Tiling Spaces and the Expanding Universe: Bridging Quantum Mechanics and Cosmology

Abstract: We propose a heuristic model of the universe as a growing quasicrystal projected from a higher-dimensional lattice. This quasicrystalline framework offers a novel perspective on cosmic expansion, where the intrinsic growth dynamics naturally give rise to the observed large-scale expansion of the universe. Motivated by this model, we explore the Schr\"odinger equation for a particle in a box with time-dependent boundaries, representing the expanding underlying space. By introducing a constraint that links microscale quantum phenomena with macroscale cosmological quantities, we derive an equation resembling the Friedmann equation, providing potential insights into the Hubble tension. Our model incorporates phonons and phasons-quasiparticles inherent in quasicrystalline structures-that play critical roles in cosmic-scale dynamics and the universe's expansion. This framework suggests that the necessity for an inflationary period may be obviated. Furthermore, phonons arising from the quasicrystalline structure may serve as dark matter candidates, influencing the dynamics of ordinary matter while remaining largely undetectable through electromagnetic interactions. Drawing parallels with crystalline matter at atomic scales, which is fundamentally quantum in nature, we explore how the notion of tiling space can support continuous symmetry atop a discrete structure. This provides a novel framework for understanding the universe's expansion and underlying structure. Consequently, our approach suggests that further development could enhance our understanding of cosmic expansion and the universe's structure, bridging concepts from quantum mechanics, condensed matter physics, and cosmology.