## Small scale effects of expansion, and 'bound' objects

I often hear something to the extent of,
1) "despite cosmological expansion, small-bound objects do not expand."
and further,
2) "things like galaxies will aways remain bound, and will not expand."

Pertaining to 1)
Because cosmological expansion is a coordinate property, don't small scale objects still expand? I.e. at redshift ~ 1, was a ruler half the size? an electron's orbital separation?

Pertaining to 2)
If expansion is accelerating, and if---in particular---that expansion is beginning to grow exponentially (Friedman's equation for cosmological-constant dominated universe), won't all objects become unbounded eventually?

Is there a(n easy) way of expressing coordinate expansion as an effective force?
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 Quote by zhermes Pertaining to 1) Because cosmological expansion is a coordinate property, don't small scale objects still expand? I.e. at redshift ~ 1, was a ruler half the size? an electron's orbital separation?
Think about how expansion arises from the Friedmann Equations: the energy density must be sufficiently homogeneous in order to obtain the FRW solution. Does this solution hold on the scale of, say, an atom?

 Pertaining to 2) If expansion is accelerating, and if---in particular---that expansion is beginning to grow exponentially (Friedman's equation for cosmological-constant dominated universe), won't all objects become unbounded eventually?
It is possible for expansion to eventually dissociate bound objects, but I believe this only occurs in models with phantom energy, for which $\dot{\rho} > 0$. The reason is that the phantom energy density grows until even the space around local, nonlinear inhomogeneities becomes uniform.

 Quote by bapowell Think about how expansion arises from the Friedmann Equations: the energy density must be sufficiently homogeneous in order to obtain the FRW solution. Does this solution hold on the scale of, say, an atom?
Awesome, that makes a lot of sense. Do you happen to know what the Friedmann equation analog looks like for non-vacuum solutions?

 Quote by bapowell It is possible for expansion to eventually dissociate bound objects, but I believe this only occurs in models with phantom energy, for which $\dot{\rho} > 0$. The reason is that the phantom energy density grows until even the space around local, nonlinear inhomogeneities becomes uniform.
I don't follow why you would need an extra source of energy; I was thinking that with exponentially accelerating expansion, you'll rapidly be able to overcome any geometrical attractive force.

Recognitions:

## Small scale effects of expansion, and 'bound' objects

 Quote by zhermes Do you happen to know what the Friedmann equation analog looks like for non-vacuum solutions?
I'm not sure I follow you. The Friedmann equations for a homogeneous source are
$$H^2 = \frac{8\pi G}{3}\rho$$
and
$$\frac{\ddot{a}}{a}= -\frac{4\pi G}{3}\left(\rho + 3p\right)$$
for a flat universe, where a is the scale factor, H is the Hubble parameter, and $\rho$ and p are the energy density and pressure of the source.
 I don't follow why you would need an extra source of energy; I was thinking that with exponentially accelerating expansion, you'll rapidly be able to overcome any geometrical attractive force.
It's not an extra source of energy, but the energy that's sourcing the accelerated expansion!

 Quote by bapowell It's not an extra source of energy, but the energy that's sourcing the accelerated expansion!
... so you mean dark energy. But that has $\rho =$constant.
 Recognitions: Gold Member To take a less formal approach to it, I like the analogy that says the effect of dark energy on bound systems is exactly the same as the effect of an ant pushing on a tank. It's not that the result is tiny, it's that there IS no result.

 Quote by phinds To take a less formal approach to it, I like the analogy that says the effect of dark energy on bound systems is exactly the same as the effect of an ant pushing on a tank. It's not that the result is tiny, it's that there IS no result.
Thanks for that illuminating analogy. It seems to beg the question, however, why is that the case?

Recognitions:
Gold Member
 Quote by zhermes Thanks for that illuminating analogy. It seems to beg the question, however, why is that the case?
Well, because the ant isn't very strong and no matter how hard it pushes on the tank, it hasn't enough strength to move the tank against the friction of it sitting on the ground.

Similarly, dark energy is staggeringly weak, to the point of triviality on small scales and can't even BEGIN to have any effect against the gravitation of bound systems.

Another analogy that I like is this: Although it is true that the universe is expanding, it will still be very hard to find a parking place.

The point behind this one is that if you were to go way out in intergalactic space, far away from bound objects, and magically paint parking place lines in space, it would take approximately a billion years before they got farther enough apart to make room for another car.

In local terms, dark energy is essentially nonexistent. Over vast distances, it DOES add up to quite a bit.

 Quote by phinds Well, because the ant isn't very strong and no matter how hard it pushes on the tank, it hasn't enough strength to move the tank against the friction of it sitting on the ground.
Yes, I understand why the ant can't move the tank... why does expansion have 'no' effect on bound systems?

 Quote by phinds The point behind this one is that if you were to go way out in intergalactic space, far away from bound objects, and magically paint parking place lines in space, it would take approximately a billion years before they got farther enough apart to make room for another car.
This is fundamentally different. Previously you said there was no effect; in this example there is a small effect.... which is it?

Recognitions:
 Quote by zhermes ... so you mean dark energy. But that has $\rho =$constant.
Dark energy is actually more general than that: it is any source that satisfies $p<-\rho/3$ (with the cosmological constant, a very special case, satisfying $p = -\rho$, i.e. $\rho =$const. as you have written) Phantom energy is characterized by an increasing energy density as the universe expands, leading to an even greater rate of inflation than ordinary de Sitter expansion (this is sometimes called superinflation in the literature.)

