Problem 3. Consider a flat, single component universe. 1. For a light source at redshift z that is observed at time to, show that z changes at a rate Ho(1+2) - Ho(1+2)³(1+w)/2 dz dto (2.1) 2. For what values of w does the observed redshift increase with time? 3. Assuming the single component is matter and Ho = 68 km/s/Mpc, you observe a galaxy at z = 1. Using Equation 2.1, determine how long you will have to keep observing the galaxy in order to see its redshift change by 1 part in 106.

Problem 3. Consider a flat, single component universe. 1. For a light source at redshift z that is observed at time to, show that z changes at a rate Ho(1+2) - Ho(1+2)³(1+w)/2 dz dto (2.1) 2. For what values of w does the observed redshift increase with time? 3. Assuming the single component is matter and Ho = 68 km/s/Mpc, you observe a galaxy at z = 1. Using Equation 2.1, determine how long you will have to keep observing the galaxy in order to see its redshift change by 1 part in 106.

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Question

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Problem 3. Consider a flat, single component universe.
1. For a light source at redshift z that is observed at time to, show that z changes at a rate
dz
dto
=
= Ho(1 + 2) — Ho(1+2)³(¹+w)/2
(2.1)
2. For what values of w does the observed redshift increase with time?
3. Assuming the single component is matter and Ho = 68 km/s/Mpc, you observe a galaxy at z = 1. Using
Equation 2.1, determine how long you will have to keep observing the galaxy in order to see its redshift
change by 1 part in 106.

Expert Solution

Step 1

Given a flat single-component universe.

1.)

The Hubble parameter is defined by

$H (t) = \frac{1}{a (t)} \frac{d a (t)}{d t} . . . (i)$

And the relation between a, and z is given by

$1 + z = \frac{a (t_{0})}{a (t_{e})} . . . (i i)$

Let $d t_{0} a n d d t_{e}$ to the time of observation and the time the light was emitted.

Now taking $\frac{d z}{d t_{0}}$ we get

$\frac{d z}{d t_{0}} = \frac{d a (t_{0})}{d t_{0}} \frac{1}{a (t_{e})} - \frac{a (t_{0})}{a^{2} (t_{e})} \frac{d a (t_{e})}{d (t_{0})} \frac{d z}{d t_{0}} = \{\frac{1}{a (t_{0})} \frac{d a (t_{0})}{d t_{0}}\} \frac{a (t_{0})}{a (t_{e})} - \frac{a (t_{0})}{a (t_{e})} \{\frac{1}{a (t_{e})} \frac{d a (t_{e})}{d t_{e}}\} \frac{d t_{e}}{d t_{0}} u \sin g e q n (i) a n d e q n (i i) w e g e t \frac{d z}{d t_{0}} = H (t_{0}) (1 + z) - H (t_{e}) \frac{a (t_{0})}{a (t_{e})} \frac{d t_{e}}{d t_{0}} . . . (i i i)$