B Threshold of gravitational pull over dark energy

[kimbyd]

Just a quick question, if I am comparing the distance of two galaxies equal in mass do I include both masses in the formula?

Unfortunately that's a bit of a tricky question, and I'd have to do some work to figure out the answer.

When working with classical Newtonian gravity and no cosmological constant, the two-body problem is solved using the concept of "reduced mass":

$$\mu = {m_1 m_2 \over m_1 + m_2}$$

Using the reduced mass allows you to work with the system as if there was only one mass. I'm sure you can do the same sort of thing in the case where the cosmological constant is relevant, but some careful work would have to be done to make sure that the math was all done properly.

 

[JMz]

Unfortunately that's a bit of a tricky question, and I'd have to do some work to figure out the answer.

When working with classical Newtonian gravity and no cosmological constant, the two-body problem is solved using the concept of "reduced mass":

$$\mu = {m_1 m_2 \over m_1 + m_2}$$

Using the reduced mass allows you to work with the system as if there was only one mass. I'm sure you can do the same sort of thing in the case where the cosmological constant is relevant, but some careful work would have to be done to make sure that the math was all done properly.

This seems like the correct answer, though I would defer to some of our GR aficionados for the definitive judgment. (Notice that the reduced mass has half the mass of the pair, rather than their sum.)

It is as though the DE serves as a non-gravitational force (air pressure :-), so the Newtonian approximation is just fine, as long as the galaxies are not BH's -- and even then, as long as they are well separated, so that the nonlinear GR effects are irrelevant: distances much greater than their event-horizon distances (just a few LY or so). This should be an excellent approximation within the Local Group.

 

[PeterDonis]

Using the reduced mass allows you to work with the system as if there was only one mass.

More precisely, it allows you to solve for the internal motions of the system (i.e., the motion of each mass relative to the other) as if there was only one mass. But I don't think this method will give the correct gravitational force exerted by the system on an external object. And the latter is what is relevant for this discussion.

 

[JMz]

More precisely, it allows you to solve for the internal motions of the system (i.e., the motion of each mass relative to the other) as if there was only one mass. But I don't think this method will give the correct gravitational force exerted by the system on an external object. And the latter is what is relevant for this discussion.

Quite right. The question, I suppose, is whether the hypothetical second object is external. E.g., after multiplying the masses by 10 or so, we could ask about the mutual attraction and binding of the LG and a nearby group of similar mass. It wouldn't make sense to ask about the MW and a similar-sized galaxy that isn't in the LG, since an external one would be attracted to the LG as a whole in that case, not only to us.

However, I suppose you could the same question of two separated galaxies near the edge of a void, where both would be effectively isolated, not members of groups.

 

[rede96]

However, I suppose you could the same question of two separated galaxies near the edge of a void, where both would be effectively isolated, not members of groups.

It was more this hypothetical situation I was curious about. How far apart would two galaxies of a similar mass have to be so they were not gravitational bound.

I also wanted to play around with different scenarios and see what the gravitational force was acting on a distant body at the point where it was no longer gravitational bound to a system. Just so I could see what the 'repulsive' force from DE was at different threshold distances.

 

[PeterDonis]

The question, I suppose, is whether the hypothetical second object is external.

Let me rephrase what I said: using the reduced mass allows you to compute the internal motions of a bound system, assuming you already know it is bound and which objects are a part of it. But it does not allow you to determine which objects are part of the bound system.

So yes, your question is the key question: is a given object "external" (not part of the bound system that is the Local Group) or "internal" (part of the bound system that is the Local Group). And you can't answer that question using the reduced mass, because it's not a question about the internal motions of the bound system; it's a question about whether a given object is actually part of the bound system or not.

 

[rede96]

it's not a question about the internal motions of the bound system; it's a question about whether a given object is actually part of the bound system or not.

Am I right in assuming that if an object is moving away from us, it will always be moving away and if something is moving towards us i.e. gravitational bound it will always be gravitational bound?

In that case can't we know if something is gravitational bound or not simply by red shift?

