81018 Toy Model

A Discrete Geometric Toy Model for Scale-Dependent Cosmological Expansion

by Bruce E. Camber with ChatGBT and a synthetic peer review

Abstract
^{We introduce a discrete geometric toy model in which large-scale cosmological expansion emerges from iterative growth at the Planck scale subject to sub-extensive constraints. The model defines a sequence of configurations generated by a binary growth rule with a correction term, interpreted as geometric or interaction-induced inefficiencies. We construct a mapping between discrete iteration steps and effective cosmological observables, including a scale factor and Hubble-like parameter. A concrete choice of the correction term yields scale-dependent expansion, providing a minimal mechanism for deviations from constant expansion. While highly simplified, the framework serves as a heuristic laboratory for exploring how hierarchical structure and expansion may arise from discrete microphysical rules.}

1. Introduction

Reconciling Planck-scale discreteness with the smooth spacetime description of cosmology remains a central open problem in theoretical physics. General relativity successfully describes large-scale dynamics, while quantum theory governs microscopic phenomena, yet a unified description is incomplete.

In this work, we present a discrete geometric toy model designed to explore whether large-scale expansion can emerge from simple growth rules defined at the Planck scale. The model is not intended as a replacement for established frameworks, but rather as a minimal and explicit construction for examining scale hierarchies bridging quantum and cosmological domains.

2. Model Definition

We define a discrete sequence of configurations $S_n$Sn​, where $n \in \mathbb{N}$n∈N represents a discrete iteration step. Each configuration consists of a collection of identical fundamental units, interpreted as Planck-scale volumetric elements.

2.1 Binary Growth Rule

In the idealized case, the system evolves via:

$$N(n) = 2^n$$

where $N(n)$N(n) is the number of fundamental units at step $n$n.

2.2 Geometric Interpretation

Each unit is assigned a characteristic volume $v_P \sim l_P^3$vP​∼lP3​, where $l_P$lP​ is the Planck length. The total volume is:

$$V(n) = N(n) \, v_P$$

Assuming an effective spherical geometry, we define a radius:

$$R(n) \sim V(n)^{1/3} \sim N(n)^{1/3} l_P$$

2.3 Extended Growth Dynamics

To incorporate geometric and interaction constraints, we generalize the growth rule:

$$N(n+1) = 2N(n) – \epsilon(n)$$

where $\epsilon(n) \ge 0$ϵ(n)≥0 represents deviations from ideal doubling due to effects such as packing inefficiencies, boundary limitations, or local exclusion rules.

A natural and minimal parametrization is sub-extensive scaling:

$$\epsilon(n) = \kappa \, N(n)^{\alpha}, \quad 0 < \alpha < 1, \; \kappa > 0$$

This form preserves monotonic growth while introducing scale-dependent deviations from exponential behavior.

3. Effective Cosmological Quantities

3.1 Discrete Scale Factor

We define an effective scale factor:

$$a(n) = \frac{R(n)}{R(0)} \sim N(n)^{1/3}$$

3.2 Time Mapping

We map discrete steps to physical time via:

$$t = n \, \tau$$

where $\tau$τ is a characteristic time scale, taken heuristically to be of order the Planck time.

3.3 Effective Expansion Rate

We define a discrete analogue of the Hubble parameter:

$$H(n) = \frac{1}{a(n)} \frac{\Delta a}{\Delta t}$$

In the ideal case ($\epsilon = 0$ϵ=0):

$$H_0 \sim \frac{1}{\tau} \left(2^{1/3} – 1\right)$$

With the generalized growth rule, the expansion rate acquires scale dependence. To leading order:

$$H(n) \approx H_0 – c \, \frac{\epsilon(n)}{N(n)}$$

for some geometric constant $c \sim \mathcal{O}(1)$c∼O(1).

Substituting the sub-extensive form yields:

$$H(n) = H_0 – c \, N(n)^{\alpha – 1}$$

Since $\alpha < 1$α<1, the correction term decreases with scale, implying an evolving effective expansion rate.

4. A Minimal Test Case

To illustrate the model, consider:

$$\epsilon(n) = \kappa \, \sqrt{N(n)} \quad (\alpha = 1/2)$$

Then:

$$H(n) = H_0 – c \, N(n)^{-1/2}$$

This yields:

• Early steps (small $n$n): correction is significant
• Late steps (large $n$n): correction diminishes

Thus, the model exhibits scale-dependent expansion, with deviations from constant expansion naturally suppressed at large scales.

5. Scaling Behavior and Interpretation

The model exhibits exponential or near-exponential growth in volume and power-law growth in radius. The inclusion of $\epsilon(n)$ϵ(n) introduces a hierarchy of behaviors:

• Ideal exponential expansion ($\epsilon = 0$ϵ=0)
• Constrained expansion ($\epsilon > 0$ϵ>0)
• Scale-dependent deviations

This suggests a mechanism by which effective smooth expansion may emerge from discrete underlying dynamics with local constraints.

6. Relation to Standard Cosmology

The model may be compared heuristically with Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology:

• The scale factor $a(n) \sim N(n)^{1/3}$a(n)∼N(n)1/3 mimics exponential expansion
• The effective Hubble parameter is constant in the ideal limit
• Sub-extensive corrections introduce scale dependence

However, the model does not derive from Einstein’s equations and does not incorporate curvature, matter content, or relativistic dynamics.

7. Limitations

This model is highly simplified and subject to several limitations:

• No relativistic invariance
• No coupling to matter or radiation fields
• No action principle or field equations
• Geometric assumptions (e.g., spherical symmetry) are imposed
• Discrete-to-continuum mapping is heuristic

8. Testable Implications (Exploratory)

The model suggests several exploratory directions:

• Scale-dependent expansion rates arising from discrete constraints
• Deviations from purely exponential growth
• Possible connections to scale-dependent cosmological observations

Further development is required to produce quantitative predictions.

9. Conclusion

We have presented a discrete geometric toy model in which cosmological expansion emerges from iterative growth at the Planck scale with sub-extensive constraints. The introduction of a correction term $\epsilon(n)$ϵ(n) yields scale-dependent expansion in a minimal and explicit framework. While highly idealized, the model provides a concrete starting point for exploring how discrete microphysical rules may give rise to large-scale cosmological behavior.

Future work may extend this framework to incorporate interactions, curvature, and connections to established physical theories, as well as explore implications for emergent constants, dark energy analogues, and entropy-like structures.

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