Scientists have derived explicit formulas to describe the propagation of spherical Dirac waves in an expanding universe. This work addresses how solutions to the Dirac equation, which describes spin-1/2 particles like electrons, behave in a dynamic spacetime modeled by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric with de Sitter-like expansion.

The study allows for modeling the evolution of specific initial wave functions, such as those of a hydrogen-like atom or a spherical wave in Minkowski space, as the universe expands. This is crucial for understanding how the quantum properties of matter are affected by cosmological expansion and how general relativity influences the quantum mechanics of fundamental particles.

The obtained formulas provide a theoretical tool for investigating phenomena at the interface between quantum mechanics and cosmology. Although the study is theoretical in nature, its implications could be relevant for understanding the evolution of quantum systems in the early universe or in cosmological environments where expansion is a dominant factor.