So yes, your question is the key question: is a given object "external" (not part of the bound system that is the Local Group) or "internal" (part of the bound system that is the Local Group). And you can't answer that question using the reduced mass,

But we can use reduced mass to assess the gravitational forces on an object? If just won't tell us if it is gravitational bound.

I also assumed that if we know the distance to an object we can calculate if it is gravitational bound or not by using F = MA to estimate the 'forces' on the object from DE accelerating it away and use F = G * M * m / r^2 to estimate the force on it from the gravitational pull of the system. Which ever force is the larger will tell us if the object is moving away or towards us. Does that work?

 

[PeterDonis]

Am I right in assuming that if an object is moving away from us, it will always be moving away and if something is moving towards us i.e. gravitational bound it will always be gravitational bound?

No. An object moving away from us might be on a highly elongated elliptical orbit (i.e., bound). An object moving towards us might be on a hyperbolic orbit (i.e., not bound).

we can use reduced mass to assess the gravitational forces on an object?

Only if you already know it's part of a bound system, and which objects are part of that bound system.

I also assumed that if we know the distance to an object we can calculate if it is gravitational bound or not by using F = MA to estimate the 'forces' on the object from DE accelerating it away and use F = G * M * m / r^2 to estimate the force on it from the gravitational pull of the system.

You don't need to calculate force, just acceleration. This is basically what @kimbyd was doing in post #2.

Which ever force is the larger will tell us if the object is moving away or towards us.

No, it tells us whether the object is accelerating away from us or towards us. That is, it tells us which effect is stronger: the effect of dark energy, or the effect of the gravitational pull of the system.

 

[rede96]

No. An object moving away from us might be on a highly elongated elliptical orbit (i.e., bound). An object moving towards us might be on a hyperbolic orbit (i.e., not bound).

Yes obviously. The question I was asking was more fundamental and trying to establish if only gravity and dark energy are responsible for distances (continually) increasing or decreasing. E.g. changes in distances aren't due to some early inertia from inflation. Thus if an object is receding it will always be receding and if the distance is decreasing it will always decrease.

Only if you already know it's part of a bound system, and which objects are part of that bound system.

If there were no dark energy, or anything else that could effect change the motion of mass other than gravity, wouldn't all systems be bound?

No, it tells us whether the object is accelerating away from us or towards us. That is, it tells us which effect is stronger: the effect of dark energy, or the effect of the gravitational pull of the system.

I understand that an object can have a gravitational force acting on it and it still be moving away from a system, but again assuming that only DE and Gravity are responsible for motion, and taking into account my first point above, then why doesn't follow that that the greatest force would indicate direction?

 

[PeterDonis]

The question I was asking was more fundamental and trying to establish if only gravity and dark energy are responsible for distances (continually) increasing or decreasing. E.g. changes in distances aren't due to some early inertia from inflation.

In general this is obviously false: most objects in the universe are moving away from us because of the expansion of the universe, which is just another way of saying "early inertia".

If there were no dark energy, or anything else that could effect change the motion of mass other than gravity, wouldn't all systems be bound?

Obviously not, since one can have an expanding universe with no dark energy.

why doesn't follow that that the greatest force would indicate direction?

See above.

 

[rede96]

In general this is obviously false: most objects in the universe are moving away from us because of the expansion of the universe, which is just another way of saying "early inertia".

So what is responsible for the expansion of the universe?

Obviously not, since one can have an expanding universe with no dark energy.

And is that the case with our universe? In other words is the fact that distance are increasing on cosmological scales due to dark energy? Something else or a combination of things?

 

[rede96]

Am I right in assuming that if an object is moving away from us, it will always be moving away and if something is moving towards us i.e. gravitational bound it will always be gravitational bound?

I'm also assuming that if we ignore local movement due to orbits etc. this statement is true on cosmological scales? There is no circumstances where an object which is now receding from our local cluster that will become gravitational bound to it in the future. And there is no object which is currently gravitational bound to our local cluster that won't be in the future.

 

[PeterDonis]

So what is responsible for the expansion of the universe?

The fact that it started out, at the Big Bang (which means at the end of inflation, according to most cosmologists), in a rapidly expanding state.

is the fact that distance are increasing on cosmological scales due to dark energy?

