Now for the promised derivation of the cosmological redshift from the Doppler effect by considering the continuous Doppler shift as EMR propagates across the continuum of comoving frames making up expanding universe (Li-38--39):

First in the expanding universe, the recession velocity v can be treated as an ordinary velocity v relative to the an observer (e.g., us) in the observer's comoving frame if v is sufficiently small. We assume it is sufficiently small.
Hubble's law for the expanding universe is
```
  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".
We now apply EMR 1st-order Doppler effect formula in Hubble's law to get
```
  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.
We integrate the differential equation dλ/λ = da/a to get
```
  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/a 
```
and cosmological redshift
```
  z = (λ_0 - λ)/λ = a_0/a - 1  or z+1 = a_0/a  QED.  
```
So the cosmological redshift can be derived---maybe NOT rigorously, but yours truly wouldn't know---from the EMR 1st-order Doppler effect formula plus other effects. So in a sense, the cosmological redshift can be called a Doppler effect if you really want to---but yours truly doesn't really want to.

It is remarkable fact that cosmological redshift z (which is an easily measured direct observable) gives us directly the scaling up of the expanding universe (i.e., a_0/a) since the cosmic time a light started its journey to us.

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.

File: Cosmology file: cosmological_redshift_doppler_shift_4.html.