  A detection of B-mode polarization of the Cosmic Microwave Background (CMB)
anisotropies would confirm the presence of a primordial gravitational wave
background (GWB). In the inflation paradigm this would be an unprecedented
probe of the energy scale of inflation as it is directly proportional to the
power spectrum of the GWB. However, similar tensor perturbations can be
produced by the matter fields present during inflation, breaking this simple
relationship. It is therefore important to be able to distinguish between
different generation mechanisms of the GWB. In this paper, we analyse the
detectability of a new axion-SU(2) gauge field model using its chiral,
scale-dependent tensor spectrum. We forecast the detectability of the resulting
CMB TB and EB cross-correlations by the LiteBIRD satellite, considering the
effects of residual foregrounds, gravitational lensing, and for the first time
assess the ability of such an experiment to jointly detect primordial TB and EB
spectra and self-calibrate its polarimeter. We find that LiteBIRD will be able
to detect the chiral signal for $r_*>0.03$ with $r_*$ denoting the
tensor-to-scalar ratio at the peak scale, and that the maximum signal-to-noise
for $r_*<0.07$ is $\sim 2$. We go on to consider an advanced stage of a
LISA-like mission, and find that such experiments would complement CMB
observations by providing sensitivity to GWB chirality on scales inaccessible
to the CMB. We conclude that in order to use the CMB to distinguish this model
from a conventional vacuum fluctuation model two-point statistics provide some
power, but to achieve high statistical significance we would require higher
order statistics which take advantage of the model's non-Gaussianity. On the
other hand, in the case of a spectrum peaked at very small scales, inaccessible
to the CMB, a highly significant detection could be made using space-based
laser interferometers.
