Relativity and Cosmology

Physicist, I apologized to you for the confusion that I caused you. (I have already apologized to StardardsGuy.)

After Jorrie's comments, I clarified some issues in my head and I have modified the drawing. Please, see my post #122 (and my post #118). Any further comments are wellcome.

You are right about the whole "point B" issue and your argument about the lack of homogeneity concerning the cmb. But as you will see in post #122, B is actually a circle (instead of a point) -as the "edge" of the heart is now located in the centre (BB)-. (In the real world it's a sphere and represents the cosmic horizon.)

Hi George,

Fyz is right that your diagrams do not represent correct cosmology. The way to do such diagrams is to work back from a=1 (now) to a => 0 (BB), integrating the Friedman equations, using accepted values for the parameters. Then trace a photon's path backwards along the properly shrinking hypersphere and find out when and where it reaches z~1100, the largest redshift (optically) observable (or whatever other redshift you desire).

The radius of curvature of the universe is at least 100 billion light years (could be much larger). Give your sphere that radius and you will find your 'heart' is squashed so that your photon paths drop rather quickly towards the center, not going very far around the sphere. Your 'top' and 'bottom' photons could only trave an angle ~26° each around the sphere and your 'heart' must lie within this cone. Make the radius of curvature larger and virtually nothing of the 'heart' remains. This is one reason why I stopped using this representation.

-J

Hi G.K.,Am I misinterpreting your drawing in #122? If so, please explain.
I had taken it to represent a plane that passes through the origin of the universe and through the observer's location A. In that case the intersection of the plane with the sphere B at the limits of observation will be just two points; and if we accept the Jorrie's interpretation of the cosmologists' current view** of the evolution of the growth of the universe, these will both be on the right hand side of the origin.

Regards

Fyz

**Some other sources give somewhat different values - but the principle is unchanged

I've found a nice equation to help understand the cosmological constant without tensors.

Take the gravitational force due a spherical mass M given in the equation below:

g = -GM/r² + c²Λr/3

Is this equation ok Jorrie? I find it much easier to understand since it's easy to see that close to a massive object gravitational force dominates, but far enough away the cosmological term dominates, which I think explains what you mean by tethering.

Here's the source where I found it

Ever since I saw the above equation this morning I've been thinking about the cosmological constant.

It's that equation:

g=-GM/r² + c²Λr/3

The second term reminded me of something, especially the dividing by 3 term. So I talked to some people and then I remembered, that second term reminded me of the following equation for the gravitational force felt by an object dropped through a hole in the Earth

which yields:

which is of the same form but the sign is wrong. Ignoring the sign problem for a moment, can we make the cosmological constant problem look this way?

Well, I found the following expression for the energy density of a vacuum as it's related to the cosmological constant:

ρ_energyvacc²=Λc⁴/8πG

Using E=mc² to give me ρ_energyvac=c²ρ_massvac I substitute into the above and cancel terms getting:

8πGρ_massvac = Λ

Now lets look at that second term I was talking about. If you sub in the above for Λ

and rearrange you get:

(4/3)πρ_massvac G2c²r

which looks really similar to the hole in the Earth problem if you do the following:

M_vac(r)=(4/3)πρ_massvac r³

So that you can rewrite that second term as:

2c²(M_vac(r)G/r²)

There's some serious problems with what I've done above. If anyone is interested we can discuss it. First of all the sign is wrong, so I'm missing a negative somewhere (negative energy?, antigravity?). Secondly I have an extra 2c² term, third and most importantly, I'm essentially saying that any point can be considered the center of the Universe. I'm not sure, but I think this is actually true in relativity, so that may be ok.

The above essentially says that space is a substance that exerts a gravitational force (repulsive). Any point can be viewed as a displacement from the center of the universe, which means it sees an asymetric gravitational field which applies a force. The force is repulsive, in other words, it pushes you away rather than pulls you towards it. I have no idea how the 2c² fits in, but I solved the problem classically and don't know the relativistic solution to the hole in the Earth problem. That may add the 2c² term.

So imagine you have a 1 kg round mass in the center of the universe. As you move away from it (lets say to the right), there is now more space on the left than on the right exerting a repulsive force on the object, which means there is a net force to the right. Of course, gravitational attraction is to the left, so this force always opposes gravitational attaction.

Having had a little more time to thing about this I've come up with the following.

What if I substituted R_s (Schwarzschild Radius)?

Rs=2GM/c² so

This means that the second term from that earlier post can be written as:

c⁴r/R_s

or in other words,

g=-GM/r² + c²Λr/3
=-GM/r² + c⁴(r/R_s)

or if M=0 (vacuum only) we get

g=c⁴(r/R_s) (for a vacuum)

with a corresponding force on a massive particle of

F=ma=mg= mc⁴(r/R_s)

I don't know, that means nothing to me. So since I've got nothing to lose here (and I'm long past the point of reasonableness) I'm going to do something ridiculous.

Rs=-c²dtr_s

giving me

F=(-E/dt)(r/rs)

I will not stop this nonsense and go back to the model Jorrie was doing. For anyone not sure, this answer is junk. What you just witnessed was a stream of consciousness that went nowhere. That answer is BS.

Hi Roger, the equation that you gave:

"Take the gravitational force due a spherical mass M given in the equation below:

g = -GM/r² + c²Λr/3"

is not quite for gravitational force, but rather an incomplete version of Einstein's gravitational filed strength in the low field approximation (i.e., for quasi-Newtonian gravity).

I do not know the notation used (that "caret-vector-r" involved). I think it involves a tensor, but I'm not sure.

However, it does show correctly that the cosmological constant (Lambda) has the equivalent of a repulsive force component. What it does not tell us is that Lambda also has a normal gravity component, e.g., the positive energy of the hyper-surface of our balloon, which grows with the expansion. I suppose it can be assumed to be part of a time-variable M, which also becomes terribly superficial.

-J

As I so far understand the thought experiment, we are to create a good drawing of a "finite" universe.

So far the Balloon is not working for me, and I wake up thinking of dirigibles.

Negative and positive vacuum fail me when I look at the universe.

The Balloon model of a universe is meant, to me, to make understandable a finite set of constants.

There is matter, antimatter, or dark matter, and then there is something more, like light.

Earlier in my life I had a great explanation for why the infinity circle on my camera lens, was so perfect.

I did talk to a woman who said for 30 bucks an hour she would help me get to understand algebra 1 and II. However she did say that what we are talking about, she couldn't really help me with.

She said we were getting into an area of quantum mechanics, and that there were things there that made what she could teach me, of little help.

I thought she was lying and that she simply liked my friend better than she liked me.

However I did not think she was lying all out, but just a little off the edges.

Take the Balloon and twist it, so it is not round, and then we have it?

God I love a camera lens with all its numbers that turn into a twisted balloon.

Hi Transcendian, you wrote: "As I so far understand the thought experiment, we are to create a good drawing of a "finite" universe. So far the Balloon is not working for me, and I wake up thinking of dirigibles. Negative and positive vacuum fail me when I look at the universe."

Yes, we are trying to 'design' a finite, expanding balloon that has the mathematical characteristics of the standard cosmological model. Sadly, that introduces something like negative pressure. However, we should not be too perturbed, because what is negative pressure after all? Just depends on what you take as reference. We habitually talk of negative pressure as anything below atmospheric (or general environmental) pressure. Check what I wrote to Roger in #36.

-J

The words and the images are where I have some strengths.

I have read closely with great interest this thread.

The balloon has a skin.

The skin is matter and antimatter, light, and is finite.

The universe is moving.

It is equal, and unequal, and has compression and decompression.

If I substitute these words for negative pressure, I come closer in my mind to a universe composed of matter, and antimatter.

As an "Image" I end up twisting the balloon.

When I twist the balloon, there is a place in the center, as in the center of the Infinity circle, of positive pressure.

All spreading from there is less, and therefore moving to the negative.

To have an image of the universe, I want a balloon I can twist into two, as we do those long balloons, as opposed to the round balloons.

I still have one balloon. That appears to have two parts, but shares the same skin; or does it?

Say it is that we throw in the chaos theory, and there will always be some little thing, that changes everything, and it is impossible to twist the balloon into equal halves?

Would that help us move to feel the flow of negative and the positive pressure within and without the one balloon, attempted at equality between the two halves, all confused at the twist?

"Negative pressure" is only problematic for gases. Liquids and solids have no such constraint (tensile strength is a good engineering concept). Even the gaseous Sun is not exactly flying apart, by which I am trying to show that (if you include gravity in your laws) even gases can support negative pressure. (which I believe to be the sense in which tension is used in this discussion).

N.B. Attractive as is the idea of combining engineering insights into cosmology, I don't really want to get drawn further into the basis of this discussion. It's much too distant from what I can understand, model, or measure. In additon, I suspect that many of the concerns will in due course be found to be spurious - as based on extensions of models and theories beyond the regions where they can possibly apply.

