  Motivated by the string gas cosmological model, which predicts a blue tilt of
the primordial gravitational wave spectrum, we examine the constraints imposed
by current and planned observations on a blue tilted tensor spectrum. Starting
from an expression for the primordial gravitational wave spectrum normalized
using cosmic microwave background observations, pulsar timing, direct detection
and nucleosynthesis bounds are examined. If we assume a tensor to scalar ratio
on scales of the CMB which equals the current observational upper bound, we
obtain from these current observations constraints on the tensor spectral index
of $n_{T} \lesssim 0.79$, $n_{T} \lesssim 0.53$, and $n_{T} \lesssim 0.15$
respectively.
