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Free-streaming neutrinos imprint a two-part phase shift on the 21-cm spectrum: the known BAO density phase plus a newly computed VAO velocity phase about 1.5 times larger, blended in a redshift-dependent way at cosmic dawn.

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2026-08-05 10:49 UTC pith:M2QIIU3F

load-bearing objection New VAO phase-shift template is a solid result; the 21-cm prediction rests on a manual imprinting shortcut that needs a self-consistent N_eff check. the 3 major comments →

arxiv 2509.03595 v1 pith:M2QIIU3F submitted 2025-09-03 astro-ph.CO hep-phhep-th

Tracing the Neutrino-Induced Phase Shift in the 21-cm Spectrum

Gabriele Montefalcone , Hector Afonso G. Cruz , Julian B. Munoz , Ely D. Kovetz , Marc Kamionkowski This is my paper
classification astro-ph.CO hep-phhep-th
keywords 21-cm cosmologycosmic dawnneutrino phase shiftfree-streaming neutrinosvelocity acoustic oscillations (VAOs)baryon acoustic oscillations (BAOs)effective number of relativistic specieslight relics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that the phase shift free-streaming neutrinos imprint on acoustic oscillations — a signature already measured in the CMB and galaxy surveys — appears in the cosmic-dawn 21-cm spectrum in a new, two-component form. The 21-cm signal contains two oscillation families with the same frequency but different physics: baryon acoustic oscillations (BAOs) driven by density fluctuations, and velocity acoustic oscillations (VAOs) driven by the supersonic relative motion of baryons and dark matter. The paper computes the neutrino phase shift in VAOs for the first time and shows it is distinct from the BAO phase, about 1.5 times larger in amplitude and delayed to a transition scale about three times larger, because VAOs arise from a four-point correlation of the velocity field rather than a two-point density correlation. The net 21-cm phase is then an amplitude-weighted blend of the two templates whose composition changes with redshift and scale, marking the phase of 21-cm oscillations as a new probe of light relics and of early star formation.

Core claim

Free-streaming neutrinos, decoupled near 1 MeV and moving nearly at the speed of light, gravitationally pull the pre-recombination photon-baryon plasma ahead of its sound horizon, adding a coherent, scale-dependent phase to its acoustic oscillations. The paper shows this phase reaches the 21-cm brightness-temperature spectrum through two channels that respond differently. Density-sourced BAOs inherit the standard phase, fitted by the template f_phi(k) = phi_inf / [1 + (k_star/k)^xi] with phi_inf ≈ 0.27, k_star ≈ 0.04 Mpc^-1, xi ≈ 0.74. Velocity-sourced VAOs, whose source spectrum P_eta is effectively a four-point function of the baryon-dark-matter relative velocity, carry a phase with phi_in

What carries the argument

The central object is P_eta(k), the power spectrum of η ≡ v_cb²/⟨v_cb²⟩ − 1, the normalized square of the baryon-dark-matter relative velocity, the field that modulates early star formation and sources VAOs. Rotational invariance forces physical effects to depend on v_cb², so P_eta is effectively a four-point correlation of the underlying velocity field. In Fourier space each k is assembled from all pairs of velocity modes (k1, k2) with k1 + k2 = k, weighted by the geometric factor (k² − k1² − k2²)²/(k1k2) = k1k2 cos²φ, which favours aligned configurations. This convolution makes the phase of P_eta a mode-mixed, amplified version of the phase carried by the two-point velocity spectrum: in th

Load-bearing premise

The paper's central signature rests on the assumption that the neutrino phase shift passes linearly into the 21-cm spectrum as a weighted sum of the BAO and VAO templates — a step it verifies by manually imprinting those fitted phases into the source spectra, not by running the full 21-cm calculation end-to-end with a different number of neutrino species.