Hi Fyz, I understand your position, but as I replied to S, maybe we can benefit from a 'design' that is more understandable (engineering-wise) than the standard cosmological model (LCDM), yet mimics the standard model closely.

I do agree that we must not take analogies too far, because they tend to lead to wrong conclusions. However, I do believe that there are ways to prevent that and get more benefits than drawbacks. Only time will tell, I guess.

-J

Hi Jorrie,

This is interesting, but I don't know how to proceed now. Can you throw out some ideas and I'll see if I can run with any of them (as you can see, getting me to run isn't the problem once I get some P_r going )

Roger

Hi Roger, you said: "This is interesting, but I don't know how to proceed now."

I'm still hoping that we can improve/correct the specification before we jump into the 'design phase'. As an example: point 3 of the spec (vacuum) is quite complex and somewhat prescriptive. Is 3 (ii), the requirement for negative pressure of the surface really required? Should it not rather be replaced by an expansion dynamics like: as the balloon expands, there shall be a radius at which the expansion rate starts to increase (accelerated expansion). It is then left to the designer to decide how to implement it.

I can imagine a design of a balloon with a double skin, where the small space between the skins is kept constant (i.e., R_outer - R_inner = constant). If a fluid is pumped into this 'cavity' and regulated to be at constant positive pressure, will the balloon not inflate? If we call this cavity the surface, will it not also solve spec point 3 (i)? After all, the mass of the compound surface should then grow with expansion, as more fluid is pumped in.

Or, we may throw out the requirement: "no gas pressure inside (or outside) the balloon required or allowed" and put a lesser pressure outside the balloon that inside it, with the difference regulated. You see the sort of possible routes? There are obviously many more...

-J

Hi Jorrie,

Regarding the balloon skin. From a design standpoint, there is no need for two skins and something inbetween (if we are trying to be efficient).

Just make the balloon out of a material that experiences phase transitions when stretched. Those phase transitions will lead to the changes in the properties of the expansion as the universe grows.

I don't know if you're following me here but one example of a material we could use is a polymer with liquid crystals imbedded in it. As the polymer stretches, the liquid crystals attached to the polymer backbone would start to align, at some critical stretching point of the polymer, the liquid crystal would change phase from isotropic to nematic. The polymer would now stretch differently due to this phase change in the embedded liquid crystals.

That's my suggestion for the balloon material. Let me know what you think.

Hi Roger.

Hmm... That material of yours may do just fine for both the inner and outer skins.

The reason for a double skin is that the surface must not only stretch, but it must grow in mass as it stretches. Add to that the fact that we are not (yet) allowed to put anything inside or outside the balloon...

OK, the double skin may be cheating a bit, but remember, we never said that the skin must have zero thickness (L_Planck is the minimum in any case). We also said it must have its own mass and constant density.

I think I must update the specification to allow a double skin of some sorts and remove the negative pressure requirement (of the 'surface') for now. Then we can propose designs and see how it stands up against the spec.

-J

Jorrie,

Keeping that mass density fixed is difficult with a single layer of material, as you've said.

Our matter is sitting in this balloon skin right? What if we have some of the matter get converted to space? Matter mass density is much higher than space mass density.

Matter could be spinkles in our skin that under special circumstance become the polymer-liquid crystal skin. Like little solid bits that melt? What do you think?

Roger

Hi again Roger.

"Our matter is sitting in this balloon skin right?"

Just remember that there are differences between the matter (mass) of the skin and the normal mass-energy that rides on the skin in the form of free particles, including photons. The skin mass increases with expansion (constant energy density), while the total mass of the free particles should remain constant (reducing energy density).

"What if we have some of the matter get converted to space?"

Yea, but remember that matter of the skin is our space! If the skin can absorb energy coming from somewhere in the right proportions and expand as a result, we might be in business.

-J

There seems like there are two properties we are talking about here and I don't want to confuse them.

1. As the balloon expands, the energy density (mass density) of it's skin doesn't change.

2. Something is making the skin expand.

My goal is to make these properties unique to the skin, in other words, find a way to avoid needing two skins or negative pressure. The way I see it, negative pressure is just a way to "describe" space that wants to expand, but that doesn't have to be the way our model does it.

My proposals are:

1. Put little sprinkles on the skin that as the skin expands "melt" and become more skin, thus the energy density remains the same. These sprinkles can be rolled up polymer-liquid crystal balls in solid state.

2. This I can't think of at the moment.

Tell me what you think, Jorrie. If you want this to go in a different direction, feel free to guide it, I won't mind.

Roger

Hi Roger, you wrote: "Put little sprinkles on the skin that as the skin expands "melt" and become more skin, thus the energy density remains the same."

But why would this increase the total mass-energy of the surface (which is required to keep the density constant as more volume is created)?

One would need matter from somewhere outside the system. In the real universe, it comes from the vacuum itself (virtual particles of the larger space). So we would need a reservoir of 'sprinkles' and a method of getting them onto the surface.

-J

You Wrote:"One would need matter from somewhere outside the system. In the real universe, it comes from the vacuum itself (virtual particles of the larger space)."

I'm sorry Jorrie, could you explain that further. It was my understanding that they don't know how the energy-density of space stays the same while it expands, but you said that it's virtual space? Can you explain further (or provide a link that explains further).

Roger, you wrote: "It was my understanding that they don't know how the energy-density of space stays the same while it expands, but you said that it's virtual space?"

Not "virtual space", but normal vacuum with virtual particle pairs popping in and out of existence. I'm sure you know more about them than what I do.

In theory, this creates a mass-energy that exceeds the comological value by the factor 10¹²⁰ that S wrote about. I had my bit to say in this reply.

-J

The link you posted here was one on my posts, so you must have made a mistake.

I forgot to ask in my last post do you intend to simulate inflation, or start after that occurred? It seems unimportant to the overall goal, and is to some a far-fetched theory. BTW, do you intend to build a mechanical model, a computer model, a mathematical model, or just keep it as a 'thought experiment'? How will we know if 'we' succeed?

Hi again S, thanks, I've posted a correction of the link (can't edit that reply).

I would like the model to be able to handle something like inflation, but it is not a strong requirement. We can view inflation as just a huge cosmological constant for a brief period, without worrying too much about how and why.

As for a mechanical model - not quite, but I would like to build a computer model/simulation, with the inevitable math in it. We will have succeeded if the model reproduces the standard cosmic expansion profile AND is visualizable/understandable due to its connection with standard engineering principles, whatever that may mean!

-J

"I would like to build a computer model/simulation, with the inevitable math in it. We will have succeeded if the model reproduces the standard cosmic expansion profile AND is visualizable/understandable due to its connection with standard engineering principles, whatever that may mean!"

Great! Although I don't have a lot of time for it, I would be willing to assist with the programming if we share a common language. Are you a programmer? If so, what language? I am fairly proficient in HP BASIC for Windows. I also have Quick BASIC (for DOS), and have dabbled in GW BASIC and Assembly language. I'm looking forward to see how all of this plays out.

-S

Hi S, you said: "Although I don't have a lot of time for it, I would be willing to assist with the programming if we share a common language. Are you a programmer? If so, what language?"

I normally do things on a spreadsheet first-up. If I want more accuracy (i.e., more, smaller steps in the integration), I normally revert to either Visual Basic (VB) or Java-script (like the Cosmo-Calculator). I have also used QB previously.

Actually, the most cost-effective way to achieve good accuracy in a friendly environment seems to be a spreadsheet with VB-macros, which is just like functions in VB or QB. I'm sure we can collaborate on the functions.

-J

"Actually, the most cost-effective way to achieve good accuracy in a friendly environment seems to be a spreadsheet with VB-macros, which is just like functions in VB or QB. I'm sure we can collaborate on the functions."

I haven't used VB or JS. I think we can communicate with Excel via email for starters. Ultimately I think we want to run a program that draws circles and makes graphs dynamically. We want all CR4 users to be able to see it. QB can compile, so it makes executable files that can run on windows. Can a file be attached to a post? Anyway, I think we can work it out when the time comes.

-S

Hi again S, "QB can compile, so it makes executable files that can run on windows. Can a file be attached to a post? Anyway, I think we can work it out when the time comes."

I also loved QB - so much easier to (programmer-) interface to than VB! (Although the syntax is much the same).

I usually put the files on my website and provide a link. One issue is viruses/worms etc; that's probably why CR4 does not allows general attachments.

-J

Correction of link given in reply #67. Should read: "In theory, this creates a mass-energy that exceeds the cosmological value by the factor 10¹²⁰ that S wrote about. I had my bit to say in reply #22.

-J

Rogers 53 would offer compliance with the theory of the cosmological theory as it incorporates "local irregularities" or "density fluctuations", from what I can tell.

