# Cosmological Red Shift: How Does z Change With Time?

• Magister
So if z increases by 0.259 over the course of 4 billion years, then z would be 0.379 at the present epoch.f

#### Magister

I know that the variation of cosmological red shift with the distance is given by

$$z= H_0 l (1+\frac{1}{2} (1+q_0) H_0 l)$$

Where $l$ is the luminosity distance, $H_0$ is the Hubble parameter at the corrent epoch and $q_0$ is the deceleration parameter.

I would like to know how does $z$ changes with time? (I supose it does changes with time...)

I know this formula that relates redshift and luminosity distance $d_L$:

$$d_L = \frac{1}{H_0} \left(z + \frac{1}{2} (1 - q_0) z^2 + ...\right)$$

which is valid for small z, see (8.78) in Sean Carroll's Lecture Notes on General Relativity.

The relation between redshift and look-back time for a general cosmological model is not an analytic one. To find this relation you should start with the first Friedmann equation and express the terms of matter, curvature and cosmological constant related to the critical density today. Substituting for the scaling law of each density with the scale factor you will obtain the required relation between scale factor and time, which leads immediately to the relation between redshift and time. This is a lot of work in writing everything in LaTeX, but if you have interest I can help you to go through these steps.

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I would like to know how does $z$ changes with time? (I supose it does changes with time...)

If that is what you would like to know, why don't you use this:

http://www.astro.ucla.edu/~wright/DlttCalc.html

This is an online calculator where you simply enter the LIGHT TRAVEL TIME
and it computes the REDSHIFT for you

=================
there is no simple analytical formula, so you need some program that does numerical integration
you can use Wright's to make a plot of redshift versus time, if you want, and see what the curve looks like.

This particular calculator is a new feature at Wright's website, just in the past year.
Before that he just had a calculator with would do the opposite: you put in Z and it finds the light travel time for you and a few other things.
His older calculator is here
http://www.astro.ucla.edu/~wright/CosmoCalc.html

If you didnt know Wright before, here he is
http://www.astro.ucla.edu/~wright/intro.html
http://www.astro.ucla.edu/~wright/cosmolog.htm

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$$z=\frac{R(t_0)}{R(t_1)} - 1$$

$$\frac{dz}{dt_0} = \frac{R'(t_0)-R'(t_1)}{R(t_1)}$$

Expanding R(t) I get

$$\frac{dz}{dt_0} = \frac{R''(t_0) \triangle t}{R(t_0)} + O(\triangle t^2)$$

Using

$$\triangle t=\frac{z}{H_0}$$

I get

$$\frac{dz}{dt_0} \simeq \frac{R''(t_0) z}{R(t_0) H_0}$$

$$\frac{dz}{z} \simeq - q_0 H_0 dt_0$$

Isnt this the variation of z with time? If I observe a galaxy for $\triangle t$ I will get a "z-shift" of

$$\frac{\triangle z}{z} \simeq - q_0 H_0 \triangle t_0$$

Am I wrong?
Thanks for the references!

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I do not understand how you proceed and I get confused with your use of the subindices. Let $a$ be the scale factor. Starting with:

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

You can compute:

$$\frac{dz(t)}{dt} = - \frac{a(t_0)}{a^2(t)} \dot a(t)$$

With $H = \dot a / a$, this is equal to:

$$\frac{dz(t)}{dt} = - \frac{a(t_0)}{a(t)}H(t)$$

I do not see how to progress here to find a relation between redshift and lool-back time.

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What I have done is

$$\frac{dz}{dt_0}=\frac{R'(t_0)}{R(t_1)} - \frac{R(t_0)R'(t_1)}{R(t_1)^2} \frac{dt_1}{dt_0}$$

Using

$$\frac{dt_1}{R(t_1)}=\frac{dt_0}{R(t_0)}$$

I get

$$\frac{dz}{dt_0}=\frac{R'(t_0)-R'(t_1)}{R(t_1)}$$

I am deriving z with repect to $t_0$ not to $t$.

