Expansion of space vs stuff just moving away

[imsmooth]

NOTE: I am not a cosmologist, so if any of my statements are not correct please tell me.

When we observe distance galaxies we can measure how fast they move away using the red-shifting of their light. So how do we know space itself is expanding vs the galaxies are just moving away relative to ours. That is, if the galaxies are cars on a road, how do we know the cars are driving away from each other vs the road is just getting bigger between them?

 

[bapowell]

We cannot. Both interpretations are equally valid.

 

[imsmooth]

If that is the case, why are all the recent stories about inflation talk about space expanding and not about objects moving away at near-light speed?

 

[phinds]

If that is the case, why are all the recent stories about inflation talk about space expanding and not about objects moving away at near-light speed?

Because space IS expanding. Things are not traveling at > c in our reference frame, they are receding from us at > c. I'll leave it to bpowell to discuss his response.

Things at the edge of our observable universe, for example, are receding at about 3c. They most certainly are not actually moving at 3c from any frame of reference. Nothing moves at speeds >c but recession velocities have no limit and break no speed limits.

Let's assume an object X at the edge of our observable universe and an object Y that is half way between X and us and posit that Y is co-moving with us.

To us X is receding at 3c and Y is stationary. According to Y, we are stationary and X is receding at considerably less than 3c because recession is distance dependent. If X were MOVING at 3c relative to us, it would also be moving at 3c according to Y. An object that was comoving with us but positioned out next to X would see X standing still.

I am glossing over very slight proper motions of objects relative to us because the magnitudes are insignificant compared to the recession velocity.

Google "metric expansion" for more discussion.

 

[Drakkith]

Actually, I think the galaxies really are receding from us at a velocity of 3c as viewed from our own frame of reference here on Earth. Also, galaxy Y is co-moving with us, which would require that galaxy X also be moving at 3c relative to it.

 

[imsmooth]

I appreciate the replies on how fast the edge of space is receeding, but this does not answer the question: how do we know space itself is expanding and the objects are not just moving away really fast? Can the red-shift show a >3c expansion? Can we measure a greater-than-light recession, and therefore conclude space is expanding?

 

[phinds]

Actually, I think the galaxies really are receding from us at a velocity of 3c as viewed from our own frame of reference here on Earth.

Yes, that's what I said. They just aren't MOVING relative to us (except for a small proper motion)

Also, galaxy Y is co-moving with us, which would require that galaxy X also be moving at 3c relative to it.

No, that can't be right. If Y were NOT commoving with us, but receding, it would be receding at well under 3c AND it would see X receding at well under 3c (and us receding at the same speed in the opposite direction). X is NOT "moving" relative to us it is RECEDING relative to us and it will recede at a different rate for Y because Y is closer to it.

 

[phinds]

I appreciate the replies on how fast the edge of space is receeding, but this does not answer the question: how do we know space itself is expanding and the objects are not just moving away really fast? Can the red-shift show a >3c expansion? Can we measure a greater-than-light recession, and therefore conclude space is expanding?

All galaxies outside the local cluster are moving away from us. AND they are red shifted. AND the farther away they are the more redshirted they are. Only expansion can explain all that. This is not the only reason. Did you google "metric expansion" ?

 

[Mordred]

NOTE: I am not a cosmologist, so if any of my statements are not correct please tell me.

When we observe distance galaxies we can measure how fast they move away using the red-shifting of their light. So how do we know space itself is expanding vs the galaxies are just moving away relative to ours?

There is one vital detail being overlooked. The above statement implies a preferred reference frame. All non gravitationally bound move away from each other equally. In simpler words take any number of coordinates. 3 or more, regardless of which coordinate you choose as the reference frame. You will measure the same rate of expansion to any other coordinate. However as Phinds pointed out the further away a coordinate is from the referencd frame you will measure a greater redshift.

Unfortunately I am typing from my phone so I cannot post a specific post. However this related thread has an article "Redshift and Expansion" that will provide further details on the distance dependant recessive velocity Phinds mentioned with his 3c example.

https://www.physicsforums.com/showthread.php?t=742950

Another key detail is that when one describes an object as moving. You are stating that the object has inertia. In expansion the objects themselves have no inertia. The space between them is simply increasing.

Edit there is one vital detail in regards to the chosen reference frame I forgot to add. The chosen reference frame must be at rest. A relativistic reference frame would not measure the universe as being homogeneous and isotropic. As his own movement would be a preffered loacation and direction

 

[marcus]

...
When we observe distance galaxies we can measure how fast they move away using the red-shifting of their light. So how do we know space itself is expanding vs the galaxies are just moving away relative to ours. That is, if the galaxies are cars on a road, how do we know the cars are driving away from each other vs the road is just getting bigger between them?

