Time-Redshift Relations and the Age of the Universe
Last time we found the age of a flat universe. in a flat (Einstein-deSitter) universe: , , , so:
Alternatively, recall that for a matter-dominated era, Thus, .
If we have : , , then:
Assuming , this integral is solvable:
Generally, in a flat universe, . If , it will be longer.
In an open universe: , . Recall:
Thus for , and for (an empty universe).
In a closed universe: , , . Recall:
The Robertson-Walker Metric
Lorentz invariance dictates that two inertial frame and , with one moving with respect to the other at velocity , are related by:
where . Note, to give a taste of tensor forms, this all may be written as .
Remember the Lorentz invariant interval, which is conserved between frames:
Light travels a path. In tensor form, this equation looks like:
where , the metric tensor, is given by:
Look at Weinberg, Ch. 13 for full proof, but for a homogeneous, -isotropic space, the metric looks like:
where is a radial direction (in comoving coordinates), and is the differential angle seperation of two points in space. As usual, is the measure of curvature.
so we recover the Minkowski metric for flat space, using comoving coordinates.
The (closed) Model:
We get a coordinate singularity at , so this universe has a finite volume. For , we need to define “Polar Coordinates” in 4-D (to describe a 3-sphere embedded in 4-D). Here is a comparison of how we define polar coordinates for a 3-sphere in 4-D versus for a 2-sphere in 3-D:
Take a line element on a 2-sphere:
Changing variables for :
Then , so rewriting our line element, we get:
For a 3-sphere,
where . Again, using a change of variables so that , , we get that:
This is what Robertson-Walker showed.