<p>Well during those periods we can assume universe is matter dominated. Thus, we can write <span class="math-container">$a(t) \propto t^{2/3}$</span>. By also using <span class="math-container">$1+z = a^{-1}$</span>, we can write.</p>
<p><span class="math-container">$$\frac{1+z_{\rm CMB}}{1 + z_{\rm HD1}} = \frac{a_{\rm HD1}}{a_{\rm CMB}}$$</span></p>
<p>Thus,</p>
<p><span class="math-container">$$\frac{1+z_{\rm CMB}}{1 + z_{\rm HD1}} = \frac{t^{2/3}_{\rm HD1}}{t^{2/3}_{\rm CMB}}$$</span></p>
<p>By putting your values we obtain</p>
<p><span class="math-container">$$\frac{1+z_{\rm CMB}}{1 + z_{\rm HD1}} = \frac{(3.3 \times 10^8)^{2/3}_{\rm HD1}}{(3.8 \times 10^5)^{2/3}_{\rm CMB}} = 91.02$$</span></p>
<p><span class="math-container">$$\frac{1+z_{\rm CMB}}{1 + z_{\rm HD1}} = \frac{1101}{14} = 78.64$$</span></p>
<p>Since we did not account DE and Radiation we didnt obtain the exact ratio but as you can see two values are really close.</p>
<blockquote>
<p>Why has the redshift decreased by so much in only a few hundred million years? As far as I'm aware, the universe's rate of expansion has undergone only relatively minor changes since the inflationary epoch, so why has the CMB been stretched so much?</p>
</blockquote>
<p>It's due to fact that <span class="math-container">$z \propto t^{-2/3}$</span> (for Matter dominated universe).</p>
<p><strong>Edit w.r.t comment.</strong></p>
<p>There's more easy and subtle answer that I can give you. As I have said earlier before we use <span class="math-container">$$1+z \propto a^{-1}$$</span> relation in cosmology.</p>
<p>When we set <span class="math-container">$t = 0$</span> to the Big Bang, the scale factor and the redshift is set to be <span class="math-container">$a=0$</span> and <span class="math-container">$z=\infty$</span> respectively. Meanwhile <span class="math-container">$t=t_0$</span> (current time) corresponds to <span class="math-container">$a=1$</span> and <span class="math-container">$z=0$</span>.</p>
<p>So, the universe has an age about <span class="math-container">$13.8 \times 10^9\rm{y}$</span>. Thus the we are scaling <span class="math-container">$t=(13.8 \times 10^9\rm{y}, 0)$</span> to <span class="math-container">$z=(0, \infty)$</span>.</p>
<p>From this perspective, you can see that as you go in the past the change in the <span class="math-container">$z$</span> will increase dramatically w.r.t. current change. For instance consider the change in <span class="math-container">$z$</span> due to <span class="math-container">$\Delta t = 1\rm{y}$</span>.
You'll see that as you approach the big bang, due to this scaling relation, the change in <span class="math-container">$z$</span> will increase dramatically.</p>

CC BY-SA 4.0
2022-05-23T17:36:14.563Z
Thanks for your answer! It seems like I was a bit confused about what the term "rate of expansion" is really referring to. The intuitive conception of this term is it's just the rate of change of z w.r.t; by that conception, the universe really was expanding a lot faster in the recombination epoch than it is now. However, it seems like the formal definition of that term is the rate of change of the scale factor w.r.t. time, and using this definition, the difference in speed of expansion is a lot smaller. I think the root of my confusion is the fact that the formal definition isn't intuitive.
Max
CC BY-SA 4.0
2022-05-23T19:00:44.430Z
@Max see my edit
seVenVo1d
CC BY-SA 4.0
2022-05-23T15:12:55.807Z
2022-05-24T17:40:48.107Z
2022-05-24T17:40:48.107Z
seVenVo1d
<p>It is pretty well-established that the CMB was originally emitted around 380,000 years after the Big Bang, at a redshift of ~1100.</p>
<p>The most distant known object is HD1, the light from which was emitted 330,000,000 years after the Big Bang, at a redshift of only around 13.</p>
<p>Why has the redshift decreased by so much in only a few hundred million years? As far as I'm aware, the universe's rate of expansion has undergone only relatively minor changes since the inflationary epoch, so why has the CMB been stretched so much?</p>

CC BY-SA 4.0
2022-05-24T10:14:42.887Z
Minor quibble that doesn't change the fact that it's a great question: HD1 isn't spectroscopically confirmed, and could thus easily be a lower-redshift interloper. The most distant confirmed object is still [GN-z11](https://en.wikipedia.org/wiki/GN-z11) at a redshift of 11.1 (meaning that its light was emitted ~100 Myr later).
pela
<cosmology><expansion><redshift><early-universe><cosmic-microwave-background>
Why is the CMB's redshift so high?
Max
# Why is the CMB's redshift so high?

