Guest Post: Max Tegmark on Cosmic Inflation

Max TegmarkMost readers will doubtless be familiar with Max Tegmark, the MIT cosmologist who successfully balances down-and-dirty data analysis of large-scale structure and the microwave background with more speculative big-picture ideas about quantum mechanics and the nature of reality. Max has a new book out — Our Mathematical Universe: My Quest for the Ultimate Nature of Reality — in which he takes the reader on a journey from atoms and the solar system to a many-layered multiverse.

In the wake of the recent results indicating gravitational waves in the cosmic microwave background, here Max delves into the idea of inflation — what it really does, and what some of the implications are.


Thanks to the relentless efforts of the BICEP2 team during balmy -100F half-year-long nights at the South Pole, inflation has for the first time become not only something economists worry about, but also a theory for our cosmic origins that’s really hard to dismiss. As Sean has reported here on this blog, the implications are huge. Of course we need independent confirmation of the BICEP2 results before uncorking the champagne, but in the mean time, we’re forced to take quite seriously that everything in our observable universe was once smaller than a billionth the size of a proton, containing less mass than an apple, and doubled its size at least 80 times, once every hundredth of a trillionth of a trillionth of a trillionth of a second, until it was more massive than our entire observable universe.

We still don’t know what, if anything, came before inflation, but this is nonetheless a huge step forward in understanding our cosmic origins. Without inflation, we had to explain why there were over a million trillion trillion trillion trillion kilograms of stuff in existence, carefully arranged to be almost perfectly uniform while flying apart at huge speeds that were fine-tuned to 24 decimal places. The traditional answer in the textbooks was that we had no clue why things started out this way, and should simply assume it. Inflation puts the “bang” into our Big Bang by providing a physical mechanism for creating all those kilograms and even explains why they were expanding in such a special way. The amount of mass needed to get inflation started is less than that in an apple, so even though inflation doesn’t explain the origin of everything, there’s a lot less stuff left to explain the origin of.

If we take inflation seriously, then we need to stop saying that inflation happened shortly after our Big Bang, because it happened before it, creating it. It is inappropriate to define our Hot Big Bang as the beginning of time, because we don’t know whether time actually had a beginning, and because the early stages of inflation were neither strikingly hot nor big nor much of a bang. As that tiny speck of inflating substance doubled its diameter 80 times, the velocities with which its parts were flying away from one another increased by the same factor 2^80. Its volume increased by that factor cubed, i.e., 2^240, and so did its mass, since its density remained approximately constant. The temperature of any particles left over from before inflation soon dropped to near zero, with the only remaining heat coming from same Hawking/Unruh quantum fluctuations that generated the gravitational waves.

Taken together, this in my opinion means that the early stages of inflation are better thought of not as a Hot Big Bang but as a Cold Little Swoosh, because at that time our universe was not that hot (getting a thousand times hotter once inflation ended), not that big (less massive than an apple and less than a billionth of the size of a proton) and not much of a bang (with expansion velocities a trillion trillion times slower than after inflation). In other words, a Hot Big Bang did not precede and cause inflation. Instead, a Cold Little Swoosh preceded and caused our Hot Big Bang.

Since the BICEP2 breakthrough is generating such huge interest in inflation, I’ve decided to post my entire book chapter on inflation here so that you can get an up-to-date and self-contained account of what it’s all about. Here are some of the questions answered:

  • What does the theory of inflation really predict?
  • What physics does it assume?
  • Doesn’t creation of the matter around us from almost nothing violate energy conservation?
  • How could an infinite space get created in a finite time?
  • How is this linked to the BICEP2 signal?
  • What remarkable prize did Alan Guth win in 2005?
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51 Responses to Guest Post: Max Tegmark on Cosmic Inflation

  1. kashyap Vasavada says:

    Max: My understanding is that BICEP2 ( assuming it is correct) shows that one can get by with classical general relativity and quantum field theory , separately, until about 10^16 GeV. But it seems that some prominent physicists like Guth and Wilczek are saying that it proves quantum gravity. What is your opinion? In what sense it says anything about QG and perhaps string theory?

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  2. Max Tegmark says:

    Good question, Kashyap. The point is that the BICEP2 signal is due to quantum fluctuations not in any old quantum field, but in the *gravitational* field, which proves that gravity is in fact quantized. Most researchers had of course assumed this to be the case, but it’s important to test our assumptions, and this is the first experimental evidence of this fact. So now we know that there is in fact such a thing as quantum gravity.

