Are Primordial Black Holes Dark Matter? CMB μ-Distortion Limits & QCD-Epoch Formation

The Subsolar Merger S251112cm and GWTC-5.0 Implications

1. Event Kinematics and LVK Parameters

   In November 2025, the LVK collaboration recorded S251112cm, an unprecedented gravitational-wave transient characterized by a remarkably low chirp mass. Bayesian parameter estimation places the chirp mass definitively within the 0.1 to 0.87 M☉ range at greater than 90% confidence, localized to a luminosity distance of approximately 90 Mpc. Crucially, the component masses fall firmly below the Chandrasekhar limit, occupying a regime where standard stellar evolution mechanisms and core-collapse supernovae cannot easily populate the macroscopic black hole mass function. The absolute absence of any electromagnetic counterpart, combined with the extreme subsolar nature of the primary and secondary components (>99% statistical probability of both being strictly subsolar), severely disfavors a double neutron star or white dwarf binary origin. From a theoretical standpoint, this kinematic profile heavily elevates the primordial hypothesis, suggesting the inspiral of two dark sector macroscopic objects formed in the very early universe.

2. The Magaraggia & Cappelluti Analysis

   Following the release of the GWTC-5.0 catalog, which now comprises over 390 highly vetted gravitational-wave events, Magaraggia & Cappelluti (ApJ, 2026) conducted an exhaustive population synthesis analysis of the current dataset. Their findings reveal a statistically significant deviation from standard isolated binary evolution models at the extreme low-mass boundary. By injecting S251112cm-like synthetic events into their hierarchical Bayesian framework, they demonstrated that known astrophysical channels require highly non-physical, fine-tuned mass-transfer efficiencies to produce such a binary in the local universe. Consequently, the subsolar merger rate directly inferred from GWTC-5.0 implies a non-negligible primordial contribution to the overall merger density. This necessitates a rigorous re-evaluation of early universe formation mechanisms—specifically those governed by the Friedmann equations during radiation domination—that are natively capable of generating a highly clustered population of subsolar mass black holes without violating existing large-scale structure bounds.

QCD-Epoch Formation Dynamics

1. Equation of State Softening

   The macroscopic formation of PBHs is exquisitely sensitive to both the background expansion rate and the thermodynamic properties of the primordial plasma. During the QCD phase transition at a cosmic temperature of T ≈ 150 MeV, the chiral symmetry breaking and subsequent confinement of free quarks and gluons into hadrons induce a transient softening of the cosmological equation of state, w(T). This critical deviation from a pure non-interacting radiation fluid (where w = 1/3) alters the critical density contrast, δ_c, required for a super-horizon Hubble patch to undergo complete gravitational collapse against local radiation pressure upon horizon reentry.

   w(T) = p / ρ = (1/3) [ 1 − α_c exp(−(T − T_c)² / ΔT²) ]

   The localized minimum in the sound speed, c_s² = w(T), dynamically lowers the effective Jeans mass. Consequently, primordial density fluctuations entering the horizon during this specific epoch experience a profoundly reduced restorative pressure gradient, leading to an exponentially enhanced collapse probability. This mechanism naturally produces a pronounced peak in the PBH mass function around the solar and subsolar mass scales, precisely encompassing the component masses inferred for the S251112cm event.

2. Press-Schechter Formalism and the Lognormal Mass Function

   To rigorously quantify the abundance of PBHs generated during the QCD epoch, we employ the Press-Schechter formalism. The mass fraction of the universe collapsing into black holes of mass M at the exact time of horizon entry, β(M), is derived from the integration of the primordial density perturbation probability distribution function, assuming strictly Gaussian initial conditions seeded by inflation. The variance of the density field smoothed on a comoving scale R, denoted as σ²(M), governs the high-sigma tail of this distribution.

   β(M) = ∫_δ_c^∞ (2π σ²(M))^−1/2 exp(−δ² / 2σ²(M)) dδ = (1/2) erfc(δ_c / √2 σ(M))

   The resulting fractional contribution of Primordial Black Holes to the total dark matter density, f_PBH, is subsequently obtained by integrating this production rate over the relevant mass spectrum. Because of the direct non-linear coupling between the scale-dependent primordial power spectrum and the thermal history of the equation of state during the phase transition, the predicted macroscopic PBH mass distribution is typically well-described by a lognormal function.

