How does the expansion of the universe work?

[Orodruin]

The dark energy density ΩΛΩΛ\Omega_{\Lambda}, OTOH, is, as far as we can tell, actually constant everywhere in the universe.

I agree with the "as far as we know", but going from there to the assertion that it really is a cosmological constant seems like a big leap of faith to me. Do you have any references on the homogeneity of dark energy?

 

[PeterDonis]

I agree with the "as far as we know", but going from there to the assertion that it really is a cosmological constant seems like a big leap of faith to me.

It might be, yes; I wasn't intending to imply that I have any special knowledge on this. I was basing my remarks on the fact that the current "best guess" model attributes it to a cosmological constant. If it turns out that it isn't--that the density of dark energy does vary--then that would make language about a "force" even more misleading, since both kinds of density involved (dark energy and matter) would be variable and both densities appearing in the math would only be large scale averages--neither would be usefully described as producing a "force" on smaller distance scales.

 

[Jorrie]

So I don't entirely agree with your exposition, because you are lumping together two things--matter density and dark energy density--that actually behave very differently on small distance scales. Your exposition obscures that difference, which is crucial to the discussion we have been having. It is the reason I have been saying that expansion, in and of itself, does not cause any "force" at all on bound systems on small scales, whereas dark energy does cause a tiny "force" on those scales.

OK, I agree that we should cut out the "force" idea, but I'm not convinced that the "cosmic tidal acceleration" plays no role in structures.

I also agree that expansion per se (non-accelerating/decelerating) will have zero effect on bounded structure sizes (essentially the orbital radii).
In the case of accelerating expansion (irrespective of which dark energy model used), I think the consensus is that the orbital radii of components of large scale bound structures are a little larger than what they would have been in a ('neutral') coasting phase of expansion.

In the case of decelerating expansion, the issue is whether the orbital radii of components of large bound structures will be a little smaller than what they would have been in a coasting phase of expansion (irrespective of the non-homogeneity model used). The 'Solution to the tethered galaxy problem' of Tamara Davis et. al seems to suggest that it does, as I interpret them. When a long tether between two galaxies is cut during a decelerating phase, the galaxies will swap positions and eventually join the Hubble flow on opposite sides of the initial middle point, even when ignoring the gravitational attraction between the two.

I understand that the effect will be totally swamped by the gravitational field of the cluster, but since we are saying that we must look at the math to understand the effect, we must be able to answer the question "does decelerating expansion cause any tidal effect of compression, however small, on gravitationally bound clusters?" Or something to this effect...

 

[rede96]

I think you need to read this article by Sean Carroll:

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

Thanks for the link. I did already have a very basic understanding that energy is not conserved in an expanding universe and that the lost energy from the photon is translated into kinetic energy of expansion. Which is why I didn't ask where the energy went, I asked what process causes the photon to lose energy, or in other words what process causes redshift.

As you have said it is not due to inertia of empty space (as there in no such thing.) it is not due to dark energy or a scalar field, so just what does cause redshift? I know the standard answer is 'expansion', but then we go around in circles and ask what causes expansion and just what is 'expanding'

So the point I was trying to make was that there is more going on with expansion that just inertia. Which hopefully I've understood that correctly.

His statements here are frustratingly vague. I can't tell for sure whether or not he means the effect of dark energy when he talks about a very, very tiny force--or even if he means the same thing each time he talks about it. He doesn't actually show the math that he's referring to, and he mentions dark energy several times but also mentions expansion several times without mentioning dark energy. If he did show the math, I don't think there is anything in the math that he could point to that would show a very, very tiny force in the absence of dark energy. (In the presence of dark energy, of course, there is one, and towards the end of the 5-minute segment you refer to, he does talk about the tiny effect that dark energy has on an atom.)

Yes, his comments are very ambiguous. Hence why I might have interpreted them as I did. But he had gone through the math in previous lectures. Well at least how the FRW equations were derived from Newtonian cosmology and from energy conservation.

Anyway, in short what I am really struggling to understand is just what causes two bodies to recede from each other. What I mean by that is if for example I place two bodies sufficiently far apart, in such a way that they are at rest wrt each other, and the forces acting upon them to keep them together are less than the 'pressure' of expansion, then as I understand it, they will start to recede from each other at the Hubble rate. That's basically what I understand from the FRW equations, as you can't have a static universe. So they can only do one of two things, move together or move apart. Moreover I could have done this experiment at any point in time in the past and got the same result.