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  3. John Gordon says:

    The only Google hits today on “Hawking/Unruh quantum fluctuations” are to this article. Is this a new term? Can you explain relationship of those quantized gravity fluctuations to Hawking and Unruh?

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  4. Hidden due to low comment rating. Click here to see.

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  5. Max Tegmark says:

    Here you go, John: http://en.wikipedia.org/wiki/Unruh_effect
    Stephen Hawking realized that black holes evaporate, because quantum fluctuations cause radiation to emerge from its event horizon. Bill Unruh realized that analogous radiation emerges from *any* event horizon, not merely the kind surrounding black holes. During inflation, you have an event horizon around you that you find yourself inside, and the Unruh radiation from it includes the gravitational waves seen by BICEP2.

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  6. Agron says:

    Hidden due to low comment rating. Click here to see.

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  8. Nick says:

    Hi,

    I read the article, it was very interesting.

    But my question would be, is the theory of eternal inflation almost certain to be true if inflation is true? Are there models of inflation without internal inflation?

    In other words have we discovered the origin of the universe??

    Thank you,
    Nick

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  9. MeOldChina says:

    Squeezing an infinite amount of space into a finite space is a neat trick!

    Are we causally connected to the other volumes of space kicked off by other inflation ending events? Are we screened off by our event horizon or can the radiation coming off that horizon carry information about them?

    Would these other volumes of space have the same fundamental parameters as ours?

    Can we possibly tell, and would it be interesting to know, which generation of inflation decays our local volume of space came from?

    Cheers!

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  10. Max Tegmark says:

    Hi Nick,
    > is the theory of eternal inflation almost certain to be true if inflation is true?
    As I explain in the chapter (linked above), Alex Vilenkin and others have shown that inflation is generically eternal, in the sense that that’s what typical inflation models predict. The basic reason is that for inflation to work in the first place, the amount of inflating substance has to double faster than its half-life (the time it takes half of it to stop inflating), so the total volume of inflating stuff just keeps growing forever.

    > have we discovered the origin of the universe?
    No: We’ve discovered what caused our Big Bang, but we still haven’t figured out what, if anything, caused inflation to start.

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  11. Gordon D. Munro says:

    Among “elite” theoretical physicists what percentage still believe “mutiversality” of any kind is “not even wrong?”

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  12. Max Tegmark says:

    Squeezing an infinite amount of space into a finite space is a neat trick!

    Hi MeOldChina,

    > Are we causally connected to the other volumes of space kicked
    > off by other inflation ending events?
    No, we’re no longer causally connected to those other volumes, because space between here and there is still inflating, stretching faster than we could ever pass through them.

    > Would these other volumes of space have the same
    > fundamental parameters as ours?
    Inflation alone can’t answer that question. It depends on whether the fundamental theory of physics, whatever it is (string theory, loop quantum gravity, something we haven’t yet thought of, etc.) has one or more solutions for homogeneous space. If it has many solutions (like the equations governing homogeneous water has the three solutions ice, steam and liquid), then inflation will create volumes realizing each of these solutions, and some parameters describing measurable properties can then differ. If there’s only one solution, then the parameters will be the same everywhere.

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  13. Ben Goren says:

    Max,

    Thanks much for taking the time to help us mere mortals get a better understanding of this amazing Cosmos we happen to find ourselves in.

    This would seem to be a good place to ask a couple questions I’ve been wondering about. I haven’t yet read the book excerpt, so feel free to tell me to just go read it if you’ve already answered these there.

    First, it seems to me that, especially with the Fermi-LAT (yet-to-be-confirmed) discovery of ~35 GeV WIMPs, we now have all the major empirical pieces of the puzzle that were missing just a short while ago: the Higgs, quantum gravity, inflation, and dark matter — and with specific value ranges (some very precise, others not so) for all. What, if anything else, do you theorists need from the experimentalists to finally crack the big nuts of quantum gravity, cosmogenesis, and perhaps a GUT? (And quite likely, of course, they’re a package deal.) Are you making any bets on when those will be solved?