When we observe a redshift, does that tell us how fast the source is moving away NOW? Or does it tell us how fast it WAS moving away back when it emitted the light?
(According to standard model of cosmos, the redshift does not correspond in a straightforward doppler way to either of those two speeds.)

I think the choice of interpretation is partly a matter of making sense of the whole picture. For example we do not only observe galaxies, we observe the Ancient Light ("cosmic microwave background"). The physical explanation of that light is that it has to have been emitted when glowing hot gas filling space had cooled to around 3000 kelvin, because any hotter and ionization of the gas would dazzle and scatter the light. That 3000 kelvin is like a "threshold of transparency" when the fog clears.

We know what mix of wavelengths gas emits at 3000 kelvin. So we measure the wavelengths NOW and we see that the CMB ancient light has been redshifted by a factor of 1000 (more exactly estimated 1090, but roughly 1000). Wavelengths are that much longer.

The temperature and density of the CMB light NOW squares with the model that distances have increased 1000-fold and volumes increased 1000³-fold, and temperature decreased 1000 fold, since the moment the hot gas filling space cleared (so light could pass freely).

The simplest way to model the CMB is with an expanding distances model. What could be moving away, and when? to produce just that redshift of 1000? It's hard to think of an imaginary set-up that would reproduce what we observe with MOTION. Expanding distance works better (plus it is a prediction of GR, the version called Friedman equation that cosmologists use, and GR has been tested a lot.)

Individually, just looking at galaxies (not the whole overall picture), you can go thru some mathematical gymnastics (there was a famous paper by Bunn and Hogg that did this) and explain the redshift as a series of a large humber of intervening doppler shifts. But it's simpler to just equate it to the factor by which the distance has grown while the light was in transit.

It's not mathematically WRONG to look at redshift as the cumulative effect of hundreds of intervening little doppler shifts. It is just clumsy and inconvenient. And then you still have the CMB to explain.

So maybe it just comes down to convenience---astronomers wanting the simplest best-fit model.

 

[Drakkith]

I appreciate the replies on how fast the edge of space is receeding, but this does not answer the question: how do we know space itself is expanding and the objects are not just moving away really fast? Can the red-shift show a >3c expansion? Can we measure a greater-than-light recession, and therefore conclude space is expanding?

There is no single, easy answer. The current model of the universe is known as the ΛCDM (Lambda CDM) Big Bang model. In this model, the universe is described as expanding from a once very dense, very hot state to the cooler, less dense state that it is in now.

Now, it's very important to realize that scientific models are not built upon only one or two pieces of evidence. Such is the case for the ΛCDM Big Bang model. Each piece of evidence, when viewed alone, could have a great many explanations. Only by looking and trying to explain ALL the evidence (or at least as much as possible) can we build the predictive models that science strives for.

For example, a measurement of redshift, by itself, does not let us know whether or not galaxies are moving "through" space or whether space itself is expanding. Instead it is a combination of many different pieces of evidence combined to form a model that predicts and explains what we can and should see to a very, very high degree of accuracy.

Per Ned Wright's Cosmology FAQ: http://www.astro.ucla.edu/~wright/cosmology_faq.html#BBevidence

The evidence for the Big Bang comes from many pieces of observational data that are consistent with the Big Bang. None of these prove the Big Bang, since scientific theories are not proven. Many of these facts are consistent with the Big Bang and some other cosmological models, but taken together these observations show that the Big Bang is the best current model for the Universe. These observations include:

The darkness of the night sky - Olbers' paradox.
The Hubble Law - the linear distance vs redshift law. The data are now very good.
Homogeneity - fair data showing that our location in the Universe is not special.
Isotropy - very strong data showing that the sky looks the same in all directions to 1 part in 100,000.
Time dilation in supernova light curves.

The observations listed above are consistent with the Big Bang or with the Steady State model, but many observations support the Big Bang over the Steady State:

Radio source and quasar counts vs. flux. These show that the Universe has evolved.
Existence of the blackbody CMB. This shows that the Universe has evolved from a dense, isothermal state.
Variation of TCMB with redshift. This is a direct observation of the evolution of the Universe.
Deuterium, 3He, 4He, and 7Li abundances. These light isotopes are all well fit by predicted reactions occurring in the First Three Minutes.
Finally, the angular power spectrum of the CMB anisotropy that does exist at the several parts per million level is consistent with a dark matter dominated Big Bang model that went through the inflationary scenario.

To add to this, I would say that we also have a theory of gravity, General Relativity, which is perfectly able to have an expanding universe in which space is expanding instead of galaxies moving through space. GR is a theory of geometry, and a basic explanation of how this is possible is that on the largest scales of the universe, geometry itself is "dynamic", meaning that it can change. This changing geometry is a very "intuitive" way of explaining expansion, meaning that expanding space falls very naturally right out of GR.

