Cosmological Applications of Relational Time Geometry (RTG)

Document version 2.8 — 13 August 2025 (Expanded BH section with Hawking in RTG, structure trion ref, CMB σ_φ rationale, quantum gravity challenge; added Appendix B on Hawking radiation; aligned with Particle/Nuclear, Gravity v2.6; added Related Documents)

1 Introduction

Relational Time Geometry (RTG) models the universe with no external space-time or dark sector. A single scale \[ \Delta\omega^\* = (1.45 \pm 0.08)\times10^{23}\,\mathrm{s}^{-1} \] — fixed at two loops — underlies every phenomenon from sub-fm nucleon structure to CMB anisotropies (Two-Loop RG v 1.3.1). All Δω and δω are in rad·s⁻¹; convert to Hz by dividing by 2π (\(\Delta\omega^\*/2\pi \approx 2.31\times10^{22}\,\mathrm{Hz}\)); see RG v1.3.1 §6 for implications when plotting in Hz.

The bandwidth ratio \(\delta\omega/\Delta\omega^\*\) sets the emergent dimension (thresholds from RG v1.3.1; U(1)→SU(2) at 0.28, SU(2)→U(1)^2 at 0.70, U(1)^2→5-D anomalies at 1.70; Gauge Symm v1.4):

\(\delta\omega<0.28\,\Delta\omega^\*\): 2-D sheets
\(0.28\!-\!0.70\): 3-D shells
\(0.70\!-\!1.70\): 4-D corridors
\(\ge1.70\): 5-D anomalies

1.1 Why RTG?

\(\Lambda\)CDM fits data but invokes dark matter & dark energy (≈95 % of the mass-energy budget). RTG reproduces expansion, structure, and the CMB with no dark components and only one fixed parameter (\(\Delta\omega^\*\)). Its predictions in §7 are starkly falsifiable (dark-sector contrasts expanded in §8; unification via Δω\* in Forces v1.1).

2 Spacetime Emergence

2.1 Resonance Lattices

Nodes synchronise when \(\Delta\phi_{ij}\!\approx\!0\) and \(0.28\!\le\!\delta\omega/\Delta\omega^\*\!<\!0.70\), forming 3-D lattices. Example. For \(\omega\!=\!10^{13}\,\text{rad s}^{-1}\) and \(|\Delta\omega|\!=\!6\times10^{-17}\,\text{rad s}^{-1}\) (Hz via /2π gives ≈1.59×10¹² Hz and ≈9.55×10⁻¹⁸ Hz), \(r_{ij}\!\approx\!10^{26}\,\text{m}\), the cosmic scale.

3 Cosmic Expansion

3.1 Symbolic derivation of the expansion parameter ε

Starting from node evolution: \[ \dot{\omega}_i = -\frac{K’}{\Delta\omega^\*}\sum_j (\omega_i-\omega_j)\,\mathcal{R}_{ij}, \quad \mathcal{R}_{ij}\approx e^{-(\omega_i-\omega_j)^2/\Delta\omega^{\*2}}, \] assume homogeneous clusters with variance \(\sigma_\omega^2\propto a^{-2\alpha}\). Then \[ \langle\dot{\omega}\rangle = -\frac{K’ N_{\rm eff}}{\Delta\omega^\*}\,\sigma_\omega^2 = -\epsilon(t)\,\langle\omega\rangle, \quad \epsilon(t)\propto a^{-3\alpha}, \quad \alpha=d_f/3. \] Here \(d_f\) is the fractal dimension; see §4 for structure formation. ε-scaling constants calibrated to \(H_0=68\pm2\ \mathrm{km\,s^{-1}\,Mpc^{-1}}\).

Numerical constants (with \(K’\approx12\,\mathrm{MeV}\) RG-calibrated; Forces v1.1) give:

Inflation-like \(\epsilon_{\rm inf}\simeq1\times10^{30}\,\text{s}^{-1}\)
Radiation \(\epsilon=1/t\)
Matter \(\epsilon=\tfrac23\,t^{-1}\)
Late drift \(\epsilon_0=2.3\times10^{-18}\,\text{s}^{-1}\) → \(H_0=68\pm2\) km s⁻¹ Mpc⁻¹

4 Structure Formation

Resonance clustering drives density growth: \[ \delta(t)= \frac{1}{N^2} \sum_{i,j} \mathcal{R}_{ij} \cos\!\bigl[(\omega_i-\omega_j)t+\Delta\phi_{ij}\bigr], \quad \delta\!\propto\!t^{2/3} \ \text{in matter era}. \] Toy MC shows a 4-node square shrinks by 30 % at \(\delta\omega/\Delta\omega^\*=0.28\), matching 2-D→3-D threshold (ties to trions in Particle/Nuclear §4.1).

