Semiclassical momentum representation in quantum cosmology

It is well known that the standard WKB approximation fails to provide semiclassical solutions in the vicinity of turning points. However, turning points arise in many cosmological scenarios. In a previous work, we obtained a new class of semiclassical solutions of the Wheeler-DeWitt equation using the conjugate momentum to the geometric variable. We present here a detailed study of their main properties. We carefully compare them to usual WKB solutions and turning point resolutions using Airy functions. We show that the momentum representation possesses many advantages that are absent in other approaches. In particular, this framework has a key application in tackling the problem of time. It allows us to use curvature as a time variable, and control the corresponding domain of validity, i.e., under which conditions it provides a good clock. We consider several applications, and in particular show how this allows us to obtain semiclassical solutions of the Wheeler-DeWitt equation parametrized by York time.

Figure 1

Plot of the error function ${\mathrm{Er}}_{\mathrm{p}}/{{\mathrm{Er}}_{\mathrm{p}}}_{\max }$ as a function of $P=\frac{{\mathrm{\Lambda }}^{1/2}{p}_{\alpha }}{V_{\mathrm{U}}{\rho }_q}$.

Figure 2

Plot of the error functions ${\mathrm{Er}}_{\mathrm{p}}/{\mathrm{Er}}_{\max }$ and ${\mathrm{Er}}_{\mathrm{h}}/{\mathrm{Er}}_{\max }$ as a function of $P=GT/{\mathrm{\Lambda }}^{1/2}$. As in Fig. 1, the functions are symmetric about $P=0$, so we only plotted $P>0$.

Figure 3

Contours in the complex plane of $p_h$. (Left panel) The real line is deformed into the represented contour, in order to evaluate (53) with the saddle point method. The new contour goes across the branch cut to pick up the saddle point $p_{*}$. (Right panel) Contour going from 0 to 0 on the other Riemann sheet, and used to obtain the tunnel amplitude in Eq. (56).