Dynamical analysis of loop quantum $R^2$ cosmology

The effective dynamics of loop quantum $f(R)$ cosmology in the Jordan frame is considered by using the dynamical system method and numerical method. To make the analysis in detail, we focus on the $R^2$ model since it is simple and favored from observations. In classical theory, $(\varphi =1,\dot{\varphi }=0)$ is the unique fixed point in both contracting and expanding states, and all solutions are either starting from the fixed point or evolving to the fixed point, while in loop theory, there exists a new fixed point (a saddle point) at $(\varphi \simeq 2/3,\dot{\varphi }=0)$ in the contracting state. We find that two of the critical solutions which start from the saddle point cause the solutions starting from the fixed point $(\varphi =1,\dot{\varphi }=0)$ to bounce at small values of the scalar field in $0<\varphi <1$. All other solutions, including the large field inflation solutions, have the history with $\varphi <0$, which we think of as a problem of the effective theory of loop quantum $f(R)$ theory. Another difference from loop quantum cosmology with the Einstein-Hilbert action is that there exist many solutions that do not have a bouncing behavior.

Figure 1

Phase portrait with $H=H_{+}$ for the Starobinsky model, where $x\equiv 1-\frac{1}{\varphi },y=\frac{1}{\mu }\frac{\dot{\varphi }}{{\varphi }^{3/2}}$.

Figure 2

($\sigma \equiv \frac{\mu }{\sqrt{{\rho }_c}}=0.3$) The global behavior of critical solutions $\pm g_{\pm }$ from or to the saddle point $(\varphi \simeq \frac{2}{3},{\varphi }^{\prime }=0)$. The two solutions $\pm g_{+}$ restrict the solutions starting from the source (1,0) so they can only bounce at $0<\varphi <1$. We also show the initial points of the three classes of solutions (a1), (a2), and (a3), whose examples are shown in Fig. 3.

Figure 3

($\sigma \equiv \frac{\mu }{\sqrt{{\rho }_c}}=0.3$) Three examples of solutions (a1), (a2), and (a3) whose initial values have $\cos b<0$. The solutions in case (a1) bounce at large field $\varphi >0.344$, while the solutions in the cases of (a2) and (a3) all bounce at small field $0<\varphi <0.344$.

Figure 4

($\sigma \equiv \frac{\mu }{\sqrt{{\rho }_c}}=0.3$) Two critical solutions: one starts from ${\varphi }^{\prime }={\varphi }_{\max }^{\prime }$ and bounces at $\varphi =2.808$; the other one starts at ${\varphi }^{\prime }=3.745$ and bounces at $\varphi ={\varphi }_{\max }$. We also show the initial points (at $\varphi =0$) of the two classes of solutions (b1) and (b2), whose examples are shown in Fig. 5.

Figure 5

($\sigma \equiv \frac{\mu }{\sqrt{{\rho }_c}}=0.3$) Two examples of the solutions of classes (b1) and (b2). The solution of (b1) approaches to $b=\frac{\pi }{2}$ twice (the first one is not a bouncing point and the second one is) while the solution of (b2) never bounces.

Figure 6

($\sigma \equiv \frac{\mu }{\sqrt{{\rho }_c}}={10}^{-6}$) Critical solutions in $(u,v)$-space: $\pm g_{\pm }$, which give the classes of solutions (a1), (a2), and (a3). $u$, $v$ are defined in Eq. (34).

Figure 7

($\sigma \equiv \frac{\mu }{\sqrt{{\rho }_c}}={10}^{-6}$) Critical solutions which give the classes of solutions (b1) and (b2).

Figure 8

Some general solutions in cases (a1), (b1), and (b2) which can give large field inflation solutions. The slow-roll solution is also shown in the figure. We can see that when ${\varphi }^{\prime }$ is small and negative and $0<b<\pi /2$, the slow-roll solution works well, which justifies the slow-roll approximations.