Probing the Universe's Topology through a Quantum System?
Evangelos Achilleas Paraskevas, Leandros Perivolaropoulos
TL;DR
This work investigates whether global flat cosmic topology can influence local quantum bound states by analyzing a particle bound by a 3D Dirac delta potential on topologies $E_1$ (3-Torus) and $E_2$ (Half-turn space). Using a renormalized coupling and topology-adapted Fourier mode sums, the authors derive energy-eigenvalue equations and extract large-$L$ asymptotics, showing $|\tilde{E}| \simeq \frac{1}{2 g_R^2}\left(1 + C_\Gamma \frac{2 g_R}{L} e^{-L/g_R}\right)$ with $C_\Gamma=6$ for $E_1$ and $C_\Gamma=4$ for $E_2$. They validate the method in 1D and demonstrate that topological effects deepen binding relative to infinite space, with exponentially suppressed corrections at the present epoch due to the enormous particle horizon. In the early Universe, when the horizon size approaches the system's length scale $L$, these spectral shifts could become measurable, offering a theoretical pathway to connect cosmic topology with quantum phenomena. The work opens avenues to study additional flat topologies and other quantum systems in cosmological settings.
Abstract
The global topology of the Universe could, in principle, affect quantum systems through boundary condition constraints. We investigate this connection by analyzing how compact, flat, cosmologically inspired topologies, specifically the $3-$Torus ($E_1$) and half turn space ($E_2$), influence the energy eigenvalues of a quantum particle in the bound state of a 3D Dirac delta potential. Using rigorous renormalization techniques, we derive the equations satisfied by the energy eigenvalues in each topology and develop a systematic method to compute spectral shifts. Our results reveal that each topology induces characteristic deviations in the energy spectrum. In the large$-L$ limit ($L >> g_R$), to leading order, the energy eigenvalues for both the $E_1$ and $E_2$ spaces can be written in the unified form $E\simeq -\frac{\hbar^2}{2mg_R^2}(1 + C_Γ\,\frac{2g_R}{L}\,e^{-L/g_R})$, where the topology dependent coefficient is $C_Γ= 6$ for the $E_1$ space and $C_Γ= 4$ for the $E_2$ space, $g_R$ is the characteristic length scale of the quantum system, and $L$ is the side physical length of the fundamental cubic region. Using the three dimensional Dirac potential as a toy model, we show that at the current cosmic epoch ($a=1$), these topological effects are exponentially suppressed, rendering direct observation infeasible. However, such effects may become measurable in the early Universe, when the physical size of the particle horizon is comparable to the characteristic scale of the quantum system. While immediate experimental verification remains impractical, our work offers theoretical insight into how global cosmic topology might manifest in quantum bound states and may inform future studies of early Universe quantum phenomena.