Plugging known RST values into the substrate

Plugging known RST values into the substrate–cosmology picture

📚 Introduction

This post uses the available Reactive Substrate Theory (RST) values — w_S ≈ −0.95, z_t ≈ 0.34, and c = 3.0 × 10⁸ m/s — to explore their implications for cosmic evolution if substrate tension acts as the dominant late‑time driver. We derive scaling relations, infer present‑day density fractions from the transition redshift, compute the current deceleration parameter, and interpret how the substrate field equation’s nonlinear term shapes wave dynamics without assigning a numeric value to β yet.

📈 Scaling laws from the equation of state

For a homogeneous component with equation of state \(p = w \rho\), the energy density scales with redshift as:

\[ \rho(z) = \rho_0 \, (1+z)^{3(1+w)}. \]

Matter (dust):
\[ \rho_m(z) = \rho_{m0} \, (1+z)^3. \]
Substrate tension (RST component):
\[ \rho_S(z) = \rho_{S0} \, (1+z)^{3(1+w_S)} = \rho_{S0} \, (1+z)^{0.15}, \] since \(3(1+w_S) = 3(1-0.95) = 0.15\). This extremely slow dilution is consistent with a near–dark‑energy behavior.

🔍 Inferring present‑day density ratio from the transition redshift

Define the transition redshift \(z_t\) by the equality \(\rho_S(z_t) = \rho_m(z_t)\). Using the scaling relations above:

\[ \frac{\rho_{S0}}{\rho_{m0}} = (1+z_t)^{3 - 3(1+w_S)} = (1+z_t)^{2.85}. \]

Plugging \(z_t = 0.34\) gives:

\[ \frac{\rho_{S0}}{\rho_{m0}} \approx 1.34^{2.85} \approx 2.30. \]

Assuming a typical matter fraction \(\Omega_{m0} \approx 0.30\), this implies:

\[ \Omega_{S0} \approx 2.30 \times \Omega_{m0} \approx 0.69, \] which aligns closely with the late‑time energy fraction driving accelerated expansion in standard cosmology, but here interpreted as substrate tension rather than a cosmological constant.

🚀 Expansion history and acceleration today

In a spatially flat Friedmann model containing matter plus the RST substrate tension component, the Hubble rate is:

\[ H^2(z) = H_0^2 \left[ \Omega_{m0} (1+z)^3 + \Omega_{S0} (1+z)^{0.15} \right]. \]

The deceleration parameter at \(z=0\) is:

\[ q_0 = \frac{1}{2}\Omega_{m0} + \frac{1+3w_S}{2}\Omega_{S0}. \]

With \(w_S = -0.95\), \(\Omega_{m0} \approx 0.30\), and \(\Omega_{S0} \approx 0.69\):

\[ q_0 \approx \frac{1}{2}\cdot 0.30 + \frac{1 - 2.85}{2}\cdot 0.69 = 0.15 - 0.638 \approx -0.49. \]

A negative \(q_0\) of roughly −0.49 indicates robust present‑day acceleration consistent with the substrate tension component dominating the expansion.

🧭 Consistency check around the transition

Evaluating the densities at the transition \(z = z_t = 0.34\):

Matter amplification:
\[ \frac{\rho_m(z_t)}{\rho_{m0}} = (1.34)^3 \approx 2.41. \]
Substrate amplification:
\[ \frac{\rho_S(z_t)}{\rho_{S0}} = (1.34)^{0.15} \approx 1.04. \]
Equality condition: The equality \(\rho_m(z_t) = \rho_S(z_t)\) is consistent with the inferred present‑day ratio \(\rho_{S0}/\rho_{m0} \approx 2.30\). Matter dilutes as \((1+z)^3\), while substrate tension barely changes, so the substrate component naturally overtakes matter at low redshift.

🌊 Wave dynamics from the substrate field equation

The substrate field equation used across RST posts:

\[ \partial_t^2 S - c^2 \nabla^2 S + \beta S^3 = \sigma(x,t)\, F_R(C[\Psi]) \]

Linear regime: For small amplitudes \(S\), waves propagate at speed \(c\), enforcing a relativistic signaling limit and elastic Lorentz symmetry.
Nonlinear regime: The cubic term \(\beta S^3\) introduces amplitude‑dependent stiffness. A heuristic crossover scale satisfies:
\[ \beta S^2 \sim c^2 k^2, \] indicating when nonlinearity modifies dispersion. While \(\beta\) is not yet numerically calibrated, the cosmological effective behavior \(w_S \approx -0.95\) suggests a negative‑pressure background consistent with slow dilution and late‑time acceleration.

