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The Perfect CR4 Cosmic Balloon

Posted June 02, 2009 5:00 PM by Jorrie
Pathfinder Tags: Balloon analogy cosmology

I invite all interested readers to help design the 'perfect cosmic balloon', compatible with all of cosmology - if not, then at least with most it! The recent debate on this Blog, pitching the kinematical against the hyper-spherical (balloon) cosmic models, made it clear that there are lots of uncertainties about what the balloon model can and cannot represent.[1]

First, let's do the system engineer's thing and write a brief high level specification for the perfect cosmic balloon, trying to keep it "as simple as possible, but not simpler" (A. Einstein).

1 Specification

1.1 General

The two-dimensional balloon surface (black) shall represent all of 3D-space. This means we disregard the third spatial dimension. The extra inside/outside dimension (pink) shall represent hyperspace. It is not accessible to us - just a visualization aid. It simply provides a direction for the surface to expand and curve into.

Formally, the hyper-radius (R) may tend to infinity, or even have an imaginary value (iR), but for this exercise we will stick to real, finite values of R. Our observable universe (blue) is a limited circular patch on the surface with a radius RO, determined by how far light could have traveled since the balloon blow-up started.

1.2 Specifics

1.2.1 Mass

The surface of the balloon shall be able to hold massive particles (green) in a frictionless manner, i.e., they shall be able to move freely across the surface, but never be able to 'fly' off the surface, even if the radius should suddenly shrink. These particles shall represent the ordinary and dark matter of the cosmos.

1.2.2 Radiation

The surface shall also be able to hold photons (red), always traveling at the speed of light along the local surface and hence have energy that depends on wavelength. These photons shall represent cosmic radiation energy and hence also the cosmic microwave background (CMB).

1.2.3 Vacuum (Change note: reply #243)

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

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 + Ωm/a3 + Ωr/a4 + ΩΛ]

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

The operating equation is then:

(ΔR/dt)2 = R2H2 = R2H02 [(1-Ω0)/a2 + Ωm/a3 + Ωr/a4 + ΩΛ]

'Guest' has proposed the following neat little high level algorithm for this system in reply #242:

1. Set gas flow direction valve for inflating the balloon.

2. Continuously measure the radius R of the balloon and calculate (ΔR/dt)2 = R2H2 = R2Ho2 ((1-Ω)/a2 + Ωm/a3r/a4 + ΩΛ), with all the constants given and a = R/Ro, where Ro is a value that corresponds to the R for which the parameters are given.

3. Calculate switch = (1-Ω)/a2 + Ωm/a3r/a4 + ΩΛ. If switch goes negative, even temporarily, change the gas flow direction valve for permanently deflating the balloon.

4. Measure ΔR/dt, square the result and compare it with the calculation of (ΔR/dt)2.

5. If squared result is smaller than the calculation, increase gas flow rate.

6. If squared result is larger than the calculation, decrease gas flow rate.

7. Repeat from 2 until exit condition is reached.

3 Tests

First a general discussion is given and then specific simulations and 'tests' (to follow).

3.1 General

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.

3.2 The de Sitter model

The de Sitter expansion curve for a flat (Ωm = Ωr = 0, ΩΛ = 1) universe is obtained by integration of the following simplified form of the above expansion equation:

ΔR/dt = RH0/978

where the factor 1/978 is a conversion of Ho from km/s/Mpc to 1/Gy.

This is also called case (0,0,1), as on tthe figure below. It is clearly an exponential expansion curve. A Hubble constant of 72 km/s/Mpc was used for the curve.

The age of such a universe would have been about 105 Gy, read off where the curve intersects the 100 Gly radius line (which is taken as a=1). De Sitter did not intend this to be a model of the real universe, but as a tool to investigate expansion dynamics.

3.3 The Einstein-de Sitter model

The first workable attempt to model the real universe came when Einstein and de Sitter made the assumption that the universe is flat and for all practical purposes contains only matter, i.e., Ωm = 1, Ωr = ΩΛ = 0. This means that the Friedman equation reduces to: (ΔR/dt)2 = (RHo/978)2 Ωm/a3, giving the parabolic curve below, again for Ho = 72 km/s/Mpc.

The curve intersects the 100 Gly radius line at around 9 Gy age, making such a universe uncomfortably young! It used to be no problem when Ho was still believed to be around 50 km/s/Mpc, but not any more.

3.4 The Lambda-cold-dark-matter (LCDM) model

This is the 'standard' model at the present time, comprising about 26% matter (ordinary plus dark matter), a tiny amount of radiation energy, with the bulk of the energy (74%) made up of vacuum energy (the cosmological constant). The full equation must be used here: (ΔR/dt)2 = (RHo/978)2 ((1-Ω)/a2 + Ωm/a3r/a4 + ΩΛ)

This caused an expansion curve that started out with a decreasing rate, slowly turning over to an increasing expansion rate at around 7 Gy.

This model universe has a present age of around 13.7 Gy, which is quite comfortable.

4 Conclusion

While it may be impossible to 'design and build' a laboratory sized cosmic balloon that will 'automatically' have the properties of the real cosmos, it is definitely possible to construct one that can follow the Friedman equations, at least for a short period of time. Section 2 (Design) describes such a device and its method of operation.

Such a balloon can serve as a 'crutch' to lean on in discussions of cosmological principles. At least it is a little more 'tangible' than the presumed dark matter and dark energy of the real cosmos. In a next Blog entry I will attempt to use it to show how some cosmological issues can be explained.

Jorrie

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#1

Re: The Perfect CR4 Cosmic Balloon

06/03/2009 11:02 AM

Jorrie,

Forgive me if my questions are simplistic but I'd like to form clear picture in my mind.

In general (when all effects are considered), is the balloon easier to blow up as you go, the same, or harder?

Roger

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#2
In reply to #1

Re: The Perfect CR4 Cosmic Balloon

06/03/2009 10:04 PM

Hi Roger.

You asked: "In general (when all effects are considered), is the balloon easier to blow up as you go, the same, or harder?"

This is not such an easy question to answer! Like in a normal balloon, it is that first movement that is toughest, because the force of gravity is the strongest when the cosmic balloon is smallest. Gravity 'gets weaker' as the surface area gets larger, but one needs a lot more 'gas' to keep on blowing the the balloon larger.

According to our present knowledge, the blowing-up agent is vacuum energy (Lambda), which increases linearly with the volume of space (constant energy density). This caused the blow-up rate (dR/dt) to first decrease, because vacuum energy was insignificant in the beginning. Lately dR/dt increased, as vacuum energy became significant. We expect dR/dt to further increase in the future.

There is some uncertainty as to what caused the initial huge dR/dt (cosmic inflation), but I think Lambda is also a good bet for that.[1] It is only that it had to be many orders of magnitude larger than is is today.

-J

[1] I describe this loosely in chapter 15 of Relativity 4 Engineers. The chapter can also be downloaded from my website.

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#5
In reply to #2

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 12:17 AM

Jorrie,

Can we add a rule that the ease at which to blow up the balloon is constant and explore the consequences of that, especially with regards to the Cosmological Constant. I think this might be an interesting tangent.

Roger

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#7
In reply to #5

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 1:06 AM

Hi Roger.

For the case "the ease at which to blow up the balloon is constant", it essentially means an empty universe (no gravity or vacuum energy, approximately the Milne Model). I doubt if it will give much insight.

It obviously depends on what we mean by "ease at which to blow up the balloon". In my 'specification' the thing is self-inflating and it is difficult to define the 'ease'.

-J

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#8
In reply to #7

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 10:05 AM

Couldn't Vacuum Energy be varied to compensate? I don't think the universe has to be empty if we allow Vacuum Energy to change over time.

What I'm proposing is that in this model or a parallel model we allow Vacuum Energy to change over time with the goal of keeping the "ease at which the Universe expands" constant.

I think this would result in no rapid inflationary periods, just a constant expansion. What do you think? I think this is worth exploring.

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#9
In reply to #7

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 10:29 AM

Ok, scratch my suggestion. I'll go along with the model you proposed and if I see an opportunity to introduce my idea where it will be more clear, I'll do that. Disregard my previous comment about vacuum energy.

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#11
In reply to #9

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 10:44 AM

Hi Roger, no need to "scratch" the suggestion! Variable vacuum energy is a hot topic today, but a little above my comprehension as well. For simplicity, let's first stick to Einstein's Lambda, the constant density vacuum energy. As you suggest, we may get back to it later.

-J

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#12
In reply to #11

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 10:47 AM

"For simplicity, let's first stick to Einstein's Lambda, the constant density vacuum energy"

Yeah, that's what I was thinking and why I changed my mind.

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#3

Re: The Perfect CR4 Cosmic Balloon

06/03/2009 11:01 PM

Specification update. This change will be referenced to this reply - a simple form of configuration management.

