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Special thread for answers to Mathbrain's questions

  1. Nov 28, 2011 #1

    marcus

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    A newcomer named Mathbrain posed some questions/comments that need response, but posted in the balloon model sticky thread where discussion would likely be off-topic.
    I don't want to overload the balloon model thread with a possibly lengthy discussion so I'm starting a special thread for Mathbrain here.

    Here are his/her first two posts. Hopefully others will help respond.


     
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  3. Nov 28, 2011 #2

    marcus

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    Mathbrain, I don't understand where you get this, or what you mean by it:

    Furthermore the baloon analogy is incapable of showing the curvature of space, as an extra dimension is require to express curvature in a geometric manner.

    I was saying that you need an extra dimension to represent curvature.​

    But the spaces that geometers study do not have to be embedded in higher dim. in order to be represented mathematically! In particular, in order to represent curvature mathematically you do not need an extra spatial dimension. This was taken care of in the first half of the 19th century by people like Gauss and Riemann around 1820-1850. You can take a simple 2D plane and make it curved by declaring a different distance function on it.
     
  4. Nov 28, 2011 #3

    marcus

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    Mathbrain, I don't understand your question #1. It seems to me that the answer would be "it depends".
    You would need to specify some spatial geometry. Typically moving bodies that start off parallel do not continue moving parallel.

    A spatial geometry is typically specified by a metric---a distance function. Calculating the separation at some time in the future, for two bodies moving on geodesics, would be done using the metric.

    Your question is probably better to ask in the RELATIVTY FORUM. Or general physics. Basically it is the kind of thing you learn about in a college class in "Differential Geometry". Sophomore level would be OK.

    Here is your question #2


    2 - Are all instances of CMB from the Big Bang? If so, how is it that we can constantly sense CMB? They would need to be moving at different speeds, or bouncing off something.

    By "instances" I guess you mean photons? The universe (as far as we know) is full of the ancient light that originated about 380,000 years after the start of expansion.

    I don't see why anything would "need to be moving at different speeds".

    Look at the little movie of the balloon model. The photons are the wiggly things. They all move at the same constant speed. To get the movie, google "wright balloon model".

    Since the U is full of CMB light, a galaxy sitting still in some location will constantly be bathed in that light. There is always some more light coming towards you from every direction in the sky.

    BTW as a side note: the CMB light we are getting now from all directions started towards us from matter that was, at that time, about 41 million lightyears away from our matter (if you could have stopped the expansion process right then and measured :biggrin:).

    That is less than a thousandth of the distance that that same matter is now---the matter that emitted the CMB light that we are now seeing is now about 45 billion ly away (over a thousand times the 41 million ly distance I mentioned.)
     
    Last edited: Nov 28, 2011
  5. Nov 29, 2011 #4
    I get my own thread. =)

    The point of question #1 is to understand whether the universe actually expands, or if the galaxies are moving further apart. I did not mean to bring up the possibilities in abstract geometrical space, I meant in the physical world.

    To rephrase question 1: Does space expand at a predictable rate, if so is it related to the Hubble constant? AND How do we know how far Galaxies are from the Milky Way if space is expanding? I understand the concept of the Redshift, and how we are able to calculate how long light has traveled, but what is its relationship to the current position of an observed Galaxy. Do we know where an observed Galaxy is now? If so how can we know this with an expanded space?

    WRT #2: If all CMB photons are from the Big Bang, then how are they coming at Earth from all directions, for all moments in time? Do they reflect off other surfaces, or are bent by the curvature of space and the existance of dark matter?

    WRT Balloon Analogy: Marcus said "do not have to be embedded in higher dim. in order to be represented mathematically" I understand and agree with you, but the Balloon analogy is incapable of representing that curvature, because it has no mathematics. It is a 3D representation of 4D phenomenon. Furthermore the balloon analogy has no way of modeling the curvature of space as any mathematical representations must be geometrically represented in the balloon analogy.
     
  6. Nov 29, 2011 #5
    Marcus,

    Thanks for your help and you work on "An Effort..." you really should get paid for repeating yourself a million times!
     
  7. Nov 29, 2011 #6

    marcus

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    Your question #2 offered the idea that the ancient light is coming from all directions because it is BENT AROUND. Well that is one possible way to think of it. The finite balloon model way.
    The light follows the curve of the 3D hypersphere. In the infinite picture it is even simpler. Space is infinite and full of light. How could the light NOT be constantly coming from all directions?

    The main thing is to realize that all space, whether finite or infinite, is roughly uniformly filled with the same light and the same largescale average density of material. This is basic to standard mainstream cosmology.

