The Expanding Universe: Understanding Hubble's Law

[jbutcher]

As I understand things, the following are true:

• 
  The further away in space we observe, the further back in time we are looking

• The further away in space we look, the faster galaxies are receding from us

If these are true, doesn't it indicate that the rate of expansion of the universe is in fact decreasing? (ie expansion was faster further back in time).

From what I understand, this is contrary to current thinking about an increase in expansion rate and dark energy.

 

[tasman2010]

I think i get where you are going with this but please do elaborate...

 

[nicksauce]

No. Any expanding universe will meet these criteria. It says nothing about the rate of expansion.

To see this, we need just examine the elementary derivation of Hubble's law. Of course, it is fuzzy what we mean by "the distance to a galaxy" in cosmology, but for close enough objects we can use the proper distance... that is the distant that we would measure to the galaxy with a ruler if we could freeze time. The proper distance is $d_p = ar$ where a is the scale factor of the universe at that time, and r is the comoving coordinate of the galaxy. The rate at which the galaxy appears to be moving away is then
$$v = \frac{d}{dt}d_p$$

$$v = \dot{a}r$$

$$v = \frac{\dot{a}}{a}d_p$$

$$v = H_0d_p$$

(H is measured to be about 72 km/s/Mpc). The point is that we have shown that the apparent recession speed of galaxies is proportional to the distance from them. We have made no connection to the acceleration or deceleration of the expansion to derive this fact.

Disclaimer: What I have said here is only "valid" for objects that are close enough to it. Corrections due to the accelerated expansion are visible for further away objects. That is why high redshift supernovae were needed to infer the accelerated expansion.