Fermions in a loop quantum cosmological spacetime

TL;DR

We investigate Dirac fermions propagating on a loop-quantized, closed FLRW background within LQC, formulating a mode-by-mode Hamiltonian for fermionic perturbations expanded in spinor harmonics on $\mathbb{S}^3$. In the test-field limit, each fermionic mode evolves on a dressed quantum background, with massive modes perceiving a time- and space-dressed metric, while massless modes sample a conformal class of backgrounds. Extending to the Born–Oppenheimer regime, we show a mode-dependent rainbow metric arises from fermionic backreaction, where each mode's energy shifts the background constraint and delays or advances the quantum bounce; at large volumes a massive-mode backreaction mimics an emergent cosmological constant, whereas massless modes do not. These results reveal qualitative differences between fermionic and bosonic backreaction in quantum cosmology, and predict rainbow-like, mode-specific corrections to early-universe dynamics with potential observational signatures. Future work includes numerical simulations of the full backreaction dynamics, inclusion of gauge fields, and exploration of alternative quantization schemes for spinor fields.

Abstract

We present a detailed Hamiltonian treatment of an inhomogeneous fermionic perturbation propagating on a closed FLRW spacetime quantized via LQC. Expanding the fermion in spinor harmonics on spatial 3-sphere and truncating at quadratic order, we derive a decoupled, mode-by-mode Hamiltonian, where each mode behaves as a time-dependent Fermi oscillator. This framework naturally facilitates a Schrodinger-picture evolution for fermionic perturbations on a quantum-corrected background. Under the test-field approximation, each massive mode sees its own dressed metric, akin to bosonic perturbations, but with distinctive Planck-scale modifications in both temporal and spatial components. Massless modes, by contrast, experience an equivalent class of conformal backgrounds: quantum corrections drop out of the spatial sector, while the temporal component alone is dressed by quantum gravity corrections. Extending beyond the test-field regime via a Born-Oppenheimer approximation, we incorporate fermionic backreaction self-consistently. Each mode's energy (depending on the either vacuum or pair states they occupied) sources a mode-dependent shift of the background Hamiltonian, resulting in a rainbow metric. In the deep Planck regime, vacuum occupation yields a positive fermionic perturbation term in the Hamiltonian, raising the effective minisuperspace potential and delaying the bounce to higher density in the contracting branch. Excited occupation flips the perturbation sign, lowering the barrier and advancing the bounce to lower density. At large volumes, massive fermion backreaction settles into a constant energy density -- an emergent cosmological constant -- capable of driving late-time acceleration, whereas massless modes remain dynamically inert postbounce. These effects break the exact time-reversal symmetry of the quantum bounce and generate asymmetric pre- and postbounce trajectories.