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Einstein-deSitter Expansion Questions

  1. Sep 18, 2007 #1
    I calculate that both in the matter-dominated Einstein-deSitter universe, and in the present universe in which the cosmological constant dominates the expansion rate, the observable universe expands at exactly the "escape velocity" of its total mass energy. At present, this expansion is about 985 million radial meters/second, generating about 2.34e+63 cubic meters/second of new vacuum. As the domination of the expansion rate by the cosmological constant continues, the future universe will expand geometrically, approaching the rate of a simple deSitter universe.

    I think it is also correct that the future ACCELERATION of expansion will be driven by the addition of "new" vacuum space in the future, not by the vacuum space that is already here. If (hypothetically and contra-factually) one posited that future vacuum space will not bring a cosmological constant with it, but the existing vacuum retains its cosmological constant, then the future expansion rate would be at a (declining) Einstein deSitter rate.

    Which brings me to my first question: What is the motivating source of expansion in an Einstein-deSitter universe? I would appreciate if you would direct me to whatever you see as the most compelling explanation. I have read that an Einstein deSitter universe is analogous to a "cannonball" shot off in a single impulse. That impulse may be "inflaton energy" left over from Inflation. I've also seen it described vaguely as "kinematics". Frankly, I don't see where this massive energy artifact could be hiding in the universe, and continuing to create new vacuum space for us in perpetuity (assuming the universe is flat). I understand the "negative pressure" theory for dark energy, but I haven't seen that described as the source of the original Einstein-deSitter expansion.

    I note that if the mass/energy of the universe were considered at the level of tiny units (such as atoms), the "escape velocities" of those individual units would result in a total volumetric expansion rate for the universe which exactly equals the observed expansion rate. This raises a second question: is there anything in General Relativity that treats the gravity of mass/energy as ITSELF a motivating source of spatial expansion? For example, if gravitation imposes a stress-energy tensor on the surrounding space, causing geometric curvature, could the same mechanism also cause spatial expansion?

    I understand that in the Einstein-deSitter model, gravity is treated as a restraint on expansion rather than as a cause of it. But if GR explains gravitation as a cause of expansion, then some important pieces would fit neatly in place. For example, by definition it would cause the universe to perpetually maintain itself exactly at "critical mass", maintaining perfect geometric flatness.

    If gravitational expansion isn't consistent with GR, might it perhaps be consistent with recent speculation on quantum gravity?
    Last edited: Sep 19, 2007
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  3. Sep 18, 2007 #2


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    I don't understand what you've calculated. Hubble's law says that the expansion rate is proportional to the distance. H is usually expressed as 70 (km/sec) / megaparsec, which can be converted to 1 / (14 billion years). There is no way that I know of to get from this value of H to the numbers you present. Are you perhaps assuming that the Hubble radius is the "size" of the universe? Note that the usual view is that if the universe is flat (which it appears to be), it is also infinite, and furthermore it was infinite at the time of the big bang. You seem to be assuming that the universe is finite (you even give a volume), but the basis for this assumption is not clear to me, I can only guess that you might be considering the Hubble radius to be the "size" of the universe, perhaps?

    Nasa's WMAP cosmology page says, for instance

    Thus, I think the place to start this discussion is to go over the details of your calculations and see if they look right.
  4. Sep 18, 2007 #3
    Thanks for responding Perv!

    Here is a quick summary of my calculations (I have a spreadsheet):

    I have the co-moving radius of the observable universe at 4.35e+26 meters, the volume at 3.45e+80 cubic meters, the density at 9.16e-27, and the total mass at 3.16E+54 Kg.

    When the Hubble Constant is converted to meters/second/meter (instead of kilometers/Gy/Megaparsec), it is 2.26e-18. This yields a simple calculation that the total radius of the observable universe is expanding at 9.85e+8 meters/second. Which yields a volume growth rate of 2.34e+63.

    That also happens to be the same volume growth rate that is calculated as the total "escape velocity" the mass/energy of the observable universe. The escape velocity formula being (2Gm/r)^1/2.