Hi Jorrie,

Well you may not be trying to design a universe, but you are trying to simulate one (with one too few dimensions). The double skin makes no sense to me, in fact the skin has to be infinitely thin to match reality. Energy has to be added to the skin somehow to make it grow in diameter. I don't know how you will accomplish this unless you pressurize the balloon. There can't be zero pressure outside. It makes no difference which pressure you change to simulate lambda.

-S

Hi S, when you wrote: "There can't be zero pressure outside", did you mean "inside"?

You said: "It makes no difference which pressure you change to simulate lambda."

It seems that you feel that provided we can manage a specified expansion profile (e.g. by programming the machine pumping fluid into the balloon), it does not matter what dynamics the surface (our 3D space) follows.

I would like something more like: switch off the pump and the balloon will perform just like the Einstein-de Sitter model with matter only (no Lambda). I don't quite know how to do that, but maybe its worth a try. The double skin is an attempt towards that.

-J

Hi Jorrie,

"There can't be zero pressure outside", did you mean "inside"?

No, I meant that if you pressurize the inside, there has to be some pressure outside, or the balloon will get infinitely large immediately, as I think was discussed before, unless I missed the point. So I meant that you can control the expansion either with a vacuum pump outside or a pressure pump inside, or both. The amount the pump runs would have a formula to match the observations of space of the real universe.

So I take it that the double skin is the only way you think you can simulate mass (a problem of dealing with one or more fewer dimensions?). Is this what you have in mind?

-S

Hi S, you wrote:

"No, I meant that if you pressurize the inside, there has to be some pressure outside, or the balloon will get infinitely large immediately, ..."

Not quite, because the surface has mass and hence momentum. A constant pressure differential will accelerate the expansion, which is more or less what we want! I think we can hence let the external space be unpressurized.

I am starting to lean over towards pressurizing the inside, as you proposed, because it is simpler than the double skin idea. I am still thinking on whether this can produce the correct dynamics, though.

-J

Jorrie,

"Not quite, because the surface has mass and hence momentum. A constant pressure differential will accelerate the expansion, which is more or less what we want! I think we can hence let the external space be unpressurized

Wait a minute, this brings up comments and questions. In the inflation theory, space expanded a factor of 10^25 in about 10^-30 second, much faster than the speed of light. I think we can rule out momentum then. If the speed of light doesn't apply to space, why would momentum? If it does apply, how did the space (vacuum) change from having no momentum to having some?

Without outside pressure it may be harder to control the expansion, but I wouldn't know how in either case. I wasn't actually proposing the pressurization, but I think it may the simplest way.

-S

You asked a difficult question, S!

"In the inflation theory, space expanded a factor of 10^25 in about 10^-30 second, much faster than the speed of light. I think we can rule out momentum then. If the speed of light doesn't apply to space, why would momentum?"

The hypersherical model allows any speed (dR/dt) in that extra (hyper) dimension, as you said. The Friedman equations (which is a specific solution to Einstein's field equations) tells us that there is a 'kinetic energy of expansion' coupled to the cosmological constant. Since the cosmological constant translates to the energy of the vacuum, it follows that expanding empty space has momentum.

Even during inflation, the expansion rate (dR/dt) did not go from zero to large instantaneously, but just very quickly, as you stated. This is thought to be caused by a huge cosmological constant (negative pressure) that quite rapidly fizzled out due to a phase transition, like from water to ice (or rather from virtual particles to real particles in this case?). How and why, nobody knows.

I have though about applying an internal pressure to the balloon, but I cannot quite get the momentum of expanding space right. The outer layer of gas will have an outward momentum at a rate dr/dt equal to dR/dt, where r is the hyper-distance of the layer from the center, but inner layers will have a lower dr/dt and the Friedman equation does not balance.

The double skin, with the constant gap or cavity (R_outer - R_inner) into which fluid is being pumped has more or less the right characteristics. Total mass increases with expansion while density and pressure can readily be kept constant. The skin material does not have to be exotic, just able to stretch by a huge factor without tearing (this is to cope with inflation). If we only want to model from the 'CMB release time' to the present, the stretch factor is 'only' 1090 times.

My one remaining problem is that a 3D cavity with constant gap (R_outer - R_inner) also does also not fit the bill; I need a 4D cavity to balance the 'Friedman books'. Still working on that...

-J

Hi S, despite what I said in this reply, I am back at the double-skinned balloon. Here is a 'design' and some rationale behind it.

Consider a balloon with a double skin, where the two concentric skins are kept at a constant separation ΔR=R_o-R_i, e.g., by short, thin pipes between them so that they cannot separate in the radial direction. Call the volume between the skins 'the cavity', the volume outside the outer skin 'the exterior' and inside the inner skin 'the interior'. The pipes also ensure equal interior and exterior (atmospheric) pressure.

Now pressurize the cavity to a fixed level above atmospheric pressure and regulate it at that pressure differential. If the balloon does not burst, an equilibrium size will obviously be achieved due to the elasticity of the skins. If the skins had no resistance against stretching, this balloon would have inflated for ever, but at an ever increasing rate!

The fact that we do this in air does complicate things a bit, but there should be no objection if we do the same thing in the vacuum of space. In fact, if we put no extra (fixed) mass and/or radiation into the cavity, we have a near-perfect de Sitter cosmological model (zero matter, just cosmological constant). What is more, if we have the right capacity reservoir (and pump), we can even mimic the inflation epoch and the big rip. Obviously, if we also put the correct density of normal/dark matter particles and radiation onto the surface, we have a near-perfect LCDM model.

One may ask: why not just a normal balloon with a single skin, being pressurized in the same way? The reason is that we need all the mass of the air inside the cavity to be accelerated outward (in the hyperspace direction), which cannot be done with a single-skinned balloon. Otherwise, the Friedman books do not balance.

One snag is: where do we get the gas or fluid to keep the pressure of the growing cavity constant? It might be from the vacuum, but let's first debate the double-skin model for its own merits.

-J

I agree with Physicist? if I understood him. You are designing a universe to match the present theories. What if they are wrong? What would the universe model look like if you were designing it from scratch? (For instance, would you have a universe that is expanding linearly, or maybe static?)

Regardless, I am curious what the actual "designing" phase will be. I am ready for that.

-S

Hi S, you wrote: "You are designing a universe to match the present theories. What if they are wrong?"

We are not trying to design a universe! The idea is to see if we can 'specify and make' a balloon that operates as closely as possible to the (present) standard cosmic model, using standard engineering principles. Or, maybe we just want to see how close we can come to a balloon that has the dynamics of the standard model. If we succeed, the model (of the 'hardware') may be a valuable pedagogical tool, readily understandable by engineers. If we don't, we would hopefully have learned something! Does it then matter if the present standard model turns out to be way off the mark?

You said: "Regardless, I am curious what the actual "designing" phase will be."

What I foresee is that we try to implement (on paper) a balloon that has the expansion characteristics of the standard model. The specification was intended to put constraints on this design, broadly saying what it must do and what is not allowed. I still expect that the specification will have to be modified in order to be implementable, but maybe we should do a 'rapid prototype' to learn if it is more or less achievable.

-J

Hi again Jorrie,

I agree that a frictionless surface is the right answer.

You also have to think about the design of the model ant that marches across the surface.

"Ant Version 1.0" needs to get traction on the surface in order to continue propelling himself at a constant rate. With no friction there's no traction. Neither this traction nor the ongoing expenditure of energy by the marching ant has any direct analog in the real universe.

"Ant Version 2.0" is more like the puck in an air hockey game. You push him to get him started, and then he continues moving at a proper surface velocity (peculiar velocity) which decays at 1/a as he first accelerates his proper velocity on the way to the origin, and then later as he enters new neighborhoods with ever faster proper Hubble rates relative to the origin. This guy's advantage is that he requires neither traction nor ongoing energy expenditure.

Ant V2.0 is more logically consistent, but perhaps not as easy to visualize, as Ant v1.0.

By the way, we designate the origin by marking a dot at an arbitrary location on the balloon's surface. We would need to tether Ant V2.0 to the origin at some distance and then untether him. But with Ant V1.0, we don't need to actually tether him at all. We just place him initially at some location on the balloon, facing the origin, where his surface marching velocity exactly matches the rate at which the origin is stretching away from him.

Jon

Hi again Jon, you said: "You also have to think about the design of the model ant that marches across the surface."

I would prefer to have no 'ants with traction', but just two types of particles: (i) photons that always move at c relative to the local surface and (ii) massive particles that keep a constant speed (peculiar velocity) v < c relative to the local surface, at least for a uniformly expanding balloon (dR/dt = constant). Remember that R is the radius of curvature, not a distance.

Of course, v = 0 is the natural state for massive particles in the FLRW model, so we essentially only have particles at v=c and particles at v=0.

I presume that what you think of as 'traction' is actually a natural consequence of changes in expansion rate (i.e. acceleration d²R/dt² <> 0), which changes the speed of a moving, massive particle relative to the local (spherical) surface due to 4-momentum considerations. I see no traction effect here.

