QTT vs the Dark Sector: Vacuum Capacity and Renewal Dust Instead of Dark Matter & Dark Energy

Reference: https://doi.org/10.5281/zenodo.17594186

In the standard cosmological model (ΛCDM), two mysterious components dominate the universe:

Dark energy, usually modelled as a cosmological constant or exotic fluid.
Dark matter, usually modelled as new, cold particles (WIMPs, axions, etc.).

Quantum Traction Theory (QTT) takes a different path. It does not add a new particle dark sector. Instead, it replaces:

Dark energy with a vacuum–capacity law fixed by the Planck four–cell (no tuning), and
Dark matter with a combination of
- extra gravitational terms from the endurance (renewal) mechanics, and
- a natural MOND–like low–acceleration scale from the two–clock geometry, plus a purely gravitational “renewal dust” component.

Below, I’ll walk through the key boxed equations and then highlight the concrete tests that distinguish QTT from ΛCDM.

1. Dark energy → vacuum law from the Planck four–cell

1.1. Unified Equilibrium Law (UEL): Planck 4–cell capacity

QTT treats the Planck 4–cell as the primitive “capacity unit” of spacetime. The elementary four–volume is

$V^{(4)}_{\text{quant}} = 4\pi \ell_P^4.$

One such four–cell carries the Planck energy. QTT writes the Unified Equilibrium Law as

$\boxed{ E_P := m_P c^2 = \hbar \omega_P = \rho^{(4)}\,\bigl(4\pi \ell_P^4\bigr) }$

Here $\rho^{(4)}$ is a universal four–density (capacity per four–volume). This ties energy, Planck mass, Planck frequency, and four–volume into a single capacity relation.

1.2. Vacuum energy as a thin slice of Planck capacity

QTT reads the observed vacuum density $\rho_\Lambda$ as a thin “slice” of that same four–density, with a geometric factor $\kappa = 1/3$ and a small amplitude $\varepsilon$ :

$\boxed{ \rho_\Lambda = \kappa\,\varepsilon\, \frac{\hbar c}{4\pi \ell_P^4}, \qquad \kappa = \frac{1}{3}. }$

Equivalently, in terms of the Planck density $\rho_P = \hbar c/\ell_P^4$ ,

$\frac{\rho_\Lambda}{\rho_P} = \frac{\kappa\,\varepsilon}{4\pi} = \frac{\varepsilon}{12\pi}. $

So the notorious $\sim 10^{-122}$ suppression is encoded as a dimensionless weight $\varepsilon/(12\pi)$ multiplying a single geometric normalisation.

1.3. FRW consistency fixes $\varepsilon$ (no tuning)

Matching this QTT vacuum density to the FRW dark–energy density $\rho_\Lambda = \Lambda c^4/(8\pi G)$ fixes $\varepsilon$ in terms of the de Sitter Hubble rate $H_\Lambda$ and the Planck time $t_P$ :

$\boxed{ \varepsilon = \frac{3}{2\kappa}(H_\Lambda t_P)^2, \qquad \frac{\rho_\Lambda}{\rho_P} = \frac{3}{8\pi}(H_\Lambda t_P)^2. }$

Once $\hbar, c, G$ are fixed, $\rho_\Lambda$ is not a free parameter: it is determined by the tiny ratio $(H_\Lambda t_P)^2$ .

1.4. Cosmological constant in Planck units

The same match yields the cosmological constant in Planck units:

$\boxed{ \Lambda = \frac{8\pi G}{c^4}\,\rho_\Lambda = \frac{2\kappa\,\varepsilon}{\ell_P^2} = \frac{2\kappa\,\varepsilon}{c^2 t_P^2} = \frac{3H_\Lambda^2}{c^2}. }$

In QTT, “dark energy” is therefore vacuum capacity of Planck four–cells with a small amplitude $\varepsilon$ fixed by cosmic expansion, not a separate dark fluid.

