Relaxion Stars and Their Detection via Atomic Physics

The work addresses how light scalar dark matter, specifically the relaxion that mixes with the Higgs, can be detected via time-varying fundamental constants in precision atomic experiments. Ultralight scalar dark matter (DM) can form a coherently oscillating field, and the relaxion’s Higgs mixing induces both scalar and pseudoscalar couplings to Standard Model fields, leading to temporal modulation of quantities such as the electron mass and fine-structure constant. Existing experiments have not yet probed the most motivated regions, especially at higher oscillation frequencies. The paper proposes that if the relaxion forms self-gravitating compact objects (boson stars) or becomes gravitationally bound to massive bodies (Earth or Sun) forming halos, the local scalar field amplitude—and hence experimental signals—can be dramatically enhanced. The study explores conditions for formation, stability, encounter rates with Earth, and the detectability of such structures, aiming to identify parameter space where current or near-future atomic physics experiments can test physical relaxion DM models.

Previous studies have explored ultralight scalar DM effects on atomic clocks and fundamental constants, including dilaton-like scenarios and Higgs-portal couplings, with bounds from optical cavities, trapped ions, and atomic-clock comparisons (e.g., dynamic decoupling, Doppler-free spectroscopy, hyperfine–optical comparisons). Fifth-force and equivalence-principle tests often provide stronger constraints at higher frequencies. Formation of boson stars has been studied extensively for axions: miniclusters (if PQ symmetry breaks after inflation) can undergo gravitational relaxation to form self-gravitating axion stars; simulations indicate soliton-like cores in fuzzy DM scenarios. Transient signatures with terrestrial networks (e.g., GNOME) have been proposed for axion stars and Q-balls under phenomenological mass–radius assumptions. However, analogous formation and fluctuation spectra for relaxions have not been comprehensively investigated; this work extrapolates insights from axion studies and focuses on their experimental implications for relaxion stars and halos.

• Model and couplings: Consider a light scalar field φ (relaxion benchmark) with effective interactions L ⊃ g_e ē φ e + g_γ φ F_{μν}F^{μν}. Higgs mixing parameterizes couplings, with g_e ≈ sinθ and g_γ ≈ (α/4π v) sinθ in relaxion/Higgs-portal-like models. Naturalness suggests g ≲ 4π m/A.
• Variation of constants: Oscillations induce δm_e/m_e = g_e φ and δα/α = g_γ φ. For coherent background DM of local density ρ_local ≈ 0.4 GeV/cm^3, φ(t) has amplitude |φ| ≈ √(2ρ_local)/m and oscillation frequency ω ≈ m.
• Boson stars (relaxion stars): Treat self-gravitating, free-field configurations supported by gradient pressure. Use mass–radius relation R ≈ M/(m^2 M_p), with M_p the Planck mass. The overdensity relative to local DM δ ≡ ρ/ρ_local is estimated from approximate profiles as δ ≈ (2M)/(7π m^2 R^3 ρ_local), yielding numerical benchmarks (e.g., δ ~ 7×10^21 (10^−10 eV/m)^2 (10^5 km/R)^3). The field amplitude scales as √δ relative to the background.
• Encounter rates: Assume an O(1) fraction η of local DM in bound states with geometric cross-section σ ≈ πR^2 and virial speed v ~ 10^−3. The encounter rate with Earth is Γ ≈ η σ n_ρ v, leading to Γ ≈ 2×10^−18 yr^−1 (m/10^−10 eV)^2 (R/10^5 km)^3 under benchmark assumptions. More generally, Γ ≈ 0.05 yr^−1 δ^{−3/4} (m/10^−10 eV). Stability boundaries from self-interactions are considered (see Supplementary), mapping allowed regions in m–f and mass–radius space.
• Relaxion halos bound to external masses: Consider halos supported by the gravitational potential of an external body (Earth or Sun), with M_r ≪ M_ext. The halo radius is set by external potential: R_r ≈ [(M_r M_ext)/(4π ρ_ext m^2)]^{1/4} for R_r > R_ext (point-mass approximation; exponential profile), and R_r ≈ (3M_r/4πρ_ext)^{1/3} for R_r ≤ R_ext (constant-density sphere; Gaussian profile). Self-gravity is kept subdominant by requiring M_r ≤ M_ext/2 and consistency checks. Use these profiles to compute the scalar amplitude at Earth’s surface relevant for experiments.
• Gravitational constraints on halo mass: Derive upper limits on M_r(m) from precision dynamics: lunar laser ranging for Earth-bound halos and planetary ephemerides for Solar halos, ensuring consistency with orbital data. Use these maxima to set conservative upper bounds on experimental signals.
• Coherence properties: The halo’s coherence length is ≈ R_r, and coherence time τ_c ~ 1/(m v^2) with v set by the halo’s potential (τ_c ~ 10 s × (10^−6 eV/m) for Earth halos at R_r ≳ R_⊕; Solar halos give ≥ two orders of magnitude longer τ_c).
• Experimental projections: Translate φ amplitudes into predicted δm_e/m_e and δα/α at Earth’s location and compare with present/prospective sensitivities across m. Consider four cases: Solar halo constraints via δm_e/m_e and δα/α (sensitivities 10^−16–10^−18), and Earth halo constraints via δm_e/m_e and δα/α (10^−14–10^−18). Compare to fifth-force/equivalence-principle bounds, Higgs-portal naturalness (Λ ~ 3 TeV), and ranges motivated by relaxion models (with Yukawa benchmark and LHC limits).