So there must be some component of expansion that is sort of 'pressure' acting upon these bodies that has always been present. Even if it hasn't always been the dominant component of expansion. And I understand it, we don't need dark energy in the FRW equations to make this prediction. So this pressure, what ever it is, must be different than dark energy.

 

[tedbmoss]

So if I want atoms to "fly apart", due to the "expanding" universe, entropy or age; I am free to design my own cosmology?

 

[PeterDonis]

I'm not convinced that the "cosmic tidal acceleration" plays no role in structures.

The "cosmic tidal acceleration" you refer to is based on looking at a particular family of geodesics, the "comoving" worldlines. But there are other families of geodesics besides the "comoving" ones, and most objects in the universe are traveling on one of those other, non-comoving worldlines. The math you showed only applies to "comoving" worldlines; that's why I said that, at least with regard to the matter in the universe, the math you showed only applies to large scale averages, not to individual objects.

This doesn't mean that the overall spacetime geometry of the universe has no effect on those other worldlines. It just means we have to look at its actual effect on those worldlines, rather than assuming that the effect is the same as on "comoving" worldlines.

In the case of accelerating expansion (irrespective of which dark energy model used), I think the consensus is that the orbital radii of components of large scale bound structures are a little larger than what they would have been in a ('neutral') coasting phase of expansion.

First, I think the consensus assumes that, whatever dark energy model is used, the density of dark energy does not vary very much. If dark energy "clumped" the way ordinary matter does, I think the consensus argument for it exerting a tiny force on all bound structures would not go through, nor would the consensus argument that dark energy causes accelerated expansion! Instead, I think we would expect to see a large force on bound structures (because dark energy would be clumping inside them), and a smaller effect on large distance scales.

Second, I'm not sure the comparison is with "neutral" expansion--I think the comparison is with no expansion. That is, I think the (at least implicit) comparison being made in talking about the size of bound structures is with a model of a bound structure as an isolated system embedded in a background asymptotically flat spacetime.

I'll look at the paper you linked to for the decelerating case.

 

[PeterDonis]

just what does cause redshift?

Here's what I think is the key point: the redshift is not a property of the photon by itself. It's a property of the system consisting of the emitter, the photon, and the receiver.

This is easy to see in the case of ordinary Doppler shift in flat spacetime. An emitter emits a photon. An observer moving towards the emitter sees the photon blueshifted; an observer moving away from the emitter sees it redshifted; an observer at rest with respect to the emitter sees no shift.

What I have just described is the usual way of describing the Doppler shift. But there is another way to look at it, which, unlike the way I just gave, generalizes to any spacetime. When the photon is emitted, it has a certain 4-momentum vector, which carries information about the emitter's 4-velocity (i.e., which direction the emitter is "pointing" in spacetime). The photon's 4-momentum is parallel transported (which is a technical term, but basically means "transported unchanged") along the photon's path through spacetime, until it reaches the receiver. The receiver then compares the photon's 4-momentum with its own 4-velocity (i.e., which direction it is "pointing" in spacetime) to determine whether there is any redshift/blueshift.

So in effect, what is happening is that the directions in spacetime of the emitter and the receiver are being compared by the photon. If the directions are parallel, there is no shift. If the directions diverge, there is redshift. If the directions converge, there is blueshift. What makes the flat spacetime case simple is that "parallel", "diverge", and "converge" have direct physical interpretations as "at rest relative to each other", "moving away from each other", and "moving towards each other".

The cosmological redshift is the same sort of thing, except that the geometry of spacetime isn't flat, it's curved, and the curvature isn't static, it changes with time. So there isn't a simple way to translate the comparison of "directions in spacetime" into physical interpretations as I described for flat spacetime above. But the general method I described still works fine, and you can still work out a correspondence between the comparison of "directions in spacetime" of the emitter and receiver and properties of the universe. When you work it out for the case of a photon traveling between two "comoving" objects in a universe that might be expanding, contracting, or static, using our standard cosmological models, it turns out that the correspondence works like this: "parallel" directions in spacetime for emitter and receiver corresponds to "the universe did not expand or contract during the photon's travel"; "diverging" directions means "the universe expanded during the photon's travel"; and "converging" directions means "the universe contracted during the photon's travel".

