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

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 $z \approx 0$ to $z \approx 2$ , 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 $a_{0,\tau}(z) = \dfrac{c\,H_\tau(z)}{2\pi}$ , so $a_{0,\tau}/H_\tau = c/(2\pi)$ is strictly constant.
Observers, however, measure a projected quantity $a_{0,t}(z) = a_{0,\tau}(z) / [\cos\alpha(z)\,F_{\rm drift}(z)]$ , where $\alpha(z)$ and $F_{\rm drift}(z)$ encode the mapping between cosmic and lab clocks.
A constant observed $a_{0,t}$ is therefore allowed (and even natural) if the projection factor $\cos\alpha(z)\,F_{\rm drift}(z)$ grows roughly like $H_\tau(z)$ .

What do the data say?

BTFR/RAR analyses from $z \approx 0$ to $z \approx 2$ find that the effective “knee” acceleration $a_{0,t}(z)$ is approximately constant at $\sim 10^{-10}\,\text{m s}^{-2}$ , 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:

$a_{0,\tau}(z) = \dfrac{c\,H_\tau(z)}{2\pi} \quad\Rightarrow\quad \dfrac{a_{0,\tau}(z)}{H_\tau(z)} = \dfrac{c}{2\pi} = \text{constant}.$

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

$a_{0,t}(z) = \dfrac{a_{0,\tau}(z)}{\cos\alpha(z)\,F_{\rm drift}(z)}.$

Here

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

So:

$a_{0,t}(z) = \dfrac{c\,H_\tau(z)}{2\pi} \cdot \dfrac{1}{\cos\alpha(z)\,F_{\rm drift}(z)}.$

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

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

so that

$a_{0,t}(z) \propto H_{\rm obs}(z).$

That is falsified by the data.

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

$\dfrac{a_{0,t}(z)}{H_{\rm obs}(z)} \propto \dfrac{H_\tau(z)}{H_{\rm obs}(z)\,\cos\alpha(z)\,F_{\rm drift}(z)}.$

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

$\cos\alpha(z)\,F_{\rm drift}(z) \propto H_\tau(z)$

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 ( $z \approx 0$ )
SPARC and related samples give a very tight RAR with a characteristic acceleration $a_0 \approx 1.2\times10^{-10}\,\text{m s}^{-2}$ 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 ( $z \approx 0.6$ –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 $g\sim 10^{-10}\,\text{m s}^{-2}$ . (arXiv:1703.04310)
Hubble parameter $H(z)$ over $0 \le z \le 2$
Cosmic chronometer and BAO analyses (Moresco, Borghi, Tomasetti; BOSS/eBOSS) show that the Hubble rate increases by a factor of $\gtrsim 2$ –3 between $z = 0$ and $z \approx 1.5$ –2. (MNRASL 450, L16)

So empirically:

$a_{0,t}(z) \approx \text{const}$ (with at most mild, as-yet-uncertain evolution). (MNRAS 526, 3342)
$H_{\rm obs}(z)$ 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)

a_{0,\tau}(z) = \dfrac{c\,H_\tau(z)}{2\pi}

Model	Prediction for $a_0(z)$	Observational findings (same data as before)	Verdict for QTT-style prediction
QTT (two-clock)	Fundamental identity: $a_{0,\tau}(z) = \dfrac{c\,H_\tau(z)}{2\pi}$ , so $a_{0,\tau}/H_\tau = c/(2\pi)$ is strictly constant in τ-time. The lab-measured value is $a_{0,t}(z) = a_{0,\tau}(z)/[\cos\alpha(z)F_{\rm drift}(z)]$ . A constant observed $a_{0,t}$ is obtained if $\cos\alpha(z)F_{\rm drift}(z) \propto H_\tau(z)$ . No extra free parameter if $\alpha(z)$ and $F_{\rm drift}(z)$ are already fixed by the QTT ledger.	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 $z \approx 2$ . The inferred $a_{0,t}$ remains of order $10^{-10}\,\text{m s}^{-2}$ with small scatter, while $H(z)$ clearly increases by a factor $\gtrsim 2$ –3 over the same range.	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 $\cos\alpha\,F_{\rm drift}$ .
MOND (original)	Takes $a_0$ as a genuine constant (no redshift dependence): usually $a_0 \approx 1.2\times10^{-10}\,\text{m s}^{-2}$ , with the RAR/BTFR knee fixed in time.	Observations from $z \approx 0$ to $z \approx 2$ 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.	PASS. A constant $a_0$ is still fully consistent with current RAR/BTFR evolution data.
ΛCDM	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 $a_0/H(z)$ .	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.	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:

What the data say:
- RAR/BTFR knee in the lab frame is approximately constant, $a_{0,t}(z) \approx \text{const}$ , from $z \approx 0$ to $z \approx 2$ .
- Hubble rate $H_{\rm obs}(z)$ grows strongly over the same interval.
What naive (single-clock) QTT demanded:
$a_0 \propto H_{\rm obs}$ , so observed $a_0/H_{\rm obs}$ should be constant.
Since it isn’t, that version was falsified.
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 $a_{0,t}(z) = \dfrac{c\,H_\tau(z)}{2\pi}\,\dfrac{1}{\cos\alpha(z)\,F_{\rm drift}(z)}$ .
- Current data imply $a_{0,t}(z)\approx \text{const} \quad\Rightarrow\quad \cos\alpha(z)\,F_{\rm drift}(z) \propto H_\tau(z)$ .
This is not a fine-tuning knob if and 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.
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 $\cos\alpha\,F_{\rm drift}$ . 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 $0 \le z \le 2$ , then:

The functional form of the RAR, $g_{\rm obs} = \nu\!\left(\dfrac{g_{\rm bar}}{a_{0,t}}\right) g_{\rm bar},$ 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 $z \approx 2$ — 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 $\cos\alpha\,F_{\rm drift}$ 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 $\cos\alpha\,F_{\rm drift}$ .
A roughly constant lab-frame with a rising observed is fully compatible with:
- QTT (with $a_{0,\tau}\propto H_\tau$ and $\cos\alpha\,F_{\rm drift}\propto H_\tau$ ), 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 ( $\alpha(z)$ , $F_{\rm drift}(z)$ ) from other QTT observables, so that Test 3 fixes or breaks the remaining degeneracy rather than absorbing it.

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

0. Test, prediction, and outcome

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

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

3.2. Observational inputs (same as before)

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

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

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

3.6. Updated synthesis for Test 3

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0. Test, prediction, and outcome

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

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

3.2. Observational inputs (same as before)

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

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

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

3.6. Updated synthesis for Test 3

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