# Expanding Universe

#### freyar

My understanding is that the rate of expansion appears to be increasing?
Yes, that's right. There are a couple of different ways to measure that now.

But, I have always had the same question: At what scale does expansion occur? Uniform expansion implies odd effects in bound systems. On the other hand, non-uniform expansion seems to imply curvature between the expanding regions and the non-expanding regions.
There is curvature anywhere there's matter (or energy), so that's not really an issue. When we say the universe is flat (in space), we mean on average over very very large regions. But there is curvature, for example, around galaxies. In any case, while the average universe is expanding, bound objects, like galaxies are stuck together --- space is not expanding between the stars of our galaxy, for example.

Also, the scale at which gravity still has an effect, say, for a galaxy, is quite large, and never quite goes away - it's just overwhelmed at a distance by the fields from other masses. I don't understand how there wouldn't be an an expansion within some bound system.
You have to be careful what you mean by a bound system, and the statement is a little vague. Consider a galaxy: the stars are "bound" in the sense that they are not moving quickly enough to escape from each other (in the same way that a rocket has to be moving at a certain speed to escape the earth). And there are small galaxies close to our galaxy that are bound in the same way. Even if you applied Hubble's law on these scales, the expansion speed you'd find would be very small compared to the speeds of the bound objects, so it is irrelevant. You'd expect the structure of spacetime in a galaxy to just be that for a bunch of matter in a non-expanding universe.

However, if you are talking about two far-apart galaxies, yes, they can be gravitationally bound in the sense that they will eventually collapse together but still have some expansion in between. For example, our galaxy is expected to eventually fall into (and merge with) the galaxies of the Virgo Cluster, and, in fact, we are moving closer to them. But the overall expansion of the universe isn't entirely negligible when we look at our motion relative to those galaxies, either.

#### tomBitonti

##### Explorer
Yes, that's right. There are a couple of different ways to measure that now.

There is curvature anywhere there's matter (or energy), so that's not really an issue. When we say the universe is flat (in space), we mean on average over very very large regions. But there is curvature, for example, around galaxies. In any case, while the average universe is expanding, bound objects, like galaxies are stuck together --- space is not expanding between the stars of our galaxy, for example.

You have to be careful what you mean by a bound system, and the statement is a little vague. Consider a galaxy: the stars are "bound" in the sense that they are not moving quickly enough to escape from each other (in the same way that a rocket has to be moving at a certain speed to escape the earth). And there are small galaxies close to our galaxy that are bound in the same way. Even if you applied Hubble's law on these scales, the expansion speed you'd find would be very small compared to the speeds of the bound objects, so it is irrelevant. You'd expect the structure of spacetime in a galaxy to just be that for a bunch of matter in a non-expanding universe.

However, if you are talking about two far-apart galaxies, yes, they can be gravitationally bound in the sense that they will eventually collapse together but still have some expansion in between. For example, our galaxy is expected to eventually fall into (and merge with) the galaxies of the Virgo Cluster, and, in fact, we are moving closer to them. But the overall expansion of the universe isn't entirely negligible when we look at our motion relative to those galaxies, either.
Hi,

For curvature, I was meaning curvature not explainable by gravity, showing up, say, as an extra lensing effect around galaxies, since the transition from no-expansion to expansion seems to occur when moving outward from a galaxy (basically away from clumps of matter to "flat" intergalactic space).

If expansion is allowed between two distant infalling galaxies, won't that add energy? With expansion, when the galaxies finally reach each other, they will be moving faster than if there were no expansion.

Thx!

TomB

#### Umbran

Staff member
For curvature, I was meaning curvature not explainable by gravity
Freyar may correct me slightly but...

No such thing. The expansion we are talking about is a gravitational effect.

Think of it this way - imagine as if the Universe was filled with a very, very thin fog of stuff* that had gravitational repulsion, rather than attraction. Near massive objects (planets, stars, galaxies) you don't notice. The stuff is so very thin, that compared to the local normal mass it just doesn't show up. But out in the deeps, the real deeps, where there's nothing else, that thin fog of stuff dominates. And there is a whole lot of deeps. Lots and lots of deeps. So, overall, in the universe, this thin stuff dominates in the long run.

