  Assuming that inflation is followed by a phase where the energy density of
the Universe is dominated by a component with a general equation of state, we
evaluate the spectrum of primordial gravitational waves induced in the
post-inflationary Universe. We show that if the energy density of the Universe
is dominated by a component $\phi$ before Big Bang nucleosynthesis, its
equation of state could be constrained by gravitational wave experiments
depending on the ratio of energy densities of $\phi$ and radiation, and also
the temperature at the end of the $\phi$ dominated era. Also, we discuss the
impact of scale dependence of tensor modes on the primordial gravitational wave
spectrum during the $\phi$-domination. These models are motivated by beyond
Standard Model physics and scenarios for non-thermal production of dark matter
in the early Universe. We also constrain the parameter space of the tensor
spectral index and the tensor-to-scalar ratio, using the experimental limits
from gravitational wave experiments.
