  Recently, BICEP2 measurements of the cosmic microwave background (CMB)
$B$-mode polarization has indicated the presence of primordial gravitational
waves at degree angular scales, inferring the tensor-to-scalar ratio of $r=0.2$
and a running scalar spectral index. In this {\em Letter}, we show that the
existence of the fluctuations of cosmological birefringence can give rise to
CMB $B$-mode polarization that fits BICEP2 data with $r<0.11$ and no running of
the scalar spectral index. Thus, it might be too hasty to conclude that many
inflation models with small $r$ are ruled out based on BICEP2 result.
