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