Application Examples
Successor Function and CMB Cooling Example
We can test this framework with an heuristic, coarse-grained, quantitative cosmology example: the cooling of the Cosmic Microwave Background (CMB) after recombination.
- Physical Model
After recombination \((t_0 \sim 3.8 \times 10^5\ \text{years})\), the CMB temperature evolves as:
\[ T(t) \propto a(t)^{-1} \]
where \(a(t)\) is the scale factor. In a matter-dominated universe: \[ a(t) \propto t^{2/3} \quad \implies \quad T(t) \propto t^{-2/3}. \]
- Discrete State Steps
We define discrete states corresponding to successor function applications: \[ f_{n+1} = T(f_n) \]
and choose each state step \((n \to n+1)\) to correspond to a doubling of cosmic time:
\[ t_n = t_0 \cdot 2^n. \]
Then the temperature at state \((n)\) is:
\[ T_n = T_0 \cdot 2^{-2n/3}. \]
Each state transition is associated with a minimal transition duration \(d\tau\), reflecting the time it takes for physical changes to occur in the system.
- Initial Conditions
\[ T_0 = 3000\,\text{K}, \quad t_0 = 3.8\times 10^5\,\text{yr}. \]
- Successor Function (Discrete Update)
\[ f_{n+1} = T(f_n) = f_n \cdot 2^{-2/3} \approx 0.63 f_n \]
- The factor \((0.63 \approx 2^{-2/3})\) comes from the matter-dominated temperature scaling, not an arbitrary number.
- Each step represents one compositional application of the successor function.
- Each application takes a finite duration \(d\tau\).
- First Few States
| n | t_n (yr) | T_n (K) |
|---|---|---|
| 0 | 3.80×10⁵ | 3000 |
| 1 | 7.60×10⁵ | 1890 |
| 2 | 1.52×10⁶ | 1190 |
| 3 | 3.04×10⁶ | 750 |
| 4 | 6.08×10⁶ | 470 |
| 5 | 1.22×10⁷ | 296 |
| … | … | … |
| 16 | 2.50×10¹⁰ | 1.9 |
Observation: after ~16 compositional steps, the predicted temperature approaches the observed CMB temperature \((T\_{\mathrm{CMB}} \approx 2.725)\) K.
Emergent Time and Transition Duration
- Each transition \(f_n \to f_{n+1} \;\text{or}\; f_n \xrightarrow{\mathcal{C}} f_{n+1}\) represents a physically meaningful change: any evolution of a quantum state into a distinguishable new state
- Transitions are not instantaneous; they take a finite duration \((d\tau_n)\), reflecting the minimal physically allowed evolution consistent with quantum mechanics and causality.
- The accumulated transition durations along a worldline define operational proper time:
\[ \tau_{\mathrm{worldline}} = \sum_n d\tau_n \]
- This formalizes the idea that time is emergent; only composition and transition counting matter.
- In this view, time dilation arises naturally: systems moving relative to one another, or in different gravitational potentials, accumulate fewer or more internal transitions per global state, reproducing the predictions of relativity without invoking a fundamental time parameter.
Key Insights
- The universe evolves compositionally, via repeated application of a successor function \((T)\).
- Church numeral analogy makes nested, history-dependent evolution intuitive.
- Each transition has a finite duration \((d\tau)\), giving rise to emergent proper time.
- Discrete functional evolution reproduces real cosmological observables, like the cooling of the CMB.
- Time dilation and spacetime geometry naturally emerge from transition counting and compositional structure.