Hi Jon, yea, at least we agree on something.

You wrote: "I think you're just saying that the galaxy has a peculiar velocity inward. I agree that its velocity is peculiar relative to the untethered galaxy's local Hubble flow at its own location, but it is not peculiar relative to the origin's local Hubble flow."

Careful with your terminology; peculiar velocity is always defined in terms of what we measure relative to the Hubble flow at the position of interest, never relative to us (except for the moment when the test galaxy eventually flies past us at close range).

Your: " If the peculiar velocity relative to the origin is zero, then by definition the galaxy cannot cross comoving coordinates in order to reach the origin, which is at a different comoving coordinate ."

Wrong definition of peculiar velocity, as stated above. It's only the proper velocity of the test galaxy that is initially zero. That proper velocity will change as soon as the expansion rate at the starting position changes, without any forces acting on the test galaxy. My 3.4 stands, I'm afraid. I will elaborate more in response to other statement below.

Your: " You are not distinguishing between comoving and proper motion here, so you're coming up with a mishmash (albeit a clever mishmash). "

I was not referring to comoving velocity here, just the movement of the galaxy through local space, as measured by a set of local comoving observers being passed. Only when the recession velocity exactly balances the local velocity will there be zero proper velocity. Only a (0,0) universe can maintain this condition.

Your: " If the proper velocity of a particle that starts with peculiar motion (relative to the background Hubble flow) actually increased as the Hubble rate drops (or even if the proper velocity remained constant), then peculiar velocities would not decay at exactly 1/a in comoving coordinates, the math would calculate a different result. So we know that any such notion is conclusively wrong. "

But it is only a negative proper (radial) velocity of the galaxy that increases in magnitude (getting more negative) as the Hubble rate drops. When it is going in any positive radial direction, the proper velocity decreases and peculiar velocity decays.

Your: " 3.5 In the absence of gravitational acceleration, the proper distance from the origin would never change, so this point is moot. "

I think you are dead wrong!

Your: " Tamara goes on to say (p.47): "Thus all terms in [dot R] cancel and we conclude that the expansion, [dot R] > 0 does not cause acceleration, [double dot D] > 0. Thus, the expansion does not cause the untethered galaxy to recede (or to approach) but does result in the untethered galaxy joining the Hubble flow (v_pec → 0)." You say that the expansion does cause the galaxy to approach, so clearly you disagree with Tamara. "

You are quoting (slightly) out of context from section (3-1.1), which is about "Expansion makes galaxies join the Hubble flow". This subsection only shows that any expansion, whether constant, accelerating or decelerating, makes galaxies with peculiar motion eventually join the Hubble flow. It appears that Barnes, Francis, James & Lewis (Sep 2006) dispute this analysis for the general case - espaceially the last paragraph on page 9. However, it is not a pear-reviewed paper and we will have to see how it stands up.

The strictly correct context is the in next section (3-1.2). Davis writes: " Thus the [uniform] expansion does not 'drag' the untethered galaxy away from us, even though the untethered galaxy does end up joining the Hubble flow. Only the acceleration of the expansion can result in a change in distance between us and the untethered galaxy. We have shown that the direction of that change is not always outwards. "

Clearly, change in expansion rate does the trick, as I have said before and still hold.

Have you contemplated how 'funny' a gravitational force (or gravitational acceleration) profile would be required in order to create the (1,0) curve of Davis' Fig. 3.2? The test galaxy 'almost immediately' (in cosmo-terms) acquires a proper velocity of ~ -0.05c and then essentially maintains that proper velocity for the next few hundred Gy, moving through the origin with no further appreciable acceleration or deceleration. If you can show me the math (or a reference) that produces that velocity profile from standard gravitational accelerations, I'll concede that your interpretation must be equally valid to mine. Until that time, I will consider it flawed.

As far as I can tell, the Peebles, Davis, etc. analyses and math are not based on gravitational accelerations according to the Shell Theorem. They are all based on the standard universal expansion considerations of the FLRW metric and Friedman equations.

-J