#
The Expanding Universe

**1917:** **Willem
de Sitter** used general relativity to describe an expanding
universe.
Einstein favored a static model which neither expanded nor contracted,
by inserting an ad-hoc term he called the **cosmological
constant**.

**1929:** Hubble
measures
the expansion of the Universe

**v=H**_{0}d,
where H_{0} = 72 km/s/Mpc (+/- a handful) is the
**Hubble constant.**

(Because of historic
uncertainty in the Hubble constant, in the past, astronomers
often defined a parameter **h=H**_{0}/100,
and write distance dependant results as (for example), d=200h^{-1}
Mpc.)

## The Cosmological Redshift

Remember the redshift:

Using normal Doppler equation (v<<c), v=cz,
and
d=cz/H_{0}.

But it is important to realize here that the
cosmological
redshift is not really due to galaxies moving through space at high
velocities.
Galaxies are actually "embedded" in space and space itself is
expanding.
The light emitted from the galaxies is "stretched" as it makes its way
towards us, and we see it shifted to longer wavelengths.

**Measuring distances in an
expanding
universe:**

The distance between two objects can be
defined
as d = R(t) x r, where
- d =
**proper distance**
- R =
**dimensionless
scale
factor**
- r =
**comoving
coordinate**

For two galaxies **r only
changes
due to peculiar motion** of the galaxies (ie due to gravity in
a cluster, etc). The change in proper distance due to the expansion of
space is entirely contained in R(t). We define R(now) = 1.

Note that on
cosmological scales, this makes the distance between two objects an
ambiguous concept. There are different ways of defining distances,
which give different answers.

Go back to the redshift. What does that tell us about
the
expansion?
So if we observe a quasar at a redshift of z=3, at
the
time the light left the quasar, the universe was 1/4 its current size.

## The Age of the Universe

As z gets large, R gets small. At some point, R=0!
When did that happen?

Homer
solves the age of the Universe.

Let's see. Space is expanding in such a way
that
d=v/H_{0}, then to get to a distace d moving at velocity v
would
take time t_{0}=d/v=1/H_{0}. *So
the inverse of the Hubble constant is a (rough) measure of the age*.
This is called the **Hubble time**.
If H_{0} = 72 (km/s)/Mpc, this is ~ 72
(pc/Myr)/Mpc, which is 72x10^{-6} Mpc/Myr/Mpc = 7.2x10^{-5}
Myr^{-1}.

Then t_{0} = 1/H_{0} =
1.39x10^{4} Myr
= **13.9 billion years**.

*There's a pretty big assumption built into
this -- what is it? *