Defining the Big Bang

“As far as I see, such a theory [of the Big Bang] remains entirely outside any metaphysical or religious question. It leaves the materialist free to deny any transcendental Being. He may keep, for the bottom of space-time, the same attitude of mind he has been able to adopt for events occurring in non-singular places in space-time… Science has not to surrender in face of the Universe and when Pascal tries to infer the existence of God from the supposed infinitude of Nature, we may think that he is looking in the wrong direction.” -Georges Lemaître

Recently, David Dilworth has been getting some attention for his assertion that the Big Bang has no universally agreed-upon definition. In particular, the International Astronomical Union, of Pluto-demoting infamy, has no definition of it. He argues that the Big Bang is, perhaps, not even a scientific theory because of this, and quotes — out of context (Update 07/20/2011: not out-of-context; that was the complete statement) — famed cosmologist Jim Peebles.

David, the IAU, and anyone else reading, allow me. (And for those of you wondering about my pedigree, Jim Peebles was the Ph.D. supervisor of this excellent cosmologist, who was in turn my Ph.D. supervisor a generation later.)

We know, from many different observations, that the Universe is expanding. Not like an exploding grenade, mind you, where the pieces of it would fly apart, expanding away through an already-existing space.

Not like this.

Rather, thanks to general relativity, we understand that what’s going on is that our Universe itself is expanding. The space-time that makes up our Universe expands in proportion to the amount of stuff-per-unit-volume, or energy density, of our Universe. And this means something remarkable for the radiation in our Universe.

Because one of the defining properties of all radiation is its wavelength, and the Universe is expanding, this means that the expanding Universe gets cooler over time!

As the image above shows, as space expands over time, the wavelength stretches, or gets longer. This necessitates that the radiation (or light) reaches lower and lower energies as the Universe ages. But it also means that in the past, the Universe was hotter, and at higher energies!

Now, remember the first image I showed you?

This one. This was the original conception of what the “Big Bang” was. Originally given names like the “primeval atom” (in reference to the tininess of space early on), or the “primeval fireball” (in reference to the ultra-hot temperatures), the term “Big Bang” was actually coined in 1949 by the theory’s biggest detractor, Fred Hoyle.

But this description of the Universe — with the aforementioned tie between energy density to expansion — predicts some amazing things. Imagine, if you will, everything you know of in the Universe when it was younger. You know it was hotter, closer together (denser), and was expanding away from everything else at an increased, faster rate than it is today. What would it look like?

Image credit: Lancaster University.

Well, you might be tempted to go all the way back in time to what you think of as the very beginning. You might want to know what happens when you extrapolate all the way back to all of the matter and energy in the Universe being at a single, isolated point. I’m not going to lie; it’s very tempting to do. And if you do it, you’ll miss the Big Bang. It is the single greatest source of the public’s misunderstanding of what the Big Bang is, and I am here to try and clear things up. Let’s take a look at what we’d run into if we naïvely extrapolated back in time, to arbitrarily higher and higher temperatures and densities.

Image credit: ESA.

Before gravity did what it does, and collapsed matter into stars, planets, galaxies, and clusters of galaxies, the Universe was much hotter, and much more uniform. At some point in the past, it was so hot and dense that there were enough high-energy photons zipping around that it was too hot to form neutral atoms! So instead of atoms, we had a Universe made up of atomic nuclei, electrons, and radiation.

The radiation we see left-over from the time that electrons and nuclei first came together is the famous Cosmic Microwave Background: the observation that confirmed the Big Bang and ruled out the alternatives. We know many more details than this today, but we’ve known about this part of it since the early 1960s.

Image credit: V. Travieso, N. Turok, and WMAP/NASA.

But, of course, we can go back farther in time, to higher temperatures and more exotic happenings. Back to when it was hot enough that individual nuclei were blasted apart, into lonesome protons and neutrons, by the tiny-wavelengthed, incredibly hot radiation bath.

Image credit: CERN photo.

Back to when there was so much energy density to the Universe that all the particles we make in high-energy accelerators — such as unstable quarks, muons, neutrinos, and high-energy bosons — existed in great abundance.

And although that’s the energy limit of what we’ve studied, we know that there are even more things that had to have happened in our Universe.

We live in a Universe, today, with much more matter than anti-matter; at least 99.9% of the normal matter in the Universe is, well, normal matter and not anti-matter. For the very early Universe, that means that there was a very, very tiny overabundance of matter as compared to antimatter; a difference of only one extra baryon for every 10 billion baryon/anti-baryon pairs.

(What’s a baryon? A proton and a neutron are each examples of baryons. It comes from the Ancient Greek: bareus, which means heavy. If you were an Ancient Greek and wanted to call someone a drunk, you called them oinobareus, which means “heavy with wine.” Protons and neutrons contain nearly all the mass of atoms, and that’s why they call them baryons. Now, you’ll never forget it.)

So you have to have some way of making more baryons than antibaryons. This is an exciting sub-field of cosmology, known as baryogenesis. We have ways of making it work at a bunch of different energy scales, but we know it happened after the Big Bang, and before the physics we’ve presently discovered.

(And it could have happened at the electroweak scale, courtesy of The University of Heidelberg.)

And then there are the unknown things. Were the weak, electromagnetic, and strong forces all unified together at some high energy, after the Big Bang, that our Universe actually reached?

Image credit: CERN, Beyond the Standard Model.

Signs point to “no” right now, but we could be wrong about that. The important thing, in all of this, is that the Big Bang doesn’t go back infinitely far! Why not?

Because the Big Bang describes this hot, dense, matter-and-radiation-filled expanding state. But something happened to set that up.

Image credit: Ned Wright, with a possible contribution from Andy Albrecht.

Inflation! There was a time where the Universe wasn’t full of matter and radiation, where it wasn’t cooling as it expanded, and where the expansion rate didn’t drop at all, and that’s what we call the period of cosmic inflation.

Image credit: NASA/WMAP science team.

We don’t know what caused/came before inflation, and we don’t know how long inflation lasted (save to say, long enough). But inflation was the thing that happened before the Big Bang, and the Universe could not have been as small as a singular, collapsed point at that moment of transition.

So, with all of that in mind — which is the story every physical cosmologist knows — what can we define as the Big Bang?

The Big Bang is the first moment in the history of the Universe where we can describe it as a hot, dense, expanding state, full of matter, antimatter and radiation. It has a temperature of at least a quadrillion Kelvin (but no more than 1029 Kelvin), and it coincides with the time where inflation ends and the Universe’s expansion rate is dominated by the matter and radiation density.

