[Cosmology] Red Shift Problem

[ajclarke]

Homework Statement

The redshift of a galaxy is measured at the present epoch t0 and again at the infnitesimal future epoch t0 + δt0. Show that the rate of change of z with t0 is given by

$$\frac{dz}{dt_{o}}=(1+z)H_{o}-H(z)$$

Homework Equations

Hint - Start From:

$$\frac{\delta z}{\delta t_{o}}=\frac{\delta z}{\delta t_{o}}+\frac{\delta z}{\delta t}\frac{dt}{dt_{o}}$$

The Attempt at a Solution

Haven't a clue tbh. Don't even know where to begin :/

 

[fzero]

How does redshift depend on the expansion parameter?

 

[ajclarke]

[PLAIN]http://img141.imageshack.us/img141/8478/screenshot20110130at114.png

 

[fzero]

That's the Friedmann equation, written in terms of the redshift using the fundamental relationship

$$z(t) = \frac{a(t_0)}{a(t)} -1.$$

This is the formula you need to derive the relationship in your OP.

 

[ajclarke]

I think I'm being dense here

$$z(t)=\frac{a(t_{0})}{a(t)}-1= \frac{H_{o}}{H(z)}-1$$

$$\frac{\delta z}{\delta t_{o}}= \frac{\delta}{\delta t_{o}}(\frac{H_{o}}{H(z)}-1)$$$$\frac{\delta}{\delta t}= \frac{\delta}{\delta t}(\frac{H_{o}}{H(z)}-1)$$

$$\frac{dt}{dt_{o}}$$

However there is no time involved in them. Ho and H(t) are just constants. I understand the principle that they are specific to time but I don't understand how to perform the differentiation to get the answer

 

[fzero]

The Hubble parameter is

$$H(t) = \frac{\dot{a}(t)}{a(t)},$$

so the RHS of your 1st line is incorrect. In your problem the time derivatives act on the scale factors. You only rewrite it in terms of $$H(z)$$ as a last step.

 

[ajclarke]

That terminology has confused me somewhat. The $$\dot{a}$$ is the scale factor at some unknown time and a alone is the scale factor at the present epoch?

Or the other way around?

I'll have a tinker. I still feel a bit lost. Maybe writing some stuff down will help me out some

Thanks =]

 

[fzero]

No, the dot means derivative. So

$$\dot{a}(t) = \frac{da(t)}{dt},$$

while

$$\dot{a}(t_0) = \frac{da(t_0)}{dt_0}.$$

 

[ajclarke]

So far I have:

$$z(t)=\frac{a(t_{o})}{a(t)}-1 = \frac{1}{a(t)}-1$$

Since a(to)=1 by definition,

Thus:

$$\frac{\delta z}{\delta t_{o}} = 0$$

$$\frac{\delta z}{\delta t}=-a(t)^-2$$

$$\frac{dt}{dt_{o}}$$

And now I am at another brick wall.

 

[fzero]

So far I have:

$$z(t)=\frac{a(t_{o})}{a(t)}-1 = \frac{1}{a(t)}-1$$

Since a(to)=1 by definition,

You shouldn't try to set $$a(t_0)=1$$ when you're going to be varying things with respect to $$t_0$$.

Thus:

$$\frac{\delta z}{\delta t_{o}} = 0$$

$$\frac{\delta z}{\delta t}=-a(t)^-2$$

$$\frac{dt}{dt_{o}}$$

And now I am at another brick wall.

To compute $$dt/dt_0$$ you might go back to the derivation of the redshift formula to see how the proper time interval depends on the scale factor.