From the the Robertson-Walker metric and the Einstein field equations, an
"equation of motion" of the universe can be derived:
Where H is Hubble's constant, a is the cosmic scale
factor, 
is the density in matter
and radiation, 
is the cosmological
constant, and k is the curvature constant. This is commonly
referred to as Friedmann's equation (originally written without the
cosmological constant term, it was actually Lemaitre who brought about
its general recognition). Friedmann's equation describes the time
evolution of the cosmic scale factor, or the "size" of the universe.
With the definitions:
the equation of motion becomes:
Where the subscript o represents quantities measured at the present time, and a is normalized such that it is equal to one today. The subscript m refers to the contribution of matter. (The contribution of radiation to the density has been dropped since it is of negligible size today (10-5). The inclusion of radiation is important only in the evolution of the very early universe.) With the above definitions is is clear that:
It is useful to define the quantity:
tells at a glance the spatial geometry of a given universe. If < 1 the universe is spatially open, if = 1 the universe is spatially flat, and if > 1 then the universe is spatially closed.
The diagram below shows the behavior of cosmological models in the 
, plane.
This plot is useful as a "finding" plot for universes with different cosmological parameters, but it takes some time to get oriented with it. The blue line at = 1 is where spatially flat universes are located. Open universes lie to the left of the blue line, and closed universes to the right. The black dotted line is the 
= 0 line, universes with negative values for the cosmological constant lie to the left of this line and universes with positive cosmological constants lie to the right. The green line separates universes that will expand forever from universes that are fated to recollapse sometime in the future. The red line is the "loitering" universe line, universes located below this line are "bouncing" universes. To see the behavior of universes for the points indicated on the plot click on them. Click here for a listing of the available points and their coordinates. We most likely live in a universe which is close to one of three universes represented by the points above: (4), (5), or (8).
A negative cosmological constant adds to the attractive gravity of matter, therefore universes with a negative cosmological constant are invariably doomed to recollapse (1). A positive cosmological constant resists the attractive gravity of matter due to its negative pressure. For most universes, the positive cosmological constant eventually dominates over the attraction of matter and drives the universe to expand exponentially (8). For some universes, ones with large and small (the thin wedge in the upper right corner of the plot), the cosmological constant never dominates and the matter wins out, collapsing the universe after some finite time (12). For large positive cosmological constants universes exist which actually experience no big bang. These universes collapse from an infinite size, turn around, and then expand to infinity again. These universes are appropriately called "bouncing" universes (11). Since the scale factor for bouncing universes never gets smaller than a certain size, objects will only be observed to have redshift less than some maximum redshift (and indeed some objects may even have blueshifts.) Universes just outside the bouncing region spend a large amount of time deciding whether to collapse due to their matter or to expand due to their cosmological constant. These universes are called "loitering" universes, and are located on the red line in the finding plot (10). These universes spend a long time at a nearly constant scale factor, and thus have a characteristic loiter redshift. Click here to see a plot showing how the loiter redshift depends on (the bounce redshift of bouncing universes obeys a similar dependence on ).
The values of and have a large effect on the age of the universe. The cosmological constant either aids or resists the attraction of matter, this can lead to younger or older universes, respectively, than universes with the same amount of matter and no cosmological constant. Below is a plot in which constant age contours are overlain on the finding plot. Since the actual age of our universe depends on the value we measure for 
, the contours are in units of 
. In units of pure time is believed around 14 Gyr, but its value is widely debated. Even though is not known with certainty, the contours are useful in showing the relative ages of universes.
As per the solution to the age problem, it can easily be seen that a flat universe with a positive cosmological constant can be significantly older than a flat, = 0 universe.
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