    The whole discussion here rests on the 1915 Law of Gravity. You should look up "Einstein Field Equation" on Wikipedia to see how simple it is. It is actually a Law of Geometry, or how geometry relates to matter and how geometry evolves. Geometry is specified on the LHS and matter on the RHS of the equation. As matter moves around, so geometry has to change.

    To do cosmology you make a simplifying assumption that at very large scale stuff is distributed roughly uniformly on average, and derive a simplified version of the EFE.
    This is called the Friedmann Equation(s) there are actually two.

    They describe how the Hubble parameter changes with time. Also the Friedmann equations describe the evolution of the distance scale factor. Often written a(t). The Hubble parameter at any moment in time H(t) is the fractional growth rate in a(t). So they are very closely related. One equation takes care of both of them and determines how they change.

    The logical structure of cosmology is built on that foundation.

    1. The EFE is the established law of gravity which has been repeatedly tested in many ways at many scales. It keeps passing all the tests we can devise, with "flying colors" in the sense of amazing precision. It still could be wrong and fail at very high density (or some other extreme scale) but it works fine for cosmology.

    2. If you buy the EFE then you have to buy the Friedmann model of cosmology. And by adjusting 3 or so parameters it can be made to fit the observational data remarkably well.
    It too seems to fail at very high density, but works fine for ordinary cosmology. It is derived from the EFE.

    3. So the standard cosmic model "LambdaCDM" is just the Friedmann equation with some parameters plugged in like .73 for Einstein's cosmological constant (Lambda) and .04 for ordinary matter and .23 for dark matter (CDM).

    It fits millions of datapoints---masses and masses of observations: of counts of galaxies at all different redshifts, of supernovae surveys, of microwave temp at all different parts of the sky, of chemical elements seen in spectra, of how things looked when you look back in time, the physical way stars and stuff evolved, of gravitational lensing...

    Basically if you buy the EFE law of gravity, which apart from quantized versions that quantum gravity people are working on is pretty much the only law of gravity we've got, then you buy what is derived from it. And that means the LambdaCDM, which turns out to be one of mankind's great triumphs.

    Note that it is a MATHEMATICAL MODEL based on a CENTRAL EQUATION. There will always be trouble explaining in words because that involves translation from one language to another. But we can certainly try.

    In the most common LCDM versions, space is either infinite with infinite matter roughly uniform throughout (and expansion began with infinite volume) or else space is a very large hypersphere. The 3D analog of the balloon surface. Analogous distance metric defined on it.
    Not embedded. No "inside or outside of the balloon".
    This makes good mathematical sense and is routine differential geometry.
    It turns out not to matter very much for the numbers whether you think infinite 3D space or a very large 3D hypersphere where our observable patch is nearly flat, imperceptibly curved.

    So we don't worry about whether it is infinite with zero largescale curvature, or merely very large finite with almost zero but slightly positive largescale curvature. Someday with better measurements of curvature we might find out which, but for now we don't waste time worrying about that.

    =====================

    About your question #1 distances are told in various ways. It's interesting and you should learn about. Certain recognizable types of variable stars, & types of supernovae serve as standard candles (known brightness)
     
    Last edited: Nov 29, 2011
  8. Dec 1, 2011 #7

    Jorrie

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    I had the same type of question on another forum and it dawned upon me that we must perhaps not quite relate the curvature of the surface of the balloon to spatial curvature. Is it not so that in any small curvature expansion, the curvature gets larger over time (deviates away from zero), while the balloon's (positive) surface curvature gets smaller over time (larger radius, less curvature)?

    If so, it is in the 'wrong direction' and is a limitation of the balloon analogy.
     
  9. Dec 1, 2011 #8

    marcus

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    I don't think that's right Jorrie. I believe that I know what you are remembering but (if so) that you are misinterpreting it. Could you be referring to something you read about Omega drifting away from 1? Omega-1 directly concerns matter and only indirectly the geometry.

    To be specific, could you be thinking about what for example you see on page 11 of Lineweaver's 2003 article
    http://arxiv.org/pdf/astro-ph/0305179
    ==quote charley equations (33-35)==
    To summarize:
    0.95 < Ωo(z = 0) < 1.05 (33)
    0.99995 < Ω(z = 103) < 1.000005 (34)
    0.9999999999995 < Ω(z = 1011) < 1.0000000000005 (35)
    ==endquote==

    The gist is that spatial flatness is unstable because of MATTER considerations. Matter criticality is unstable. Like the pencil balanced on its point.
    In order for MATTER to be near critical now, it must have been VERY near critical back at the ("recomb...") moment she became transparent and VERYVERY near critical back even closer to go.