Now, I know this isn't quite a direct answer to your question, but the fact is that the issue is, as far as I know, a little too complicated to explain with just a few pieces of evidence and ignoring the current model as a whole.

Yes, that's what I said. They just aren't MOVING relative to us (except for a small proper motion)

Okay, if you use "recede" and "moving" as two different words, then sure.

No, that can't be right. If Y were NOT commoving with us, but receding, it would be receding at well under 3c AND it would see X receding at well under 3c (and us receding at the same speed in the opposite direction). X is NOT "moving" relative to us it is RECEDING relative to us and it will recede at a different rate for Y because Y is closer to it.

That isn't possible. If Y is co-moving with us then X MUST be receding at 3c relative to both of us. I think the problem here is that it isn't possible for Y to be co-moving with us because of the expansion of space.

 

[Mordred]

Roflmao... You have no idea how much of a headache it was to include Bunn and Hoggs work into the Redshift and expansion article. Thankfully PAllen's assistance and extreme patience stepped me through it lol.

 

[bahamagreen]

Another key detail is that when one describes an object as moving. You are stating that the object has inertia. In expansion the objects themselves have no inertia. The space between them is simply increasing.

Could this be clarified?

What is the relationship between the object and the expanding space that compels the object to move with or "stick to" the expanding space? How is inertia of objects no longer independent, but now dependent on the space metric?

You are suggesting that objects subject to expansion offer no inertial resistance to the expansion, which is also a net acceleration, and a local accelerometer would show no indication... very similar to free fall in a gravitational field...?

 

[phinds]

That isn't possible. If Y is co-moving with us then X MUST be receding at 3c relative to both of us. I think the problem here is that it isn't possible for Y to be co-moving with us because of the expansion of space.

So you think that recession velocity is independent of distance? That is just flat wrong.

As a thought experiment, there is nothing wrong with having Y commoving with us and since recession velocity IS dependent on distance, X's recession velocity from Y will not be the same as X's recession velocity from us.

 

[Mordred]

Could this be clarified?

What is the relationship between the object and the expanding space that compels the object to move with or "stick to" the expanding space? How is inertia of objects no longer independent, but now dependent on the space metric?

You are suggesting that objects subject to expansion offer no inertial resistance to the expansion, which is also a net acceleration, and a local accelerometer would show no indication... very similar to free fall in a gravitational field...?

Objects at rest require a force to act upon them to get them moving and a force to act upon it to stop that motion. The three laws of inertia apply on this case. Now consider the geometry involved in expansion metrics. All objects are increasing in distance from each other equally. (not gravitationally bound)
This alone tells cannot be explained as the result of a kinetic explosion, you would have a centralized source. It also cannot be explained as a higher pressure/temperature flowing to a lower pressure/temp as this too would give a preferred location.

In order for expansion to be explained by inertia would require some very unusual mechanism to be both homogeneous and isotropic. After all what force could act upon those objects that would fit a homogeneous and isotropic expansion? I certainly can't think of any.

In expansion however the geometric distance between objects is simply put increasing. This increase in volume is both homogeneous and isotrophic. Think of it simply as an increase in geometric volume.

google metric expansion for more detail

 

[bahamagreen]

The problem is applying the laws to pairs of objects... individual objects may appear to be each locally inertially at rest, but the distances between pairs are not simply increasing; the distances are accelerating. The mutual acceleration of distance between massive objects typically implies force.

Acceleration is absolute, yet with expansion, two objects whose separation distance is accelerating are inertially at rest. What is the mechanism for that?

I'm not seeking to explain expansion by inertia; I'm trying to see how expansion overcomes the inertia of objects. Saying that the objects don't move but rather recede is at the heart of it - that is saying objects resist changes in movement because of their inertia, but objects don't resist recession, which is an acceleration of distance.

How do two objects accelerating their distance apart distinguish movement from recession, and so offer inertial resistance to the former but not the later?

 

[Drakkith]

So you think that recession velocity is independent of distance? That is just flat wrong.

No, that's not what I'm saying.

As a thought experiment, there is nothing wrong with having Y commoving with us and since recession velocity IS dependent on distance, X's recession velocity from Y will not be the same as X's recession velocity from us.

I don't think its possible for a galaxy to be co-moving with us when it's at a distance where the recession velocity due to the expansion of space is greater than c. That would seem to imply that the galaxy would need to be moving "through" space at a speed greater than c in order to stay co-moving with us.

 

[russ_watters]

I appreciate the replies on how fast the edge of space is receeding, but this does not answer the question: how do we know space itself is expanding and the objects are not just moving away really fast? Can the red-shift show a >3c expansion? Can we measure a greater-than-light recession, and therefore conclude space is expanding?

Yes, that was all already answered yes.

 

[Cosmobrain]

Your question is pretty interesting, actually. It has been answered already and I don't think I can add much more.

