Notes on Relativity and Cosmology - Physics Department, UCSB

10.4. OBSERVATIONS AND MEASUREMENTS 303
1 − 8πGρ
3H 2 −
Λ
3H 2 + kc2
H 2 = 0. (10.7)
a2 Now, the three interesting cases are k = −1, 0, +1. The middle case is k = 0.
Look at what happens then: the quantity 8πGρ
3H
+ Λ 2 3H
, which measures the
2
overall density of stuff (matter or cosmological constant) in ‘Hubble units’ must
be one! So, this is a convenient reference point. If we want to measure k, it is
this quantity that we should compute. So, cosmologists give it a special name:
Ω ≡ 8πGρ
3H 2 + Λ
3H2. (10.8)
This quantity is often called the ‘density parameter,’ but we see that it is slightly
more complicated than that name would suggest. In particular, I should point
out that (like the Hubble ‘constant’) Ω will in general change with time. If,
however, Ω happens to be exactly equal to one at some time, it will remain
equal to one. So, to tell if the universe is positively curved (k = +1), negatively
curved (k = −1), or [spatially] flat (k = 0), what we need to do is to measure
Ω and to see whether it is bigger than, smaller than, or equal to one.
By the way, cosmologists in fact break this Ω up into two parts corresponding
to the matter and the cosmological constant.
Ω matter ≡ 8πGρ
3H 2
Ω Λ ≡
Λ
3H 2 (10.9)
Not only do these two parts change with time, but their ratio changes as well.
The natural tendency is for Ω Λ to grow with time at the expense of Ω matter
as the universe gets larger and the vacuum energy becomes more important.
Anyway, when cosmologists discuss the density of matter and the size of the
cosmological constant, they typically discuss these things in terms of Ω matter
and Ω Λ .
So, just how does one start looking for matter in the universe? Well, the place
to start is by counting up all of the matter that we can see – say, counting
up the number of stars and galaxies. Using the things we can see gives about
Ω = 0.05.
But, there are more direct ways to measure the amount of mass around – for
example, we can see how much gravity it generates! Remember our discussion
of how astronomers find black holes at the centers of galaxies? They use the
stars orbiting the black hole to tell them about the mass of the black hole.
Similarly, we can use stars orbiting at the edge of a galaxy to tell us about the
total amount of mass in a galaxy. It turns out to be much more than what
we can see in the ‘visible’ matter. Also, recall that the galaxies are a little bit