Chirality of tensor perturbations for complex values of the Immirzi parameter

In this communication, we generalize previous work on tensor perturbations in a de Sitter background in terms of Ashtekar variables to cover all complex values of the Immirzi parameter γ (previous work was restricted to imaginary γ). Particular attention is paid to the case of real γ. Following the same approach as in the imaginary case, we can obtain physical graviton states by invoking reality and torsion-free conditions. The Hamiltonian in terms of graviton states has the same form whether γ has a real part or not; however, changes occur for the vacuum energy and fluctuations. Specifically, we observe a γ-dependent chiral asymmetry in the vacuum fluctuations only if γ has an imaginary part. Ordering prescriptions also change this asymmetry. We thus present a measurable result for CMB polarization experiments that could shed light on the workings of quantum gravity.