Caption: A comparison of the Doppler effect (or shift) and the cosmological redshift for electromagnetic radiation (EMR). The observer is the Sun.
Features:
The Doppler effect is a shift in wavelength/frequency depending on the relative motion of two observers in a single inertial frame. It's treatment when the two observers are accelerated with respect to each other can be somewhat tricky (see Wikipedia: Transverse Doppler effect).
The expansion of the universe is a growth of space itself, NOT an ordinary motion.
So the cosmological redshift is NOT a Doppler effect though it is a related phenomenon. One can, in fact, derive the cosmological redshift from the Doppler effect by considering the continuous Doppler shift as EMR propagates across the continuum of local inertial frames making up expanding universe: i.e., the comoving frames of the expanding universe (Li-38--39). So there is an argument for calling the cosmological redshift a Doppler effect, but yours truly doesn't buy it.
We do the derivation from the Doppler effect below. There is also a direct general relativity derivation (Li-127--129).
Two observers at rest in their respective comoving frames observe the cosmological redshift. If they are in motion relative to their comoving frames, there are superimposed Doppler shifts.
Yours truly thinks this 1st-order agreement shows that growth of space and free-fall motion in the classical limit do become the same thing somehow in the classical limit. Yours truly could be wrong.
Δλ/λ = v/c ,where λ is initial or final wavelength (there is no 1st-order difference), Δλ is the change in wavelength, v is the line-of-sight relative velocity counting increasing/decreasing separation as positive/negative relative velocity, and vacuum light speed c = 2.99792458*10**8 m/s.
The 1st-order formula is valid in the limit that v/c << 1.
Note the 1st-order formula is NOT a differential equation and CANNOT be used for recover the full relativistic Doppler effect (see Wikipedia: Relativistic Doppler effect: Relativistic longitudinal Doppler effect).
v = Hr = [(da/dt)/a]r ,where v is recession velocity, H is the Hubble parameter, r is proper distance, "a" is the cosmic scale factor, and (da/dt) is the cosmic time derivative of "a".
Δλ/λ = v/c = Hr/c = [(da/dt)/a]r/c .We now let Δλ and r become differentials (i.e., infinitesimally small) and get
dλ/λ = [(da/dt)/a]dr/c = [(da/dt)/a]dt = da/a ,where dt = dr/c is the differential flight time of light across dr. The last equation is, in fact, a differential equation and is infinitesimally valid everywhere as light propagates across the expanding universe.
Actually, exactly why dλ/λ = da/a is a differential equation and Δλ/λ = v/c is NOT a differential equation takes an argument that still eludes yours truly.
ln(λ_0/λ) = ln(a_0/a) ,where the subscript 0 indicates the present cosmic time t_0 (i.e., right now where we live). Exponentiating the natural logarithms with base exponential e, we get
λ_0/λ = a_0/aand cosmological redshift
z = (λ_0 - λ)/λ = a_0/a - 1 or z+1 = a_0/a QED.
But what we do NOT know is when the light started its journey to us. We know a(z), but not a(t).
Alas, there is no way to determine directly the cosmic time evolution of a(t) (since we can't determine cosmic time t directly for cosmologically remote astronomical objects). If we knew a(t), we'd know a whole lot more than we do about the evolution of the observable universe since the Big Bang.