What would settle it

Run the paper's 21-cm pipeline twice, once with the fiducial neutrino content and once with a different effective number of relativistic species, keeping the baryon density, matter-radiation equality, and sound-horizon scales fixed, with no manual phase imprinting, and extract the peak/trough mode shifts of the 21-cm spectrum across z ∈ [10, 30]. If the extracted phase does not match the weighted-superposition prediction of Eq. (17) — VAO-like at large scales during Lyman-alpha coupling and BAO-like at late times and small scales — the central claim fails. A lighter observational test: measure

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • The phase of acoustic oscillations in the 21-cm power spectrum becomes a new, independent probe of free-streaming light relics at cosmic dawn, complementary to the CMB and galaxy-survey BAO phases.
  • Because the two-point velocity spectrum P_vcb carries the standard BAO phase while the four-point spectrum P_eta carries the modified one, a phase measurement directly tests the four-point origin of velocity-induced statistics.
  • The phase shift tracks the BAO/VAO amplitude balance, so its redshift evolution is simultaneously sensitive to astrophysics: enhanced Pop III star formation sustains VAO dominance longer and keeps the large-scale phase elevated above the BAO template.
  • Whatever the astrophysics, small-scale modes (k ≳ 0.15-0.3 Mpc^-1) stay anchored to the BAO phase because VAOs are damped there, giving the signal a robust asymptotic limit.
  • The extracted phase templates support a fitting framework in which the phase amplitude is a free parameter, the 21-cm analogue of BAO phase fitting in galaxy surveys, enabling direct constraints on the effective number of relativistic species.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The two-point versus four-point phase distinction is general: any observable built from a squared or otherwise nonlinear function of a phase-shifted field should inherit an amplified, mode-mixed phase. The same logic could be checked in other quadratic statistics, such as CMB lensing or 21-cm bispectra.
  • If Eq. (17) survives a fully self-consistent N_eff run, phase measurements alone cannot separate neutrino physics from astrophysics, since the blend weights A_BAO/A_VAO are themselves astrophysical; joint fits of phase and peak amplitudes would be needed to break the degeneracy.
  • The largest difference between the two templates sits in the transition region k ≈ 0.05-0.15 Mpc^-1 at z ≈ 23-27, so that region is where a future cosmic-dawn 21-cm interferometer measurement would most decisively confirm or reject the predicted VAO phase contribution.
  • The paper leaves open the natural testable extension of a quantitative sensitivity forecast: folding in instrumental noise, foregrounds, and sample variance to compute whether the predicted large-scale phase offset is detectable with upcoming arrays, and with what integration time.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the phase shift imprinted by free-streaming neutrinos on the 21-cm power spectrum during cosmic dawn (z∼10–30). It first recovers the established BAO phase-shift template from CLASS power spectra, then applies the same methodology to the power spectrum Pη(k) of the squared baryon–DM relative velocity that sources velocity acoustic oscillations (VAOs). The resulting VAO template (Table II, Fig. 2) has a larger asymptotic phase (φ∞≈0.36 vs 0.27), a larger transition scale (k*≈0.09 vs 0.04 Mpc^-1), and a smoother scale dependence (ξ≈1.40 vs 0.74). The paper then proposes that the 21-cm phase shift is a weighted superposition of BAO and VAO phases, Eq. (17), with weights determined by the relative BAO/VAO amplitudes, and tests this by manually imprinting the fitted templates into zeus21 source spectra (Sec. IVC, Figs. 3–4). The claimed result is a redshift- and scale-dependent phase that interpolates between VAO-dominated behavior at early times/large scales and BAO-dominated behavior at late times/small scales.

Significance. The new VAO phase-shift template is a genuinely interesting and plausible result: because Pη is a four-point function of the relative-velocity field, it is natural that the phase accumulates differently than in the two-point density spectra, and the CLASS-based extraction with fixed ωb, aeq, rs, and θd follows the established phase-shift literature. If the 21-cm phase shift is indeed the weighted average claimed in Eq. (17), this would provide a new observable that is sensitive both to free-streaming relics and to the astrophysics controlling BAO/VAO amplitudes. The paper is transparent about its modeling choices and provides uncertainty ranges and a direct test inside zeus21. However, the central predictive claim is not yet fully supported: the 21-cm phase shift is obtained by manual imprinting of fitted templates rather than by a self-consistent Neff variation, and the validation of Eq. (17) is in part circular because the same templates are used as input and their weights are measured from the output. These issues are addressable and do not invalidate the underlying physical idea, but they currently prevent the paper from making a fully convincing first-principles prediction.

