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Chapter Summary 
A model of the universe includes a mathematical description of how the scale factor R(t) evolves with time. In this chapter we develop some models of the universe. As a first approximation, consider the analogy of the Newtonian ball of selfgravitating particles. Gravity acts to try to pull the ball together. If the ball is expanding with sufficient velocity it can resist this collapse. We obtain a simple equation to describe the evolution of this Newtonian ball. One of the most important consequences of this analysis is the realization that gravity permits three possibilities for the evolution of the universe: it could expand forever; it could stop expanding at infinite time; or it could stop expanding at some finite point in time and recollapse. Remarkably, the fully general relativistic solution for a universe consisting of smoothly distributed matter has the same form as the Newtonian solution. The equations that describe the evolution of the universe under the influence of its selfgravity are called the Friedmann equations. Models in which only gravity operates (i.e. zero cosmological constants) and massenergy is conserved are called standard models . (They are also widely known as FriedmannRobertsonWalker models, or FRW models, but we do not refer to them as such in the text.) The three possible fates of the universe were seen to correspond to the three basic geometry types studied in Chapter 8. The hyperbolic universe expands forever; the flat universe expands but ever more slowly, until it ceases expanding at infinite time; and the spherical universe reverses its expansion and collapses in a "big crunch." From the Friedmann equations we can derive a large number of important parameters:
The standard models of cosmology assume zero cosmological constant (Lambda). The only force acting is gravity. Then we have the following possible models: Standard Model Summary Table
The special case of the flat (k = 0, Omega = 1), matteronly universe is called the EinsteindeSitter model. Various numerical parameters such as the age of the universe, the lookback time to distant objects, and so forth, are easiest to compute in the EinsteindeSitter model, so it provides a convenient guide for estimation of some cosmological quantities. Adding a nonzero cosmological constant provides a number of new possible models. Of these nonstandard models, the de Sitter, the Steady State, and the Lemaitre models are the most significant. The cosmological constant acts as an additional force, either attractive (negative lambda) or repulsive (positive lambda). Instead of decreasing in strength with distance like gravity, the "lambda force" increases with scale factor. This means that any nonzero cosmological constant will ultimately dominate the universe. An attractive "lambda force" will cause a recollapse and big crunch regardless of the model's geometry. However, the possibility of an attractive lambda in the physical universe is ruled out by observations. A repulsive lambda force has more interesting possible effects. The details depend upon the model, but eventually all such models expand exponentially.


Points to Ponder 
What is the fate of each of the universe models? How do their present ages relate to the Hubble time? How could we determine which model best describes the physical universe? How do the standard models change if we add a cosmological constant term (Lambda)? Which "Lambda" models will end in a crunch? Which with heat death? What will those ultimate fates be like? 

Potential Pitfalls  
Questions & Answers 
Questions and Answers related to Chapter 11 . If you are running a Javaenabled browser, try Solving the Friedmann Equation. 

Web World 
A hypertext tutorial on the big bang can be reached from the Violence in the Cosmos page. Have astronomers found new evidence for a cosmological constant? Read about it at NASA's astronomy picture of the day web site. 