    Next…well, without even reaching for the envelope, it would seem to be safe to suggest that the Schwarzschild Radius of the mass of the entire universe is just a wee bit larger than the radius of our own Sun (which is what I get from that envelope as the initial conditions of the post-Inflation Big Bang). I’m guessing that the obvious types of confusion that would naïvely arise from this sort of observation mostly don’t apply. For example, it seems clear that you can’t use the same approach to describe the Universe as an whole as you would to stuff within it, and I think you’ve already indicated that Inflation is an ongoing process, continuing to this day. Still, could you perhaps spare a moment or two to elaborate on what it means for so massive a Universe to be so small, and how that differs from compressing parts of the universe to such densities today?

    Thanks again!

    b&

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  14. Nick says:

    “Hi Nick,
    > is the theory of eternal inflation almost certain to be true if inflation is true?
    As I explain in the chapter (linked above), Alex Vilenkin and others have shown that inflation is generically eternal, in the sense that that’s what typical inflation models predict. The basic reason is that for inflation to work in the first place, the amount of inflating substance has to double faster than its half-life (the time it takes half of it to stop inflating), so the total volume of inflating stuff just keeps growing forever.

    > have we discovered the origin of the universe?
    No: We’ve discovered what caused our Big Bang, but we still haven’t figured out what, if anything, caused inflation to start.

    Thank you for the explanation Doctor Tegmark.

    I was wondering because I had heard Lawrence Krauss say that the universe may have started from absolutely nothing, yet with an inflation. If I understand correctly what Krauss says, he is talking about just our universe, not an eternally inflationary universe.

    I don’t know whether Lawrence Krauss’s ideas could be relevent to explain the start of an eternal inflation.

    Nick.

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  15. MeOldChina says:

    Hi Max,

    Thanks for the response.

    So, the quantum fluctuations give rise to volumes with slightly different initial conditions, differences that then get amplified, but does eternal inflation predict the different volumes of space all initially having low entropy, like own observable region? If so, is this because the initial inflationary patch is low entropy?

    Cheers!

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  16. Antonio Sanchez says:

    Max, many thanks for sharing the chapter of your new book. I am going to buy it and read it anyway. As you mentioned in several interviews, I agree that math, laws, physical patterns… are not just useful tools to describe reality, they constitute reality itself.

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  18. @Forrest: “As far back as we can see the universe appears to have the same density, or appears to have less density, than the close-by universe. “

    Simply not true. No-one claims this, except you, as far as I know.

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  19. @Ben Goren: “it would seem to be safe to suggest that the Schwarzschild Radius of the mass of the entire universe is just a wee bit larger than the radius of our own Sun”

    Why?

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  20. Ben Goren says:

    Phillip, the Schwarzschild Radius of Sagittarius A* is about a tenth of an AU, more than an order of magnitude bigger than the Sun’s radius and about a third of the way to Mercury’s orbit. I don’t think I need a calculator to know that the mass of an hundred billion such black holes plus all the visible matter is going to have just a wee bit larger Schwarzschild Radius still.

    Cheers,

    b&

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  21. UncleMonty says:

    Max, thanks – a couple of things clicked for me that never clicked before, when reading your chapter. Could you tell me, if inflation is still going on in some regions of the universe, and if during inflation the density of the initial blob does in fact get smeared out a little with each doubling, then when inflation ends in those regions will they necessarily have very different properties from ours because the ur-stuff had time to get more diluted before inflation came to an end? Or is it thought instead that the “trigger” for inflation coming to an end occurs when the density is sufficiently smeared out, so that in fact all regions will be similar when their own “hot big bang” begins–they just took a different amount of time during inflation to reach that critical density?

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  22. ” I don’t think I need a calculator to know that the mass of an hundred billion such black holes plus all the visible matter is going to have just a wee bit larger Schwarzschild Radius still.”

    Right. So what is the point of your original comment?

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  23. Ben Goren says:

    Phillip, my point was that, at least naïvely, it would seem that the Universe is a black hole. And pursuing that line with similarly naïve reasoning leads to questions such as, “If we’re in a supermassive black hole, why haven’t we been crushed into oblivion?”

    As I noted in my original question, that’s clearly not the way things actually are and the naïve approach fails spectacularly. I was hoping Max or Sean or somebody could expand upon why that should be so.

    Cheers,

    b&

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  24. Joan Hendricks says:

    Dr. Tegmark, Thank you for this blog. I read all of the Chapter 5 you included the link to. Reading that chapter almost blows my mind. It seems very different from other things I have read about the Big Bang. Significantly, the Big Bang after inflation. Positive and negative gravity sound like the ultimate perpetual motion machine if we could capture that energy. And, you are writing this without detailed mathematical formulas that I probably couldn’t comprehend.