So expansion does cause redshift. But on this interpretation, it causes it by making the "directions in spacetime" of comoving objects diverge, not by "stretching" photons. The photon isn't changed at all during its travel (see "parallel transport" above). The expansion affects the emitter and receiver, not the photon. The photon just provides a physical link between emitter and receiver that allows their "directions in spacetime" to be compared.

 

[PeterDonis]

if for example I place two bodies sufficiently far apart, in such a way that they are at rest wrt each other, and the forces acting upon them to keep them together are less than the 'pressure' of expansion, then as I understand it, they will start to recede from each other at the Hubble rate

No. They will start to recede from each other, but if there is a force between them, they won't recede at the Hubble rate at first. Their recession rate will gradually approach the Hubble rate as time goes on. At least, according to the standard cosmological model, they will; but that model doesn't really apply on small distance scales (see below).

The reason this will happen, in the model, has nothing to do with "space expansion" causing a "pressure". It has to do with the rest of the matter in the universe affecting the two objects. Remember that, in the FRW model, the matter in the universe is a continuous fluid, with the same density everywhere. And the "flow lines" of this fluid are expanding--objects at rest relative to the fluid at different spatial locations will be moving apart. So when you put two objects at rest relative to each other into this fluid, at least one must be moving relative to the fluid. And if there is not enough force between the two objects, then the flow of the fluid will pull them apart, because each one is entrained in the local fluid flowing past it. But if there is at least some force between the two objects, the fluid won't pull them apart at the same rate as the fluid itself is flowing--the objects will pull on each other and counteract some of the fluid's effect. But as the objects get farther apart, the force between them will weaken, so their rate of recession will gradually approach the flow rate of the fluid in general (the Hubble rate).

But, as I've said before, there is a huge problem with all of this if you try to apply it to bound systems on small scales: the matter in the universe is not a continuous fluid! It's not even close. The intuitive model of fluid flow sweeping objects apart is simply wrong on small distance scales; that isn't what's happening. To see what might happen on small distance scales, we need a model that applies at those scales.

Here's one such model. On average, as viewed from any point, the matter in the universe is spherically symmetric. In particular, if we pick out a bound system, such as the solar system, and draw a boundary around it (say a sphere one light-year in radius centered on the Sun), the matter outside that sphere will be, on average, spherically symmetric. And there is a theorem called the "shell theorem" which says that, if the matter distribution outside some spherical shell is spherically symmetric, it has no effect on anything inside the shell--we can ignore it completely and just focus on the matter inside the shell when determining what will happen inside the shell. So we can ignore the rest of the matter in the universe when determining what the structure of the solar system will be; we only need to consider the Sun and planets and whatever other objects are large enough to be significant.

The model I just described is what I have been using in this thread. And note that in this model, if the density of dark energy really is constant everywhere, then there is dark energy inside the sphere that bounds the solar system, so it will indeed exert a tiny force and have a tiny effect on the solar system's structure. But the rest of the matter in the universe is not a continuous fluid of the same density everywhere; there isn't any "cosmological fluid" inside the solar system, so it will not exert any force inside the solar system. The only ordinary matter we have to worry about is the ordinary matter we already know is inside the solar system.

 

[PeterDonis]

So if I want atoms to "fly apart", due to the "expanding" universe, entropy or age; I am free to design my own cosmology?

Sure, you can try, but you'll need to make sure it is consistent with all the evidence we already have, and you will need it to make predictions that get verified by experiment. Good luck.

 

[rede96]

Here's what I think is the key point: the redshift is not a property of the photon by itself. It's a property of the system consisting of the emitter, the photon, and the receiver.

Thank you very much for your detailed explanation. I still have to study up on this, but that really helped. Thanks.

 

[rede96]

No. They will start to recede from each other, but if there is a force between them, they won't recede at the Hubble rate at first.

Ah ok, yes of course. And again, thanks for the detailed explanation.

The reason this will happen, in the model, has nothing to do with "space expansion" causing a "pressure". It has to do with the rest of the matter in the universe affecting the two objects.