My note 2: "Normal mass-energy has positive pressure, but the vacuum can only have positive energy if it has negative pressure. See e.g. Ned Wright's Piston pulling on the vacuum. When the dynamics of the expanding balloon (increasing R) are included, you can get a 'self-pulling piston', causing a positive feedback loop that can give an outward acceleration to the piston." is confusing.

Ned Wright's 'Piston pulling on the vacuum' is not representative of what might be going on in the false vacuum of the universe. He just demonstrated why the pressure of the false vacuum (inside the cylinder) is negative. In the balloon, it may be the other way around: normal vacuum on the surface (which represents 3D space, remember) and the false vacuum with negative pressure outside the balloon, causing the surface to have an outward force component. This may perhaps also not be a very good way of putting it, but it is less confusing than the original.

I propose a changed wording for note 2:[1]

[2] Normal mass-energy has positive pressure, but the vacuum can only have positive energy if it has negative pressure. See e.g. Ned Wright's Piston pulling on the vacuum. [Start change] In the case of the balloon, one can think of the negative pressure (false vacuum) to be on the outside, while the surface represents the true vacuum, with zero pressure. If the surface is expanding, the negative pressure 'outside' will cause a larger 'suction force' and hence attempt to increase dR/dt. [End change] Don't worry if this is as clear as mud - later, when we design this mechanism, the how it works may become clearer, if not quite the why it works.

-J

[1] Strictly, I should now wait for approval of the change, but I'll rather limit the confusion straight away and update.

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#4

Re: The Perfect CR4 Cosmic Balloon

06/03/2009 11:34 PM

Hi Jorrie,

I'm not sure what I can add, but I'll think about it. Most of this I have heard before from our discussions except this:

[3] The negative pressure material of the surface must be even weirder than described above. If it has any mass and you compress a finite piece of it, it shrinks and loses mass (its energy density remains constant under compression). If it starts to shrink, the negative pressure makes it shrink even faster on its own accord. Quite bizarre, but this apparently models the real universe closely!

Can you give references for this phenomena?

-S

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#6
In reply to #4

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 12:56 AM

Hi S.

The phenomenon is well described in formal textbooks, but the link I gave to Ned Wright's page has the math and some discussion on it, e.g. under the heading "Einstein's Greatest Blunder":

"However, there is a basic flaw in this Einstein static model: it is unstable - like a pencil balanced on its point. For imagine that the Universe grew slightly: say by 1 part per million in size. Then the vacuum energy density stays the same, but the matter energy density goes down by 3 parts per million. This gives a net negative gravitational acceleration, which makes the Universe grow even more! If instead the Universe shrank slightly, one gets a net positive gravitational acceleration, which makes it shrink more! Any small deviation gets magnified, and the model is fundamentally flawed."

Note that when Wright speaks of "net positive gravitational acceleration", it is the same as my d2R/dt2 < 0, because he (correctly) takes gravitational acceleration to be negative.

-J

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#21
In reply to #6

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 11:20 PM

Hi Jorrie,

Nice link, I failed to open it before. There is probably nothing there that I haven't heard before, but I don't claim to understand it all. It doesn't answer my question in the way I intended. I was looking for empirical evidence, not a thought experiment. In the same regard, has anyone tried this: "If the vacuum is trying to pull the piston back into the cylinder, it must have a negative pressure"? If someone tried it and it didn't work, then we would learn something. It would need to be done in outer space, or the effects of atmospheric pressure deducted from the readings.

Here's something I would like your comment on:

"For the highest reasonable elementary particle mass, the Planck mass of 20 micrograms, this density is more than 1091 gm/cc. So there must be a suppression mechanism at work now that reduces the vacuum energy density by at least 120 orders of magnitude."

and this:

"The cosmological constant will also cause a precession of the perihelion of a planet. Cardona and Tejeiro (1998, ApJ, 493, 52) claimed that this effect could set limits on the vacuum density only ten or so times higher than the critical density, but their calculation appears to be off by a factor of 3 trillion."

Should I have warm fuzzies when calculations are off by 3 trillion and when Bayesian statistics are worse than damn lies?

-S

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#22
In reply to #21

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 1:37 AM

Hi S.

The only evidence that I know of is the cosmological measurements and the Casimir effect between two plates. None of these are very convincing, though...

On your quote: "So there must be a suppression mechanism at work now that reduces the vacuum energy density by at least 120 orders of magnitude."

Here's a very interesting observation that I've stumbled upon while toying with the balloon analogy. (I hope my sums are right - please check!)

If an "incompressable" pressure vessel with an internal gas pressure p is placed inside a pressure chamber that maintains exactly the same internal gas pressure p, there will be a net inward stress on the vessel, due to the lesser inner surface area relative to its outer surface area. Now replace the positive pressure everywhere with the negative pressure p = -ρc2 of the vacuum and there will be a net outward stress on the vessel, not so? (Fyz will probably show me why this whole line of thinking is flawed!) Nevertheless...

For a large spherical vessel with outer radius Ro and inner radius Ri, with identical inner and outer negative pressures (p = -ρc2), the net outward force (ΔFo) on the sphere (pressure multiplied by the areas and subtracted) is:

ΔFo = 4Πρc2 (Ro2 – Ri2) = 8Πρc2 R ΔR

where R = (Ri+Ro)/2, the 'average radius' of the vessel, for simplicity.

In the case of the cosmic balloon, we are not allowed to have any thickness into the hyper-radial direction, so ΔR = 0 and hence ΔFo = 0. But quantum physics prevents us from having a length less than the Planck length, so maybe ΔR = L_Planck ~ 10-35 m. Making this assumption, the net outward force of the vacuum is around

ΔFo ~ 8Π 10-26 1015 1028 10-35 ~ 10-17 N

where (according to latest WMAP results) the radius of curvature of the universe is ~1028 m (or more) and the cosmic density ~10-26 kg/m3.

If only the negative pressure on the 'outside' is considered (with zero on the 'inside', like quantum physicists apparently do), the outward force would have been:

Fo ~ 8Π 10-26 1015 (1028) 2 ~ 1046 N.

This is a factor 1063 higher than ΔFo calculated above. Cosmologists and quantum physicists reckon that the energy density of the vacuum should be a factor 10120 higher than what cosmological measurements indicate. With this variant of the balloon model, one can achieve that sort of factor. Just make the radius R ~ 1085 m, which is just on the closed side of being precisely 'flat'; it is for all practical purposes a balloon with a radius "approaching infinity".

This is just "too good [NOT] to be true!" Are physicists missing a trick here? They call their 120 orders of magnitude discrepancy the "most embarrassing prediction of theoretical physics". Why won't they just listen to engineers?

-J

PS: The truth is that they can probably shoot this balloon so full of holes that it will vanish into the vacuum, never to surface again, at least not his side of eternity. I've never seen this type of argument published, so it is probably just so much wishful thinking. It does not deminish the value of the hyperspherical (balloon) model, though...

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#47
In reply to #22

Re: The Perfect CR4 Cosmic Balloon

06/06/2009 3:45 PM

Hi Jorrie,

I'd like to understand this line of reasoning, but I don't.

First, in the paragraph you cite (whatever its source is), I don't understand the conclusion that there is any net inward or outward stress on the inner vessel. The way I look at it is that all of the surface area of the outer vessel is a unified whole, including the surface of the inner vessel. The pressure tries equally to push against every square inch of the total surface. So with positive pressure, the PSI squeezing against the inner vessel is identical to the PSI pushing against the outer vessel's outer wall. I would expect the PSI pushing inward on the inner vessel is the same as the PSI pushing back out of the inner vessel, resulting in no net stress. Obviously if you instantly disintegrated the inner vessel itself, but left the inner vessel's pressure-creating contents in place inside the larger vessel, you wouldn't expect a net flow of material into or out of the region formerly enclosed by the inner vessel.

It's no different than a scenario where there's no inner vessel, but a positive external pressure is applied against only a portion of the vessel's exterior, with zero external pressure applied against the remaining portion of the vessel's exterior. There should be outward stress on the portions of the exterior wall which confront zero external pressure, and no net stress on the portions confronted with an evenly matched pressure.

I also get confused about translating the idea of pressure to the balloon scenario. I like the translation of Lambda into a negative pressure outside the balloon, but it seems to me it needs to put in the context of the positive pressure inside the balloon which drives the original expansion. A balloon wouldn't expand at all if the air pressure inside were zero.

When you blow a fixed amount of air into the balloon in a (true) vacuum, how it behaves depends on the characteristics you specify for the balloon material. If the material can stretch infinitely without resisting expansion at all, then in a vacuum the balloon would continue expanding forever with its initial fixed amount of air. (This would be like an unenclosed bubble of air released into empty space.) There would continue to be a pressure gradient across the balloon surface which would asymptotically approach zero as the radius approaches infinity, so the rate of increase in r would decay over time. I think that in order to keep the balloon expanding at a constant rate of Δr you would need to increase the volume inflow rate by a factor of 2/3 Δr3, (i.e. ΔV/Δr), requiring an ever-faster rate of pumping.