If you go back to this picture, the GUT era may or may not be something that happened in our Universe since the Big Bang; the Planck era certainly is not. The Big Bang does not include inflation nor anything else that happened before it; it also did not occur at one particular place (although it did occur at one particular time, everywhere in space). It occurred roughly 13.7 billion years ago, and it occurred in a Universe that had the same temperature everywhere (to just a few parts in 105) and was (and still is) spatially flat. In the context of inflationary cosmology, the Big Bang coincides with the end of inflation and the cosmic reheating of the Universe.

And our Universe — everything that’s come since — is a consequence of that one event that started it all: the Big Bang! And that’s how I, in my capacity as a physical cosmologist, define the Big Bang.

Comments

  1. #1 Sphere Coupler
    June 6, 2011

    Excellent, but doesn’t the Big Bang start it all?

    Ha HA just kidding,don’t hit me…really no hitting

    *runs away*

    I would like to hear the ideas of what caused Cosmic Inflation,I know it’s driven by a negative-pressure vacuum energy density,I know it’s a inflaton scalar field, I know what I think, as many of you know what I think (lack of curvature/gravity) enabling a quantum fluctuation but what do others in the field think?

    Oh and someone asked me what energy was the other day and the answer I gave was;

    Energy creation is the opposite of singularity creation.

    Any action within the boundaries of the expanded Universe of the non singularity mass induced phenomena that can be located in a coordinate system constitutes energy.

    Perhaps someone can show me where this is lacking in explanation, and if it’s in the wiki article I already know about it.

    Ethan, If you want to make people WAKE UP, your gonna have to do something drastic…might I suggest adding to your blog title? STARTS WITH A BANG Cosmic Inflation

  2. #2 Andrew
    June 6, 2011

    I’m confused – according to Wikipedia, cosmic inflation came shortly after the big bang. Obviously I’m not suggesting Wikipedia is the ultimate authority on cosmology, but I’m surprised that something so fundamental as this is incorrect, or have I misunderstood?

  3. #3 Andrew
    June 6, 2011

    I’m confused – according to Wikipedia, cosmic inflation came shortly after the big bang. Obviously I’m not suggesting Wikipedia is the ultimate authority on cosmology, but I’m surprised that something so fundamental as this is incorrect, or have I misunderstood?

  4. #4 Andrew
    June 6, 2011

    Just discovered some of the other posts which address this point, so feel free to ignore my (double-posted) question above.

  5. #5 fancyflyer
    June 7, 2011

    So…huff huff HUFF POP! Is that it?

  6. #6 Petter Häggholm
    June 7, 2011

    This post, and others, and many other resources on cosmology all refer to the GUT phase, where the fundamental forces were united. But what does this mean? I have a layman’s vague idea of what the electromagnetic and weak nuclear forces are: but what would the electroweak force be? And what about the “GUT” force? Not least puzzling when electromagnetism can be repulsive whereas, as I understand it, the other forces are all attractive.

  7. #7 Bjoern
    June 7, 2011

    @Peter Häggholm: First, the easy part: no, the other forces are not “all attractive”; they can be repulsive as well. The only force which is all attractive is gravity. (well, if we ignore dark energy ;-) )

    Second, the hard part: how to picture this unification? I think it’s easier if you don’t think directly about the forces, but instead about the particles on which they act. For example, electrons interact electromagnetically and by the weak force, whereas neutrinos interact only by the weak force; that’s one way (beside the masses) in which we can distinguish these particles. At the unification scale of the electromagnetic and the weak force, this distinction breaks down: electrons and neutrinos react in exactly the same way to each other then.

    Does that help?

  8. #8 The Tim Channel
    June 7, 2011

    We live in a Universe, today, with much more matter than anti-matter; at least 99.9% of the normal matter in the Universe is, well, normal matter and not anti-matter.

    Is this the same 99.9% of ‘normal’ matter that makes up the five percent of stuff we can see….as opposed to the ‘abnormal matter’ of dark energy and matter that supposedly makes up 95% of the universe? I guess I’m a bit confused here because what I understand from a layman’s viewpoint is that, at the end of the day, astrophysics is missing 95 percent of what their math insists MUST be there, yet there is NO experimental or observational proof of this new-age ether. My point is only that it’s scientific jujitsu to refer to what we can see and measure of the universe as ‘normal’, when the same science depends on the existence of a HUMONGOUS dark energy fairy to keep the undeniable cosmological observations consistent with (e.g.) known gravitational mathematics.

    I’m not smart enough to know why the bulk of the research seems to favor searching for this mysterious dark force over the search for a model of malleable gravitation. I’ll bet you do.

    Enjoy.

  9. #9 Bjoern
    June 7, 2011

    @The Tim Channel: Before writing nonsense about “new-age ether” and asserting that “there is NO experimental or observational proof”, try reading Ethan’s article series on the topics of Dark Matter and Dark Energy. Ignorance is curable in general – arrogant, proud ignorance usually isn’t…

  10. #10 John M
    June 7, 2011

    Thanks, Ethan, for all your great work. I have learned so much from you.

    I still have trouble really understanding the nature of the universe’s expansion. If space-time is expanding that means any unit of measure is “expanding”, too. But our perception is embedded in that space-time: a meter is still a meter and should always look like the same meter to an embedded observer. So space-time is expanding *in relation to what?* And how does red shift occur? Light was emitted from a distant star with a specific wavelength measured in meters, and a meter is still a meter to us observers, so how did the wavelength shift? It seems like there is some other frame of reference I’m missing.

  11. #11 The Tim Channel
    June 7, 2011

    @ Posted by: Bjoern | June 7, 2011 6:50 AM

    I read it a couple days ago. It said that galaxies would rotate apart without some mysterious force many factors stronger than gravity holding them together. The mysterious force/ether is not identified (even if the mystery ether is named)

    Maybe in places where they vowels are optional, they’ve already figured everything out about his mysterious ether, but from what I read, they haven’t exactly been able to pin it down yet. Some folks who aren’t as proudly ignorant as me (as in proper astrophysicists) aren’t in agreement either, so you can’t get off my balls ok Bjooeane.

    Because your first name indicates a possibility that the best English you speak is likely math, I guess I ought to point out that I’m using ‘ether’ as a satirical element in these posts. I’m guessing….northern country….cold and sad….filled with people of no humor.