    Guess I’d better get your book and not be satisfied with just one chapter!

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  25. @Ben Goren:

    The universe is not a black hole. A black hole is a static solution of Einstein’s equation in a background metric while the universe is expanding and has no background as such. Yes, if you plug in the numbers, the radius of curvature of the universe is of the order of its Schwarzschild radius, but that is essentially just dimensional analysis.

    However, even if we were inside a large black hole, we wouldn’t necessarily be crushed to oblivion (although, in the context of GR, it is inevitable that someday we will be). The larger the mass of a black hole, the larger the radius and the lower the density. With a really large black hole, one wouldn’t even notice crossing the horizon.

    I still don’t get why you equate the Schwarzschild radius of the Sun with that of the universe.

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  26. Sean Carroll says:

    Ben– Here is my explanation of why the universe is not a black hole:

    http://www.preposterousuniverse.com/blog/2010/04/28/the-universe-is-not-a-black-hole/

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  27. Ben Goren says:

    Sean, thanks for that link! That helps a lot.

    Tying it back with Inflation…there was a point in time when all the mass of the Universe today was in a volume much smaller than it is today. And if space today is (essentially) flat, meaning that the current volume is roughly equivalent to that of a black hole, the density at that early point would have been much, much greater.

    Again, the naïve assumption would seem to be that, at such an early date, gravity should have been enough to (eventually) stop the expansion and cause a Big Crunch (regardless of whether it’s correct to call the Universe a black hole or not). And that assumption is clearly worng.

    Would I be correct in suggesting that Inflation is a force (carried, as I recall from an earlier post of yours, by the inflaton with an associated field) sufficiently stronger than gravity such that Inflation overwhelms the gravitational attraction?

    Max’s paper explains the regime when space was growing exponentially but its density remained constant, thereby creating more “stuff.” It seems to me that there would have been a transition period after that era to today, presumably the Big Bang. I guess what I’m groping for…is whether or not space was as flat at the beginning of that transition as it is today, or if space was highly curved in the beginning and only after expanding (because of continuing inflation) has it flattened out.

    …am I making sense?

    Thanks,

    b&

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  28. @Ben Goren:

    First, just because the universe was once incredibly dense doesn’t mean that gravity must have been able to stop the expansion. Without a cosmological constant, expansion eventually stops if Omega>1. With a positive cosmological constant, it is more complicated, but in this case as well it is possible, depending on the values of Omega and lambda, that the universe expands forever and also that it collapses in the future. (If the cosmological constant is negative, the universe will always collapse.) So, yes, the assumption is clearly wrong.

    Second, the fact that the universe was denser in the past is independent of curvature. The density goes like the inverse third power of the scale factor.

    Inflation overwhelming the gravitational attraction is perhaps true in a sense, but even without inflation we would not expect the universe to collapse.

    Your last paragraph is OK: Yes, inflation makes the universe close to flat because the exponential expansion makes the scale factor much larger than the Hubble radius. (The absolute radius of curvature doesn’t matter. At the end of inflation, it was less than a meter. The important thing is that it was much, much larger than the Hubble radius at that time.)

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  29. kashyap Vasavada says:

    Sean: There is a typo in your 2010 article (http://www.preposterousuniverse.com/blog/2010/04/28/the-universe-is-not-a-black-hole/) in the equation for M. You left out ρ. It may be confusing for beginners. You have a nice argument that universe is not a black hole. I had not seen that before.

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  30. Steve says:

    Wow. The advantage of giving a full chapter as a ‘taster’ for a book is that it allows a fair assessment of a writers ability before buying the full tome. Well, I bought the book as a consequence of the free chapter and I’m devouring the rest of it. I’m a technically literate ‘lurker’ on this site (sold my soul to engineering decades ago) but love to keep up to date with the latest cosmological ideas via people like Sean, and now Max.
    Just seemed a very informative explanation for me of inflation and its merging with the Big Bang process. Keep up the good informative work chaps……..

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  31. Reynaldo Carmona says:

    I would love it if either Max or Sean answer this question:

    Max, in your book Our Mathematical Universe, you suggest that space goes through phase changes akin to matter. An article I read to day is claiming that space-time could be like a superfluid. With that in mind, could it be that dark matter is the mass of space itself? In other words, if space is of some form of matter it can call its own, wouldn’t it have an inherent mass? Wouldn’t it possibly interact with ordinary matter in the way WIMPs are said to? I suggest this because if that’s the case, then dark matter has been hiding right underneath our noses. I would also suggest that dark energy is a force inherent in space, but that’s a bit of a stretch when considering that dark energy is accelerating cosmic expansion.

