Peter Coles has issued a challenge: explain why dark energy makes the universe accelerate in terms that are understandable to non-scientists. This is a pet peeve of mine — any number of fellow cosmologists will recall me haranguing them about it over coffee at conferences — but I’m not sure I’ve ever blogged about it directly, so here goes. In three parts: the wrong way, the right way, and the math.
The Wrong Way
Ordinary matter acts to slow down the expansion of the universe. That makes intuitive sense, because the matter is exerting a gravitational force, acting to pull things together. So why does dark energy seem to push things apart?
The usual (wrong) way to explain this is to point out that dark energy has “negative pressure.” The kind of pressure we are most familiar with, in a balloon or an inflated tire, pushing out on the membrane enclosing it. But negative pressure — tension — is more like a stretched string or rubber band, pulling in rather than pushing out. And dark energy has negative pressure, so that makes the universe accelerate.
If the kindly cosmologist is both lazy and fortunate, that little bit of word salad will suffice. But it makes no sense at all, as Peter points out. Why do we go through all the conceptual effort of explaining that negative pressure corresponds to a pull, and then quickly mumble that this accounts for why galaxies are pushed apart?
So the slightly more careful cosmologist has to explain that the direct action of this negative pressure is completely impotent, because it’s equal in all directions and cancels out. (That’s a bit of a lie as well, of course; it’s really because you don’t interact directly with the dark energy, so you don’t feel pressure of any sort, but admitting that runs the risk of making it all seem even more confusing.) What matters, according to this line of fast talk, is the gravitational effect of the negative pressure. And in Einstein’s general relativity, unlike Newtonian gravity, both the pressure and the energy contribute to the force of gravity. The negative pressure associated with dark energy is so large that it overcomes the positive (attractive) impulse of the energy itself, so the net effect is a push rather than a pull.
This explanation isn’t wrong; it does track the actual equations. But it’s not the slightest bit of help in bringing people to any real understanding. It simply replaces one question (why does dark energy cause acceleration?) with two facts that need to be taken on faith (dark energy has negative pressure, and gravity is sourced by a sum of energy and pressure). The listener goes away with, at best, the impression that something profound has just happened rather than any actual understanding.
The Right Way
The right way is to not mention pressure at all, positive or negative. For cosmological dynamics, the relevant fact about dark energy isn’t its pressure, it’s that it’s persistent. It doesn’t dilute away as the universe expands. And this is even a fact that can be explained, by saying that dark energy isn’t a collection of particles growing less dense as space expands, but instead is (according to our simplest and best models) a feature of space itself. The amount of dark energy is constant throughout both space and time: about one hundred-millionth of an erg per cubic centimeter. It doesn’t dilute away, even as space expands.
Given that, all you need to accept is that Einstein’s formulation of gravity says “the curvature of spacetime is proportional to the amount of stuff within it.” (The technical version of “curvature of spacetime” is the Einstein tensor, and the technical version of “stuff” is the energy-momentum tensor.) In the case of an expanding universe, the manifestation of spacetime curvature is simply the fact that space is expanding. (There can also be spatial curvature, but that seems negligible in the real world, so why complicate things.)
So: the density of dark energy is constant, which means the curvature of spacetime is constant, which means that the universe expands at a fixed rate.
The tricky part is explaining why “expanding at a fixed rate” means “accelerating.” But this is a subtlety worth clarifying, as it helps distinguish between the expansion of the universe and the speed of a physical object like a moving car, and perhaps will help someone down the road not get confused about the universe “expanding faster than light.” (A confusion which many trained cosmologists who really should know better continue to fall into.)
The point is that the expansion rate of the universe is not a speed. It’s a timescale — the time it takes the universe to double in size (or expand by one percent, or whatever, depending on your conventions). It couldn’t possibly be a speed, because the apparent velocity of distant galaxies is not a constant number, it’s proportional to their distance. When we say “the expansion rate of the universe is a constant,” we mean it takes a fixed amount of time for the universe to double in size. So if we look at any one particular galaxy, in roughly ten billion years it will be twice as far away; in twenty billion years (twice that time) it will be four times as far away; in thirty billion years it will be eight times that far away, and so on. It’s accelerating away from us, exponentially. “Constant expansion rate” implies “accelerated motion away from us” for individual objects.