    Enjoy.

  12. #12 Bjoern
    June 7, 2011

    @The Tim Channel:

    I read it a couple days ago. It said that galaxies would rotate apart without some mysterious force many factors stronger than gravity holding them together.

    Since that is not what Ethan’s articles where actually saying, we are left with two choices: (1) You haven’t actually read the articles, or (2) you have severe reading comprehension problems. Which is it?

    The mysterious force/ether is not identified (even if the mystery ether is named)

    That we still don’t know what Dark Matter consists of is no reason at all to call it “ether” (even not satirically) or a “mysterious force” (you don’t even know what “force” actually means in physics – right?). And it doesn’t change the fact that there are many lines of evidence that it actually exists – as Ethan has explained in great detail. (rotation curves of galaxies is only one of many pieces of evidence – that you only mention that single one, and botch even that badly, says a lot about your knowledge…)

    Oh, and that you go on by using ad hominems says even more about you… (a little help for you: the guess “northern country” was wrong)

  13. #13 John M
    June 7, 2011

    @Bjoern: ever heard of a troll? He’s not worth it.

  14. #14 Bjoern
    June 7, 2011

    @John M: Thanks for the hint. Sometimes, the line between genuine trolls and simply people with strange ideas is rather thin… ;-)

  15. #15 rob
    June 7, 2011

    Ethan wrote: “David, the IAU, and anyone else reading, allow me. (And for those of you wondering about my pedigree, Jim Peebles was the Ph.D. supervisor of this excellent cosmologist, who was in turn my Ph.D. supervisor a generation later.)”

    classic argument from authority! lol.

    great post!

  16. #16 Mu
    June 7, 2011

    One question so on this timeline, where does the separation of baryonic matter and dark matter happen? Dark energy, when seen as a form of vacuum pressure per unit space, is actually easier to understand than the appearance of dark matter somewhere without totally throwing off the energy content of the universe.

  17. #17 Bjoern
    June 7, 2011

    @Mu:

    One question so on this timeline, where does the separation of baryonic matter and dark matter happen?

    That depends on what dark matter actually is. If it consists of neutralinos, then the “separation” happens at the super-symmetry-braking scale, which should be somewhere between 100 GeV and 1 TeV.

    (oh, and I don’t understand what you mean with “totally throwing off the energy content of the universe”)

  18. #18 Randy Owens
    June 7, 2011

    So, when will we see a post about “the best bang since the big one”?

  19. #19 Mu
    June 7, 2011

    Bjoern, at some point in the timeline, the universe drops 80% of the available energy into the dark matter hole if the 5% baryonic 20% dark is right. That should show up somewhere.

  20. #20 Ethan Siegel
    June 7, 2011

    Mu & Bjoern,

    This is an interesting back-and-forth the two of you are having. (Don’t stop on my account.)

    I have the next 3 or 4 posts planned out, but I will take on your question, of where dark matter comes from in this picture, after that.

    Best,
    Ethan

  21. #21 John M
    June 7, 2011

    A friend points out that Ethan tried to answer my question above about measurements and frames of reference and expansion in a previous post: http://scienceblogs.com/startswithabang/2009/08/distances_position_and_motion.php.

    I’m still confused, though. In the expanding Earth metaphor, New York and Los Angeles *are* moving – away from the center of the Earth. They are “moving” (changing their space and time coordinates) in the coordinate system of the Earth at the starting time of the measurement.

  22. #22 OKThen
    June 7, 2011

    I’m not sure that anything has a “universally agreed-upon definition”. But the big bang theory is the standard model (give or take a nuance).

    “standard model.. of cosmology signifies (that it) is a theory that is considered well established because it has survived non trivial tests… The use of the word “model” is appropriate because the picture is known to be an incomplete approximation to what is really happening, and there certainly is the chance that there is something very wrong with the picture.” Peebles, Principles of Physical Cosmology 1993, pg xvi.

    Yes the standard model (i.e. the big bang theory) of cosmology has withstood some very tough (i.e. non trivial) test. Furthermore, it is a focal point driving important research. Yes the definition is moving and wiggling and not perfect; but in an active field of knowledge what definition isn’t shifting. Definitions petrify only when the idea is dead.

  23. #23 daedalus2u
    June 7, 2011

    Ethan, Could the excess of matter over antimatter be just a fluctuation that got inflated so that is all we see?

  24. #24 Kristofer Bergstrom
    June 8, 2011

    I have never fully understood why the universe’s expansion causes frequency shift in light. If matter is being stretched along with light, wouldn’t the light appear the same to the stretched observer? I mean, if my ruler stretches along with the thing being measured, how do I measure a change?

    I assume the answer has to do with the fixed maximum speed of light but I can never quite wrap my head around this.

    Thank you for all the great writing, Ethan!

  25. #25 Bjoern
    June 8, 2011

    @Kristofer Bergstrom: The answer is simple – matter isn’t “stretched along with light”. Only the distances between large agglomerations of matter (galaxy clusters) is stretched.

  26. #26 Bjoern
    June 8, 2011

    @OKThen: If I understood Ethan correctly, this post was not about defining “big bang theory”, but about defining “big bang”. (and BTW: he uses a definition which I haven’t seen anywhere else so far)

  27. #27 Bjoern
    June 8, 2011

    @Mu:

    Bjoern, at some point in the timeline, the universe drops 80% of the available energy into the dark matter hole if the 5% baryonic 20% dark is right.

    I don’t understand what you mean with that. You seem to think that Dark Matter appears suddenly in the evolution of the universe out of nowhere, or what? That simply isn’t right! Dark Matter particles are there right from the start, alongside the normal matter. (well, if they are neutralinos; this could be different in more exotic models with which I’m not familiar)

  28. #28 Mu
    June 8, 2011

    I don’t think DM comes out of nowhere, that’s the point. At some point between GUT and nucleosynthesis, 80% of the available energy condenses into DM, and that is the step I’m trying to get an feel for how the model accounts for that. Unless DM doesn’t obey E=mc^2, in which case the energy balance is unaffected.

  29. #29 Bjoern
    June 8, 2011

    @Mu: I’m not sure what you mean with “the energy condenses”. The particles are already there all the time, they don’t appear suddenly at a “point between GUT and nucleosynthesis”.

    And even if you were right and the energy really “condenses” at some point, I still don’t understand why you think that would affect the “energy balance” in any way. The energy which is already there simply takes another form; what’s your problem with that?

