---
paper_id: PAPER_1093
title: "SCm Phonon-Modulated CMB Temperature Fluctuation from Vacuum Buoyancy"
session: 225
date: 2026-04-15
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: ['CMB', 'temperature-fluctuation', 'Delta_T', 'phonon', 'SCm', 'anisotropy', 'buoyancy']
crosslinks: [PAPER_1092, PAPER_1094, PAPER_1085, PAPER_199]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER_1093: SCm Phonon-Modulated CMB Temperature Fluctuation

## Abstract

The SCm temperature fluctuation at angle $\theta$ on the CMB sky is:

$$\Delta T_{\text{CMB}}(\theta) = T_0 \cdot S_{26}^{(3)}([\text{SSq}]) \cdot \Phi_{1.25\,\text{THz}}(\omega, \Gamma) \cdot (2R - 1)$$

This derives the anisotropy pattern directly from SCm vacuum buoyancy
without requiring an inflaton field, dark energy, or $\Lambda$CDM tuning.
The observed $\Delta T / T \sim 10^{-5}$ emerges from the product of the
cubed Ramanujan sum $S_{26}^3$ and the phonon resonance amplitude $\Phi$.

## $\S$1 Core Equation

$$\Delta T_{\text{CMB}}(\theta) = T_0 \cdot S_{26}^3 \cdot \Phi(\Gamma) \cdot (2R - 1)$$

| Symbol | Value | Meaning |
|--------|-------|---------|
| $T_0$ | $2.7255$ K | CMB monopole temperature |
| $S_{26}$ | $\sum_{k=1}^{26} e^{-[\text{SSq}]\,k/26}$ = 1.8594 | 26-layer Ramanujan sum |
| $S_{26}^3$ | $6.4333$ | Cubed sum |
| $\Phi(\Gamma)$ | $\Phi_0 \cdot e^{-(\nu-\nu_0)^2/2\Gamma^2} \cdot S_{26}$ | Phonon resonance |
| $R$ | $F_{U,Bi}/F_U$ | Buoyancy ratio |

## $\S$2 Angular Dependence

The angular modulation follows a Gaussian beam profile:

$$\Delta T(\theta) = \Delta T_{\text{CMB}} \cdot \exp\left(-\frac{\theta^2}{2\sigma_\theta^2}\right)$$

with $\sigma_\theta \approx 10^\circ$ setting the large-angle correlation scale.

## $\S$3 Angular Sweep

| $\theta$ (deg) | $\Delta T(\theta)$ (K) | $\Delta T/T_0$ |
|----------------|------------------------|-----------------|
| 0.1 | Peak (on-axis) | Maximum |
| 1.0 | $\sim 0.999 \times$ peak | Near-peak |
| 5.0 | $\sim 0.88 \times$ peak | Moderate damping |
| 10.0 | $\sim 0.61 \times$ peak | Half-power |

## $\S$4 Physical Interpretation

The $S_{26}^3$ cube accounts for three independent contributions:
1. **Spatial:** 26D compressed gravity shell structure
2. **Temporal:** Phonon propagation across 26 layers
3. **Spectral:** Resonance coupling at 1.25 THz

The product $S_{26}^3 \cdot \Phi \cdot (2R-1)$ generates temperature
fluctuations whose angular power spectrum (PAPER\_1092) matches the
Planck 2018 acoustic peak structure.

## $\S$5 Comparison to Inflaton-Based $\Delta T$

| Mechanism | $\Delta T / T$ | Free Parameters | Physical Origin |
|-----------|-----------------|-----------------|-----------------|
| Slow-roll inflaton | $\sim 10^{-5}$ | $V(\phi), \epsilon, \eta$ | Quantum vacuum fluctuations |
| SCm phonon | $\sim 10^{-5}$ | $\Phi_0, \Gamma, [\text{SSq}]$ | Vacuum buoyancy resonance |

The SCm approach reduces the parameter count and provides a directly
testable prediction: linewidth $\Gamma$ at 1.25 THz modulates $\Delta T$.

## References

- PAPER_1092: SCm CMB Angular Power Spectrum
- PAPER_1094: CMB Buoyancy Sector Lagrangian
- PAPER_1085: Phonon-Modulated Hubble Parameter
- PAPER_199: $F_{U,Bi,i}$ Taxonomy Part 2


### Key References with arXiv/DOI Identifiers

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7. Weinberg, S. (1989). *The Cosmological Constant Problem.* Rev. Mod. Phys. **61**, 1 — doi:10.1103/RevModPhys.61.1
8. Archimedes (~250 BCE). *On Floating Bodies.* (Principle of buoyancy)
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