Inflationary tensor fluctuations, as viewed by Ashtekar variables and their generalizations

Abstract

We investigate tensor modes in inflationary scenarios from the point of view of Ashtekar variables and their generalizations labeled by Immirzi parameter $\ensuremath{\gamma}$, which we will assume imaginary. Being careful to properly define the classical perturbed Hamiltonian by taking several subtleties into account, we reproduce, on shell, the usual expression found in cosmological perturbation theory. However the quantum Hamiltonian displays significant differences, namely, in the vacuum energy and fluctuations of the various modes. We can identify combinations of metric and connection variables representing graviton states, noting that before reality conditions are imposed there are gravitons and antigravitons. It turns out that half of these modes have negative energy but after defining the inner product we conclude that they are nonphysical and should be selected out. We are left with the usual graviton modes but with a chiral asymmetry in the vacuum energy and fluctuations. The latter depends on $\ensuremath{\gamma}$ and on the ordering prescription (namely in the Hamiltonian constraint). Such an effect would leave a distinctive imprint in the polarization of the cosmic microwave background, thus finally engaging quantum gravity in meaningful experimental test.