Observables for a Friedmann-Robertson-Walker model with a cosmological constant in the framework of loop cosmology

We consider a flat cosmological model with a free massless scalar field and the cosmological constant $\Lambda$ in the framework of loop quantum cosmology. The scalar field plays the role of an intrinsic time. We apply the reduced phase space approach. The dynamics of the model is solved analytically. We identify elementary observables and their algebra. The compound physical observables like the volume and the energy density of matter field are analyzed. Both compound observables are bounded and oscillate in the $\Lambda <0$ case. The energy density is bounded and oscillates in the $\Lambda >0$ case. However, the volume is unbounded from above, but periodic. The difference between standard and nonstandard loop quantum cosmology is described.

Figure 1

Plot of $\rho (\varphi )$ for $\delta =-1$ and ${\mathcal{O}}_2=0$. Here, the period of oscillations is equal to $\Delta \varphi \simeq 0.427$.

Figure 2

Plot of $v(\varphi )$ for $\delta =-1$ and ${\mathcal{O}}_2=0$. Here, the period of oscillations is equal to $\Delta \varphi \simeq 0.427$.

Figure 3

Plot of $\rho (\varphi )$ for $\delta =0.5$ and ${\mathcal{O}}_2=0$. Here, $\Delta \varphi \simeq 0.604$.

Figure 4

Plot of $v(\varphi )$ for $\delta =0.5$ and ${\mathcal{O}}_2=0$. Here, $\Delta \varphi \simeq 0.604$.