Your explanation of shell theory got me thinking. If in the example I gave we draw a boundary around the two object's centre of mass, so there was nothing inside this boundary except the two objects placed at rest. Does this mean that everything outside this shell would have no effect? So what would happen then?

 

[PeterDonis]

If in the example I gave we draw a boundary around the two object's centre of mass, so there was nothing inside this boundary except the two objects placed at rest. Does this mean that everything outside this shell would have no effect?

Yes, assuming everything outside the shell was distributed in a spherically symmetric manner (at least to a good enough approximation).

So what would happen then?

Well, you have two massive objects in an otherwise empty space, and they are at rest relative to each other at some instant, and there is nothing else affecting their motion. What do you think would happen?

 

[rede96]

Well, you have two massive objects in an otherwise empty space, and they are at rest relative to each other at some instant, and there is nothing else affecting their motion. What do you think would happen?

I'm not sure as I guess it depends on the initial conditions.But assuming an expanding universe under today's Hubble constant then they would either attract due their gravitational pull being greater than the pull of the Hubble flow or they would recede. But if the recede, I don't know how it could be due to the rest of matter in the universe affecting the two objects as you said as there are only those two objects.

Hence why I keep thinking there must be some 'pressure' acting upon them from expansion.

 

[PeterDonis]

I'm not sure as I guess it depends on the initial conditions.

Um, what? You gave the initial conditions: the two masses are at rest relative to each other at some instant of time, there is empty space between them, and no other interactions are relevant.

assuming an expanding universe under today's Hubble constant then they would either attract due their gravitational pull being greater than the pull of the Hubble flow or they would recede.

There is no "pull of the Hubble flow". None of the rest of the matter in the universe is relevant. See above and my previous posts.

if the recede, I don't know how it could be due to the rest of matter in the universe affecting the two objects as you said as there are only those two objects.

Exactly.

Hence why I keep thinking there must be some 'pressure' acting upon them from expansion.

In other words, you agree that the only relevant interaction is between the two objects; but somehow, instead of accepting the obvious conclusion that the two objects will fall towards each other through their gravitational attraction, you think there must somehow be "pressure from expansion" acting on them? Why? Where would it come from, since we've agreed the rest of the matter in the universe has no effect?

This sort of confusion is why I keep on insisting (and I've done this in a number of threads), that "expansion", in and of itself, exerts no force. The misconception that it does is what leads to confusion like that which you are exhibiting--it makes people unwilling to accept their common sense intuition (that two masses will attract each other) in a situation where the common sense intuition is actually correct! There's enough counterintuitive stuff in relativity and cosmology already; no need to make it worse.

 

[rede96]

Um, what? You gave the initial conditions: the two masses are at rest relative to each other at some instant of time, there is empty space between them, and no other interactions are relevant.

Yes, initial conditions do matter as I didn't think I'd clearly specified them. E.g. how massive are the two objects, how far apart they are and do we assume dark energy or not. Changing those conditions would lead to a different outcome.

In other words, you agree that the only relevant interaction is between the two objects;

No: If there is dark energy present then that is an additional interaction.

you think there must somehow be "pressure from expansion" acting on them? Why? Where would it come from

Space (which includes dark energy)

it makes people unwilling to accept their common sense intuition (that two masses will attract each other) in a situation where the common sense intuition is actually correct!

Not at all, I agree that there will always be an attraction between the two objects in proportion to the inverse square law. But that attractive force does not tell me anything about the relative motion between the two bodies.

But I do agree with you that in the absence of dark energy then the only logical conclusion is that the two bodies would move towards each other BUT I am now thinking that we shouldn't talk about expansion without dark energy, as it is a real part of 'space'. That for me is where the confusion starts. One of the other big confusion factors is when people talk about recession as distant bodies being at rest wrt each other and the space growing between them.

 

[PeterDonis]

how massive are the two objects, how far apart they are

Since it is specified that there is no other matter between them, just empty space, this doesn't make any difference qualitiatively; all it affects is how fast the objects will fall together. Of course that specification might not be realistic for large enough separations.

If there is dark energy present then that is an additional interaction.

I thought we had ruled out dark energy for this particular scenario. You said there is "nothing inside the boundary except those two objects". That means no dark energy. Obviously the presence of dark energy will change things--but the separation would have to be very, very large (tens to hundreds of millions of light years) for it to change things significantly.

