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.
Anthony Aguirre said: “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.”
Yes, I teach that too. BUT I stress to the students that “ignoring the exterior” means that you are NOT dealing with a homogeneous, isotropic universe like ours. When I teach this stuff I emphasise that what we are really doing is proving that FRW *cannot* be deduced at all from Newtonian gravity. In more technical language: no [3-]vector theory of gravity can give a sensible cosmological model, because a non-trivial 3-vector cannot be isotropic.
In fact, the nice thing about cosmology is that it immediately makes it clear that all talk of “gravitational force” must be wrong — forces cannot be isotropic. This is why SC’s campaign to rid us of pressure is such a good idea: we have to teach people to get away from the whole concept of “gravitational force”.
SC’s idea of stressing the constancy of the dark energy density is an excellent one. He just needs to find an intuitive way of explaining geodesic deviation, as applied to the geodesic worldlines of galaxies in an isotropic homogeneous universe.
“SC’s idea of stressing the constancy of the dark energy density …..”.
According to me the constancy of the energy in the universe is quod erat expectandum.
…energy density…
@vmarko: Thanks for the reply to my question. I see your main point. However, I am not sure if the difference between relativistic and non-relativistic thermodynamics is just due to CC (I am not sure if you are saying that.). Creation of additional space (expansion) does require additional energy which has to come from somewhere. Also the universe will keep on expanding for a while even without CC, although eventually it has to collapse because of normal gravity. It seems that such problems have been ignored in cosmology.
“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.”
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Exactly. Since the energy density of the cosmological constant is, errm, constant, while that of matter is inversely proportional to volume (radiation is more complicated, but taking it into account introduces only small corrections), at early times the universe is matter-dominated and at late times, if the cosmological constant is positive, approaches exponential expansion. So, a relatively complicated outcome, if you like, from a simple premise—like most of the universe.
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“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?”
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Well, a Nobel Prize was awarded for it not long ago. The theory behind the Nobel Prize is literally textbook stuff, but that is not what it was awarded for. It was awarded for the enormous observational effort and the attention to detail to make sure that what are believed to be standard candles (or can be corrected to behave as such) really are. (This is common in astronomy. To measure the Hubble constant, in theory all you need is one distance and the corresponding redshift. The devil is in the details.) Yes, it does fit—remarkably well. In fact, for the first time we have a standard cosmological model which fits essentially all the data. (Caveat for experts: Initial analysis of PLANCK results confirms some discrepancies at low l but, considering how good the fit is at high l, I suspect that the solution here will be instrumental and/or involve the CMB and keep the “cosmological standard model” intact.
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“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.”
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No, it is quite good. It is very good. It looks like a pure cosmological constant (w=-1). While one should be open to the possibility that it is not a pure cosmological constant, at the moment there is not one shred of evidence for this and a pure cosmological constant fits all observations (with the same value of the cosmological constant, of course). In other words, “dark energy” (as Sean pointed out a while back, this is a terrible name; it’s a shame his “smooth tension” didn’t catch on) is the cosmological constant. A rose, by any other name…but I prefer to call it the cosmological constant, or lambda.
@ Sean: I think these controversies again bring up the point that it is hopeless to try to understand theoretical physics without equations. Remember your write up “most embarrassing graph in physics” ? All these efforts are turning futile because we are trying to put these complex non classical phenomena in human languages which are based on our everyday life which is classical!!!
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Phillip Helbig: Thank you, I’m beginning to really enjoy this conversation.
I have been following the observational effort–the subject of the recent Nobel prize–more closely than the theoretical background and am therefore more acquainted with it, qualitatively, though.
About the agreement between the CC-comprising model and the best, up-to-date, observational results (“…it is quite good. It is very good.”) what I would like not to miss is seeing the actual comparison and getting elated over it myself. I’d appreciate if you post the relevant (free-access) link/s. Thank you again.
@ DEL:
“About the agreement between the CC-comprising model and the best, up-to-date, observational results (“…it is quite good. It is very good.”) what I would like not to miss is seeing the actual comparison”
You can start here:
http://en.wikipedia.org/wiki/Lambda-CDM_model
Most importantly, look at the table of parameters and their values. The error-bars can give you a feeling of how well the experimental data fits the theory.
HTH, 🙂
Marko
The expanding universe is a serious glitch. So I’ve had to revise my Maths Free (fantasy) theory of how the universe works.
As I now (choose to) see it:
The universe is made up of 2 types of energy, positive energy and negative energy. Everything that is visible is made of positive energy. Space is filled with negative energy which is both the Higgs field and Dark energy (same thing). Negative energy is attracted proportionally to positive energy but repels itself to form a homogeneous energy field filling the universe. Negative energy forms a dense energy shell around positive energy effectively encapsulating it. This denser negative energy pocket is repelled by the general body of negative energy expelling it towards any other dense negative energy pocket, carrying the encapsulated positive energy with it. This effect is what we experience as gravity. So gravity is in fact the effect of things being pushed together rather than being pulled together. So the Higgs Field creates gravity by pushing matter together and at the same time pushing accumulations of matter further apart.