📌 Implications and next steps

Energy fractions: With \(w_S \approx -0.95\) and \(z_t \approx 0.34\), we infer \(\Omega_{S0} \approx 0.69\) and \(\Omega_{m0} \approx 0.30\), matching the observed split but attributing it to substrate tension rather than Λ.
Acceleration today: The deceleration parameter \(q_0 \approx -0.49\) reflects strong acceleration without requiring a true cosmological constant.
Dispersion predictions: Curvature‑dependent gravitational‑wave dispersion can emerge from the \(\beta S^3\) nonlinearity once \(\beta\) is fixed; supernovae/BAO fits could help calibrate \(\beta\) and refine \(w_S\).
Model development: Formalizing the mapping from the PDE to background cosmology (effective fluid description) will allow direct confrontation with data while keeping RST’s elastic interpretation intact.

🧮 Quick reference: parameters used

Symbol	Description	Value
c	Substrate wave speed (relativistic limit)	3.0 × 10⁸ m/s
w_S	Equation‑of‑state parameter for substrate tension	≈ −0.95
z_t	Transition redshift (substrate overtakes matter)	≈ 0.34
Ω_m0	Present‑day matter fraction (assumed)	≈ 0.30
Ω_S0	Present‑day substrate fraction (inferred)	≈ 0.69
q₀	Present‑day deceleration parameter (computed)	≈ −0.49
β	Nonlinear elasticity constant	Symbolic (numeric calibration pending)

Note: The table fixes the numeric values used in the derivations above; \(\beta\) remains symbolic until calibrated by observational fits or microphysical modeling.

🧭 Closing

Using only the currently available RST values, we obtain a self‑consistent late‑time cosmology: substrate tension dilutes very slowly, overtakes matter near \(z_t \approx 0.34\), yields \(\Omega_{S0} \approx 0.69\), and drives present acceleration with \(q_0 \approx -0.49\). The substrate field’s nonlinearity provides natural avenues for testable dispersion effects once \(\beta\) is numerically pinned down. This forms a solid baseline for your next RST post and for a future parameter‑fitting study that can translate elastic microphysics into cosmological observables.

In Short: What the Values Tell Us About RST

Plugging the available values into the Reactive Substrate Theory (RST) framework shows that it can mimic the successes of standard cosmology without invoking a cosmological constant. The following points summarize the implications:

Substrate tension behaves like dark energy: With w_S ≈ −0.95, it dilutes extremely slowly, just like the observed dark energy component.
Transition redshift: z_t ≈ 0.34 naturally gives a present‑day split of Ω_S0 ≈ 0.69 and Ω_m0 ≈ 0.30, which matches the observed matter–dark energy balance.
Acceleration emerges correctly: The computed deceleration parameter q₀ ≈ −0.49 is consistent with current measurements of cosmic acceleration.
No infinities or singularities: RST’s elastic substrate avoids the paradoxes of singularities, offering finite density cores and bounce cosmology.
Predictive potential: The nonlinear term (βS³) hints at testable effects like gravitational‑wave dispersion once β is calibrated.

👉 Takeaway

These results suggest RST is viable in principle as a cosmological model. It reproduces the key observational features (dark energy fraction, transition redshift, acceleration) with the values we have. Its viability now depends on whether future work can assign numeric values to constants like β and show unique, testable predictions that distinguish it from standard ΛCDM.

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 (w_S ≈ −0.95, z_t ≈ 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 J^1/2·m^−3/2
Baseline energy density: ρ_S0c² ≈ 5.3 × 10⁻¹⁰ J/m³
Equation of state: w_S ≈ −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 w_S ≈ −0.95.

🔧 Matter Soliton Parameter (σ)

Role: Encodes localized knots of substrate tension (particles).
Background average: ⟨σ⟩·⟨F_R(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: γ_m ~ 0.1–1, to be fit against growth data.

🚀 Effective Background Pressure and Sound Speed

Pressure today: p_S0 ≈ −5.0 × 10⁻¹⁰ J/m³
Effective sound speed: c_s,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:

k_NL: ≈ 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
w_S	Equation‑of‑state parameter	≈ −0.95
z_t	Transition redshift	≈ 0.34
Ω_S0	Present‑day substrate fraction	≈ 0.69
Ω_m0	Present‑day matter fraction	≈ 0.30
S₀	Field amplitude today	≈ 2.3 × 10⁻⁵ J^1/2·m^−3/2
β	Nonlinear elasticity constant	≈ 1 × 10⁻²⁶ m⁻²·J⁻¹
σ	Matter soliton parameter	δS/S₀ ≈ 10⁻⁵–10⁻⁴
γ_m	Phenomenological coupling	~0.1–1 (to be fit)
p_S0	Effective pressure today	≈ −5.0 × 10⁻¹⁰ J/m³
c_s,S	Effective sound speed	≈ 0.97c
k_NL	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.

Conspiranon