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We study the effect of primordial scalar curvature perturbations on thepropagation of gravitational waves over cosmic distances. We point out thatsuch curvature perturbations deform the isotropic spectrum of any stochasticbackground of gravitational waves of primordial origin through the (integrated)Sachs-Wolfe effect. Computing the changes in the amplitude and frequency of thepropagating gravitational wave induced at linear order by scalar curvatureperturbations, we show that the resulting deformation of each frequency bin ofthe gravitational wave spectrum is described by a linearly biased Gaussian withthe variance $\sigma^2 \simeq \int d\ln k \Delta_{\mathcal R}^2$, where$\Delta_{\mathcal R}^2(k)$ denotes the amplitude of the primordial curvatureperturbations. The linear bias encodes the correlations between the changesinduced in the frequency and amplitude of the gravitational waves. Taking intoaccount the latest bounds on $\Delta_{\mathcal R}^2$ from primordial black holeand gravitational wave searches, we demonstrate that the resulting ${\mathcalO}(\sigma)$ deformation can be significant for extremely peaked gravitationalwave spectra. We further provide an order of magnitude estimate for broadspectra, for which the net distortion is ${\mathcal O}(\sigma^2)$.

Deformation of the gravitational wave spectrum by density perturbations