After some 'slogging', Jonmtkisco and myself almost came to an agreement that the 4D hypersphere^{[1]} and the 3D kinematic^{[2]} cosmic models are quite equivalent. Then, in post 84 of that thread (and what followed), we again left it hanging in the air, at least to some degree. I have now worked an example that shows them to give equivalent results, at least for a matteronly, flat universe (the 1,0 case, referring to 100% matter and 0% dark energy in the literature).^{[3]}
On the left are the plots for hypothetical cosmic travelers (immortal beings!) that fly far away, turn around and come home again (the black curve), almost 'twinparadox' style.^{[4]} The only differences here are the vast cosmological distances and timescales and that the speeds are not very relativistic.
At time zero, the cosmic travelers fire their superpropulsion system for a rapid, but modest Δv of 0.0234c relative to their home galaxy, which sits at the origin with zero peculiar velocity. The acceleration lasts for a time that is negligible compared to the gigayear (Gy) timescales involved and needs only modest gforces. They then coast inertially for exactly 15 Gy of cosmological time, while their proper speed gradually decreases, as shown by the red curve. This is due to the decelerating expansion of the (1,0) universe that couples to their proper velocity.
They turn around at 15 Gy by means of a negative Δv of double the original magnitude (Δv = 0.0468c). Their speed now slowly increases (getting more negative) until they arrive home after another 13 Gy, for a total round trip time of 28 Gy. Due to relativistic time dilation, their onboard clock will read some 78 years less, but that is of no importance here (and utterly negligible compared to the Gy timescales).
The decelerating expansion of the universe creates an interesting negative acceleration profile for the travelers, as indicated by the blueish curve. So far the plots used the hypersphere model with the Friedman equations for the expansion rate. A comparison plot of the acceleration curve was then done for a kinematic model, using a Newtonian shelltheorem calculation of the traveler's acceleration relative to the origin, together with the Friedman equations for density. The acceleration curves proved to be identical  the pink 3D kinematic curve sits on top of the blue curve from the 4D hypersphere model.
The result shows that for at least the matteronly universe, one is free to use the 4D hypersphere or the 3D kinematic model. This equivalence is a hot topic in cosmology today. Some authors use the matteronly hyperspherical model to advocate that "space is expanding", while others use the 3D kinematic model to advocate that "space is not expanding". If they are genuinely equivalent, it is a meaningless debate, not so? What do you think?
J
Notes:
[1] The 4D hypersphere model (also loosely termed the balloon analogy) is broadly described in this post of the previous Blog topic. Some more details are available on my website and in the eBook (http://www.relativity4engineers.com). It is effectively a five dimensional spacetime (four space and one time) and uses the FLRW solution to general relativity as a basis. One can usually ignore most of the dimensions: a simple expanding twodimensional (one space dimension and one
hyperspace dimension) wire ring with frictionless beads strung
uniformly along it does the trick very nicely.
[2] The 3D kinematic model is based on the low field limit of general relativity and strives to do all the dynamics only in the usual 4 dimensional spacetime. It also uses FLRW solution to general relativity as a basis, e.g. to calculate the changing energy density of the universe. Using the Shell Theorem for a spherical, uniform density body of frictionless gas, the acceleration of a test particle at any distance from the center can be calculated.
[3] I'm not sure if the LCDM universe (with dark energy) is also compatible with both approaches, but my guess is that it is. Some more about that in a future post (unless Jon has solved that one already).
[4] I've been prompted to these calculations by a draft paper: 'Cosmological Radar Ranging in an Expanding Universe' by Geraint F. Lewis et al. (arXiv:0805.2197v1 [astroph] 15 May 2008). I disregarded their accelerating rockets and just used an 'instantaneous Δv', which is OK due to the immense timescales involved. Modeling of the acceleration of the rocket burns just complicates the issue for no real benefit or insight.
The authors use the fact that their inbound trip takes a shorter time than the outbound trip to suggest that expanding space must be a flawed concept, e.g. they wrote: "However, it is the presence of matter that necessitates the inclusion of gravitational forces upon the motion of the rocketeers and it is this  the changing gravitational inﬂuence of matter in the universe on the rocketeers  that causes the increasing asymmetry moving down the panels in Figure 2, not that space physically expands."
The expanding 4D hypersphere seems to suggest otherwise. E.g., it sports expanding space and it gives equally valid results; in certain cases even more so. What's more, it models standard cosmology, easily calculated and easily visualized, provided one ignores some dimensions.
Some of the problems I have with the 3D kinematic model:
 It explains the fact that clocks tick at equal rates everywhere in a homogeneous universe by the relativistic time dilation of comoving particles moving through space relative to the origin of an arbitrary coordinate system.
 It runs into trouble when distances are large enough to give apparent recession velocities exceeding c.
 It may have problems incorporating the various forms of hypothetical dark energy.
AFAIK, the 4D hypersphere model does not suffer from any of these issues. (It probably has others, though...)

Re: 3D versus 4D Kinematics in Cosmology