Relativity Lite

Cosmology | 77 Such a visualization led to the idea of the Big Bang—that an initial singularity of in- finitely dense, hot matter somehow exploded outward and is continuing to fly outward to this day, cooling as it expands. AN OPEN AND SHUT CASE What would cause the shape of the universe to curve in on itself, as in figure 1, or flop open, as in figure 3, or to take on perfect flatness? One may profitably ask a parallel question, “What causes an umbrella to close?” Well, you cause it to close by pulling down on the ring surrounding the handle, which in turn tugs inward on the spokes that attach to the cloth-­ covered ribs. There are a few umbrellas that I have had a pretty hard time closing because I could not exert sufficient force to counter the outward pressure of the spring. Suppose you had a universe that had enough mass that the inward gravitational force (in Newtonian language) could slow the outward expansion, bring it to a halt, and then cause it to recollapse in a Big Crunch. This is the k = +1 case. Suppose you had a universe that had insufficient mass to counter the outward momentum of the explosion of galaxies that is the Big Bang. That universe would not curl in on itself into a sphere but stay open like a floppy umbrella. This is the k = −1 case. There would, then, be some critical density of matter, ρ π c H G = 3 8 0 2 , just sufficient to halt the outward expansion at infinite time. This is the k = 0 case. * We saw in the last section that running the closed, spherical-universe, k = +1 case back- ward in time led to an infinitely dense pinpoint singularity from which the Big Bang ex- panded outward. What about the other two, the open-and critical-universe cases where our universe is presently a saddle shape of infinite extent or flat and infinite? As one looks at figure 5, one sees in either case a universe at present at the top becoming denser and denser as one proceeds downward, back in time. No matter how far to the left the galaxies move, as we go backward in time to create increasing density, there will always be an infinite number of additional galaxies to the right moving into our frame of view. Thus, we can have an infinitely dense universe that is infinitely large just before the Big Bang occurred. So in the k = 0 or k = −1 cases, the universe would not all be stuffed within a single point, as it would be if k = +1 , but the entire infinite universe is nevertheless infinitely dense. In these cases, the Big Bang did not expand from a single point but from everywhere in the universe, and the scale of the universe, R , has been increasing ever since. * Before rescaling r to give k unit magnitude, k is given by k R G c c 0 2 2 0 8 3 = − ( ) π ρ ρ where ρ π c H G = 3 8 0 2 is the critical density, ρ 0 is the present density of the universe, R 0 is its present size, H 0 is the Hubble constant, and G is the universal gravitational constant.