Cosmological Times

One thing that comes out of all the different models is the age of the Universe, t₀. t₀ depends on H₀, Omega_M, and Lambda, so we can't uniquely determine cosmology just from measuring the age of the Universe. But we can place constraints. How?

The Ages of Globular Clusters

Obviously, globular clusters can't be older than the Universe. We measure the age of a globular cluster by measuring the main sequence turnoff in the color-magnitude diagram of GCs: By measuring the luminosity and color of the turnoff, and comparing to models of stellar evolution, we can determine the ages of the globular clusters. Typical numbers are 10-16 billion years (see, eg, the analysis by Chaboyer and Krauss 2003). M92 is a typical example: t=14.5 +/- 2.5 Gyr (Grundahl etal 2000).

Ages of Lambda=0 Universes

H₀ Omega_M t₀ (Gyr)

A 65 1 10

B 40 1 15

C 65 0.3 12

D 65 0.1 15

E 75 0.1 13

F 50 0.1 19.5

We can see from this table that some models are "ruled out":

it's extremely hard to envision an Omega_M=1 universe.

it's extremely hard to envision a high H₀ universe (ie H₀ > 80)

The Cosmological Constant may be real

Ages of H₀=65 Universes

Omega_M Omega_Lambda t₀

A 1 0 10

B 0.3 0 12

C 0.3 0.7 13.5

D 0.1 0.9 19.5

Ages of Lambda=0 Universes
	H₀	Omega_M	t₀ (Gyr)
A	65	1	10
B	40	1	15
C	65	0.3	12
D	65	0.1	15
E	75	0.1	13
F	50	0.1	19.5

Ages of H₀=65 Universes
	Omega_M	Omega_Lambda	t₀
A	1	0	10
B	0.3	0	12
C	0.3	0.7	13.5
D	0.1	0.9	19.5

The Ages of High Redshift Objects

When we look at distant objects, we are seeing the young universe. How old are objects in a young universe? If we measure an object that is 3 Gyr old at a time when the Universe was only 2 Gyr old, that's bad...

Now we need to introduce a few concepts. When we look at an object at a given redshift, we can define (in a model-dependent way):

Lookback time t_L(z): How far back in time we are looking.
Cosmic age t(z): the age of the universe at that redshift

Let's calculate this for one model: Omega_M=1, Lambda=0

For this model, we had (from last time) an expression for how scale factor grows with time:

from this we solve for t:

We also know how scale factor and redshift are related:

Plugging in, we get

so the way we have defined things, lookback time can be calculated this way:

so

or

analagous but messier equations exist for other cosmologies as well...

(Note: this is for Lambda=0 universes only...)

Recently, an elliptical galaxy was found at high redshift (z ~ 1.5) which looks to be at least 3.5 billion years old. How can this constrain cosmology? You tell me....