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This means that R is proportional to t2/3. The universe always expands, but the rate of expansion becomes smaller and smaller.

We now look at the case k > 0. Since the first term on the right-hand side of equation (20.19) becomes smaller when R increases, a point will be reached eventually, at some finite R, where R = 0. The expansion stops at some maximum scale factor Rmax. We can find Rmax as the value of R, which makes R = 0 in equation (20.19). That is,

3 Rmax

Solving for Rma

0. Equation

It should be noted that R is always positive, but approaches zero as R approaches infinity.

After Rmax is reached, the universe starts to collapse. We say that the universe is closed.

We next look at the case k < 0. If k is negative then — k is positive, and the right-hand side of equation (20.19) is always positive. As R gets very large, the first term on the right-hand side approaches zero, and R2 approaches — k. (Remember, — k is a positive number.) This means that R approaches (— k)1/2. The expansion continues forever, and we say that the universe is open.

We can think of an analogous situation of throwing a ball up in the air. If the total energy is negative, the ball will return to Earth. If the total energy is positive, the ball will escape, and its speed will remain positive. If the total energy is zero, the ball will reach infinity, but its speed will approach zero. We can think of k as being related to the energy of the spherical region of the universe that we are following.

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