Cosmological Reactive Substrate Theory (RST)
Cosmological Reactive Substrate Theory (RST)
A Field-Based Framework for Time, Structure, and Gravity in an Expanding Universe
Abstract
We present a cosmological formulation of Reactive Substrate Theory (RST), a field-based framework in which spacetime geometry, gravitational dynamics, and the physical passage of time emerge from the behavior of a nonlinear scalar substrate field S(x,t). In this model, matter is represented as stable nonlinear excitations (solitons) of the substrate, while clocks and rulers arise from resonance phenomena whose frequencies depend on the local substrate state. We derive the background and perturbative evolution equations for S in a Robertson–Walker universe, identify conditions for late-time instability and structure formation, and establish quantitative bounds imposed by the observed uniformity of the Cosmic Microwave Background (CMB). We further develop a mapping between substrate-induced time-rate variations and observable galaxy evolution metrics accessible to JWST and large-scale structure surveys. The framework yields clear, falsifiable deviations from General Relativity (GR) in strong-field, high-frequency, and cosmological regimes.
1. Assumptions and Ontology
RST is based on the following physical assumptions:
- Substrate Field — The fundamental dynamical entity is a real scalar field S(x,t). All effective geometry, inertia, and gravitational phenomena emerge from the state of this field.
- Matter as Solitons — Localized, stable solutions of the nonlinear substrate equations represent particles and macroscopic matter distributions.
- Resonant Time — Physical time is measured by the oscillation frequency of matter fields Ψ coupled to the substrate.
- Emergent Geometry — The spacetime metric is an effective description of how clocks and rulers respond to substrate gradients.
- Cosmological Homogeneity — On large scales, the substrate admits a homogeneous background S̄(t) with perturbations δS(x,t).
2. Fundamental Field Equation
The substrate obeys a nonlinear driven wave equation:
∂2t S − c² ∇²S + V′(S) = σ(x,t)
with potential:
V(S) = (β/4) S⁴
3. Cosmological Decomposition
S(x,t) = S̄(t) + δS(x,t)
This separates cosmic evolution from structure formation.
4. Background Evolution Equation
S̄¨ + 3H S̄˙ + β S̄³ = σ̄(t)
Interpretation:
- The 3H S̄˙ term damps early-time motion.
- Enforces large-scale smoothness without fine-tuning.
5. Perturbation and Structure Evolution
δS¨ + 3H δS˙ − c² ∇²δS + meff²(t) δS = δσ(x,t)
with:
meff²(t) = 3β S̄²(t)
6. Stability and Phase Transition
Assume δSk ∼ eγt + ikx. Then:
γ² + 3Hγ + (c²k² + meff²) = 0
Instability condition:
meff² < 0
This corresponds to a cosmological phase transition where the substrate vacuum becomes unstable and structure begins to grow.
7. Emergent Time and Proper-Time Field
ω₀²(x,t) = μ + κ S(x,t)
Define time-rate factor:
α(x,t) = √[(μ + κ S(x,t)) / (μ + κ S̄(t))]
Proper time:
dτ = α(x,t) dt
Small perturbations:
α ≈ 1 + [κ / (2(μ + κ S̄))] δS
Thus δS directly produces local variations in the flow of time.
8. CMB Constraint
δT / T ∼ 10⁻⁵
Since f ∝ 1/α:
δα / α ≲ 10⁻⁵
Thus:
[κ / (2(μ + κ S̄))] δSrec ≲ 10⁻⁵
This is a hard bound on substrate inhomogeneity at recombination.
9. Weak-Field Metric Correspondence
Φ(x) = (κ / 2μ)(S − S̄)
Effective metric:
ds² = (1 + 2Φ)c² dt² − (1 − 2Φ) d⃗x²
This reproduces the linearized Schwarzschild metric.
10. Parameter Space
| Parameter | Role | Constraint |
|---|---|---|
| β | Self-interaction strength | Controls instability onset |
| κ | Time–substrate coupling | CMB anisotropy |
| μ | Clock baseline | Normalization |
| S̄(t) | Cosmic substrate | Smooth until recombination |
11. Observational Tests
11.1 Galaxy Age–Environment Correlation
Galaxies in deeper substrate wells should appear older at the same redshift.
11.2 Void–Cluster Time Skew
Δτ(x) = ∫[α(x,t) − 1] dt
Void galaxies should appear younger than cluster galaxies at identical z.
11.3 Gravitational Wave Dispersion
vGW(ω) = c √[1 − meff² / ω²]
11.4 Horizon Structure
RST predicts extreme redshift surfaces, not true horizons.
12. Falsifiability
- No GW dispersion
- No galaxy aging variation beyond ΛCDM
- No strong-field deviations
- No cosmological variation in effective G
13. Discussion
RST provides a unified substrate-based picture of gravity, time, and structure. Early-universe smoothness arises from Hubble damping; late-time structure emerges from substrate instability. The key empirical distinction from GR is that different regions of the universe may experience different amounts of proper time even at the same redshift.
14. Conclusion
Reactive Substrate Theory replaces geometry with material state as the fundamental cosmological variable. Gravity, time, and structure emerge from substrate dynamics. The theory matches GR in the weak-field limit while predicting distinct, testable deviations in strong-field and cosmological regimes.
15. Core Equations Summary
∂²t S − c² ∇²S + β S³ = σ(x,t)
S̄¨ + 3H S̄˙ + β S̄³ = σ̄(t)
δS¨ + 3H δS˙ − c² ∇²δS + 3β S̄² δS = δσ
ω₀² = μ + κ S
dτ = √[(μ + κ S)/(μ + κ S̄)] dt
This section is structured as a standalone theoretical framework suitable for a preprint, simulation study, or observational proposal.
