Measure problem in slow roll inflation and loop quantum cosmology

We consider the measure problem in standard slow-roll inflationary models from the perspective of loop quantum cosmology (LQC). Following recent results by Ashtekar and Sloan, we study the probability of having enough $e$-foldings and focus on its dependence on the quantum gravity scale, including the transition of the theory to the limit where general relativity (GR) is recovered. Contrary to the standard expectation, the probability of having enough inflation, that is close to 1 in LQC, grows and tends to 1 as one approaches the GR limit. We study the origin of the tension between these results with those by Gibbons and Turok, and offer an explanation that brings these apparent contradictory results into a coherent picture. As we show, the conflicting results stem from different choices of initial conditions for the computation of probability. The singularity-free scenario of loop quantum cosmology offers a natural choice of initial conditions, and suggests that enough inflation is generic.

Figure 1

Three sets of trajectories are plotted for different values of the critical density. Note that near the origin, all trajectories approach the attractor.

Figure 2

Here we are plotting trajectories for three different values of the critical density ${\rho }_{\mathrm{crit}}$. In each case, we have a boundary of the trajectories in the $(\dot{\varphi },\varphi )$ plane, corresponding to the bounce, that are depicted as ellipsoids. The smallest ellipsoid can be taken as the LQC one, and the larger ones are closer to the GR limit. The “critical” curves that separate the region of enough $e$-foldings, as determined by the LQC scale, are then plotted for the three different values of the critical density ${\rho }_{\mathrm{crit}}$. One can see that, as the critical density increases, the intersection with the ellipsoid of critical density comes closer to the $\dot{\varphi }$ axis.

Figure 3

For a fixed value of ${\rho }_{\mathrm{crit}}$, we plot the exterior, critical density surface and a surface of constant density ${\rho }_{\mathrm{GT}}\ll {\rho }_{\mathrm{crit}}$ (not drawn to scale, of course) on the $(\dot{\varphi },\varphi )$ plane. Trajectories with a uniform distribution at the LQC bounce ellipsoid are plotted. Note that trajectories for which there is enough inflation get funneled into a small region in the smaller ${\rho }_{\mathrm{GT}}$ ellipse. Near this surface, the GR and LQC dynamics almost coincide.