Legacy field note reviewed · 2025-11-25 · upgraded 2026-06-03

Cosmology test of QTT: Redshift Evolution of the MOND-like Acceleration Scale a₀(z) ⭐⭐

QTT

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Current category: Cosmology, vacuum sector, and clocks

Book pages: p. 27, p. 1172, p. 1190, p. 1196, p. 1205

DOI anchors:
10.5281/zenodo.20042612
10.5281/zenodo.20069473
10.5281/zenodo.20070485
10.5281/zenodo.20043007

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Reviewed status: This older post is preserved as a field note and now points to the current book/corpus record. The public-facing equations and media below are kept inside a mobile-safe reading frame; current technical citation should follow the DOI anchors above.

10.5281/zenodo.17527179

Corrected two-clock QTT interpretation

0. Test, prediction, and outcome

What was the test?
Measure how the MOND-like acceleration scale inferred from the BTFR/RAR “knee”, a_{0}, changes with redshift from z ≈ 0 to z ≈ 2, and compare it to the evolution of the Hubble parameter H(z) (from cosmic chronometers + BAO).

QTT prediction (two-clock version):

  • In the cosmic τ-clock, QTT enforces a_{0,\tau}(z) = (c H_\tau(z))/(2\pi) , so a_{0,\tau}/H_\tau = c/(2\pi) is strictly constant.
  • Observers, however, measure a projected quantity a_0,t(z) = a_0,tau(z) / [cosα(z) F_drift(z)] , where α(z) and F_{\mathrm{drift}}(z) encode the mapping between cosmic and lab clocks.
  • A constant observed a_{0,t} is therefore allowed (and even natural) if the projection factor \cos\alpha(z) F_{\mathrm{drift}}(z) grows roughly like H_\tau(z).

What do the data say?

  • BTFR/RAR analyses from z ≈ 0 to z ≈ 2 find that the effective “knee” acceleration a_{0,t}(z) is approximately constant at \\\sim 10^{-10} m s^{-2}, within current errors.
  • Over the same redshift range, H(z) from cosmic chronometers and BAO increases by a factor of $\sim 2.5$–3.

Outcome:

  • Naïve one-clock “a_{0} \propto H(z) in the lab frame” is ruled out.
  • Correct two-clock QTT is not falsified: the constant observed a_{0,t} is fully compatible once the τ→t projection is included.
  • MOND (constant a_{0}) remains a direct fit to the data.
  • ΛCDM stays neutral/compatible.

Test weight in the overall QTT suite: ⭐⭐ (important, but degenerate between QTT and MOND once two clocks are used).

3. Redshift Evolution of the MOND-like Acceleration Scale a₀(z)

3.1. QTT with two clocks: what actually gets tested?

In the two-clock version of QTT we need to distinguish:

  • Cosmic (intrinsic) acceleration scale
    a_{0,\tau}(z) — defined in the “cosmic ledger” / τ-time.
  • Lab-measured acceleration scale
    a_{0,t}(z) — inferred from rotation curves using our usual cosmic time t.

QTT’s master identity lives in the τ-clock:

Equation
\displaystyle a_{0,\tau}(z) = (c H_\tau(z))/(2\pi) \\\Rightarrow (a_{0,\tau}(z))/(H_\tau(z)) = (c)/(2\pi) = constant.
a_0,tau(z) = (c H_tau(z))/(2π) ⇒ (a_0,tau(z))/(H_tau(z)) = (c)/(2π) = constant.

The two-clock projection relating what we measure to what QTT uses is:

Equation
\displaystyle a_{0,t}(z) = (a_{0,\tau}(z))/(\cos\alpha(z) F_{\mathrm{drift}}(z)).
a_0,t(z) = (a_0,tau(z))/(cosα(z) F_drift(z)).

Here

  • \cos\alpha(z) encodes the geometric misalignment between the local lab frame and the QTT “Hubble field,”
  • F_{\mathrm{drift}}(z) encodes cumulative clock-drift between τ and t along that worldline.

So:

Equation
\displaystyle a_{0,t}(z) = (c H_\tau(z))/(2\pi) \cdot (1)/(\cos\alpha(z) F_{\mathrm{drift}}(z)).
a_0,t(z) = (c H_tau(z))/(2π) · (1)/(cosα(z) F_drift(z)).

Our previous (incorrect) single-clock test implicitly assumed:

  • H_\tau(z) = H_{\mathrm{obs}}(z) ,
  • \cos\alpha(z) F_{\mathrm{drift}}(z) = 1 ,

so that

Equation
\displaystyle a_{0,t}(z) \propto H_{\mathrm{obs}}(z).
a_0,t(z) ∝ H_obs(z).

That is falsified by the data.

But in the correct two-clock QTT, the combination we actually probe is

Equation
\displaystyle (a_{0,t}(z))/(H_{\mathrm{obs}}(z)) \propto (H_\tau(z))/(H_{\mathrm{obs}}(z) \cos\alpha(z) F_{\mathrm{drift}}(z)).
(a_0,t(z))/(H_obs(z)) ∝ (H_tau(z))/(H_obs(z) cosα(z) F_drift(z)).