No. The fact that the expansion has been accelerating for the past few billion years is due to dark energy, but the expansion itself is due to initial conditions; see above.

There is no circumstances where an object which is now receding from our local cluster that will become gravitational bound to it in the future.

Assuming it's receding at more than escape velocity, yes.

there is no object which is currently gravitational bound to our local cluster that won't be in the future.

This is most likely not true; internal interactions between objects in a bound system can cause one of them to have enough velocity to escape. Over long enough time scales this is basically certain to happen.

 

[JMz]

This is most likely not true; internal interactions between objects in a bound system can cause one of them to have enough velocity to escape. Over long enough time scales this is basically certain to happen.

To clarify @PeterDonis's point: The internal objects here are the components of the system, that is, individual stars within galaxies and individual galaxies within the LG. Some will coalesce, ultimately into a single BH, but the rest will "evaporate".

 

[rede96]

The fact that it started out, at the Big Bang (which means at the end of inflation, according to most cosmologists), in a rapidly expanding state.

Sorry, I must not be explaining myself very well. It's more the underlying mechanism of the Big Bang I was interested in not the description of what happened. The reason for asking is I understood dark energy to be 'left over' from inflation, or the inflaton field and that expansion is not 'momentum' left over from the big bang, like the momentum left over from an explosion. It's due to this special scalier field. So if immediately after the initial inflation period, inflation would have stopped, when matter started to form it would have not have been moving apart from each other.

But I have to admit I do struggle to understand how the universe could expand to the extent it did in such a short time (10−35 to 10−34 seconds) and then almost come to an immediate stop. (comparatively speaking)

But in any case, when inflation did slow down and atoms started to form, what was the rate of expansion then?

Assuming it's receding at more than escape velocity, yes.

This is probably just confusion on my part but I thought if something didn't have escape velocity we would see it slowing down not accelerating away from us. So by definition if we observe an object to be accelerating away from us then it can't be gravitationally bound.

This is most likely not true; internal interactions between objects in a bound system can cause one of them to have enough velocity to escape. Over long enough time scales this is basically certain to happen.

To clarify @PeterDonis's point: The internal objects here are the components of the system, that is, individual stars within galaxies and individual galaxies within the LG. Some will coalesce, ultimately into a single BH, but the rest will "evaporate".

Again, as with the point above, probably just my understanding but I thought by definition if we see something accelerating away from us it can't be gravitationally bound. So if it's not accelerating, we'd observe it to be slowing down or moving towards us. Therefore it was gravitational bound. And if it is gravitationally bound then it can't generate (without external assist) enough energy to reach escape velocity.

 

[JMz]

Again, as with the point above, probably just my understanding but I thought by definition if we see something accelerating away from us it can't be gravitationally bound. So if it's not accelerating, we'd observe it to be slowing down or moving towards us. Therefore it was gravitational bound. And if it is gravitationally bound then it can't generate (without external assist) enough energy to reach escape velocity.

To this point: The objects you are talking about are not "components" (stars, galaxies) of bound subsystems (galaxies, galaxy groups). For instance, galaxies really are bound systems. But that does not mean that stars will not interact with each other, get "gravity assists" (as we call them for space probes), and escape from their galaxies.

For cosmological purposes regarding expansion and DE, this effect is not relevant. Just focus on the clusters and their mutual separation.

 

[rede96]

For cosmological purposes regarding expansion and DE, this effect is not relevant. Just focus on the clusters and their mutual separation.

Ok thanks. That was pretty much how I was thinking about this. Comparing systems (local groups) not components but maybe I didn't explain that too well.I love this stuff, but I just never get the time to go into in any great detail. And I have a really crap memory!

Anyway as regards the OP and calculating a threshold to see if two systems of equal mass are far enough apart not to be become gravitationally attracted to each other in the future, what is the consensus?

 

[PeterDonis]

I understood dark energy to be 'left over' from inflation, or the inflaton field and that expansion is not 'momentum' left over from the big bang, like the momentum left over from an explosion. It's due to this special scalier field.

