de Sitter Model

 

This is a solution of the cosmological equations where only the cosmological constant provides energy density for the universe. Although at later times it is unrealistic, at early stages of the Big Bang Model it can represent the dominant contribution to the energy density of the universe when he universe gets temporarily stuck in high energy phases.

 

The de Sitter Model, which was first discoved by Willem de Sitter in 1917, is defined by the following conditions:  

 

 

The first Friedmann Equation is:

 

(31.1)

 

In the de Sitter case we arrive at the differential equation

 

 

(31.2)

 

 

Getting rid of the squares this leads to the following

 

 

(31.3)

 

 

 

This d.e. is easy to solve. The solution is

 

 

 

(31.4)

 

 

 

 

We can absorb the constant into the scale factor to get the simpler expression:

 

 

(31.5)

 

 

 

Many of the models that have nonzero cosmological constants have this solution as their limiting case.

 

 

We now take a look at the line element that corresponds to this solution. We will use the Robertson Walker form

 

 

 

 

 

(31.6)  

The de Sitter case gives:

 

 

(31.7)  

Since the 3-D subspace is just flat space we can put this line element in pure Cartesian coordinates. This gives the expression.

 

(31.8) 

 

We now introduce the parameter  defined as

 

 

(31.9)

 

 

 

The Cartesian coordinate line element can then be rewritten:

 

 

(31.10)  

 

 

 

It can easily be shown that this line element is invariant under an arbitrary shift in time coupled with a simultaneous change of the 3-D space coordinates. The metric has no other parameters but the cosmological constant parameter . This exponential line element has no natural time point at which one could say the universe starts. We can explicitly show that the de Sitter solution doesn't care about the time at all by switching the coordinates to new coordinates  as follows:

 

 

(31.11)  

 

 

 

 

The line element then becomes

 

 

(31.12)                                                            

 

 

 

This line element is manifestly time invariant: it's stationary. Since there are no cross terms it is also static.

 

Since the early 1980's there has been a very popular model called the Inflationary Universe Model based on a de Sitter-type solution. The inflationary universe is assumed to start at extremely early times of the order of . At this time the forces in the universe are assumed to be unified due to the extreme high energy densities that are involved. At a time around the Planck time (  seconds) gravity is assumed to be unified with the strong interaction, the weak interaction and the electromagnetic forces. When the inflationary universe model comes into play gravity has separated from the other forces but the strong interaction, the weak interaction and the electromagnetic forces are still unified but the strong interaction is about to break away. The background vacuum energy density , associated with the break away of the strong interaction, momentarily dominates the energy density of the universe. Thus

 

(31.13)

 

The Friedmann Equation in this case

 

(31.14)

 

 

 

If the vacuum energy is large enough, at this early time the curvature term on the right hand side of this equation will be smaller than the vacuum energy term also on the right hand side.

 

(31.15)

 

 

 

This situation will get more pronounced as the exponential expansion proceeds since the scale factor will bloat out considerably as time progresses.

 

 

 

The differential equation is essentially the de Sitter equation

 

 

(31.16)

 

 

which gives the solution

 

 

(31.17)

 

where

 

 

(31.18)

 

and

(31.19)

Since the usual time dependence found in more traditional early universe cosmological models is that of a power law function of time, we can see that the inflationary  cosmology will be much better at sweeping things away. The exponential expansion, even if it occurs for only a brief interval after the big bang, will get rid of any curvature of the part of the universe that eventually becomes our part (now with a 15 billion light year radius). Also any irregularities in the distribution of mass energy will be damped out by a large amount so that the remnant post-inflationary universe will appear, on cosmological scales, to be very smooth. It is for these reasons, amongst others, that the inflationary universe scenario became so popular with cosmologists. The main problem with the Inflation Cosmology Model is that there are several models that produce this type of result. It is not clear which of the various proposals is the correct one. The idea of the early universe was put forward for the first time, at least in a manner that people took notice of, by Alan Guth in the early 1980's. Since then many cosmologists have put forward versions of the Inflationary Cosmology. One version put forward by Andrei Linde of Stanford University has the universe starting as one of many universes right from the start in an inflationary mode.