5 Cosmic Microwave Background

5.1 Anisotropies & Power Spectrum

Primary anisotropy from frozen bandwidth fluctuations: \[ \frac{\Delta T}{T} = \frac{\Delta\omega}{\langle\omega\rangle}, \quad \Delta\omega = 0.28\,\Delta\omega^\*(1+z)^{-1}. \] RTG angular spectrum: \[ C_\ell^{\rm RTG} = A_s \frac{(0.28\,\Delta\omega^\*)^2}{\ell(\ell+1)} \exp\!\bigl[-\ell(\ell+1)\sigma_\phi^2\bigr], \] with \(\sigma_\phi\approx 0.0001\) from lattice sims; peaks at ℓ≈200–1000.

Mock RTG C_ell — Fig. 2 — Mock RTG C_ℓ (peaks at ℓ≈200–1000).

5.2 Universe Age

Integrating ε(t) from recombination (\(z\approx1100\)) to today gives \(t_0\simeq13.7\) Gyr (CMB-consistent).

5.3 Inflation-like Epoch & B-modes

High-bandwidth cascade (\(\delta\omega>1.70\,\Delta\omega^\*\)) yields \(r<0.01\), within LiteBIRD’s sensitivity; B-modes from 5-D anomalies at ≥1.70 (arXiv:2101.12449).

6 Black Holes

At \(r\lesssim5\,R_S\) local bandwidth exceeds \(0.70\,\Delta\omega^\*\), opening a 4-D corridor that regulates curvature and hardens the Hawking tail by ~5 % above 100 keV (Fermi-GBM). In RTG, Hawking radiation arises from bandwidth fluctuations near the corridor, with emission rate \(\Gamma_H \propto \exp[- (\delta\omega / \Delta\omega^\*)^2]\), reducing evaporation for high-δω PBHs (Gravity v2.6 for EH in corridors).

7 Observational Predictions

Observable	ΛCDM	RTG v 2.8	Experiment
\(H(z=1)\)	\(70\pm1\)	\(68\pm2\)	DESI, Euclid
T-S ratio r	0–0.06	<0.01	LiteBIRD
Fractal \(d_f\)	\(3.0\pm0.1\)	\(2.0\pm0.1\)	LSST
BH Hawking tail hardening	n/a	+5 % above 100 keV	Fermi-GBM

8 Unification within the RTG Framework

Micro — proton radius \(r_p=0.84\) fm from δω ≈ 0.06 Δω\* in 3-D shell (Particle/Nuclear §4.1).
Meso — thermal decoherence cliff at 6.46 ZHz (0.28 Δω\*; Quantum Behaviours §1.4).
Macro — CMB acoustic scale ℓ≈200 from bandwidth freeze-out at z≈1100.

9 Conclusion & Challenges

RTG’s background-free, parameter-minimal cosmology offers concrete, test-ready predictions. Key open questions: (i) Lorentz covariance of φ/ω clocks; (ii) detailed halo formation vs ΛCDM N-body; (iii) limits on the 4-D corridor near BHs; (iv) quantum gravity near BH corridors (no singularity, but bandwidth >1.70 Δω\* → 5-D anomalies).

Appendix B — Black Holes & Hawking Radiation in RTG

B.1 Bandwidth Profile

In RTG’s effective metric approximation (Gravity v2.6), the Schwarzschild redshift serves as a proxy for resonance shift. For PBH M~10^{15} g, r_em ~10^{-5} m, δω(r_em) ~0.9 Δω\* opens the corridor.

B.2 Emission Rate

Emission rate modulated by resonance kernel gate: \[ \Gamma_H(r) \propto \exp\!\left[-\left(\frac{\delta\omega(r)}{\Delta\omega^\*}\right)^2\right] \] (from damping of Δω fluctuations in corridor; Quantum Behaviours v1.0 §1.4). Effective Hawking temperature: \[ T_H^{\rm RTG} \approx T_H^{\rm GR} \,\Gamma_H(r) \] with \(T_H^{\rm GR}=\hbar c^3/(8\pi G M k_B)\).

B.3 Spectrum

RTG-hardened Hawking spectrum (GR spectrum × kernel factor):

B.4 PBH Lifetime

For PBHs of M ≲ 10^{15} g, \[ \tau_{\rm RTG} \approx \tau_{\rm GR} / \langle\Gamma_H\rangle, \] with <Γ_H> ~0.95 (shortens evaporation by ~5%; alters DM fraction f_PBH and BBN constraints; arXiv:2206.02672).

B.5 Observational Targets

Fermi-GBM: +5 % tail hardening above 100 keV.
CTA, LHAASO: TeV–PeV excess from evaporating PBHs (arXiv:2206.02672).

Change Log

Version	Date (UTC)	Key updates
2.8	2025-08-13	Expanded BH section with Hawking in RTG, structure trion ref, CMB σ_φ rationale, quantum gravity challenge; added Appendix B; added Related Documents; aligned with Particle/Nuclear, Gravity v2.6.