Propagation of Gravitational Waves in an Inhomogeneous Universe

In this talk, we shall present our recent studies on gravitational waves (GWs) produced by remote compact astrophysical sources. To describe such GWs properly, we introduce three  scales, 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, we show that such GWs can be well approximated as high-frequency GWs. To simplify the field equations, we show that the spatial, traceless, and Lorentz gauge conditions can be imposed simultaneously, even when the background is not vacuum, as long as the high-frequency GW approximation is valid. Applying the general formulas, along with the geometrical optics approximation, we calculate the gravitational integrated Sachs-Wolfe effects due to the presence of the  cosmological scalar and tensor perturbations, whereby the dependences of the amplitude, phase and luminosity distance of the GWs on these two kinds of perturbations are read out explicitly.