# RedShift-scale factor (FRW)

1. May 2, 2014

### ChrisVer

I checked on wikipedia:
http://en.wikipedia.org/wiki/Redshift
and I found that the scale factor is related to the red shift, in FRW model, by:
$1+z(t) = \frac{a(t_{0})}{a(t)}$
How is that derived?

Also intuitively could you check this reasoning of mine?
Intuitively I can understand this relation, since the scale factor in the past was smaller, then the wavelengths were practically more compressed, so:
$z=\frac{a(t_{0})}{a(t)}-1$ was much bigger than 0 (I don't know why it happens to be 0 for today)
while as time passes, and the scale factor is raising more than a(t0) the redshift is going to get "smaller"...in fact there will be always new "normalization" on the nominator, just to keep it falling "asymptotically" to 0... (the wavelengths are stretched)

Last edited: May 2, 2014
2. May 2, 2014

### Mordred

The scale factor is a function of time which represents the relative expansion of the universe. It relates the proper distance which can change over time.

$$d(t)=a(t)d_o$$

d(t) is the proper distance, tdo is the reference time at to, where ato=1 for the present age of the universe 13.78 billion years

these relations may help understand the form you have

$$1+Z=\frac{\lambda}{\lambda_o}$$

$$a(t)=\frac{1}{1+Z}$$ you should be able to derive the equation you have with this

edit forgot to add:particularly since they give
$$1+z=\frac{a_ (now)}{a_ (then)}$$ and how its solved from the FLRW metric and it relation to wavelength on that same page you posted,,, grr can't recall how to subscript a word within latex but you get the idea lol

Last edited: May 2, 2014
3. May 3, 2014

### phsopher

The derivation is given in the Wiki entry you linked to under the section 3.2.1.

Additionally, redshift is always calculated by comparing the past to the scale factor today because today is when we are receiving the light. It doesn't make sense to speak about redshift for larger scale factors because we don't receive light from the future.