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)?
[I asked this question in Astronomy SE earlier.]
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 $\boxed{v=c}$ now. The distance that fulfils this condition is known as current Hubble Distance, (or Hubble Radius, or Hubble Length):
$$d_{H_0}=\frac{c}{H_0}$$
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 $z=1.474 \approx 1.5$
Best regards.
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