There are three types of redshift:

Doppler redshift: This is caused by an object moving away from us. It's maximum value is 1.4. For a greater redshift, the object would have to be traveling faster than light or...
Cosmological redshift: The redshift can be caused by the expansion of space itself that stretches the electromagnetic wave. That's how Hubble can measure redshifts higher than 8
Gravitational redshift: This is caused by an object that stretches the wave due to its gravitational pull. This is only significant when the body is very dense, like a neutron star. This doesn't contribute much to the measured redshift of the galaxies

correct me if I'm wrong

 

[Drakkith]

Your question is pretty interesting, actually. It has been answered already and I don't think I can add much more.

There are three types of redshift:

Doppler redshift: This is caused by an object moving away from us. It's maximum value is 1.4. For a greater redshift, the object would have to be traveling faster than light or...

Doppler shift resulting from relative velocity is indistinguishable from other types of redshift and can have any value. Using the relativistic formula instead of the classical one gives you the correct values.

 

[Mordred]

The problem is applying the laws to pairs of objects... individual objects may appear to be each locally inertially at rest, but the distances between pairs are not simply increasing; the distances are accelerating. The mutual acceleration of distance between massive objects typically implies force.

per m³ the cosmological constant has a low energy density, its influence is easily overpowered. Why dark energy has such a large influence is the sheer amount of overall volume. As that volume increases the total amount of energy the cosmological constant gains becomes exponential. The density per volume is constant, what changes is the volume. hence the acceleration.

Acceleration is absolute, yet with expansion, two objects whose separation distance is accelerating are inertially at rest. What is the mechanism for that?.

treat the cosmological constant as a negative pressure influence among a positive matter influence, however as stated above its easily overpowered so its influence is seen mainly in the spaces between objects

I'm not seeking to explain expansion by inertia; I'm trying to see how expansion overcomes the inertia of objects. Saying that the objects don't move but rather recede is at the heart of it - that is saying objects resist changes in movement because of their inertia, but objects don't resist recession, which is an acceleration of distance..

again the amount of influence per locality environment dictates the amount of measurable influence the cosmological constant has. One key note is that as the cosmological constant is homogenous and isotropic, (the same at any location/uniform) an object would have the same amount of influence on all sides and remain at rest. However the volume between two objects can increase.
[/QUOTE]

 

[bahamagreen]

Mordred,

Your answers don't seem to address my questions; that may be because I'm having trouble making sense of them.

I didn't ask the source of the acceleration, I asked about the effect of that acceleration on objects (the mechanism by which the acceleration of space acts to separate objects) - specifically why the objects being relatively accelerated apart from each other don't experience inertial effects... why is absolute acceleration not apparent?

What do you mean by "...so its influence is seen mainly in the spaces between objects."? What is being influenced by an acceleration if not massive objects?

If "The density per volume is constant..." then what does "...the amount of influence per locality environment dictates the amount of measurable influence the cosmological constant has." possibly mean?

"...an object would have the same amount of influence on all sides and remain at rest." Yet two objects would also have the same amount of influence on all sides, yet recede (accelerate) from each other? What is the mechanism for that?

"However the volume between two objects can increase." A restatement; is it simply that nobody knows the mechanism? Is there a purpose to using the word "volume" rather than "distance" to describe what is between two points?

 

[Nugatory]

Could this be clarified?

What is the relationship between the object and the expanding space that compels the object to move with or "stick to" the expanding space? How is inertia of objects no longer independent, but now dependent on the space metric?

You are suggesting that objects subject to expansion offer no inertial resistance to the expansion, which is also a net acceleration, and a local accelerometer would show no indication... very similar to free fall in a gravitational field...?

The comparison with gravity is pretty good.

Inertia says that an object wants to move in a straight line, which seems simple enough; it satisfied most classical physicists most of the time for most of several centuries. However, if you poke enough at it, you'll realize that we've just shifted the discussion to what a straight line is and how straight lines are related to one another.

We can describe the attractive gravitational force in terms of initially parallel straight worldlines eventually intersecting... But suppose they diverged instead? Then we'd have the expansion effects that we observe at cosmological scales.

 

[Mordred]

fair enough, let's try another tack.I didn't ask the source of the acceleration, I asked about the effect of that acceleration on objects (the mechanism by which the acceleration of space acts to separate objects) - specifically why the objects being relatively accelerated apart from each other don't experience inertial effects... why is absolute acceleration not apparent?

lets look at this portion first.

according to the laws of inertia f=ma in order for an object to accelerate for rest the object must have a force act upon it, and in order to come to rest from being in motion it will also require a force to act upon it.

now let's treat the cosmological constant as a force surrounding a star, or galaxy. Let's treat this as being in a region with no other influence such as gravity from other stellar objects.

The energy density of the cosmological constant is the same everywhere, even in gravationally bound regions. Roughly 6*10^-10 joules per m³.

