  We discuss a method to constrain the fraction density $f$ of the relativistic
gas in the radiation dominant stage, by their impacts on a relic gravitational
waves and the cosmic microwave background (CMB) $B$ polarization power
spectrum. We find that the uncertainty of $f$ strongly depends on the noise
power spectra of the CMB experiments and the amplitude of the gravitational
waves. Taking into account of the CMBPol instrumental noises, an uncertainty
$\Delta f=0.046$ is obtained for the model with tensor-to scalar ratio $r=0.1$.
For an ideal experiment with only the reduced cosmic lensing as the
contamination of $B$-polarization, $\Delta f=0.008$ is obtained for the model
with $r=0.1$. So the precise observation of the CMB $B$-polarization provides a
great opportunity to study the relativistic components in the early Universe.
