## M

4q2dq

Integrating equation  with a table of integrals leads to a physical outcome radically different from that for the recontracting model Universe:

This curve is plotted in Figure 5, along with curves for the other cases.

The third model of the Friedmann model Universe lies between the cases of contraction and eternal expansion. This third model describes expansion at a steadily decreasing rate that tends to zero in the limit of large radius. If the mass parameter takes on a critical value, which we call Mcrit/ the result obeys the equation m -

QUERY 13 Expansion rates for a flat Universe. Examine what equation  says qualitatively about the time development of the Universe. What happens to the expansion rate dR/dt for R very small and R very large? Is there any value (or limiting value) of R for which the expansion rate goes to zero? Does this model Universe expand forever?

Write equation  in a form similar to that of equation . Interpret the resulting equation in terms of Newtonian energy. In this case is the total energy of the dust particle positive, negative, or zero?

QUERY 14 Single equation for all three Friedmann models. Write down a single equation for dR/dt for all three Friedmann models of the Universe. Use the parameter k, defined in the paragraph following equation  on page G-6.

QUERY 15 Radius R{t) for flat Universe. Integrate equation  from radius 0 to R and from time 0 to t. Show that the result can be written

The resulting curve is plotted in Figure 5, along with curves for the other two cases.

The three cases we have been describing are often distinguished using the parameter Q (capital omega, the last letter of the Greek alphabet).

Then the three models are characterized by different values of £2 as follows:

£2 > 1. The mass parameter is large enough to recontract the Universe. The result is called a closed Universe and is said to be a Universe with overall positive spatial curvature. The constant k has the value +1 in the metric . 