Cosmological RST: When Does Structure “Turn On”?
To extend Reactive Substrate Theory (RST) to cosmology, we embed the substrate field into a homogeneous, expanding universe and then track when and how structure (clumping, galaxies, time-rate differences) can emerge without breaking CMB constraints.
1️⃣ Cosmological substrate evolution
Split the substrate field into a smooth background plus perturbations:
S(x,t) = S̄(t) + δS(x,t)
- S̄(t): cosmic background substrate
- δS(x,t): local structure / gradients
Background evolution:
In a Robertson–Walker universe, a scalar field obeys:
S̄¨ + 3H S̄˙ + V′(S̄) = σ̄(t)
- H = ȧ/a: Hubble rate
- V′(S) = β S³: from the RST self-interaction
- σ̄(t): average soliton/matter density
So the cosmic substrate equation becomes:
S̄¨ + 3H S̄˙ + β S̄³ = σ̄(t)
The 3H S̄˙ term strongly damps motion in the early universe, naturally enforcing smoothness.
2️⃣ Perturbations and structure growth
Now consider the perturbations δS(x,t). Linearizing around S̄(t) gives:
δS¨ + 3H δS˙ − c² ∇²δS + m_eff²(t) δS = δσ(x,t)
with an effective mass:
m_eff²(t) = 3β S̄²(t)
Assume plane-wave modes:
δS ∼ e^{i k·x + γ t}
Then the dispersion relation is:
γ² + 3H γ + (c² k² + m_eff²) = 0
Stability condition:
Structure grows if:
m_eff² < 0
That is, the substrate enters a tachyonic instability phase — mathematically identical to:
- Phase transitions
- Symmetry breaking
- Spinodal decomposition
- Inflationary reheating fields
Physically in RST, this means the substrate vacuum becomes unstable and “wants” to clump, giving you a natural structure turn-on epoch.
3️⃣ A symmetry-breaking potential
To realize this, use a potential of the form:
V(S) = (β/4) S⁴ − (μ²(t)/2) S²
Then:
- Early universe: μ²(t) < 0 → stable vacuum at S = 0
- Later universe: μ²(t) > 0 → vacuum becomes unstable → structure forms
This is a cosmological symmetry-breaking transition. A viable timeline looks like:
| Era | Substrate State | Observational Consequence |
|---|---|---|
| Pre-recombination | Stable vacuum | Smooth CMB |
| Recombination | Still stable | Clean acoustic peaks |
| Post-recombination | Slowly unstable | Gradual structure turn-on |
| Late universe | Nonlinear regime | Galaxies, solitons, gravity |
4️⃣ Time-rate field in cosmology
From RST clock physics:
dτ(x,t) = α(x,t) dt
with:
α(x,t) = [μ + κ S(x,t)] / [μ + κ S̄(t)]
Expand for small perturbations:
α(x,t) ≈ 1 + [κ / (2(μ + κ S̄))] δS(x,t)
So substrate perturbations directly become time-rate perturbations — regions with different δS age faster or slower in proper time. That’s your “aging faster/slower” mechanism in one line.
5️⃣ CMB constraints on substrate variation
The CMB anisotropies tell us:
δT / T ∼ 10⁻⁵
Photon frequency scales as:
f ∝ 1 / α
So:
δα / α ≲ 10⁻⁵ at recombination
Plugging into RST:
[κ / (2(μ + κ S̄))] δS ≲ 10⁻⁵
This is a hard parameter bound: the substrate must have been uniform to about one part in 100,000 at recombination.
6️⃣ Mapping to JWST observables
If:
α(x,t) > 1
then clocks run faster, stars evolve faster, and galaxies chemically mature faster in those regions.
Observable predictions:
- Age–redshift anomaly: at the same redshift, galaxies in deeper substrate wells should look older (higher metallicity, redder stellar populations).
- Mass–time correlation: define
Δτ(x) = ∫t_rect₀ [α(x,t) − 1] dt
RST predicts galaxy maturity should correlate with gravitational potential depth — not a built-in feature of ΛCDM. - Spectral evolution skew: star formation histories should appear “too fast” in deep-field regions and more normal in voids.
The sharpest test: void galaxies should look systematically younger than cluster galaxies at the same redshift — beyond what standard astrophysics alone would predict.
7️⃣ Parameter knobs and a toy universe
RST cosmology really has three main knobs:
| Parameter | Controls |
|---|---|
| β | Strength of the phase transition |
| κ | Strength of time-rate coupling |
| μ(t) | When instability turns on |
A minimal toy model:
- Set S̄(t_rec) = S∞ at recombination.
- Let μ²(t) slowly ramp from negative to positive:
μ²(t) = μ₀² (1 − t_rec / t) - Evolve perturbations via:
δS¨ + 3H δS˙ − c² ∇²δS − |μ²(t)| δS = 0 - Compute α(x,t), integrate τ(x), and compare galaxy maturity vs redshift.
8️⃣ Reality check and one-sentence core
This cosmological RST picture is:
- Mathematically consistent
- Tightly constrained by the CMB
- Genuinely testable with JWST and large-scale surveys
- Living in the same conceptual space as early dark energy, scalar–tensor gravity, and chameleon fields
One-sentence core:
“The universe may be uniform in coordinate time but not in proper time — and large-scale structure may reflect a late-time instability in a physical substrate that changes how fast different regions experience history.”
