Derivation 8: High-Redshift Test of a₀(z) = cH(z)/(2π)

Derivation 8: High-Redshift Test of a₀(z) = cH(z)/(2π)#

The Prediction#

The sync_cost framework (Derivation 03) derives the MOND acceleration scale as the threshold where local gravitational synchronization cost equals cosmological mean-field maintenance cost:

a₀ = cH₀ / (2π) ≈ 1.04 × 10⁻¹⁰ m/s²

(observed: 1.2 × 10⁻¹⁰ m/s²; ratio 1.15, attributed to the exact form of the Kuramoto frequency distribution g(ω)).

Because the derivation ties a₀ to the instantaneous Hubble parameter, not to its present-day value, the framework predicts:

a₀(z) = c H(z) / (2π)

where H(z) = H₀ √(Ω_m(1+z)³ + Ω_Λ) in flat ΛCDM.

Numerical predictions (Planck 2018: H₀=67.4, Ω_m=0.315, Ω_Λ=0.685)#

z

H(z) [km/s/Mpc]

a₀(z) [m/s²]

a₀(z)/a₀(0)

Lookback [Gyr]

0

67.4

1.04 × 10⁻¹⁰

1.00

0.0

0.5

89.1

1.38 × 10⁻¹⁰

1.32

5.2

1.0

120.7

1.87 × 10⁻¹⁰

1.79

8.0

2.0

204.3

3.16 × 10⁻¹⁰

3.03

10.5

3.0

307.7

4.76 × 10⁻¹⁰

4.57

11.7

5.0

558.7

8.64 × 10⁻¹⁰

8.29

12.6

Key result: At z = 2, a₀ should be ~3× its present value. This is a large, in-principle-detectable effect over a redshift range now accessible to JWST and ALMA.

What this means observationally#

The baryonic Tully-Fisher relation (BTFR) has the form:

M_b = A × V_flat⁴

where the normalisation A ∝ 1/a₀ in MOND. If a₀(z) increases with z, then at fixed V_flat, the inferred baryonic mass should be smaller — or equivalently, at fixed M_b, the rotation velocity should be higher — at high z compared to today. Specifically:

  • The BTFR zero point should shift by ~0.5 dex in log(M_b) between z=0 and z=2.

  • Rotation curves should transition from Newtonian to MOND behaviour at a higher acceleration (larger radius for a given mass) at high z.

Competing Predictions#

Framework

a₀ at z=2

BTFR evolution

Standard MOND

a₀ = const

No evolution

sync_cost: a₀=cH/2π

a₀ ~ 3× today

Zero point shifts ~0.5 dex

ΛCDM (Magneticum)

a₀ ~ 3× today

Effective a₀ emerges from baryonic physics

Xu (2022) power law

a₀ ∝ (1+z)^¾

Similar at z<3, diverges at z>3

Notable: the sync_cost prediction and ΛCDM simulations give similar a₀(z=2)/a₀(0) ratios (~3). The discriminating lever arm is at z > 3, where the functional forms diverge. The sync_cost prediction (a₀ ∝ H(z) ∝ √(Ω_m(1+z)³ + Ω_Λ)) grows faster than (1+z)^(3/4) at high z and has a specific functional form set by cosmological parameters with no free parameters.

Observational Status (as of early 2026)#

Data that exist#

  1. Nestor Shachar et al. (2023, ApJ 944, 78) — “RC100” 100 massive star-forming galaxies at z = 0.6–2.5 with Hα/CO rotation curves from VLT SINFONI/KMOS and ALMA. Dark-matter fractions within R_e decline with redshift: f_DM ~ 0.38 at z~1, ~0.27 at z~2. Half of z~2 galaxies are maximal disks. This declining DM fraction is qualitatively consistent with a larger a₀(z) pushing the MOND transition to smaller radii.

  2. McGaugh et al. (2024, ApJ 976, 13) — “Accelerated Structure Formation” Binned the RC100 data on the BTFR by redshift (0.6 < z < 2.5). Found no clear evolution of the BTFR zero point. However, the RC100 sample is heavily biased toward massive, fast-rotating galaxies (V > 200 km/s) that are deep in the Newtonian regime — exactly where a₀ shifts are least visible.

