Note the increase in wavelength is NOT a forced stretching. Energy is NOT being put into the EMR. It's actually being taken out. For an explication, see topic Cosmological Redshift and Energy Conservation below.
The cosmic scale factor for cosmic present t_0 (equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018)) is conventionally set to 1 and given symbol a_0. Note subscript 0 indicates cosmic present by usual convention. Thus, a(t=t_0) = a_0 = 1.
Cosmic time is measured from Big Bang which happened at lookback time the age of the observable universe = 13.797(23) Gyr (Planck 2018) (see Planck 2018: Age of the observable universe = 13.797(23) Gyr).
Note from our understanding of general relativity (GR), the scaling up with cosmic scale factor a(t) is a literal growth of intergalactic space.
From our understanding of the cosmic scale factor (which we further explicate below in section The Cosmic Scale Factor), the wavelength of EMR signal (or, in the quantum mechanical perspective, a photon) propagating through intergalactic space just scales with a(t): i.e.,
λ_0 = [a_0/a(t)]λ ,where λ is the initial wavelength of emission, t is the cosmic time of emission, and λ_0 is the wavelength at cosmic present t_0 (equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018)).
λ_0 - λ a_0 z = ------- = ---- - 1 . λ a(t)We see that the cosmological redshift is the relative change in wavelength from emission to observation.
a_0 ---- = z + 1 . a(t)Since cosmological redshift z is a direct observable, the relative cosmic scale factor a_0/a(t) is a direct observable. Just add 1 to z and we get it.
But we do NOT get cosmic time t.
The fact is that a(t)'s functional dependence on cosmic time t is NOT known observationally: it is a function dependent on the cosmological model. So the scaling-up formula does NOT give us t or lookback time which is t_0-t.
If we did, we'd easily be able to determine the cosmic scale factor a(t) throughout most of cosmic time and cosmology would be a lot easier.
If only galaxies had clock faces on them that told cosmic time, but they don't.
In fact, solving for cosmic scale factor a(t) from the Friedmann equation of cosmology (which is derived from general relativity along with certain assumptions) is one of the main goals of cosmology.
With the atom or molecule identified, one does know all the intrinsic line wavelengths (i.e., the λ's) and then the cosmological redshift z can easily be calculated from the above formula z = (λ_0 - λ)/λ using the observed wavelengths (i.e., the λ_0's).
They can only be known in general from a cosmological model fitted to the observable universe.
Since circa 1998 to circa 2020????, the standard model of cosmology (SMC) has been the Λ-CDM model which fits all observations of the observable universe pretty well. It may need revision or replacement in the near future, but even so its predictions for physical distance and lookback time will probably still be pretty accurate.
Thus, astronomers would usually say galaxy X is at redshift z_X.
They are DIFFERENT, though closely related, phenomena and their general formulae are DIFFERENT.
Doppler shift is the shift in wavelength due to relative velocity in one inertial frame or between two inertial frames in relative acceleration. The situation is actually a bit complex in general for EMR: see Wikipedia: relativistic Doppler effect.
Cosmological redshift, on the other hand is a shift wavelength due to propagation through a continuum of inertial frames that are separating due to the expansion of the universe.
Note the cosmological redshift can be derived from the Doppler shift (see Li-38), but in the opinion of yours truly that does NOT make it Doppler shift in any simple sense---to repeat, the general formulae are DIFFERENT.
However, many people are less finicky and call the cosmological redshift a Doppler shift.
The peculiar velocities give real Doppler shifts that have to be "subtracted off" the actually observed z to get the cosmological redshift z.
For large cosmological redshifts, the Doppler shifts are negligible, but when you get very close to the Milky Way, the Doppler shifts dominate and a correction is needed to get the cosmological redshift z.
Inside the Milky Way, it is usually pointless to consider the cosmological redshifts at all.
In intro astronomy courses, we do NOT worry about the complication of Doppler shift correction, but it is important.
z = Δλ/λ = v/c << 1 ,where Δ = (λ_0-λ), λ is either of λ_0 or λ since they are the same to 1st-order, and v is relative velocity along the line of sight) between emitter and observer.
Solving for it from the Friedmann equation of cosmology (which is derived from general relativity along with certain assumptions) is one of the main goals of cosmology.
Image 4 below presents the cosmic scale factor a(t) for fiducial cosmological models.
But where does that lost energy go. We believe in energy conservation (as exemplified by an ideal Newton's cradle), right?
This ideal Newton's cradle exhibits exact conservation of mechanical energy since NO mechanical energy is dissipated to waste heat by friction, air drag, or any other process. Mechanical energy is the sum of kinetic energy and gravitational potential energy in this case.
It just vanishes (see Car-120). It's somewhat distressing that energy conservation in the ordinary sense does NOT hold necessarily according to general relativity. But all is NOT lost. General relativity gives us a generalization of ordinary energy conservation: the general-relativity energy-momentum conservation equation (see also Car-120). Which equation we will NOT go into here.
We just have to note that in certain contexts (which do NOT occur in everyday life NOR in most of astrophysics) ordinary energy conservation does NOT hold.
The cosmological redshift is one of those cases.
Another case is gravitational waves. Gravitational waves do convey energy and momentum across spacetime, but there is NO general proof that they conserve these quantities as they propagate across spacetime though there is a special case proof for energy conservation and momentum conservation which gives what is called the Bondi-Sachs energy-momentum conservation law (see Roger Penrose, The Road to Reality, 2004, p. 467--468).
But what about gravitational waves in an expanding universe. Surely, their wavelength increases as they propagate, but do they lose energy thereby? We need a great expert to tell us.
Another interesting point about gravitational waves is that they make NO contribution to the energy-momentum tensor T_ij of the Einstein field equations. So though they convey energy and momentum across spacetime, there is NO way to assign so much energy density to any point in the gravitational waves. Somehow the energy is coded into the gravitational waves in a non-local way (see Roger Penrose, The Road to Reality, 2004, p. 467--468).