major comments (3)
  1. [Sec. IVC and Fig. 3] The 21-cm phase-shift prediction is generated by manually imprinting the fitted BAO/VAO templates into the fiducial zeus21 source spectra, as stated in Sec. IVC and the Fig. 3 caption. This assumes that under a real Neff variation the phase shift is the only modification relevant to the 21-cm peak positions, and that the weights ABAO(k,z) and AVAO(k,z) extracted from the fiducial model are Neff-independent. Because the paper fixes ωb, aeq, and rs while varying Neff, a consistent variation also changes ωm and hence the expansion/growth/star-formation/heating/ionization history, which can alter BAO/VAO amplitudes and the smooth component. No test of this assumption is provided. I request at least one self-consistent zeus21 run with Neff varied, or a quantitative estimate of the resulting changes in ABAO/AVAO, before the extracted phase can be claimed as the physical 21-cm neutrino-induced
  2. [Eq. (17), Sec. IVC, Fig. 4] The numerical validation of Eq. (17) is partially circular: the same fitted templates are imprinted into the source spectra, and the amplitude weights are measured from those very outputs. The agreement therefore demonstrates that the weighted-sum algebra of Eq. (17) holds for the templates inside zeus21, but it does not test whether a physical Neff variation produces the same phase. The small-phase derivation of Eq. (17) is a useful mathematical identity for a sum of two sine waves, but it does not by itself turn the fitted templates into an independent prediction. Please validate against a full Neff run, or clearly reframe the central result as a template model whose physical validity remains to be established.
  3. [Sec. IVB and Table II] The VAO phase-shift template is presented as the paper's primary theoretical contribution, but the values in Table II (φ∞=0.36±0.02, k*=0.09±0.01, ξ=1.40±0.08) are best-fit parameters of the same empirical functional form used for BAOs, extracted from CLASS spectra with Neff∈[0,3.3]. The analytic argument around Eq. (16) explains the qualitative direction of the differences (larger φ∞, larger k*, smoother transition), but it is not a first-principles calculation of those values. The paper should explicitly distinguish the fitted template from an analytic derivation, or supplement the fit with a quantitative convolution estimate.
minor comments (4)
  1. [Sec. II, after Eq. (15)] The sentence 'governed in fφ(k) by the parameters k⋆ and ξ<0' conflicts with Eq. (15) and Table II, where ξ>0; with ξ<0 the template would not approach φ∞ at high k. Please correct the sign.
  2. [Sec. III.B, Eqs. (5)–(8)] The text refers to 'correlation functions ψ0(r), ψ1(r) in Eq. (5)', but Eqs. (7) and (8) define ψ0 and ψ2; ψ1 is never defined. Also, Eq. (5) writes Pη(k) in terms of an r integral while the surrounding text says 'configuration space'—please harmonize the notation.
  3. [Sec. IVC and Fig. 4 caption] The main text says uncertainties are obtained by sampling BAO/VAO parameters within 95% confidence intervals, while the Fig. 4 caption says 1σ. Please use one convention consistently.
  4. [References [10] and [26]] Reference [10] is cited as 'in preparation' for the BAO/VAO decomposition underlying Eq. (12) and the isolation procedure. If this companion paper is not yet public, please include the essential equations or a preprint number so that the reader can verify the decomposition. Reference [26] is also 'in preparation'.

Circularity Check

2 steps flagged

The 21-cm phase-shift 'prediction' is produced by manually imprinting the fitted BAO/VAO templates into zeus21, so the output reduces, by construction, to the fitted inputs; the central decomposition additionally leans on an unpublished companion paper by the same authors.

specific steps
  1. fitted input called prediction [Sec. IVC (paragraph after Eq. 17; Fig. 3 caption)]
    "The 21-cm spectra are computed here using zeus21 [36, 37, 46], assuming our fiducial ΛCDM cosmology (Table I) and manually imprinting the neutrino-induced phase shifts in the source density and velocity spectra according to their respective best-fit templates. This approach cleanly isolates the phase effects in the resulting 21-cm spectrum by avoiding the complications that would arise from self-consistently varying Neff as a cosmological parameter. By applying this manual imprinting method across different redshifts and extracting the resulting mode shifts, we can directly derive the phase-sh"

