  We investigate constraints on the spectral index of primordial gravitational
waves (GWs), paying particular attention to a blue-tilted spectrum. Such
constraints can be used to test a certain class of models of the early
Universe. We investigate observational bounds from LIGO+Virgo, pulsar timing
and big bang nucleosynthesis, taking into account the suppression of the
amplitude at high frequencies due to reheating after inflation and also
late-time entropy production. Constraints on the spectral index are presented
by changing values of parameters such as reheating temperatures and the amount
of entropy produced at late time. We also consider constraints under the
general modeling approach which can approximately describe various scenarios of
the early Universe. We show that the constraints on the blue spectral tilt
strongly depend on the underlying assumption and, in some cases, a highly
blue-tilted spectrum can still be allowed.
