In this final 'application' of the cosmic balloon, the effect of expansion on 'rigid bodies' will be investigated. Two buttons are separated by proper distance D on the surface of the cosmic balloon. We tether the two buttons to each other by means of a semirigid tether and then inflate the balloon.
Gravitational tidal forces^{[1]} attempt to either compress or stretch objects in a gravitational field. It has to do with the different spacetime curvatures at different parts of an extended object. Cosmic tidal forces have similar effects on extended objects, but it has to do with the expansion dynamics and not directly with gravity.
Cosmic tidal forces
Fig. 1 (right) shows the cosmic situation  two galaxies (red buttons) on the surface of the balloon, with a tether keeping them at a constant proper distance (D) apart. To determine the cosmic tidal force (if any), we measure the force on the tether, using strain gauges or some other practical method, taking into account that the forces can be either stretching or compressing.
It is reasonable to expect that cosmic tidal forces will depend on the inflation profile of the balloon. As it happens, it feels intuitively correct (and it is in fact easy to show) that a constant inflation rate (dR/dt = constant) causes no tidal force on the tether (apart from a transient when we start the inflation). With the balloon in uniform expansion, the buttons will essentially move inertially across the surface of the balloon. This is equivalent to a phase of the actual universe, around half of the present age, when the expansion rate was essentially constant (see this post).
If we blow up the balloon with a constant gas flow rate, the expansion rate will slow down over time and free buttons would tend to move towards each other, because they would have had initial momentum towards each other. The semirigid tether will prevent that and we should measure a compression force on the tether. This is equivalent to a phase in the actual universe before 7 Gy age, when the expansion rate was decreasing under the dominant matter density.
If we control the gas flow rate so that the expansion rate of the balloon increases over time, equivalent to the now dominant vacuum energy phase, one would expect the buttons to separate. Again, the tether will prevent the separation and we should measure a stretching force in the tether. This is all rather intuitive, so let's try to quantify these forces on a cosmological scale.
Our Local Group
Rather than working with billions of light years (as usual), let's keep it 'local' and make the tether 10 million light years long. This is about the diameter of our Local Group of galaxies (Andromeda, the Milky Way, Triangulum and some 27 dwarf galaxies). It will be interesting to find the magnitude of the cosmological tidal force on this scale (ignoring the local gravitational effects at first). Fig. 2 (right) plots the accelerative tidal force exerted on the 10 million light years long tether. The timescale starts some 10 billion years ago and ends far into the future. For the relevant equations, see note [3].
For the first 7 Gy, the tidal force was negative (compressing) and then it became stretching, as expected. At the present time (~13.6 Gy), the stretching tidal force over 10 million light years has an order of magnitude (OOM) 30 femtog,^{[2]} or 300 femtoNewton per kg mass. This is small, really small. However, when compared to the gravitational acceleration at the edges of the Local Group, it may not be so negligible.
Gravitational acceleration
Let's get an idea of how much effect the cosmic expansion may have on the structure of our Local Group of galaxies. It has a total mass of ~ 1.3 × 10^{12} Sols, or M ~ 3 x 10^{42} kg^{[4]} and a radius r ~ 5 x 10^{22} meter. If we include dark matter (at ratio 6:1), it puts the mass of the Local Group at M ~ 2 x 10^{43} kg. The rough gravitational acceleration of a particle just outside of the Local Group is: a ~ 5 x 10^{13} m/s^{2}, (50 femtog).^{[5]}
If my calculations are correct,^{[6],[7]} this is of the same OOM as the 30 femtog tidal stretching force at that distance. It may mean that this very feeble tidal stretching force has some influence on the structure of things like galactic clusters  and even more so on superclusters. It probably only means slightly larger orbits for the galaxies right at the edge of clusters. Even so, never again 'sneeze' at an acceleration of a few tens of femtog!
I have made a table of our family of gravitationally bound structures, using my equations and data from ref [7]:
I have included dark matter in the masses of the structures, although it does not make a huge difference. The last three columns are: g_acc is gravitational acceleration of a particle at radius r from the gravitational center of the structure (in femtog), t_acc is the tidal acceleration at that same place; r/r_crit is the ratio of r to r_crit, the 'critical radius' where the two forces precisely balance each other.^{[3] eq. (6)}
It is clear that the Milky way is gravitationally bounded very well, our Local Group cluster is marginally bounded, the Virgo Cluster is bounded very well (it has a great density of galaxies) and the Virgo Supercluster is marginally unbounded. Above the size of superclusters, we get the 'filaments' and 'great walls', which are not gravitationally bounded at all.
