Cosmology Lecture 04

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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:

So today:

Thus for , and for (an empty universe).

In a closed universe: , , . Recall:

Thus, today:

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.

The Model:

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.