On the modification of the cosmic microwave background anisotropy spectrum from canonical quantum gravity

We evaluate the modifications to the cosmic microwave background anisotropy spectrum that result from a semiclassical expansion of the Wheeler-DeWitt equation. Recently, such an investigation in the case of a real scalar field coupled to gravity has led to the prediction that the power at large scales is suppressed. We make here a more general analysis and show that there is an ambiguity in the choice of solution to the equations describing the quantum gravitational effects. Whereas one of the two solutions describes a suppression of power, the other one describes an enhancement. We investigate possible criteria for an appropriate choice of solution. The absolute value of the correction term is in both cases of the same order and currently not observable. We also obtain detailed formulas for arbitrary values of a complex parameter occurring in the general solution of the nonlinear equations of the model. We finally discuss the modification of the spectral index connected with the power spectrum and comment on the possibility of a quantum-gravity induced unitarity violation.

Figure 1

The various curves correspond to values of $\zeta =0.5$, 2, 5, $2\pi$, 10 and are ordered so that on the left part of the figure (say $\beta$ close to 0) $\zeta$ increases. In other words, the lower curve corresponds to $\zeta =0.5$, whereas the upper curve to $\zeta =10$.

Figure 2

The quantity $\ln (C_k)$ is plotted as a function of $H^2/(k^3{m}_{\mathrm{P}}^2)$ [see Eq. (5.8)].

Figure 3

Plot of the imaginary part of $\mathrm{Ei}(1,2i\xi +2)\equiv E_1(2\mathrm{i}\xi +2)$.

Figure 4

Plots of $f_1$, $f_2$, $f_3$, $f_4$ (i.e., fixed values of $\rho =1$, 2, 3, 4) as functions of $\theta$. As $\rho$ increases, the curves show two enhanced peaks at around $\theta =0$ and $\theta =2\pi$.

Figure 5

Plots of $g_1$, $g_2$, $g_3$, $g_4$ (i.e., fixed values of $\rho =1$, 2, 3, 4) as functions of $\theta$. As $\rho$ increases the curves show two enhanced peaks at around $\theta =0$ and $\theta =2\pi$.

Figure 6

Plot of the real part of the solution of (5.22) when the amplitude $\rho$ of the complex variable $z=1+\frac{\mathrm{i}}{\xi }$ equals 1, 2, 3, 4.

Figure 7

Plot of the imaginary part of the solution of (5.22) when the amplitude $\rho$ of the complex variable $z=1+\frac{\mathrm{i}}{\xi }$ equals 1, 2, 3, 4.