You don't see any notable different lensing effect. The fog is pretty darned evenly distributed across everywhere - it doesn't clump up with or away from the normal matter, there is no boundary or transition layer we can point to of "here there is expansion, here there is not". You may not be too wrong to really think that the dark energy that does this isn't a separate thing at all, but is associated with space itself. Or you could say that it is as if space expands by default. That's what it does - expands faster and faster, forever. It is only locally around masses where it *doesn't* expand like that, where there is something that halts this larger scale process.

*This is not a fog of physical material, or normal matter with exotic properties. This is just an analogy to grant a visualization that may help with the concept.

#### tomBitonti

##### Explorer
The maths get complicated rather quickly, and I'm hardly following them, but the contribution that the cosmological constant makes to Einstein's equation of general relativity seems to enter into the equation in a different place than the mass energy tensor. That affects the metric (using, say, the Robertson-Walker line element), and has an effect which is similar to that produced by the mast energy tensor, but still seems to be a different contribution.

I guess we need to clarify what we mean by "gravity" as opposed to the metric.

But, if the metric equation applies everywhere, won't that produce an effect even at small scales (since the equation applies everywhere). Though, since the effect is so very small at those small scales, does that mean that there is no effect (truncate to zero), or does the effect occur probabilistically, with there being a chance of the effect occurring as planck scale event just about everywhere, with an average contribution which adds up to match the overall effect implied by the metric equation?

Hopefully I've not butchered that too terribly.

Thx!

TomB

Edit: To make a crude analogy, would this be like putting a rubber band on a rubber surface, then stretching the rubber surface? The band will be pulled slightly as the surface stretches beneath it, with the pull being detectable as an extra force and perhaps a slight stretch of the band, but, since the band quickly reaches a steady state and does not move any further, no energy is added. This presumes some slight friction between the rubber band and the surface, and not that the band is glued to the surface.

Making this more concrete: A neutron orbiting a neutron star at a distance, with the orbit of the neutron slightly larger than it should be assuming just a contribution to the metric from the mass of the neutron star. This would appear as a slight change to the solution to the geodesic equation for the neutron, because the metric is not exactly that given by the star.

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#### Umbran

Staff member
I guess we need to clarify what we mean by "gravity" as opposed to the metric.
As my professors would have said, "gravity", when all is said and done, is the effect on objects. Something curves space-time, objects moving in space-time move in accordance to that overall curvature - that is gravity. So, anything that changes the curvature exerts some gravitational effect. It doesn't matter how many terms we are adding up to come up with that final curvature, or where they enter - gravity is what you get when you add them all up!

Thus, the cosmological constant produces a gravitational effect.

You go into speaking about Planck scale effects. I would advise against that. "Planck scale" is a quantum concept. The cosmological constant is a classical concept. We have a real b*tch of a time getting these to work together.

There's an error in thinking about the rubber-band on the rubber sheet - an error of scale. If you are using the rubber-sheet analogy for the expanding universe, then material objects like galaxies are point objects, not extended things lying on on the surface of the sheet.

Making this more concrete: A neutron orbiting a neutron star at a distance, with the orbit of the neutron slightly larger than it should be assuming just a contribution to the metric from the mass of the neutron star. This would appear as a slight change to the solution to the geodesic equation for the neutron, because the metric is not exactly that given by the star.
Well, remember - space can expand without the dark energy. Dark energy affects the rate of change of expansion. So, it isn't that the neutron would be at point A with Dark energy, but point B without it. It is about how the position of that neutron would change over time. Right now, the orbit is at point A. At some time later, the neutron would be at Point B or Point C, depending on the rate of expansion.

Freyar and I seem to differ a bit on this point - gravitationally bound systems are not a vague concept to me, but a very specific one. "Gravitationally bound" and "gravitationally interacting" are not equivalent. Bound systems don't expand, by definition. The cosmological constant does not change this - if the things are bound, the distance between them isn't going to increase with time.

The thing to remember is that a bound system doesn't just have the mass-energy of the objects, and the kinetic energy of motion. It also has a binding energy, just as an electron in an atom has an energy of binding to the nucleus. That biding energy, like any energy (like, say "dark energy") *changes* the spacetime metric locally, such that expansion does not occur. That's why I say it is part of the definition of a bound state.