  30. #30 John M
    June 8, 2011

    @Bjoern: You replied “Only the distances between large agglomerations of matter (galaxy clusters) is stretched.” Ethan said something similar but I thought he meant the effect occurs at small distance scales but is not *noticeable*. Are you saying expansion is discontinuous across distance scales?

    And regardless of that, it still doesn’t answer the question @Kristofer Bergstom and I asked: if the coordinate system itself is expanding and distant galaxies are not “moving” away from us, how can redshift occur?

  31. #31 Mu
    June 8, 2011

    Ok, lets try to describe it a different way; the model was established without DM. DM was introduced not due to problems with the early development of the universe but due to problems in model vs observation today. So s the model was able to absorb a 500% increase in energy at t=0, before there were particles, and still develop in the same way as before. Now, you can postulate the excess energy gets removed from the equation very rapidly by what I called condensation into DM since the DM doesn’t interact for the most part. When you get a second issue, with DM you have 5 times the gravity for the same amount of radiation pressure driving the particles apart, with gravity acting at very short distances and being extremely powerful that way.
    It might be intuitive to you and not a problem whatsoever, to me it looks like that the development of the universe is independent from the starting energy.

  32. #32 Bjoern
    June 8, 2011

    @John M:

    Are you saying expansion is discontinuous across distance scales?

    Essentially, yes. The usual argument goes like that: small objects (say, a table) are hold together by electric forces, large objects (planets, solar systems, galaxies) by gravitational forces – hence these things don’t expand. Only on really large scales (galaxy clusters) the counter-acting forces are so small that expansion takes the lead.

    if the coordinate system itself is expanding and distant galaxies are not “moving” away from us, how can redshift occur?

    Sorry, I don’t understand your problem. When space expands, the wavelength of radiation is stretched, that’s all.

  33. #33 Bjoern
    June 8, 2011

    @mu:

    Ok, lets try to describe it a different way; the model was established without DM.

    That’s a bit vague – what “model” do you mean exactly? An open universe with about the amount of “normal” matter we see today and nothing else there, or a flat universe with only normal matter, or what?

    So s the model was able to absorb a 500% increase in energy at t=0, before there were particles, and still develop in the same way as before.

    No, that’s wrong. The development of the universe depends on its energy content. If one considers a universe with only the known amout of normal matter, the development is totally different than the actual universe. I don’t know where you got the idea from that the universe “still develop[s] in the same way as before” (whereas “before” probably refers to the model without dark matter, i. e. essentially an open universe)

    Now, you can postulate the excess energy gets removed from the equation very rapidly by what I called condensation into DM since the DM doesn’t interact for the most part.

    I don’t understand why you think that the “excess energy get removed” by “condensation into DM”. If the energy “condenses” into DM, it is still there (only in another form), it is not “removed”!

    When you get a second issue, with DM you have 5 times the gravity for the same amount of radiation pressure driving the particles apart, …

    Well, yes. That’s why models with DM behave totally different than models with only the known normal matter!

    It might be intuitive to you and not a problem whatsoever, to me it looks like that the development of the universe is independent from the starting energy.

    Sorry, I still don’t understand how you arrive at that conclusion.

  34. That depends on what dark matter actually is. If it consists of neutralinos, then the “separation” happens at the super-symmetry-braking scale, which should be somewhere between 100 GeV and 1 TeV.

  35. #35 JH
    June 8, 2011

    OK, amateur hour here:

    If the universe has a limit that is described similarly to the surface of an expanding balloon, it is relative simple to comprehend what is inside the balloon (everything about which we are aware).

    So what, exactly, does theory say about what exists on the other side of the balloon’s surface?

    Or if this is a question that causes professionals to burst into laughter, can someone let me in on the joke?

    JH

  36. #36 Sphere Coupler
    June 8, 2011

    JH, Your taking the analogy too far, All the analogy shows you is the topographical expansion where matter is separated by the expanding rubber, It is not meant to describe inside and outside.

    You are suppose to see the topological exspansion and relate that to all space.

    Remember analogies are not meant to describe a phenomena in totality, if they were…they would not be analogies, they would be literal explanations.

    No joke.

  37. #37 Mark
    June 8, 2011

    @Bjoern

    I understand what John is asking regarding redshift. Look at the diagram showing the balloons, where the wave is shown blue at small scales, then red at the largest.

    The point is: it’s still passed through the same number of squares on the spacetime grid (in the diagram: three). As far as the photon is concerned, nothing changed.

    Is the diagram lacking clarity there?

    Still can’t get my head around why the expansion doesn’t just result in the same-looking blue light taking longer to reach us than it otherwise would have done in a static Universe – nor where the extra energy goes as it redshifts.

    I shall now mine more of Ethan’s posts and find out!

  38. #38 Harlequin
    June 8, 2011

    @Mark

    The point is: it’s still passed through the same number of squares on the spacetime grid (in the diagram: three). As far as the photon is concerned, nothing changed.

    That’s one way to think about it. But at the same time, the size of, say, the Earth isn’t changing as the spacetime grid expands. So the ruler used on the Earth (the wavelength of light given out by certain atomic transitions, for example) and the ruler seen by the photon (distance between non-gravitationally-bound galaxies) are getting more and more out of sync. That’s why the photon looks redder to us.

  39. #39 Bjoern
    June 9, 2011

    @Mark:

    The point is: it’s still passed through the same number of squares on the spacetime grid (in the diagram: three). As far as the photon is concerned, nothing changed.

    It’s difficult picturing this with photons – just look at the wave. The wavelength is obviously longer, hence the light is redder. That’s all.

    Still can’t get my head around why the expansion doesn’t just result in the same-looking blue light taking longer to reach us than it otherwise would have done in a static Universe

    That’s an additional effect: it takes the light indeed longer to reach us.

    nor where the extra energy goes as it redshifts.

    If you look at this from a Newtonian standpoint, you could say that the light has to travel up a potential well (against gravitational forces of the galaxy cluster where it came from), and hence its energy goes to gravitational potential energy. But that explanation isn’t very good (even in Newtonian gravity, in a homogeneous universe, there shouldn’t be any gravitational field in the mean!). The standpoint of the General Theory of Relativity is simple this: energy is not conserved in this case!

    http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

  40. #40 Bjoern
    June 9, 2011

    @JH:

    If the universe has a limit that is described similarly to the surface of an expanding balloon, …

    I think you misunderstand the analogy here: the surface of the balloon is not analogous to a “limit” of the universe; the surface is the universe (at a certain time).

