May 2026
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A Discrete Geometric Toy Model
with Sub-Extensive Constraints for
Scale-Dependent Cosmological Expansion



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 construction for
examining scale hierarchies bridging quantum and cosmological domains.

2 Model Definition

We define a discrete sequence of configurations Sn, where n ∈ N represents a discrete iteration step. Each configuration consists of identical fundamental units interpreted as Planck-scale volumetric elements.

2.1 Binary Growth Rule

In the idealized case:

_______________N (n) = 2n____________________(1)

where N (n) = 2n number of units.


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2.2 Geometric Interpretation

Each unit has volume vP ∼ l3P . The total volume is:

_______________(n) = N (n)vP __________________(2)

We define an effective radius:

______________R(n) ∼ V (n)1/3 ∼ N (n)1/3lP___________(3)

2.3 Extended Growth Dynamics

We generalize the growth rule:

________________N (n + 1) = 2N (n) − ϵ(n)___________(4)

where ϵ(n) ≥ 0 represents geometric or interaction constraints.
A sub-extensive parametrization is:

________________ϵ(n) = κN (n)α, 0 < α < 1________–__ (5)

This preserves monotonic growth while introducing scale-dependent deviations.

3 Effective Cosmological Quantities

3.1 Scale Factor

___________________a(n) ∼ N (n)1/3 __________ ___(6)

3.2 Time Mapping

____________________t = nτ______________ __ __(7)

3.3 Expansion Rate

In the ideal case:

With constraints:

Substituting:
_______________H(n) = H0 − cN (n)α−1 _____________ ___(11)

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4 Minimal Test Case

Let:

Then:

_______________H(n) = H0 − cN (n)−1/2_______________(13)

This yields scale-dependent expansion.

5 Interpretation

The model exhibits exponential or near-exponential growth modified by constraints, suggesting a mechanism for emergent large-scale expansion from discrete dynamics.

6 Relation to Cosmology

The model resembles FLRW expansion qualitatively but does not derive from Einstein equations.

7 Limitations

  • No relativistic invariance
  • No matter coupling
  • No action principle
  • Heuristic assumptions

8 Conclusion

We presented a discrete toy model in which sub-extensive constraints produce scale-dependent expansion. This provides a minimal framework for exploring connections between discrete micro-physics and cosmological behavior.

References

  • Ambjørn, J., Jurkiewicz, J., & Loll, R. (2005). Reconstructing the universe.
  • Rovelli, C. (2004). Quantum Gravity.
  • Van Raamsdonk, M. (2010). Building up spacetime with quantum entanglement.
  • Weinberg, S. (2008). Cosmology.
  • Barab’asi, A.-L. (2016). Network Science.

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