So, for now, I would rather ignore the traction issue, until such time as we have a working model in which we can tether particles to give them your "marching velocity" relative to the surface.

-J

Hi Jorrie,

What I mean by traction is that if a "marching" ant tries to march on a frictionless surface, like trying to drive a car on very slick ice, he won't make any headway, he'll just spin his wheels. (There's a mixed metaphor!)

Anyway, I prefer Ant v2.0 who doesn't need traction.

Jon

Hi Jon. OK, traction-less ants it will be!

On your concern about the 10¹²⁰ factor. No, I do not think the balloon model will make the problem worse, but may rather lessen the issue - as I wrote in #38 (and before that) to S, it may perhaps be 'solved' by the "factor of L_Planck/R_curvature ... - who knows?" Anyway, we should not be overly concerned about it, because we can ignore the quantum physicists for now and just try to please the cosmologists!

On the pressure inside the balloon. The spec as it stands only says that we shall not apply any pressure to the inside (or the outside), making it very tough! It will probably be better to put the whole thing in empty space, where there are only virtual particles. Drat! Then we have to please the particle physicists...

-J

I don't think there need be any traction with the surface itself - just forces between different particles that reside on the surface.

(That's already more drawn in than intended - perhaps I should just de-register from the discussion)

Hi Fyz. What did you read in my post that looks like traction? Jon may (or may not) have such an issue, but I don't.

I wrote: "I presume that what you think of as 'traction' is actually a natural consequence of changes in expansion rate (i.e. acceleration d²R/dt² <> 0), which changes the speed of a moving, massive particle relative to the local (spherical) surface due to 4-momentum considerations. I see no traction effect here."

There are some counter-intuitive dynamics though, like that free particles with a peculiar velocity will eventually, without being influenced by anything other than a changing expansion rate, loose the peculiar velocity and join the 'Hubble flow'. To some people, this may look like 'traction'.

-J

Hi Phys,

Don't go away. I'm barely more than an observer myself. We appreciate your inputs even if they seem insignificant to you.

-S

Spec 3. Vacuum presently reads:

"The surface material of the balloon (black) shall have some peculiar properties: (i) if it has any mass and we stretch a finite piece of the material, it shall gain mass in order to keep its mass density (ρ) constant; (ii) if ρ > 0, it shall have a negative pressure (p = -ρc²), so that while being stretched, it tends to self-stretch even faster. This is obviously not easy to understand, but it should become clearer as we go forward.^[2]

In short, the balloon shall be able to self-inflate under certain conditions - no gas pressure inside (or outside) the balloon required or allowed. It might be a nearly impossible material to make, but it shall represent Einstein's cosmological constant (vacuum energy) faithfully. "

The issue of "a negative pressure (p = -ρc²)" is hardly implementable on a small scale. A "self-inflating" balloon may also be a bit of a dubious concept. Allowing for a balloon surface with non-zero thickness (it is implied, because it may have mass and density, after all), this should not rule out (or specifically require) a double skin configuration with some matter between the two skins.

I also think that the "no gas pressure inside (or outside) the balloon required or allowed" is overly restrictive and should be dropped.

I propose that point (i) above be left as is and that point (ii) be deleted up to the note [2] reference, with the last paragraph simplified, so that the whole 3. Vacuum simply reads:

"The surface material of the balloon (black) shall have some peculiar properties:

(i) if it has any mass and we stretch a finite piece of the material, it shall gain mass in order to keep its mass density (ρ) constant;

(ii) the balloon material shall allow Einstein's cosmological constant (vacuum energy) dynamics to be followed as faithfully as possible."

Do you think this is OK, or should it be stated better without choosing/describing a specific design?

-J

PS: I think notes [2] and [3] should possibly stay for traceability, but just not referenced in the body.

I think that makes sense.

Since #56, a lot of trial and error has happened and the 'final' design happened to be quite an ordinary balloon, with the expansion energy (vacuum and momentum) supplied by a pump or pressure reservoir that simply inflates/deflates the balloon to simulate the Friedman equations. All other efforts were hampered by either exotic fluids or very difficult concepts to fathom. The present 'best buy' is at least 'engineering-friendly' in that we can even contemplate building it!

1 Specification

It is proposed that spec. 1.2.3 to 1.2.5 be replaced by:

1.2.3 Vacuum (Change note: this post)

Vacuum energy (cosmological constant) shall be included in any inflating/deflating mechanism of the balloon.

1.2.4 Total Energy

It may be assumed that the energy of expansion is included in the algorithm that inflates/deflates the balloon.

1.2.5 Momentum

It may be assumed that the momenta of the skin and all energy on it are included in the algorithm that inflates/deflates the balloon.

2 Design

For the Design part, it is proposed that the 'special fluid' design be scrapped and the 'ordinary balloon' of reply #228 be inserted. Here is a summary:

Assume a balloon material that retains its elasticity over a reasonable range of balloon radii (at least enough range to illustrate the principles). A sensor system measures the balloon radius (R) directly and also determines the rate of change (ΔR/dt). A pump/reservoir/valve system supplies or withdraws gas to/from the balloon at a rate that will keep ΔR/dt = R H, where H is a function of R and the energy density makeup of the cosmos to be simulated. H, the time variable Hubble parameter, is obtained from:

H² = H₀² [(1-Ω₀)/a² + Ω_m/a³ + Ω_r/a⁴ + Ω_Λ]

where H₀ is the (present) Hubble constant, Ω₀ = Ω_m + Ω_r + Ω_Λ is the present total energy density parameter, a=R/R₀ the expansion factor, Ω_m the present matter energy density parameter, Ω_r the present radiation energy density parameter and Ω_Λ the present vacuum energy density parameter.

The operating equation is then:

(ΔR/dt)² = R²H² = R²H₀² [(1-Ω₀)/a² + Ω_m/a³ + Ω_r/a⁴ + Ω_Λ]

'Guest' has proposed a neat little high level algorithm for this system in reply #242. In its final form, it should be part of the design section.

3 Tests

Since the balloon is ensured to follow the Friedman equations (at least theoretically), paper 'tests' would not be very meaningful. It is proposed that the 3 cases (de Sitter, Einstein-de Sitter and Lambda-cold-dark-matter (LCDM)) be simulated and the results shown here for comparison to other simulations.

Of more interest may be specific 'applications' of the simulation, e.g., the 'cosmic teardrop', 'tethered galaxy' and redshift:distance experiments.

-J

I have updated the opening post with the info in change note: reply #243, plus the neat algorithm supplied by Guest in reply #242, slightly modified.

I think we should stand with this design for now and see what can be learned from using it.

-J

I have now tidied up the entire opening Blog post, with brief conclusions drawn.

Unless there are some more discussions coming forth, I will close this thread and start a new one on 'applications'. The 'cosmic heart' may be one of the applications...

-J

All things being equal, all things are unequal.

We cannot have one universe, without having another.

Take the balloon and twist it as you did, when you were a kid, and were given a balloon that wasn't round.

actually what I thought I was doing was applying string theory to the balloon model, making two universes out of one balloon, applying chaos at the twist, then skinning the model with two different constants for the speed of light created at the imperfect balance of the two universes, that are one.

I am trying to get some help with Calculus, which is where I am most incompetent and weakest as far as your excellent thread question thought experiment.

Apparently I have a friend whose wife's sister? - is married to a guy who is big in the Haldor? Haldron? Collider.

He consented to passing on my request that his semi relative take a look at your thought experiment.

P.S. You could take my balloon model and hang it off a string from the ceiling, and then bat at one side or the other to make it spin. If someone knows how to make an equation for that image, it would probably help me understand, that which I do not yet have full capacity to understand. If nothing else if I can get this guy on the forum, I expect Jorries GA numbers to go up, since I figure they are low since few of us can tell here a good answer from a bad one.

Here is a design concept for the CR4 Cosmic Balloon to consider.

Make a balloon with a double skin, where the two concentric skins are kept at a constant separation ΔR=Ro-Ri, e.g., by short, thin pipes between them so that they cannot separate in the radial direction. Call the volume between the skins 'the cavity', the volume outside the outer skin 'the exterior' and inside the inner skin 'the interior'. The pipes shall ensure equal interior and exterior (atmospheric) pressure.

Now pressurize the cavity to a fixed level above atmospheric pressure and regulate it at that pressure differential (Δp). The balloon will inflate and if does not burst first, an equilibrium size will be achieved due to the elasticity of the skins. If the skins had no resistance against stretching (and could stretch forever), this balloon would have inflated forever, at an ever-increasing rate!

The fact that we do this in air does complicate things a bit, but there should be no objection if we do the same thing in the vacuum of space. It would also be better if we use a special frictionless, collisionless cavity fluid, so that it does not interfere with the radiation and other free particles that we later want to put there.

