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 ${\mathcal O}(\sigma^2)$.

中文翻译：

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密度扰动引起的引力波谱变形

我们研究了原始标量曲率扰动对引力波在宇宙距离上的传播的影响。我们指出，这种曲率扰动通过（综合）萨克斯-沃尔夫效应使原始引力波的任何随机背景的各向同性谱变形。计算由标量曲率扰动以线性顺序引起的传播引力波的幅度和频率的变化，我们表明引力波谱的每个频率仓的最终变形由方差为 $\sigma 的线性偏置高斯描述^2 \simeq \int d\ln k \Delta_{\mathcal R}^2$，其中 $\Delta_{\mathcal R}^2(k)$ 表示原始曲率扰动的幅度。线性偏差对引力波的频率和幅度引起的变化之间的相关性进行编码。考虑到来自原始黑洞和引力波搜索的 $\Delta_{\mathcal R}^2$ 的最新边界，我们证明了由此产生的 ${\mathcal O}(\sigma)$引力波谱。我们进一步提供了宽光谱的数量级估计，其净失真为 ${\mathcal O}(\sigma^2)$。