Foundations of Modern Cosmology

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Chapter 11: Modeling the Universe

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
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 in which only gravity operates (i.e. zero cosmological
constants) and mass-energy is conserved are called standard models
. (They are also widely known as Friedmann-Robertson-Walker 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 critical density and the Omega parameter

the deceleration parameter q

the relationship of the curvature k to q and Omega

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

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.

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.

Lambda negative - attractive force and all models recollapse

Lambda positive - repulsive force; open and flat models eventually
expand at an accelerating rate, q < 0

Lambda critical - the value that just balances gravity in the
Einstein static spherical model

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 Java-enabled 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.