If we have just this fluid with constant pressure in the cavity, we have a near-perfect de Sitter cosmological model (zero matter, just cosmological constant). What is more, if we have the right capacity reservoir (and pump), we can even approximate the inflation epoch, the big rip, etc. If we add the right density mix of normal/dark matter particles and photons to the cavity, we can have a near-perfect standard LCDM model.

One may ask: why not just a normal balloon with a single skin, being pressurized in the same way? The reason is that we need all the mass of the air inside the cavity to be accelerated outward (in the hyperspace direction), otherwise the Friedman books do not balance. This cannot be achieved with a single-skinned balloon.

Let's first investigate the empty (de Sitter) cosmology, with its constant vacuum energy density, to see if the balloon model will stand up to it. The de Sitter cosmos has a relatively simple dynamics - the acceleration of expansion is directly proportional to the scale factor. In balloon terminology, it means that the acceleration of the expansion rate grows with the radius of the balloon.

Since the cavity is kept at a constant ΔR, the volume of the cavity grows with the square of the radius, or Volume proportional to R². Since the density of the cavity fluid is constant, the gravitational mass of the fluid is also directly proportional to R². However, the gravitational pull is inversely proportional to R², so the two effects cancel out and the contraction effect caused by gravity does not depend on the radius of the double-skinned balloon. As far as acceleration is concerned, it is as if the cavity has a constant mass, i.e., a = -F/M = constant.

Now, it is very easy to show that the net expansion force on the balloon, caused by the positive pressure Δp = ρ c² inside the cavity, is given by

F = 4 Π ρ c² (Ro²-Ri²) = 8 Π ρ c² R ΔR

where R = (Ri+Ro)/2, the `average radius' of the balloon. Since ΔR is fixed, it means that the expansion force grows directly proportional to the radius of the balloon (F proportional to R). Since the `effective mass' of the cavity is constant, the balloon will have an outward acceleration a=F/M proportional to R, just like for the de Sitter cosmic model.

One important question that must be answered is: does the fact we have suppressed one spatial dimension make any difference to this conclusion? If we add the third spatial dimension, the mass of the fluid will be proportional to R³, while the gravitational pull will still be inversely proportional to R². This means that the contraction effect caused by gravity will increase directly proportional to the radius of the double-skinned balloon. However, with the extra spatial dimension, the expansion force caused by the constant pressure of the fluid will increase directly proportional to R². The net effect is the same: the balloon will still have an outward acceleration a=F/M proportional to R, just like for the de Sitter cosmic model. Hence, we can say that the pumped cavity fluid has a similar effect than vacuum energy in the de Sitter cosmic model.

Next we take the matter-only case (the Einstein-de Sitter model), where there is no further vacuum energy after inflation ended. Radiation energy is also ignored. We now have a fixed number of free particles of matter, static and evenly spread in the cavity, with no cavity fluid. These particles represent the normal matter and dark matter of the cosmos. Directly after inflation (at time t) they have a density ρ_m representing a mass M=4 Π R³ρ_m, with the balloon expanding at an initial rate dot_R. There is no fluid pumped into the cavity, so Δp = 0. The balloon surface will coast outward just like a cannon ball shot straight up from the surface of the moon.

The mutual gravity of the particles will decrease the expansion rate and if the initial expansion rate exactly equaled the 'escape velocity': dot_Re = √[GM/R], the balloon will expand forever, but at a rate approaching zero asymptotically. The particle density will obviously drop inversely with the volume of the cavity, i.e., ρ_m proportional to 1/ R². When we add in the neglected third spatial dimension, the mass becomes M=4/3 Π R³ρ_m. The particle density will now drop inversely with the volume of space, i.e., ρ proportional to 1/ R³. This is exactly what the (flat) Einstein-de Sitter model predicts.

Now look at the case of radiation energy only. It is the same as for matter only, except that the particles are now photons traveling at c along the cavity. Let the radiation have an energy density ρ_r, which is diminished by two factors: (i) the expansion of the of the balloon, i.e., ρ_r proportional to 1/ R² and (ii) by the red shifting ('stretching' of the wavelengths) of the photons by a factor 1/ R as they move along the expanding surface of the balloon. This gives a net reduction in energy density of the photons: ρ_r proportional to 1/ R³. However, add in the neglected third spatial dimension and the radiation energy density drops as ρ_r proportional to 1/ R⁴.

This corresponds to the 'Radiation-Dominated Era' in cosmic models. The radiation era started after most leptons and anti-leptons are annihilated, 3 minutes after the BB, with the annihilation energy converted to photons. It ended after about 60,000 years, when matter started to dominate proceedings. We should perhaps start the balloon model at the radiation era, because before that things were very 'fuzzy' due to the quantum mechanics operating. We surely do not want to contaminate our system with a 'quark-gluon plasma'!

Not the ultimate 'design', but (maybe) getting closer! Any comments before I 'publish' it as part of the OP?

-J

Hi Jorrie,

Here are the relevant equations as I understand it (Source):

Doesn't that first equation say that the energy density is changing in time? Am I misreading that because I thought you said the energy density is constant.

In what way does the paragraph I sited here:

You Wrote:"One may ask: why not just a normal balloon with a single skin, being pressurized in the same way? The reason is that we need all the mass of the air inside the cavity to be accelerated outward (in the hyperspace direction), otherwise the Friedman books do not balance. This cannot be achieved with a single-skinned balloon."

Relate to the two equations above?

Hi Roger.

It is only the vacuum energy density that does not change over time. Matter and radiation energy density must obviously change, because there is more or less a constant amount of energy, being spread over a larger volume.

The discussion around the quote from my post considered vacuum energy only, because that was the context. Remember that that 'cavity fluid' being pumped in represents only vacuum energy. Matter and radiation are just fixed energy quantities (hence decreasing densities) that live in the expanding cavity (3D space).

You quoted:

The first equation does not separate matter and vacuum energies, hence ρ is changing due to the matter density change over time. The second one considers matter (the ρ and p) and vacuum energy (Λ) separately, but is clear that changes in density do not couple through to Λ.

-J

Hi Jorrie,

You wrote:"The first equation does not separate matter and vacuum energies, hence ρ is changing due to the matter density change over time. The second one considers matter (the ρ and p) and vacuum energy (Λ) separately, but is clear that changes in density do not couple through to Λ."

The ρ(rho) in the 1st equation is the same as the ρ(rho) in the second equation.

Perhaps if you showed me how to separate out the mass density from the vacuum energy density I would understand better?

Roger

Hi Roger, you asked: "Perhaps if you showed me how to separate out the mass density from the vacuum energy density I would understand better?"

I must apologize: I did a very poor job of explaining in #82. The equation:

is proportional to the momentum of expansion (it's a direct function of expansion rate or hyper-velocity a-dot). There is no difference between the momentum of matter, radiation and vacuum energy, as far as the dependency on density is concerned: ρ = ρ_m + ρ_r + ρ_Λ . Multiply by volume and you get mass. This explains better what I meant in #78: "The reason is that we need all the mass of the air inside the cavity to be accelerated outward (in the hyperspace direction), otherwise the Friedman books do not balance." Better stated, it should have read: "The reason is that we need all the mass of the air that we pumped into the cavity to have the same hyper-velocity (in the hyperspace direction), otherwise the Friedman books do not balance."

The equation:

is obviously the acceleration of expansion. It shows that all the energy (contained in ρ = ρ_m + ρ_r + ρ_Λ) has the normal negative gravitational effect (decrease of positive hyper-velocity). The cosmo-constant Λ has a positive effect on acceleration and that component is not a function of ρ at all (ρ is constant and built into Λ).

This fact is behind the common statement (mostly poorly motivated) that 'vacuum energy works both for and against expansion'. I hope it is clearer now, because realizing this is one of those AHA! moments in cosmology.

-J

Thanks for posting this. I just got home and have to go to bed, but I'm eager to look at this in the morning. I've learned a lot on this thread both from you and through reading and research, it's great.

I'll post a response once I've had a chance to read and digest this post.

BR

Roger

Hi Jorrie,

I wish I could say it made it clearer, because you've gone to a lot of effort to explain it to me, I'm just having a mental block.

Here's how I understand things.

Basically Friedman et al. assumed an isotropic homogeneous universe and used a perfect fluid to model it. They then got a equation of state:

.

and so



...because....

which is the thermal velocity of the ideal fluid, which for matter in the universe is slow compared to light (at least now it is) so

Then something occurs which I don't know but involves solving Einstein's Equations for this model and you get (feel free to show these steps if you have them handy):

Which if you make rho=0 gives you the vacuum components. The K in the equation above is curvature (of space) so for a flat Universe you make that 0 and you are left with the cosmological constant term (which causes expansion of space)

Two things are confusing to me here.

