Light reaching us from a distant galaxy was emitted when the scale factor was (with today)

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Astrophysics & Cosmology · Einstein's field equations

The Geometry of Gravity, Ch FRW cosmology as an exact solution

Problem

Light reaching us from a distant galaxy was emitted when the scale factor was

a_{\rm em}=0.5

(with

a(t_0)=1

today), so the cosmological redshift is fixed by

1+z = a(t_0)/a(t_{\rm em})

. Someone argues instead: "At the moment of emission the galaxy's proper distance was

d=3000

Mpc, so by Hubble's law it was receding at

v=H_0 d

. With

H_0=67.7

km s

^{-1}

Mpc

^{-1}

, that gives

v=2.03\times10^5

km s

^{-1}=0.677c

. Since this is the recession velocity, the redshift follows from the special-relativistic Doppler formula

1+z=\sqrt{\dfrac{1+\beta}{1-\beta}}

with

\beta=v/c

." ((a)) Evaluate this proposed formula for

z

. ((b)) Evaluate the correct redshift from

a_{\rm em}=0.5

. ((c)) Explain what is wrong with the reasoning quoted above.