Do we have unitary and (super)renormalizable quantum gravity below the Planck scale?

We explore how the stability of metric perturbations in higher-derivative theories of gravity depends on the energy scale of the initial seeds of such perturbations and on a typical energy scale of the gravitational vacuum background. It is shown that, at least in the cases of specific cosmological backgrounds, the unphysical massive ghost which is present in the spectrum of such theories is not growing up as a physical excitation and remains in the vacuum state until the initial frequency of the perturbation is close to the Planck order of magnitude. In this situation, the existing versions of renormalizable and super-renormalizable theories can be seen as very satisfactory effective theories of quantum gravity below the Planck scale.

Figure 1

The plots for the flat-space case. $k$ is measured in units of $M_P$, and the growing modes appear for $k$ close to 1. Oscillations here mean that the eigenvalues with both positive and negative real parts have imaginary components. For smaller values of $k$, the amplitude greatly increases, but asymptotically it goes to zero (out of the plot).

Figure 2

Some plots of $h(t)$ for the exponential case, $a(t)=a_0{e}^{H_{0}t}$. The solution with growing modes appears only starting from $k=0.036$.

Figure 3

Graph for $h(t)$ perturbation as a function of time analyzed for radiation, when $a(t)=a_0{t}^{1/2}$. Starting from $k\sim 0.50$, the solutions become “violent”, as one can see on the last plot. However, below this value, there are no growing modes.

Figure 4

Graph for $h(t)$ perturbation as a function of time analyzed for matter, when $a(t)=a_0{t}^{3/2}$.

Figure 5

Once again, growing modes appear only close to the Planck scale.