The Neutrino Acoustic Phase Shift: ACT DR6 Constraints on Light Relics

The Standard Model Baseline and Relativistic Formalism

1. Relativistic Energy Density and Expansion

   During the radiation-dominated epoch, the expansion rate of the universe is exquisitely sensitive to the total relativistic energy density. In standard cosmological perturbation theory, this energy density is parameterized by the effective number of relativistic species, N_eff, which normalizes the energy density of neutrinos and other potential light relics relative to the photon energy density. The Friedmann equation linking the Hubble parameter to these components illustrates how any deviation from the Standard Model directly alters the background expansion history.

   H² = (8πG/3) [ ρ_m + ρ_γ ( 1 + (7/8) (4/11)⁴/³ N_eff ) ]

   By definition, N_eff isolates the non-photonic relativistic contribution. Extracting N_eff algebraically from the energy density ratio explicitly demonstrates its dependence on the thermal history of neutrino decoupling. The canonical Standard Model prediction, N_eff = 3.044, accounts for non-instantaneous neutrino decoupling, finite temperature QED corrections, and flavor oscillations.

   N_eff = (8/7) (11/4)⁴/³ (ρ_ν / ρ_γ)

2. Thermal Floors for Light Relics

   Any hypothesized particles that decoupled from the primordial plasma after the QCD phase transition will contribute a minimum amount to the effective number of relativistic species. These theoretical minimums, often referred to as thermal floors, are strictly defined by the spin statistics and the degrees of freedom of the interacting particles. If a light relic was in thermal equilibrium with the Standard Model bath at any point, the conservation of comoving entropy density enforces a strict lower bound on its present-day energy density.

   For a single scalar degree of freedom (spin-0), the minimum contribution is ΔN_eff ≥ 0.027. A Weyl fermion (spin-1/2) introduces a larger floor of ΔN_eff ≥ 0.047, while a gauge vector boson (spin-1) generates a floor of ΔN_eff ≥ 0.054. The precision of ACT DR6 places immense pressure on extensions to the Standard Model, as the allowed window for hidden sectors and early dark energy models narrows towards these fundamental thermal floors.

The Free-Streaming Boltzmann Hierarchy

1. Collisionless Propagation and Supersonic Speeds

   To accurately model the impact of dark radiation and active neutrinos on the CMB, one must track their phase-space distribution using the collisionless Boltzmann equation. Because these species decouple well before recombination, they do not participate in the acoustic oscillations of the strongly coupled photon-baryon plasma. Instead, they free-stream at the speed of light. The evolution of the neutrino density contrast, δ_ν, and velocity divergence, θ_ν, in the synchronous gauge is intrinsically linked to the metric perturbations.

   ∂_τ δ_ν + (4/3) k θ_ν = 4 ∂_τ h

   This Euler-equivalent equation demonstrates how the metric perturbation h sources the velocity divergence of the free-streaming fluid. Unlike the photon-baryon fluid, which is restricted to a sound speed of c_s = 1/√3, free-streaming neutrinos propagate at c = 1. This supersonic velocity allows the free-streaming particles to traverse ahead of the acoustic wavefront, creating a gravitational wake that pulls on the tightly coupled plasma.

2. Metric Perturbations and Higher Multipoles

   The full description of this free-streaming radiation requires an infinite hierarchy of multipole moments. The higher-order moments (ℓ ≥ 2) represent anisotropic stress, which actively sources the tensor and scalar metric perturbations in the Einstein field equations. The truncation of this Boltzmann hierarchy at a finite multipole is a standard approximation in modern Einstein-Boltzmann solvers like CAMB and CLASS, but the physical implications of the unsuppressed anisotropic stress are profound.

   It is precisely this anisotropic stress that dampens the amplitude of the CMB acoustic peaks while simultaneously altering their phase. Any extra relativistic species that is collisionless will exhibit this behavior, whereas self-interacting dark radiation would lack anisotropic stress, providing a clean phenomenological distinction between free-streaming N_eff and interacting dark radiation (N_idr).

The Neutrino Acoustic Phase Shift

1. Supersonic Drag and Phase Modulation

   The gravitational interaction between the supersonic free-streaming neutrinos and the photon-baryon acoustic waves culminates in a distinct observable: the neutrino acoustic phase shift. As mathematically formalized by Bashinsky and Seljak (2004), the density perturbations of the free-streaming species induce a temporal shift in the oscillation phase of the acoustic modes. This shift is remarkably insensitive to the precise details of recombination, making it a robust signature of early-universe physics.

   Δθ_ℓ ≈ - 0.19 π ( ρ_ν / ( ρ_γ + ρ_ν ) )

   This analytic approximation reveals that the phase shift Δθ_ℓ is directly proportional to the fractional energy density of the free-streaming radiation. An increase in N_eff from collisionless relics will linearly enhance this phase shift, shifting the positions of the CMB acoustic peaks to slightly higher multipoles (smaller angular scales).

2. CMB and BAO Observational Detections

   The existence of the cosmic neutrino background, and its specific free-streaming nature, is not merely a theoretical construct. The temporal phase shift predicted by the Bashinsky-Seljak formalism was robustly isolated in the primary CMB temperature and polarization anisotropies by Follin et al. in 2015. Their analysis confirmed that the acoustic peaks are shifted exactly as expected for a standard cosmic neutrino background.

   Furthermore, because the baryon acoustic oscillations (BAO) are the late-time, low-redshift imprint of these same primordial sound waves, the phase shift must also survive into the spatial distribution of galaxies. Baumann et al. (2019) successfully detected this signature in the clustering of large-scale structure, providing an independent, low-redshift verification of the free-streaming nature of N_eff. Both detections serve to rigidly anchor the parameter space now being probed by ACT DR6.

ACT DR6 Constraints and Interacting Dark Radiation

1. High-Resolution N_eff Measurements

   The ACT DR6 data release (Calabrese et al. 2025) represents a monumental leap in the precision characterization of the Silk damping tail and the high-ℓ CMB polarization. By combining high-fidelity measurements of the TE and EE power spectra, ACT DR6 derives a baseline value of N_eff = 2.86 ± 0.13. When combined with Big Bang Nucleosynthesis (BBN) primordial abundance data, the constraint tightens to N_eff = 2.89 ± 0.11.

   These values sit remarkably close to, yet slightly below, the Standard Model expectation of 3.044. While statistically consistent with the standard paradigm within roughly 1.5σ, this slight downward pull significantly restricts the viability of many light relic models. Any hidden sector that successfully thermalized in the early universe is tightly squeezed by the upper bounds of this new measurement, practically ruling out large families of fully decoupled relativistic species.

2. Silk Damping and N_idr Limits

   Beyond free-streaming relics, ACT DR6 profoundly constrains interacting dark radiation (N_idr). Models of interacting dark radiation postulate a fluid of dark particles that are self-interacting or coupled to dark matter, thus possessing a negligible anisotropic stress. Because they do not free-stream, they do not induce the characteristic neutrino acoustic phase shift, but they still alter the expansion history and modify the Silk damping scale.

   The enhanced expansion rate from N_idr increases the Hubble parameter at recombination, which subsequently modifies the diffusion length of photons. The precise measurement of the Silk-damping tail suppression by ACT yields an exceptionally stringent upper bound of N_idr < 0.134 at 95% confidence. This limit severely restricts dark sector models proposing dark acoustic oscillations or strongly coupled fluid dynamics in the early universe, confirming that the radiation content of the cosmos is predominantly standard and free-streaming.

Future Prospects and Conclusion