Loop cosmological dynamics and dualities with Randall-Sundrum braneworlds

Abstract

The discrete quantum geometric effects play an important role in dynamical evolution in the loop quantum cosmology. These effects which are significant at the high energies lead to the quadratic energy density modifications to the Friedmann equation, as in the Randall-Sundrum braneworld scenarios but with a negative sign. We investigate the scalar field dynamics in this scenario and show the existence of a phase of super-inflation independent of the inverse scale factor modifications as found earlier. In this regime the scalar field mimics the dynamics of a phantom field and vice versa. We also find various symmetries between the expanding phase, the contracting phase and the phantom phase in the loop quantum cosmology. We then construct the scaling solutions in the loop quantum cosmology and show their dual relationship with those of the Randall-Sundrum cosmology.

Inflationary universe in loop quantum cosmology

“…It is remarkable that the origin of ρ crit is purely quantum, since in the classical limit → 0 one has ρ crit → ∞. In addition, it should be noted that, interestingly, ρ 2 modifications also appear in string inspired braneworld scenarios and it has been shown that there exist interesting dualities between the two frameworks [16]. Such modifications in braneworlds, however, are usually positive definite so that a bounce is absent, unless the existence of two timelike extra dimensions is assumed [20,21].…”

“…Though the issue of super-inflation in LQC has been discussed widely in the literature (see, e.g., [16,18,23,24]), we still make some analyses here for maintaining this paper selfcontained. Besides, we discuss the effective behavior of a quintom matter in the inflation process in this subsection.…”

“…According to the definition of the parameter x, x = ρ/ρ crit , we have x = (a/a crit ) −3(1+w) , then we obtain 3.15) where N is the number of e-foldings from the bouncing point (corresponding to x = 1) to somewhere in inflation stage (labeled by x). Therefore, for cases of w > −1/3, the total number of e-foldings can be evaluated directly, 16) where N tot represents the total e-folding number, and x w is given by Eq. (3.10).…”

“…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.…”

Qualitative study in loop quantum cosmology

“…The general formula of loop quantum gravity, which takes into account the discrete nature of space-time, expresses the Hamiltonian in terms of the holonomies h j (λ) ≡ e −i λβ 2 σj , where σ j are the Pauli matrices [45,50] …”