This doesn't take gravity into effect yet. To model gravity, you would let the air pressure in the balloon decline over time. My off-the-wall guess that increasing the volume inflow rate by a factor of Δ2r2 (???) (i.e. ΔSurface Area/Δr) might achieve a realistic result of a decreasing expansion rate. (We are modeling Lambda as a separate external effect so we don't need to account internally for its effect.)

An alternative model would specify a more realistic balloon elastic material, which has some coefficient of resistance against stretching. Then if a balloon in a vacuum is inflated with a fixed quantity of air, it will inflate to some size and then asymptotically stop stretching at some equilibrium point where the air pressure equal's the elastic material's resistance to stretching. I think it's too messy to try to calculate the rate of air pumping into such a balloon to model the effects of the expansionary momentum, the gravitational contraction and the effect of Lambda negative pressure.

Jon

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#49
In reply to #47

Re: The Perfect CR4 Cosmic Balloon

06/06/2009 4:35 PM

Hi Jon, thanks for joining in!

If your "First, in the paragraph you cite (whatever its source is)" refers to my para starting with: "If an "incompressable" pressure vessel with an internal gas pressure p ...", then it is my own wording, not a quote. With "stumbled upon" I meant I thought about it while working on something else.

I think the confusion may arise because I described two pressurized items: the spherical vessel, with a definitive wall thickness, sitting inside a pressure chamber of arbitrary size and shape, but with the same gas pressure than inside the spherical vessel.

The wall thickness of the spherical vessel results in the inner surface area being less than the outer surface area of that vessel. In the case of identical positive pressures inside and outside the spherical vessel, is there any reason why there would not be a net inward force on the vessel? After all the outer surface area multiplied by ρ must be larger than the inner surface area, multiplied by the same ρ.

I'll come back to the rest of your post if I'm sure we understand my setup the same way.

-J

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#50
In reply to #49

Re: The Perfect CR4 Cosmic Balloon

06/06/2009 5:00 PM

Hi Jorrie,

OK I get your point. I can see how there would be a compressive stress on the inner vessel's wall material, but since the inner vessel is incompressible, that pressure would not be felt by the material inside the inner vessel.

If the inner vessel were made entirely of compressible material, then presumably it would compress as much as possible, causing the thickness to decrease, and the remaining differential in surface area would cause the inner vessel to contract a bit, such that it's higher pressure offsets its smaller surface area.

If you apply the same concept to a balloon, I think you still have to account for the positive pressure inside the balloon.

Also, in the real universe, I thought the problem is that most quantum theories predict 10120 times too much energy for it to be the source of the cosmological constant. Doesn't your theory make the energy disparity even greater instead of reducing it?

Jon

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#38
In reply to #21

Re: The Perfect CR4 Cosmic Balloon

06/06/2009 1:56 AM

Hi S.

Another piece of speculation on the factor 10120 discrepancy. Recall that I wrote in the 'specification' that the mass of the hyper-surface (the vacuum) itself must be added to the mass of the other energies that 'sits' on the surface. The discrepancy means that not only is the negative pressure a factor 10120 too large in magnitude, but the mass of the surface is also too large by the same factor.

Now this may look like a nice canceling out, but the problem is that such a mass would utterly mask radiation and matter energy, which is not what is observed. In the data, we can 'see' the effect of the early radiation energy dominance and matter dominance after that. It is only later on that vacuum energy started to dominate.

I first thought the mass discrepancy may also be cleared by a hyper-surface of only one Planck length thick. However, realizing that we calculate the particle mass on the surface as if it is a 3D space, we must do the same for the virtual particles of the vacuum. Sadly, the mass discrepancy seems to remain.

However, there may perhaps come a surprising conclusion out of quantum gravity that inserts a multiplication factor of LPlanck/Rcurvature into the equations - who knows?

-J

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#10

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 10:33 AM

You Wrote:

4. Total Energy

The sum of the energies in points 1 to 3 (matter, radiation and vacuum) shall constitute the static surface energy of the balloon. Ignoring the possible negative pressure,[3] this energy shall attempt to gravitationally collapse or shrink the balloon, i.e., it shall work like normal gravitating energy, but in the hyperspace direction only.[4]"

I'm confused by this. You wrote that "this energy shall attempt to shrink the balloon". I thought the Vacuum Energy pushed outward. Am I incorrect? So in other words, the Gravity and Radiation are trying to keep the balloon from blowing up while vacuum energy is trying to get it to blow up (expand). If that isn't the case, what is providing the expansion, where does it come from?

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#13
In reply to #10

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 10:52 AM

Hi again Roger.

You asked: "I thought the Vacuum Energy pushed outward. Am I incorrect? So in other words, the Gravity and Radiation are trying to keep the balloon from blowing up while vacuum energy is trying to get it to blow up (expand). If that isn't the case, what is providing the expansion, where does it come from?"

Vacuum energy does two things: it has positive mass-energy and negative pressure. The positive energy of the vacuum causes normal gravity (like any mass), while the negative pressure creates an expansive force - the difficult one to get to grips with! Check the modified note 2 in the OP.

-J

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#14
In reply to #13

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 10:57 AM

Jorrie,

I see. How does dark energy fit into this?

Roger

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#16
In reply to #14

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 4:00 PM

Hi Roger.

Vacuum energy (Lambda or cosmological constant) is one type of dark energy. The other other well-know type of dark energy is quintessence, a dynamic quantity whose energy density can vary in time and space.

-J

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#17
In reply to #16

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 4:13 PM

So, again in the spirit of keep it simple in the begining, I propose we leave quintessence out and focus only on the energies you've identified.

My instinct, as a physicist, is to look for something that remains constant in our model. Is there anything that is constant in this model? I don't think the static energy you mentioned is. This would be usefull knowledge.

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#18
In reply to #17

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 4:24 PM

Roger, you asked: "Is there anything that is constant in this model?"

Only the vacuum energy density (cosmological constant) remains constant, AFAIK. One can perhaps say that the total amount of matter in the model remains constant, but that's not very useful, because the density of the matter in the model changes over time.

I must go for now. Will be back in some 6 hours or so.

-J

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#19
In reply to #18

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 4:39 PM

Ok. I'm just doing some thinking out loud now and am going to throw out some ideas (they may make no sense, but I think that's the point of this). I really don't like a system that doesn't have some sort of global constant.

Usually that constant is energy, but apparently the amount of energy of this system is increasing in time. Before you read anymore, is that correct? If not, disregard the rest of this.

If it is true, than I propose that we make up another, otherwise unknown energy that is equal and opposite to the static energy so that the sum of these energies are zero. This energy will be the "resevoir" from which the static energy comes from. We don't need to worry about what the "unknown energy" is yet, lets create is and see what it looks like. I just feel if we are going to get anywhere we need a constant.

Tell me what you think (in 6 or so hours),

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#20
In reply to #19

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 11:07 PM

Hi Roger, you said: "I really don't like a system that doesn't have some sort of global constant."

If it will help, the constants c and G are obviously featuring in any gravitational system. Relativists and cosmologists often choose all units so that c=G=1 and then don't show them in their equations.

"Usually that constant is energy, but apparently the amount of energy of this system is increasing in time. Before you read anymore, is that correct?"

Correct. As long as there is expansion, total vacuum energy is increasing; it is only the density (and hence the negative pressure) of vacuum energy that remains constant.

Another quantity that remains constant is the total mechanical energy defined as kinetic energy plus potential energy. Kinetic energy is proportional to the expansion rate Ho (squared) and potential energy is negative and inversely proportional to the radius of curvature. The sum of the two is constant - actually it is precisely zero for a flat universe. This is just like the mechanical energy of an orbit, where the flat case is represented by escape velocity (a parabolic trajectory). Newton's specific orbital energy (energy per unit orbiting mass) is given by: ε = ½ M v2 - 2GM/r.

So, I'm not sure that we need another constant, but maybe you are right.

-J

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#23
In reply to #20

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 7:20 AM

Jorrie,

The ever increasing energy bothers me. Most of physics is based on conservation laws and the problem here is we don't seem to have a conserved quantity. It's hard to get any useful information about a system without conservation because it's basically an open ended system.

It would be my preference if this ever increasing static energy was coming from someplace, an energy reservoir. I realize we would have to make this thing up, but I think we need it for our model.

Also, I have a separate question. Does vacuum have momentum? You don't have to have mass to have momentum. Radiation has momentum. Does vacuum?

Just in case you're not sure what I'm talking about (I'm pretty sure you are), here is the momentum of radiation (light)

Do we have something like that for vacuum, because the way vacuum behaves, it sure seems like we should.

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#25
In reply to #23

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 8:46 AM

I guess what I'm trying to say here. If an engineer were to look at the design above, and we didn't tell him it was the universe, rather a device we were making. I think he'd ask where the power supply was.