    You could think of the stuff “inside” the balloon as all things that happened in the past and the stuff “outside” as all thing that will happen in the future – but that already means taking the analogy quite far…

  41. #41 Suzanne
    June 9, 2011

    What, no mention of the M-theory??

  42. #42 OKThen
    June 9, 2011

    Bjoern
    Regarding definition; yes, yes. But I think Ethan is trying to limit “big bang” to the astronomical range of observable phenomenon and leave the more speculative unobservables (e.g. inflation) out of the term “big bang”. Even if Ethan’s definition is new; he is not redefining phenomenon; he is trying to clarify. Ethan’s definition seems in sync with the standard model (nitpicking aside). Either his definition will catch on or not.

    As you know, I have put my skepticism aside for a while. The important thing is to
    1) understand current phenomenon clearly within the context of various theories
    2) seek decisive new phenomenon suggested by various theories.

    As a skeptic, I care most about new phenomenon which may overturn current theories.
    e.g. are the extra-dimensions of string theory testable? or
    e.g. can gravity resonance spectroscopy with neutrons be done; thus giving deeper insight into quantum gravity without high energy experiments
    e.g. which astronomical phenomenon are informed by the gravitomagnetic clock concept of general relativity. or
    e.g. is time an emergent phenomenon

    Until then..

  43. #43 Mark
    June 9, 2011

    Great answers, thank you guys.

    (Total amateur stumbles on actual genuine mystery in Physics…? My life is complete… :) )

    So here’s my problem. I understand that gravity rules at small scales, but what does that actually mean? Does it prevent expansion entirely, or just compensate for it after the fact?

    In the grid-on-balloon analogy, gravity could be acting like a particularly thick patch of rubber, keeping its own grid smaller than the surroundings.

    Or the grid could be expanding uniformly, everywhere, which would mean physical objects are literally taking up less units of space as time progresses.

    (Not sure I like either analogy, but I hope you get my meaning.)

  44. #44 Sphere Coupler
    June 9, 2011

    The initial momentum velocity energy of cosmic inflation is converted into the big bang and is sustained by a slight deficiency of virtual particles, this deficiency being the negative-pressure vacuum energy density of space.

    My conjecture is that;
    These virtual pairs are created within a black hole(of a mass greater than) at the point of singularity attempt and are propagated by quantum tunneling only to appear as a quantum fluctuation in an attempt to equalize space expansion, which is not possible due to the momentum of space itself and since space continues to expand and not equalize, a negative pressure of the vacuum energy density is set up again and while it is equal for that briefest of moments the singularity attempt is tried again, only to create more virtual particles to fill the newly created space and thus this is the engine that continues the great expansion that we see today.

    In this scenario it is mass, that is the fuel that feeds and reduces the expansion of space.
    The rate of virtual particle creation = (is almost equal to)accelerated Dark Energy appearance.
    The reason how space expands in all direction is because we judge expansion by detecting mass and it is that mass (BH) that emit virtual particles to reduce the expansion.

    Then since Dark energy virtual particle creation is finite (in this scenario), the Universe, due to rapid increasing negative- pressure vacuum energy density upon Black hole depletion, might once again inflate wildly and then………………..BANG

    This scenario leads to some really amazing realities, I think there is more to blackholes than even the greatest of physicist can tell us…Hmmmm

  45. #45 Bjoern
    June 10, 2011

    @Mark: The usual analogy to galaxies etc. is coins which are glued to the surface of the balloon. Does that help?

  46. #46 Jon
    June 10, 2011

    Is it correct to view the Cosmic Microwave Background as “light” that has “redshifted” so far down the electromagnetic spectrum that it now presents as the faintest whisper of radiation? And we can know nothing empirically of what lies beyond because any light there may have been is too far away to reach us? So all the models beyond that horizon are an imaginative stew of physics and speculation? If so, how much faith should we put in them? Can they reasonably be called science? I’m not suggesting that the calculations are senseless, but all they really yield is a model that tells us about our present paradigm, and not insight into a past that is beyond our reach. Or so it seems.

  47. #47 Bjoern
    June 10, 2011

    @Jon:

    And we can know nothing empirically of what lies beyond [the Microwave Background radiation]because any light there may have been is too far away to reach us?

    You neglect to consider that there are other possibilites than only light for making observations. One method we already have is studying the abundance of light elements – that takes us back to a few minutes after the Big Bang (whereas the CMBR was emitted 380 000 years after the Big Bang!). Another possibilites is studying the “ripples” in the CMBR and trying to analyze where these came from; that takes us back to small fractions of a second.

    Other possibilites which we so far not have, but which probably will be available in the future, are neutrinos from that early time, and gravitational waves.

  48. #48 Mark
    June 10, 2011

    @bjoern

    Yes, that analogy does nicely. If they’re glued on, then that suggests gravity is actually stopping the local space expanding, and not pulling matter into ever-decreasing volumes. Fascinating stuff.

    Maybe the big rip will have some edges.

  49. #49 steve
    June 11, 2011

    We know, from many different observations, that the Universe is expanding. Not like an exploding grenade, mind you, where the pieces of it would fly apart, expanding away through an already-existing space.
    ————————–
    How can you be sure that there is no space outside our universe if it is impossible to ever see outside our universe?
    Why not just say you don’t know what if anything is outside? Can you explain why our universe must contain all possible space,in other words.

  50. #50 Jon
    June 11, 2011

    Point taken, Bjoern, we can “see” a bit further than the light horizon. But are you not at all troubled by the belief that the cosmos began just where our data points drop off? Or is the technology such that we could gather data from further off/longer ago if there were any data to gather? Put another way, in a half million years or so (a mere wink by cosmic standards) will the origin of the universe have become undetectable? if so, the lack of remark on that truly bizarre fact is in itself interesting, no?

  51. #51 Bjoern
    June 11, 2011

    @Jon:

    Put another way, in a half million years or so (a mere wink by cosmic standards) will the origin of the universe have become undetectable?

    ??? Why on Earth should it?!? We will always be able to see the CMBR, which comes from 380 000 years after the BB. We can always look at the abundance of light elements, which arose some minutes after the BB. We can always study the ripples in the CMBR, which arose a fraction of a second after the BB. And so on. I really don’t understand your question…!