    I have a formula for estimating the volume growth of a shell on a sphere, which I use for the volumetric calculations. Let me know if you want that.
  5. Sep 19, 2007 #4


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    4.35e26 meters is about 46 [correction] billion light years. Referring to figure 1 of
    Lineweaver and Davis, this is about right for the particle horizon.

    (Open question: Does this diagram (figure 1) take into account inflation? I would guess not, I think it's only for the lambda-cdm model. I don't know if there is any information on the location of the particle horizon that takes early inflation into account. I'm not sure how much this matters in a practical sense, as the universe was fairly opaque for the first 400,000 years of its existence- I don't think we'll be able to see anything further away than the CMB of the big bang, which was emitted when the universe first became reasonably transparent.)

    Continuing on, the velocity of recession at 46 billion light years (via Hubble's law) is well above c - you calculate this, in fact, as 9.85e8 meters per second, i.e about 3 times the speed of light.

    So I don't see what the physical significance is of calculating an "escape velocity" via a Newtonian formula, i.e. (sqrt(2GM/r) is, especially when the result is greater than the speed of light!

    There are also some concerns about the figure you quote for mass and volume. Most GR textbooks warn readers about the issues involved with defining the "mass" of the universe. While most of these remarks are aimed at the "mass of a closed universe", the comments also apply to computing the "mass" of the observable universe.

    You've definitely calculated a number by defining a particular commonly used coordinate system to calculate the volume (and then multiplying by the density) but this number isn't covariant, i.e. it depends on you adopting a particular coordinate system, comoving coordinates - so it's physical significance is rather questionable.

    I'm going to take the liberty of moving this thread to the cosmology forum with a redirect, I think it's a slightly better fit, and perhaps we can get some more comments there.
    Last edited: Sep 19, 2007
  6. Sep 19, 2007 #5


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    The particle horizon is at 46 billion light years.

    There is actually a coincidence when computing recession speeds and "escape velocities" that applies for every flat cosmological model. The calculation goes as follows. The density of a flat universe is equal to the critical density:

    [tex]\rho_c = \frac{3H^2}{8 \pi G}[/tex]

    The Schwarzschild radius for a mass [itex]M[/itex] is:

    [tex]R_s = \frac{2GM}{c^2}[/tex]

    For a spherical mass with density equal to the density of the universe with radius [itex]r[/itex] and mass [itex]M = 4/3 \, \, \pi r^3 \rho_c[/itex]:

    [tex]R_s = \frac{r^3H^2}{c^2}[/tex]

    The value for which the radius of the contained mass is equal to its Schwarzschild radius is:

    [tex]r = \frac{r^3H^2}{c^2}[/tex]

    [tex]r = \frac{c}{H}[/tex]

    This is the Hubble radius. This means, the distance at which the objects recede at the speed of light is equal to the distance at which the "newtonian escape velocity" would be equal to [itex]c[/tex]. This seams to be a mere coincidence.
    Last edited: Sep 19, 2007
  7. Sep 19, 2007 #6
    Hi Hellfire,

    Thanks for laying out the applicable formulas for expansion and escape velocity.

    Actually, it is no coincidence that escape velocity and Einstein-deSitter expansion yield the same result. The underlying concept of the Einstein-deSitter model is that the universe was hurled outwards like a cannonball, and the gravity of its mass/energy contents restrains the expansion rate. If the universe is flat (meaning that it is exactly at critical mass), then the expansion rate declines over time, approaching zero but never reaching zero. The model is intended to be analagous to a cannonball shot (say from the moon, where there is no atmospheric friction) at exactly escape velocity; the speed of the cannonball will decrease over time, approaching but never quite reaching zero.

    I don't want to confuse anybody that the connection between these two equations is a new insight. It's not at all.

    However, the idea that the gravitation of mass/energy itself is the ONGOING CAUSE of spatial expansion strikes me as incredibly elegant and powerful. In one stroke it explains why the universe continued expanding after inflation ended, and how (against all odds) it can be geometrically flat. It also demonstrates that the accelerated expansion caused by the cosmological constant is another application of the same mechanism. If vacuum has a built-in mass/energy value, as it appears to have, then the simple addition of the vacuum's mass/energy to the existing matter/radiation mass/energy explains the present boost to the expansion rate. There's no need to invent "dark energy" or "quintessence".

