Probability of inflation in loop quantum cosmology

We discuss how initial conditions for cosmological evolution can be defined in loop quantum cosmology with a massive scalar field and how the presence of the bounce influences the probability of inflation in this theory, compared with General Relativity. The main finding of the paper is the existence of an attractor in the contracting phase of the Universe, which results in a special probability distribution at the bounce, quite independent of the measure of initial conditions in the remote past, and hence a very specific duration of the inflationary stage with the number of $e$-foldings about 140.

Figure 1

GR vs LQC inflationary solutions. Reproduced from Ref. [25].

Figure 2

Asymptotic solution of cosmological equations of LQC with the massive scalar field in the limit $\rho \to 0$ corresponding to Eqs. (38) and (40). The origin is denoted by the white circle. (a) Only the term $t^{-2}$ is kept. The blue circle shows the boundary ${\rho }_0$ where initial conditions are set. (b) Full asymptotic solution of the same equations corresponding to Eq. (40). Red curves denote repulsive separatrices.

Figure 3

Mass is given in Planck units. (a) Dependence of the peak of probability density distribution on scalar field mass. (b) Dependence of the standard deviation in the probability density distribution on scalar field mass.

Figure 4

Initial probability distributions and probability distributions at the bounce for all three cases considered. (a) Probability distributions for initial conditions in the remote past given by: Eq. (41) - orange, Eq. (35) red, narrow distribution - blue. Vertical lines denote the values of the $\delta$ variable used in the calculations. (b) Probability distribution of the rescaled field Values at the bounce.

Figure 5

Solutions of the cosmological equations of LQC in the $\varphi -\dot{\varphi }$ diagram. All solutions start in the distant past in the origin, pass through the oscillatory phase with increasing amplitude of oscillations, and end up at the bounce shown by the thick external circle. The blue internal circle (obtained setting $H=m$) represents the region where most solutions deviate from the repulsive separatrix (located along the horizontal axis, at which the Hubble parameter is nearly constant). This region shrinks with decreasing mass (the value of the mass in this figure is $m=0.1M_{\mathrm{P}\mathrm{l}}$, selected for better clarity). The red curve shows the complete most probable solution originating from the remote past (including both the contraction and expansion phases). This solution has no exponential contraction phase, but it has a successful inflationary phase.