I'm sure most of you have heard of the twin paradox "in which a twin makes a journey into space in a high-speed rocket and returns home to find he has aged less than his identical twin who stayed on Earth." This paradox has been worked out for special relativity in Minkowski spacetime. Recently, Boblest et al. worked out the details using general relativity for an expanding universe. (de Sitter spacetime.)
First a review of the standard Minkowski version: In this case the whole universe is flat Minkowski spacetime and can therefore be handled with special relativity.
Now consider an expanding universe: For an expanding de Sitter spacetime, the universe is no longer flat and so the authors have to appeal to general relativity. The Hubble expansion parameter, quantifying how fast the expansion is happening, is denoted by H. For our universe, H = H0 = 71 km/s/Mpc. Larger H means faster expansion.
Now For A Proper Round Trip. Now let us demand Tina accelerates and decelerates in such a way that Tina is able to return home in 20 years. Now Tina must turn around before the 10 year mark. Here the authors compare three "trips" where the constant acceleration/deceleration is greater for each trip. IE, Tina accelerates faster for trip 3 than trip 2 which is faster than trip 1. In each case, H = 109 H0.
Conclusion: This paper shows how the twin paradox is altered for an expanding universe. Interestingly, the expansion of the universe lessons the distance Tina can travel in 20 years and makes it so that Tina's change in age changes much more closely to how Eric's age changes. However, it should also be noticed that to see significant differences compared to the special relativity case the universe must be expanding significantly faster than our own universe is.
Sebastian Boblest, Thomas Müller, & Günter Wunner (2010). Twin Paradox in de Sitter Spacetime E-Print arXiv: 1009.3427v1