 Quote by bapowell Dark energy is actually more general than that: it is any source that satisfies $p<-\rho/3$ (with the cosmological constant, a very special case, satisfying $p = -\rho$, i.e. $\rho =$const. as you have written) Phantom energy is characterized by an increasing energy density as the universe expands, leading to an even greater rate of inflation than ordinary de Sitter expansion (this is sometimes called superinflation in the literature.)
Okay, totally didn't know that---thanks.
But back to the main point; even constant density (i.e. cosmological constant) leads to accelerating expansion----shouldn't that be sufficient to eventually overcome any 'binding'?

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Gold Member
 Quote by zhermes ... But back to the main point; even constant density (i.e. cosmological constant) leads to accelerating expansion----shouldn't that be sufficient to eventually overcome any 'binding'?
Zhermes, I think Brian is on the East coast so it is 11:30 PM for him. I will step in, but he might still answer, which takes precedence since your conversation is with him.

However to interject, just limiting to the case of a cosmological constant (pretty much agreeing with observation) as you said, the answer is no. Stuff like your refrigerator does not eventually fly apart.

Our galaxy stays bound. Of course our solar system stays bound. the earth. our cars and refrigerators etc.

Probably our local grouping of galaxies is destined to coalesce in the long run, into one blobby big galaxy. But larger collections of galaxies will be drawn apart.

Your intuition is correct, except for the scale. Our local SUPERCLUSTER of galaxies IS large enough to be drawn apart. At some point we will no longer be able to see the Virgo cluster (the large collection in our neighborhood).

Some things are big enough, and weakly-enough bound, to be dissipated by the accelerating expansion.

 Quote by zhermes This is fundamentally different. Previously you said there was no effect; in this example there is a small effect.... which is it?
Not fundamentally different at all.
"if you were to go way out in intergalactic space, far away from bound objects,"

Imaginary parking space lines are not real tanks. They will be subject to expansion, there will be an effect albeit a very tiny one. The planet earth however is too strongly bound, it would need to be torn into pieces violently to be subject locally to expansion.
 Recognitions: Gold Member Science Advisor That is the basis for the 'big rip' scenario.
 Correct, and as we are not discussing that scenario in the context of this thread and as it is not the generally mainstream accepted theory, such a thing is not what would happen (the earth being violently ripped apart that is, that implies that the energy is very intense which as this thread is explaining couldn't be further from the truth). Zhermes you may find some insight in this article or some of the articles linked within it: http://en.wikipedia.org/wiki/Scale_factor_(Universe))

 Quote by marcus However to interject, just limiting to the case of a cosmological constant (pretty much agreeing with observation) as you said, the answer is no. Stuff like your refrigerator does not eventually fly apart.
Thanks for your reply Marcus; still no one has attempted to explain why....

Assuming this answer is the correct one (I believe all of you), and looking at the equation, I guess the idea is that if $\rho_\Lambda =$ const, then for objects we consider 'bound', it will always be true that $\rho_m > \rho_\Lambda$, and thus no expansion of that region of space. This seems to have two consequences,
1) It has nothing to do with an object being 'bound' or not; its purely a question of density (albeit considering the future density---and thus objects which remain together, i.e. bound, are requisite). From an order of magnitude estimate, it looks like even galaxy clusters are far far too dense to ever be torn apart, It looks like only objects greater than ~100 Mpc would become Lambda dominated.

2) This suggests that overdense regions will be subject to collapsing solutions of Friedmann's equations....

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Gold Member
 Quote by zhermes Thanks for your reply Marcus; still no one has attempted to explain why.... ...
I'll make a provisional attempt to explain why. Brian will probably get back to this sometime today or tomorrow and can correct me or beef up the explanation if it's flakey.

You know that by definition H(t) = a'(t)/a(t), the time deriv. of the scalefactor divided by the scalefactor.

By simple calculus that means if a(t) grows exponentially then H(t) must be constant. So we can infer that H(t) is currently declining, because growth is not yet pure exponential due to effect of matter. And we can infer that H(t) will gradually decline and level off at some asymptote. It will become essentially constant in the future.

For definiteness, let's say that the present value H(now) = 71 km/s per Mpc and that the future near constant value will be around 61 km/s per Mpc.

Now one interpretation of your WHY? question is to ask how can bound structures adapt?

Well H tells us about the distance between two observers both at CMB rest. It says that if you have two observers each stationary with respect to the CMB and at this moment one Mpc apart then the distance between them is growing 71 (or in future 61) km/s.

So let's say we have a very massive galaxy that is at CMB rest and a stationary observer who is holding a small cannonball at 1 Mpc distance from galaxy who wants to create a bound structure. EASY! He just has to hurl the cannonball directly at the galaxy at exactly 71 km/s.

Then the cannonball will start by seeming to remain at a constant distance from galaxy. But then because the galaxy is so very massive it will ever so slowly begin to fall towards it. And it will be in a kind of YO-YO periodic orbit. Falling thru the galaxy and coming out the other side and then falling back thru---as long as it does not bump anything.

This is not a complete answer, for starters I just want us to understand that bound structures can exist when there is a constant Hubble parameter---a constant growth rate of distances-between-stationary-observers.

You can imagine more interesting orbits. They just have to be adjusted to compensate for the fact that distances (between objects at CMB rest) tend to increase slightly.

In our example the guy did not need to set up an orbit where the cannonball plunges straight in and out the other side. He could have given it a slight sideways push too, so it would go into an elliptical orbit.

I'll post this just as a tentative explanation. Powell may want to re-explain from scratch. Or others may. My take on it is that the smaller structures (less than Mpc scale) that we see as bound are already adapted to 71 km/s per Mpc. And so a future 61 has no terrors for them. They, including the Milkyway and its local group, will do just fine. 61 km/s per Mpc is no big deal.

 Tags cosmology, expansion, friedmann equations, general relativity

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