Figure S Computer plot. Time development of three versions of the Friedmann model Universe composed of pressure-free dust■ the open Universe, the flat Universe, and the closed Universe. The open Universe continues to expand forever The flat Universe also continues to expand forever, but the rate of expansion gradually approaches zero. The closed Universe expands to a maximum size, then contracts to a final crunch. The parameter omega fi is the ratio of the mass parameter M of each of these models to the critical mass parameter Mcrit that leads to a flat Universe When the mass parameter has a value equal to McnP then £i = 1 A value of Q less than unity corresponds to the open, ever-expanding Universe, whereas a value of £2 greater than unity leads to a closed Universe.
£2 = 1. The mass parameter has the critical value Mcrit such that the Universe continues to expand, but the rate of expansion approaches zero. This result is called a flat Universe and is said to be a Universe with overall zero spatial curvature. The constant k has the value 0 in the metric , yielding a space part that is flat. (It is only the space part that is flat.)
£2 < 1. The mass parameter is small enough that the rate of expansion of the Universe decreases but never reaches zero. This result is called an open Universe and is said to be a Universe with overall negative spatial curvature. The constant k has the value -1 in the metric .
Do any of these three models of expansion describe the Universe we live in? There are many other models of the Universe, some more realistic than our Friedmann pressure-free dust models. Cosmology has made great strides in recent years, but as yet no one model of the Universe has won general acceptance. All matter in the Universe for which we have direct evidence adds up to only 10"2 to 10_1 of the critical mass Mcrit. Until recently the most popular cosmological theories assumed, nevertheless, that the total mass was equal to Mcrit, which led theorists to look for so-
called dark matter, matter of new and exotic kinds, to account for the needed mass (so far observed only indirectly). In the meantime, recent observations have been interpreted as evidence that the rate of expansion of the Universe is actually increasing with time, as described in the box on page G-20.
How much of the Universe can we see right now? Are some galaxies so far away that light from them has not had time to reach us since the Big Bang? And what can these questions possibly mean, since all parts of the Universe were together at the Big Bang itself?
We cannot answer these questions until we have decided what model Universe correctly represents the one in which we live. In the meantime, we can answer the questions for one of our simplified model universes. We choose the closed model Universe, for which the parameter k- 1 in the metric . This metric (for two spatial dimensions) becomes
This equation can be simplified still further by converting from time increment dt to the angle increment dr\ using equation .
QUERY 16 Time as an angle. For dt in equation  substitute from equation  to
show that the metric can be written
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