IN our standalone star/galaxy example the cosmological constant will act upon the star/galaxy equally from all sides of the star/galaxy. So the net balance of the force acting upon that object will be zero. As the net balance of forces acting upon said object is zero, the object will remain at rest.

So given what I just described does the object gain inertia? The correct answer is no it doesn't, it will remain at rest. This will be true for all objects in space. As the cosmological constant is a uniform force,(vacuum pressure) its net balance of force on any object will be zero.

So why do objects move apart?

to answer that we have to look at the volume. If for example the volume of the universe stays static and cannot increase. IE a closed container (this an example only and not a descriptive of the universe). Then no object would separate from each other.

However in our universe the volume is influenced by the relation of the cosmological constant and gravity. If gravity is stronger the universe collapses. If the cosmological constant is greater the universe expands.

So the reason why objects appear to move apart is due to changes in volume, not due to momentum.

If you think about the above those properties would be the same as two molecules inside a closed container.. The forces due to pressure on any molecule would be the same on all sides of the molecule. So the net balance of forces acting upon the molecule is zero. Therefore the molecule is at rest. If you increase the volume of the container though, then the molecules will separate to a uniform distribution.

keep in mind the universe has no container walls or boundary lol however the same principles apply.

The above should also answer why I chose to use volume instead of distance.

hope this clarifies why we simply state the expansion of the universe in terms of changes in volume/distance as opposed to objects moving apart due to momentum.edit had to add a change, wrong value for energy density above.

 

[oldGhost1]

I’m so glad you’re having this particular discussion bahamagreen. This is one of the issues I’ve been mulling over for a week or so now, trying to find that missing piece. For me the pertinent question was

What is the relationship between the object and the expanding space that compels the object to move with or "stick to" the expanding space?

In other words, “in order for the distance between our galaxy and a distant one to increase, the distant galaxy must somehow ‘grip’ onto the underlying space as it expands.” And if I’m reading it correctly this post from Mordred has just answered it.

One key note is that as the cosmological constant is homogenous and isotropic, (the same at any location/uniform) an object would have the same amount of influence on all sides and remain at rest.

You see it’s all too easy to limit our thinking on this problem to just the two galaxies. But as Mordred implies, these galaxies do not exist in isolation. The same mechanism of expansion is also happening on the other side of the distant galaxy, holding it in position (as indeed it is on our own). Roughly speaking, “the pressure of expansion doesn’t move a galaxy because the pressure is being applied to the galaxy from all directions”. I hope that’s basically correct and that it helps (although I see now that Mordred has posted a more detailed explanation of this point).

Finally I can free myself of considering space as fabric-like... Thanks all round

 

[Mordred]

“the pressure of expansion doesn’t move a galaxy because the pressure is being applied to the galaxy from all directions”.

Correct, much of cosmology can be described as a form of gas, and by its thermodynamic relations in terms of a perfect fluid. All perfect fluids can be described by the ideal gas laws.

http://en.wikipedia.org/wiki/Equation_of_state_(cosmology)
http://en.wikipedia.org/wiki/Ideal_gas_law

 

[bahamagreen]

I was with you until "So the reason why objects appear to move apart is due to changes in volume, not due to momentum."

I don't see how that is explanatory; what is the mechanism?

The two molecules example makes no sense to me. It is unclear if the molecules are initially moving. You suggest "The forces due to pressure on any molecule would be the same on all sides of the molecule.", but pressure forces come from contact with the containment, which will act in one direction at the time it happens. There is no "same on all sides of the molecule" force. Then you say the molecules are at rest... so there are no containment contacts, so no pressure forces are even happening and the volume of the container (the distance to the containment boundary) is irrelevant... so what is the force such that the molecules will separate?

"Therefore the molecule is at rest. If you increase the volume of the container though, then the molecules will separate to a uniform distribution." This sounds like you are assuming the molecules are in motion... Again, you just make the statement without indicating any mechanism.

I'm asking how changes in motion (accelerating separation of masses) are being done in such a way that the individual masses don't measure an acceleration, an explanation for how the inertial properties of the masses (resistance to acceleration) is absent during this kind of separation.

 

[Mordred]

Please read the equation of state link I posted the, relation of energy density to pressure is determined by the equation of state

$w=\frac{P}{\rho}$, where w is the Dimensionless ratio of that relation.

 

[Nugatory]

I'm asking how changes in motion (accelerating separation of masses) are being done in such a way that the individual masses don't measure an acceleration, an explanation for how the inertial properties of the masses (resistance to acceleration) is absent during this kind of separation.

This is basically the same question as how two isolated masses near one another will accelerate towards one another (as measured by the shrinking separation between them) under the influence of gravity without measuring any acceleration. The only difference is that the worldlines are diverging instead of converging.

