Mathematical Steps in RST Cosmology
Reactive Substrate Theory Cosmology: Emergent Time and Early Structure Formation
We consider a cosmological framework based on Reactive Substrate Theory (RST), in which the universe is permeated by a continuous scalar field S(x,t) (the substrate). Both spacetime geometry and the flow of proper time are emergent phenomena determined by the local state of this field.
1. Emergent Proper Time
Local proper time τ is related to global coordinate time t via:
dτ = dt · [ 1 + α · δS(x,t) ]
where δS(x,t) = S(x,t) − ⟨S⟩ denotes deviations from the mean substrate, and α is a coupling constant characterizing the influence of the substrate on local clock rates.
In regions of enhanced substrate, dτ/dt > 1, leading to accelerated local evolution of physical processes.
2. Density Perturbation Growth
The standard linear growth equation for matter density contrast δ = δρ / ρ in ΛCDM cosmology is:
d²δ/dt² + 2H(t) · dδ/dt = 4πG ρm δ
In RST, this equation is expressed in terms of local proper time τ:
(1 + α δS)² · d²δ/dτ² + α · (d(δS)/dt) · dδ/dτ + 2H(t)(1 + α δS) · dδ/dτ = 4πG ρm δ
For small α δS, a linearized form yields:
d²δ/dτ² + 2H(t) · dδ/dτ · (1 − α δS) ≈ 4πG ρm δ · (1 − 2α δS)
This formalism explicitly demonstrates that regions with higher substrate values undergo accelerated growth of density perturbations, thereby providing a natural explanation for the appearance of early, evolved cosmic structures at high redshift without invoking modifications to dark matter or other exotic physics.
3. Implications for Observations
- CMB Anisotropies: Temperature fluctuations may encode both standard density perturbations and variations in accumulated proper time.
- High-Redshift Galaxies: Apparent over‑mature galaxies are a consequence of differential local clock rates induced by substrate fluctuations.
- Testability: The framework predicts correlations between CMB features and the maturity of early structures, offering potential observational signatures distinct from ΛCDM.
1️⃣ Mathematical Steps in RST Cosmology
Step 1: Define the Local Proper Time
RST introduces proper time τ as emergent from the substrate field S(x,t):
dτ = dt · [ 1 + α · δS(x,t) ]
Terms explained:
- dτ → infinitesimal proper time in a local region (the “clock experienced” locally)
- dt → global coordinate (cosmic) time in standard cosmology
- δS(x,t) = S(x,t) − ⟨S⟩ → local deviation of the substrate field from its spatial average
- α → coupling constant controlling how strongly substrate variations affect local proper time
If δS > 0, time runs faster locally; if δS < 0, time runs slower. This is the key mechanism for accelerating or decelerating local evolution of structures.
Step 2: Standard Linear Growth Equation in Cosmology
In standard ΛCDM, the density contrast δ = δρ / ρ evolves as:
d²δ/dt² + 2H(t) · dδ/dt = 4πG ρm δ
Terms explained:
- δ = δρ / ρ → fractional density perturbation
- H(t) → Hubble parameter (expansion rate)
- G → gravitational constant
- ρm → background matter density
- t → cosmic coordinate time
This is the standard linear perturbation growth equation.
Step 3: Transform the Growth Equation to Proper Time
RST replaces coordinate time t with local proper time τ:
d/dt = (dτ/dt) · d/dτ = (1 + α δS) · d/dτ
Similarly:
d²/dt² = (1 + α δS)² · d²/dτ² + α · (d(δS)/dt) · d/dτ
Interpretation of terms:
- (1 + α δS)² · d²/dτ² → accelerates the second derivative of δ in regions where substrate is high
- α · d(δS)/dt · d/dτ → correction term due to time‑variation of the substrate (usually small)
Step 4: Modified Linear Growth Equation
Substituting into the standard growth equation gives:
(1 + α δS)² · d²δ/dτ² + α · (d(δS)/dt) · dδ/dτ + 2H(t)(1 + α δS) · dδ/dτ = 4πG ρm δ
Interpretation:
- High substrate regions (δS > 0) → effective acceleration of structure growth because dτ/dt > 1
- Early galaxy formation emerges naturally because regions with higher S evolve faster than the average cosmic clock
- The equation reduces to standard ΛCDM when α → 0
Step 5: Linearized Approximation
For small α δS, the equation can be linearized:
d²δ/dτ² + 2H(t) · dδ/dτ · (1 − α δS) ≈ 4πG ρm δ · (1 − 2α δS)
This makes explicit how local substrate deviations modulate growth without changing the underlying form too much.