    But Omega - 1 does not translate directly into curvature. The relation between geometry curvature and matter criticality is subtle.

    I should emphasize that I'm not an authority, which of course you already realize but i don't want any stray passerby to misunderstand. Some of the others will I hope catch any errors. But that said, I think the balloon analogy is truthful about this. If you have a U which is spatially hypersphere, and it expands, the radius of curvature keeps on increasing, and the curvature keeps on decreasing.

    Just like the balloon case.

    But even though the curvature is decreasing as the 3-sphere expands, OMEGA will be ever so gradually creeping away from 1. It is the ratio of the actual energy density to the critical energy density and it is unstable. So Omega - 1 will be creeping away from zero and going
    0.00000000001 to 0.00001 to 0.01 (the upper end of today's confidence interval).

    Anyway, that's my immediate reaction to your comment. Let me know if what you were thinking about (noncriticality automatically getting aggravated) is fairly represented by that passage from Lineweaver. If it is not then I don't understand and we should ask one of the others.
     
    Last edited: Dec 1, 2011
  10. Dec 1, 2011 #9

    Jorrie

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    Ah, thanks Marcus, I think I see it now.
    Even in the case of a Lambda-dominated far-future, with H tending to a constant, the hypersphere curvature (1/R) will continue to decrease, while Omega will evolve away from 1, as per Lineweaver equation 14.

    [tex] (1-\Omega)H^2R^2 = constant[/tex]

    Good old balloon analogy!
     
  11. Dec 6, 2011 #10
    Question 1 has been answered. Just to make sure I understand it correctly: The special metric implied by the Cosmological Principle is used to calculate the disance form Earth to an arbitrary Galaxy. Thus when we say that Galaxy X is Y light years away, it is with regards to this metric.

    For Question 2 I'm having trouble understanding how all CMB comes from a common source, but it hits all point of space from all directions for all moments of time. This seems impossible to me given that we don't know if the universe is finite or infinite. How does a finite amount of radiation from a single origin continue to pass through all point in space?
     
  12. Dec 6, 2011 #11

    marcus

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    The common source of the CMB that is hitting US is a spherical shell surrounding us which at the time the light was emitted (in year 380,000) had a radius of about 41 million LY.

    Our matter was in the center of this sphere and matter was uniformly distributed throughout the universe both inside this small sphere and outside it.

    The photons started on their way towards us from matter which was, as I say, at that time about 41 million LY away.

    Now that matter is about 1100 times farther away----about 45 billion LY. and the photons are just arriving.

    That spherical shell (of radius 41 million THEN and 45 billion NOW) is called the "surface of last scattering". It is where the matter is that emitted the light we get today as CMB. We ALREADY GOT the light from the matter inside that shell. The shell is of the matter that emitted the CMB we are getting now.

    Every other galaxy in the universe has its own surface of last scattering. There are galaxies that condensed from matter in our SoLS which are now receiving CMB emitted from OUR matter. We, the matter of which we are composed, at that time emitted light which is now arriving as somebody else's CMB.

    I don't find this at all hard to imagine. I don't see any evidence that other people find it hard either. Maybe you are trying to imagine the wrong thing, and that is why it is hard.
    For example no one ever said that all the CMB that every galaxy is now receiving came from a single point in currently existing space! :biggrin: I hope that is not what you meant when you said "comes from a common source."
     
  13. Dec 6, 2011 #12
    Most cosmologists try to avoid saying that "galaxy X is Y light years away." The problem is that there are a lot of reasonable definitions for "distance".

    http://en.wikipedia.org/wiki/Distance_measures_(cosmology)

    All of those different definitions give you the same answer for close objects, but they can give wildly different answers for distant objects. For example "lookback distance" is quite different from "brightness distance".

    If you are talking about galaxy evolution, you are interested in "lookback distance" (i.e. the distance calculated by how long it takes light to reach you). If you are talking about cosmic expansion, you are interested in "brightness distance" (i.e. the distance calculated by the dimming of an object).

    It doesn't come from a single source.....

    Right now, the universe is filled with cold gas. As you go back in time, the universe is more compressed so the gas that makes up the universe is hotter and hotter. If you go back to the time of the CMB, then the entire universe is filled with gas that is roughly 3000 K and that's what you are seeing when you see the CMB.

    The problem is that people think of the big bang as an explosion from a point. That's not a good way of thinking about it. The big bang is the entire universe expanding, and at some point in the past, everything in the universe was 3000K, and that's what we are seeing.
     
