A primordial cosmic microwave background B mode is widely considered a ‘smoking gun’ signature of an early period of inflationary expansion. However, competing theories of the origin of structure, including string gases (SG) and bouncing cosmologies, also produce primordial tensor perturbations that give rise to a B mode. These models can be differentiated by the scale dependence of their tensor spectra: inflation predicts a red tilt (nT < 0), SG and loop-quantum cosmology predict a blue tilt (nT > 0), while a non-singular matter bounce gives a zero tilt (nT= 0). We perform a Bayesian analysis to determine how far |nT| must deviate from zero before a tilt can be detected with current and future B-mode experiments. We find that Planck in conjunction with Q/U Imaging Experiment II will decisively detect nT≠ 0 if |nT| > 0.3, too large to distinguish either single-field inflation or SG from the case nT= 0. While a future mission like CMBPol will offer improvement, only an ideal satellite mission will be capable of providing sufficient Bayesian evidence to distinguish between each model considered.