2. Dark matter → endurance gravity, MOND scale, and renewal dust

2.1. Gravity from endurance: Newtonian sector

From the Law of Endurance, each rest mass $M$ consumes space quanta at rate

$\frac{dN_{\rm SQ}}{dT} = \frac{M}{\tilde m}\,\frac{1}{\tilde t}, \qquad V_{\rm SQ} = 4\pi \tilde\ell^3. $

This defines a four–volume sink rate

$\boxed{ \frac{dV^{(4)}_{\rm sink}}{dT} = \frac{M}{\tilde m}\,\frac{1}{\tilde t}\,\bigl(4\pi \tilde\ell^4\bigr). }$

QTT interprets this as an endurance current $J_{\rm end}$ . Its divergence is proportional to the mass density $\rho$ , and the Newtonian gravitational field is identified as

$g = \frac{c}{\tilde\ell}\,J_{\rm end}.$

In the continuum limit this yields

$\boxed{ \nabla\!\cdot g = -4\pi G \rho, \qquad G = \frac{\tilde\ell^2 c^3}{\hbar}. }$

So QTT reproduces standard Newtonian gravity, but expresses $G$ purely in terms of tick/step quantities $(\tilde\ell,\tilde t)$ .

2.2. MOND–like acceleration scale from the two clocks

The same two–clock geometry that sets the cosmic expansion also yields a natural low–acceleration scale. QTT predicts a MOND–like acceleration parameter:

$\boxed{ a_0(z) = \frac{c\,H(z)}{2\pi}. }$

At $z=0$ , $a_0 = cH_0/(2\pi)$ matches the empirical MOND acceleration scale that governs galaxy rotation curves and the radial acceleration relation.

Interpretation:

No new particle species is introduced at galactic scales.
Instead, two–clock kinematics implies that when $g \lesssim a_0(z)$ , the effective relation between baryonic mass and acceleration crosses over from Newtonian to a MOND–like regime driven by the global Hubble rate.

This already replaces a large fraction of what ΛCDM attributes to cold dark matter.

2.3. Renewal dust as the remaining “dark matter”

QTT still allows an extra gravitational component, but it is not a new Standard Model–coupled particle. It is a purely gravitational renewal dust sourced by the endurance mechanism.

Its stress–energy tensor is that of pressureless dust:

$\boxed{ T^{\mu\nu}_{\rm RD} = \rho_{\rm RD}\,u^\mu u^\nu }$

It has no direct interaction with Standard Model fields:

$\boxed{ \mathcal L^{\rm int}_{\rm RD} = 0 }$

i.e. it only gravitates.

The full Einstein equation in QTT then reads schematically:

$\boxed{ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\Bigl(T^{\rm SM}_{\mu\nu} + T^{\rm RD}_{\mu\nu}\Bigr). }$

Here $\Lambda$ and $G$ are given by the QTT vacuum and endurance relations above. Dark matter is replaced by:

a low–acceleration modification $a_0(z) = cH(z)/(2\pi)$ from the two clocks, and
a non–SM interacting, purely gravitational renewal dust component originating from endurance flow, not a new WIMP/axion sector.

3. Time–plane tilt: geometric age gap and “dark energy” effects

QTT also rewrites part of dark–energy phenomenology as a geometric effect of the time–plane tilt between the Absolute Clock $T$ and the laboratory clock $\tau$ :

Two–clock relation: $d\tau = \mathcal N(x,v)\,dT.$
In cosmology, the misalignment angle $\theta(a)$ between the lab time axis and the absolute time plane drifts with scale factor $a$ as matter and vacuum content evolve.
This leads to a projection factor between lab age $t_0$ and absolute age $\tau_0$ :

$\tau_0 \simeq \dfrac{t_0}{\cos\theta_{\rm eff}}.$

Part of what looks like a “dark energy age tension” is therefore reinterpreted as geometry of the time plane, not exotic physics.

4. Boxed “dark sector replacement” summary

For quick reference, the core QTT replacement of the dark sector can be summarised in a single multi–line boxed relation:

