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

TL;DR

This paper introduces a quasicrystalline model of the universe's expansion, linking quantum phenomena with cosmological observations, and proposes phonons as potential dark matter candidates, challenging traditional inflationary theories.

Contribution

It presents a novel quasicrystalline framework for cosmic expansion that integrates quantum mechanics and cosmology, offering new insights into dark matter and the inflationary period.

Findings

Derives a Friedmann-like equation from quantum boundary conditions.

Suggests phonons as dark matter candidates influencing cosmic dynamics.

Proposes a universe model that may eliminate the need for inflation.

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.