    For people who aren't familiar with the Friedmann expansion model, I want to remind them that after Inflation (supposedly) ended in the first tiny fraction of a second, the universe continued to expand very rapidly. Between about the 9.45Gy mark and the present (13.7Gy), the expansion rate has been dominated by the cosmological constant. Before 9.45Gy, the cosmological constant was NOT the dominant source of expansion. In fact, I am astounded by the dearth of cosmology writing analyzing WHAT was the cause of the early expansion. As I said, it is variously attributed to "initial conditions" (whatever that means), "kinetic energy" (so where is this kinetic energy hiding?), or a single residual impulse of "inflaton energy" left over from Inflation. Before Alan Guth invented Inflation, the early expansion seems to have been generally viewed simply as residual momentum of the explosive energy of the Big Bang.

    I am a bit disappointed that this post was redirected out of the Relativity forum. Although playing around with the escape velocity and Friedmann expansion formulas is interesting, the answer to my questions doesn't lie there. If my guess is right, spatial expansion needs to be considered as a part of, or an extension to, General Relativity. Perhaps it could just be added as another factor in the Einstein Field Equations.
    Last edited: Sep 19, 2007
  8. Sep 19, 2007 #7


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    Nowhere in my derivation I did assume that the universe was Einstein-deSitter ([itex]\Omega_m = 1[/itex], [itex]\Omega_{\Lambda} = 0[/itex]). I only assumed it is flat. It could be flat with a cosmological constant ([itex]\Omega_{\Lambda} \neq 0[/itex]) so that the expansion does not decline over time and the result would still hold. I really do not see any physical interpretation of the result beyond the coincidence.
    Last edited: Sep 19, 2007
  9. Sep 19, 2007 #8


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    Oops, yes, thanks for pointing that out. (I corrected that in my original post


    I don't see any physical significance to the result either, but note that the Hubble radius is about 15 billion light years, not the 45 billion that the OP was using. I believe he's using the particle horizon rather than the Hubble radius in his calculation.

    So perhaps this "escape velocity" thing works from any radius?

    If we multiply the radius by n, the mass increases by n^3, and the newtonian escape velocity goes as sqrt(2 G n^3 M / n R) = n sqrt(2 G M / R), so the escape velocity is proportional to n, we could also say that escape velocity is proportional to the radius.

    Hubble's law also says that expansion is proportional to the radius.

    Since they are equal at the Hubble radius, as you've shown, they are always equal.

    But I can't see sensibly applying the formula when v_escape > c.
    Last edited: Sep 19, 2007
  10. Sep 19, 2007 #9
    Hi Hellfire and Pervect:

    Hellfire: I appologize profusely if you feel I implied that you disbelieve in the cosmological constant. I did not mean that. As you can see from what I've written, the model I'm using embraces the existence of a cosmological constant, so we're completely together on that. My point was simply that, during almost all of the first 9.45Gy of the universe, the expansion rate was dominated by matter, and the contribution of the cosmological constant was immaterial. So during that time period (the period my questions focus on), the universe was for all practical purposes a pure Einstein-de Sitter universe. Right?

    I'm curious why you still suggest that the direct relationship between the Einstein-deSitter expansion formula and the escape velocity formula is a "coincidence". As I pointed out, it most definitely is not. There is no lack of scholarly support for this point. For example, here's what the esteemed Prof's Peebles and Ratra have to say on the matter:

    "...This simplifies the expansion rate equation to what has come to be called the Einstein-de Sitter model, ˙a2/a2 =/3πGρM, where ρM is the mass density in non-relativistic matter; here M = 8πGρM/(3H^2) is unity. The left side is a measure of the kinetic energy of expansion per unit mass, and the right-hand side a measure of the negative of the gravitationalpotential energy. In effect, this model universe is expanding with escape velocity." http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Aastro-ph%2F0207347 [Broken]

    Prevert: First, you are correct that "this 'escape velocity' thing" works from any radius. Thus it is irrelevant whether we focus on the observable universe or a subset or superset of it.

