Effective dynamics of loop quantum Kaluza-Klein cosmology

The five-dimensional loop quantum Kaluza-Klein cosmology is constructed based on the symmetric reduction of the connection formulation of the full theory. Through semiclassical analysis, the effective scalar constraint for the cosmological model coupled with a dust field is derived, incorporating the quantum fluctuations of geometry as a subleading order correction. It demonstrates that the quantum model has the correct classical limit. The explicit solutions to the equations of motion show that the big bang and past big rip singularities in the classical model are avoided by a quantum bounce and a quantum collapse respectively in the effective model. In a particular scenario, the dynamical compactification of the extra dimension is realized, while the observable four-dimensional universe transitions through three distinct epochs: (i) a super-inflationary phase generating 55 e-folds, (ii) a decelerated expansion era, and (iii) a late-time accelerated expansion phase driven by quantum fluctuations. These results suggest that both cosmic inflation and dark energy may originate from the interplay between the compact extra dimension and quantum geometric effects.