  3. Übler, Nestor Shachar et al. (2024, A&A) — TFR at 0.6 ≤ z ≤ 2.5 Stellar TFR slope α = 3.03 ± 0.25. “Subtle deviation” from local studies. Modest evidence for evolution; not yet decisive.

  4. JADES / JWST rotation field (2025, MNRAS 538, 76) Distribution of galaxy rotation in JWST Advanced Deep Extragalactic Survey. Provides kinematic classifications but not yet precision rotation curves for low-mass systems.

What the data do NOT yet constrain#

  • Low-mass galaxies at z > 1.5 (V_flat ~ 50–120 km/s): These are the systems where MOND effects dominate and where a₀ shifts would be most visible. Current high-z rotation curve samples are biased toward the most massive disks.

  • The BTFR at z > 2.5: Beyond the RC100 redshift range. Only a handful of individual kinematic measurements exist (e.g., Neeleman et al. 2020 at z = 4.26, a single cold rotating disk).

  • Precision at the 0.2 dex level: Current high-z V_flat measurements have ~20–40% uncertainties, comparable to the predicted signal.

What Needs to Be Measured#

To test a₀(z) = cH(z)/(2π) at 3σ significance:

1. Target selection#

Low-mass disk galaxies at z = 1.5–3, with V_flat ~ 50–120 km/s. These are faint (H ~ 25–27 mag) but detectable with JWST behind lensing clusters or in deep fields.

2. Rotation curves#

  • JWST NIRSpec IFU at R ~ 2700–4000 for Hα (rest-frame) at z ~ 1.5–3.

  • Spatial resolution < 1 kpc required — gravitational lensing magnification of 5–10× or JWST diffraction limit (0.1” ~ 0.8 kpc at z = 2) may suffice.

  • ALMA CO or [CII] lines for independent kinematic cross-check.

3. Baryonic masses#

  • Stellar mass from JWST NIRCam multi-band photometry + SED fitting (rest-frame near-IR at z ~ 2 maps to observed 4–5 μm).

  • Gas mass from ALMA dust continuum (Band 6/7) or molecular line emission.

  • Target accuracy: 0.2 dex in total baryonic mass.

4. Sample size#

  • Predicted signal: ~0.5 dex shift in BTFR zero point between z = 0 and z = 2.

  • With intrinsic scatter ~0.3 dex, need N > 20 galaxies per redshift bin to beat systematics (beam smearing, pressure support, inclination).

  • Minimum programme: ~60 galaxies across z = 0–1, 1–2, 2–3 bins.

5. Discriminating between models#

The sync_cost prediction diverges from the (1+z)^(3/4) power law at z > 3:

z

cH(z)/(2π) ratio

(1+z)^(3/4) ratio

2

3.0

2.3

3

4.6

3.0

5

8.3

4.6

A single well-measured BTFR at z ~ 5 (feasible with JWST [CII] + ALMA) would strongly discriminate.

Relation to Other Predictions#

  • Derivation 03: This test directly probes the a₀ = cH₀/(2π) relation by checking its redshift extension.

  • Derivation 05 (Two Forces): If a₀(z) varies, the effective dark matter fraction within galaxies should also be z-dependent in a calculable way — providing a second, correlated observable.

  • Galaxy clusters: The cluster-scale anomaly noted in Derivation 03 (multi-body synchronization threshold) may also evolve with z. Cluster dynamics at z > 1 (accessible via Sunyaev-Zel’dovich + X-ray with SPT-3G and eROSITA) provide an independent test channel.

Status#

Prediction: Concrete, parameter-free, and falsifiable. a₀(z) is fully determined by H₀, Ω_m, and Ω_Λ — all independently measured.

Data: Tantalisingly close but not yet decisive. The RC100 sample covers the right redshift range but the wrong mass range. JWST Cycle 3+ programmes targeting low-mass lensed disks at z > 1.5 are the critical next step.

Timeline: Feasibility demonstration with ~10 galaxies could come from existing JWST archival data (JADES, GLASS, UNCOVER lensed fields) within 1–2 years. A definitive test (N > 60, z = 0–3) likely requires a dedicated JWST programme (Cycle 5+, ~100 hours) or ELT first light (~2028+).

Computation#

See a0_high_z.py in this directory for the full numerical computation and detailed observational comparison.