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)=cH0∫z0dx√ΩR0(1+x)4+ΩM0(1+x)3+ΩK0(1+x)2+ΩΛ0
The Hubble-Lemaître Law:
v=H0⋅d
We want v=c now. The distance that fulfils this condition is known as current Hubble Distance, (or Hubble Radius, or Hubble Length):
dH0=cH0
Combining both, we obtain the condition:
∫z0dx√ΩR0(1+x)4+ΩM0(1+x)3+ΩK0(1+x)2+ΩΛ0=1
For ΩR0≈0ΩK0≈0ΩM0≈0.31ΩΛ0≈0.69
The condition is:
∫z0dx√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≈1.5
Best regards.
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