[Edit: Also see reply to Roger below (effect of cosmic tidal forces on galaxy formation).]
Jorrie
Notes:
[1] Gravitational tidal forces are 'real', coordinate independent forces, measurable by means of strain gauges or accelerometers.
[2] It is convenient to work in nano, pico or femtog, because my calculations output the acceleration in units of 1/Gy, which is roughly one nanog. An acceleration of 1 ly/y^{2} is roughly one g (9.8 m/s^{2}), because the radius of spacetime curvature at Earth's surface is roughly one light year. One nanog is equivalent to a gravitational radius of curvature of one Gly. (Can you think why this must be so?)
[3] Some relevant cosmological equations (read with the equations of the previous Blog):
The acceleration of expansion:
d^{2}a/dt^{2} = a H_{0}^{2}(Ω_{Λ}  Ω_{m}/(2a^{3})) (1)
(From Peebles 1993, eq. 13.3, p 312, where one must read H_{0} as implying the usual H_{0}/978 Gy^{1}, in order to be compatible with our units convention. The factor Ω_{m}/(2a^{3})  Ω_{Λ} has historically been called the deceleration parameter q. It has the present value q ~ 0.6 (negative, because the deceleration is negative, i.e., it's an acceleration ).
The tidal acceleration over a fixed distance D follows exactly the same profile, but scaled to D. Any change in a causes the same % change in the proper distance between two free particles that were momentarily at rest relative to each other. Hence:
d^{2}D/dt^{2} = D H_{0}^{2}(Ω_{Λ}  Ω_{m}/(2a^{3})) (2)
At the present time, with a=1, it means
d^{2}D/dt^{2} = D H_{0}^{2}(Ω_{Λ}  Ω_{m}/2) (3)
Assuming the present values: H_{0} = 72/978 Gy^{1}_{, }Ω_{Λ} = 74% and Ω_{m} = 26% of the critical density, this works out to a present proper tidal acceleration of:
d^{2}D/dt^{2} = 0.00334 D Gy^{1}, (4)
or about 3.3 femtog per Mly of proper distance (about 10 femtog per Mpc).
If we work with radius r instead of with diameter D, we can find the critical radius of spherical structures (r_crit) where tidal and gravitational forces have the same magnitude. From equation (3) we can write:
GM/r_crit^{2} = r_crit H'_{0}^{2}(Ω_{Λ}  Ω_{m}/2) (5)
where H_{0} must be expressed as inverse seconds (s^{1}) in order to keep the (SI) units (m/s^{2}) of the two sides compatible, i.e., H'_{0} ~ H_{0} /( 3 x 10^{16}) s^{1}. (Coming from 1 Gy ~ 3 x 10^{16} seconds, together with the usual H_{0}/978 conversion to Gy^{1}). Hence:
r_crit = [GM / (H'_{0}^{2} (Ω_{Λ}  Ω_{m}/2))]^{1/3} meters (6)
We can 'cosmologize' it to (say) Mly: divide by 10^{22}, roughly the meters in a million light years.
[4] http://en.wikipedia.org/wiki/Local_Group
[5] The gravitational acceleration is obtained as: a ~ (6.67 x 10^{11} ) (2 x 10^{43}) / (5 x 10^{22})^{2} ~ 5 x 10^{13 }m/s^{2}, or a ~ 50 femtog. This is not strictly correct for a very nonhomogeneous mass concentration like the Local Group, but the OOM should be about right.
[6] The only applicable reference that I managed to find so far is a quite technical paper by Gregory S. Adkins et. al (2006): 'Cosmological perturbations on local systems'. Their conclusion is that matter dominated phases do not perturb the orbits of local systems, but for the vacuum energy dominated phase they say: "However, the cosmological repulsion becomes more important for more extended clusters, and would make a contribution comparable to that of the gravitational term for clusters that are only marginally bound."
Unfortunately, they do not give quantified results (at least not in a form that I can understand).
[7] I have now found a very accessible paper: "On the influence of the global cosmological expansion on the local dynamics in the Solar System" by Matteo Carrera and Domenico Giulini (2006), which seems to confirm my approach. They use a different, more general approach than myself, but the results seem to be compatible.
J

Re: Cosmic Balloon Application V: Cosmic Tidal Force