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#28
In reply to #23

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 1:26 PM

Hi Roger, you wrote: "It would be my preference if this ever increasing static energy was coming from someplace, an energy reservoir."

The ever-increasing vacuum is where the growing energy is coming from. Recall that vacuum energy density is constant, so larger vacuum, more energy.

You asked: "Does vacuum have momentum? You don't have to have mass to have momentum. Radiation has momentum. Does vacuum?"

Sure vacuum has momentum. The vacuum has positive energy, which is equivalent to normal mass. Remember that we add vacuum energy to matter and radiation energy to get the 'static energy' of the surface. Vacuum momentum can only work in the hyper-radial dimension, because vacuum energy does not clump (as far as we know) and so cannot move in normal 3D space.

-J

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#29
In reply to #28

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 1:41 PM

Thanks Jorrie regarding momentum. That's what I suspected.

Is it safe to say that in the entire system (our ball) that momentum is conserved? I mean, if matter and radiation are homogeneous, and the expansion is radial and equal in all directions, then the net momentum is zero? Is that correct.

Also, I understand that the energy comes from the vacuum. My point is, we have a system here with ever increasing energy. That doesn't happen in nature. Anytime something is increasing in energy, something else is giveing it up. If a ball is falling, then its kinetic energy is increasing, but it's potential energy is decreasing.

What I'm asking for, and I think we're trying to approach this as a model that is similiar to the universe,not exactly it. What I'm asking for is an energy reservoir so there is conservation of energy. I realize that this doesn't exist (as far as we know) in the real world. I just would like one because I think things make more sense with one.

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#32
In reply to #29

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 2:47 PM

Hi again Roger. Yes, I think one can say momentum is conserved in the way that you stated. However, just like in an orbit, specific orbital energy and specific angular momentum are conserved, not momentum per se. One may say that the Earth-satellite system's momentum is conserved, but not the momentum of the satellite itself.

You wrote: "My point is, we have a system here with ever increasing energy. That doesn't happen in nature."

But, this is how we think the universe works! The vacuum apparently gives up a tiny fraction of its energy in order to increase the mechanical energy of expansion, while keeping the kinetic expansion energy and the potential energy of the expansion in balance.

If you look at the vacuum model that I speculated on in reply #22 to S, you will notice that the outer vacuum, with its pressure p = -ρc2 may actually be of infinite (or at least undetermined) size. That ρ represents positive energy density, implying a limitless reservoir of energy. (If we could just find a way to tap it!)

-J

PS: I'll reply to your posts 26, 27 and 31 later - have some other commitments to follow up.

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#33
In reply to #32

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 3:09 PM

Jorrie,

Don't bother responding to posts 26,27, and 31 unless you see something you specifically wanted to address. That was just a stream of consiousness on my part. I'm still interested, but it definitely hit a dead end and I need to think about it more.

Roger

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#34
In reply to #32

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 3:23 PM

You Wrote: "If you look at the vacuum model that I speculated on in reply #22 to S, you will notice that the outer vacuum, with its pressure p = -ρc2 may actually be of infinite (or at least undetermined) size. That ρ represents positive energy density, implying a limitless reservoir of energy. (If we could just find a way to tap it!)"

I would prefer a self-contained system. This idea of infinite energy that exists just outside our universe that only manifests itself by expanding space-time seems........unlikely and overly convenient (I half expect Jon Lovitz to say "yeaaah, that's the ticket". That may certainly be an interpretation, maybe even a popular one, but it's not a useful interpretation in my opinion. At that point we might as well say that expansion is "magic". Unless the change in the rate of expansion can somehow be tied to the surface area of a hypersphere (r3 dependence)?

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#36
In reply to #34

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 9:58 PM

Roger, you wrote: "I would prefer a self-contained system. This idea of infinite energy that exists just outside our universe that only manifests itself by expanding space-time seems........unlikely and overly convenient"

I would prefer the same, if it is possible. That's why I wrote in the spec: "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. "

So, that's still the quest! The diversion into the negative pressure 'inside and outside' the balloon just addressed certain comments informally.

You wrote: "At that point we might as well say that expansion is "magic". Unless the change in the rate of expansion can somehow be tied to the surface area of a hypersphere (r3 dependence)?"

This is certainly possible in terms of math. How to present it visually is a huge challenge, but the balloon may still come to the rescue.

We are not 'designing the system' yet, but when we do, we may find that we can't stay out of the interior or exterior. E.g., we may perhaps just put the balloon in a huge chamber with pressure regulated to be always lower than the changing pressure inside the expanding balloon. If we then also include the suction device and its energy source into 'the system', we have a sort-of closed system.

-J

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#15

Re: The Perfect CR4 Cosmic Balloon

06/04/2009 11:45 AM

There is another "alternative" shape for the universe: the toroidal shape (like a donut... yiami... ...) But it is supposed that this kind of universe must have derived from a circular (ring) singularity... I (personally) find no reason for such a (1 dimensional) ring singularity as it is much simpler and probable to start from a (0 dimensional) point singularity. So, I really like the (much simpler) hyperspherical universe. (But who knows???... Maybe we are wrong and this is not the right shape...)

I don't want to comfuse things and I'll stick on the "Hyperspherical Universe"... I have some other thoughts about this whole concept but I have no time to discuss them now (a lot of work... lack of time...) I'll try to write sth on th next days... Keep in touch...

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#24

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 7:37 AM

The above figure (I made this drawing) represents the hyperspherical universe as it is expanded in different time periods (t0, t1, t2, e.t.c.). The * in the middle (at t0) represents the initial singularity (BB). The blue sphere is how an observer standing outside the universe -i.e. in the 4th dimensional hyperspace (maybe God???)- sees the evolution of the universe. We (on the right) observe the borders of the observable universe (e.g. point A1). A1 is related to a previous time period t4 (while we are in the time period t5) due to the fact that it takes so long for the light (from A1) to reach us. So, we actually observe the point A1 instead of the point A2. Hence, we actually observe a smaller universe (the red sphere) and not the real universe (blue sphere). We should take this into account. BTW for us it seems that the BB singularity is not in the centre of our hyperspherical universe (it seems to be eccentric).

These are some of my thoughts. Any comments???

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#30
In reply to #24

Re: The Perfect CR4 Cosmic Balloon

06/05/2009 2:21 PM

Hi George. Nice picture. Just note that the observable universe (your pink sphere) is not a sphere, but actually has a heart shape (the "cosmic heart"). There are two reasons for this: i) if it was not for the fact that the cosmos was initially opaque, we would have been able to observe photons from just after inflation, i.e., virtually from the center and ii) the expansion rate was much faster directly after inflation than it is today.

If you trace the path of any photon back from us to the center, moving along smaller and smaller spherical surfaces, it reaches the center and you get a heart shape. The exact shape of of the heart depends on the expansion law followed, i.e., it is slightly different for a matter+radiation only universe than for one that also includes vacuum energy (non-zero cosmological constant).

-J

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#79
In reply to #30

Re: The Perfect CR4 Cosmic Balloon

06/09/2009 11:29 AM

Thanks for the reply. This new drawing will clarify things.

I had, also, (initially) in mind sth like in Fig.A. But then (erroniously) I thought that Fig.A and Fig.B were (geometrically) equivalent. (That's why I represented the universe -as we observe it- like a smaller sphere.) But after a second thought I considered that these two figures are not (geometrically) equivalent (am I right???)... So, I end up with the Fig.A as the proper shape.

Do you agree???

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#81
In reply to #79

Re: The Perfect CR4 Cosmic Balloon

06/09/2009 2:38 PM

Hi George. Yes, that 'cosmic heart' is more like it - nice picture!

The exact shape obviously depends on the ratio of matter to vacuum energy density. The one you have showed is more or less the "over-dense matter-only" universe. In the case of a vacuum energy (Lambda) dominated expansion, the heart is more pointed at the 'time now' side.

-J

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#92
In reply to #81

Re: The Perfect CR4 Cosmic Balloon

06/10/2009 7:06 AM

Just a question: Let's suppose that the universe is no longer expanded (i.e. stable size)... In this case the cosmic horizon is expanded as the time passes (we see even deeper in the space and much more back in time). Obviously, there will be a moment that the horizon will include the whole universe, meaning that we (at point A) will be able to observe the point B. How close to the BB the point B will be??? (Which time period of the universe is related to the point B???... What will we observe then, around us...???)

(I know that this question is rather philisophical as it is known that the universe is expanded and -moreover- this expansion is accelerated.)

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#95
In reply to #92

Re: The Perfect CR4 Cosmic Balloon

06/10/2009 5:02 PM

Any answer is possible, depending on the history of this altered universe.

In principle we should be able to see around the universe and back our own location (whatever that might mean) as often as we might desire. Whether the physical laws of a universe with ever-increasing entropy would allow that is another matter.

