B Threshold of gravitational pull over dark energy

[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?

 

[rede96]

M⊙M⊙M_\odot is one solar mass. mmm and MMM are the two masses of the objects in the system.

Ah ok, got it. Thanks again.

 

[rede96]

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

Only for the purposes of understanding some basic underlying principles. If it helps to me to understand, nothing wrong with that. For example I was treating a system and all parts of that system as just a single mass, so to speak, so I could work out in today's DE dominated universe just where the threshold distance was that another system would no longer become gravitationally bound.

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.

And what is density... Mass per unit volume. And as distances increase volume goes up the mass/radiation density goes down. It's the same thing. But the point was that it is inertia from the big bang that caused expansion not dark energy as you've said. And my personal way of thinking about it is that only when the distances between comoving objects became great enough could dark energy take over and cause accelerated expansion. So very crudely the critical size of the universe was just over 2/3 of what it is now. (As I thought I read DE started to dominate about 4 billion years ago)

You really, really need to be careful about the distinction between moving apart and accelerating apart.

I'm not sure what context you mean this in? If something is accelerating apart it is moving apart. If the rate of change isn't important to the context, moving apart is fine. Or did I miss something?

In any case I found this thread really useful and helped to straighten a few things I hadn't understood properly. So just wanted to say thanks to everyone for their help.

 

[PeterDonis]

If it helps to me to understand, nothing wrong with that.

But I'm not sure it's helping you to understand, since you keep making incorrect statements.

And what is density... Mass per unit volume. And as distances increase volume goes up the mass/radiation density goes down. It's the same thing.

No, it isn't, because this naive intuitive reasoning doesn't explain why dark energy density doesn't also decrease as "distances increase". Nor does it explain why the density of matter (more precisely, non-relativistic matter) goes like the inverse cube of the scale factor, while the density of radiation (more precisely, relativistic matter and radiation) goes like the inverse fourth power of the scale factor.

In other words, "distances increase" is not the fundamental thing that is going on here. The fundamental thing is that the densities of dark energy, matter, and radiation are behaving differently along comoving worldlines.

the point was that it is inertia from the big bang that caused expansion not dark energy as you've said

Inertia from the big bang is why comoving objects are moving apart, yes. But it's not why they are accelerating apart. The latter requires dark energy.

my personal way of thinking about it is that only when the distances between comoving objects became great enough could dark energy take over and cause accelerated expansion. So very crudely the critical size of the universe was just over 2/3 of what it is now.

But this number, 2/3 of the current scale factor, if you try to derive it as we have been doing in this thread, should not be a universal number; it should depend on the masses of the comoving objects, since the distance at which two objects are just on the threshold of not being gravitationally bound depends on their masses.

If we calculate the critical scale factor on the basis of average densities, however, it is a universal number: heuristically, it's the scale factor at which the average matter density just equals the average dark energy density (radiation density can be neglected here because by a few billion years ago it was already much smaller than the matter density, because radiation density decreases faster).

If the rate of change isn't important to the contex

But it is important, which is why the distinction between "moving apart" and "accelerating apart" is important. Failing to recognize this has already tripped you up a couple of times in this thread; that's why I emphasized it.

 

[rede96]

Inertia from the big bang is why comoving objects are moving apart, yes. But it's not why they are accelerating apart. The latter requires dark energy.

Yes I know, where did I imply this wasn't the case?

No, it isn't, because this naive intuitive reasoning doesn't explain why dark energy density doesn't also decrease as "distances increase". Nor does it explain why the density of matter (more precisely, non-relativistic matter) goes like the inverse cube of the scale factor, while the density of radiation (more precisely, relativistic matter and radiation) goes like the inverse fourth power of the scale factor.

Actually it does, at least for two of them. I'll have to go back to my notes on FRW to understand why relativistic matter and radiation goes like the inverse fourth power of the scale factor as I can't remember.

But the reason non-relativistic matter goes like the inverse cube of the scale factor as distances increase is the same principle as a cube expanding with a finite amount of matter in it. That's simple and how we model expansion anyway. All I am saying is what is caused that 'cube' to expand initially was inertia from the big bang. And eventually the matter / radiation density reached a critical point, which will correspond to a specific scale factor / distance, and dark energy took over and accelerating expansion started.

As for dark energy how I understand it is it doesn't expand as its part of the structure of space / spacetime and is a kind of geometry not necessarily a density per se. Although modeling it on a constant energy density does work in the FRW equations.

But how I've come to think about it is like having a huge ball somewhere on the surface of the Earth and I put two small toy cars at the top of the ball and give them a push apart so they are traveling apart in opposite directions. They move apart slowly at first as there is some friction and the gradient isn't very big over small distances, but as I have given them a sufficient push, they continue to move apart until a point where the gradient increases sufficiently, gravity takes over and they start to accelerate apart. That's sort of how I imagine dark energy to work. As distances increase, the attractive force of gravity reduces sufficiently to allow expansion to accelerate.

Now I know my dark energy analogy isn't technically correct and there is a lot more for me to understand. But as a basic principle to explain things to a layman like me I don't see anything wrong with it.

 

[PeterDonis]

the reason non-relativistic matter goes like the inverse cube of the scale factor as distances increase is the same principle as a cube expanding with a finite amount of matter in it.

In a universe that is spatially flat, yes, this intuitive reasoning gives you the right answer. But it does not explain why the matter density still goes like the inverse cube of the scale factor in a universe that is spatially closed or spatially open, i.e., not spatially flat (which means the geometry of a spacelike 3-surface of constant time is not Euclidean).

I know my dark energy analogy isn't technically correct and there is a lot more for me to understand. But as a basic principle to explain things to a layman like me I don't see anything wrong with it.

The question is whether it leads you to make correct predictions.

 

[rede96]

In a universe that is spatially flat, yes,

Yes, of course. All of my thought process is based on the assumption of a specially flat universe.

The question is whether it leads you to make correct predictions.

That's why I am trying to get more into the Math as conceptually my understanding is stuck I think. It's just finding the time as I will have to go back to basics.

Anyway, thanks again for your time and patients.

 

[PeterDonis]

All of my thought process is based on the assumption of a specially flat universe.

Ok, fair enough. Since our best current model of our actual universe is that it's spatially flat, this is not an unreasonable approach. As long as you're aware that you won't necessarily be developing an understanding of the most general underlying laws, but only of one particular solution to those laws (the one that describes our actual universe).