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#### tomBitonti

##### Explorer
As my professors would have said, "gravity", when all is said and done, is the effect on objects. Something curves space-time, objects moving in space-time move in accordance to that overall curvature - that is gravity. So, anything that changes the curvature exerts some gravitational effect. It doesn't matter how many terms we are adding up to come up with that final curvature, or where they enter - gravity is what you get when you add them all up!

Thus, the cosmological constant produces a gravitational effect.

You go into speaking about Planck scale effects. I would advise against that. "Planck scale" is a quantum concept. The cosmological constant is a classical concept. We have a real b*tch of a time getting these to work together.

There's an error in thinking about the rubber-band on the rubber sheet - an error of scale. If you are using the rubber-sheet analogy for the expanding universe, then material objects like galaxies are point objects, not extended things lying on on the surface of the sheet.

Well, remember - space can expand without the dark energy. Dark energy affects the rate of change of expansion. So, it isn't that the neutron would be at point A with Dark energy, but point B without it. It is about how the position of that neutron would change over time. Right now, the orbit is at point A. At some time later, the neutron would be at Point B or Point C, depending on the rate of expansion.

Freyar and I seem to differ a bit on this point - gravitationally bound systems are not a vague concept to me, but a very specific one. "Gravitationally bound" and "gravitationally interacting" are not equivalent. Bound systems don't expand, by definition. The cosmological constant does not change this - if the things are bound, the distance between them isn't going to increase with time.

The thing to remember is that a bound system doesn't just have the mass-energy of the objects, and the kinetic energy of motion. It also has a binding energy, just as an electron in an atom has an energy of binding to the nucleus. That biding energy, like any energy (like, say "dark energy") *changes* the spacetime metric locally, such that expansion does not occur. That's why I say it is part of the definition of a bound state.
About plank scale ... I was implying a quantum effect. If the effect of expansion is too small to manifest on small scales, because the size of the effect rounds down to zero, must there still be an effect, and if so, how could it manifest, except as small distributed effects that add up in average to the expected effect? Admittedly, totally made up. I'm trying to reconcile the scale at which expansion is measurable with the need to have the expansion have a continuous effect.

For the neutron ... the path that it takes when not subject to forces other than gravity, that is determined by the metric equation. If the metric equation has values which are slightly different because of the cosmological constant, won't the path be different than the path found when the constant is zero?

Thx!

TomB

#### Umbran

Staff member
About plank scale ... I was implying a quantum effect.
Yes, I know. But surely, the way you talk, you understand that we have no recognized and solid transition from general relativity to quantum mechanics!

If the effect of expansion is too small to manifest on small scales, because the size of the effect rounds down to zero, must there still be an effect, and if so, how could it manifest, except as small distributed effects that add up in average to the expected effect?
Ah. You're under the impression that the way to model the macro-scale effect is by a large number of micro-scale events that add up. We don't generally try to handle it that way at the energy levels of normal matter.

If the metric equation has values which are slightly different because of the cosmological constant, won't the path be different than the path found when the constant is zero?
The cosmological constant does not directly affect the curvature. It affects the rate of change of expansion - which itself does not necessarily affect the curvature. You can have perfectly flat spacetime expanding at a constant rate, or at an increasing rate, and in both the objects will be moving as if the space is flat! Expanding does not directly imply changing curvature.

But, in any case - bound states do not expand.

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#### freyar

Sorry for the delay, haven't been able to get back to EN World.

I'm going to agree with Umbran here with just a few re-iterations and additional comments, I think, since he's painted what seems like a pretty clear picture to me.

For curvature, I was meaning curvature not explainable by gravity, showing up, say, as an extra lensing effect around galaxies, since the transition from no-expansion to expansion seems to occur when moving outward from a galaxy (basically away from clumps of matter to "flat" intergalactic space).
Just to repeat what Umbran said as quickly and straightforwardly as possible: in Einstein's general relativity and theories based on it, spacetime curvature = gravity. That's it. Anything else is some other force.

And, now I am called away again. Will finish this later.