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#97
In reply to #92

Re: The Perfect CR4 Cosmic Balloon

06/10/2009 7:23 PM

I'm having a hard time wrapping my mind around this picture. If we are at point A, what does the point B represent? Is there a link that would explain it?

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#103
In reply to #97

Re: The Perfect CR4 Cosmic Balloon

06/11/2009 9:23 AM

I don't know any link or book or article... These are some thoughts of mine... (I have the sense that some others had similar thoughts... including Jorrie...)

Point B represents exactly the "opposite side" of the universe, the more faraway point (from us) in space and time.

Please, read my post 101 for further explanations.

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#113
In reply to #103

Re: The Perfect CR4 Cosmic Balloon

06/11/2009 8:44 PM

Hi GK,

"Point B represents exactly the "opposite side" of the universe, the more faraway point (from us) in space and time."

Since the universe has no center point, it seems to me that there are an infinite number of "opposite sides". I think that is what is bothering me. If you plot them all, they would "smear out" to become a sphere? A sphere is what link #1 in #99 says.

-S

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#119
In reply to #113

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 6:16 AM

I think you can see that a point on a circle has an opposing point, and also that a point on a sphere has an opposing point.

You can readily define the opposing point mathematically - if a point on the surface of a 4-hypersphere with centre (0,0,0,0) has coordinates (a,b,c,d), the opposing point has coordinates (-a,-b,-c,-d).

If we go away from the hyperspherical model, the number of "opposing points" will depend on the complexity of the surface shape and the definition of opposing* (there can be 1, 2, or 4 on a toroid, depending on position and definitions). Currently, there are no proposed models for the universe that result in a continuum of such points (such a model would also require singularities of a completely different order to those involved in black holes).
*Some of these definitions are not always reciprocal - for example, a point is not necessarily the furthest point from the point that is furthest from it.

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#120
In reply to #119

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 7:04 AM

As far as an ant crawling on the surface of a 3D sphere is concerned there is no centre of its 2D "world", all points are equivalent (ignoring gravity etc). But, when you can observe the sphere in 3D space it has a very obvious centre, and points have clear antipodes. You (I mean StandardsGuy, not you Fyz) need to extend this analogy to the 4D hypersphere.

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#121
In reply to #120

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 10:09 AM

"All points are equivalent"...

I think it's worth being clear that the relativities would be observable from within the surface, assuming that there was sufficient time to make observations.

For example, in the absence of gravity and friction, momentum would define great circles (the closest approach to straight lines). The point half way around a great circle is the ant-IPod.

Or in principle the ant could light a match - another ant at the antipode would see the light appear simultaneously from all directions.

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#125
In reply to #120

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 10:57 AM

Thanks Randall, you and the guest have helped. The problem is that we are representing one less dimension than reality. So I take it that the heart "phenomena" would apply in any direction that I looked in space. Is that correct?

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#128
In reply to #125

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 11:14 AM

You wouldn't actually see anything of the sort. What you see is a sphere surrounding you at a vast distance. If you take a plane section that passes through the centre you would see only part-way around the globe, so the picture is not yet helpful.

You can get a good idea of what is happening with a standard 3D sphere globe, and pretend you are at the north pole. The limits to your vision would be a circle at 64O latitude. If you imagine a plane that includes the earth's axis of rotation it will intersect this circle at two points. Shrink the globe (keeping the same centre) to represent the size when the light passed specific points and you have something resembling the model.

(That "guest")

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#124
In reply to #113

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 10:55 AM

Sorry for the confusion StandardsGuy. Yes, the B doesn't represent a point on the drawing. It represents a circle. In the real world it is a sphere and represents the cosmic horizon.

Read my posts #122 (and #121) for further explanations.

I had misinterpreted myself some issues but Jorrie gave me some good directions to clarify the whole issue (and modify the drawing).

Thanks for your comments and markings.

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#132
In reply to #124

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 12:00 PM

Sorry... I meant my posts #122 and #118...

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#98
In reply to #92

Re: The Perfect CR4 Cosmic Balloon

06/10/2009 11:08 PM

Hi George, you said: "Obviously, there will be a moment that the horizon will include the whole universe, meaning that we (at point A) will be able to observe the point B."

But according to your drawing, we are observing point B now! B represents an event that happened a very long time ago. We see it once and that's that - the galaxy that event B happened in may still exist, but it is not where you plotted it any more. It is now also on the circle. It there is no expansion for a very long time (static cosmos), photons simply move on the circumference of the circle and can go around the universe, in principle many times (as Guest has said in #95).

"How close to the BB the point B will be??? (Which time period of the universe is related to the point B???... What will we observe then, around us...???)"

In the expanding scenario, point B can be viewed as the time of the CMB release (last scattering), which happened around 400 million years after the BB. In that case, B has a redshifft of about 1100 and is situated at a point about 1/1100th of the radius of the 'outer circle', where we are.

-J

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#99
In reply to #98

Re: The Perfect CR4 Cosmic Balloon

06/11/2009 12:21 AM

Jorrie,

We can't see past the "last scattering". According to this link, its surface is a sphere. Are they wrong? I think you made a slight error in this post. Look at this link and see if you can tell what it is. I assume point B is where our point in space was a long time ago, and I don't understand the heart shape.

-S

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#100
In reply to #99

Re: The Perfect CR4 Cosmic Balloon

06/11/2009 12:56 AM

Hi S, I see nothing in the links that conflicts with what I said in #98.

I suspect that you misunderstand George's 'cosmic heart', which is understandable - it can be confusing.[1] Point B is not our position a long time ago, but some object that emitted a photon (an event) a long time ago. At that time, our position was directly inwards from our present position, at the same radius as B. The 'heart' is bit of a diversion from the model, so I won't go deeper into it here.

-J

[1] I also used the 'cosmic heart' when I started to write the cosmology section of Relativity 4 Engineers, but discarded it in favor of the 'cosmic teardrop', for its ease of understanding (section 13.2, pp. 175 - quite illuminating, I hope!). The 'teardrop' is just the 'heart' drawn in Cartesian instead of circular (or spherical) coordinates. It is also commonly used in the literature. I'll see if I can find a reference for you.

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#112
In reply to #100

Re: The Perfect CR4 Cosmic Balloon

06/11/2009 8:31 PM

Hi Jorrie,

"I see nothing in the links that conflicts with what I said in #98."

Link 2 in #99 says that the last scattering was at 300,000 years ABB, you said 400 million in #98. Were you talking of something else?

Two of your links at the top of this section are broke (at least to me). Check them out.

-S

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#114
In reply to #112

Re: The Perfect CR4 Cosmic Balloon

06/11/2009 10:30 PM

Hi S, sorry, should be 0.4 million, not 400 million, because it is the time of last scattering at z=1088. I read 0.3887 from the cosmo-calculator and misquoted it as 0.4 Gy. Well spotted!

Could you please point me to the broken links? I can't figure which...

-J

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#123
In reply to #114

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 10:42 AM

"Could you please point me to the broken links?"

Sorry, I thought the second one was coming back to the same place, but it wasn't, and the index was in a different form than I expected. They are OK.

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#101
In reply to #98

Re: The Perfect CR4 Cosmic Balloon

06/11/2009 9:11 AM

I redesigned my initial drawing in order to be correct:

For StandardsGuy: The * shows the Big Bang (at t0). The concentric spheres shows "snapshots" of the evolution (expansion) of the universe in several time periods (t1, t2, e.t.c.). (In fact these are hyperspheres as the shape of the universe is hyperspherical.) Now we are at point A at time t5. We observe the point A1 which is located on the edge of the cosmic horizon (for now we can't see further than this). Because of the travel of light (at c) we see this point not as it is now (at t5 i.e. point A2) but as it was long time ago (somewhere between t4 and t5 as you see). Hence, we observe a smaller universe -or (if you like) a greater curvature for our universe- (see pink shape) than the actual universe is (see blue sphere). [Although we cannot see beyond the cosmic horizon, we can conclude (or imagine) the overall shape of the universe by just observing the curvature of the observable part of the universe. This shape seems to be a "heart shape" (or I should say hyperheart).] This is what I trying to show in my initial post.

For Jorrie (about my philosophical question): I'm not sure that I understood your thought. I didn't mean an "always steady" universe. [In this case it's obvious that we could observe "ourselves" all around us (related to previous time periods though... again and again...).] I meant this: imagine that the expansion of the universe stops and from now on its size remains steady.

Another post follows with this scenario.

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#106
In reply to #101

Re: The Perfect CR4 Cosmic Balloon

06/11/2009 11:08 AM

Hi George.

Your 'cosmic heart' is still somewhat misleading. You call A1 the 'cosmic horizon', which implies it is where the CMB photons originated. That point must be on a hypersphere with radius of just 1/1100th of your outer (now) sphere (if you are using radius for scale factor a). It is even smaller if you use radius for time (some 1/34000). To cover the observable universe, your 'heart' must effectively start at/inside the star in the center of your figure.