 

[Mordred]

As added reading this article will help show the mathematics of the FLRW metric in terms of distance. This is more related in terms of geometry relations to flat or curved universes. However the metrics are handy in terms of how galaxies separate in a flat universe.

Universe geometry

The origins of the universe is unknown in cosmology. The hot big bang model only covers the history of the universe from 10^-43 seconds forward. Prior to that is described as a singularity. However its important to note that the singularity is not a black hole style. Instead singularity in this case simply means a point in time where our mathematics can no longer accurately describe it. Numerous youtube videos and pop media articles would have you believe our universe exploded from some super particle. This was never predicted by the hot big bang model.

The observable universe which is the portion we can see is a finite, sphere with a radius of 46 Gly, which is equal to 46 billion light years. The 46 Gly particle horizon refers to the today's distance of objects, whose radiation emitted in the past we receive today. The overall size of the universe is not known, it could be infinite or finite. If its infinite now then it would be infinite in the past, a finite value can never become infinite. So why is geometry so important to cosmology if we know the size of the observable universe? The answer to that question lies in how geometry affects the following aspects, Light paths, rate of expansion or collapse and overall shape.

In regards to light paths and geometry a closed universe described as a sphere will have two beams of light emitted at different angles eventually converge. An open hyperbolic universe such as a saddlebag will have those same two light beams diverge. A flat universe will have parallel light paths (provided the beams at emission were parallel to begin with)
You will notice on each image there is a triangle, this triangle represents how the geometry affects our measurements. In a flat curvature the three angles of a equilateral triangle will add up to 180⁰. A positive curvature will add up to greater than 180⁰, a negative curvature will add up to less than 180⁰

Image from http://universeadventure.org

The topography of the universe is determined by a comparison of the actual density (total density) as compared to the critical density. The critical density is represented by the following formula

$\rho_{crit} = \frac{3c^2H^2}{8\pi G}$

P=pressure
c=speed of light
G= gravitational constant.

density is represented by the Greek letter Omega $\Omega$ so critical density is $\Omega crit$
total density is

$\Omega$total=$\Omega$_{dark matter}+$\Omega$_baryonic+$\Omega$_radiation+$\Omega$_{relativistic radiation}+${\Omega_ \Lambda}$

$\Lambda$ or Lambda is the value of the cosmological constant often referred to as "dark energy" more accurately it is the vacuum pressure that attributes to expansion.
the subscript "₀"for $\Omega$ shown in the image above denotes time in the present.

Energy-density is the amount of energy stored per unit volume of space or region. Energy per unit volume has the same physical units as pressure, and in many circumstances is an exact synonym.

$\Omega=\frac{P_{total}}{P_{crit}}$
or alternately
$\Omega=\frac{\Omega_{total}}{\Omega_{crit}}$

Geometry in 2D
In developing a theory of space-time, where curvature is related to the mass-energy density, Scientists needed a way of mathematically describing curvature. Since picturing the curvature of a four-dimensional space-time is difficult to visualize. We will start by considering ways of describing the curvature of two-dimensional spaces and progress to 4 dimensional spaces.
The simplest of two-dimensional spaces is a plane, on which Euclidean geometry holds.
This is the geometry that we learned in high school: parallel lines will go off to infinity
without ever crossing; triangles have interior angles that add up to 180. Pythagoras’
theorem which relates the lengths of the sides of a right triangle also holds:
c² = a² + b²
where c is the length of the hypotenuse of the right triangle, and a and b are the
lengths of the other two sides. One can generalize the Pythagorean theorem to three dimensions as well:
c²= a² + b² + c²
see image 2.0 below


On a plane, a "geodesic" is a straight line(shortest distance between two points). If a triangle is constructed on a flat 2 dimensional plane by connecting three points with geodesics. The curvature can be represented in 2D, if you establish each angle of a equilateral triangle with
$\alpha$,$\beta$,$\gamma$ for a flat geometry this follows the relation

$\alpha$+$\beta$+$\gamma$=$\pi$.

image 1.0

On a plane, (shown above) we can set up a cartesian coordinate system, and assign to every point a coordinate (x; y). On a plane, the distance ds between points (dx and dy) is given by the relation
$d{s^2}=d{x^2}+d{y^2}$

If a triangle is constructed on the surface of the sphere by connecting the angles will obey the relation

$\alpha$+$\beta$+$\gamma$=$\pi+{AR^2}$

image 1.1
where A is the area of the triangle, and R is the radius of the sphere. All spaces in which
$\alpha$+$\beta$+$\gamma$>$\pi$ are called positively curved" spaces. It is a space where the curvature is homogeneous and isotropic; no matter where you draw a triangle on the surface of a sphere, or how you orient it, it must always satisfy the above equation.
"On the surface of a sphere, we can set up polar coordinates "north pole" and "south pole" and by picking a geodesic from the north to south pole to be the "prime meridian". If r is the distance from the north pole, and $\theta$ is the azimuthal. angle measured relative to the prime meridian,"⁽¹⁾ then the distance ds between a point (r; $\theta$) and another nearby point (r+dr+$\theta$+d$\theta$) is given by the relation

