Saturday, 4 October 2014

cosmology - At what cosmological redshift z, does the recession speed equal the speed of light? How is it calculated?


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)=cH0z0dxΩR0(1+x)4+ΩM0(1+x)3+ΩK0(1+x)2+ΩΛ0


The Hubble-Lemaître Law:


v=H0d


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:


z0dxΩR0(1+x)4+ΩM0(1+x)3+ΩK0(1+x)2+ΩΛ0=1


For ΩR00ΩK00ΩM00.31ΩΛ00.69


The condition is:


z0dx0.31(1+x)3+0.69=1


Searching by trial and error, we find that the value of redshift that fulfils the condition is z=1.4741.5


Best regards.


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