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The signal strength of primordial gravitational waves experiencing an epochof early scalar domination is reduced with respect to radiation domination. Inthis paper, we demonstrate that the specific pattern of this reduction issensitive to the coupling between the dominant field and gravity. When thiscoupling is zero, the impact of early matter domination on gravitational wavesis solely attributed to the alteration of the Hubble parameter and the scalefactor. In the presence of non-zero couplings, on the other hand, the evolutionof primordial gravitational waves is directly affected as well, resulting in adistinct step-like feature in the power spectrum of the gravitational wave as afunction of frequency. This feature serves as a smoking gun signature of thismodel. In this paper, we provide an analytical expression of the power spectrumthat illustrates the dependence of power spectrum on model parameters andinitial conditions. Furthermore, we provide analytical relations that specifythe frequency interval in which the step occurs. We compare the analyticalestimates with numerical analysis and show they match well.

Signature of Non-Minimal Scalar-Gravity Coupling with an Early Matter Domination on the Power Spectrum of Gravitational Waves