# Is cosmological redshift at a given time has a constant value?

On one hand, people say that various cosmic events occurred at various redshift values. For example, recombination happened at $$z=1100$$. This would imply that at any given point in time, in the history of the Universe, the cosmological redshift $$z$$ has a definite value and only changes as time passes. On the other hand, the measurement of $$z$$ values for different astrophysical sources will give different values at a given time! What is really going on?

If the value of cosmological redshift has a fixed value at a given point in time how is that value determined? For example, today and ten thousand years ago! If cosmological redshift is not fixed at a given time how can we make sense of the statement that "recombination happens at $$z=1100$$." and translate redshifts with cosmic time?

• You need to define what you mean by 'at a given time', because that term has no useful meaning as it stands. – tfb Oct 8 '18 at 11:26
• I am not certain to understand your point. Let me summarize. First, the question is meaningful within a given cosmological model. The minimum requirement is Robertson-Walker geometry. In it a cosmic time is defined, and each event in Universe's life gets a label $t$ denoting this cosmic time. Now consider a physical event, like e.g. a supernova explosion. It has a very small extension, both in space and in time, so that we can treat it as an event. And it has a cosmic time of its own. (to be continued) – Elio Fabri Oct 8 '18 at 16:20
• (end of comment) Given $t$, and assuming the scale parameter $a(t)$ as a function of $t$ is known, it can be shown that $z=1/a(t)-1$. This is how a redshift is assigned to every event in the past. Of course this does not mean that we could see that explosion just now: this requires a relation is satisfied between $z$ and the proper distance: the redshift-distance equation. – Elio Fabri Oct 8 '18 at 16:21

## 3 Answers

$$\def\ns#1#2{#1_{\rm#2}} \def\te{\ns te} \def\tr{\ns tr} \def\10#1#2{#1\cdot10^{#2}} \def\qy#1#2{#1\ {\rm#2}}$$ $$\let\Om=\Omega \def\Omm{\ns\Om m} \def\OmL{\ns\Om\Lambda}$$ I would suggest you to correct the recombination redshift to $$z=1100$$.

Generally speaking, cosmological redshift involves two times: emission time $$\te$$ and reception time $$\tr$$. Usually $$\tr$$ is present time $$t_0$$.

In Robertson-Walker geometry there is a relation betweeen cosmic time $$t$$ and scale factor $$a$$: $$a=a(t)$$. In terms of scale factor the redshift parameter $$z$$ has a simple expression: $$1+z = {a(\tr) \over a(\te)} = {1 \over a(\te)}$$ as we are usually interested in reception at present time, and $$a(t_0)=1$$ by definition.

For different emission events you will have different $$\te$$, $$a(\te)$$ and $$z$$.

So your question amounts at finding $$\te$$ given $$z$$. You have $$a(\te)=1/(1+z)$$ and the function $$a(t)$$ (or better its inverse) is needed. This requires a physical model (cosmological model) about which kinds of matter there are in the Universe, and in what proportions. In $$\Lambda$$CDM model analytical expressions can be given: $$a(t)=\left(\!{\Omm \over \OmL}\!\right)^{\!\!1/3}\!(\sinh p\,t)^{2/3}$$

where $$p=\frac32\,H_0\sqrt{\OmL}\,$$

$$H_0\,t = {2 \over 3\,\sqrt\OmL}\> \log\!\left(\!\sqrt{{\OmL \over \Omm}\,a^3} + \sqrt{{\OmL \over \Omm}\,a^3 + 1}\right)$$