• Transient boson star encounters: For self-gravitating relaxion stars, although the overdensity can be enormous (e.g., δ ~ 7×10^21 for illustrative parameters), geometric encounter rates with Earth are typically extremely small. Achieving both δ > 1 and Γ ≳ 1 yr^−1 is only possible for m ≳ 10^−8 eV (MHz-scale frequencies). At lower masses, either Γ is far below 1 per year or the star’s density falls below ρ_local, making encounters unhelpful for near-term experiments. A more general rate scaling, Γ ≈ 0.05 yr^−1 δ^{−3/4} (m/10^−10 eV), quantifies this trade-off.
• Relaxion halos bound to Earth/Sun: If substantial relaxion mass is gravitationally bound as a halo around Earth or the Sun, the local amplitude can be greatly enhanced, leading to strong signals in precision atomic experiments. The halo size is set by the external gravitational potential, with exponential (R_r > R_ext) or Gaussian (R_r ≤ R_ext) profiles, and coherence times from ~10 s up to ≥10^3 s (Earth halo) and longer for Solar halos.
• Gravitational constraints: Upper limits on halo mass M_r(m) derived from lunar laser ranging (Earth halo) and planetary ephemerides (Solar halo) delineate allowed parameter space. These bounds are used to compute conservative maximum signal amplitudes at Earth’s surface.
• Experimental reach: With Solar halos, projected sensitivities in δm_e/m_e and δα/α at the 10^−16–10^−18 level can probe regions approaching technical naturalness in g_e and g_γ. With Earth halos, sensitivities at 10^−14–10^−18 can reach and surpass naturalness limits and test parameter space of physically motivated relaxion models (including ranges tied to Higgs mixing and LHC constraints). Even current or near-future experiments can test coherent relaxion DM scenarios if halos exist.
• Practical detection: The oscillatory nature enables strategies such as synchronized, distributed sensor networks and space-based clocks to map spatial profiles and phases, enhancing discovery potential.

The findings show that while isolated, self-gravitating relaxion stars are unlikely to be encountered frequently enough to aid terrestrial detection (except at relatively large masses m ≳ 10^−8 eV), gravitationally bound relaxion halos around the Earth or Sun can significantly amplify oscillatory signals in fundamental constants at Earth. This directly addresses the challenge of probing realistic relaxion parameter space with atomic physics: enhanced local amplitudes allow surpassing or complementing fifth-force and equivalence-principle constraints in key mass ranges. The work underscores the utility of oscillatory signal phase coherence over macroscopic distances, suggesting coordinated measurements (e.g., synchronized networks or repeated runs) and space-based platforms to characterize halo distributions. Consequently, atomic spectroscopy and clock comparisons emerge as powerful probes of light scalar DM under halo scenarios, potentially testing natural and model-motivated couplings.

The paper introduces and analyzes relaxion stars and, crucially, relaxion halos bound to external gravitational potentials (Earth or Sun) as mechanisms to enhance detectable signals of light scalar dark matter in atomic physics experiments. It derives encounter rates, stability and density estimates for boson stars, formulates halo structural relations set by external gravity, applies stringent gravitational constraints to bound halo masses, and translates these into projected experimental sensitivities. The main conclusion is that relaxion halos can render current and near-future atomic experiments sensitive to physically motivated relaxion models across wide mass ranges, reaching naturalness limits and beyond. Future work should address the detailed formation mechanisms and mass functions for relaxion structures (including baryonic effects), refine halo profile modeling beyond simple exponential/Gaussian approximations, and develop dedicated synchronized terrestrial and space-based measurement strategies to map and detect potential halos.

• Formation and mass distribution of relaxion structures are not derived from dedicated simulations; the relaxion fluctuation spectrum and halo/star formation history are assumed by analogy to axion scenarios.
• Relaxion halo profiles are approximated by exponential (point-mass potential) or Gaussian (constant-density sphere) forms; deviations due to realistic mass distributions are not modeled.
• Encounter-rate estimates for boson stars assume geometric cross-sections, virial velocities, and an O(1) fraction of DM in bound objects; actual fractions and distributions may differ.
• Self-interaction effects and stability limits are treated via benchmark criteria; detailed self-interaction potentials and collapse dynamics are deferred to supplementary analyses.
• Gravitational constraints (lunar laser ranging, planetary ephemerides) bound halo masses but rely on current ephemeris models and assumptions; more complex scenarios (e.g., mini-halos within Earth) are only briefly noted.
• Experimental projections assume target sensitivities in δm_e/m_e and δα/α and do not include all instrument systematics or network-phase analysis gains.