  14. Dec 6, 2011 #13
    One poetic (and scientifically accurate) way of thinking about it is that the CMB is a "wall of fire". If you take ordinary gas at room temperature, it's clear. You can see through it. Now if you heat the gas to 3000K, you can't see through it. At 3000K, the protons and electrons separate and the gas starts absorbing light. So if you have a room full of 3000K gas, then you can't see through it. All you see is fire.

    At one point the entire universe was made of fire, and that's what you see when you see the CMB.
     
  15. Dec 7, 2011 #14
    Hi Marcus, Mathbrain, all,
    I came across a similar concept a while ago when someone (on another forum) used a degenerate triangle to describe this type of 3D to 2D transformation on Lie Groups.

    http://en.wikipedia.org/wiki/Lie_group
    http://en.wikipedia.org/wiki/Pompeiu's_theorem
    http://mathworld.wolfram.com/PompeiusTheorem.html
    The first problem was that they failed to mention P at all and described A--B--C as a 2D representation of a degenerate triangle. I called it a good example of a Pea and three cup trick because the P was never declared and would always end up where the conjuror wanted it to end up while the P could also be in contact with 1 or 2 cups, but never 3, at any one time in the 2D plane.
    The second problem was that the process described by them masked the time axis and created a deterministic result set that was reliant of having all the true universal information available at each discrete point along the (masked) time axis by discarding the false universal data (that could only be determined as false at the end of the universe).
    The third problem was that they had a weird representation of the transformation from Minkowski space to Euclidian space where they used the imaginary unit i as both integral limits and a function component at the same time! When you consider that the Wick Rotation (multiply t by i to get it) can be used to transform Minkowski space into Euclidian space you can understand what they were trying to do but an imaginary unit as integral limits i.e. area under the curve from -i to 0 and 0 to i is just plain wrong.

    Anything that avoids these three pitfalls would be really informative.
     
  16. Dec 7, 2011 #15
    We can't see the present, only the past. The further away we look, the more past. The furthest we can see is the year 300,000 after the big bang. We see that in all directions. That light has been in transit to our eyes through empty space for 13.5 billion years. We are in the center of a sphere 13.5 billion years old.

    The light is much lower frequency than it was because space has expanded. It's like a Slinky: space stretches so the Slinky stretches, and the wavelength increases. Another way to look at it is that the energy per volume goes down as the energy stays the same but the volume increases.

    You may have noticed research on gravity waves and neutrinos. With those we can "see" before 300,000 years.
     
  17. Dec 7, 2011 #16
    So as I understand it, it is possible to see light from all points because space is expanding faster than the speed of light. With expansion being that high, seeing CMB from all moments in time would be impossible. I think I understand why we can't pinpoint the center of the universe and why CMB is everywhere at all times now. Please let me know if my understanding is flawed.

    Laurie AG - Thanks for your response. I'm still an undergrad math student, but am somewhat familiar with Lie Groups. As I understand it the transforms that you're talking about are difficult at best to represent geometrically, especially since a two-dimensional complex space would require four dimensions to actually represent. The core of my argument about the baloon analogy is reliant on the idea that we need a visual representation of the curvature of space. It seems related to what you're talking about, but it's a little over my head.

    Twofish - Is it safe to say that "Galaxy X is Y light years away" is more of a pop-cosmology interpretation?

    This board has been extremely helpful! At the risk of pushing my luck:
    Qusetion 3 - It seems that CMB and the existance of certain elements only verifies the Big Bang theory if it is true. Is there any observable data that proves the Big Bang theory is correct? I'm looking for something as strong as P implies Q. Or am I thinking about this in too much of a mathematical sense? Do we have a P implies Q connection for Newton's Laws?
     
  18. Dec 8, 2011 #17

    marcus

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    No physical theory is ever proven correct. No matter how many observational tests they pass, they can always fail the next test.
    Newton F = ma was shown to be incorrect in 1905 (special rel.) so of course there is no P implies Q. Energy conservation does not hold in expanding space. Physics laws have limits on their applicability, typically.

    It is safe to say "the proper distance to Galaxy X is now Y lightyears." this is not pop.
    A lot of cosmo is done using the concept of proper distance (what you would measure by conventional means if you could stop expansion process to give your self time to measure, by radar say, or tapemeasure.)
    The standard cosmic model (Friedman equation) uses proper dist. The Hubble law v=Hd is written in terms of it. So you should know about it. Don't confuse it with "lookback time" :biggrin: My attitude differs strongly from Twofish on this. I do not consider "all distance measures equal". Some are more basic and widely used. You should also learn to think in terms of redshift, since in a lot of cases it is the basic measurement from which distance estimates are derived.