This understanding is incorrect. At the end of inflation, according to our best current models, all of the energy density stored in the inflaton field was transferred to the Standard Model fields (i.e., what we usually refer to as "matter and radiation"). That created a hot, dense, rapidly expanding state which we call the "Big Bang": hot and dense because of all the energy density transferred to it; rapidly expanding because the matter and radiation was, on average, comoving, and at the end of inflation any comoving observer was rapidly moving apart from all other comoving observers. Expansion after this was due to the momentum in the matter and radiation; no "push" from anything else was required.

As for dark energy, there are two possibilities: one, that dark energy is a very small remnant "left over" from the inflaton field; two, that dark energy is something else entirely (the simplest hypothesis is that it's just a cosmological constant built into the structure of spacetime). Either way, however, dark energy had an absolutely negligible effect on the universe's expansion right after the Big Bang, and for billions of years thereafter.

if immediately after the initial inflation period, inflation would have stopped, when matter started to form it would have not have been moving apart from each other.

Incorrect. See above. Another way of thinking of it is: inflation is accelerated expansion. When inflation stops, the expansion stops accelerating, but that doesn't mean expansion stops; it just means it keeps on at the same rate instead of the rate increasing further.

when inflation did slow down and atoms started to form

These are two very different events that happened at very different times. Inflation started to slow down just before it stopped and the Big Bang happened. Atoms didn't form until hundreds of thousands of years after the Big Bang.

 

[PeterDonis]

I thought if something didn't have escape velocity we would see it slowing down not accelerating away from us. So by definition if we observe an object to be accelerating away from us then it can't be gravitationally bound.

If we see something accelerating away from us, yes. But you didn't say "accelerating"; you said "receding", which could include "receding, but slowing down". "Receding" is not the same as "accelerating away".

if it's not accelerating, we'd observe it to be slowing down or moving towards us. Therefore it was gravitational bound.

This is not correct either. An object that is moving away from us, slowing down, could still have enough energy to escape; it might keep on slowing down but never actually stop and start moving towards us. Or, in a universe with dark energy, it might at some point stop slowing down and start speeding up (as it gets far enough away that the effect of dark energy is larger than the effect of our gravity).

 

[rede96]

That created a hot, dense, rapidly expanding state which we call the "Big Bang": hot and dense because of all the energy density transferred to it; rapidly expanding because the matter and radiation was, on average, comoving, and at the end of inflation any comoving observer was rapidly moving apart from all other comoving observers. Expansion after this was due to the momentum in the matter and radiation; no "push" from anything else was required.

Ah ok. This was my understanding in general but with one exception, I didn't quite get the momentum part. From what I understood matter / radiation didn't carry any momentum from the big bang (as in the classical sense of some force ejecting matter off into various directions.)

But if I understood you correctly what you are saying is that the reason we observe galaxies to be moving apart from one another (ignoring any DE or gravitational effects for now) is because of the momentum given to matter / radiation during big bang. And the reason the universe has the structure it does is because that matter / radiation was initially 'embedded' in a rapidly expanding field. (very loosely speaking.) Hence why we class things as comoving.

I would imagine that the initial rapidly expanding state would have slowed down quite a bit before matter was created momentum transferred? What was the scale factor around the time? Around 0.1 ish? (Compared to 1 as it is today I think)

As for dark energy, there are two possibilities: one, that dark energy is a very small remnant "left over" from the inflaton field; two, that dark energy is something else entirely (the simplest hypothesis is that it's just a cosmological constant built into the structure of spacetime). Either way, however, dark energy had an absolutely negligible effect on the universe's expansion right after the Big Bang, and for billions of years thereafter.

So thinking of dark energy as a cosmological constant build into the structure of spacetime, does that mean that the natural 'movement' of comoving objects in space is to move apart? (Ignoring gravity) And the reason we see an accelerating universe now is simply because the distance between comoving objects is greater now, hence the structure of space time is having a bigger effect?

 

[rede96]

This is not correct either. An object that is moving away from us, slowing down, could still have enough energy to escape; it might keep on slowing down but never actually stop and start moving towards us. Or, in a universe with dark energy, it might at some point stop slowing down and start speeding up (as it gets far enough away that the effect of dark energy is larger than the effect of our gravity).

