In this paper, we systematically study gravitational waves (GWs) first produced by remote compact astrophysical sources and then propagating in our inhomogeneous Universe through cosmic distances, before arriving at detectors. To describe such GWs properly, we introduce three scales, $\lambda$, $L_c$, and $L$, denoting, respectively, the typical wavelength of GWs, the scale of the cosmological perturbations, and the size of the observable Universe. For GWs to be detected by the current and foreseeable detectors, the condition $\lambda \ll L_c\ll L$ holds. Then, such GWs can be approximated as high-frequency GWs and be well separated from the background ${\gamma }_{\mu \nu }$ by averaging the spacetime curvatures over a scale $\ell$, where $\lambda \ll \ell \ll L_c$, and $g_{\mu \nu }={\gamma }_{\mu \nu }+\epsilon h_{\mu \nu }$ with $h_{\mu \nu }$ denoting the GWs. In order for the backreaction of the GWs to the background spacetimes to be negligible, we must assume that $|h_{\mu \nu }|\ll 1$, in addition to the condition $\epsilon \ll 1$, which are also the conditions for the linearized Einstein field equations for $h_{\mu \nu }$ to be valid. Such studies can be significantly simplified by properly choosing gauges, such as the spatial, traceless, and Lorenz gauges. We show that these three different gauge conditions can be imposed simultaneously, even when the background is not a vacuum, as long as the high-frequency GW approximation is valid. However, to develop the formulas that can be applicable to as many cases as possible, in this paper we first write down explicitly the linearized Einstein field equations by imposing only the spatial gauge. Then, applying these formulas together with the geometrical optics approximation to such GWs, we find that they still move along null geodesics and its polarization bivector is parallel transported, even when both the cosmological scalar and tensor perturbations are present. In addition, we also calculate the gravitational integrated Sachs-Wolfe effects due to these two kinds of perturbations, whereby the dependences of the amplitude, phase, and luminosity distance of the GWs on these perturbations are read out explicitly.

中文翻译：

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引力波宇宙学：高频近似

在本文中，我们系统地研究了引力波 (GW)，首先由远程致密天体物理源产生，然后在到达探测器之前通过宇宙距离在我们的非均匀宇宙中传播。为了正确描述这些 GW，我们引入了三个尺度，$\lambda$, ${升}_C$， 和 $升$，分别表示 GW 的典型波长、宇宙学扰动的尺度和可观测宇宙的大小。对于当前和可预见的检测器检测到的 GW，条件$\lambda \ll {升}_C\ll 升$持有。然后，这样的 GW 可以近似为高频 GW，并且可以很好地与背景分离${\gamma }_{\mu \nu }$ 通过在一个尺度上平均时空曲率 $\ell$， 在哪里 $\lambda \ll \ell \ll {升}_C$， 和 $G_{\mu \nu }={\gamma }_{\mu \nu }+\epsilon H_{\mu \nu }$ 和 $H_{\mu \nu }$表示 GW。为了使 GW 对背景时空的反向反应可以忽略不计，我们必须假设$|H_{\mu \nu }|\ll 1$，除了条件 $\epsilon \ll 1$，这也是线性化爱因斯坦场方程的条件 $H_{\mu \nu }$有效。通过正确选择量规，例如空间量规、无痕量规和洛伦兹量规，可以显着简化此类研究。我们表明，即使背景不是真空，只要高频引力波近似有效，就可以同时施加这三种不同的规范条件。然而，为了开发适用于尽可能多情况的公式，在本文中，我们首先通过仅施加空间规范来明确地写出线性化的爱因斯坦场方程。然后，将这些公式与几何光学近似应用于此类 GW，我们发现即使存在宇宙学标量和张量扰动，它们仍然沿零测地线移动，并且其偏振双向量平行传输。此外，