Sorry but I cannot make any sense of this. The variable is $t$. The $t_0$ is not a variable, but a constant value for the time at which the light emitted at $t$ is observed with redshift $z$, which is the current epoch (we are talking about redshifts measured today).

But if we are making a acquisition for a period of time $t_0$ and $t_1$ will be changing.
I think I have saw this result, or similar, somewhere, but unfortunately I can remember where. I think it was in Weinberg's book...

$t_1$ can change depending on the redshift of the light source, this is what I have called $t$, but $t_0$ is fixed.

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Am I wrong?
Thanks for the references!

As per the references, you are wrong unless your math reproduces this relation between time and redshift. So here is a way to check your math: compare with this.

Code:
light travel time in billions of years       redshift z
1                                                 0.077
2                                                 0.162
3                                                 0.259
4                                                 0.369
5                                                 0.497
6                                                 0.650
7                                                 0.835
8                                                 1.069
9                                                 1.378
10                                                 1.815
11                                                 2.501
12                                                 3.809
13                                                 7.875
13.5                                              21.384

The figures I'm leaving unrounded need to be rounded off.
This is about what you should get, using the mainstream concordance model, flat LambdaCDM, with current Hubble parameter 71 and dark energy fraction 0.73

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Sorry. I believe that I didn't make my shelf clear at first.
I am not asking for the variation of z with time travel but with the time of acquisition.
Supose that we observe a comoving light source for a period of time, what is the variation of z that we will see?
I supose that if you acquire for a long enough period z will change.
Probably that would not be much pratical because z vary too slow, but...

Weird. 'the time of acquisition' thing blew me away. Sounds very ATM.

Ok, it seams that we misunderstood your question.

For the general case you should proceed in the same way that I told you before with the Friedmann equation. This will allow you also to compute the future value of the scale factor (and redshift) depending on time.

For the case of small variations of $z$ I guess you can proceed in a similar way than you have written, taking $t_0$ as a variable. But then it seams to me that $t_1$ is fixed. It is a fixed cosmological epoch when the light was emitted and the value of the scale factor at this epoch does not vary (you take scale factor = 1 for the present epoch, lower values for past and greater for future).

Code:
light travel time Gyr       redshift z
1                             0.077
2                             0.162
3                             0.259
4                             0.369
5                             0.497
6                             0.650
7                             0.835
8                             1.069
9                             1.378
10                            1.815
11                            2.501
12                            3.809
13                            7.875
13.5                         21.384

Now that I understand a little better what you are asking about, I'd like to take an example. Let's say we watch a certain z=1 object for 100 years and we want to estimate how much z should change during that time. If it is exactly z=1 as close as we can measure, today, then will it be measurably any different a hundred years from now?

AFAICS at the end of 100 years we will be observing an object which is about 50 years older. So there are these little time differences like 100 years at our end and 50 years at the other end. But the whole travel time for the light is 8 billion years. Percentagewise 50 or 100 years is very small by comparison.

Moreover z is a slowly changing function of travel time. And the relation between z and travel time is itself slowly changing-----hellfire has discussed this, i.e. how you run the Friedmann equation model out into the future.

I don't think I have anything quantitative to say that hasn't already been mentioned, but just looking at the numbers my impression is that you'd have to observe something quite a bit longer than 100 years in order to have a detectable change in z.

Unless perhaps that thing you are observing is the CMB at a particular point in the sky. That might eventually become interesting to observe for an extended period to see how temperature changed and I speculate that the observer might then conceivably wish to compensate for changing z. With light from early epochs the z is not such a slowly changing function of travel time. And with CMB one is not looking at a fixed object, every year one sees deeper and deeper because the time of last scattering was longer and longer ago.

(The figures are for the mainstream concordance model, flat LambdaCDM, with current Hubble parameter 71 and dark energy fraction 0.73.)

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