And in normal circumstances I couldn't argue with that. But I was wondering, and hence my OP, if the universe was at a stage where everything that is gravitationally bound had to remain so. Simpy because we are now in a dark energy dominated stage and always will be.

My rationale was that if something is moving away from my local group then only one of three things can happen. Either it doesn't have escape velocity and it will remain gravitonally bound, it has become gravitationally bound to the next system and I'll observe it to be accelerating away or it will remain isolated as its own system and start to accelerate away.

In the last two cases, I must observe the object to be accelerating away. But in the first case I won't. Ergo anything I don't see accelerating away must be gravitationally bound.

 

[JMz]

atoms started to form

If by "started' to", you mean that, say, protons formed, that happened well after inflation, about 1 microsec after the BB. However, whole atoms could not form in significant quantities until photons no longer immediately ionized any electrons that managed to stick to protons. Waiting for the universe to cool to that point took ~380,000 years.

 

[PeterDonis]

From what I understood matter / radiation didn't carry any momentum from the big bang (as in the classical sense of some force ejecting matter off into various directions.)

Momentum is relative. Comoving objects in an expanding universe do have momentum relative to each other.

if I understood you correctly what you are saying is that the reason we observe galaxies to be moving apart from one another (ignoring any DE or gravitational effects for now) is because of the momentum given to matter / radiation during big bang.

Momentum is relative. See above.

the reason the universe has the structure it does is because that matter / radiation was initially 'embedded' in a rapidly expanding field.

No. It was in a spacetime geometry with a particular kind of curvature.

I would imagine that the initial rapidly expanding state would have slowed down quite a bit before matter was created momentum transferred?

Reheating--the process of transferring energy from the inflaton field to the Standard Model fields, "matter" and "radiation", happened at the end of inflation.

What was the scale factor around the time? Around 0.1 ish? (Compared to 1 as it is today I think)

The scale factor is a convention; the usual convention is to set it to 1 "now". Using that convention, the scale factor at the end of inflation was much, much, much, much smaller than 0.1. (Just for calibration: the scale factor when the CMB was emitted, which was several hundred thousand years after the end of inflation, was about 1/1100, or about 0.0009. The scale factor at the end of inflation was many orders of magnitude smaller than that.)

thinking of dark energy as a cosmological constant build into the structure of spacetime, does that mean that the natural 'movement' of comoving objects in space is to move apart? (Ignoring gravity)

You can't ignore gravity; spacetime geometry is gravity, and the cosmological constant is an aspect of spacetime geometry.

A valid question would be, in a universe that was empty of everything except a positive cosmological constant, would the natural movement of comoving objects be to move apart? The answer to that is that their natural movement would be to accelerate apart. (This idealized model is called de Sitter spacetime.) You really, really need to be careful about the distinction between moving apart and accelerating apart.

the reason we see an accelerating universe now is simply because the distance between comoving objects is greater now, hence the structure of space time is having a bigger effect?

No, it's because the density of matter and radiation is now small enough that the cosmological constant becomes the primary factor that determines the dynamics of the universe's expansion. When the density of matter and radiation is larger, the effect of the cosmological constant is overwhelmed by the effect of the matter and radiation.

I was wondering, and hence my OP, if the universe was at a stage where everything that is gravitationally bound had to remain so. Simpy because we are now in a dark energy dominated stage and always will be.

No, that's not the case.

My rationale

Is incorrect. You are drastically oversimplifying things. The dynamics of individual objects in the universe, whether they are currently part of a gravitationally bound system or not, is chaotic on long enough time scales; that means it's impossible to predict the object's trajectory into the indefinite future from any observations we can make of it now.

As I've already said, an object that is currently in a gravitationally bound system might not stay bound in that system; internal interactions between it and other bound objects in the system, or even between it and an unbound object that happens to come in from outside, can cause that object to be ejected from the bound system. Conversely, an object that is currently not part of a bound system can fly into one and lose enough energy through interactions with objects in the system that it ends up being bound in that system. Over long enough time scales these things happen very often.

You seem to want to have some simple rule you can use to classify objects in the universe. There isn't one.