$\boxed{ \begin{aligned} \textbf{Vacuum law:}&\quad \rho_\Lambda = \kappa\,\varepsilon\,\frac{\hbar c}{4\pi\ell_P^4}, \quad \varepsilon = \frac{3}{2\kappa}(H_\Lambda t_P)^2, \\[0.4em] &\quad \Lambda = \frac{8\pi G}{c^4}\rho_\Lambda = \frac{2\kappa\varepsilon}{\ell_P^2} = \frac{3H_\Lambda^2}{c^2}. \\[0.7em] \textbf{Gravity from endurance:}&\quad \nabla\!\cdot g = -4\pi G\rho, \quad G = \frac{\tilde\ell^2 c^3}{\hbar}, \\[0.4em] &\quad \frac{dV^{(4)}_{\rm sink}}{dT} = \frac{M}{\tilde m}\frac{1}{\tilde t}\,(4\pi\tilde\ell^4). \\[0.7em] \textbf{MOND scale from two clocks:}&\quad a_0(z) = \frac{c H(z)}{2\pi}. \\[0.7em] \textbf{Renewal dust:}&\quad T^{\mu\nu}_{\rm RD} = \rho_{\rm RD}u^\mu u^\nu, \quad \mathcal L^{\rm int}_{\rm RD} = 0, \\[0.4em] &\quad G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\bigl(T^{\rm SM}_{\mu\nu}+T^{\rm RD}_{\mu\nu}\bigr). \end{aligned} }$

5. Concrete tests against ΛCDM

QTT’s dark–sector replacement is falsifiable. Some sharp tests are:

Vacuum law test. Measure $\rho_\Lambda$ and $H_\Lambda$ (from SN Ia, BAO, CMB). QTT predicts

$\dfrac{\rho_\Lambda}{\rho_P} = \dfrac{3}{8\pi}(H_\Lambda t_P)^2$

with no free parameter. Any robust disagreement falsifies the QTT vacuum law. MOND scale evolution. QTT predicts

$a_0(z) = \dfrac{c H(z)}{2\pi}.$

Test with high–redshift rotation curves and lensing: does the characteristic acceleration in the radial acceleration relation track $H(z)$ this way? Renewal dust non–interactions. Renewal dust has $\mathcal L^{\rm int}_{\rm RD} = 0$ with the Standard Model. Direct detection should keep seeing nothing; any positive dark–SM coupling at the expected densities contradicts QTT. Age–Hubble relations. The time–plane tilt implies specific age–Hubble identities (e.g. $H_0 T_0 \approx 1$ in the T–ledger) plus a geometric enhancement of absolute age over lab age. Compare precision stellar–chronometer ages with Hubble–rate determinations. Structure formation. N–body simulations with baryons + renewal dust + the $a_0(z)$ modification must reproduce the matter power spectrum and CMB lensing without cold dark matter. Systematic mismatch at linear scales would challenge the QTT picture.

In short, QTT does not hide new particles in the dark; it rewires the dark sector into vacuum capacity, endurance–driven gravity, two–clock kinematics, and renewal dust—all governed by the same Planck–scale capacity laws that control the rest of the theory.

QTT vs the Dark Sector: Vacuum Capacity and Renewal Dust Instead of Dark Matter & Dark Energy

1. Dark energy → vacuum law from the Planck four–cell

1.1. Unified Equilibrium Law (UEL): Planck 4–cell capacity

1.2. Vacuum energy as a thin slice of Planck capacity

1.3. FRW consistency fixes $\varepsilon$ (no tuning)

1.4. Cosmological constant in Planck units

2. Dark matter → endurance gravity, MOND scale, and renewal dust

2.1. Gravity from endurance: Newtonian sector

2.2. MOND–like acceleration scale from the two clocks

2.3. Renewal dust as the remaining “dark matter”

3. Time–plane tilt: geometric age gap and “dark energy” effects

4. Boxed “dark sector replacement” summary

5. Concrete tests against ΛCDM

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1. Dark energy → vacuum law from the Planck four–cell

1.1. Unified Equilibrium Law (UEL): Planck 4–cell capacity

1.2. Vacuum energy as a thin slice of Planck capacity

1.3. FRW consistency fixes (no tuning)

1.4. Cosmological constant in Planck units

2. Dark matter → endurance gravity, MOND scale, and renewal dust

2.1. Gravity from endurance: Newtonian sector

2.2. MOND–like acceleration scale from the two clocks

2.3. Renewal dust as the remaining “dark matter”

3. Time–plane tilt: geometric age gap and “dark energy” effects

4. Boxed “dark sector replacement” summary

5. Concrete tests against ΛCDM

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1.3. FRW consistency fixes $\varepsilon$ (no tuning)