Energy-momentum pseudotensor of relic gravitational waves in an expanding universe

We study the energy-momentum pseudotensor of gravitational waves, and examine the one introduced by Landau-Lifshitz for a general gravitational field and the effective one recently used in literature. In the short-wavelength limit after the Brill-Hartle average, both lead to the same gauge-invariant stress tensor of gravitational waves. For relic gravitational waves in the expanding universe, we examine two forms of pressure, $p_{\mathrm{gw}}$ and ${\mathcal{P}}_{\mathrm{gw}}$, and trace the origin of their difference to a coupling between gravitational waves and the background matter. The difference is shown to be negligibly small for most cosmic expansion stages starting from inflation. We demonstrate that the wave equation is equivalent to the energy-conservation equation using the pressure ${\mathcal{P}}_{\mathrm{gw}}$ that includes the mentioned coupling.