# Redshift (Ryden 5.2), confusing step.

1. Mar 4, 2012

### Clever-Name

1. The problem statement, all variables and given/known data
A light source in a flat, single-component universe has a redshift z when observed at a time $t_{0}$. Show that the observed redshift changes at a rate

$$\frac{dz}{dt_{0}} = H_{0}(1+z) - H_{0}(1+z)^{3(1+w)/2}$$

2. Relevant equations
$$H_{0} = (\frac{\dot{a}}{a})|_{t = t_{0}} = \frac{2}{3(1+w)t_{0}}$$

$$(1+z) = \frac{a_{t_{0}}}{a_{t_e}} = \left( \frac{t_{0}}{t_{e}}\right )^{2/3(1+w)}$$

$$\frac{dz}{dt_{0}} = \frac{dz}{da_{0}}\frac{da_{0}}{dt_{0}} + \frac{dz}{da_{e}}\frac{da_{e}}{dt_{e}}\frac{dt_{e}}{dt_{0}}$$

$$t_{e} = \frac{t_0}{(1+z)^{3(1+w)/2}}$$

w is the component index (matter, radiation, lambda), not sure what its formal name is.
3. The attempt at a solution

Everything goes ok following my work below until the last step:

$$\frac{dz}{dt_{0}} = \frac{2}{3(1+w)t_{0}}\left( \frac{t_{0}}{t_{e}} \right)^{2/3(1+w)} - \frac{2}{3(1+w)t_{e}}\left( \frac{t_{0}}{t_{e}} \right)^{2/3(1+w)}\frac{dt_{e}}{dt_{0}}$$

$$= H_{0}(1+z) - \frac{2}{3(1+w)t_{e}}(1+z)\frac{dt_{e}}{dt_{0}}$$

Here is where I am stuck. Using the definition for $t_{e}$, as given in the chapter, we get:

$$\frac{dt_{e}}{dt_{0}} = (1+z)^{-3(1+w)/2} = \frac{t_e}{t_{0}}$$

But when subbing this in we end up with 0!

I have no idea how to proceed.

Last edited: Mar 4, 2012