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  32. Tony says:

    In the many worlds scenario I have an exact copy of myself living exactly the same life, maybe an infinite number of exact copies. Poor guys! I wonder then, what makes me, me, and not them? What is it that constitutes the individual that no other person can possess, no matter how exact the copy?

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  33. @Forrest: “As far back as we can see the universe appears to have the same density, or appears to have less density, than the close-by universe. “

    (your quote) “Simply not true. No-one claims this, except you, as far as I know.”

    Hi Phillip, I didn’t see your comment earlier :)

    Phillip, I think you misunderstood my statement ? My statement was that: I’ve never seen a claim that the universe was more dense in the past based upon an observational study, which would seem to be an obvious requirement for an expanding universe model, no?

    If you have seen such an observational claim concerning an astronomical density study I will be glad to consider it. I think one could post topic related links to such papers on this site?

    If you find such a link or claim I will recognize it on this thread, but it might be better to comment on it in detail in another venue of your choice since some may consider the density of the universe unrelated to the Inflation hypothesis, or a distraction to Max’s interesting perspectives, the topic of this thread.

    what do you think? Forrest

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  34. piledHighAndDeep says:

    here’s hoping that Max or Sean are still following this thread: hence a possibly stupid question.

    I understand how inflation solves the horizon “problem”, but I am having difficulty understanding what the problem is in the Big Bang model since at the “beginning” everything was at the same point. In Max’s invitation example, if all the friends started from the same town, then why couldn’t they all agree to send the invitation simultaneously? I guess I don’t understand how the situation is any different between inflation and big bang with respect to the horizon problem.

    thanks in advance,

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  35. John Ramsden says:

    As the expansion rate during inflation is so fast by our reckoning, and we’re told continues indefinitely here and there, one wonders if this rubrik can’t be taken a stage further and a similar mechanism be still operating all round us, as a sort of “operational” form of the Universe splitting of Everett’s Many Worlds picture.

    As Max explains, to an external observer an infinite universe appears subatomic sized. So maybe, fantastic as it sounds, each elementary event manifests as a kind of “inflation in reverse” which takes a faithful snapshot of the universe essentially instantly shrunk from its infinite size to a Planck size or below but intrinsically still of identical structure to its progenitor save for exactly one difference!

    After all space “shrinkage”, for want of a better word, seems generally in keeping with progress into the future, as a black hole interior attests.

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  36. @piledHighAndDeep (you of cartoon fame?)

    First, although classically everything was at one point, in reality there was probably not a singularity, so although everything was very close together, it was still too far apart for communication. Second, the above applies if the universe is finite. If it is infinite, then it was infinite even at the big bang.

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  37. @Forrest: There is loads of such evidence. Of course, no-one can travel back in time to the early universe and directly measure the local density. But we can count galaxies per volume, say, and so determine the density. Big-bang nucleosynthesis tells us something about the density at that time. We can measure the CMB temperature and hence radiation density at high redshift. And so on.

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  38. Ray says:

    @Forrest

    Here’s a good summary for the evidence against the steady state model (i.e. for the universe being denser in the past) including source counts: http://www.astro.ucla.edu/~wright/stdystat.htm .

    Relevant text:

    ” The Steady State model makes some definite predictions. The first one to be tested involved the number of faint radio sources. In the 1950’s astronomers found that radio sources were typically much more distant than typical optical galaxies, so modifications to the usual source count law due to cosmology were expected. For the standard Big Bang model the counts were expected to fall below the usual “8 times more sources for 4 times fainter limit” law by an amount given approximately by 1/(1+z)4 where z is the redshift of the sources. This law assumes that radio sources are conserved, so a given section of the Universe has the same number of radio sources at all times. Because the volume of the section was smaller by a factor of (1+z)3 at early times, the actual density of radio sources was higher by a factor of (1+z)3. The density was constant in the Steady State model, of course, so the count correction factor would be given by 1/(1+z)7. The diagram below shows what was expected and actually seen:
    Radio source count schematic

    The Big Bang should have a deficit of faint sources, the Steady State should have an even bigger deficit, but the observations showed a surplus of faint sources. The Steady State model has no adjustable parameters to correct for this error, but the Big Bang does. The assumption of conserved radio sources (CRS) can be dropped in favor of an excess of radio sources 1-3 Gyr after the Big Bang. Thus the Steady State failed the radio source count test, while the Big Bang passed by “winning ugly” – introducing a new parameter to describe a new datum. See Maran’s review of Hoyle’s book, Galaxies, Nuclei, and Quasars. Maran describes the birth and death of the Steady State theory without reference to the microwave background.”

