$ Mpc/$h$. The velocity variance $\sqrt{\langle \bm{v}_{Xc}^2 \rangle}$ decreases with increasing $m_X$, and its time evolution is nearly constant for $z < 1$.

Figure 2: Left: Velocity variance $\sqrt{\langle \bm{v}_{Xc}^2 \rangle}$ and expansion parameter $\langle y_i \rangle$ as functions of $m_X$ for $F_X = 10^{-2}$ (solid) and $F_X = 10^{-5}$ (dashed). Right: Relative velocity evolution for $m_X = 0.1$, $1$, $10$, and $100$ eV.

The mass scaling index $n$ for $\sqrt{\langle v_k^2 \rangle}$ transitions from $-0.5$ to $-2$ across the relevant scales.

Figure 3: Upper: Mass scaling behavior of velocity dispersion, $\sqrt{\langle v_k^2 \rangle} \propto m_X^n$. Lower: Power index $n$ as a function of $m_X$.

Comparison with Other Probes

The CGF method is complementary to Lyman-$\alpha$ and CMB $\Delta N_{\rm eff}$ constraints. Lyman-$\alpha$ is insensitive to $m_X$ in the relevant range, yielding a flat $F_X < 0.1$ bound. CMB constraints rapidly weaken for $m_X > 0.1$ eV. The CGF approach provides superior sensitivity for $0.1$–$100$ eV, especially in the $1$–$10$ eV window.

Implications and Future Directions

The results demonstrate that CGF is a robust, model-independent probe of minor light DM components, capable of constraining energy fractions down to $10^{-3}$ for $1$–$10$ eV masses. This sensitivity is unattainable with current Lyman-$\alpha$ or CMB methods. The approach is readily extendable to future spectroscopic and photometric surveys (DESI, LSST, WFIRST, Euclid, CSST), which will further improve constraints and enable detailed mapping of DM substructure.

Theoretically, the CGF formalism provides a unified framework for analyzing mixed DM scenarios, bridging free-streaming and clustering regimes. The analytic treatment of phase space distributions and velocity scaling is directly applicable to other non-thermal relics, including sterile neutrinos and axion-like particles.

Conclusion

Cosmic gravitational focusing offers a powerful, gravitationally-driven method to probe minor light DM components in the eV mass range. The projected sensitivity from DESI galaxy surveys exceeds existing bounds by up to two orders of magnitude for $1$–$10$ eV DM, with analytic and numerical results confirming the scaling behavior across free-streaming and clustering limits. The CGF effect is poised to become a key tool in the search for multi-component DM, with significant implications for both cosmology and particle physics.