I’ve made several efforts to provide a simple description of how the cosmological redshift works. Each has been closer to the mark than its predecessor, and now I think I’ve got it nailed. The premise underlying the redshift is simply that a photon must travel at a local velocity of c through each and every infinitesimal local inertial frame along its worldline from the distant receding emitter to the stationary observer. Accumulated Doppler shift Each local frame can be thought of as containing its own fundamental observer moving exactly with the local Hubble flow. Since each such local frame has a different Hubble velocity HD relative to the ultimate observer, the photon in effect must change its coordinate velocity (relative to the ultimate observer) at every local frame crossing. Each such local frame crossing results in an infinitesimal Doppler shift (relative to the ultimate observer’s frame), and the accumulation of those Doppler shifts over the entire worldline yields the total cosmological redshift. As Peacock says at p. 87 of his textbook: "One way of looking at this issue is to take the rigid point of view that 1+z tells us nothing more than how much the universe has expanded since the emission of the photons we now receive. Perhaps more illuminating, however, is to realize that, although the redshift cannot be thought of as a global Doppler shift, it is correct to think of the effect as an accumulation of infinitesimal Doppler shifts caused by photons passing between fundamental observers separated by small distances..." One way to understand this accumulating Doppler shift is to think of it as an accumulation of physical stretching of the proper distance between successive wave crests of traveling light. The first wave crest always has an inward velocity (toward the ‘stationary’ observer) which is faster than the second wave crest’s velocity, because the first wave crest's proper distance from the observer is smaller than the second wave crest's. Therefore the Hubble velocity HD (relative to both the emitter and the origin) that the first wave crest must match at a local velocity of c is always is contemporaneously greater (less negative) than the HD the second wave crest is required to match. It's like a line of evenly-spaced joggers following each other at a constant peculiar velocity over a series of moving sidewalks which are moving in the opposite direction the joggers are running -- with each successive sidewalk having a lower 'negative' velocity than the prior one. The line of joggers will progressively stretch apart. This is true even if the speed of every such sidewalk is simultaneously reduced over time by the same proportion. Note that the outcome does not depend on any paradigm of ‘space itself’ stretching (or a stretching hypersphere.) Even if ‘space itself’ does not expand, two successive wave crests will progressively separate due simply to the different contemporaneous local Hubble velocities HD each of them is required to match with a local velocity of c. It’s like a long train with stretchable couplers between cars, where the engine is running on a part of the track with a faster speed limit than where the caboose is. Note that the radial distance between photons in a discrete pulse of light increases in the same proportion as their wavelength, and for the same reason. Another way to think of this Doppler shift is as an accumulation of losses of locally-measured momentum by the photon as it is required to adjust its local velocity c to match progressively higher (less negative) Hubble velocities HD as it approaches the observer. Peacock and Peebles both comment that it is appropriate to think of the cosmological redshift as a progressive loss of momentum by the photon. The accumulated Doppler shift equation The equation for the accumulated Doppler shifts is the multiplicative series (1+H1dt)(1+H2dt)...(1+Htdt). This equation is equivalent to Peacock’s equation 3.67. In units of c=1, the light travel time dt is an easier to use substitute for the light travel distance dr, since dt=dr. Note that Ht = da/(atdt), and (1+da/a1)(1+da/a2)…(1_da/at) = 1/at, because z = z/at-1. So the accumulated Doppler shift yields exactly the same result as the standard cosmological redshift formula. This equation for the Doppler shift does not include any element of SR time dilation. This is because no time dilation occurs as between fundamental comovers in the FRW metric. To illustrate this point, consider the empty Milne model. In that case, SR Minkowski coordinates are converted to FRW coordinates by applying a Lorentz length expansion gamma factor 1/[tex]\gamma[/tex] (see Peacock p. 88), which offsets the SR time dilation gamma factor [tex]\gamma[/tex] and yields a constant cosmological time shared by all fundamental comovers. So (1+Htdt) can be thought of as classical Doppler shift with a stationary observer, (1+v/c), or if that offends relativistic sensibilities, it can be thought of as a relativistic Doppler shift with the time dilation element eliminated by the transformation to FRW coordinates. The role of gravity in the cosmological redshift In FRW coordinates, gravity plays only an indirect role in the cosmological redshift: it acts to reduce the Hubble rate Ht over time. As the Hubble rate progressively diminishes, the rate of accumulation of incremental Doppler shift diminishes in the same proportion, because the difference between the Hubble velocities HD that successive wave crests must contemporaneously match (each at a local velocity of c) decreases. This temporal change in the Hubble rate is already incorporated in the accumulated Doppler shift equation stated above, so no additional correction for gravity is required or allowed. It is popular to think of gravity as applying a blueshift factor to photons approaching an observer, but applying that as a separate factor would result in an incorrect calculation. Returning to the idea of the distance between wave crests increasing due to their differential local Hubble velocities HD, it is obvious that there is no place for reduced stretching (i.e. blueshift) between the wave crests due to gravity. Artificially inserting such a gravitational stretching reduction factor would make it impossible for both successive wave crests to maintain their local velocities at c (unless an additional offsetting Doppler shift is also inserted). Or if one thinks of the redshift as a loss of local momentum, that momentum loss is already fully accounted for by the accumulated Doppler shift alone, without any separate gravitational factor. The tethered galaxy exercise shows that an inward gravitational force vector exists, which causes the untethered galaxy to move inward (if Lambda=0) toward the observer. However, while this is true for non-relativistic particles, it is not true in the case of relativistic photons. The velocity of a photon cannot be accelerated inward because its local velocity is constrained to remain at c regardless of any acceleration force. And as described above, the effect of gravity is already fully incorporated into the accumulated Doppler shift formula as a reduction in the Hubble rate over time. In other words, gravity causes exactly the same increase in the photon's inward momentum as it causes in the inward (less outward) momentum of the fundamental observers the photon passes along its worldline. So gravity causes no 'additional' blueshift with respect to those freefalling fundamental observers, in FRW coordinates. (As Bunn & Hogg have written, gravity can be viewed as creating its own blueshift if certain non-FRW coordinates are used). It helps to keep in mind that gravitational time dilation occurs only as a result of a clock rate differential between an emitter and observer, and not literally as a result of the photon ‘gaining energy’ from the gravitational ‘pull’. Since the clocks of all FRW fundamental observers run at the same rate, there is no opportunity for gravitational time dilation for a photon moving between them.