    The output — the 21-cm phase-shift template shown in Figs. 3, 4, and 7 — is generated by taking the already-fitted BAO and VAO phase templates (Eqs. 13–15, Table II) and imprinting them into the zeus21 source spectra. The 'extracted' 21-cm phase shifts therefore contain, by construction, exactly the phase information that was inserted. The later validation of the weighted-sum formula, Eq. (17), uses amplitude weights measured from those same imprinted spectra ('we extract the amplitude ratios A_BAO(k,z) by measuring the relative peak heights between the isolated BAO component and the full 21-cm oscillatory spectrum'), so the agreement is a consistency check of linear superposition inside the imprinting scheme, not an independent prediction of how a self-consistent change in Neff would affe

  2. self citation load bearing [Sec. IIIB and Sec. I; Ref. [10]]
    "The individual BAO and VAO oscillatory contributions are obtained by computing the 21-cm power spectrum with different combinations of the source spectra, namely using the total baryon and matter auto- and cross-correlation spectra with the no-wiggle η spectrum for O21,BAO(k,z), while O21,VAO(k,z) uses the full η spectrum with the no-wiggle density spectra [10]."

    Ref. [10] is 'H. A. G. Cruz, G. Montefalcone, J. B. Munoz, E. D. Kovetz, and M. Kamionkowski, in preparation' — an unpublished companion paper by the same five authors. The paper's central decomposition of the 21-cm signal into additive BAO and VAO components (Eq. 12), the relative amplitude weights used in Eq. (17), the evaluation of the velocity transfer function at z=50 'rather than at kinematic decoupling', and the updated zeus21 code are all attributed to this self-citation. The claim that the 21-cm phase shift is a weighted superposition of separately computed BAO and VAO templates therefore rests on the authors' own unpublished framework, not on an externally available, machine-checked, or independently benchmarked result. This is load-bearing self-citation rather than a peripheral

full rationale

The paper has two genuinely new components: the first computation of a neutrino phase-shift template for the scalar velocity spectrum Pη (the VAO template, Table II) and the proposal that the 21-cm phase shift interpolates between BAO and VAO behavior during cosmic dawn. The VAO template is a fit to CLASS spectra computed with Neff varied while ωb, aeq, rs, and θd are fixed; fitting a template to a Boltzmann code is standard practice and not itself circular. The circularity enters at the 21-cm stage. The paper does not perform a self-consistent zeus21 run with Neff varied. Instead it takes the fitted BAO and VAO phase templates as inputs and manually imprints them into the fiducial source spectra; the resulting 21-cm spectra are then used to 'derive' the 21-cm phase-shift template. Because the input phases and output phases are the same fitted functions, the central claim that the 21-cm phase shift is a weighted average of the BAO and VAO templates is, in the numerical demonstration, true by construction. The paper's analytic derivation of Eq. (17) from Eq. (18) is internally consistent, but Eq. (18) is an assumed sinusoidal ansatz and the additive decomposition of Eq. (12) is imported from the authors' unpublished companion paper (Ref. [10]). Those imports are load-bearing because without them there is no independent support for the relative weights or for the claim that the phase shift propagates linearly through the 21-cm transfer functions without additional mode coupling or Neff-dependent amplitude effects. The paper itself flags the key limitation: the imprinting method was chosen 'to cleanly isolate the phase effects without the complications that would arise from self-consistently varying Neff.' That stated limitation is exactly the untested assumption that makes the 21-cm phase-shift signature a model construction rather than a prediction from a self-consistent cosmology. Weighing all of this, the score is 6: some of the central 'predictions' reduce by construction to the fitted inputs, and the framework leans on an in-preparation self-citation, but the VAO template itself and the general idea that BAO/VAO weights redshift-evolve retain independent content.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

All free parameters are fits to CLASS-derived spectra; the VAO template parameters are new fits from the P_eta spectrum. The main axioms are standard cosmological assumptions plus the paper-specific manual imprinting ansatz. No new particles or forces are introduced.

