 We now turn our attention to some cosmological observations. The way cosmologists judge any model for the universe is to compare the model's predicted outcomes with what we actually observe. You may recall from our video book on general relativity that curved space has different volume implications than we have for Euclidean flat space. So one way to determine if the universe is flat, spherical, or hyperbolic is to count galaxies at different distances, i.e. different redshifts. If we look out into a flat universe, we would see the number of galaxies vary with the volume. If we look out into a spherical space, we would see the number of galaxies increase more slowly than for flat space, reach a maximum, and then come back down. And if we look out into a hyperbolic space, we would see the number of galaxies increase dramatically faster than for flat space. With our modern technologies, we can see galaxy populations out to around 10 billion light years. And as far as we can see, the number of galaxies increases according to the flat space model. Observations of galaxy diameters and luminosity distances also show a flat universe. Large scale space looks completely flat. But if the universe is large enough, say with a radius of curvature at around 200 billion light years, it is possible for it to look flat to us, examining such a small part of it. So it is still possible that we live in a three-dimensional sphere with a huge radius of curvature. But for the rest of our study of the benchmark model, we'll assume we exist in flat space.