  52. #52 Sphere Coupler
    June 12, 2011

    Bjoern, Can you clarify? Can you source your response?

    http://www.youtube.com/watch?v=7ImvlS8PLIo

    Save time…Skip to 50:46

  53. #53 Bjoern
    June 12, 2011

    @Sphere Coupler: You probably refer to Krauss’ argument that the CMBR won’t be able to permeate galaxies in the distant future (because it will then have redshifted so much that it is below the plasma frequency)? Well, that seems a sensible argument – I was not aware of that, sorry. But nevertheless, the CMBR will still be measurable outside of galaxies even then…

  54. #54 Sphere Coupler
    June 12, 2011

    Not that I consider Krauss’ or anyone else as the final authority, I think it is the scientific concensus that given time and the continued expansion a void will develope around the super clusters of galaxies, this does not mean that some manner of detection won’t be possible (I think)only the known detection abilities that we now possess, I am unaware of any detection possibilities of forgone redshifted mass, other than perhaps some type of Dark Energy density compensation status, and even that would be years from developement,

    “But nevertheless, the CMBR will still be measurable outside of galaxies even then…”

    I’ll take this to mean you are leaving possibilities open?

  55. #55 Sphere Coupler
    June 12, 2011

    Mu,
    Com 31
    as for your comment on Dark Energy reduction by possible Dark Matter induction, (which is a very interesting thought) One would have to take into consideration the continued momentum of space expansion to also momentarily create the negative-pressure vacuum energy density/Dark Energy reduction.

    You said;
    “It might be intuitive to you and not a problem whatsoever, to me it looks like that the development of the universe is independent from the starting energy.”

    This is a profound statement, one that I do not disagree with,However I do think that the momentum(cosmic inflation event) of the separation of mass at the start of the Actual Universe, (not the start of the Apparent observable Universe/the Big Bang).

    Let me start again…However I do think that the momentum of the separation of mass at the start of the Actual Universe is the primary function that gives energy it’s meaning.

    John M
    Com 30 and Bjoern response 32

    “if the coordinate system itself is expanding and distant galaxies are not “moving” away from us, how can redshift occur?”

    Bjoern’s response,(which I agree with)
    “Sorry, I don’t understand your problem. When space expands, the wavelength of radiation is stretched, that’s all.”

    I think the confusion comes from the poorly drawn picture of the four balloons by Addison Wesley in which the size of the squares is increased along with the red shifting of the wave length, I believe this picture does not express the continues Dark Energy Cosmological Constant view of today, if it did, then there would be more squares progressivly in the 2nd 3rd and 4th picture and the squares would be the same size as in the first picture, thus not only showing the stretching of the waveform, but also showing the expansion of space and the increase of the coordinate system.

    The coordinate system expands and Dark Energy fills it,so that the coordinate system in effect increases by adding space in the form of more squares, in other words the energy density tries to remain consistent, the squares in the picture do not really stretch…they multiply.

    The waveform is streched as it has to move through more coordinate zones than it did previously.

  56. #56 Sphere Coupler
    June 12, 2011

    John M,
    The distant galaxies are actually moving through space yet it is trivial/not much compared to expansion, yet they gain greater distance from one another due to increase of space between them.

    Now let me muddy the waters a little bit, Andromeda and the Milky way are actually moving towards each other in an expanding space, this is due to proximity (their is not enough space between the two to counteract the merging).

    So distant galaxies gain distance by the expansion of space because of their distance their is more space to expand and thus becomes a primary mover, rather than mass attraction in a local merger with less space.

  57. #57 Sphere Coupler
    June 12, 2011

    There are many type of singularities concepts in our cognition abilities their are mathematical, geometrical, Mechanical, and Gravitational to name just a few.

    It has been stated that the mathematical singularity of a Black hole leads to a undefinable product, I think this is a failure of mathematics and not an indication that their is no action at this point of singularity attempt.

    if (and that’s a very small if) virtual particles are created at the point of attempted singularity of a Black hole and if they propagate to the easiest path to the weakest point (space) then Dark Energy is a singularity of the combined influence of attempted Black hole singularity and the momentum of space.

    This would then explain where the DE comes from and it would tie the expansion rate to the coalescence rate and it would explain why the rate is increasing with time (more black holes).

    A singularity comes in several forms, it can be point like or it can be a homogeneous quality, As one would travel from the vastness of space towards the Attempted singularity of a Black hole, he would realise that the curvature of space becomes extreme and forces matter to be infinitely compressed (that is to say that it can not be more compressed than at this point), Mathematical this point becomes undefinable though at every neighboring point it is definable, it is that comparison to the neighboring states the attempted singularity is not the same, it is distorted from the neighboring states, it becomes a state where all defining qualities such as color, spin, charge and many others become Same and undefinable to non-singularity matter.

    Dark Energy consist of at least two qualities momentum and a resulting overall negative derivation due to the lag of the ability of the overall system to compensate the expanded space with the induction of virtual particles/DE.

    This may be the limit of physical mathematical capabilities

    The Universe is a system that is driven by it’s momentum, mass moves, all mass moves all the time, when you slow mass it becomes energy, as an example when an atomic bomb releases energy from mass it is from the attempted singularity, only for atoms, close proximity is good enough, when mass is forced to slow down energy is released and the more mass the greater the difference in states, a fire cracker is held tightly in a confined space, if too loose it will fizzle, yet more intensive energy will result the tighter(the closer to singularity attempt)it is wrapped.

    A black hole singularity attempt is tightly wrapped, it is the densest,tightest, place that can exist in the system, the energy produced has abilities that enable it to tunnel through matter seeking the weakest point in the system in an attempted equilibrium and would stop the expansion were it not for momentum and the coupling aspect between matter and energy.

    Black hole virtual particle possible production increase should be almost equal to the increase in rate of expansion.

    Dark Energy comes from BH Singularity attempt.

    Where should we look?
    We should look at the weak points within our galaxy, points where energy is held at the minimum.

    How does mainstream science confirm or reject this?

  58. #58 Sphere Coupler
    June 12, 2011

    OK, one more,

    Take Andromeda and the Milky Way for example, they are on a path to merge in the distant future, their Black holes produce DE, why would they not be pushed apart, because the amount of expanded space between them is a low power compared to the close proximity of two extreme curvatures/gravity. They still produce DE, but the weakest point does not lie between them due to space pressure, the DE creation points must be greatest between the most distant objects, this is why the further objects recede the fastest, another way of saying the weakest point that is likeliest for DE to be induced in is the greater distances of space.