#### freyar

If expansion is allowed between two distant infalling galaxies, won't that add energy? With expansion, when the galaxies finally reach each other, they will be moving faster than if there were no expansion.
If the galaxies are already infalling (moving toward each other), the expansion can slow that down a little. In the case of our galaxy, we are currently moving away (on average) from the galaxies of the Virgo cluster, but the references I've seen say that (in standard cosmology) we will eventually turn around and fall back into the Virgo cluster due to the gravitational attraction to all the galaxies there. Not sure if that answers your question, but I didn't quite follow it entirely.

Hate to post and run again...

#### Chronikoce

##### Villager
Just noticed I'd missed a page in my reading and my points had already been brought up.

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#### freyar

The maths get complicated rather quickly, and I'm hardly following them, but the contribution that the cosmological constant makes to Einstein's equation of general relativity seems to enter into the equation in a different place than the mass energy tensor. That affects the metric (using, say, the Robertson-Walker line element), and has an effect which is similar to that produced by the mast energy tensor, but still seems to be a different contribution.
Maybe it's the math getting in your way, but the cosmological constant comes into the Einstein equation in precisely the same way as the energy-momentum tensor. It is just another kind of energy (potential), like mass and radiation. That's why it is often called "dark energy." The big question is exactly where that energy comes from or if (big if) the law of gravity is modified (rather than there being extra energy).

Just to skip through some of the other discussion, I don't really think issues of quantum gravity are too relevant to how space expands in the presence of a cosmological constant or other dark energy (though they may be important in understanding why, say, the cosmological constant takes the value it does). The cosmological constant is present in our universe everywhere and shows up in Einstein's equation at each point. In the voids between galaxies, it is the most important type of energy; in galaxies, it is negligible.

The cosmological constant does not directly affect the curvature. It affects the rate of change of expansion - which itself does not necessarily affect the curvature. You can have perfectly flat spacetime expanding at a constant rate, or at an increasing rate, and in both the objects will be moving as if the space is flat! Expanding does not directly imply changing curvature.
I'll nitpick a little. Flat spacetime does not expand. Flat space can expand, which gives spacetime nonzero curvature. And the cosmological constant will certainly contribute to that curvature.

But, in any case - bound states do not expand.
So what is your take on systems like our local group of galaxies falling into the Virgo Cluster?

#### Chronikoce

##### Villager
So what is your take on systems like our local group of galaxies falling into the Virgo Cluster?
I believe that in cases of clusters "colliding" they are actually moving through space towards each other. They do not fall into each other due to expansion (expansion would keep them from falling together).

#### Umbran

Staff member
So what is your take on systems like our local group of galaxies falling into the Virgo Cluster?
Well, as I said - gravitationally attracting or interacting does not imply gravitationally bound. Most usually, "gravitationally bound" means "in orbit around/with". Are we *orbiting* the Virgo cluster? No. Then we probably aren't bound to it.

Could some or all of our local group's galaxies end up bound in that cluster? Possibly. But we aren't there yet.

#### freyar

I believe that in cases of clusters "colliding" they are actually moving through space towards each other. They do not fall into each other due to expansion (expansion would keep them from falling together).
I don't quite follow your meaning. When a cluster of galaxies collide, the galaxies making them up rarely collide themselves. (The intergalactic gas does pile up and heat up quite a lot.) But, while you're right that the expansion of the universe pushes two clusters apart, they are gravitationally attracted.

Well, as I said - gravitationally attracting or interacting does not imply gravitationally bound. Most usually, "gravitationally bound" means "in orbit around/with". Are we *orbiting* the Virgo cluster? No. Then we probably aren't bound to it.

Could some or all of our local group's galaxies end up bound in that cluster? Possibly. But we aren't there yet.
I'll have to quibble, I guess. I don't think we can say if we're "orbiting" the Virgo cluster since the universe hasn't lived long enough for us to complete a full orbit yet, I believe. But, if our local group can be projected to "fall into" the Virgo cluster, that means we are in an orbit of it. A highly elliptical orbit but certainly an orbit. So am I right in understanding your concern about us and the Virgo cluster just that we have to project forward using a model of cosmology rather than watching something that has already happened?

#### Chronikoce

##### Villager
[MENTION=40227]freyar[/MENTION] Yes that is what I mean. The colliding is in quotes because they don't really collide per say. What I was trying to indicate was that it was due to gravitational attraction and movement through space that galaxies approach each other rather than any effects of expansion.

