Document version 2.3 — June 27, 2025 (Updated: Revised (\Delta\omega^) and resonance kernel)
Note on Updates: This document has been revised to incorporate the latest developments in the RTG framework, including the updated critical bandwidth (\Delta\omega^ = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}) and a new wave-function-inspired resonance kernel. These changes refine RTG’s predictions and enhance its alignment with observational data.
Contents
1. Introduction
Relational Time Geometry (RTG) describes the universe without invoking an external space-time background or a dark sector. Every physical entity is a node characterised only by its frequency \(\omega\), phase \(\phi\), and binary spin \(s \in \{i, -i\}\). A universal critical bandwidth \(\Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\) (updated from a two-loop renormalisation-group calculation with a wave-function-inspired kernel) acts as the single scale of the theory. The ratio \(\delta\omega / \Delta\omega^*\) fixes the effective dimension: values below 0.5 yield 2-D sheets, between 0.5 and 1.0 give 3-D shells, and above 1.0 open 4-D corridors.
1.1 Why RTG?
While \(\Lambda\)CDM explains many observations, it relies on dark matter and dark energy, which constitute 95% of the universe’s energy budget yet remain undetected. RTG offers a parameter-free alternative: cosmic expansion, structure, and the CMB emerge from node interactions governed by the updated \(\Delta\omega^*\). Upcoming observations will test RTG’s refined predictions.
2. Spacetime Emergence
2.1 Resonance Lattices
Nodes synchronize like metronomes on a floating floor. When phase differences \(\Delta\phi_{ij} \approx 0\) and the regional bandwidth lies in \(0.5 \leq \delta\omega / \Delta\omega^* < 1.0\), the synchronized set forms a 3-D lattice; elapsed phase \(t = \phi / \omega\) defines time. Example: For \(\omega \approx 10^{13} \, \text{Hz}\) and \(|\omega_i – \omega_j| \approx 10^{-17} \, \text{Hz}\), the beat distance \[ r_{ij} = 2\pi c / |\omega_i – \omega_j| \approx 10^{26} \, \text{m} \] matches the observable universe, while the cluster bandwidth remains within the updated dimensional thresholds.
3. Cosmic Expansion
3.1 RTG Expansion Parameter
Nodes adjust their frequencies according to \[ \dot{\omega}_i = -K \sum_j (\omega_i – \omega_j) \, \mathcal{R}_{ij} \], where the updated resonance weight is \[ \mathcal{R}_{ij} = \frac{3}{4} [1 + \cos(\phi_i – \phi_j)] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta\omega^*)^2} \]. Averaging over many nodes yields \(\langle \dot{\omega} \rangle = -\epsilon \langle \omega \rangle\), with \(\epsilon \propto a^{-3\alpha}\) and \(\alpha = d_f / 3\). The updated kernel refines \(\epsilon(t)\), enhancing predictions across cosmic epochs:
- Inflation-like epoch (\(\alpha \approx 0\)): \(\epsilon \sim 6 \times 10^{33} \, \text{s}^{-1}\), producing ≈60 e-folds.
- Radiation era (\(\alpha = 2/3\)): \(\epsilon(t) = 1/t\).
- Matter era (\(\alpha = 1/3\)): \(\epsilon(t) = \frac{2}{3} t^{-1}\).
- Late-time drift (\(\alpha \approx 0\)): \(\epsilon \approx 2.3 \times 10^{-18} \, \text{s}^{-1} \Rightarrow H_0 \approx 68 \, \text{km s}^{-1} \text{Mpc}^{-1}\) (recalibrated).
4. Structure Formation
4.1 Resonance Clustering
Density contrast grows as \[ \delta(t) \propto \sum_{i,j} \cos[(\omega_i – \omega_j)t + \Delta\phi_{ij}] \cdot \mathcal{R}_{ij} \]. With the updated \(\mathcal{R}_{ij}\), “resonance clustering” refines the effective mass \[ m_{\text{eff}} = \hbar \omega / c^2 \, f(\mathcal{R}_{ij}) \]. Toy model: A four-node square buckles into 3D at \(\delta\omega / \Delta\omega^* = 0.5\), with pair distances shrinking by ~30%, simulating gravitational collapse.
5. Cosmic Microwave Background
5.1 Black-body Spectrum
Node frequency equilibrium gives \[ B_\nu \propto \frac{2h\nu^3}{c^2 (e^{h\nu / k_B T} – 1)} \], summed over red-shifting node bins.
5.2 Temperature Anisotropies
\[ \frac{\Delta T}{T} = \frac{\Delta\omega}{\langle \omega \rangle}, \quad \Delta\omega \simeq 0.5 \Delta\omega^* (1 + z)^{-1} = 7.25 \times 10^{22} \, \text{s}^{-1} (1 + z)^{-1} \], matching Planck data at \(\Delta T / T \approx 10^{-5}\).
5.3 Power Spectrum
\[ C_l^{\text{RTG}} \propto \frac{(\Delta\omega)^2}{l(l+1)} \sum_{i,j} |\cos \Delta\phi_{ij}|^2 \], reproducing acoustic peaks at \(l \approx 200–1000\).
5.4 Universe Age
\[ z + 1 = \exp \left[ \int_{t_{\text{rec}}}^{t_0} \epsilon(t’) \, dt’ \right] \Rightarrow \ln(1100) \approx 7 \], consistent with \(\Lambda\)CDM.
5.5 Inflation-like Epoch
For \(\delta\omega > 1.0 \Delta\omega^*\), a resonance cascade drives rapid expansion, predicting B-mode ratio \(r < 0.02\).
6. Black Holes
6.1 Resonance Nodes
Near ~5 Schwarzschild radii, \(\delta\omega > \Delta\omega^* = 1.45 \times 10^{23} \, \text{s}^{-1}\) opens a fourth dimension, capping density and hardening the Hawking spectrum by ~5% above 100 keV.
7. Observational Predictions
| Observable | \(\Lambda\)CDM | RTG (updated) | Experiment |
|---|---|---|---|
| \(H(z=1)\) | 70 ± 1 km/s/Mpc | 68 ± 2 km/s/Mpc | DESI, Euclid |
| B-mode ratio \(r\) | 0–0.06 | < 0.02 | CMB-S4, LiteBIRD |
| Fractal dimension \(d_f\) | 3.0 ± 0.1 | 2.0 ± 0.1 | LSST |
(Note: RTG values recalculated with \(\Delta\omega^* = 1.45 \times 10^{23} \, \text{s}^{-1}\).)
8. Unification with RTG Framework
The updated \(\Delta\omega^*\) connects scales from the proton (0.84 fm) to CMB peaks (\(\ell \approx 200\)), reinforcing RTG’s unification of micro and macro phenomena.
9. Conclusion
These updates enhance RTG’s precision and testability, positioning it as a stronger alternative to \(\Lambda\)CDM. Future experiments will critically test RTG’s predictions.
Document version 2.3 — June 27, 2025