# Cosmology Lecture 04

### From AstroBaki

## Contents |

### 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: *k* = 0, Ω_{m} = 1, Ω_{Λ} = 0, so:

Alternatively, recall that for a matter-dominated era, Thus, .

If we have : *k* = 0, Ω_{0,M} + Ω_{0,Λ} = 1, then:

Assuming , this integral is solvable:

Generally, in a flat universe, . If , it will be longer.

In an open universe: *k* < 0, Ω_{0} < 1(Ω_{0,Λ} = 0). Recall:

So today:

Thus for , and for Ω_{0} = 0 (an empty universe).

In a closed universe: *k* > 0, Ω_{0} > 1, (Ω_{0,Λ} = 0). Recall:

Thus, today:

### The Robertson-Walker Metric

Lorentz invariance dictates that two inertial frame (*x*,*y*,*z*,*t*) 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 *d**s*^{2} = 0 path. In tensor form, this equation looks like:

where *g*_{αβ}, the metric tensor, is given by:

Look at Weinberg, Ch. 13 for full proof, but for a homogeneous, γ-isotropic space, the metric looks like:

where *r* is a radial direction (in comoving coordinates), and *d*Ω = *d*θ^{2} + sin^{2}θ*d*^{2}φ is the differential angle seperation of two points in space. As usual, *k* is the measure of curvature.

The *k* = 0 Model:

so we recover the Minkowski metric for flat space, using comoving coordinates.

The *k* > 0 (closed) Model:

We get a coordinate singularity at , so this universe has a finite volume. For *k* > 0, 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.