Curvature of the Universe
The galaxies spread throughout the universe are moving away from each other.  To describe this mathematically, cosmologists use a universal scale factor called a(t) which grows as the universe expands, along with a coordinate system that expands along with the galaxies.  In this coordinate system, called co-moving coordinates, the galaxy positions do not change.

Said another way, the position of some far away galaxy with respect to our galaxy is given by the vector R(t).  We write this vector as

R(t) = a(t) · X

where we have factored all of the time dependence of R(t) into a(t), leaving the vector X with no time dependence, representing the co-moving distance between the two galaxies.  The common convention is to set a(t) = 1 at the current age of the universe, such that the physical vector R(t) is equal to the co-moving vector X today.

The history and ultimate fate of the expansion is described by the function a(t).  If the universe contained a lot of dark matter, called a closed universe, a(t) would rise, reach a maximum, and then fall.  The universe would suffer a "Big Crunch" in which the galaxies would all collide at some point in the future.  If the universe contained very little dark matter, called an open universe, a(t) would rise forever.  Galaxies would forever be moving away from each other, with expansion never stopping.  If the universe contained precisely the critical density of matter and energy, called a flat universe, a(t) would rise forever but would approach an asymptote at some final value.  Galaxies would still be forever moving away from each other, but the expansion of the universe would approach zero.  The following plot shows these cases.


The following animations show three examples of an expanding universe drawn in physical and co-moving coordinate systems.  The physical system on the left side of each video shows the motions of three galaxies that lie on the vertices of an equilateral triangle.  In the co-moving system, the coordinate grid expands along with the matter in the universe, so the galaxies are shown at rest.

The following animation shows galaxies in a low density universe.  At the time light from the bottom two galaxies is emitted, the galaxy on top will be moving away.  When the light reaches this galaxy, an observer will see the two galaxies not where they are at the end of the animation, but where they were when the light was first emitted.  The angle between the galaxies will be less than 60 degrees.  Observers in all three galaxies will see the same angle between the other two.  Since this is a low density universe, there is no coordinate system curvature in the physical system, and the observers ascribe the angular deficit to the motions of the galaxies.  The co-moving system, however, has a negative curvature which causes this deficit.

Physical Coordinates                       Co-moving Coordinates                                                                   
                                                                                        

The next animation shows galaxies in a critical density universe.  Light rays emitted by the bottom galaxies will curve because of matter between the three galaxies in the physical system.  Observers in the top galaxy will see the bottom galaxies separated by precisely 60 degrees.  Because of this, there is no curvature to the light paths in the co-moving system, and the grid lines that form the system are straight.

Physical Coordinates                       Co-moving Coordinates                                                                     

                                                         

The final animation shows galaxies in a universe with a density higher than critical.  In the physical system, light rays from the bottom two galaxies will be curved so much by the matter in between the galaxies that an observer in the top galaxy will see an angle greater than 60 degrees.  In the co-moving system, this creates a positive curvature which causes the change in observed angle.  In the spherical graph on the right, note how the angles of the triangle appear greater than in the plot above with no curvature.

Physical Coordinates                       Co-moving Coordinates                                                                    
                                                     

When cosmologists say the universe is flat, they mean that the curvature in co-moving coordinates is zero.  In physical coordinates, such a universe has positive curvature.  It contains precisely the critical density of matter and energy.