  The signal strength of primordial gravitational waves experiencing an epoch
of early scalar domination is reduced with respect to radiation domination. In
this paper, we demonstrate that the specific pattern of this reduction is
sensitive to the coupling between the dominant field and gravity. When this
coupling is zero, the impact of early matter domination on gravitational waves
is solely attributed to the alteration of the Hubble parameter and the scale
factor. In the presence of non-zero couplings, on the other hand, the evolution
of primordial gravitational waves is directly affected as well, resulting in a
distinct step-like feature in the power spectrum of the gravitational wave as a
function of frequency. This feature serves as a smoking gun signature of this
model. In this paper, we provide an analytical expression of the power spectrum
that illustrates the dependence of power spectrum on model parameters and
initial conditions. Furthermore, we provide analytical relations that specify
the frequency interval in which the step occurs. We compare the analytical
estimates with numerical analysis and show they match well.
