<p>From Friedmann Equation, distance as a function of redshift is:</p>

<p><span class="math-container">$$d(z)=\frac{c}{H_0}\int_0^z \frac{dx}{\sqrt{\Omega_{R_0}(1+x)^4+\Omega_{M_0}(1+x)^3+\Omega_{K_0}(1+x)^2+\Omega_{\Lambda_0}}}$$</span></p>

<p>The Hubble-Lemaître Law:</p>

<p><span class="math-container">$$v=H_0 \cdot d$$</span></p>

<p>We want <span class="math-container">$\boxed{v=c}$</span> The distance that fulfils this condition is known as Hubble Distance, (or Hubble Radius, or Hubble Length):</p>

<p><span class="math-container">$$d_H=\frac{c}{H_0}$$</span></p>

<p>Combining both, we obtain the condition:</p>

<p><span class="math-container">$$\int_0^z \frac{dx}{\sqrt{\Omega_{R_0}(1+x)^4+\Omega_{M_0}(1+x)^3+\Omega_{K_0}(1+x)^2+\Omega_{\Lambda_0}}}=1$$</span></p>

<p>For <span class="math-container">$\Omega_{R_0}\approx 0 \quad \Omega_{K_0}\approx 0 \quad \Omega_{M_0}\approx 0.31 \quad \Omega_{\Lambda_0}\approx 0.69$</span></p>

<p>The condition is:</p>

<p><span class="math-container">$$\int_0^z \frac{dx}{\sqrt{0.31(1+x)^3+0.69}}=1$$</span></p>

<p>Searching by trial and error, we find that the value of redshift that fulfils the condition is: <span class="math-container">$$z=1.474 \approx 1.5$$</span></p>

<p>I hope I am not breaking any rules by repeating here the solution I wrote on <a href="https://physics.stackexchange.com/questions/518543/at-what-cosmological-redshift-z-does-the-recession-speed-equal-the-speed-of-l">Physics StackExchange</a>
I did it because the creator of the thread asked me to do it, there.</p>

<p>Best regards.</p>

CC BY-SA 4.0
2019-12-10T10:19:22.817Z
Thank you! I am sure no rules are broken by trying to help the community :)
Ritesh Singh
CC BY-SA 4.0
2019-12-10T08:04:01.697Z
2019-12-10T08:16:00.913Z
2019-12-10T08:16:00.913Z
Albert
<p>At what cosmological redshift <span class="math-container">$z$</span>, does the recession speed equal the speed of light?</p>

<p>What equations are used to calculate this number (since at large redshifts, <span class="math-container">$z=v/c$</span> won't apply)?</p>

<p>[The interested reader may refer to answers on <a href="https://physics.stackexchange.com/questions/518543/at-what-cosmological-redshift-z-does-recession-speed-equal-the-speed-of-light">Physics SE</a>.]</p>

CC BY-SA 4.0
2019-12-07T23:41:44.600Z
@uhoh Thanks for the link. However, the equations for relativistic redshift do not apply to cosmological redshift, where $v \ge c$ is quite well known and commonly observed.
Ritesh Singh
CC BY-SA 4.0
2019-12-07T23:45:19.340Z
Oops, I missed that, thanks!
uhoh
CC BY-SA 4.0
2019-12-09T09:30:44.650Z
Duplicate at Physics SE and it has an answer there https://physics.stackexchange.com/questions/518543/at-what-cosmological-redshift-z-does-recession-speed-equal-the-speed-of-light
Alchimista
CC BY-SA 4.0
2019-12-09T09:34:16.743Z
@Alchimista The answer clearly doesn't show how is the number $z \sim 1.5$ arrived at.
Ritesh Singh
CC BY-SA 4.0
2019-12-09T09:37:05.037Z
It is in there. You can't further simplify things. You have equation and various plots depending on parameters.
Alchimista
CC BY-SA 4.0
2019-12-09T23:36:48.653Z
I'm voting to close this question as off-topic because the OP asked the [same question on Physics.SE](https://physics.stackexchange.com/questions/518543/at-what-cosmological-redshift-z-does-recession-speed-equal-the-speed-of-light) where it has been answered. Cross-posting the same question on different SE sites is discouraged.
Chappo Hasn't Forgotten
CC BY-SA 4.0
2019-12-10T03:14:11.003Z
@ChappoSaysReinstateMonica Even when one of the questions doesn't get an answer in 60 hours?
Ritesh Singh
CC BY-SA 4.0
2019-12-12T00:59:33.347Z
Ritesh, please take the [Tour] of our site and read the guidance available on the [Help] page. If you have multiple questions, you should ask them separately: too broad a post is grounds for closure. In any case, if you're unsatisfied with the answer on [Physics.se], why did you accept it?
Chappo Hasn't Forgotten
CC BY-SA 4.0
2019-12-14T04:42:33.527Z
@ChappoSaysReinstateMonica I never said I was unsatisfied with the answer on Physics SE. I asked this question here first and when it didn't an answer in 2 days I asked it there.
Ritesh Singh
<expansion><redshift><hubble-constant>
At what cosmological redshift $z$, does the recession speed equal the speed of light? How is it calculated?
Ritesh Singh
# At what cosmological redshift $z$, does the recession speed equal the speed of light? How is it calculated?

