The socalled 'tethered galaxy' thought experiment has created a lot of cosmic interest in the past. It is a rather complex issue with significant pedagogical value, but it is surprisingly easily and simply handled by the cosmic balloon analogy.
Button on a Balloon
The best balloon variant of the 'tethered galaxy' thought experiment has been suggested by StandardsGuy before. I use it with a slight modification here. Pick any spot on a partially inflated cosmic balloon's surface and attach one end of a tether (string) there. Call this spot the origin of the coordinate system. At the other end of the tether, attach a button lying frictionless on the surface  the button is now 'tethered' to the origin at a distance D from it.
Increase the inflation of the balloon from radius R_{0} to R in time Δt, as in Fig. 1 (right). Assuming that the tether does not stretch with the skin of the balloon, the button will follow the red spacehyperspace vector, being pulled across the surface of the expanding balloon. Without a tether, the expanding balloon would have taken the button along the blue spacehyperspace vector and hence increased the distance from the origin at the Hubble velocity (H D). The starting angle between the blue and the red vectors represents a negative peculiar velocity (relative to the skin), although the button stays at a constant proper distance (D) from the origin  in other words, the tethered button has zero proper velocity. (These are important terms and I shall be asking questions later.)
The Curves
Now cut the tether, so that the button (galaxy) becomes 'untethered'. Wait some time and observe what happens to the proper distance of the button. We need no more than the 'decay of particle momentum' of the previous Blog
entry to predict what will happen to the button relative to the origin.
It depends on how the balloon is being blown up. For simplicity, let's first control the gas input rate so that it keeps the expansion rate (dR/dt) of the balloon constant. Starting the simulation at the present age of the cosmos, the curves in Fig. 2 (right) result. For clarity, let's take things step by step, starting with the graph title.
(i) Case (0,0,0) means zero matter, zero radiation and zero vacuum energy density. This gives a constant expansion rate (dR/dt
= constant) over all of time. Although not a realistic case, it is a
good, simple starting point for understanding the various terms and
dynamics.
(ii) The blue D_proper curve is essentially constant at a distance 0.5 Gly (note the yscale Gly/10). It means that the button's peculiar velocity (across the balloon surface) towards the origin is canceled by the balloon's expansion that is trying to carry it away from the origin. The slight apparent drop over time is just a numerical integration
error (only 1000 steps over the 500 billion year time span were used).
(iii) The V_Hubble curve drops down in a typical inverse of time fashion, because for a constant expansion rate the Hubble constant (H) changes proportional to 1/t. Think of a very large balloon that expands at the same rate (dR/dt) than a very small balloon. Since the Hubble constant is recession rate divided by distance, the large balloon will have a much smaller H.
(iv) V_proper is zero in this case, because the proper distance is constant. Proper distance and proper velocity are measures of the instantaneous distance and recession speed of an object. It is as if we use a tape measure and synchronized clocks to find the distance and separation rate between two points on the balloon's surface over a very small time interval.
(v) The green V_peculiar curve represents the (negative) velocity of the button relative to the local balloon surface. The conservation of angular momentum (discussed in the previous Blog) causes the curve to approach the time axis asymptotically. Because the expansion rate is constant in this case, V_peculiar also changes proportional to 1/t. V_peculiar + V_Hubble = V_proper = 0 in this case, so V_peculiar is a mirror image of V_Hubble around the time axis. This is not generally true for more realistic cases, though, as will become clear later.
I suggest that interested readers first 'digest' this information and ask questions as required, before we move on to slightly more complex cases. These Blog posts are necessarily compact, cryptic issues for discussion, so do not be afraid to ask questions. There is no thing like a 'dumb question'  the only 'dumb thing' is not to ask!
Due to some questions asked, here is the (presently) realistic case.
Realistic
Expansion
For the more realistic case with
present matter energy 26% and vacuum energy 74% of the critical energy density,
the balloon is blown up with an accelerated expansion (dR/dt gets larger
with time at present). In Fig. 1 above, the distance D remains constant.
Here it is to be expected that the fastergrowing expansion will drive the
button farther from the origin, despite its peculiar velocity towards the
origin. V_peculiar now decreases, while V_Hubble quickly starts
to increase.
Fig.
