Cosmological inflation driven by holonomy corrections of loop quantum cosmology

At the level of heuristic effective dynamics, we investigate the cosmological inflation with holonomy corrections of loop quantum cosmology in the $k=0$ Friedmann-Robertson-Walker model with a single inflaton field subject to a simple potential. In the symmetric bouncing scenario of loop quantum cosmology, the condition for occurrence of the quantum bounce naturally and uniquely fixes the initial conditions at the bouncing epoch. Around the quantum bounce, the Universe undergoes a short superinflationary phase, which drives the inflaton field to its potential hill and thus sets the proper initial conditions for the standard slow-roll inflation. Between the superinflation and the standard inflation, there is a noninflationary phase, which violates the slow-roll condition. The violation of slow roll is expected to give some suppression on the low angular power spectrum of the cosmic microwave background and different orders of holonomy corrections shall yield different suppressions.

Figure 1

Evolutions for the cases of $n=0$ (solid curve) and $n=100$ (dashed curve) with $m_{\varphi }={10}^{-8}{M}_{\mathrm{Pl}}$ and $\gamma =\ln 2/(\sqrt{3}\pi )$. (a) Growth of $a=L^{-1}p$ with $a_0=L^{-1}{p}_0=a(t=0)$. The final $e$-foldings are about $\ln {10}^{32}$. (b) Evolution of $\varphi$. After $\varphi$ reaches ${\varphi }_{\max }$, it starts to roll down slowly. In the end of the slow roll, $\varphi$ oscillates. (c) Evolution of the Hubble rate $H$. We can see a short period of the superinflationary phase ($\dot{H}>0$ or $H$ is increasing) close to the quantum bounce. (d) Evolution of $\dot{a}/a_0$. We can see a noninflationary phase ($\ddot{a}<0$ or $\dot{a}$ is decreasing) between the superinflation and the standard inflation.

Figure 2

Zoom in of Fig. 1 around the quantum bounce (with the prebounce side included). (a) Evolution of $a$. (b) Evolution of $\varphi$. (c) Evolution of $H$. We can see that the superinflationary phase ($\dot{H}>0$) is shorter and more abrupt for larger $n$. (d) Evolution of $\cos (\overline{\mu }c)$, signaling how significant the quantum effects are. $\cos (\overline{\mu }c)\to \pm 1$ when the Universe is classical.