The most incomprehensible thing about the world is that it is comprehensible. Albert Einstein

A model of the universe is 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 self-gravitating 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 self-gravity are called the Friedmann equations. Models of the universe derived from this equation are called Friedmann-Robertson-Walker models, or FRW models. The three possible fate of a universe containing only ordinary mass density 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."

These cosmological models assume zero cosmological constant (Lambda). The only force acting is gravity. They can be summarized thusly:

### Standard Model Summary Table

Model Geometry k Omega qo Age Fate
Closed Spherical +1 >1 > 1/2 to < 2/3 tH Recollapse
Einstein- deSitter Flat 0 =1 = 1/2 to = 2/3 tH Expand forever
Open Hyperbolic -1 <1 < 1/2 2/3tH < to < tH Expand forever

The special case of the flat (k = 0, Omega = 1), matter-only universe is called the Einstein-deSitter 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 Einstein-deSitter model, so it provides a convenient guide for estimation of some cosmological quantities. For example, the age of the universe in the Einstein-deSitter model is 2/3rd the Hubble time.

Adding a nonzero cosmological constant provides a new possibilities. 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.