3 (right) shows the same curves as for Fig. 2 above, but for the realistic scenario, again
starting at the present cosmic age. What is surprising is how
"quickly" (in cosmological terms) the proper velocity (red) of the button starts to 'follow the Hubble
flow', i.e., how quickly the peculiar velocity of the button decays to
near zero.
Some
other salient points on this chart:
(i)
The V_Hubble line first dips a little, because the Hubble constant
initially decays faster with time than the away movement of the button happens.
In this scenario, the Hubble constant drops from ~ 74 km/s/Mpc and it settles
to a constant ~ 63 km/s/Mpc in another 13 billion years or so. Hence, the
Hubble constant eventually becomes a true constant.
(ii)
V_peculiar of the button (relative to the balloon skin) remains
negative, i.e., towards the origin. The increasing expansion rate does however
quickly carry the button farther from the origin. Conservation of angular
momentum relative to the center of the ever largergrowing balloon decays the
peculiar velocity to zero over time (approaching zero as time tends to infinity).
The above case untethers the galaxy at the present cosmic age. The curves become a bit more complex if the untethering is done much earlier, as shown below.
Multiple
epoch case
Here
we postulate some 'early universe astronomer' that untethered the galaxy when
the cosmos was less than a billion years old. Matter density still largely
dominated the cosmic expansion (until vacuum energy took over) and the expansion
rate first decreased and later increased. It results in the quite complex (but
very interesting) curves of Fig. 4 (right).
At
cosmic time t = 0.2 Gy after the BB, the tethered galaxy was at D =
–0.1 Gly.^{[1]} At that stage, the expansion rate was
decreasing under the dominant (99.96%) influence of matter density of the time.
The Hubble velocity at D=0.1 Gly was a whopping –0.335c and hence the
peculiar velocity of the tethered galaxy at that time was 0.335c towards
the origin. But, due to the decreasing expansion rate, the Hubble velocity
quickly diminished for that proper distance and hence when untethered, the
galaxy started to 'fall' rapidly towards the origin. It 'fell through' the
origin at some 1.5 billion years and then continued to move in the positive D direction.
The
proper velocity (red) started at zero (because the galaxy was tethered). When
untethered proper velocity increased rapidly until the galaxy passed through
the origin and then the proper velocity started to decrease – that is for as
long as matter density dominated and the expansion rate slowed down.
At around
7 Gy, the acceleration of expansion caused by the constant vacuum energy
density more or less balanced out the deceleration of expansion caused by the
decreasing matter density. During this time the expansion rate remained more or
less constant and the proper velocity of the galaxy also remained constant.
After 8 Gy age the vacuum energy started to win the 'tugofwar' and the expansion rate started
to increase (and so did the proper velocity of the galaxy). During all this, the
peculiar velocity (green) continuously decayed and will keep on doing that. The proper velocity eventually joins the Hubble velocity, as before.
Jorrie
Notes:
[1] The
tether here had to be a solid rod, not a string, because for a decreasing
expansion rate, the tether must actually have keep the galaxy from 'falling'
towards the origin. This is essentially a 'cosmic tidal force' at work.
More about that in a future Blog post.
[2] Here are some of
the equations used (for those who just cannot live without them). Actually,
it's good to have a compact set of reference equations for the curves (for my
own selfish purposes).
V_{0}
= H_{0}D_{0} (1)^{[a]}
ψ_{0}
= D_{0}/R_{0} (2)
L
= R_{0} V_{0}/√[1V_{0}^{2}/c^{2}] =
constant (3)
V_peculiar
= L/√[L^{2} + R^{2}] (4)
D
= R ψ + V_peculiar * Δt (5)
ψ
= ψ + V_peculiar Δt/R (6)
H
= H_{0} √[Ω_{m}/a^{3} + Ω_{v}]
(7)^{[b]}
da/dt
= a H (8)^{[c]}
V_Hubble
= H D (9)
V_proper
= V_Hubble + V_peculiar (10)^{[d]}
Footnotes:
[a] V_{0}
of Eq. (1) is the initial peculiar velocity of the button and is specific to
the tethered galaxy scenario.
[b] H is
the time varying Hubble constant, also sometimes denoted H(t).
[c] a
is the expansion factor, R/R_{0}.
[d] This proved to not require the relativistic addition of velocities equation, due to the way the various velocities are defined in cosmology.
J

Re: Cosmic Balloon Application IV: Tethered Galaxy