    As I understand it, your primary concern is that the formula isn't sensible above the speed of light. I have two reactions to that:

    1. I assume you agree that the formula is sensible below the speed of light. If so, then do you agree with my major point, within a radius where escape velocity < c?

    2. As you well know, very distant (high redshift) objects in the present universe are expanding away from us far faster than the speed of light. This is considered to be of no significance, since the objects are not actually moving, rather the distance between us is increasing by means of spatial expansion. The same principle clearly would apply if spatial expansion were caused by the gravitation of mass/energy. Spatial expansion is spatial expansion, regardless of what causes it.

    Is there anything more I can do to satisfy you guys about the soundness of the physics I applied, before we start grappling with my actual questions?

    Thanks, Jon
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  11. Sep 20, 2007 #10


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    Ok I understand what you mean with the result for the escape velocity for the Einstein-deSitter model, but I would agree with pervect that one cannot sensibly apply it for every situation.
    Sorry but after reading again your previous posts I did not have the impression you were asking something. Could you state your questions again?
  12. Sep 20, 2007 #11
    Hi Hellfire,

    Sure, here are my questions:

    1. During the matter-dominated and radiation-dominated eras (before the cosmological constant began dominating expansion around 9.45Gy), was expansion caused by (a) residual inertial momentum leftover from an Inflation period that had already burned itself out, (b) a one-time energy impulse at the end of Inflation, or (c) ongoing exertion of a perpetual energy source?

    2. If the answer to #1 is (a), then which one possesses this leftover momentum, matter or space? It's unlikely to be matter, because that implies that expansion occurs as matter moves THROUGH space. I have not heard of any mechanism whereby the mere inertial momentum of a mass causes the space it passes through to expand. If instead, the momentum were possessed by space itself, then that raises obvious additional questions. Can a quantum of space possess normal inertia and momentum, and "move through space", as if it were matter or radiation? Or alternatively, does space possess a "moment of inertial expansion" whereby it continues stretching perpetually as the summation of every energetic perturbation in its ancient history?

    3. If the answer to #1 is (b), then was the single impluse:(i) a mini-grand-finale burst of inflation, or (ii) a blast of kinetic energy (from what source)?

    4. If the answer to #1 is (c), then is the energy source: (i) residual inflatons which survived the inflaton anihilation at the end of Inflation, (ii) kinetic energy (if so, what is the source and why don't we detect it directly?), (iii) dark energy (from what source?), or (iv) some inherent property of thermodynamics (perfect fluid), such as pressure?

    5. Is there any known principle explaining why the magnitude of the (one-time or ongoing) expansion energy happened to be exactly or almost exactly the perfect amount to maintain the expansion rate at escape velocity (and thereby preserve geometric flatness)? No anthropic principle, please!

    6. Isn't it suspiciously complex to invent two entirely different expansion mechanisms, one for the cosmological constant, and another for the period preceding the domination of the cosmological constant? Include Inflation and maybe we can have 3 mechanisms!

    7. Do you share my dissatisfaction with the current state of physics theory regarding the early expansion period?

    8. Even if my idea, that the gravitation of mass/energy causes expansion, is too radical for your taste, can you please help me design an element that could be added to the Einstein Field Equations, whereby gravity generates radial expansion at escape velocity?

    Thanks again, Jon
    Last edited: Sep 20, 2007
  13. Sep 20, 2007 #12


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    I would say the answer is 1a, or may be more or less between 1a and 1b. A simple way to model inflation is as a very short period of exponential expansion [itex]a = \exp(Ht)[/itex], with constant Hubble parameter [itex]H[/itex]. The acceleration of expansion is [itex]\ddot a = H^2 \exp(Ht)[/itex]. The longer inflation lasts, the more the expansion becomes accelerated. As soon as inflation ends, radiation starts dominating and the expansion starts decelerating [itex]\ddot a < 0[/itex]. Inflation itself is assumed to have been a very short event, a kind of impulse to the expansion of space in the very early universe.

