Fermions in a loop quantum cosmological spacetime

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.