${ds^2} = {dr^2} + {R^2} {sin^2}(r/R)d\theta^2$

"An example of a negatively curved two-dimensional space is the hyperboloid, or saddle-shape. A surface of constant negative curvature. The saddle-shape has constant curvature only in the central region, near the "seat" of the saddle."⁽¹⁾. David Hilbert proved that a constant negative curvature cannot be constructed in a Euclidean 3D space. Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices $\alpha$
$\beta$,$\gamma$ obey the relation $\alpha$+$\beta$+$\gamma$=$\pi-{AR^2}$.

${ds^2} = {dr^2} + {R^2} {sinH^2}(r/R)d\theta^2$

image 1.2

A negative curvature is an open topography

If a two-dimensional space has curvature or flat which is homogeneous and isotropic, its geometry can
be specified by two quantities k, and R. The number k, called the curvature constant, R is the radius

k = 0 for a flat space,
k = +1 for a positively curved space,
k = -1 for a negatively curved space

Geometry in 3D
A two dimensional space can be extended to a three-dimensional space, if its curvature is homogeneous and isotropic, must be flat, or have uniform positive curvature, or have
uniform negative curvature.

The 3 possible metrics for homogeneous and isotropic 3D geometries can be represented in the form ds²=dr²=S_k(r)²dΩ²

where
dΩ²=dθ²=sin²d$\phi$²


$$S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}$$

If a three-dimensional space is flat (k = 0), it
has the metric

ds² = dx² + dy² + dz² ;

expressed in cartesian coordinates or

${ds^2} = {dr^2} +{r^2}[d\theta^2 + {sin^2} d\phi^2]$

If a three-dimensional space has uniform positive curvature (k = +1), its
metric is

${ds^2} = {dr^2} +{R^2}{sin^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]$

A negative curvature in the uniform portion has the metric (k=-1)

${ds^2} = {dr^2} +{R^2}{sinH^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2]$

Geometry in 4D

Thus far we have discussed the 2 and 3 dimensional components. The Friedmann-Lemaitre-Robertson-Walker metric (FLRW) can be used to describe the 4D dimensions with the use of a(t). a(t) is the scale factor. See the redshift and expansion article for more information or the cosmocalc link on the main page. Scale factor in a homogeneous and isotropic universe describes how the universe expands or contracts with time.
The FLRW metric can be written in the form

$d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]$

references
(1)"Introductory to Cosmology" Barbera Ryden"
images 1.0,1.1 and 1.2 (see (1))
(2)"Modern Cosmology" Scott Dodelson
(3)"lecture notes, Introductory to Cosmology" Dr. Ka Chan Lu

 

[Drakkith]

I was with you until "So the reason why objects appear to move apart is due to changes in volume, not due to momentum."

I don't see how that is explanatory; what is the mechanism?

Here's my understanding. Someone tell me if I'm wrong.

The answer you're seeking is geometry. First, consider that gravity itself is the result of the geometry of spacetime and that an observer cannot use an accelerometer to find out whether they are in free fall due to gravity or whether they are motionless in space. An inertial frame and one in free fall under the influence of gravity are identical in accordance with the equivalence principle of General Relativity. In other words, objects in free fall are not accelerating, where acceleration here means "proper acceleration", which is what accelerometer measures. This is very similar to what happens to galaxies far apart under the influence of expansion.

A proper understanding requires us to discuss what a "metric" is.
Per wiki: A metric defines how a distance can be measured between two nearby points in space, in terms of the coordinate system. Coordinate systems locate points in a space (of whatever number of dimensions) by assigning unique positions on a grid, known as coordinates, to each point. The metric is then a formula which describes how displacement through the space of interest can be translated into distances.

Metrics are used in something called a metric tensor which is used to describe the overall properties of a surface of any number of dimensions. (Even normal geometry itself is a 2d metric tensor)

Metric tensors are required in order to find the shortest distance between two points. A good example is Earth's surface. Consider that the shortest distance between two points on the Earth's surface is not a straight line, but by a curved line. In addition, the standard rules of flat geometry don't apply. For example, if two observers start out from different points on the equator and head north, they will eventually run into each other at the north pole. In other words, two lines that were initially perpendicular to the equator and parallel to each other end up crossing.

Similarly, the presence of energy and mass causes the geometry of spacetime to shift away from the flat euclidean geometry we are used to. And just like we can have planets of different sizes, which would require changes in the metrics describing them, the metric of space changes under the influence of mass and energy.