    Yes, you can take the analogy of a rocket escaping from earth. You can give impulse to it to accelerate it and allow it to escape. Afterwards it will decelerate. After a time decelerating, you could accelerate it again.

    A violent exponential expansion can create flatness, as well as homogeneity, out of any previous initial condition.

    "Suspiciously complex" sounds suspiciously subjective to me. Anyway, complex or not, that's the current best fit model to the observations.

    I do not feel that there are many other alternatives to explain homogeneity and flatness in a big bang model. However, high energy physics may tell us something interesting about the mechanisms that triggered inflation.

    In an homogeneous an isotropic cosmological model you can have acceleration of expansion with positive pressure. For this to happen you need a scalar field with high potential energy, or something that behaves as a scalar field in such a state (a cosmological constant, for example).
    Last edited: Sep 20, 2007
  14. Sep 21, 2007 #13
    Hi Hellfire,

    Thanks for being specific in your response. I would like to discuss two points, the "Moment of Inertial Expansion", and inflation-related flatness.

    1. Vacuum's Moment of Inertial Expansion: I'm not aware that this subject has been explored in any depth, although I'd appreciate a reference to any article about it. I think we can assume that it acts like normal momentum, in the sense that force is required to speed it up, and an equal force to slow it down. It has no inherent energy content. The momentum will continue forever unless changed by a force like gravity.

    As a thought experiment, let's visit deep space and collect two samples of vacuum, one which has experienced strong gravity and another which has not. Although both samples are identical by all measures, the 2nd one has a Moment of Inertial Expansion and the first does not. The 2nd sample will expand forever, while the 1st will not expand. However, by simply placing the 2nd sample briefly in the presence of a gravitating mass, we can instantly transform it to be like the first sample. Even after the gravitating mass is removed, that sample of space will never expand again unless it is subjected to a new expansionary force. (I'm ignoring the cosmological constant for the moment).

    It's as if the vacuum experienced Traumatic Stress Disorder in its youth, causing a condition whereby it gains weight for all eternity. We can cure it instantly by waving a magic wand of mass. We can't slim it back down, but we can stop it from gaining any more weight. It sounds like the old traveling salesman's magnetic cure for headaches.

    At present I assume we have no idea where in the DNA or memory bank of the vacuum this Moment of Inertial Expansion is stored, nor can we detect it directly. Of course, if we bring this puzzle to our neighborhood quantum mechanic, he can quickly diagnose whether our vacuum has suffered a tilt in its particle orbits, color fading, or some other quantum trauma. However, in the end I predict he will prescribe yet another scalar field and an accompanying new mediating particle, the "elasticon". Shouldn't this particle be added to the list, along with the inflaton and graviton, for study by the new European accelerator?

    Sorry for the (good natured) sarcasm, but if I had waltzed into this forum with such a theory, I would be banned for life!

    [I'm adding this paragraph as an edit to this post. After more thought, it occurs to me that the Moment of Inertial Expansion can go negative as well as positive. If we expose our sample vacuum to a large enough mass (or perhaps a smaller mass with longer exposure) it should stop expanding and then start shrinking. If we can intensify the gravitational exposure enough, the vacuum sample might disappear entirely as we watch! This is spooky stuff!]

    2. Inflation-Related Flatness. I am familiar with the concept that the violent expansion of inflation stretched all curvature out of space, leaving it very flat. It strikes me, though, that this is an imprecise description. I understand the effect of stretching, but I cannot see how the SPEED of the stretching is relevant. For example, an expanding 4-sphere of any given diameter will have the same average surface curvature, regardless of whether it expanded quickly or slowly. If hypothetically the full universe has a present radius 1000 times the radius of the observable universe, its overall curvature doesn't depend on whether it took 10Gy or 30Gy to expand to that size. If the universe is infinite, then by definition its curvature apparent to us will always be exactly zero. So whether or not inflation occurred is irrelevant, except that it was one of the contributors to the absolute size of the present universe. The rule is simply that really big universes always appear flat to little people like us, and only a miniature universe could have detectable curvature. A microbe sized creature will never know if she was standing on the Hindenburg or the deck of an aircraft carrier.