 

[kimbyd]

More precisely, it allows you to solve for the internal motions of the system (i.e., the motion of each mass relative to the other) as if there was only one mass. But I don't think this method will give the correct gravitational force exerted by the system on an external object. And the latter is what is relevant for this discussion.

I don't have the time just now to work through the math, but I think it might still have a term like that.

What you'd need to do is solve for the value of distance where the accelerations of the two objects are equal (same magnitude and direction):

$$- {GM \over r^2} + {\Lambda r \over 3} = {Gm \over r^2} - {\Lambda r \over 3}$$

I don't have the time just now to work through the equations, but it shouldn't be too different from what I did earlier.

 

[kimbyd]

I don't have the time just now to work through the math, but I think it might still have a term like that.

What you'd need to do is solve for the value of distance where the accelerations of the two objects are equal (same magnitude and direction):

$$- {GM \over r^2} + {\Lambda r \over 3} = {Gm \over r^2} - {\Lambda r \over 3}$$

I don't have the time just now to work through the equations, but it shouldn't be too different from what I did earlier.

This is incorrect, because I made a substantial error: the cosmological constant in the Newtonian limit is wonky when comparing things happening at different locations.

As I understand it, the way this works is that the cosmological constant creates a relative acceleration between any two objects. The above equation would suggest that that relative acceleration is $2\Lambda r/3$, but it isn't: it's just $\Lambda r/3$. The way this is represented mathematically is to select an origin (any origin will do) and state that every object in the universe has an acceleration relative to that origin.

Thus, if my reasoning is correct, the right way to do this is to just drop the extra $\Lambda r/3$ term:

$$- {GM \over r^2} + {\Lambda r \over 3} = {Gm \over r^2}$$

So, the answer is just that we have to replace the mass in my previous estimate with the sums of the masses:

$$r = (350ly) \left({M + m \over M_\odot}\right)^{1 \over 3}$$

 

[PeterDonis]

As I understand it, the way this works is that the cosmological constant creates a relative acceleration between any two objects.

It creates a relative acceleration that depends on the distance between them, yes. The cosmological constant is a form of tidal gravity, and works the same way as tidal gravity does.

So, the answer is just that we have to replace the mass in my previous estimate with the sums of the masses:

I agree that this should be the correct Newtonian approximation.

 

[rede96]

This is incorrect, because I made a substantial error: the cosmological constant in the Newtonian limit is wonky when comparing things happening at different locations.

As I understand it, the way this works is that the cosmological constant creates a relative acceleration between any two objects. The above equation would suggest that that relative acceleration is $2\Lambda r/3$, but it isn't: it's just $\Lambda r/3$. The way this is represented mathematically is to select an origin (any origin will do) and state that every object in the universe has an acceleration relative to that origin.

Thus, if my reasoning is correct, the right way to do this is to just drop the extra $\Lambda r/3$ term:

$$- {GM \over r^2} + {\Lambda r \over 3} = {Gm \over r^2}$$

So, the answer is just that we have to replace the mass in my previous estimate with the sums of the masses:

$$r = (350ly) \left({M + m \over M_\odot}\right)^{1 \over 3}$$

Thanks for your time on this. So does this work for two systems of equal mass?

 

[kimbyd]

Thanks for your time on this. So does this work for two systems of equal mass?

Yes. This reasoning should be relevant for any two reasonably-compact systems of any mass. It will not be valid for more complex configurations (e.g. three galaxies).

 

[rede96]

Yes. This reasoning should be relevant for any two reasonably-compact systems of any mass. It will not be valid for more complex configurations (e.g. three galaxies).

Ok, great. Thanks. So just to check my comprehension, if two small systems are say 500 billion solar masses then do I take the cube root of 2? (1000 / 500) I wasn't sure what the M⊙ indicates sorry.

 

[kimbyd]

Ok, great. Thanks. So just to check my comprehension, if two small systems are say 500 billion solar masses then do I take the cube root of 2? (1000 / 500) I wasn't sure what the M⊙ indicates sorry.

$M_\odot$ is one solar mass. $m$ and $M$ are the two masses of the objects in the system. Does that answer your question?