---
paper_id: PAPER_1085
title: "Phonon-Modulated Hubble Parameter H(t,Gamma) via F_U and E_net Coupling"
session: 225
date: 2026-04-15
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: ['Hubble', 'phonon', 'F_U', 'E_net', 'modulation', 'expansion', 'cosmology']
crosslinks: [PAPER_1084, PAPER_1073, PAPER_897]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER_1085: Phonon-Modulated Hubble Parameter $H(t, \Gamma)$ via $F_U$ and $E_{\text{net}}$ Coupling

## Abstract

We derive the phonon-modulated Hubble parameter:

$$H(t, \Gamma) = H_0 \left(1 + \frac{\Phi}{F_U} \cdot E_{\text{net}}\right)$$

This couples the phonon resonance flux $\Phi_{1.25\,\text{THz}}$ to the
unified gravitational field $F_U$ and the net vacuum energy $E_{\text{net}}$,
producing a dynamical Hubble rate that amplifies during periods of strong
phonon-vacuum coupling. Unlike PAPER_1084 (which derives the inflationary
$H$ from $\rho_{\text{SCm}}$), this formulation connects $H$ to the
present-epoch observables $F_U$, $E_{\text{net}}$, and $\Phi$.

## §1 Core Equation

$$H(t, \Gamma) = H_0 \left(1 + \frac{\Phi(\Gamma)}{F_U(M, r)} \cdot E_{\text{net}}(t)\right)$$

where:
- $H_0 = 2.195 \times 10^{-18}$ s$^{-1}$ (67.74 km/s/Mpc)
- $\Phi(\Gamma) = \Phi_0 \cdot S_{26}$ at resonance
- $F_U = \sum_{i=1}^{26} G M / r^2 \cdot [\text{SSq}] \cdot i / 26$ (26-layer gravity)

## §2 Net Vacuum Energy

$$E_{\text{net}}(t) = E_0 \cdot e^{(\kappa + [\text{SSq}]/26) t} \cdot S_{26} \cdot (2R - 1)$$

where $R = F_{U,Bi} / F_U$ is the buoyancy ratio:
- $R > 0.5$: expansion (positive $E_{\text{net}}$)
- $R < 0.5$: erosion (negative $E_{\text{net}}$)

## §3 Amplification Factor

The phonon amplification of $H$ is:

$$\frac{H(t, \Gamma)}{H_0} = 1 + \frac{\Phi \cdot E_{\text{net}}}{F_U}$$

For solar parameters ($M = M_\odot$, $r = 1$ AU, $R = 0.8$):
$H/H_0 \sim 10^6$, showing enormous amplification — physically this
represents the local departure from Hubble flow near massive objects.

## §4 Gamma Dependence

The phonon linewidth $\Gamma$ enters through $\Phi(\Gamma)$:

$$\Phi(\Gamma) = \Phi_0 \cdot \exp\left(-\frac{(\omega - \omega_{\text{SCm}})^2}{2\Gamma^2}\right) \cdot S_{26}$$

At resonance: $\Phi = \Phi_0 \cdot S_{26} \approx 1.45 \times 10^{21}$.
Off-resonance: $\Phi$ decreases Gaussian-sharply with linewidth.

## §5 Physical Interpretation

The formula $H(t, \Gamma) = H_0(1 + \Phi/F_U \cdot E_{\text{net}})$ encodes:
1. **Cosmic expansion base rate:** $H_0$ (Hubble constant)
2. **Local phonon enhancement:** $\Phi/F_U$ ratio (SCm vacuum coupling strength)
3. **Net energy direction:** $E_{\text{net}}$ sign (expansion vs erosion)
4. **Linewidth modulation:** $\Gamma$ controls phonon resonance sharpness

## References

- PAPER_1084: SCm Phonon-Coupled Inflationary Scale Factor
- PAPER_1073: SCm Phonon-Driven Inflation
- PAPER_897: PhononModulatedEnergyEnetPhononCalc


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