    What bothers me is that the geometric flatness of the universe (prior to domination by the cosmological constant) should depend on two seemingly independent variables: (1) the absolute amount of mass/energy, and (2) the rate of expansion. By definition, a universe is at critical density (and therefore flat) when it is expanding exactly at the escape velocity of its mass/energy. So, while I understand how the aggregate expansion of the universe up to the point in time when inflation ended could have helped to iron out any pre-existing curvature, I cannot understand what caused the rate of expansion to slow down at that point to precisely the escape velocity of this universe's mass/energy content.

    The rule seems too powerful: large, expanding universes always appear flat. Period, end of story. The FLRW metric is obsoleted. Einstein, Freidmann, Lamaitre, etc., etc., were all on a fool's errand to worry about overall geometric curvature and critical density.
    Last edited: Sep 21, 2007
  15. Sep 21, 2007 #14


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    I did not understand the point you are trying to make in 1. and I cannot comment on it. However, I might clarify something regarding point 2.

    You can rewrite the first Friedmann equation as:

    [tex]\Omega = 1 + \frac{k}{a^2 H^2}[/tex]

    For a simple model of inflation you assume a constant Hubble [itex]H[/itex] parameter and [itex]a = \exp(Ht)[/itex] with an increase of the scale factor [itex]a[/itex] by many orders of magnitude. You can see that this leads to [itex]\Omega \rightarrow 1[/itex]. This is independent of the value of the Hubble parameter, and, moreover, the fact that flatness is (nearly) reached does not imply that the expansion rate has to slow down (as the Hubble parameter remains constant in such a model).
    Last edited: Sep 21, 2007
  16. Sep 21, 2007 #15
    Hi again Hellfire,

    On my point number 1, communication would be enhanced if you were less cryptic and provided more explanation about what you don't understand. All I was doing was trying to analyze the implications of YOUR answer in your prior post, that my "Moment of Inertial Expansion" comment roughly captured your understanding on the currently accepted theory for expansion after inflation and before the cosmological constant dominated. Here's your comment I was responding to:

    Originally Posted by jonmtkisco
    Or alternatively, does space possess a "moment of inertial expansion" whereby it continues stretching perpetually as the summation of every energetic perturbation in its ancient history?

    [Your reply:] Yes, you can take the analogy of a rocket escaping from earth. You can give impulse to it to accelerate it and allow it to escape. Afterwards it will decelerate. After a time decelerating, you could accelerate it again.​

    Regarding your response to my point #2, as best I can tell you are explaining the mechanism for inflation itself. But that's not what I was talking about. I was talking about how, after the END of inflation, the expansion rate in fact did slow down to precisely the escape velocity of the universe's mass/energy content. That's the only expansion rate that could possibly allow the universe to REMAIN THEREAFTER at critical density and flat. When you point out that it wasn't REQUIRED to slow down to that rate at the end of Inflation, that's my point -- what mechanism explains the extraordinarily "lucky" outcome our universe arrived at, where it actually DID settle in at a post-Inflation expansion rate at precisely escape velocity, and thereby continues to be flat today?
  17. Sep 21, 2007 #16


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    If the universe would be exactly flat, then it would stay flat forever, regardless of the behavior of the expansion (regardless of the evolution of [itex]H[/itex] or [itex]a[/itex]). You can see this in the equation I put above setting [itex]k = 0[/itex]. However, a universe that is exactly flat is a fine-tuning case. Inflation drives the spatial curvature to zero, but does not reach it. Afterwards, the universe starts deviating again from flatness until a new accelerated expansion starts that will drive the curvature again to zero.
    Last edited: Sep 21, 2007
  18. Sep 21, 2007 #17
    Hellfire, I understand your point that, in effect, curvature can be detectably large today only if it springs from at least a tiny kernal of curvature 13.7Gy ago. But I don't think that changes my point. Maybe we're tripping up on semantics here.