#### Nagol

##### Unimportant
I don't quite follow your meaning. When a cluster of galaxies collide, the galaxies making them up rarely collide themselves. (The intergalactic gas does pile up and heat up quite a lot.) But, while you're right that the expansion of the universe pushes two clusters apart, they are gravitationally attracted.

I'll have to quibble, I guess. I don't think we can say if we're "orbiting" the Virgo cluster since the universe hasn't lived long enough for us to complete a full orbit yet, I believe. But, if our local group can be projected to "fall into" the Virgo cluster, that means we are in an orbit of it. A highly elliptical orbit but certainly an orbit. So am I right in understanding your concern about us and the Virgo cluster just that we have to project forward using a model of cosmology rather than watching something that has already happened?
If the collision is following a non-closed conic like a hyperbolic arc, you are merely colliding rather than orbiting. Sufficient energy remains in the objects to escape becoming bound, no?.

#### Umbran

Staff member
I'll have to quibble, I guess. I don't think we can say if we're "orbiting" the Virgo cluster since the universe hasn't lived long enough for us to complete a full orbit yet, I believe. But, if our local group can be projected to "fall into" the Virgo cluster, that means we are in an orbit of it. A highly elliptical orbit but certainly an orbit.
Remember that hyperbolic orbits are entirely possible - we could just be passing through. And, even if the final result will be a bound state, that state may not yet have been achieved.

I don't expect there to be enough data (what with the distances involved, and the relatively short time human kind has been able to take data) to distinguish between the bound orbit and the hyperbolic orbit cases.

#### tomBitonti

##### Explorer
A paper which may be on point:

http://arxiv.org/pdf/0707.1350v1.pdf

Cosmological expansion and local physics
Valerio Faraoni∗ and Audrey Jacques†
Physics Department, Bishop’s University
2600 College Street, Sherbrooke,
(Dated: February 1, 2008)

The interplay between cosmological expansion and local attraction in a gravitationally bound
system is revisited in various regimes. First, weakly gravitating Newtonian systems are considered,
followed by various exact solutions describing a relativistic central object embedded in a Friedmann
universe. It is shown that the “all or nothing” behaviour recently discovered (i.e., weakly coupled
systems are comoving while strongly coupled ones resist the cosmic expansion) is limited to the de
Sitter background. New exact solutions are presented which describe black holes perfectly comoving
with a generic Friedmann universe. The possibility of violating cosmic censorship for a black hole
approaching the Big Rip is also discussed.
Thx!

TomB

#### freyar

So am I right in understanding your concern about us and the Virgo cluster just that we have to project forward using a model of cosmology rather than watching something that has already happened?
If the collision is following a non-closed conic like a hyperbolic arc, you are merely colliding rather than orbiting. Sufficient energy remains in the objects to escape becoming bound, no?.
Remember that hyperbolic orbits are entirely possible - we could just be passing through. And, even if the final result will be a bound state, that state may not yet have been achieved.

I don't expect there to be enough data (what with the distances involved, and the relatively short time human kind has been able to take data) to distinguish between the bound orbit and the hyperbolic orbit cases.
I'll take that as an affirmative to my question above, in that case. I do agree that some (expanding universe generalization of) a hyperbolic orbit is possible even with a collision, though the initial conditions seem pretty nongeneric and unnatural. The initial conditions you'd expect is the formation of different galaxies all basically comoving (simply expanding along with the universe) and proper motions developing later as clusters start to form and a gravitational attraction starts. (In a slightly different context, those are basically the initial conditions used to describe cosmic string wake formation, though the existence of such things is speculative.) Anyway, that's beside the point. I think we can agree that hyperbolic orbits are possible, but saying that all systems that (1) experience some cosmic expansion and (2) will collide in the future must be on hyperbolic orbits is not what you're saying now.

I'm also not sure quite what you mean by "the final state being bound but not yet achieved." We're talking about a deterministic system; either the galaxies will be able to escape in the future (unbound) or not (bound). In Newtonian physics, we'd just ask if the total energy is positive or negative. In the absence of dissipation (which should be pretty negligible for galaxies in clusters), that's not something that changes over time.

But I'm not sure there's a lot to be gained by going over this more.

Yes, it looks relevant, but I can't really vouch for it without reading it (which I don't have time to do).