Also, if you are trying to understand what effect "expansion" has in itself, it would seem that you would want to rule out dark energy.

Space (which includes dark energy)

"Space" in itself does not have to include dark energy. It happens to in our actual universe, but we are considering thought experiments in order to understand the underlying physical principles involved. In such thought experiments it's perfectly reasonable to say there is no dark energy. "Space" without dark energy is perfectly consistent physically.

I agree that there will always be an attraction between the two objects in proportion to the inverse square law. But that attractive force does not tell me anything about the relative motion between the two bodies.

It does if it's the only force acting, which is what I thought the specification of the scenario was.

I am now thinking that we shouldn't talk about expansion without dark energy, as it is a real part of 'space'.

No, it isn't. Dark energy is something separate from "space". It happens to be present everywhere in our universe, but that is not required by the laws of physics. It's just a contingent fact about our universe. If you are trying to understand expansion in and of itself, it is perfectly reasonable to assume, for purposes of a thought experiment, that there is no dark energy; the physical model you get is perfectly consistent. See my comments above.

One of the other big confusion factors is when people talk about recession as distant bodies being at rest wrt each other and the space growing between them.

Yes, and I am saying that a good way to avoid this confusion is to understand that there is no force associated with expansion. There is a force associated with dark energy, but that is a force associated with accelerating expansion. You can have expansion without having accelerated expansion.

To put this another way, there is no force associated with "space" in and of itself. "Space" cannot expand in the sense of pushing or pulling things apart. Dark energy can push things apart, but that's because dark energy is not "space"; it's something separate from space. It can be thought of as a kind of "exotic" substance that causes repulsive instead of attractive gravity. It's not the same as empty space.

 

[rede96]

if you are trying to understand what effect "expansion" has in itself, it would seem that you would want to rule out dark energy.

Yes, you're right of course. I started to jump around a bit sorry.

It does if it's the only force acting, which is what I thought the specification of the scenario was.

Ok got it. However looking at this another way. If I take the FRW equations, assuming flat space and no dark energy, and as the radiation energy density today is negligible, then all I am left with is the matter energy density, this would give me a 'rate' of movement. But as I understand it, that in itself doesn't tell me if the universe is expanding or contracting, is that correct?

Today we model this movement on 'expansion' as that matches the observations, so the FRW equations tell us that expansion rate. But in the scenario we mentioned (ie two bodies at rest) then what does the FRW equations tell us about that situation? The rate of attraction? (I am also assuming I can't get a negative rate of expansion from the FRW equations.)

Dark energy can push things apart, but that's because dark energy is not "space"; it's something separate from space. It can be thought of as a kind of "exotic" substance that causes repulsive instead of attractive gravity. It's not the same as empty space.

Quite often I hear / read of vacuum energy being a property of empty space and we can't have totally 'empty' space. I also read a lot that vacuum energy is the same as dark energy, hence why I assumed dark energy it is a property of empty space as you can't have empty space that consists of 'nothing'

Just to clarify is vacuum energy of empty space different from dark energy? If so why isn't there a term for both energy densities in the FRW equations?

 

[PeterDonis]

If I take the FRW equations, assuming flat space and no dark energy, and as the radiation energy density today is negligible, then all I am left with is the matter energy density, this would give me a 'rate' of movement. But as I understand it, that in itself doesn't tell me if the universe is expanding or contracting, is that correct?

The first Friedmann equation has the square of the Hubble constant on the LHS, so for a given density there are two possible solutions, expanding and contracting, yes. (Note that there is also a curvature term on the LHS, but you can always just try each of the three possibilities, k = +1, 0, -1, to see which of them give possible solutions.) To distinguish between them, you need some kind of initial condition--for example, the observation that the universe right now is expanding.

in the scenario we mentioned (ie two bodies at rest) then what does the FRW equations tell us about that situation?

Nothing, because it isn't applicable. The Friedmann equations assume that the matter in the universe is a continuous fluid; it can't be applied to the case of two isolated bodies in empty space.

If we assumed we had a matter-only universe where all the matter (the continuous fluid) was at rest at some instant of time, the Friedmann equations--more precisely, the second Friedmann equation--would tell us that it would start contracting. "At rest at some instant of time" is an initial condition, and is enough to allow the equations to give a unique solution.

