Propagation of gravitational waves in a universe with slowly-changing equation of state

An exact solution for the expansion of a flat universe with dark energy evolving according to a simple model is explored. The equation for weak primordial gravitational waves propagating in this universe is solved and explored; gravitational waves in a flat cosmology possessing both a "big bang" singularity and a "big rip" singularity can be described with confluent Heun functions. We develop approximation methods for confluent Heun equations in regimes of interest to gravitational wave astronomers and predict the diminution in gravitational wave amplitude in a universe with both a Big Bang and a Big Rip.