Now, here's the key point to all this. Let's say that you take off in a spaceship and travel through the solar system. As you travel, you will notice the geometry of space changing as you move relative to other objects. So you can say that the metric changes as your position in space changes. But, what about time? If we were able to freeze everything so that nothing in the solar system was moving, we would see that the metric does not change with time. However, the situation is different when we get to the very large scale of intergalactic distance. It turns out that when the curvature of space due to gravity is small enough, the metric itself DOES change over time.

Thus, the expansion of space is a result of the metric changing with time. In other words, the very geometry of the universe is dynamic and, just like gravity, this changing geometry does not result in a proper acceleration. No force is required to hold you in place at your current location in space, nor is anything required to force galaxies to "stick" to space as the geometry changes. Remember that geometry itself is used to describe the distance, shape, and position of real objects relative to other objects, and all the math and grids are tools used to understand the real world. If you are imagining some sort of underlying "grid" that objects need to "stick" to, then that is an incorrect understanding of what geometry is.

To conclude this long post, remember that there are no forces at work here. A force would result in proper acceleration, which is measurable. We are not accelerating. Other galaxies are not accelerating. We are all afloat on our little islands of stability in an ever changing universe.

 

[Mordred]

This article has a decent coverage of dark energy in terms of perfect fluid equations and the FLRW metric.

The Cosmological Constant and Dark Energy by Peebles. keep in mind its an older article, however still informative.

http://arxiv.org/pdf/astro-ph/0207347v2.pdf

 

[bahamagreen]

Let me get caught up... see where things stand.

Mordred: "...relation of energy density to pressure is determined by the equation of state..."
This looks like another restatement. Repeating statements does not reveal the mechanism to me.

Nugatory: "The only difference is that the worldlines are diverging instead of converging."
The lack of proper acceleration of expansion looks similar to gravitational free fall, but we don't think of objects moving toward each other because the space is contracting between them. We think of the objects as translating through the space between them. This suggests to me the possibility of a different mechanism.

Mordred: "...much of cosmology can be described as a form of gas..."
The distinguishing thing to me about gas is that the objects which comprise the gas have a frequency of collisions based on their average velocity and mean free path, which for the oxygen and nitrogen molecules in my room right now is on the order of 10^10 collisions per second. Pressure is the result of these collisions. Cosmological collisions are a rarity, so I don't see the possible mechanism for a "pressure". The objects moving apart are doing so without making contact collisions.

Mordred: "As added reading..."
If you substitute "geometry", then the question becomes, "How does geometry cause the separation of objects as the space is expanded?"

Drakkith: "The answer you're seeking is geometry."
If so, I'll need to understand by what action geometry moves masses.

"Consider that the shortest distance between two points on the Earth's surface is not a straight line, but by a curved line."
I would consider that it is a line through the interior of the surface. If you stipulate that the line must curve with the sphere's surface, that is fine, but that is a longer distance.
Does this mean I won't be satisfied with a geometric answer?

 

[Drakkith]

Drakkith: "The answer you're seeking is geometry."
If so, I'll need to understand by what action geometry moves masses.

We're getting into the realm where words need to be used very carefully. I would say the word "action" suggests a force, which suggests a proper acceleration. There is no force. I expect you are using it differently.

The only thing I can say is that there is no action. At least not in any sense of the normal use of the word. No force is required to move these masses, and they never undergo proper acceleration. They also do not gain inertia. I believe I remember reading somewhere that if you could magically teleport to a galaxy 20 billion light years away, while retaining the velocity you had prior to the teleport, the galaxy would not be moving at a significant fraction of c relative to you like it is relative to us back here on Earth.

"Consider that the shortest distance between two points on the Earth's surface is not a straight line, but by a curved line."
I would consider that it is a line through the interior of the surface. If you stipulate that the line must curve with the sphere's surface, that is fine, but that is a longer distance.

Yes, I mean a line on the surface of the Earth, not through the Earth.

Does this mean I won't be satisfied with a geometric answer?

You may or may not be satisfied with the answer, yet, as far as I know, it is the correct one.

 

[Jonathan Scott]

After reading various opinions about the usefulness of the concept of expanding space (including those of J A Peacock in "A Diatribe on Expanding Space" and Sean Carroll's comments in his 2008 blog entry "Does Space Expand?") I think a simple way to understand it is to consider a universe model consisting of a cone, where the distance from the point is the time since the Big Bang and the tangential distance gives the one-dimensional separation in space.

It is then clear that there is more space as time elapses, but nothing actually stretches, so saying it is expanding could be misleading.

If two objects remain a fixed fraction of the size of the universe apart, they then diverge at a constant rate. However, if two objects are locally moving in parallel, they remain moving in parallel, at least if the rate of expansion is approximately constant.

I think this is a reasonable analogy, but this isn't something I've looked into in great depth, so feel free to correct me if I'm wrong.