Greetings--this is from Ryden's(adsbygoogle = window.adsbygoogle || []).push({}); Introduction to Cosmology, question # 5.2. Let me restate the question:

Here w is the equation of state parameter for whatever the universe is made of (e.g. w = 0 for matter, 1/3 for radiation). A light source in a flat, single-component universe has a redshiftzwhen observed at a time [tex]t_0[/tex]. Show that the observed redshift changes at a rate

[tex]\frac{dz}{dt_0} = H_0(1+z) - H_0(1+z)^{\frac{3(1+w)}{2}}[/tex]

Here is my best attempt so far:

Simplify notation by introducing [tex]y=\frac{2}{3(1+w)}[/tex]

Then (eq. 5.51 in Ryder):

[tex]1+z = \left(\frac{t_0}{t_e}\right)^y[/tex]

Where [tex]t_e[/tex] is the time at which the light was emitted.

Also (eq. 5.48 in Ryder):

[tex]t_0 = \frac{y}{H_0}[/tex]

So differentiating the first equation w/rt [tex]t_0[/tex], we get:

[tex]\frac{dz}{dt_0} = yt_0^{y-1}t_e^{-y} - y t_e^{-y-1}t_0^y \frac{dt_e}{t0}[/tex]

[tex]\frac{dz}{dt_0} = H_0(1+z) - y t_e^{-1} (1+z) \frac{dt_e}{t0}[/tex]

But [tex]\frac{dt_e}{t0} = (1+z)^{-1/y}[/tex] from our equation (5.51) above. Equation (5.52) in Ryder also tell us:

[tex]t_e = \frac{y}{H_0}(1+z)^{-1/y}[/tex]

So plugging these two in:

[tex]\frac{dz}{dt_0} = H_0(1+z) - y\left(\frac{H_0}{y}(1+z)^{1/y}\right)(1+z)(1+z)^{-1/y}[/tex]

Which becomes identically zero! Where am I making my mistake?

Thanks,

Flip

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# Homework Help: Change in Redshift vs. Obs. Time

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