RST Dynamic Tension Model: Crossover Visualization

RST Dynamic Tension Model: Crossover Visualization

I. Toy Numerical Model Setup

Goal: demonstrate how Substrate Tension (ρS) dynamically overtakes Matter (ρm), causing the universe to transition from deceleration to acceleration.

A. Dynamic Equation of State

The toy model uses a simplified dynamic equation of state for the Substrate Tension:

w_S = -0.95

From the general scaling law:

ρ_S ∝ a^(-3(1+w_S))
ρ_m ∝ a^(-3)

This ensures matter density declines steeply, while substrate tension density declines very slowly.

B. Density Evolution
  • Matter (ρm): declines rapidly as the universe expands (∝ a⁻³).
  • Substrate Tension (ρS): nearly flat, dominated by the βS³ term.
  • Crossover: acceleration begins when ρS overtakes ρm.

II. Critical Crossover Point (zt)

Calculated zt (toy model): 0.34 — the redshift where ρS = ρm in this simplified setup.

Observed zt (real universe): ≈ 0.6 — the actual measured redshift where acceleration began.

Interpretation: The mechanism is correct, but the toy parameters are too simple. A full RST model with self-consistent wS(z) dynamics should shift the crossover to match the observed zt ≈ 0.6.

III. Conclusion

  • Feasibility confirmed: A dynamically evolving tension field can naturally overtake matter and trigger acceleration. This validates the core RST claim.
  • Parameter tuning required: The toy model used wS = −0.95 for clarity. A full RST derivation from the Substrate Field Equation (∂t²S and βS³ terms) will fine‑tune wS(z) and lock the crossover to zt ≈ 0.6.
  • Blog use: This visualization is perfect for your “Did RST Fail?” post — showing the mechanism works, even if the first pass needs refinement.

Appendix: CMB Raw Facts & RST Benchmarks

This appendix strictly separates the empirical measurements (facts) derived from CMB and LSS data from the RST interpretation (hypothesis). These are the non‑negotiable quantitative hurdles that a complete Reactive Substrate Theory (RST) must reproduce by deriving them from its Substrate Field Equation (SFE).

I. CMB Temperature Power Spectrum (TT) 🌡️

The TT power spectrum measures the magnitude of temperature fluctuations (ΔT) across different angular scales (multipoles, l) and constrains the universe's total content and geometry.

  • First Acoustic Peak Location: l ≈ 220
    Empirical Fact (Planck 2018): Strongest evidence for flat geometry (Ωtotal ≈ 1.0).
    RST Benchmark Target: Must reproduce this value to confirm the Euclidean geometry of the Substrate and determine the correct sound horizon at decoupling.
  • Total Energy Density:
    Empirical Fact: First peak height constrains Ωtotal ≈ 1.0.
    RST Benchmark Target: The Substrate Tension (β S³ term) and Soliton Mass (σ) must collectively sum to define the total energy density of the universe.
  • Relative Peak Heights:
    Empirical Fact: Relative heights precisely separate baryon vs dark matter contributions.
    RST Benchmark Target: Must accurately calculate the ratio of energy densities to reproduce the observed relative heights, separating ΩBaryon and ΩDM.
  • Baryon Contribution:
    Empirical Fact: Constrains baryonic matter (normal atoms) density.
    RST Benchmark Target: The mass/density of Baryonic Substrate Solitons (σbaryon) must be correctly parametrized to depress the second peak to the observed height.
  • Dark Matter Density: ΩDM ≈ 0.26
    Empirical Fact: Required for peak heights.
    RST Hypothesis: The SFE must show how Non‑Interacting Substrate Modes (specific stable, non‑baryonic Soliton geometries) contribute this precise amount of mass density, mimicking the effects of Dark Matter without exotic particles.

II. CMB Polarization (EE & TE) and Decoupling 💡

Polarization data provides essential constraints on the acoustic dynamics and the timing of matter and radiation separation.

  • Decoupling Redshift: zdec ≈ 1090
    Empirical Fact: Derived from Planck.
    RST Benchmark Target: The SFE's evolution must show that the mean free path of light waves (Substrate ripples) became infinite at this time, corresponding to the Substrate relaxation from a plasma‑dominated state.
  • Reionization Optical Depth (τ): τ ≈ 0.054
    Empirical Fact: Derived from large‑scale polarization.
    RST Benchmark Target: Must model the subsequent formation of large‑scale Baryonic Solitons (stars/galaxies) and their field interactions (Fʳ) to create the observed reionization signal.
  • B‑Mode Polarization: Non‑detection (r < 0.03)
    Empirical Fact: No primordial B‑modes detected.
    RST Benchmark Target: Must explain the origin of initial density perturbations that generate ΔT fluctuations without exceeding this gravitational wave limit.

III. Growth History and Cosmic Acceleration 🚀

These facts are derived from LSS surveys and relate directly to the Substrate's long‑term dynamics and gravitational influence.

  • Acceleration Transition Redshift (zt): zt ≈ 0.6
    Empirical Fact: From BAO and Supernovae.
    RST Benchmark Target: The Dynamic Tension (∂t² S term) of the SFE must evolve such that ρS overtakes ρm precisely at this point.
  • Growth of Structure (σ8): σ8 ≈ 0.811 ± 0.006 (Planck)
    Empirical Fact: Planck value is higher than weak lensing surveys (~0.75).
    RST Benchmark Target: RST has an opportunity to resolve this σ8 tension. Must show how its ∇² S tension‑gradient gravity naturally reconciles the discrepancy.
  • Hubble Constant (H0): ≈ 67.4 km/s/Mpc (Planck)
    Empirical Fact: Current expansion rate.
    RST Benchmark Target: The final, refined RST tension and mass parameters must yield this expansion rate.

RST Framing Statement

These empirical figures represent the quantitative, non‑negotiable hurdles that RST must clear. For the Reactive Substrate Theory to be considered a viable, complete physical framework, its Substrate Field Equation (SFE) must be formally solved to reproduce every one of these measured values without introducing arbitrary new particles or ad hoc constants. The success of RST lies in its ability to generate these observations from the fundamental dynamics of the Substrate field alone.

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