  We consider a Friedmann-Robertson-Walker cosmological model with some exotic
perfect fluid with spin known as the Weyssenhoff fluid. The possibility that
the dark energy may be described in part by the Weyssenhoff fluid is discussed.
The observational constraint coming from supernovae type Ia observations is
established. This result indicates that, whereas the cosmological constant is
still needed to explain current observations, the model with spin fluid is
admissible. For high redshifts $z > 1$ the differences between the model with
spin fluid and the cold dark matter model with a cosmological constant become
detectable observationally for the flat case with $\Omega_{\text{m},0}=0.3$.
  From the maximum likelihood method we obtain the value of
$\Omega_{\text{s},0} = 0.004 \pm 0.016$. This gives us the limit
$\Omega_{\text{s},0} > -0.012$ at the $1\sigma$ level. While the model with
``brane effects'' is preferred by the supernovae Ia data, the model with spin
fluid is statistically admissible. For comparison, the limit on the spin fluid
coming from cosmic microwave background anisotropies is also obtained. The
uncertainties in the location of a first peak give the interval $-1.4 \times
10^{-10} < \Omega_{\text{s},0} < -10^{-10}$. From big bang nucleosynthesis we
obtain the strongest limit $\Omega_{\text{s},0} \gtrsim -10^{-20}$. The
interconnection between the model considered and brane models is also pointed
out.