There’s absolutely no reason why a non-scientist shouldn’t be able to follow why dark energy makes the universe accelerate, given just a bit of willingness to think about it. Dark energy is persistent, which imparts a constant impulse to the expansion of the universe, which makes galaxies accelerate away. No negative pressures, no double-talk.
The Math
So why are people tempted to talk about negative pressure? As Peter says, there is an equation for the second derivative (roughly, the acceleration) of the universe, which looks like this:
(I use a for the scale factor rather than R, and sensibly set c=1.) Here, ρ is the energy density and p is the pressure. To get acceleration, you want the second derivative to be positive, and there’s a minus sign outside the right-hand side, so we want (ρ + 3p) to be negative. The data say the dark energy density is positive, so a negative pressure is just the trick.
But, while that’s a perfectly good equation — the “second Friedmann equation” — it’s not the one anyone actually uses to solve for the evolution of the universe. It’s much nicer to use the first Friedmann equation, which involves the first derivative of the scale factor rather than its second derivative (spatial curvature set to zero for convenience):
Here H is the Hubble parameter, which is what we mean when we say “the expansion rate.” You notice a couple of nice things about this equation. First, the pressure doesn’t appear. The expansion rate is simply driven by the energy density ρ. It’s completely consistent with the first equation, as they are related to each other by an equation that encodes energy-momentum conservation, and the pressure does make an appearance there. Second, a constant energy density straightforwardly implies a constant expansion rate H. So no problem at all: a persistent source of energy causes the universe to accelerate.
Banning “negative pressure” from popular expositions of cosmology would be a great step forward. It’s a legitimate scientific concept, but is more often employed to give the illusion of understanding rather than any actual insight.
@vmarko: Oh that conservation stuff is a b$&@? when you get into non-zero…baby. 😀
@ kashyap vasavada:
“I am aware of the fact that many people believe energy is not conserved in GR.”
This is not a matter of belief. Conservation laws are tightly connected to global symmetries via the theorem of Emmy Noether. In particular, the conservation of energy is a consequence of the global symmetry of time translations. If this symmetry exists in a given physical system, energy is conserved. If it doesn’t exist, energy is not conserved. When the physical system is the entire universe (i.e. when studying cosmology), one can check that time translations are not a symmetry. Consequently, the energy is not conserved.
“If there are alternate derivations directly from GR (without using thermodynamics) doesn’t it lead to some conflict somewhere?”
Shortly put — there are, and it doesn’t. Basically, you specify (at initial moment) the matter content and all their nongravitational interactions, plug all that into Einstein equations, and from there you can derive a detailed evolution of both matter and geometry. Including the laws of thermodynamics. The whole thing is self-consistent, and in the appropriate special cases it reduces to what we already know (ordinary thermodynamics, Newtonian mechanics, classical electrodynamics, etc.), courtesy of the equivalence principle.
But you need to be aware that the “ordinary” thermodynamics (as you know it from non-GR physics courses) is only a special case of the “real thing”, which can be derived from Einstein equations. When you look at your tabletop experiment, the ordinary version and GR-version of the first law of thermodynamics are practically indistingushable, due to the fact that CC effects are extremely small for tabletop volumes of gases etc. But when you increase the volume of your gas to cosmological scales, the CC correction to the first law of thermodynamics becomes important, and then you can see that energy is not really conserved.
Also, there are no inconsistencies in the whole story. Most notably, the CC correction to the first law of thermodynamics still does not allow you to make a perpetuum-mobile or some such stuff… 😉
HTH, 🙂
Marko
Seems to me, if you say “constant expansion rate” means doubling in size over a constant time interval, then the acceleration is automatically built in. In fact, the only thing constant here is the acceleration itself.