(Asked by: username_0 on 2019-12-07)

At what cosmological redshift $z$, does the recession speed equal the speed of light?

What equations are used to calculate this number (since at large redshifts, $z=v/c$ won't apply)?

[The interested reader may refer to answers on [Physics SE](https://physics.stackexchange.com/questions/518543/at-what-cosmological-redshift-z-does-recession-speed-equal-the-speed-of-light).]

**Tags:** expansion, redshift, hubble-constant
> **Comments:**
> **username_0**: @uhoh Thanks for the link. However, the equations for relativistic redshift do not apply to cosmological redshift, where $v \ge c$ is quite well known and commonly observed.  
> **username_1**: Oops, I missed that, thanks!  
> **username_2**: Duplicate at Physics SE and it has an answer there https://physics.stackexchange.com/questions/518543/at-what-cosmological-redshift-z-does-recession-speed-equal-the-speed-of-light  
> **username_0**: @Alchimista The answer clearly doesn't show how is the number $z \sim 1.5$ arrived at.  
> **username_2**: It is in there. You can't further simplify things. You have equation and various plots depending on parameters.  
> **username_3**: I'm voting to close this question as off-topic because the OP asked the [same question on Physics.SE](https://physics.stackexchange.com/questions/518543/at-what-cosmological-redshift-z-does-recession-speed-equal-the-speed-of-light) where it has been answered. Cross-posting the same question on different SE sites is discouraged.  
> **username_0**: @ChappoSaysReinstateMonica Even when one of the questions doesn't get an answer in 60 hours?  
> **username_3**: Ritesh, please take the [Tour] of our site and read the guidance available on the [Help] page. If you have multiple questions, you should ask them separately: too broad a post is grounds for closure. In any case, if you're unsatisfied with the answer on [Physics.se], why did you accept it?  
> **username_0**: @ChappoSaysReinstateMonica I never said I was unsatisfied with the answer on Physics SE. I asked this question here first and when it didn't an answer in 2 days I asked it there.  

---

## 1 answers

### Answer by username_4 on 2019-12-10. Score: 2 | Accepted answer

From Friedmann Equation, distance as a function of redshift is:

$$d(z)=\frac{c}{H\_0}\int\_0^z \frac{dx}{\sqrt{\Omega\_{R\_0}(1+x)^4+\Omega\_{M\_0}(1+x)^3+\Omega\_{K\_0}(1+x)^2+\Omega\_{\Lambda\_0}}}$$

The Hubble-Lemaître Law:

$$v=H\_0 \cdot d$$

We want $\boxed{v=c}$ The distance that fulfils this condition is known as Hubble Distance, (or Hubble Radius, or Hubble Length):

$$d\_H=\frac{c}{H\_0}$$

Combining both, we obtain the condition:

$$\int\_0^z \frac{dx}{\sqrt{\Omega\_{R\_0}(1+x)^4+\Omega\_{M\_0}(1+x)^3+\Omega\_{K\_0}(1+x)^2+\Omega\_{\Lambda\_0}}}=1$$

For $\Omega\_{R\_0}\approx 0 \quad \Omega\_{K\_0}\approx 0 \quad \Omega\_{M\_0}\approx 0.31 \quad \Omega\_{\Lambda\_0}\approx 0.69$

The condition is:

$$\int\_0^z \frac{dx}{\sqrt{0.31(1+x)^3+0.69}}=1$$

Searching by trial and error, we find that the value of redshift that fulfils the condition is: $$z=1.474 \approx 1.5$$

I hope I am not breaking any rules by repeating here the solution I wrote on [Physics StackExchange](https://physics.stackexchange.com/questions/518543/at-what-cosmological-redshift-z-does-the-recession-speed-equal-the-speed-of-l)
I did it because the creator of the thread asked me to do it, there.

Best regards.

> **Comments:**
> **username_0**: Thank you! I am sure no rules are broken by trying to help the community :)  

---

