Gravitational wave cosmology I: high frequency approximation
  In this paper, we systematically study gravitational waves (GWs) produced by
remote compact astrophysical sources. 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, and such
GWs can be well approximated as high-frequency GWs. In order for the
backreaction of the GWs to the background to be negligible, we must assume that
$\left|h_{\mu\nu}\right| \ll 1$, 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, where $g_{\mu\nu} = \gamma_{\mu\nu} + \epsilon
h_{\mu\nu}$, and $\gamma_{\mu\nu}$ denotes the background. 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. However, to
develop the formulas that can be applicable to as many cases as possible, we
first write down explicitly the linearized Einstein field equations by imposing
only the spatial gauge. Applying the general formulas together with the
geometrical optics approximation to such GWs, we find that they still move
along null geodesics and its polarization bi-vector 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,
whereby the dependences of the amplitude, phase and luminosity distance of the
GWs on these two kinds of perturbations are read out explicitly.
