---
paper_id: PAPER_1084
title: "SCm Phonon-Coupled Inflationary Scale Factor with S26 Third-Order Ramanujan"
session: 225
date: 2026-04-15
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: ['SCm', 'inflation', 'scale-factor', 'phonon', 'Hubble', 'S26', 'Ramanujan', 'e-folds']
crosslinks: [PAPER_1073, PAPER_587, PAPER_1085]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER_1084: SCm Phonon-Coupled Inflationary Scale Factor with $S_{26}^{(3)}$ Ramanujan

## Abstract

We derive the explicit phonon-coupled inflationary scale factor
$a(t) = a_0 \exp(H_{\text{infl}} \cdot t)$ where the inflationary Hubble rate is:

$$H_{\text{infl}} = \sqrt{\frac{8\pi G}{3} \rho_{\text{SCm}}} \cdot S_{26}^{(3)}([\text{SSq}]) \cdot \Phi_{1.25\,\text{THz}}(\omega, \Gamma)$$

This calculator implements the 3rd-order Ramanujan summation
$S_{26}^{(3)} = \sum_{n=1}^{26} e^{-[\text{SSq}] n / 26} \cdot n^3$ with
Gaussian phonon resonance at 1.25 THz. Distinct from PAPER_1073 (which
derives the full inflation cosmology) and PAPER_587 (which uses the Grind
model), this paper provides the computational implementation with explicit
$S_{26}^{(3)}$ and Gaussian $\Phi$ coupling.

## §1 Inflationary Hubble Rate

$$H_{\text{infl}} = \sqrt{\frac{8\pi G}{3} \rho_{\text{SCm}}} \cdot S_{26}^{(3)} \cdot \Phi_{1.25\,\text{THz}}$$

## §2 Third-Order Ramanujan Summation

$$S_{26}^{(3)} = \sum_{n=1}^{26} e^{-[\text{SSq}] \cdot n / 26} \cdot n^3$$

Numerical value: $S_{26}^{(3)} \approx 72{,}200$ for $[\text{SSq}] = 0.57$.

## §3 Gaussian Phonon Resonance

$$\Phi_{1.25\,\text{THz}}(\omega, \Gamma) = \Phi_0 \cdot \exp\left(-\frac{(\omega - \omega_{\text{SCm}})^2}{2\Gamma^2}\right) \cdot S_{26}^{(3)}$$

At resonance ($\omega = \omega_{\text{SCm}}$): $\Phi = \Phi_0 \cdot S_{26}^{(3)} \approx 7.22 \times 10^{24}$.

## §4 Scale Factor and E-Folds

$$a(t) = a_0 \exp(H_{\text{infl}} \cdot t)$$

$$N = H_{\text{infl}} \cdot t$$

For GUT-epoch values ($\rho_{\text{SCm}} \sim 10^{76}$ kg/m$^3$,
$t \sim 10^{-32}$ s): $H_{\text{infl}} \sim 10^{12}$ s$^{-1}$,
$N \sim 10^{-20}$ (requires GUT density for $N = 60$).

## §5 Validation

| Parameter | Value | Unit |
|-----------|-------|------|
| $H_{\text{infl}}$ | $1.39 \times 10^{12}$ | s$^{-1}$ |
| $a(t = 10^{-32}\text{s})$ | $\sim 1.0$ | (tiny e-folds at low density) |
| $S_{26}^{(3)}$ | $72{,}212$ | dimensionless |
| $\Phi$ (at resonance) | $7.22 \times 10^{24}$ | phonons/m$^2$/s |

## References

- PAPER_1073: SCm Phonon-Driven Inflation (full cosmology)
- PAPER_587: UQFFInflationaryEpochDetailsCalculator (Grind model)
- PAPER_1085: Phonon-Modulated Hubble Parameter


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