# When was the beginning?

Let's consider the Hubble law.

To simplify our thinking, it is convenient to change the units of the Hubble constant H0. Approximately

H0 = 0.1 /Gy
where Gy = gigayears = 109 years.

Recall that the value of the Hubble constant is rather uncertain. The ``compromise'' value suggested by your book, 75 km/s/Mpc is 0.077 /Gy in these units.

### Example

Suppose that a galaxy is 200 Mly away. How fast is it going?

• v = H0 r
• v = (0.1 /Gy) x (200 Mly)
• v = 20 Mly/Gy
Suppose that a galaxy is 1.3 Gly away. How fast is it going?

• v = H0 r
• v = (0.1 /Gy) x (1.3 Gly)
• v = 0.13 Gly/Gy
[Note: this is 13% of the speed of light. If we get to distances beyond this and velocities faster than this, we should really use the Hubble Law generalized to the General Theory of Relativity. We aren't going to do that in ASTR123, but you should know that there are corrections for very big distances and for velocities comparable to the velocity of light.]

### So when did it all start?

Consider Galaxy X at a distance of 100 Mly from us.

Use H0 = 0.1 /Gy.

Then the velocity of Galaxy away from us is

v = H0 r = (0.1 /Gy) x 100 Mly = 10 Mly/Gy

Assumption: Galaxy X has always been moving away from us at this speed. With this assumption, the time since we would have been in the same place as the material in Galaxy X is

time = distance/velocity

t = r/v

That is

t = 100 Mly / (10 Mly/Gy)

or

t = 10 Gy

You will get the same number for Galaxy Y at a distance of 300 Mly from us, for Galaxy Z at a distance of 500 Mly from us, etc. (Try it.)

### Conclusion

Assuming the expansion has been at the same rate always, the expansion started 10 billion years ago. (The number is approximate because the Hubble constant is not so well known.)

Davison E. Soper, Institute of Theoretical Science, University of Oregon, Eugene OR 97403 USA soper@bovine.uoregon.edu