   f_PBH = (Ω_PBH / Ω_DM) = ∫ M_PBH⁻¹ (dn / d(ln M_PBH)) dM_PBH

CMB Anisotropy and Energy Injection

1. Bondi and Disk Accretion Energy Injection

   If subsolar PBHs constitute a macroscopic fraction of dark matter, their kinematic interactions with the ambient baryonic gas during the cosmic dark ages will inevitably leave profound imprints on the Cosmic Microwave Background. As PBHs traverse the intergalactic medium, they accrete baryons via the standard Bondi-Hoyle-Lyttleton mechanism, supplemented by disk accretion when the angular momentum of the infalling material exceeds the Innermost Stable Circular Orbit (ISCO). The seminal frameworks developed by Ali-Haïmoud, Kamionkowski, Poulin, and Serpico demonstrate that a substantial fraction of the accreted rest-mass energy is radiated away as X-rays and high-energy ultraviolet photons.

   dE/dt = ϵ_disk ṁ c² + ϵ_sph (4π G² M_PBH² ρ_b c²) / (v_rel² + c_s²)^3/2

   This continuous energy injection, governed by the radiative efficiencies ϵ_disk and ϵ_sph alongside the effective relative velocity, cascades violently through the primordial plasma. It effectively heats the baryons, excites neutral hydrogen and helium atoms, and crucially, modifies the free electron fraction, x_e(z), significantly altering the thermal history long after the epoch of standard recombination.

2. TT/EE Damping Limits and x_e(z)

   The alteration of the recombination history profoundly impacts the primary CMB temperature (TT) and polarization (EE) power spectra. A delayed or broadened recombination surface drastically increases the integrated optical depth to Thomson scattering, τ. This enhanced scattering probability systematically damps the amplitude of the acoustic peaks at high multipoles (l > 1000) while simultaneously generating additional polarization signals at large angular scales via the re-scattering of local quadrupole anisotropies. By utilizing exact numerical solutions for the energy deposition fractions and marginalizing over standard ΛCDM parameters, current Planck satellite data places stringent upper limits on f_PBH. For the 0.1 to 1.0 M☉ mass window directly relevant to the S251112cm progenitor, the TT/EE damping constraints conservatively cap f_PBH at approximately 10⁻². This critical threshold indicates that while subsolar PBHs may not presently constitute the entirety of the dark matter mass budget, they remain a highly compelling and testable sub-component of the dark sector.

μ-Distortions and Next-Generation Forecasts

1. COBE/FIRAS Bounds and PIXIE/BISOU Sensitivity

   Energy injection occurring well before the recombination epoch (5 × 10⁴ < z < 2 × 10⁶) cannot be efficiently thermalized by Compton scattering and Bremsstrahlung processes. The resulting thermodynamic imbalance permanently imprints a chemical potential μ-distortion onto the CMB Planck spectrum. The legacy COBE/FIRAS mission robustly constrained this spectral distortion to |μ| < 9 × 10⁻⁵. However, the energy dissipated by Silk damping of the small-scale density perturbations—which would have originally sourced the PBHs—provides a firm theoretical lower bound on μ. Future space-based spectral observatories, such as PIXIE and the proposed BISOU concept, aim to improve this sensitivity by three to four orders of magnitude, reaching a theoretical noise floor of μ ≈ 10⁻⁸. A detection at this precise level would offer unprecedented insight into the primordial power spectrum at scales of k ≈ 10⁴ Mpc⁻¹, effectively probing the exact inflationary dynamics required to seed S251112cm-like progenitors prior to the QCD epoch.

2. Scalar-Induced Gravitational Waves and the Einstein Telescope

   The immense scalar density perturbations required to overcome the collapse threshold δ_c inherently source a secondary background of tensor perturbations at second order in cosmological perturbation theory. These scalar-induced gravitational waves (SIGWs) present a stochastic background that permeates the universe, operating completely independent of the CMB photon diffusion limits. Upcoming observatories like the Simons Observatory and LiteBIRD will probe the largest scales of this tensor background via B-mode polarization signatures. More acutely, the terrestrial Einstein Telescope (ET) will be profoundly sensitive to the high-frequency tail of the SIGW spectrum. If the GWTC-5.0 subsolar events are genuinely primordial in nature, ET will simultaneously resolve thousands of highly redshifted PBH mergers while detecting the stochastic SIGW hum generated during their initial formation. The cross-correlation of the μ-distortion temperature map (forming a μT correlation observable) with the ET stochastic background will definitively isolate the QCD-epoch formation channel from standard astrophysical noise.

Conclusion