  59. #59 AngelGabriel
    June 13, 2011

    A long time ago two magicians argued: Why when a dead fish is placed in a full bucket of water, the bucket overflows; but when a live fish is placed in a full bucket of water, the bucket does not overflow?

    The king listened patiently to his two magician advisors arguments. Finally impatient, he ordered two buckets of water, one live fish and one dead fish. The magicians protested; but the buckets and fish were brought. The dead fish placed in one full bucket, the live fish placed in the other full bucket. Both buckets overflowed.

    May I suggest that we stick to “hypothetical fish” and “hypothetical buckets” where the “predicted overflow of water” can actually be measured or not.

  60. #60 Sphere Coupler
    June 13, 2011

    AngelGabrial
    (nice moniker)

    It is thinking like that, in your whimsical tale that tend to discourage humans from seeking the hard answers, questions and critical thinking see beyond your pessimistic fatalism, If mankind followed your logic of (this can never be measured …why try?)we would not have become as we are now, a maturing intelligent species with the ability to understand and work within our changing environment.

    Your tale and your message are deceiving and counterproductive, though I want you to understand that with the proper tools/detectors/sensors we will measure and understand the system in which we reside.

    Perhaps you could explain why you protest intellectual progress?

    “There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.”
    — Douglas Adams, Hitchhikers Guide to the Galaxy

    Do statements like this scare you?
    Have no fear…it’s only a book.

  61. #61 Jon
    June 13, 2011

    Bjoern “We will always be able to see the CMBR, which comes from 380 000 years after the BB. We can always look at the abundance of light elements, which arose some minutes after the BB. We can always study the ripples in the CMBR, which arose a fraction of a second after the BB. And so on. I really don’t understand your question.”

    I am at or beyond my ability to express the issue, but I’m trying to relate the information horizon (which I thought was when light redshifted out of the electro-magnetic spectrum – guess that is not entirely right but you catch my drift) and the weirdness of our locating the cosmic origin so close to the sphere of the knowable. No matter how impeccable the math and the physics it still seems remarkable and noteworthy yet not much remarked or noted.

  62. #62 Bhangmeter
    June 15, 2011

    The BBC documentary series Horizon recently (2008) produced a show on this subject. “The Battle for the Beginning”. It has some interviews with the people you mentioned in the article.

  63. #63 Curious George
    June 15, 2011

    “The scientific paper is available here, however, they do make an assumption here, that if it’s false, invalidates their conclusions. Their assumption? That the potential that gives rise to inflation is a polynomial of the form:

    V(x) = A + Bx + Cx2 + Dx3 + Ex4.

    Is this a good assumption? No. It’s a shame that you have to be not just a scientist, but a scientist well-versed in inflation theory to realize that a big sweeping claim like this is probably wrongheaded. But now you know something that probably only a few thousand people in the world know!” How many assumptions like this are in the big bang theory?

  64. #64 P
    June 15, 2011

    Can someone explain why the concept of “time after the big bang” is meaningful? Presumably the assertion is that space-time is equipped with a submersive projection to the real numbers (time) which is positive on the positive time cone. This seems to me like an assumption, not something measurable or provable. But without it why should the Big bang make sense?

  65. #65 Kevin Alvey
    June 16, 2011

    Fusion at its grandiose scale
    Could a black hole’s singularity be so dense that nothing really enters but if contact is made it separates the -/+ atoms as well as creating dark matter? Which, might explain our expanding universe (e.g. if all of the negative atoms showed up like –+-+–+- to infinity). Could this be the reason it would be too dense to actually swallow anything but, instead create the space around us? I would also like to state that if the world’s largest hadron collider actually gets turned on & does what is supposed to do, it might create a black hole that would not suck us in but would however be too heavy for our space. Then, it would basically just rip us apart by its massive density like an impenetrable piece of matter expanding from within. The reason that the expansion of the black hole doesn’t envelope everything is that the space it creates stops it from doing so. For example, if somebody tried to hold up an anvil with a sheet of paper it would most certainly break through the paper. As for this anvil it is infinitely dense so it would continue to swell & create the matter around it. I also think that’s why when a supernova happens a black hole is created; to help in the creation of new stars as well as the space for those stars.
    Additional Details
    *I am saying that upon the primal stages of the black hole’s birth it grabs the closest objects to them thereby destroying them in the process. Now at some point after this it will cease its gravitational pull on the surrounding objects at this point in time it will then start in the creation of dark matter.

    *This is only a theoretical question for open minded individuals.

  66. #66 AngelGabriel
    June 16, 2011

    Sphere Coupler

    I can not tell whether the king or the magician is a “pessimistic fatalist”; but I am not.

    You presume incorrectly that my “logic” is “this can never be measured …why try?”

    But you have a point; certainly Democritus asserted the existence of the atom long before there was any chance of measuring it.

    “Perhaps you could explain why you protest intellectual progress?”
    Now really sphere Coupler, where did you ever get this idea?

    Let me assure you, Sphere Coupler, I do hitchhike around the galaxy.

  67. #67 Sphere Coupler
    June 16, 2011

    By referring to the ones who question the world around them as *magicians* you lower the role and the stature of the critical thinking person (the theorist).

    By elevating experiment to the level of King you raise the importance above any other part of the scientific method.

    And this line of reasoning falls in with your choice of Moniker.

    While part of the double speak draws attention to the importance of experiment, the beginning portion of your story derails the importance of critical thinking or just basically questioning the world around oneself.

    From this one comment 60 it looks like your message is to be a sheep.

  68. #68 Sphere Coupler
    June 17, 2011

    Something to consider;

    The relativistic jets emanating from a super massive black hole are the easiest path and will be point towards the weakest point of dark energy or in other words towards the greatest distances of mass in the system.
    To prove this, one would need to measure the number of black holes, their movement, their size, their capabilities to accrete/grow(available mass/food source) and compare this to the effects of DE.

    Question; do black hole jets face towards the weakest points for Dark Energy to be deposited?

    If we find any black holes that have jets facing each other within a super cluster galaxy, that are not influenced to do so by other members in the super cluster, overriding accretion momentum, then this hypothesis is dead.

    My conjecture is that the jet emitted from a galaxy center black hole is the path the virtual particles take to escape the attempted singularity, and the alignment although influenced by the surrounding environment will point as close as possible to the weakest point in space/the farthest distance to mass available, of course this could be seen easier from an isolated galaxy.

