  In this paper we generalise previous work on tensor perturbations in a de
Sitter background in terms of Ashtekar variables to cover all complex values of
the Immirzi parameter gamma (previous work was restricted to imaginary gamma).
Particular attention is paid to the case of real gamma. 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 gamma has a real part or not; however
changes occur for the vacuum energy and fluctuations. Specifically, we observe
a gamma dependent chiral asymmetry in the vacuum fluctuations only if gamma has
an imaginary part. Ordering prescriptions also change this asymmetry. We thus
present a measurable result for CMB polarisation experiments that could shed
light on the workings of quantum gravity.