1. Everyone says that Einstein added the cosmological constant to correct for an expanding universe, but the equation above seems to only expand because of the cosmological constant (I know I'm missing something important here, if you can, please help)

2. I still don't understand how rho is supposed to contain vacuum energy. It seems like the initial assumption was only with regards to matter (matter in the universe being modeled as an ideal gas)

I'll keep reading on my own, but hopefully the above points will help you understand where I'm getting lost and hopefully you can give me some insights to get me back on track.

Roger

Hi Roger. You understand correctly, up to a point.

That (wrong turn) point is: "... and you are left with the cosmological constant term (which causes expansion of space)"

The cosmological constant Λ does not necessarily cause space to expand. The original expansion rate is attributed to inflation, which may or may not have been caused by a form of Λ. For 8 billion years, it was essentially just a conservation of momentum that made the universe to continue expanding. Only then did Λ apparently started to increase the expansion rate. To make matters even more complicated, if the universe happened to be contracting, Λ would have increased the contraction rate! But that's a long story...

The Λ part did not form part of Einstein's original field equations, but he introduced it in an attempt (futile, as it turned out) to make his theory compatible with a static universe. Without it, his model universe would either be expanding or contracting. He later removed Λ, calling it the greatest blunder of his career. The Einstein-de Sitter model expands forever, asymptotically approaching a static condition at time => infinity. This is a far cry away from a static solution for 'now'!

Your: "2. I still don't understand how rho is supposed to contain vacuum energy. It seems like the initial assumption was only with regards to matter (matter in the universe being modeled as an ideal gas)"

Yes, an ideal gas with energy density ρ_matter. The Einstein-de Sitter model had this ρ_matter equaling the critical energy density ρ_crit, but it cannot account for the observed increase in expansion rate. Only the addition of an 'ideal gas' component with w = -1, i.e., p_Λ = -ρ_Λc² could do that.

At the same time, observations put total energy density very, very close to ρ_crit. So, what are the options: the sum of all the ρ's must equal ρ_crit, or some completely new type of model must emerge. At the moment, vacuum energy making up 74% of critical density fits all the data. Add to that the fact that its negative pressure causes the expansion rate to increase, and viola!

-J

The only accepted perpetual-motion machine. Such a shame that it is believed to be fundamentally impossible to exploit it.

Hi Jorrie,

I'm sorry, I still don't understand. You do a good job explaining the concepts, but I'm not having trouble with the concepts. What I need is a derivation. Just a "they took this and this, put it into this, got this and this, and that's where we get this this and this.

I've tried looking on websites, but they all just tell a piece of the story. It's very frustrating. Maybe I need an overview or something so I know the different names and how they relate to each other.

Roger

PS. If I read that Einstein said that the cosmological constant was his greatest blunder on one more website I'm going postal.

Hi Roger,

The problem we all face is that the full development of the cosmological model from general relativity (GR) is quite involved (even ignoring the development of GR itself). As an example, prof. Peebles spent almost 20 pages in his book 'Principles of Physical Cosmology' 1993, section 4, to show the development of the first part (starting from GR). If you can lay your hands on the book, it's a very good writeup.

"PS. If I read that Einstein said that the cosmological constant was his greatest blunder on one more website I'm going postal."

The truth is even worse: Einstein's use of the cosmological constant to get a static model was not very successful either. The model he created was totally unstable...

-J

You Wrote:"The problem we all face is that the full development of the cosmological model from general relativity (GR) is quite involved (even ignoring the development of GR itself)."

I'm sure that's true. Maybe then just the equations that came in the results? I don't know a full derivation really, just all the independent equations they got as a result.

You Wrote:"The truth is even stranger: Einstein's use of the comological constant to get a static model was not very successful either. The model he created was totally unstable..."

Yes I know, every website I've read on this has told me that too. Usually they mention the pencil standing on it's tip example....grrrrrrr.....

Roger, you wrote: "Maybe then just the equations that came in the results?"

Not too much profit in that, without most of the discussions in-between.

My engineering approach is to try to understand what the final equations tell us and then use them to calculate, plot, simulate etc. I propose that you hold your horses a bit. If we succeed in 'making' an engineer-friendly model that faithfully reproduce the results of the Friedman equations, we will be a long way towards understanding them.

Presently, I'm still battling to get such a model to reproduce the Friedman dynamics for the cosmological constant without artificial 'tweaking'. The two-skin balloon does not look too promising anymore. (A few readers will probably be delighted with this state of affairs!)

The single skin balloon does also not quite fit the bill either, but I'm still trying! If a 'tweak' is required, I want it to be as innocent as possible, i.e., any 'leap of faith' must be fairly credible.

-J

Jorrie,

I'm sorry, but I don't see how I can contribute if I don't at least have a general idea about the equations. It's an interesting thread, I'll sit back and watch.

Roger

Roger,

But you do have a good idea of how the equations work, I'm sure! The most important one for understanding the LCDM cosmology is undoubtedly the cosmic acceleration:

d²a/dt² = a [-4/3 Π G (ρ + 3p) + Λ/3]

The negative term inside the square brackets is just the standard Poisson solution for Newtonian gravitational acceleration at the surface of a homogeneous ball of ideal fluid with radius a. The positive Λ term obviously comes from Einstein. By equating the two terms, Einstein built his original static, unstable cosmic model. By setting the Λ term just slightly higher, we find an accelerated expansion, fitting present data.

BTW, I may have confused you earlier by stating that the energy density of the vacuum is included in the ρ of the first term - it is not. My apologies. In my own defense, cosmologists often include it in density terms like ρ and Ω, but I have established that it is not the case here - that ρ and p are just for matter.

Like most cosmologists, I prefer to work with the highly normalized version of the Friedman equations, expressed in terms of dimensionless density parameters (the Ω's), giving the fractional densities relative to the critical density. However, for the balloon model I try to work with the raw densities, because that forces one to understand the dynamics better.

-J

Hi Jorrie,

I don't think anyone is going to debate this with you. How to implement it all could take some doing. From a previous post I got the impression you want to simulate different models. That could be done with a "Menu" choice or just by changing a variable near the start of the program. Any thoughts on how the program should run?

-S

Trying out de Sitter on the double skin balloon, I ran into problems. In order to get the acceleration of expansion to change proportional to the radius, I found that the mass inside the cavity must have some constant value, requiring a decreasing density. This defeats the object. So, the double skin balloon has effectively been punctured by de Sitter! Change proposal (reply #78) must hence be retracted.

De Sitter

Here's a brief overview of the de Sitter universe, which has no matter, just a positive cosmological constant. This causes the expansion to be exponential, starting very slowly, accelerating perpetually with an acceleration increasing proportional to the expansion factor (a), where a is equivalent to the normalized radius of the balloon. The curve was plotted using the Friedman equation and current cosmological parameters, except for matter and radiation densities, which have both been set to zero. If this was the case, the universe would have been 110 billion years old now.

The equation integrated is: da = a*Ho/978*√[(1-Ω)/a² + Ω_Λ]*dt, with Ho = 72 km/s/Mpc, Ω_Λ = 0.74 and Ω = Ω_Λ, because there is zero matter in this universe.

Cosmic Balloon

Back to the single skinned balloon! Here is a somewhat promising attempt that may still require some 'tweaking'. By the Shell Theorem, the acceleration of gravitational contraction at the surface of a large sphere of gas in free space is:

d²R/dt² = -4/3 G Π ρ R

Following Einstein's cosmological constant trick, the gas will be static if a negative pressure creates an equal but opposite acceleration of:

4 Π ρ c² R² / M' = 4/3 G Π ρ R

where M' is an effective mass that needs to be accelerated by a force 4 Π ρ c² R², which is the pressure multiplied by the surface area. Hence:

M' =(4 Π ρ c² R²) / (4/3 G Π ρ R) = 3 R c²/ G.

This gives the acceleration caused by the pressure as:

d²R/dt² = 4 Π ρ c² R² / ( 3 R c²/ G)= 4/3 Π G ρ R

which is equivalent to half of Einstein's cosmological constant's contribution (Λ/3=8/3 Π G ρ). Double the pressure, keep it constant and the balloon will inflate by the required law. Or not?

-J

Hi Jorrie,

Nice chart. You said "The curve was plotted using the Friedman equation and current cosmological parameters, except for matter and radiation densities, which have both been set to zero." If the current cosmological parameters are based on the LCDM model, then is this really fair? Do we need to worry about the de Sitter model which we know to be false?

-S

Hi S, thanks.

"Do we need to worry about the de Sitter model which we know to be false?"

De Sitter cosmology deals solely with vacuum energy, which is the tough part, also for the balloon model. It nicely separates the vacuum issue from matter (simple to model) and radiation, also fairly straightforward. That's why I started with de Sitter and, although I know how to model it cosmologically, it is not easy in the balloon analogy. I'm still battling a bit.

The present cosmos actually seems to be closer to the original de Sitter model than to the original Einstein-de Sitter (matter-only) model, which is also 'false'! What's more, as time goes on, it may be getting more and more de Sitter-like - eventually, vacuum energy may mask the influence of matter completely. I say 'may' and not 'will', because the present ideas of the cosmologists may still prove to be wrong.