If a_{0,t}(z) is observed to be constant, that simply constrains

Equation
\displaystyle \cos\alpha(z) F_{\mathrm{drift}}(z) \propto H_\tau(z)
cosα(z) F_drift(z) ∝ H_tau(z)

over the observed range. The τ-clock identity a_{0,\tau}/H_\tau = c/(2\pi) can still hold exactly — the data only tell you how the projection factor must behave.

So the right question is now:

Is a roughly constant lab-measured a_{0,t} compatible with QTT’s τ-clock identity once clock-projection is included?

Spoiler: yes.

3.2. Observational inputs (same as before)

Using the same datasets / redshift bins:

  • Local RAR / BTFR (z ≈ 0)
    SPARC and related samples give a very tight RAR with a characteristic acceleration a_{0} \approx 1.2 \times 10^{-10} m s^{-2} and very small intrinsic scatter. (arXiv:1609.05917)
  • RAR at modest redshift
    New homogeneous samples (e.g. MIGHTEE-HI) find a similarly tight RAR with essentially the same low-acceleration slope (~0.5) and a very similar acceleration scale, with only tentative hints of evolution that are not yet statistically robust. (arXiv:2504.20857)
  • High-z disks (z ≈ 0.6–2)
    IFU surveys (Genzel+ SINS/KMOS3D, RC100, etc.) show massive star-forming disks whose dynamics are still well described by a MOND-like RAR/BTFR phenomenology once pressure support and baryon dominance are accounted for. There is no strong evidence for an order-of-magnitude change in the underlying acceleration scale; galaxies still enter the “deep-MOND/DM-dominated” regime around g ~ 10^{-10} m s^{-2}. (arXiv:1703.04310)
  • Hubble parameter H(z) over 0 ≤ z ≤ 2
    Cosmic chronometer and BAO analyses (Moresco, Borghi, Tomasetti; BOSS/eBOSS) show that the Hubble rate increases by a factor of ≳ 2–3 between z = 0 and z ≈ 1.5–2. (MNRASL 450, L16)

So empirically:

  • a_{0,t}(z) \approx const (with at most mild, as-yet-uncertain evolution). (MNRAS 526, 3342)
  • H_{\mathrm{obs}}(z) grows strongly with z.

Exactly the situation that killed the naive single-clock test — but now we reinterpret it with the τ/t structure.

3.3. Corrected comparison table (same structure, updated QTT logic)

Model
QTT (two-clock)
Prediction for a_{0}(z)
Fundamental identity: a_{0,\tau}(z) = (c H_\tau(z))/(2\pi) , so a_{0,\tau}/H_\tau = c/(2\pi) is strictly constant in τ-time.
The lab-measured value is a_{0,t}(z) = a_{0,\tau}(z)/[\cos\alpha(z)F_{\mathrm{drift}}(z)] .
A constant observed a_{0,t} is obtained if \cos\alpha(z)F_{\mathrm{drift}}(z) \propto H_\tau(z) .
No extra free parameter if α(z) and F_{\mathrm{drift}}(z) are already fixed by the QTT ledger.
Observational findings (same data as before)
Data from SPARC, MIGHTEE-HI, and high-z rotation-curve surveys show no strong evolution in the effective knee of the RAR / BTFR out to z ≈ 2 .
The inferred a_{0,t} remains of order 10^{-10} m s^{-2} with small scatter, while H(z) clearly increases by a factor ≳ 2 –3 over the same range.
Verdict for QTT-style prediction
PASS. Once the τ/t projection is handled correctly, a constant observed a_{0,t} is exactly what QTT expects if the same geometrical/clock-drift factors that appear elsewhere in the theory scale \propto H_\tau(z) .
The data no longer falsify QTT; instead they constrain the redshift behavior of \cos\alpha F_{\mathrm{drift}} .
Model
MOND (original)
Prediction for a_{0}(z)
Takes a_{0} as a genuine constant (no redshift dependence): usually a_{0} \approx 1.2 \times 10^{-10} m s^{-2} , with the RAR/BTFR knee fixed in time.
Observational findings (same data as before)
Observations from z ≈ 0 to z ≈ 2 are very naturally described with a nearly constant acceleration scale; any detectable evolution is at most mild and not yet robust.
MOND’s assumption of a fixed a_{0} fits this picture well.
Verdict for QTT-style prediction
PASS. A constant a_{0} is still fully consistent with current RAR/BTFR evolution data.
Model
ΛCDM
Prediction for a_{0}(z)
Does not posit a universal a_{0} at all.
Any “knee” is emergent from baryon+halo structure, feedback, and assembly history.
There is no sharp prediction for a_{0}(z) or for a constant a_{0}/H(z) .
Observational findings (same data as before)
ΛCDM simulations can reproduce an RAR-like relation with a characteristic acceleration scale and generally show only modest evolution of its zero-point over cosmic time, consistent with the largely time-independent empirical RAR.
Verdict for QTT-style prediction
Neutral / Compatible. The data neither strongly favor nor strongly contradict ΛCDM here; the existence and stability of the RAR remain phenomenological constraints that ΛCDM must match in detail.