Quite often I hear / read of vacuum energy being a property of empty space and we can't have totally 'empty' space.

Now you're talking quantum field theory, not classical GR. It is true that in quantum field theory, what we normally think of as "empty space" should have a nonzero vacuum energy. However, when we try to calculate this energy, we come up with an enormous answer: something like 123 orders of magnitude larger than the largest value which is compatible with our observations. So something is clearly wrong with our current understanding of how this works in quantum field theory.

is vacuum energy of empty space different from dark energy?

In terms of what the physical origin of dark energy (meaning, "whatever it is that is causing the accelerated expansion of the universe") is, we don't know; it could be vacuum energy or it could be something else like a scalar field, or it could be a combination of several such things. In terms of how vacuum energy would behave in the equations, it would behave the same as dark energy--like a cosmological constant. (At least, assuming that dark energy works the way we assume it does in our best current model--see below.) So the cosmological constant term in the equations covers both possibilities.

There are other speculations about types of "dark energy" that work differently from a cosmological constant--such as "quintessence", which causes accelerated expansion but not quite as strongly as a cosmological constant; or "phantom energy", which causes even more acceleration than a cosmological constant and leads to a "Big Rip" scenario. None of these speculations have any evidence to back them up; our best current evidence is that the accelerated expansion we observe is exactly what it should be if it were due to a very, very tiny cosmological constant.

 

[rede96]

The first Friedmann equation has the square of the Hubble constant on the LHS, so for a given density there are two possible solutions, expanding and contracting, yes. (Note that there is also a curvature term on the LHS, but you can always just try each of the three possibilities, k = +1, 0, -1, to see which of them give possible solutions.) To distinguish between them, you need some kind of initial condition--for example, the observation that the universe right now is expanding.

Thanks Peter. Again just to check my understanding, wouldn't both solutions need to be positive, so we don't have to take the square root of a negative number. And if they are both positive, then I assume these two answers would just be different rates and we'd still need observation to tell us if the universe is contracting or expanding. Is that correct?

Now you're talking quantum field theory, not classical GR. It is true that in quantum field theory, what we normally think of as "empty space" should have a nonzero vacuum energy. However, when we try to calculate this energy, we come up with an enormous answer: something like 123 orders of magnitude larger than the largest value which is compatible with our observations. So something is clearly wrong with our current understanding of how this works in quantum field theory.

Just out of interest, is the 123 orders of magnitude larger calculated for the present vacuum energy anywhere near the estimate for the energy in the initial inflaton field prior to inflation? I was just curious to see if there was any link between them.

 

[PeterDonis]

is the 123 orders of magnitude larger calculated for the present vacuum energy anywhere near the estimate for the energy in the initial inflaton field prior to inflation?

Good question. I don't think it's close, but I don't know for sure.

 

[PeterDonis]

wouldn't both solutions need to be positive, so we don't have to take the square root of a negative number. And if they are both positive, then I assume these two answers would just be different rates and we'd still need observation to tell us if the universe is contracting or expanding. Is that correct?

The first Friedmann equation reads

$$
H^2 + \frac{k}{a^2} = \frac{8}{3} \pi \rho + \frac{1}{3} \Lambda
$$

Since $H^2$ appears on the LHS, there will be two values of $H$ corresponding to any solution of this equation--the positive square root of $H^2$, and the negative square root of $H^2$. These two values correspond to an expanding and a contracting universe, with the same rate numerically in both cases, just opposite signs; so we need observation to tell us the sign. Of course $H^2$ itself must be positive, but that doesn't make both of its square roots positive.

 

[Jorrie]

These two values correspond to an expanding and a contracting universe, with the same rate numerically in both cases, just opposite signs; so we need observation to tell us the sign.

Hypothetically, if we have observed blueshifts instead of redshifts, with the same values for Lambda and matter density, would we have observed a present decelerating contraction? With no observable CMB?

 

[PeterDonis]

Hypothetically, if we have observed blueshifts instead of redshifts, with the same values for Lambda and matter density, would we have observed a present decelerating contraction?

Hypothetically, yes.

With no observable CMB?

Whether or not we would observe a CMB in this hypothetical universe would depend on what it was like in the past.