Inflationary universe in loop quantum cosmology

Abstract

Loop quantum cosmology provides a nice solution of avoiding the big bang singularity through a big bounce mechanism in the high energy region. In loop quantum cosmology an inflationary universe is emergent after the big bounce, no matter what matter component is filled in the universe. A super-inflation phase without phantom matter will appear in a certain way in the initial stage after the bounce; then the universe will undergo a normal inflation stage. We discuss the condition of inflation in detail in this framework. Also, for slow-roll inflation, we expect the imprint from the effects of the loop quantum cosmology should be left in the primordial perturbation power spectrum. However, we show that this imprint is too weak to be observed.

A quantum gravitational inflationary scenario in Bianchi-I spacetime

“…Singularity resolution is confirmed for a large set of initial conditions, in confirmation with the generic result on resolution of singularities in Bianchi-I spacetime in LQC [56]. In contrast to the isotropic inflationary models in LQC where there is a single bounce of the isotropic scale factor [57][58][59][60][61][62][63][64] always occurring at the maximum allowed value of the energy density ρ max = 0.41 ρ Pl , the resolution of singularity in Bianchi-I spacetime occurs via non-singular bounces of the three directional scale factors (a 1 , a 2 , a 3 ) occurring at a range of values of energy density and anisotropic shear scalar σ 2 , bounded by ρ max (same as in the isotropic model) and σ 2 max = 11.57/l 2 Pl . In the isotropic LQC, ρ = ρ max at the bounce constrains the value of the inflaton velocity at the bounce for a given initial value of the inflaton field.…”

Effective equations for isotropic quantum cosmology including matter

“…(Its value is C = (27)- (32).) The linear function has one immediate implication: It is exactly this combination which appears in the reality condition (26) implied byĴĴ † =p 2 in the loop quantization.…”

“…For the loop quantized model it is more complicated to introduce adiabaticity relative to free solutions because free quantum variables (27)- (32) are not simply proportional to powers of the expectation values as in the Wheeler-DeWitt case. The most direct possibility is to introduce the free solutions as explicitly time dependent functions G a,n V =0 (φ).…”

“…This is important to realize since the replacement has often occurred in the recent literature, see e.g. [30,6,31,32,33]. It is used either in the form of sin c in the Friedmann equation, or by using equations of motion to solve sin c in terms ofȧ, which gives rise to a correction term of energy density squared to the classical Friedmann equation [30]: starting from a Hamiltonian constraint of the form C = −µ −2 sin 2 (µc) |p|+ |p| 3/2 ρ matter (where µ may be a p-dependent function) one derives the Hamiltonian equation of motionṗ = 2µ −1 sin(µc) cos(µc) |p|.…”

Loop Quantum Cosmology: holonomy corrections to inflationary models

“…Is it possible to detect somehow the Big Bounce after the inflation? Some of these questions were discussed earlier, for instance in [8], for the flat FRW universe on the basis of the effective semi-classical theory developed in [2]. Although the results like the super-inflation phase without phantom matter are interesting, the potentially observable correction to the primordial perturbation power spectrum was found to be to weak.…”