free parameters (6)
  • φ∞,BAO = 0.27 ± 0.03
    Asymptotic BAO phase shift, fit to CLASS spectra (Table II).
  • k*,BAO = 0.04 ± 0.01 Mpc^-1
    BAO transition scale, fit to CLASS spectra (Table II).
  • ξ_BAO = 0.74 ± 0.18
    BAO steepness parameter, fit to CLASS spectra (Table II).
  • φ∞,VAO = 0.36 ± 0.02
    Asymptotic VAO phase shift, fit to CLASS P_eta spectra (Table II).
  • k*,VAO = 0.09 ± 0.01 Mpc^-1
    VAO transition scale, fit to CLASS P_eta spectra (Table II).
  • ξ_VAO = 1.40 ± 0.08
    VAO steepness parameter, fit to CLASS P_eta spectra (Table II).
axioms (6)
  • domain assumption Adiabatic initial conditions for the photon-baryon-plasma acoustic oscillations
    Sec. II, Eq. (1): the phase-shift template assumes adiabatic initial conditions; the paper states that the phase shift is a clean signature only under this assumption.
  • domain assumption Neutrinos are massless throughout
    Table I note: stated as a valid approximation for the purpose of phase-shift extraction, following ref. [25].
  • standard math The scalar velocity power spectrum P_eta is a four-point function satisfying Eq. (16)
    Sec. IVB: the Fourier-space representation is taken from Ferraro et al. 2012 and prior work; the VAO phase-shift derivation relies on this convolution.
  • domain assumption The no-wiggle separation via sine transform isolates the oscillatory component without bias
    Sec. IVA, IVB: used to define O_j(k); established methodology from refs. [28,54], but the specific implementation details are not fully specified.
  • domain assumption The 21-cm acoustic signal is well approximated by the sum of independent BAO and VAO components, Eq. (12)
    Sec. IIIB: based on the small cross-correlation between VAOs and density fluctuations, from refs. [53,66,67].
  • ad hoc to paper Manually imprinting the phase templates into source spectra is equivalent to self-consistently varying N_eff
    Sec. IVC: the paper explicitly avoids varying N_eff in zeus21 and imprints templates instead; no full-N_eff validation is provided.

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Cite this review

Pith. "Pith review of Tracing the Neutrino-Induced Phase Shift in the 21-cm Spectrum." pith.science (2026). https://pith.science/paper/M2QIIU3F

@misc{pith2026250903595,
  author       = {Pith},
  title        = {Pith review of: Tracing the Neutrino-Induced Phase Shift in the 21-cm Spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2QIIU3F}},
  note         = {Machine review of arXiv:2509.03595}
}
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read the original abstract

We study the phase shift that free-streaming neutrinos imprint on the 21-cm power spectrum during cosmic dawn, computing for the first time its effect on both density- and velocity-induced acoustic oscillations. Neutrinos are known to generate a characteristic phase shift in the acoustic oscillations of the photon-baryon plasma before recombination, a signature already detected in the cosmic microwave background (CMB) as well as the spectrum of baryon acoustic oscillations (BAOs) extracted from galaxy surveys. We show that in the 21-cm signal this phase shift is distinct from that observed in the CMB and BAO spectra, exhibiting a characteristic mode and redshift dependence arising from the additional contribution of the so-called velocity acoustic oscillations (VAOs), sourced by the baryon-dark matter relative velocities. Our results establish the phase of acoustic oscillations in the 21-cm spectrum as a promising new avenue for probing free-streaming light relics at cosmic dawn, complementary to existing CMB and BAO measurements.

Figures

Figures reproduced from arXiv: 2509.03595 by Ely D. Kovetz, Gabriele Montefalcone, Hector Afonso G. Cruz, Julian B. Munoz, Marc Kamionkowski.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the phase shift as it appears in the matter power spectrum [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Templates of the BAO and VAO phase shift [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Illustration of the phase shift as it appears in the 21-cm power spectrum at the benchmark redshift [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of the neutrino-induced phase shift in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Extracted phase-shift templates for the 21-cm spec [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Rise and Fall of Acoustic Oscillations at Cosmic Dawn

    astro-ph.CO 2026-07 conditional novelty 7.0

    Ignoring the percent-level phase offset between BAO and VAO features in the cosmic-dawn 21-cm power spectrum biases H(z) by ~2%; joint BAO–VAO templates are required for standard-ruler inference.

Reference graph

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