-J

I wrote in #117:

"This gives the acceleration caused by the pressure as:

d²R/dt² = 4 Π ρ c² R² / ( 3 R c²/ G)= 4/3 Π G ρ R

which is equivalent to half of Einstein's cosmological constant's contribution (Λ/3=8/3 Π G ρ). Double the pressure, keep it constant and the balloon will inflate by the required law. Or not?"

I think not! If we double the initial negative pressure, we also double the (positive) density and hence also the gravitational pull. This alone will not make the balloon inflate. What is required is some initial rate of expansion – somehow give this balloon an outward momentum, maintain the negative pressure and it will start to increase in radius faster and faster. Why?

The expansion causes the density to drop. The pump adds fluid to restore the density. The momentum causes the radius to overshoot, dropping the density more. The pump adds more fluid, etc. It is an unstable positive feedback loop. The same will happen in the reverse direction for an initial inward momentum.

Can this process generate an acceleration of expansion that changes directly proportional to radius? The Friedman equations say yes, provided we have a fluid with equation of state w = -1, meaning pressure p = -ρc². Where do we get that? (Roger, what sort of vapor?)

-J

I wrote in #149: "Can this process generate an acceleration of expansion that changes directly proportional to radius? The Friedman equations say yes, provided we have a fluid with equation of state w = -1, meaning pressure p = -ρc². "

This fluid sits inside the balloon and its negative pressure balances the self-gravitational 'collapse' force of the fluid itself. This is precisely equivalent to a 'suction' from outside the balloon by means of a lower pressure (negative, below vacuum in this case).

What if we have just a normal fluid inside and outside the balloon, but the outside has a lower pressure than the inside? The pressure difference must be just sufficient to prevent the balloon from gravitational collapse. Say the outside is a very large container, so that changes in the balloon size and/or pressure do not influence the outside pressure at all.

Now let a perturbation again take the balloon slightly out of equilibrium and let a pump keep the pressure difference constant by adding/withdrawing fluid to/from the inside of the balloon. Ignoring friction, will we have the same runaway inflation or deflation as predicted for the de Sitter model? It would appear so!

-J

PS: This is just an effort to get away from an exotic fluid with p = -ρc².

"What if we have just a normal fluid inside and outside the balloon, but the outside has a lower pressure than the inside? The pressure difference must be just sufficient to prevent the balloon from gravitational collapse. Say the outside is a very large container, so that changes in the balloon size and/or pressure do not influence the outside pressure at all.

Now let a perturbation again take the balloon slightly out of equilibrium and let a pump keep the pressure difference constant by adding/withdrawing fluid to/from the inside of the balloon. Ignoring friction, will we have the same runaway inflation or deflation as predicted for the de Sitter model? It would appear so!"

We talked about this earlier, but I didn't think it would be that simple. You actually have a spreadsheet that shows this to work? If so, can I get a copy or a link to it?

-S

Hi S, you wrote about my 'normal fluid and pump' effort: "We talked about this earlier, but I didn't think it would be that simple. You actually have a spreadsheet that shows this to work?"

You are right: it's not that simple! I think the normal fluid would not work unless one adds at least one knob to 'tweak' the pressure (like Roger does with his vapor pressure). It does however work automatically if you use the special fluid with pressure p = -ρc². You then need zero pressure outside the balloon and a constant negative interior pressure (essentially the 'false vacuum' of quantum mechanics, or so it seems to me).

Putting normal gas inside and outside (with a low pressure outside) causes the mass of the outside gas to reduce the expansion rate to a classical value - it has the right sense, but not the right magnitude. We need the real thing to do the trick correctly. The real vacuum apparently works like this, but the pressure is so low that it only works on vast scales - not in the lab.

About spreadsheets - I have many, but they all use the standard cosmological equations (Friedman) and not a dynamical balancing of forces. My spreadsheets essentially all comply with the p = -ρc² fluid, but that does not illustrate the dynamics of the balloon analogy as we trying it here.

A spreadsheet or BASIC program with a classical gas, a pressure differential (as I described) and normal Newtonian forces may be illuminating though! It should show the unstable equilibrium and runaway inflation/deflation, just at the wrong rate. One can also add a knob for tweaking the rate (a-la Roger). Are you interested in programming that?

-J

"About spreadsheets - I have many, but they all use the standard cosmological equations (Friedman) and not a dynamical balancing of forces. My spreadsheets essentially all comply with the p = -ρc² fluid, but that does not illustrate the dynamics of the balloon analogy as we trying it here.

A spreadsheet or BASIC program with a classical gas, a pressure differential (as I described) and normal Newtonian forces may be illuminating though! It should show the unstable equilibrium and runaway inflation/deflation, just at the wrong rate. One can also add a knob for tweaking the rate (a-la Roger). Are you interested in programming that?"

I think I would like one with the Friedman equations to see if I can program a simulation that could be for comparison with a later one that does what you want. Then we could tell if we succeeded, or be able to see if the 'knob' needs more or less adjustment. Am I making sense? Anyway, I need some way to get started if this is going to happen. Hopefully the speadsheet has a lot of notes!

-S

Hi S. Yes, it makes a lot of sense!

You are welcome to download my 'cosmo-calculator' spreadsheet if you want to take a peek (expansion.xls). No, the "speadsheet has [not got] a lot of notes!". However, it has some and the algorithm and equations are straightforward, so you should get a good start.

The defaults are set to the same value as the Java-Script cosmo-calculator and the results are roughly the same (the spreadsheet uses vastly less steps in the integration and is hence less accurate). You can set the mass and radiation Omegas to zero and Ω_v to unity for the vacuum-only solution which is the simplest case.

-J

Hi Jorrie,

I finally got it tonight. Yesterday there was a server problem, and earlier there was an internet problem here. What model is the chart in the spreadsheet? The de Sitter one in #117 makes sense if expansion factor represents the rate of expansion. For LCDM the rate of expansion should start out positive, then come down to zero in 5 to 8 GY, then go back up, right?

Don't let me hold you up on this. It is spring here, and there is a lot of yard work to do. I have also had to help one of my sons move, and had some vehicle trouble today after work! It may be some time before I have anything, so don't wait for me, OK?

Hi S, yes nice, busy time is spring!

The spreadsheet is general, so you can set it to any one model by simply changing the relative Omegas.

"For LCDM the rate of expansion should start out positive, then come down to zero in 5 to 8 GY, then go back up, right?"

Not quite! The acceleration of expansion goes to zero (and reverse) in the 'middle ages', not the rate, which stays well positive.

Take your time, I'm always 'spread-sheeting', so I can test ideas out in the meantime.

-J

"The acceleration of expansion goes to zero (and reverse) in the 'middle ages', not the rate, which stays well positive."

I was thinking acceleration, not rate when I wrote that. I want a chart in the simulation to go with the circle that I hope I can make to look like a sphere. Do you think a chart of acceleration of expansion, or rate of expansion is more useful? Can you plot one (or both) in a spreadsheet? Where will the data come from? Would one have better uncertainty than the other?

-S

Hi S, I prefer the straight expansion factor (or balloon radius if you like) against time. It gives all three: how the size change; the rate (slope) and the acceleration (change in slope). It is obviously possible to plot all three on one chart, with different scales for each.

"Where will the data come from?"

If you want to plot for the Friedman solution, all the equations are in that spreadsheet that you downloaded. They all have equal accuracy (or uncertainty, if you want).

-J

In #149 I wrote: "Can this process generate an acceleration of expansion that changes directly proportional to radius? The Friedman equations say yes, provided we have a fluid with equation of state w = -1, meaning pressure p = -ρc²."

Some more information about why the "The Friedman equations say yes, ..."

It is not quite correct to say that the "acceleration of expansion changes directly proportional to radius", because if there is no expansion, we have seen that the force of self-gravity of the fluid and the negative pressure of the vacuum cancel out, for an unstable equilibrium.

Solving both the Friedman equations together for a vacuum-only case (to keep it simple)^[1], gives another insight: the acceleration of expansion changes by the product of the radius (R) and the expansion rate at the time (H) squared, i.e.,

d²R/dt² => ±H² R

where => indicates proportionality.

If the expansion rate H > 0, the acceleration is positive, if H < 0, the acceleration is negative and if H = 0, the acceleration is zero, as expected for equilibrium.

This explains the process of accelerating expansion rate much more clearly.

-J

[1] For a flat universe, the two Friedman equations of cosmology can be written as:^[2]

H => ±√[Λ/3 + 8/3 Π G ρ_m] and d²R/dt² = R(Λ/3 - 4/3 Π G ρ_m)

For a vacuum only case (ρ_m=0), this reduce to

H => ±√[Λ/3] and d²R/dt² = R Λ/3

with the solution shown above.

[2] Principles of Physical Cosmology, P. J. E. Peebles (1993), eqs. 5-15 and 5-18.