-J

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#118
In reply to #106

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 5:28 AM

Jorrie, you said: "...You call A1 the 'cosmic horizon', which implies it is where the CMB photons originated. That point must be on a hypersphere with radius of just 1/1100th of your outer (now) sphere..." Yes, you are right that the CMB photons must originated from the specific hypersphere (with R=1/1100.R(@t5)) => t=400.000 years after the BB). The observable universe has a radius (Hubble radious) of 9.109 to 18.109 light years (depending on the cosmic model). This means that the "depth in time" that we can observe is similar to the age of the Universe (≈15.109 years). Hence we, indeed, observe this time: 400.000 years after the BB. Actually I think that we could be able to observe even earlier than this time, if the light was able, also, to be "released" from the matter earlier. But we cannot be able to observe earlier than the "end of inflation" time. That's why we can observe so back in time but not so far in space: Because of the inflation. (We must not forget that we observe just a small part of the actual space of the universe.)

You said: "... To cover the observable universe, your 'heart' must effectively start at/inside the star in the center of your figure..." Yes, I think it's more appropriate to to locate the "edge" of the heart in the center.

I'll further redesign my drawing.

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#122
In reply to #118

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 10:33 AM

This is an improvement of the drawing:

I hope this version is much more satisfactory. The "edge" of the heart is located in the centre * (i.e. the BB). B is no longer a "point" (as it was in the previous version). B is a a "circle" (actually -in the real world- is a sphere) and represents the cosmic horizon. This is related with time t1=400.000 years after the BB. From B comes the most ancient light that we can observe which is the cosmic mikrowave background (CMB). We can't see deeper in time (due to the fact that in t1 the light succeeded to be "released" from the matter).

Note that between t0 and t1 (i.e. the time duration of 400.000 years) the inflation is included. During this very short period the universe became huge. So, the size of the (hyper)sphere which is related to t1 is really huge: the volume between t0 and t1 is extremely larger than the volume between subsequent (hyper)spheres (i.e. between t1 and t2 e.t.c.). I must notice this, because I cannot make a drawing that represents the actual sizes of these several "snapshots".

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#126
In reply to #122

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 11:03 AM

"B is a sphere".

I assume that this diagram is intended to be a plane cut through the 4D hypersphere that passes through its centre "t0". If so the sphere would intersect the plane at two points B1, B2 (4D geometry results are not always obvious). According to Jorrie's numbers (based on current papers), the points B1 B2 would be on the right-hand side of the diagram, with angles B#_t0_A of at most 26O.

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#133
In reply to #122

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 12:36 PM

Hi George

I'm afraid that your hypersphere drawings are not very helpful in trying to understand/design the 'cosmic balloon', so I do not want to be totally diverged into this direction.

Your latest drawing does not repesent any known cosmic model, as far as I can see. I don't know how you draw it. Is it just a 'gut feel' artwork, or do you do calculations in the process? Recall what I wrote in #115 on integrating the Friedman equations backwards in time.

You wrote: "I must notice this, because I cannot make a drawing that represents the actual sizes of these several "snapshots"."

It will become a lot more meaningful if you would use some scale on your drawings, e.g., radius to represent either scale factor a or time. I used scale factor before and marked the actual times on the circles, like the year rings in a tree, e.g., one circle (surface) for every one Gy. But then, it will only be useful if you actually do the integration and plot 'real values' for some assumed cosmological parameters. Otherwise it is a bit too speculative and perhaps even misleading.

I do not want to discourage you from working in this direction - after all, this whole thread is a bit speculative and the 'development of ideas in full public view'. Perhaps you should start a new thread on the 'cosmic heart'.

-J

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#163
In reply to #133

Re: The Perfect CR4 Cosmic Balloon

06/15/2009 8:41 AM

Jorrie, you said: "... I don't know how you draw it. Is it just a 'gut feel' artwork, or do you do calculations in the process?..." Yes, it is a 'gut feel' artwork and nothing more. It's rather an intuitive approach of the concept, without any calculations. And my intention was a "general picture" rather than the accuration.

"...Recall what I wrote in #115..." I'm afraid that I did it... that's why I "distended" the previous drawing and I made it more "round"... But I think that I misunderstood your "instructions" (about 26° e.t.c.). After a second reading I think that you meant sth like a "teardrop" (a rather narrow one) located on the right side of the BB (with the BB at its edge). (I.e. nothing like a "heart shape".) I also took into account your comments at your post #106, so I "extended" the cosmic horizon down to the time of "400.000 years after the BB" and I placed the "edge" of the "heart" at exactly the BB (so the cosmic horizon became a "circle" -hence a "sphere" in the real world- that was very logical).

"...Otherwise it is a bit too speculative and perhaps even misleading..." When I first presented this "heart shape" I thought that (somehow) you agreed with this shape, even though you had some objections. So I had the sense that I was -more or less- in a good direction. If I was -initially- so totally wrong (i.e. if such a shape doesn't have the slightest relationship with the reality) and if I was so misleaded you should have (kindly) told me, so I wouldn't have spent my time trying to "modify" the initial drawing. (Don't forget that you are a kind of "teacher" for me -and for many of us- concerning the Cosmology issues.)

"...I do not want to discourage you from working in this direction... Perhaps you should start a new thread on the 'cosmic heart'..." If my visualization on this concept was so initially wrong, then it's not worth while... I'll leave it aside...

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#186
In reply to #133

Re: The Perfect CR4 Cosmic Balloon

06/17/2009 11:36 PM

Jorrie,

I give you a GA on this because you were kind and to the point, even though I was interested in the distraction.

-S

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#165
In reply to #122

Re: The Perfect CR4 Cosmic Balloon

06/15/2009 10:18 AM

Hi George

Here is a very rough sketch of my interpretation of what Jorrie said about the paths and positions in a plane section that passes through the bang and the present time. Some things to note:
a) Only two rays lie within the plane. Both pass through a single point "here-and-now" on the outer circle that I have used to represent the universe at present
b) The paths represent looking in precisely opposite directions. The angle to the circles may be regarded as representing the change in position of the universe's "surface" as we go back in time.
c) The transparency circle is very much too large (or I could not draw it)
d) I have extended the paths back to the bang. You can see that the only time the paths are at the same angle to the origin is now.
e) Note that some say that it may be possible to observe gravitational waves from this time - though personally I don't believe humanity will ever be able to observe the effects of such near-planar gravitational waves.
f) The angles of the paths are substantially larger than they should be for Jorrie's 26O. Otherwise it would be hard to view - again, because I had trouble drawing anything more sensible.
g) I think what you call the cosmic horizon is the limit of the part of the universe that we can in principle observe. Depending on how we define "observe", that will be a spherical surface that apparently surrounds us, whose time corresponds either to where light would have originated when the universe became transparent, or some earlier gravitational limit. Either way, the intersection of the chosen plane with that spherical surface will be just the two points at which the light rays intersect the relevant circular section that lies in the chosen plane (in other words, exactly what you can see marked in the diagram - no more and no less).

Comments?

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#166
In reply to #165

Re: The Perfect CR4 Cosmic Balloon

06/15/2009 11:23 AM

Hi Fyz

I agree with what you said. The rough heart (or rather teardrop) is also very close to what a proper plot of the rays would have yielded for present known parameter values.

-J

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#167
In reply to #166

Re: The Perfect CR4 Cosmic Balloon

06/15/2009 12:03 PM

Thanks. Unfortunately this is not advancing your alternative descriptions.

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#171
In reply to #165

Re: The Perfect CR4 Cosmic Balloon

06/16/2009 6:37 AM

Yes, Fyz. I think that it seems o.k.

Nevertheless, it's a teardrop shape not a heart shape. (I still can't understand why Jorrie insist on describing this as "heart".)

Btw I have the sense that my shape on #101 (or #122) could be -somehow- "equivalent" to this teardrop if you work on a logarithmic scale: If the succesive spheres from t0 to tnow are in a logarithmic scale then it seems that the teardrop becomes a heart.

What do you think???

(If so then it could be a good way of representation, as it is a quite "full and round" heart shape and not a "small and narrow" teardrop.)... ...

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#172
In reply to #171

Re: The Perfect CR4 Cosmic Balloon

06/16/2009 11:13 AM

However you scale it, the light will never enter the left-hand side of the diagram.

So far as I can judge, if you use a logarithmic scale for the radius the apparent angle at which the ray crosses the hypersphere will be very steep for the early universe, and gradually become flatter.

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#173
In reply to #172

Re: The Perfect CR4 Cosmic Balloon

06/16/2009 11:37 AM

Yeah, it seems that you are right. There is no way to enter the left side of the diagram.

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#142
In reply to #118

Re: The Perfect CR4 Cosmic Balloon

06/12/2009 6:04 PM

"Yes, I think it's more appropriate to to locate the "edge" of the heart in the center."

