15.4 Matter Dominance
237
scale factor the integral in (15.8a) is dominated by the matter term, so the solutions
for all k will behave like (15.9).
For nonzero values of k the integral in (15.8a) may also be evaluated easily. For
k > 0 and k0 < 0, the 3-sphere, the curvature density is negative and the we have
a
0
da
m0
a
− | k0 |
−1/2
= H 0 t
(15.10)
From integral tables we obtain
1
√
| k0 |
Dsin
−1
(
a/D) −
a(D − a)
= H 0 t,
D =
m0
k0
, , k0 < 0.
(15.11)
From this somewhat cumbersome expression we may plot the behavior of the scale
factor and see that it is qualitatively as shown in Fig. 15.2. The curve is known as
a cycloid and is explored further in Exercise 15.1. In this model the scale factor
increases to a maximum value D and then decreases to zero after a finite time; the
universe does not expand forever.
For k < 0 and k0 > 0, the 3-pseudosphere, the same manipulations give
1
√ k0
a(D + a) − Dsinh
−1
(
a/D)
= H 0 t,
D =
m0
k0
, , k0 > 0.
(15.12)
Fig. 15.2 Qualitative behavior of the scale factor for negative, zero, and positive curvature
parameter k. See Appendix 1 for comments on a mechanical analogy