Loop quantum Brans-Dicke cosmology

The spatially flat and isotropic cosmological model of Brans-Dicke theory with coupling parameter $\omega \ne -\frac{3}{2}$ is quantized by the approach of loop quantum cosmology. An interesting feature of this model is that although the Brans-Dicke scalar field is nonminimally coupled with curvature, it can still play the role of an emergent time variable. In the quantum theory, the classical differential equation which represents cosmological evolution is replaced by a quantum difference equation. The effective Hamiltonian and modified dynamical equations of loop quantum Brans-Dicke cosmology are also obtained, which lay a foundation for the phenomenological investigation to possible quantum gravity effects in cosmology. The effective equations indicate that the classical big bang singularity is again replaced by a quantum bounce in loop quantum Brans-Dicke cosmology.

Figure 1

Left and right panels present the evolution of the Hubble parameter in the Planck units as a function of the Brans-Dicke field (left panel, vacuum solution) or the massless scalar field (right panel, massless scalar field domination) for realistic values of $\beta$. Initial conditions are chosen to be ${\varphi }_{\mathrm{cr}}=1$ (left panel) and $\frac{{\dot{\varphi }}_{\mathrm{cr}}}{{\dot{\phi }}_{\mathrm{cr}}}=-\frac{{\phi }_{\mathrm{cr}}}{\beta }$, ${\varphi }_{\mathrm{cr}}-\frac{{\phi }_{\mathrm{cr}}^2}{\beta }-\frac{{\dot{\varphi }}_{\mathrm{cr}}}{{\dot{\phi }}_{\mathrm{cr}}}{\phi }_{\mathrm{cr}}=M_{\mathrm{pl}}$ (right panel).