Recent B-mode polarization observation seems to imply the tensor tilt ${n}_{T}\ensuremath{\gtrsim}1$ at large angular scale if the primordial signal is dominated. We show that for a primordial universe, which is in a slowly expanding genesis phase before the slow-roll inflation, the primordial tensor spectrum will get a large-scale cutoff, i.e., ${n}_{T}\ensuremath{\gtrsim}1$ at large scales while ${n}_{T}\ensuremath{\simeq}0$ at small scale. We find that this inflationary scenario not only may be consistent with the observation, but also predicts a large-scale anomaly in BB power spectrum, i.e., due to the large suppression of tensor perturbation amplitude we will hardly see the reionization bump at low-$l$, which may be falsified by the Planck polarization data.