[Cosmology] Red Shift Problem

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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

[tex]\frac{dz}{dt_{o}}=(1+z)H_{o}-H(z)[/tex]

Homework Equations



Hint - Start From:

[tex]\frac{\delta z}{\delta t_{o}}=\frac{\delta z}{\delta t_{o}}+\frac{\delta z}{\delta t}\frac{dt}{dt_{o}}[/tex]

The Attempt at a Solution



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

Answers and Replies

  • #2
fzero
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How does redshift depend on the expansion parameter?
 
  • #3
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[PLAIN]http://img141.imageshack.us/img141/8478/screenshot20110130at114.png [Broken]
 
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  • #4
fzero
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That's the Friedmann equation, written in terms of the redshift using the fundamental relationship

[tex]z(t) = \frac{a(t_0)}{a(t)} -1.[/tex]

This is the formula you need to derive the relationship in your OP.
 
  • #5
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I think I'm being dense here

[tex]z(t)=\frac{a(t_{0})}{a(t)}-1= \frac{H_{o}}{H(z)}-1[/tex]

[tex]\frac{\delta z}{\delta t_{o}}= \frac{\delta}{\delta t_{o}}(\frac{H_{o}}{H(z)}-1)[/tex]


[tex]\frac{\delta}{\delta t}= \frac{\delta}{\delta t}(\frac{H_{o}}{H(z)}-1)[/tex]

[tex]\frac{dt}{dt_{o}}[/tex]

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 dont understand how to perform the differentiation to get the answer
 
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  • #6
fzero
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The Hubble parameter is

[tex]H(t) = \frac{\dot{a}(t)}{a(t)},[/tex]

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 [tex]H(z)[/tex] as a last step.
 
  • #7
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That terminology has confused me somewhat. The [tex]\dot{a}[/tex] 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 =]
 
  • #8
fzero
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No, the dot means derivative. So

[tex] \dot{a}(t) = \frac{da(t)}{dt},[/tex]

while

[tex] \dot{a}(t_0) = \frac{da(t_0)}{dt_0}.[/tex]
 
  • #9
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So far I have:

[tex]z(t)=\frac{a(t_{o})}{a(t)}-1 = \frac{1}{a(t)}-1[/tex]

Since a(to)=1 by definition,

Thus:

[tex]\frac{\delta z}{\delta t_{o}} = 0 [/tex]

[tex]\frac{\delta z}{\delta t}=-a(t)^-2[/tex]

[tex]\frac{dt}{dt_{o}} [/tex]

And now im at another brick wall.
 
  • #10
fzero
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So far I have:

[tex]z(t)=\frac{a(t_{o})}{a(t)}-1 = \frac{1}{a(t)}-1[/tex]

Since a(to)=1 by definition,

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

Thus:

[tex]\frac{\delta z}{\delta t_{o}} = 0 [/tex]

[tex]\frac{\delta z}{\delta t}=-a(t)^-2[/tex]

[tex]\frac{dt}{dt_{o}} [/tex]

And now im at another brick wall.

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

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