Since its time derivative is zero, the quantity in the square brackets must be a constant. We set it equal to some arbitrary constant k, so that

As the universe expands, any given amount of mass occupies a larger volume. The density goes as 1/volume. The volume is proportional to R3, so we have

8k Gp0 3 R

Farther integration of equation (20.19) depends on whether k is zero, positive or negative. We consider each case separately. The schematic behavior of R(t) for each case is shown in Fig. 20.4(a).

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