But that's not what you did. Your #101 looks better than the last one except the cosmic horizon. I think the whole pink heart is the horizon.

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#162
In reply to #142

Re: The Perfect CR4 Cosmic Balloon

06/15/2009 4:39 AM

The surface of any circle centred on the origin is the 2D slice of the whole universe at that time.
The horizon is just the surface of a sphere at that time, and has exactly two points in this plane.

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#164
In reply to #142

Re: The Perfect CR4 Cosmic Balloon

06/15/2009 9:11 AM

"But that's not what you did."... Yes, it is... Just notice that in #101 the "edge" of the heart is located on the point B, while in #122 it is located on * (i.e. exactly on BB)...

"Your #101 looks better than the last one except the cosmic horizon. I think the whole pink heart is the horizon." Yes, in fact the cosmic horizon seems to be almost the whole pink heart (i.e. down to the circle B). Anyway, I'm not sure about any of these. These are just some (probably unsuccesfull) attempts to approach this concept. It's just a "visualization" which may be totally wrong.

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#174
In reply to #142

Re: The Perfect CR4 Cosmic Balloon

06/17/2009 4:32 AM

Maybe my reply #164 is not accurate on this:

You said: "I think the whole pink heart is the horizon." It's not the pink heart itself. The cosmic horizon (by its definition) is the virtual sphere that sorrounds all the observable universe (or the limit of the observable universe). We cannot observe further than this virtual sphere. So, in our 2D model the cosmic horizon must be represented by a virtual circle. This circle is the intersection of the teardrop (better look at the Physicist's drawing at #165, instead of mine at #122 which is wrong) and the sphere at t=400.000 years after the BB. Hence, this circular intersection is actually the cosmic horizon (which is a sphere in our 3D real world).

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#175
In reply to #174

Re: The Perfect CR4 Cosmic Balloon

06/17/2009 6:34 AM

At the risk of becoming even more boring than normal, and possibly even of being seen to shout - I repeat:
The part of that intersection with the plane that lies on the horizon (as seen from a single location is just two points. (The circle is the intersection with the entire universe at the time of first transparency).

Remember this is a 2D section of a 4D subsystem. To generate the cosmic horizon you need to rotate-sweep the plane in the two further orthogonal dimensions - and both rotations must be around the line that joins the origin (=bang) and the observer's location. Living in a three-dimensional world you can only properly visualise one of these rotations-sweeps; you can see that this creates a circle of points that lie on the cosmic horizon - the circle is orthogonal to the drawn plane and passes through the two points on the rays at the time of first transparency. The second rotation involves the direction of expansion, and creates the spherical surface that corresponds to the limits of observation (as they were at the time of transparency).

Hope that is clearer.

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#176
In reply to #175

Re: The Perfect CR4 Cosmic Balloon

06/17/2009 11:18 AM

"At the risk of becoming even more boring than normal..." Maybe I'm even more silly than normal... ...

As it is shown in the picture the intersection of the teardrop with the sphere at the time of the transparency is -obviously- a circle (the green one). These first, ancient photons (i.e. CMB photons) that were emitted almost 14 billion years ago (shown as arrows) are originated from the entire circumference of this circle (not from just two points as you said). These photons have already travelled around the whole surface of the teardrop (which represents the whole observable universe) and we observe them right now (point A).

This circle is a part of our "2D world model" (i.e. the surface of a ballon). As our universe is -actually- 3D we have to "add" one more dimension to this circle. So the circle becomes a sphere. All these CMB photons come to us from the entire surface of this sphere. This must be the cosmic horizon (due to the fact that we cannot see deeper in time).

If I'm wrong in sth please correct me... I still can't understand what you said about the "second rotation involves the direction of expansion"...

Don't hesitate to give some more explanations to me.

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#177
In reply to #176

Re: The Perfect CR4 Cosmic Balloon

06/17/2009 12:11 PM

My apologies - I misread what you wrote - and then compounded it with a comment that was so badly expressed as only to be capable of incorrect interpretation. Of course the situation is that the axis about which we are rotating runs along the local line of expansion.

The 3D solid comprising all possible teardrops does indeed pass through the whole of the observable universe, with closer points being seen at earlier times.
We can readily visualise a reduced version (as if which we live in a two-dimensional universe on the surface of an expanding sphere). We just need to rotate the rays about the axis joining us to the origin. I was trying to describe the problem of visualising the rotations about the axis. It would have been better if I had simply written that rotations about a line in 4D are closely analogous to rotations about a point in 3D - but re-reading what you actually wrote (rather than what I thought you wrote) it appears that you already took that as read.

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#196
In reply to #177

Re: The Perfect CR4 Cosmic Balloon

06/18/2009 12:11 PM

Hmm... Let's suppose -again- that the expansion of our (cosmic) balloon is stopped (meaning that the size of the balloon is steady for now on)... I think that (as the time passes) the 3d-teardrop will start to extend (i.e. it will become more and more "round"). If we just wait (e.g. hundreds of billions of years) it seems that, eventually, this teardrop will pass to the left section of the ballon... ... (or am I wrong???)... Does this mean that the shape will become sth like the 3d-heartshape in my #101 (or #122)???...

Just have a thought on this and tell me your opinion...

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#200
In reply to #196

Re: The Perfect CR4 Cosmic Balloon

06/18/2009 1:16 PM

I wouldn't like to comment on the extreme case of a stopped expansion - I'm not certain I could handle the paradoxes. But the angle of the teardrop is clearly growing. But we don't need the expansion to stop - we don't even need a constant rate of expansion - merely a rate that increases at order 0 with time*.

As an example, if the expansion rate is constant and finite we could eventually see arbitrarily far around (regardless of the actual rate, however large); the reason is that the angle (in radians) is
∫(c/R)dt = ∫(c/(dR/dt)/t)dt = [c/(dR/dt).ln(t)],
which is unbounded. So at a point in time an ideal visible universe would theoretically correspond to the total universe up to transparency**. A little later we could see some regions twice (at two different times), and I few Geop-aeons later the whole universe could become visible as many times as you might wish.

*The best-known example of a function that is unbounded but increases at order 0 would be ln(t).
**Not that this is actually an instant in time - and even if it were the variation in the microwave background tells us that it occurred in different times in different directions. On top of that, we don't believe the universe to be an ideal hypersphere. So I don't think you would actually see the antipodes appearing as if it was a sphere

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#204
In reply to #200

Re: The Perfect CR4 Cosmic Balloon

06/18/2009 9:53 PM

Hi Phys,

"we don't believe the universe to be an ideal hypersphere."

Could you expound on that?

-S

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#210
In reply to #204

Re: The Perfect CR4 Cosmic Balloon

06/19/2009 7:31 AM

As I misunderstand it, there are at the very least local distortions in the relationship between time and the 4-th dimensional radius. If correct, that would mean that light from a "uniform" time in the past will have taken different times to reach us - due to the different distances it has to travel.

Hopefully Jorrie will correct me if that is wrong.

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#213
In reply to #210

Re: The Perfect CR4 Cosmic Balloon

06/19/2009 9:02 AM

Hi Fyz, yes, I think your are essentially right.

The 'balloon' had and still has 'indentations' where the local density is higher than the average.There are both the different distances that the photons had to travel and the Shapiro time delays that differ for different paths (which is the larger effect). Photons going in and coming out of spatial indentations (gravitational wells) take longer then what they would have taken on a smooth hypersphere, although the en route Shapiro time delays tend to even out.

-J

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#217
In reply to #200

Re: The Perfect CR4 Cosmic Balloon

06/19/2009 10:38 AM

That's very interesting.

I had the sense that only in the case where the expansion was retarded and -eventually- stopped the heart could "surround" the whole hypersphere. It's amazing that this happens, also, in other cases (where the function of the angle of the teardrop -in relation with time- is unbounded). I suppose that in such cases a heart shape like in #101 appears (roughly speaking)... Am I right???...

[Btw, in the "∫(c/R)dt = ....." what represents this angle???]

About the ** I totally agree with you.

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#223
In reply to #217

Re: The Perfect CR4 Cosmic Balloon

06/19/2009 12:22 PM

The angle that I'm calculating is the angle subtended at the bang by the ends of a light path at different times. For the case in point the light "ends" at you now, and started from the time of transparency at your limits of observation.

Yes, eventually you will get a heart - but then you will be able to extend the lines backwards through the earliest point of the heart.

It's actually easiest to draw if you start one of the trajectories from the observable limit. If we take the expansion rate is constant you can then expand the curve as follows:
. If you go forwards to twice the radius, the light will go around by the same angular distance that it travelled since the universe was half the radius. Double the time and it will go the same angle again...
. When you've gone to the time in the future that is of interest, reflect the spiralabout the line joining its outer end and the bang.

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#246
In reply to #223

Re: The Perfect CR4 Cosmic Balloon

06/23/2009 11:54 AM

Thanks for the reply. I understand that the angle (that you are refering to) is the angle between the light path -at the time that light reaches us, i.e. "now"- and the line (radius) that "connects" us and the BB. As the time passes this angle is becoming wider and wider for us (Am I right???).

