Nonsingular bouncing universes in loop quantum cosmology

Abstract

Non-perturbative quantum geometric effects in Loop Quantum Cosmology predict a ρ 2 modification to the Friedmann equation at high energies. The quadratic term is negative definite and can lead to generic bounces when the matter energy density becomes equal to a critical value of the order of the Planck density. The non-singular bounce is achieved for arbitrary matter without violation of positive energy conditions. By performing a qualitative analysis we explore the nature of the bounce for inflationary and Cyclic model potentials. For the former we show that inflationary trajectories are attractors of the dynamics after the bounce implying that inflation can be harmoniously embedded in LQC. For the latter difficulties associated with singularities in cyclic models can be overcome. We show that non-singular cyclic models can be constructed with a small variation in the original Cyclic model potential by making it slightly positive in the regime where scalar field is negative.

Inflationary universe in loop quantum cosmology

“…However, an effective Hamiltonian description on a continuum spacetime can be constructed by using semiclassical states, which has been shown to very well approximate the quantum dynamics [13,14]. This analysis reveals that on backward evolution of our expanding phase of the universe, the universe bounces at a critical density (near the big bang singularity) into a contracting branch [12,18]. Thus the classical singular problem can be successfully overcome within the context of LQC by a nonsingular bounce.…”

“…On the other hand, matter with equation of state w > −1 has increasing energy density as the universe contracts and therefore a quantum bounce will occur when the quantum critical density is hit. So it is clear that generically a phantom field can not have a quantum bounce in the early universe [18]. In this paper we will only consider the "normal" matter (or canonical scalar field) without null energy condition violation, and we will focuss on the post-big-bounce stage of the universe.…”

“…The condition for the existence of a bounce is that the matter in the universe does not violate the null energy condition [18]. Let us consider the simplest case of matter with constant equation of state w (the definition of w is w = p/ρ), then from Eq.…”

“…Intriguingly, one can obtain an modified Friedmann equation from the effective Hamiltonian constraint, which can be used to investigate the role of nonperturbative quantum correction conveniently. It is remarkable that the quantum geometric effects lead to a ρ 2 modification to the Friedmann equation at the scales when ρ becomes comparable to a critical density ρ crit which is close to the Planck density (ρ crit ≈ 0.82ρ Pl ) [13,14,16,17,18]. The modified term in the Friedmann equation is negative definite, which implies a bounce when the energy density hits the critical value; this feature resolves the classical big-bang singularity problem and is in accordance with the result from the quantum evolution in LQC.…”

Covariant effective action for loop quantum cosmology à la Palatini

“…The resulting effective Hamiltonian (or the modified Friedman dynamics) describes the underlying quantum evolution extremely well at all scales for universes which grow to a macroscopic size and leads to a rich phenomenology (see for eg. [15]). …”