The Friedmann Equations, continued
Recall we had the following equations:
To close the equations, we need to relate and with an equation of state:
Note that we will generally set in this class. Combined with (2), this gives us:
Note that we’ve assumed , which is okay most of the time. Some special cases of interest are:
- Pressure-less “dust” , because volume goes as .
- Relativistic particles (photons, bosons): , because , and energy is given by .
- ()/Dark Energy: , constant in time.
To get density () as a function of time, want to solve for .
We define to be the critical density at which (and ):
Today, we measure: . Note that the mass of the sun is , and the mass of the proton is .
measures the ratio of the density of the universe to the critical density:
In general, can consist of multiple components: e.g.
is an unstable equilibrium; any perturbation from in the early universe ensures is far from 1 today. That we measure today implies that the early universe must have been extremely finely tuned.
Today (at ): . Using the Friedmann equation (1), we have:
(THE Friedmann Equation) where , and it is understood that . Each has it’s own , so really .
We’ll show in PS#1 that for any single component:
Plotting , we will find that for early , is extremely close to 1.
Evolution of : Solving the Friedmann Equation
The evolution of the expansion of the universe is governed by:
We can apply this to several models of the universe:
- The Einstein-deSitter (flat) Model: , . Using that , we have:
Thus, the rate of expansion of the universe depends on : (a) The matter-dominated era: . (b) The radiation-dominated era: . (c) The -dominated era: constant in time , where H is now actually a constant. This is exponential inflation. We used to think that this only happened early on (like seconds), but now we think that this has also been happening recently. Next time, we will do the harder two cases: open and closed.