Than you Sean.
I’ve been reading a lot of articles about this, and just been more and more confused over the different explanations. I have been trying to understand this by building different mathematical models – most dynamic simulations, and your article gives me a lot of confidence that I’m on the right way.
As I can see, nobody wants to discuss MOND. All I wanted to know, was, is the idea stlill alive or buried?
Sean’s article and dark energy have nothing to do with dark matter and MOND. Discussing MOND would only confuse the readers even more.
HTH, 🙂
Marko
Marko,
Isn’t a perpetuum mobile possible with superconductive material?
Jake
By saying that dark energy is persistent, constant throughout space and time and a feature of space itself, do you mean to say it’s of the infamous ‘cosmological constant’ variety? If so, why is this an explanation? It seems more like a just-so fudge factor constrained to preserve some feature of the equations ol’ Einstein deemed essential. And if it’s a property of space itself, one wonders how it gets its specific value and why isn’t that value derivable from other universal constants, such as the speed of light, the gravitational constant, Planck’s constant, etc.
So, the amount of stuff in the universe is fixed, hence spacetime has a fixed curvature, hence the universe has a fixed expansion rate, hence galaxies are seen to accelerate away from each other. Clear and simple, isn’t it? So how come it was such a big shocking surprise, back in 1997, when the acceleration discovery was first reported?
Also, if I recall right, the history of the cosmological expansion is quite complex, going from deceleration into acceleration about 5*10^9 years ago. Is this consistent with the constancy of the cosmological constant? Are there theories in which dark energy is a substance rather than a feature of space itself, and by which space can do all kinds of outrageous tricks?
Sean:
How is it that dark energy “persists”? In an expanding universe, how does it remain ‘constant’?
So I have a thought experiment that might help me understand this more thoroughly. Or maybe criticism of my formulating the thought experiment itself will help me.
If I have a fixed vessel filled only with space, choose any scale for this vessel. This space will expand, but it is constrained. So does it expand in any sense? If no, then placing a shell around the current universe would stop inflation?
->Marko
Right. I am a reader and I am confused :))
Now then, why is everybody so certain that space-time is a consecutive continuum? We know there are singularities, why not discontinuites? We all know the distribution of the strong force, would it really be something extravagant to suppose that gravity (space-time if you wish) behaves similarly?
I’m just trying to think out of the box; call it my crackopt theory of the week 🙂
I thought discontinuities exist in singularities as wormholes.
At least hypothetically.
“The tricky part is explaining why “expanding at a fixed rate” means “accelerating.” ”
Try this: “Suppose you have a colony of rabbits with a fixed, constant birth rate per capita. What happens to the population?”
Clearly the Universe is complex but I’m living in the real part and I like realistic explanations. Unfortunately I’m getting only imaginary solutions from most of the cosmologists…
Why the idea that somehow light dissipates its energy into the vacuum is so hard to consider and rest so obscure for the mainstream physics?! Fritz Zwicky’s “tired light” mechanism (scattering) is not a solid explanation of Hubble’s red shift but that’s not a reason to totally dismiss the hypothesis that beside gravitational and Doppler red/blue shift there’s a third phenomena that is the cause of Hubble’s red shift (red shift only).
Why nobody dares to consider there’s a vacuum “viscosity” for the propagation of light?! OK, mathematically would be equivalent with negative curvature of space so it won’t make any difference but such approach would eliminate the need for “imaginary solutions” such Big Bang.
And second, what is GR theory if not a Fluid Dynamics mathematical model without the fluid. I wonder what Ernst Mach would come up with in the light of actual information?
I am more and more convinced that Fatio and Le Sage were right and that Newton’s approach of gravity as “an intrinsic property of matter” was just a lazy solution genre “Is God’s will – so don’t worry about!”
Mathematics is God but mathematicians are (necessary) evil… Too many time we get the kind of explanations like John Wheeler’s “spacetime tells matter how to move; matter tells spacetime how to curve.”