I asked you about your equation in #200: what is the "angle" in this equation???

About you 3rd paragraph, I think that you mean the following: Every time that the light "goes" from one "time sphere" to the next one of a double radius it "travels around" always at the same angle. If so, I think that (eventually) you get a kind of heartshape.

Please confirm.

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#249
In reply to #246

Re: The Perfect CR4 Cosmic Balloon

06/23/2009 4:42 PM

The topic I am addressing is the proportion of the universal circumference that light has travelled since transparency. So the angle I am referring to is the angle between the radii from BB to now. Every time you double the radius you add the same amount to this angle. That is also the angle in the equation in #200.
(Perhaps I was naive in thinking that the fact that the angle was unbounded left no room for doubt?)

In the end you get a spiral. The angle between the line connecting light-at-start-time to (BB) and the line connecting light-at-end-time to BB increases as the log of (radius-to-start/radius-to-end). Of course of the expansion was uniform since BB that would be the same as ratio of the times - but that is not the case as we are only assuming constant expansion after some time T. If you reflect these lines about "end" you could regard the shape produced as a series of hearts with a pair of tails from the smallest heart still going inwards - but I don't see any particular significance in this shape.

I use slightly different terminology in my answer to Jorrie (#250), but the intention is the same (so if this is still ambiguous please check there also; but as I am never confident about my clarity, please don't hesitate to ask further).

Regards

Fyz

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#254
In reply to #249

Re: The Perfect CR4 Cosmic Balloon

06/25/2009 7:58 AM

Hi Fyz. I think that you mean sth like this (???):

This is in the case of constant expansion... (???) ... (light path → red line)...

Please confirm.

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#255
In reply to #254

Re: The Perfect CR4 Cosmic Balloon

06/25/2009 10:23 AM

Yes, that's it. Of course it also goes around an infinite number of times if you work backwards to BB on the same basis. Some people think that this and some additional modifiers could be equivalent to a steady-state theory, but so far as I know no-one has yet been able to fit such a model to the background radiation*.

*That's not entirely true - some "theories" assume a general creation and/or decay that generates radiation with exactly the correct spectrum, but I tend to reject anything where the theory could be fitted to an arbitrary situation. This of course is another reason I've not got too deeply involved here - I feel very uncomfortable that the model appears to requre "Einstein's greatest mistake" (it's not the so much the extra constant as the apparently arbitrary value that's worrying).

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#256
In reply to #254

Re: The Perfect CR4 Cosmic Balloon

06/25/2009 11:34 AM

Hi George, if Fyz will pardon me jumping in - I do not quite follow the meaning of the diagram. What time intervals do the intermediate circles represent? Surely not constant time intervals?

For constant expansion rate, look at my new thread. The constant expansion rate cosmos would have a 'heart' that is virtually indistinguishable from my main figure, which was drawn for a standard LCDM cosmos. The reason for this is that the cosmological constant made the expansion rate almost constant for the majority of the life of the cosmos.

I'm trying to make my spreadsheet more user-friendly and I will then post a link to it for those interested in playing around.

-J

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#257
In reply to #256

Re: The Perfect CR4 Cosmic Balloon

06/25/2009 4:41 PM

Jumping back in:

I think that George has drawn exactly what I described - going outwards each time interval is precisely twice the previous one - hence the doubling of distance between successive hyperspheres. [Simlarly, if you work inwards (backwards in time) each time interval is half the previous one. In each case the angle between successive radii is unchanged].

The main difference from the outer part of your diagram is that George has used a larger angle for each time interval than we have in the present stage of the universe. I believe his objective was clarity of presentation rather than direct correspondence with our present universe.

By the way, the diagram in the new thread looks like constant expansion rate after about 1GY, but expansion rate that is higher (reduces with time) before that. Is this intentional?

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#258
In reply to #257

Re: The Perfect CR4 Cosmic Balloon

06/25/2009 9:03 PM

Hi Fyz

Are you saying that he used approximately (say) 2, 6, 14, 30, ... Gy age circles, with every time step double the previous one?

For a constant expansion rate (dR/dt), this does not give constant angular differences, as far as I can establish. I get comoving angles 0.361, 0.215, 0.102, 0 radians for those age values, which do not represent a constant increment. What am I missing?

You wrote: "By the way, the diagram in the new thread looks like constant expansion rate after about 1GY, but expansion rate that is higher (reduces with time) before that. Is this intentional?"

Yes, this is what the development of the present 'best-fit' cosmos looks like. It represents this LCDM curve (used in the final version of the opening post of this thread) - dR/dt is only near constant in the 'middle ages'.

-J

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#259
In reply to #258

Re: The Perfect CR4 Cosmic Balloon

06/26/2009 6:15 AM

The sequence you present wouldn't result in constant angular steps. You are doubling the differential*, whereas I wrote "every time you double the radius you add the same amount to this angle". (The factor of two is of course only an example - any geometric progression of the radius would do the same).

* Though even this would tend towards constant steps as R increases. Taking your case the differentials are:
0.146, 0.113, 0.102, 0.0968, 0.0945, 0.935, 0.0929, ... (→ 0.0924)

Regards

Fyz

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#260
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Re: The Perfect CR4 Cosmic Balloon

06/26/2009 7:04 AM

OK, I see. I got side-tracked when you wrote "hence the doubling of distance between successive hyperspheres". I took that for double increments, which was not what you originally wrote - there it was clear anyway.

-J

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#251
In reply to #246

Re: The Perfect CR4 Cosmic Balloon

06/23/2009 7:16 PM

Hi G.K..

see this thread for more discussion of the heart shape.

-S

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#253
In reply to #251

Re: The Perfect CR4 Cosmic Balloon

06/24/2009 5:00 AM

Thanks StandardsGuy. I've just subsribed in the Physics Forums. There are many interesting threads there. I wish I can find some time to participate too... ...

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#247
In reply to #223

Re: The Perfect CR4 Cosmic Balloon

06/23/2009 1:03 PM

Hi Fyz, I also do not quite follow what you meant by:

"If we take the expansion rate is constant you can then expand the curve as follows:

. If you go forwards to twice the radius, the light will go around by the same angular distance that it travelled since the universe was half the radius. Double the time and it will go the same angle again..."

The angle that light travels around the circumference of the hypersphere in a small time Δt is Ψ = cΔt/R. Even for constant expansion rate, the angular rate should decrease, not so? Or did you mean by "angular distance" that the proper distance that the light travel in the same time interval is constant?

It is important, since cosmologists tend to call the angle Ψ a comoving distance in some books and papers. I prefer it multiplied by R to give the comoving distance.

-J

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#250
In reply to #247

Re: The Perfect CR4 Cosmic Balloon

06/23/2009 4:49 PM

Yes, I am writing about the co-moving distance. Of course for this condition the angular rate of increase is proportional to 1/R - but the time period I have defined for each increase is doubling, which is why the increment is the same. (That's just saying it's a logarithmic function of time - but the doubling method is a very straightforward way of showing that it is unbounded)

I use slightly different terminology in my answer to GK, but the intention is the same (so if this is still ambiguous please check there also; but as I am never confident about clarity, please don't hesitate to rattle again).

Regards

Fyz

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#179
In reply to #176

Re: The Perfect CR4 Cosmic Balloon

06/17/2009 2:59 PM

Hi George, at the risk of spreading confusion, I must point out that what you wrote:

"These first, ancient photons (i.e. CMB photons) that were emitted almost 14 billion years ago (shown as arrows) are originated from the entire circumference of this circle (not from just two points as you said)"

is not strictly correct. If we present the hyperspherical model on a flat piece of paper, it is better to stick to the two dimensions: one space and one hyperspace. In such a case, Fyz was correct in that the CMB photons originated from just two points on the teardrop. Yes, when all three spatial dimension are restored, those photons originated from a sphere, but it is perceptually confusing to go half-way with two spatial and one hyperspace dimension. That sphere actually sits around us, with us at the center, but that's impossible to represent in this form.

-J

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#195
In reply to #179

Re: The Perfect CR4 Cosmic Balloon

06/18/2009 11:35 AM

I thought that it was not so difficult to visualise a ballon or a 3d-teardrop on a flat piece of paper. This is what we used to do in stereometry in the high school, in order to visualise 3d geometrical objects as if they were placed in the real 3d space.

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#197
In reply to #195

Re: The Perfect CR4 Cosmic Balloon

06/18/2009 12:11 PM

George,

All I was saying is that your attempt to create a quasi-3D drawing of what is actually a 4D situation is more confusing than what it helps. As an example, your green circle has expanded by now so that we are sitting in its center.

The moment you try and put in all the required information to understand such a drawing, it becomes so cluttered that it is next to useless. Stick to 2D and it remains useful.

-J

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