$$H_0 = \qy{67.7}{km\ s^{-1} Mpc^{-1}} \qquad \Omm = 0.31 \qquad \OmL = 0.69.$$

• The question is whether redshift is different for two different galaxies. If that is so, what is the meaning of labelling cosmic events by a particular redshift? – mithusengupta123 Oct 8 '18 at 11:04
• I am not certain to understand your point. Let me summarize. First, the question is meaningful within a given cosmological model. The minimum requirement is Robertson-Walker geometry. In it a cosmic time is defined, and each event in Universe's life gets a label $t$ denoting this cosmic time. Now consider a physical event, like e.g. a supernova explosion. It has a very small extension, both in space and in time, so that we can treat it as an event. And it has a cosmic time of its own. (to be continued) – Elio Fabri Oct 8 '18 at 16:24
• (end of comment) Given $t$, and assuming the scale parameter $a(t)$ as a function of $t$ is known, it can be shown that $z=1/a(t)-1$. This is how a redshift is assigned to every event in the past. Of course this does not mean that we could see that explosion just now: this requires a relation is satisfied between $z$ and the proper distance: the redshift-distance equation. – Elio Fabri Oct 8 '18 at 16:25
• @Rob Jeffries You're right, an extra 0 slipped out. Thanks. I've edited. – Elio Fabri Oct 10 '18 at 9:46
• Thanks for your precious comments. So if two events (say, two different supernovae explosions) are measured to have the same $z$, we can say that they occurred at the same time in the past. How do we get the redshift $z$ corresponding to the time of recombination? How do I get $z=1100$ from your redshift formula $1+z=\frac{1}{a(t_e)}$? – mithusengupta123 Oct 10 '18 at 15:18

$$\let\Om=\Omega \def\ns#1#2{#1_{\rm#2}} \def\ar{\ns a{rec}} \def\tr{\ns t{rec}}\def\zr{\ns z{rec}} \def\Tr{\ns T{rec}} \def\OmL{\ns\Om\Lambda} \def\Omm{\ns\Om m} \def\qy#1#2{#1\,{\rm#2}}$$ mithusengupta123 wrote

How do we get the redshift $$z$$ corresponding to the time of recombination? How do I get $$z=1100$$ from your redshift formula $$1+z=1/a(\ns te)$$?

Here genuine cosmological reasoning is required. I can only give a rough idea. By recombination is meant the process by which free protons and electrons begin to "recombine" forming neutral Hydrogen atoms in an appreciable proportion. This depends on several parameters which vary with time. Mainly

• temperature
• density of particles involved.

It can be shown that temperature varies as $$1/a(t)$$, whereas densities go down as $$1/a^3$$. So if you know $$a(t)$$ you can find $$\tr$$ (through an equation I don't show). Once you know $$\tr$$, $$\ar=a(\tr)$$ is known too, then $$\zr$$.

As an exercise, let's try the reverse: given $$\zr$$, compute $$\tr$$. I wrote all relevant formulas in my first answer.

We start from $$\zr=1100$$. Then $$\ar={1 \over 1 + \zr} = {1 \over 1101} \simeq 0.0009.$$ I gave a (complex) formula to find $$t$$ from $$a$$:

$$H_0\,t = {2 \over 3\,\sqrt\OmL}\> \log\!\left(\!\sqrt{{\OmL \over \Omm}\,a^3} + \sqrt{{\OmL \over \Omm}\,a^3 + 1}\right)$$

$$H_0 = \qy{67.7}{km\ s^{-1} Mpc^{-1}} \qquad \Omm = 0.31 \qquad \OmL = 0.69.$$

$$\tr = 4.7\cdot10^5\ \rm years.$$ Not bad, given that a reliable computation is far more complex. See e.g.

[recombination]https://en.wikipedia.org/wiki/Recombination_(cosmology)

We can also find $$\Tr$$, the recombination temperature. This is easy, since $$T$$ scales as $$1/a$$. We know present temperature of CMBR: 2.7 K, for $$a=1$$. Then $$\Tr = \ns T{now} / \ar = 3000\,\rm K.$$

Maybe the key is to think about the fact that light emitted by different sources at different times can arrive at earth at the same time. This is because different sources are at different distances from us.

The redshift of a source is a measure of how big the universe was when the light which reaches us today was emitted by the source.

There is a relation between the age of the universe $$t$$ and the size $$a$$ the universe was at that age. Therefore, redshift is also a measure of how old the universe was when the light was emitted. Since light travels at the speed of light redshift is also a measure of how far away the source is from us: if it's been traveling for some particular time it must have covered a particular distance.

When people say "that galaxy is at a redshift $$z$$" they mean that the light from that galaxy that we receive today was emitted when the universe was a factor of $$1+z$$ smaller than its current size, or equivalently, that the light was emitted when the universe was a certain age. Two different sources can be at two different redshifts. They are at different distances from earth and so the light from both which reaches us now was emitted at two different times.

To get the connection between redshift and emission time you just need to integrate the Friedmann Equation to get scale factor (size of the universe) $$a(t)$$ as a function of the age of the universe. Then use the fact that redshift is related to the scale factor by $$1+z = a_0/a$$, where $$a_0$$ is the scale factor today, usually set equal to $$1$$.