    I speculate that the jet directions from AGN of super massive black holes are controlled by several inputs and short listed here, gravitational potential of super cluster galaxy, angular momentum of accreted material, weak space potential of the entire Universe.

    My focus for the cosmology and astronomy community is to show any galaxy within a super cluster of galaxies that has a jet facing towards another galaxy of equal or greater mass within the same super cluster that is not overly influenced by members of same super cluster.

    I would go on to say that the majority of these super massive black hole relativistic jets emanating from super clusters or if found independently, the jet beam will face the weakest/ farthest distance between mass/the most expanded space.

    Quasars should be farther away than Seyfert galaxies.
    The weakest DE vacuum space energy density is indicated by quasars
    The fastest expanding space will be indicated by quasars, relevant also to their distance of course.
    Quasars should show a very high red shift.
    Quasars should be more numerous than Seyfert Galaxies.
    Some observable jets should be seen to change direction(with no local input to change) indicating a dynamic changing Universe.

    I would love to see a post on relativistic jets and their positions in space…

  69. #69 Kevin Alvey
    July 13, 2011

    Fusion at its grandiose scale
    Could a black hole’s singularity be so dense that nothing really enters but if contact is made it separates the -/+ particles as well as creating dark matter/dark energy? Which, might explain our expanding universe (e.g. if all of the negative particles showed up like (- – + – + – – + – to infinity). Could this be the reason it would be too dense to actually swallow anything but, instead create the space around us? I would also guess that if the world’s largest hadron collider actually gets turned on & does what is supposed to do, it might create a black hole that would not suck us in but would however be too heavy for our space. Then, it would basically just rip us apart by its massive density like an impenetrable piece of matter expanding from within. The reason that the expansion of the black hole doesn’t envelope everything is that the space it creates stops it from doing so. For example, if somebody tried to hold up an anvil with a sheet of paper it would most certainly break through the paper. As for this anvil it is infinitely dense so it would continue to swell & create the matter around it. I also think that’s why when a supernova happens a black hole is created; to help in the creation of new stars as well as the space for those stars and that is the reason the positive always outweighs the negatives.
    Additional Details
    *I am saying that upon the primal stages of the black hole’s birth it grabs the closest objects to them thereby destroying them in the process. Now at some point after this it will cease its negative gravitational pull on the surrounding objects at this point in time start in the creation of dark matter/ dark energy = positive. I am definitely not counting out the big bang theory in anyway with this thought.

    *This is only a theoretical question for open minded individuals.

    “Let the future tell the truth, and evaluate each one according to his work and accomplishments. The present is theirs; the future, for which I have really worked, is mine.”- Nikola Tesla

    I actually meant to post this!

  70. #70 Kevin Alvey
    July 18, 2011

    Added notation to my above comment.

    *”MOND” (for modified gravity)-push pull effect.
    *You can find proof of theses black holes with radio telescopes E.G. the static noise reflected back from space.
    * Also see Friedmann-Lemaître-Robertson-Walker metric

    “Let the future tell the truth, and evaluate each one according to his work and accomplishments. The present is theirs; the future, for which I have really worked, is mine.”- Nikola Tesla

    The link below has my very first post on this subject as well as steven Hawking’s view in the matter labeled “The case for primordial black holes as dark matter”.. see attached links
    (http://scienceblogs.com/starts…_dont_understand.php)
    (http://arxiv.org/abs/1106.3875)
    Here is another attachment dated June 27 from Carnegie Mellon who leads an international team in conducting the most detailed cosmological simulation to date by incorporating the physics of black holes into a highly sophisticated model running on a powerful
    supercomputing system.
    (http://www.cmu.edu/news/archiv…e27_blackholes.shtml)

  71. #71 Johnny Dixon
    September 14, 2011

    (just so you know im 10)befour the big bang, nothing, but it is imposibel to emgin nothing, black, ee, you may say its nothing, but its black, a shade, that is somthing.
    also, you say it calsd time, so, the same thing over and over? why is there an esploshin sudenly

  72. #72 Dov Henis
    September 28, 2011

    Comprehend gravity and comprehension of the Big Bang follows: it is the start of conversion of mass into energy, its other form.

    The universe cycles between two poles:
    singularity/all-mass , and max-expanded/nearly-all-energy.

    E=Total[m(1 + D)] (D = distance travelled by mass since singularity)

    Updated Definition Of Gravity

    Per the above E,m,D relationship the essence/definition of gravity is:

    Gravity Is the propensity of energy reconversion to mass

    Dov Henis (comments from 22nd century)
    http://universe-life.com/
    http://universe-life.com/2011/09/21/the-lhc-chases-its-tail/

  73. #73 RAN
    israel
    March 31, 2013

    hey ethan, your posts are pleasure to read
    i have a question – do you have any thoughts/insights regarding the goldilock paradox? thanks for answering

  74. #74 Wow
    April 1, 2013

    “just so you know im 10″

    Even a ten-year-old should spell better, John.

    As to the remainder of your post, you just gave your own answer: you can’t imagine nothing. Therefore not being able to imagine how an “explosion” occurred in nothing is entirely explained and insufficient to propose it incorrect.

  75. #75 Wow
    April 1, 2013

    “By elevating experiment to the level of King you raise the importance above any other part of the scientific method.

    And this line of reasoning falls in with your choice of Moniker.”

    you’ll need to demonstrate the validity of that statement.

    Which is another word for experiment.

    Self-defeating statement. Please try again.

  76. #76 Wow
    April 1, 2013

    “Can someone explain why the concept of “time after the big bang” is meaningful? ”

    Because we see things happen separated by an interval. We are now experiencing time. If it were not meaningful, then you’d have to explain why you posted then I posted this response without time.

  77. #77 Sarah Marie
    January 16, 2014

    Ethan,

    I am a sophomore physics major writing a series of brochures that explain physics at a middle school level. I am currently writing a brochure about the Big Bang. Would you be able to give me permission to use the image of the orange big bang image that is included in this article?

  78. #78 Ethan
    January 17, 2014

    Sarah,

    I don’t hold the image copyright to that image, but as long as you are using this for educational and not-for-profit means in the USA, that use is protected under the Fair Use Act.

    Or you can track down the original copyright holder and ask for permission; I don’t know who that would be off the top of my head.

Current ye@r *