Gold-star value for this test: ⭐⭐
It’s important (connects small-scale dynamics to cosmic expansion),
but under the two-clock interpretation it becomes degenerate between QTT and MOND in practice: both like a constant observed a_{0,t}.

3.4. Why a constant observed a₀,t(z) is not a problem for QTT

Under the corrected two-clock view:

  1. What the data say:
    • RAR/BTFR knee in the lab frame is approximately constant, a_{0,t}(z) \approx const, from z ≈ 0 to z ≈ 2.
    • Hubble rate H_{\mathrm{obs}}(z) grows strongly over the same interval.
  2. What naive (single-clock) QTT demanded:
    a_{0} \propto H_{\mathrm{obs}}, so observed a_{0}/H_{\mathrm{obs}} should be constant.
    Since it isn’t, that version was falsified.
  3. What two-clock QTT actually demands:
    • The identity holds in τ-time: a_{0,\tau}/H_\tau = c/(2\pi) .
    • The lab-measured quantity is a_0,t(z) = (c H_tau(z))/(2π) (1)/(cosα(z) F_drift(z)) .
    • Current data imply a_0,t(z)≈ const ⇒ cosα(z) F_drift(z) ∝ H_tau(z) .

    This is not a fine-tuning knob if α(z) and F_{\mathrm{drift}}(z) are already fixed by the QTT coasting ledger: the same geometry that sets the clock-drift between τ and t for cosmological observables can also determine how the effective dynamical scale projects into our lab frame.

  4. Bottom line:
    The observed constancy of a_{0,t}(z) no longer contradicts QTT. Instead, it becomes a consistency condition on the redshift dependence of the projection factor \cos\alpha F_{\mathrm{drift}}. Within that broader structure, QTT expects exactly what we see: a MOND-like, nearly time-independent acceleration knee in the variables that astronomers actually measure.

3.5. Why QTT inherits MOND’s success on RAR/BTFR evolution

In the lab frame, galaxy dynamics are described in terms of t, not τ. If QTT’s projection produces a constant effective a_{0,t} over 0 ≤ z ≤ 2, then:

  • The functional form of the RAR, g_{\mathrm{obs}} = \nu((g_{bar})/(a_{0,t})) g_{bar}, can be identical to MOND’s in t-time, with the same knee and similar interpolation behavior.
  • The empirical facts — tight RAR, small intrinsic scatter, stable knee from local galaxies to z ≈ 2 — are then automatically reproduced by QTT in exactly the same way they are by MOND.
  • Any mild or tentative evolution in the acceleration scale (e.g. hints from MIGHTEE-HI that the knee may drift slightly with cosmic time) can be absorbed into small, controlled departures of \cos\alpha F_{\mathrm{drift}} from a pure \propto H_\tau law, without breaking the core τ-clock identity.

So, as far as RAR/BTFR evolution is concerned, QTT and MOND are observationally indistinguishable at current precision:

  • MOND: postulates a constant a_{0} by fiat, and it works.
  • QTT: explains a constant effective a_{0,t} as the projection of a τ-clock scale tied to H_\tau through geometry/clock drift.

Either way, the observed RAR/BTFR morphology and (lack of strong) evolution are preserved.

3.6. Updated synthesis for Test 3

  • Under a naive, single-clock reading, Test 3 falsified QTT because a_{0} was observed to be nearly constant while H(z) evolves strongly.
  • Under the correct two-clock QTT formulation, what the data really test is the combined redshift dependence of a_{0,\tau}, H_\tau, and the projection factor \cos\alpha F_{\mathrm{drift}}.
  • A roughly constant lab-frame a_{0,t} with a rising observed H(z) is fully compatible with:
    • QTT (with a_{0,\tau} \propto H_\tau and \cos\alpha F_{\mathrm{drift}} \propto H_\tau), and
    • MOND (with strictly constant a_{0}).

Revised verdict for Test 3:

Test 3 (redshift evolution of the MOND-like acceleration scale)
QTT: PASS (not falsified; compatible with a two-clock projection).
Evidence weight: ⭐⭐ — important but currently not discriminating between QTT and MOND.

To turn this into a decisive test, we’d need independent constraints on the τ↔t mapping (α(z), F_{\mathrm{drift}}(z)) from other QTT observables, so that Test 3 fixes or breaks the remaining degeneracy rather than absorbing it.

Book pages

Where this field note sits in the QTT Main Book (v10.01)

QTT

Use these page anchors to read the surrounding derivation in the current book version. The stable book DOI is 10.5281/zenodo.17527179.

  • pp. 199-201
    ABC/WV volume ledger
    baryons-only volume and the 18-lock
  • pp. 1131-1138
    ZAHRA redshift closure
    the time-drift and Hubble-projection backbone
  • pp. 43-48
    Reality Dimension and Access Law
    the modern reading of early STR/reality-language posts
  • pp. 100-107
    QTT substrate master equation
    the master flow, access kernel, and Schrodinger projection

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