That reminds me of a joke: Jimmy asks his physics teacher: “Why do we hear that annoying periodic noise when we ride the train?” His teacher, expert in math but with no clue of the technology used to install the rails says: “Jimmy, tell me if you know what’s the area of a circle”. “Eeeeh… Pi multiplied by the radius’s square?…” -answers Jimmy. “You see Jimmy?… Is exactly because of that square!”
Sean,
After you see the responses here, how successful do you think you have been?
“Dark energy is persistent, which imparts a constant impulse to the expansion of the universe, which makes galaxies accelerate away.”
I think I got that but I think I already understood that.
“But recently its rate of decrease has become so small that the velocity of galaxies is increasing.”
Maybe my expectation was for an explanation for that but maybe that isn’t understood yet.
“Also, if I recall right, the history of the cosmological expansion is quite complex, going from deceleration into acceleration about 5*10^9 years ago. Is this consistent with the constancy of the cosmological constant? “
I wouldn’t call that “quite complex”. Yes, it is consistent. Any introductory cosmology book which discusses the cosmological constant should cover this.
Space/time is not constant; space x time is constant. The expansion of the universe is accelerating. The reason why is invisible. That is because one does not see why.
I don’t know what happen to my previous comment…but the jist was “I have no clue why non-zero has nothing to do with expansion of the universe” and then a worried face. No explanation?
To Phillip Helbig:
You caught me–I’m neither a professional cosmologist nor an aspiring cosmology student, so I’ve missed that “Any introductory cosmology book.” (My introductory cosmology belongs to the era before the accelerated expansion was discovered.) Yet I infer from the discussion that, theoretically, matter-energy dominated the dynamics in the beginning, causing deceleration, which turned into acceleration once dark energy took over.
But if that’s correct, and if matter-energy is conserved, and if dark energy behaves as simply as an absolute constant, and if the universe composition is as found by WMAP, then it should be relatively straightforward to construct the history of the scale factor and compare it with whatever results from up-to-date supernovae observations. Was that done? If so, how good is the agreement?
Nothing would please me more than to learn the the agreement is so-so, because I think the cosmological constant is a wart on Einstein’s equations, though dark energy as a substance is even worse. (As an uncommitted amateur, who does not make a living or reputation out of papers admitted, I can afford such a stance and crave scientific revolutions.)
Edit: My bad… F$&@!
Hi Sean,
Preparing for a recent lecture, I realized you can derive the repulsive action of positive vacuum energy in pretty much the same way you can derive the Friedmann Equations from Newton’s gravity, if you treat a spherical uniform ball and ignore the exterior. If you ascribe a potential energy V[r]=-Mm/r to the a particle of mass m near the ball of mass M, and let M=M_stuff+M_vac, there M_vac is the vacuum energy density times the volume of the ball, then when you take -dV/dr for a force, you find the usual Newtonian term, then a term that goes as R and is repulsive. In words: the gravitational potential energy of normal stuff becomes less negative when you expand the ball, but the contribution from vacuum energy becomes more negative. Forces act so as to lower potential energy, so this leads to an attractive force due to the stuff, but a repulsive one due to the vacuum energy.
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What if in, your first Friedmann equation, G is not a constant at all. G is an ‘accepted value’. Why is this acceptable? My understanding is that over many years and many experiments, the measure of G varies by almost +/- 1.5%. It’s not that measurements are getting closer to the ‘real’ value; but, the measurements flutter around the +/- 1.5%. Wish you could do an study/article on the many experimental measurements of G (or what amounted to a measurement), Cavendish to today…
If G is not a constant at all, then perhaps gravity is not only a geometry but is also a physical process. After all, electro-magnetism existed as a physical process long before it was either discovered or used. In short, does ‘dark energy’ really exist or is it just an artifact of theory?
The number of beings that can compute in parallel (and compactify time) tends to increase do to intelligence being a neotonizing force (via domestication).