  Recent B-mode polarization observation seems to imply the tensor tilt
$n_T\gtrsim 1$ 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. $n_T\gtrsim 1$ at large scales while $n_T\simeq 0$ 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.
