Loop quantum cosmology from an alternative Hamiltonian

In this paper, a new Hamiltonian constraint operator for loop quantum cosmology is constructed by using the Chern-Simons action. The quantum dynamics of the $k=0$ cosmological model with respect to a massless scalar field as an emergent time is determined by a difference equation. The semiclassical analysis shows that the new quantum dynamics has the correct classical limit, and the classical big-bang singularity is again replaced by the quantum bounce. Interestingly, the inverse time evolution of the flat Friedmann-Robertson-Walker cosmological model determined by the effective Hamiltonian will be bounced to an asymptotic de Sitter universe.

Figure 1

Plots of the constraint equation ${\mathcal{H}}_{\mathrm{tot}}^{\mathrm{eff}}(b,v,p_T)=0$ for different values of $p_T$ (the similar picture in quality holds for other values of $p_T$), where $G=\hslash =1$, $\gamma =0.2375$. It shows that $v=0$ cannot be a solution.

Figure 2

Plots of the relative evolution of $v$ with respect to the scalar field $T$ given by the effective Hamiltonian constraint for different values of $p_T$ (the similar picture in quality holds for other values of $p_T$), where $G=\hslash =1$, $\gamma =0.2375$.