    I don't know what you mean by needing a "visual representation of curvature". The people in a curved world EXPERIENCE the curvature by noticing that the angles at the corners of large triangles do not add up to 180 degrees. That is what curvature means to them. It does not require an additional spatial dimension to be perceived. Gauss understood this around 1820. almost 200 years ago. Greatest mathematician who ever lived, probably. If you like math and are at all good at it you should respect Gauss's insight. Geometry can be experienced internally. You "see" by measuring angles, without needing an additional dimension to "see".

    With regard to your first comment. You ask to be told if your understanding is still flawed. Yes it appears still to be so. You say that we do not see CMB "from all moments in time" because of suchandsuch reason. According to standard cosmology all CMB comes from a SINGLE moment or era, the moment at which the glowing hot gas cooled enough to become effectively transparent. It cooled enough so it stopped scattering the light. So the light could take off in a straight direction unimpeded.

    So we do not have CMB "from all moments in time". We have it from year 380,000 or thereabouts. I don't see any logical connection with what you give as a reason, namely "with expansion being that high".

    I can describe a universe to you in which no distance is growing faster than speed of light, but in which distances are expanding according to Hubble law, and in which the CMB light is released in year 380,000 and is constantly coming to us from all directions, for the entire history of the universe It just gets more and more redshifted as the universe expands. So as regards CMB, superluminal distance expansion is not an essential feature.

    As far as we know distances in our universe are expanding > c, but this is not essential for there to be a CMB.

    I liked Twofish's "wall of fire" description, and general discussion of the CMB.
    ======================
    You asked what evidence for "Big Bang theory". I call the standard cosmic model either that or LCDM (Lambda cold dark matter) or "expansion cosmology". Or Friedman model, since Friedman wrote the differential equation for it back in 1922. A simplification of the Einstein GR equation. I don't personally call the standard expansion cosmo model the "Big Bang theory" because it gives people the wrong mental image (an explosion in a preexisting empty space) and leads to endless confusion. It is a pop-sci term.

    There are massive interlocking bodies of evidence supporting the standard cosmic model. People try and fail to invent alternative models that will fit the data comparably well. The field is littered with dead alternative models. And the supporting mass of data keeps growing.
    The main challenge now, as I see it, is to improve the model so it extends back to the start of expansion (and before) without developing infinities. That will require a quantum version of the 1922 Friedman equation. Work is under way on that. If you are an undergraduate now, as you say you are, and remain interested in cosmology then you will certainly hear more about that in the foreseeable future. Have fun:biggrin:
     
    Last edited: Dec 8, 2011
  19. Dec 8, 2011 #18
    I think you are on the way but in a muddle. I would not say that space is not expanding faster than the speed of light. If space is expanding by a constant factor, then in an infinite universe there will be areas so far away that we can CALCULATE that they are moving away faster then light.

    We do not see the CMB from all moments of time. We see it as 13.5 billion years ago. We have already had a chance to see CMB younger than that, and later will have a chance to see CMB that is older. But all the CMB we see now comes from the same time/distance.

    We are in the center of a sphere with the CMB originating from maximum visible distance in every possible direction. With special equipment we can see the year 300,000, just like we see a galaxy from the year one billion.

    The CMB was predicted by Gamow and Alphard as a consequence of the big bang. I see no other way to explain or even hint at it, so that is rather convincing. As to WHY it happened, no clue there. As for Newton's Laws, you might like to have a look at Noether's Theorem, which has to do with symmetries.
     
  20. Dec 8, 2011 #19

    marcus

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    Good to mention Emmy Noether! Her theorem can be seen as holding the key to why simple Newtonian ideas like conservation of energy and linear momentum fail in curved or expanding space. (Time translation symmetry fails, etc.) But it is a somewhat advanced topic. I think the important thing to realize first is that Newton's laws cannot be proven because they are not right. Needed correction, and the correction needs correction...etc.

    A mathematical P implies Q proof would have to be based on some not quite correct assumption P about nature. So one has to make reservatons like "only approximately or in certain limited situations where applicable..."
     
  21. Dec 8, 2011 #20

    marcus

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    In general terms I like your description very much. However it may be confusing that you say year 300,000. The usual figure is more like 380,000. You and I are saying the same thing but it could sound different because of the numerical discrepancy. I talked about the spherical "surface of last scattering" shell picture back a few posts.

    More seriously Patrick, could you please explain your figure of 13.5 billion years ago?
    What figure do you use for the age of the universe. A very common figure for people to use is 13.8.
    It looks like you have taken the figure of 13.8 and subtracted 0.3 billion years from it to get 13.5.
     
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