Appendix: Provisional RST Numeric Priors

Appendix: Provisional RST Numeric Priors

📚 Introduction

This appendix collects the current best‑effort numeric estimates for parameters in Reactive Substrate Theory (RST). These values are extrapolated from the available cosmological inputs (wS ≈ −0.95, zt ≈ 0.34, c = 3.0 × 108 m/s) and are intended as provisional priors for further study. They are not measurements, but educated guesses that make the theory predictive and testable.

🌌 Substrate Field Normalization

  • Field amplitude today (S0): ≈ 2.3 × 10−5 J1/2·m−3/2
  • Baseline energy density: ρS0c2 ≈ 5.3 × 10−10 J/m3
  • Equation of state: wS ≈ −0.95

⚖️ Nonlinear Elasticity Constant (β)

The cubic term βS3 introduces amplitude‑dependent stiffness. Dimensional analysis and cosmological balance suggest:

  • Units: m−2·J−1
  • Estimated value: β ≈ 1 × 10−26 m−2·J−1
  • Interpretation: Negligible locally, but relevant on horizon‑scale backgrounds, consistent with wS ≈ −0.95.

🔧 Matter Soliton Parameter (σ)

  • Role: Encodes localized knots of substrate tension (particles).
  • Background average: ⟨σ⟩·⟨FR(C[Ψ])⟩ ≈ 0 at zeroth order, so the cubic term dominates cosmology.
  • Perturbation amplitude: δS/S0 ≈ 10−5 to 10−4 on linear scales, consistent with CMB homogeneity and structure growth.
  • Phenomenological coupling: γm ~ 0.1–1, to be fit against growth data.

🚀 Effective Background Pressure and Sound Speed

  • Pressure today: pS0 ≈ −5.0 × 10−10 J/m3
  • Effective sound speed: cs,S ≈ 0.97c, consistent with near‑stiff response at large scales.

🌊 Gravitational‑Wave Dispersion Scale

Nonlinear elasticity modifies wave propagation. The crossover wavenumber where cubic nonlinearity matches the linear term is:

  • kNL: ≈ 7.7 × 10−27 m−1 (horizon‑scale)
  • Implication: Deviations from luminal propagation appear only on cosmological scales, safe locally but potentially testable via pulsar timing arrays or ISW effects.

📊 Parameter Table

Symbol Description Provisional Value
c Substrate wave speed (relativistic limit) 3.0 × 108 m/s
wS Equation‑of‑state parameter ≈ −0.95
zt Transition redshift ≈ 0.34
ΩS0 Present‑day substrate fraction ≈ 0.69
Ωm0 Present‑day matter fraction ≈ 0.30
S0 Field amplitude today ≈ 2.3 × 10−5 J1/2·m−3/2
β Nonlinear elasticity constant ≈ 1 × 10−26 m−2·J−1
σ Matter soliton parameter δS/S0 ≈ 10−5–10−4
γm Phenomenological coupling ~0.1–1 (to be fit)
pS0 Effective pressure today ≈ −5.0 × 10−10 J/m3
cs,S Effective sound speed ≈ 0.97c
kNL GW dispersion crossover wavenumber ≈ 7.7 × 10−27 m−1

✊ Closing Note

These provisional numeric priors demonstrate that RST can reproduce the key observational features of cosmology — dark energy fraction, transition redshift, and acceleration — while offering unique predictions tied to substrate elasticity. The next step is to refine β and σ through observational fits (supernovae, BAO, CMB, gravitational waves) to move from priors to testable predictions.

RST vs. Standard Cosmology (ΛCDM): Side‑by‑Side Comparison

📚 Overview

This chart compares Reactive Substrate Theory (RST) to the standard ΛCDM model. It highlights where RST matches observations (like cosmic acceleration and energy fractions) and where it offers different interpretations or predictions.

📊 Comparison Table

Topic RST (Reactive Substrate Theory) ΛCDM (Standard Cosmology)
Core idea Universe is an elastic substrate; matter/forces are tensions and waves. Gravity from spacetime curvature; matter and radiation in expanding space.
Driver of acceleration Substrate tension acts like a negative‑pressure fluid. Cosmological constant (Λ) or dark energy.
Equation of state for acceleration wS ≈ −0.95 (slow dilution). wΛ = −1 (constant energy density).
Transition redshift (onset of acceleration) zt ≈ 0.34. zt ≈ 0.3–0.7 (model/data dependent).
Present‑day energy fractions ΩS0 ≈ 0.69, Ωm0 ≈ 0.30. ΩΛ ≈ 0.69, Ωm0 ≈ 0.30.
Deceleration parameter today q0 ≈ −0.49 (accelerating). q0 ≈ −0.55 (typical fits; accelerating).
Singularities Avoided by elastic limits (finite cores, bounce behavior). Mathematical singularities allowed in classical GR (resolved in quantum gravity proposals).
Gravitational‑wave propagation Possible horizon‑scale dispersion from βS3 nonlinearity. Luminal propagation in GR; modifications only with new physics.
Structure formation Driven by matter solitons (σ) coupled to substrate; δS/S0 ~ 10−5–10−4. Growth from initial perturbations under gravity; σ8 ~ 0.8.
Key free parameters β (nonlinear elasticity), γm (matter coupling), wS. ΩΛ, Ωm, H0, ns, σ8, etc.
Primary tests H(z) from SN/BAO; GW dispersion on horizon scales; ISW/CMB‑lensing cross‑checks. SN/BAO/CMB background fits; growth (RSD), CMB anisotropies, lensing.
Strengths Matches energy fractions and acceleration without Λ; avoids singularities; offers new testable effects. Extremely successful global fit to data; minimal parameters; well‑tested framework.
Open challenges Calibrate β and γm; derive precise perturbation predictions; ensure consistency with CMB/BAO. Hubble tension, small‑scale structure issues; physical origin of Λ.

🧮 Reference Values Used (RST)

  • Equation of state: wS ≈ −0.95
  • Transition redshift: zt ≈ 0.34
  • Energy fractions: ΩS0 ≈ 0.69; Ωm0 ≈ 0.30
  • Deceleration today: q0 ≈ −0.49
  • Provisional nonlinearity: β ≈ 1 × 10−26 m−2·J−1

👉 Takeaway

RST lines up with the big observational features of ΛCDM (energy split, timing of acceleration, present‑day acceleration) but replaces the cosmological constant with substrate tension and predicts potential horizon‑scale wave effects. Its viability now hinges on pinning down β and the coupling to matter, and then passing precision tests with SN/BAO/CMB and large‑scale structure.

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