(Asked by: username_0 on 2022-05-23)

It is pretty well-established that the CMB was originally emitted around 380,000 years after the Big Bang, at a redshift of ~1100.

The most distant known object is HD1, the light from which was emitted 330,000,000 years after the Big Bang, at a redshift of only around 13.

Why has the redshift decreased by so much in only a few hundred million years? As far as I'm aware, the universe's rate of expansion has undergone only relatively minor changes since the inflationary epoch, so why has the CMB been stretched so much?

**Tags:** cosmology, expansion, redshift, early-universe, cosmic-microwave-background
> **Comments:**
> **username_1**: Minor quibble that doesn't change the fact that it's a great question: HD1 isn't spectroscopically confirmed, and could thus easily be a lower-redshift interloper. The most distant confirmed object is still [GN-z11](https://en.wikipedia.org/wiki/GN-z11) at a redshift of 11.1 (meaning that its light was emitted ~100 Myr later).  

---

## 1 answers

### Answer by username_2 on 2022-05-23. Score: 6 | Accepted answer

Well during those periods we can assume universe is matter dominated. Thus, we can write $a(t) \propto t^{2/3}$. By also using $1+z = a^{-1}$, we can write.

$$\frac{1+z\_{\rm CMB}}{1 + z\_{\rm HD1}} = \frac{a\_{\rm HD1}}{a\_{\rm CMB}}$$

Thus,

$$\frac{1+z\_{\rm CMB}}{1 + z\_{\rm HD1}} = \frac{t^{2/3}\_{\rm HD1}}{t^{2/3}\_{\rm CMB}}$$

By putting your values we obtain

$$\frac{1+z\_{\rm CMB}}{1 + z\_{\rm HD1}} = \frac{(3.3 \times 10^8)^{2/3}\_{\rm HD1}}{(3.8 \times 10^5)^{2/3}\_{\rm CMB}} = 91.02$$

$$\frac{1+z\_{\rm CMB}}{1 + z\_{\rm HD1}} = \frac{1101}{14} = 78.64$$

Since we did not account DE and Radiation we didnt obtain the exact ratio but as you can see two values are really close.

> Why has the redshift decreased by so much in only a few hundred million years? As far as I'm aware, the universe's rate of expansion has undergone only relatively minor changes since the inflationary epoch, so why has the CMB been stretched so much?

It's due to fact that $z \propto t^{-2/3}$ (for Matter dominated universe).

**Edit w.r.t comment.**

There's more easy and subtle answer that I can give you. As I have said earlier before we use $$1+z \propto a^{-1}$$ relation in cosmology.

When we set $t = 0$ to the Big Bang, the scale factor and the redshift is set to be $a=0$ and $z=\infty$ respectively. Meanwhile $t=t\_0$ (current time) corresponds to $a=1$ and $z=0$.

So, the universe has an age about $13.8 \times 10^9\rm{y}$. Thus the we are scaling $t=(13.8 \times 10^9\rm{y}, 0)$ to $z=(0, \infty)$.

From this perspective, you can see that as you go in the past the change in the $z$ will increase dramatically w.r.t. current change. For instance consider the change in $z$ due to $\Delta t = 1\rm{y}$.
You'll see that as you approach the big bang, due to this scaling relation, the change in $z$ will increase dramatically.

> **Comments:**
> **username_0**: Thanks for your answer! It seems like I was a bit confused about what the term "rate of expansion" is really referring to. The intuitive conception of this term is it's just the rate of change of z w.r.t; by that conception, the universe really was expanding a lot faster in the recombination epoch than it is now. However, it seems like the formal definition of that term is the rate of change of the scale factor w.r.t. time, and using this definition, the difference in speed of expansion is a lot smaller. I think the root of my confusion is the fact that the formal definition isn't intuitive.  
> **username_2**: @Max see my edit  

---

