Cosmology in Painleve-Gullstrand coordinates

Cosmology is most typically analyzed using standard co-moving coordinates, in which the galaxies are (on average, up to presumably small peculiar velocities) "at rest", while "space" is expanding. But this is merely a specific coordinate choice; and it is important to realise that for certain purposes other, (sometimes radically different) coordinate choices might also prove useful and informative, but without changing the underlying physics. Specifically, herein we shall consider the k=0 spatially flat FLRW cosmology but in Painleve-Gullstrand coordinates -- these coordinates are very explicitly not co-moving: "space" is now no longer expanding, although the distance between galaxies is still certainly increasing. Working in these Painleve-Gullstrand coordinates provides an alternate viewpoint on standard cosmology, and the symmetries thereof, and also makes it somewhat easier to handle cosmological horizons. With a longer view, we hope that investigating these Painleve-Gullstrand coordinates might eventually provide a better framework for understanding large deviations from idealized FLRW spacetimes. We illustrate these issues with a careful look at the Kottler and McVittie spacetimes.