Jorrie,

I've given it some thought and I think I understand why I oppose this and all models that describe space as a balloon.

The problem is the energy density of the vacuum. As you correctly pointed out, it doesn't change. It's not that space is being stretched, which would require a reduction in space. It's that new space is constantly being created. Energy considerations aside (where does all the energy for this new space come from?), here are my objections to the expanding balloon model:

1. When a balloon is inflated, its skin gets thinner and eventually tears, this is not like cosmological spatial expansion.

2. The curvature of a balloon decreases as it inflates. The curvature of the universe remains constant and is a function of the energy density of space which doesn't change.

That second point seems like an important problem.

If my objections above are correct, I offer the following solution.

Say we have a large sphere in a chamber. On the sphere we place a drop of viscous fluid (the rule for the fluid is that it always stays the same thickness on the sphere, so when more is added, it spreads out along the sphere to compensate). This viscous fluid acts as a nucleation point for the vapor (same fluid) in the chamber. As the vapor condenses into the fluid (only where there already is fluid), the fluid spreads out over the sphere (since there is more of it and it wants to keep it's thickness the same. Vapor pressure can be adjusted in order to cause "inflation" or slowing of the expansion, etc. Obviously matter and radiation are simply particulates in the fluid.

The k of the sphere represents the k of space.

The thickness of the fluid is analogous to it's energy density.

The vapor represents the apparent never ending reservoir of energy that's making the space.

What do you think?

Hi again Roger.

Some comments on your proposed 'vapor' solution. Taken at face value, it does not seem to be much different from my abortive double skin balloon. You condense vapor onto the skin at a constant thickness. I pumped fluid into a cavity with a constant thickness. Your 'vapor pressure' seems to be much the same as my fluid pressure.

Qualitatively, both models seem to work. As I said in #117, I could not get the correct acceleration of expansion for the double skin balloon. I'm not convinced that your scheme is any better. There seems to be more promise in a balloon filled with a fluid (or vapor) at constant density, but with a negative pressure. Controlling the negative pressure could perhaps get the correct expansion law, but I'm not sure of that either.

-J

Jorrie,

I don't think you understand my design, it's very different from your double skin design as I understood it.

In my design, the viscous fluid is space. The ballon is merely to guide the change in curvature. The expansion of the fluid is driven by two forces, the expanding balloon below it and the condensation above it (since it must maintain the same thickness it pushes outward along the surface of the balloon.

So my design has two knobs that you can adjust.

1. The first is the knob to inflate the balloon which both changes the curvature of space and expands it (space here being the fluid resting on part of the surface of the balloon).

2. The second is a knob the controls the rate of condensation of the vapor to the liquid on the balloon. This does not effect curvature, only expansion rate. You could even adjust it so that space contracted, through evaporation (surface tension mantaining the thickness), even as the balloon expanded.

The two knob approach solves a lot of problems I think. Please reread my suggestion and reevaluate your conclusion that the models are the same functionally. If you still feel that way we can go with your original model. It seems very different to me though.

Roger

Hi Roger

It then looks to me as if your "two-knobs-vapor-model" essentially combines the single and the double skin balloon. The first knob inflates the balloon by supplying interior pressure and the second one is pumping fluid into my cavity separately (the same effect as condensing more vapor, while maintaining thickness.)

There are a few issues that I can think of. One being that both your knobs affect expansion rate independently, which seems problematic. However, this happens to be an excellent brainstorm point. How can one control the curvature and expansion rate separately?

Let's take just a section of your balloon surface and condense vapor of constant thickness, so that it is forced to expand. Have a separate mechanism to maintain the curvature of the segment (which must actually be kept constant for a vacuum only model). One can even control the curvature to be zero (or negative, if you can imagine it) - hmm...

However, for a vacuum-only model, we must still have an acceleration of the expansion that is directly proportional to the expansion factor, i.e.,

d²a/dt² = 8/3 Π G ρ a (from Friedman's second equation).

This is one vicious requirement. It does not mean just turning up the 'expansion knob' more and more, but also turning it faster and faster, ad infinitum. Vacuum energy with its negative pressure apparently does that, but since we do have enough of that stuff readily available, the knob may have to do!

-J

Roger, in the last sentence of the previous post, I meant to say:

"Vacuum energy with its negative pressure apparently does that, but since we do NOT have enough of that stuff readily available, the knob may have to do!"

-J

You wrote:"Let's take just a section of your balloon surface and condense vapor of constant thickness, so that it is forced to expand. Have a separate mechanism to maintain the curvature of the segment (which must actually be kept constant for a vacuum only model). One can even control the curvature to be zero (or negative, if you can imagine it) - hmm..."

But Jorrie, this is precisely my design, I don't understand where our disconnect is. I'll write it again:

1. Chamber filled with vapor that can condense into a viscous liquid.

2. Balloon that can be inflated with a drop of liquid on it's surface (just a drop on part of it's surface)

3. Two knobs, one that controls the balloon inflation (or deflation) and one that controls the rate of consensation of the vapor onto the drop.

4. The drop is stretched by the balloon causing expansion when the balloon expands. The drop is pushed outward along the surface of the balloon as condensation occurs. (Remember, the liquid is analogous to our space)

5. As the liquid spreads out, there is more surface area, thus even without increasing the knob on the vapor pressure (which increases the rate of condensation, the condensation increases) we get accelerating expansion.

6. If we were to stop inflating the balloon at some point, the curvature becomes fixed, but the expansion continues (unless we want to slow it, stop it, or reduce it by lowering the condensation rate).

I really think this decouples the expansion from the changing curvature. You could even have contracting space with decreasing curvature with this model!!! Think about how flexible that is.

In your double skin model it seemed like the curvature and the inflation was not decoupled, is that true?

Roger

Correction:

I've been saying that inflating the balloon would cause the fluid to expand, this is incorrect. The fluids surface tension prevents any stretching, so the expanding balloon underneath only changes the curvature of the liquid sitting on it, it does not cause any expansion. Only adjusting the vapor pressure can cause expansion (by increased condensation)

I agree that we will need someone constantly turning the knobs, I nominate a computer since I'm too lazy to do it myself.

Hi Roger

You asked: "In your double skin model it seemed like the curvature and the inflation was not decoupled, is that true?"

What I had in mind is a 'cut-off' section of the balloon - the only way I can imagine the curvature to be decoupled from the expansion. If you have a proper balloon, the change of curvature is inversely proportional to the change of radius, even in your vapor model. Your model hence only deals with a segment of a balloon, as would an equivalent double-skinned balloon segment. The curvature of the segment is then controlled independently, since it is no longer a balloon in the normal sense. (I guess Transcendian will love this – think about the weird topology that he can give a balloon segment!)

Despite some apparent problems, I find the idea of 'condensation of a vapor' very interesting. If we postulate that the new condensation 'pushes' the previously condensed particles apart, we have an expanding section of space, which can move galaxies outwards with it. I'm sure it can be tweaked to have the correct characteristics.

The major problem I have with your design is that the turning of knobs makes it an arbitrary expansion law, i.e., there is no physics deciding expansion from the dynamics. My latest idea, a single skinned balloon with a fluid with positive density and negative pressure (#117, #149) mimics de Sitter's model automatically without knobs – it just needs a system to keep the pressure constant.

My special fluid is probably no more exotic than your special viscous liquid, but it avoids knobs altogether. Your idea sparked the 'balloon segment only' idea, so it has been very valuable, even if the condensation model proves unworkable. But, the jury is still out on that.

-J

Roger: "1. When a balloon is inflated, its skin gets thinner and eventually tears, this is not like cosmological spatial expansion."

I'm sure that you've heard about the possible 'big rip'?

More seriously, we are obviously not trying to make a cosmos out of the balloon. The material is simply infinitely stretchable. It's just a model...

"2. The curvature of a balloon decreases as it inflates."

Very valid objection. It means we can only use the balloon over a shortish timescale, because the balloon curvature actually goes in the wrong direction.

"The curvature of the universe remains constant and is a function of the energy density of space which doesn't change."

No, curvature is not only a function of the energy density of space (at least not presently), because there is still a significant matter component as well. In such a case, it is only zero curvature that remains constant. Eventually, when vacuum energy totally dominates, the curvature will however remain constant.

I will need to contemplate your proposal for the balloon model (tour next post) before I can reply.

-J

Jorrie,

You Wrote:"Very valid objection. It means we can only use the balloon over a shortish timescale, because the balloon curvature actually goes in the wrong direction."

Could you explain that a little more. Specifically, in a matter dominated universe (like the early universe), what does the curvature look like? Is the curvature due to matter in the opposite direction as the curvature due to vacuum energy? I thought it would be in the same direction, which would mean the balloon curvature actually goes in the right direction (from higher to lower). Is matter curving space in the direction opposite that space curves space? I'm just confused on this point.

Roger