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  39. Tony says:

    In reference to why the universe is not a black hole. Is it because the universe emerged and is growing from the inside out, while a black hole grows from the outside in?

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  40. @Ray,

    Yes, that is a good example of a mainstream explanation and criticism of Hoyle’s Steady State models. The critique does not involve any observational density studies however. I am not a fan of Hoyle’s models since they also explain redshifts by an expanding universe. In his models the density of the universe was maintained by the creation of new matter.

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  41. San Diego says:

    Max,

    My question is wether inflation (eternal or otherwise) could have to do something with the notion of ‘extension’ used by Alfred North Whitehead in Process and Reality, PART IV, THE THEORY OF EXTENSION? (New York /London 1979: THE FREE PRESS)

    I’m not a physicist but very interested in this topic.
    Please help!

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  42. Ilona Schweitzer says:

    My question is wether the phenomenon ‘inflation’ (eternal or otherwise) could have to do something with the notion ‘extension’ used by Alfred North Whitehead in Process and Reality, PART IV, THE THEORY OF EXTENSION (New York/ London 1979: THE FREE PRESS) ?

    I’m not a physicist but very interested in this topic.

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  43. Max Tegmark, you said “During inflation, you have an event horizon around you that you find yourself inside”.
    Am I a black hole looking out? (:-)

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  46. Daniel Shawen says:

    Virtual particles like the Higgs boson, even though they exist for something like 10^-21 seconds, and the photons they decay into, are both subject to the effects of gravitation.

    Being a boson superfluid (bosons, or even bosons and fermions are able to occupy the same place at the same time), these virtual particles must undergo the same time dilation effects as ordinary matter at the surface of large gravitating bodies, meaning that 10^-21 seconds gets temporally dilated too. I wish folks would stop showing the Higgs field as if it were a solid crystalline lattice (and why do they do that?), or reporters bunching up around celebrities (same place at the same time, remember?). This is wrong.

    The acceleration that derives of the Higgs mechanism (to slow down quarks, antiquarks, electrons, positrons, W and Z bosons, and even to bend photons) is ultimately the same as the acceleration due to gravity. It’s a fact. There’s no more fundamental force, higher or rolled up dimension, or free parameters (or at least, not very many). Those are the mathematical equivalents of superstitions.

    What say you would-be cosmologists now?

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  47. Daniel Shawen says:

    Time doesn’t exist without motion. The Higgs mechanism is what determines the flow of time for all inertial reference frames as well as non-inertial ones like gravitation.

    There are other tantalizing hints that this is the case, too numerous now (in my mind, at least) to mention them all here. Several articles have appeared this last week that strongly hint at the same idea. This is not original; it just so happens, I’m schooled enough in mathematical relativity to recognize a great idea (and simplification) when I see it.

    It may also explain dark matter (but not necessarily dark energy). The Higgs field interacting with gravitating bodies is going to add a lot of mass which we can’t see because it’s virtual. It will most definitely show up in experiments with gravitational lensing, for the same reason ordinary mass does.

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  48. David Brown says:

    “What does the theory of inflation really predict? What physics does it assume?” Consider the possible that the theory of Newton-Einstein inflation and the theory of Milgrom inflation might be substantially different.
    http://www.scilogs.com/the-dark-matter-crisis/2013/11/22/pavel-kroupa-on-the-vast-polar-structures-around-the-milky-way-and-andromeda/

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  49. Bruce Baumgart says:

    Thank you for the download link to the Chapter 5 pdf. Figure 5.1 in color is a great improvement over my black and white copy of Tegmark’s book. However one minor suggestion, in the 2nd paragraph of section 5.2.3 replace the phrase “less than a kilogram of mass” with “less than a gram of mass” in reference to the energy released by the Hiroshima bomb which was likely 0.6 grams for the equivalent of 15 kilotons of TNT.

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