    I'm trying to understand the connection between what happened during Inflation, and what happened immediately afterwards. According to the Inflation theory, any geometric curvature in the universe was already almost ironed out by the time Inflation ended, not perfectly to zero, but very precisely close to it by many orders of magnitude. In that last instant of Inflation, the universe was expanding at an enormous geometric rate. An instant later, Inflation ended, and the flattening process ended. However, during (at least) some significant fraction of a second after Inflation ended, the expansion rate experienced an enormous rate of deceleration. When that deceleration ended, expansion settled in at very precisely close to (probably not exactly at) the escape velocity of the universe's then existing mass/energy.

    My question is simply, what FORCE caused the enormous deceleration rate, and what caused the expansion rate to settle in at almost precisely escape velocity? I don't see how the first Friedmann equation answers that question. Obviously gravity was the only significant restraining force. But in the first instant after Inflation ended, the expansion rate was far too high for the gravity of the universe's mass/energy to reel it back in. By all rights, the momentum of expansion accumulated during Inflation should have powered expansion onward at a continuing geometric rate, only partially diminished by gravity. If at any point the universe was expanding faster than its escape velocity, it was forever doomed to "escape"!
    Last edited: Sep 21, 2007
  19. Sep 21, 2007 #18


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    The Friedmann equation must answer the question. All the dynamics of the expansion is encoded in Friedmann's equations. Consider the first Friedmann equation:

    [tex]\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2}[/tex]

    After inflation the energy density of the inflaton can be assumed to be negligible, otherwise inflation cannot be considered to be finished. The energy density is dominated by radiation. The density of radiation scales as:

    [tex]\rho \propto a^{-4}[/tex]

    We want to know how the expansion behaves after inflation in, say, a flat ([itex]k = 0[/itex]) universe:

    [tex]\left(\frac{\dot{a}}{a}\right)^2 \propto \frac{8 \pi G}{3} a^{-4}[/tex]

    You can verify that the scale factor will be:

    [tex]a \propto t^{1/2}[/tex]

    The value of the scale factor at the end of inflation only enters this equation as an integration constant or initial condition of the radiation dominated era. The acceleration of expansion will be:

    [tex]\ddot a \propto t^{-3/2}[/tex]

    The Hubble parameter will be:

    [tex]H \propto t^{-1}[/tex]

    The Hubble parameter will allow you to compute recession velocities and compare them with the escape velocity, so I assume your question boils down to how the Hubble parameter behaved after inflation... but may be I have completely misunderstood your question.
  20. Sep 21, 2007 #19
    Hellfire, buddy, I feel your pain!!!

    Obviously, the "standard" model of inflation and expansion we are discussing creates an unacceptable discontinuity in the Friedmann equation.

    The idea can't be seriously maintained, that a universe expanding far faster than its own escape velocity will spontaneously slow down to its escape velocity. By definition it MUST ESCAPE.
  21. Sep 22, 2007 #20


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    What a discontinuity? All of the quantities I mentioned above are continuous in time.

    There is a "point" at which the scale factor goes over from a [itex]exp(t)[/itex] to a [itex]\sqrt{t}[/itex] behavior, or at which the Hubble parameter goes over from a constant to a [itex]1/t[/itex] behavior.

    If you integrate properly the equations mentioned above you will make sure that the evolution is continuous. For example, for the scale factor in the radiation dominated era you will get something like:

    [tex]a = \sqrt{C (t - t_{EI}) + a^2_{EI}} [/tex]

    with the subindex EI for the value of the quantity at the End of Inflation and [itex]C[/itex] some dimensional constant. Obviosly, at [itex]t = t_{EI}[/itex] you will have [itex]a = a_{EI}[/itex], and you get a continuous evolution.

    You may argue that the evolution is not differentiable at that point. But you have to note that this was only a simplified model. Strictly speaking this "point" is not a point, since inflation does not suddenly stop and there is some smooth decreasing contribution of the inflaton energy density. But even in a more sophisticated model there will be no discontinuity of any kind.

    If this does not answer your question now I can only encourage you to formulate your point mathematically, and tell me exactly what you mean with "unacceptable discontinuity", of which quantity, and at which point. Otherwise we will certainly have a hard time to distinguish semantics from true issues.
    Last edited: Sep 22, 2007
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