In this Imaginary Universe, Dark Matter is a condensate of Dark energy and is created by Black Holes and other super dense positive energy accumulations such as stars, that is why it appears to have a gravitational effect while not reacting with matter generally, but accumulates near positive energy masses.
I am postulating that dark energy has various states likened to gas liquid and solid. The Higgs field is the “gaseous” form, dark Matter is the “liquid form” slightly denser, and the Higgs particle is the “solid” form.
Thanks, Marko.
Most of the things in the lambda-CDM Wikipedia article were known to me. What I miss, and maybe you can refer me to, is, e.g., this: a plot of the world lines of participating type-Ia supernovae progenitors according to the lambda-CDM model (pretending they all lie on a single ray and have existed since ever,) compared with the observed world events of their explosions.
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Here’s an interesting post about the explanation for dark energy:
http://m0tls.blogspot.com/2013/11/its-harmful-to-teach-wrong-physics.html
Oops. Change the first 0 to o in the above URL.
One way to understand why Sean’s claims are wrong is the following:
— The goal is to explain to the layperson the following; why can gravity be sometimes repulsive rather than attractive, in the sense that galaxies will on average accelerate away from each other rather than towards each other. We can begin by imaging a homogeneous universe that is static at some moment in time, then we let it go….will it attract or repel?
According to Sean, the answer is always to turn to the (first) Friedmann equation. Well we definitely need to include curvature, and we have:
H^2 = rho – k/a^2
where k is the curvature. Initially, let us imagine that things are at rest, so H=0, then rho-k/a^2=0 (which can be used to solve for k). The question is then: what happens next, does it collapse (which in this context is the same as attract or decelerate) or does it expand (which in this context is the same as repel or accelerate)?
An answer to this requires knowing about the SECOND TIME DERIVATE of the scale factor. And the first Friedman equation above is really hopeless at telling us what this is. Instead we need an equation for a”; which comes from the second Friedman equation and includes pressure.
Then to discover if it will accelerate or decelerate we absolutely need to know about the value of the pressure.
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It seems to me it’s only wrong because a pressure is being confused with (the effect of) a difference between two +ive pressures at a boundary there. From the POV of a test particle inside the balloon the pressure due to the gas is felt as an isotropic crushing. Negate that and the wrongness goes away, surely?
Included with an explanation of why DE makes the universe accelerate should be a summary of why it is exciting. The phenomenon of Dark Energy is one of those rare discoveries that illuminate the inadequacy of a present theory (GR). As Sean emphasized, the extra DE curvature is extraordinary because it appears to be constant in time. So either:
1) DE is just a new dimensional (1/length^2) constant Lambda whose explanation lies beyond GR and also seems to be beyond quantum field theory, or
2) Move Lambda to the right hand side of Einstein’s field equation and interpret it as an energy density. Unfortunately, a density should decrease as the universe expands, but DE is constant. Furthermore, the DE density is negative while all matter (that we know) has positive energy density.
Both alternatives lead to the conclusion that GR (and our understanding that only an energy density causes curvature) just doesn’t have the ability to explain the DE phenomenon (beyond just calling it Einstein’s cosmological constant). DE is a big clue to something new! We need a new reason both for a constant curvature, and for the curvature to have a sign opposite to that caused by normal matter !
tidal forces can compress or stretch, and (divergence of) the *deviation acceleration* of galaxies does pick up the pressure term…
“Included with an explanation of why DE makes the universe accelerate should be a summary of why it is exciting. The phenomenon of Dark Energy is one of those rare discoveries that illuminate the inadequacy of a present theory (GR). “
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Huh? There is no evidence that dark energy is anything other than the cosmological constant, and this was introduced by Einstein. So how does it somehow illuminate the inadequacy of the present theory?
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OK, we don’t know what, if anything, it is, whatever that means. But isn’t it enough to know that it exists and is described by the theory? IIRC it was Wheeler who summarized GR as “spacetime tells matter how to move; matter tells spacetime how to curve”. However, we don’t know why this is the case.
Masses are continuously absorbing or emitting energy. On balance the quantity of energy emitted is enormous but not taken account of in the calculations based on the Standard Cosmological Model and the mainstream understanding of gravity..
The consequence of what I have said in my comment on “Is time real?” is that gravity is related to mass and not to energy and momentum like pseudo-gravity.
Therefore, according to me, the question of what causes the acceleration of the expansion of the universe has nothing to do with DM/DE, pressing or stretching, whether you like it or dislike it.
Dear Prof Carrol,
In a laymen’s attempt to distinguish “apparent” velocity and “regular” velocity, whereby:
* “Apparent” velocity – the velocity you artificially associate to a distant galaxy relative say to earth.
* “Regular” velocity – velocity in the sense that fits, say, into a special relativity energy calculation
In my simplistic mind, “pushing” a galaxy till it obtains a regular velocity v – requires energy E that is a function of v.
The special relativity expression for energy explodes when you reach the speed of light.
Does this imply that: (several – not necessarily distinct – options that come to my confused mind):
* You cannot plug “Apparent” velocity into these energy calculations
* There is no relationship between the energy content in space-time and the kinetic energy i am trying to think about.
* Other…
With regards to option 2,
Using your “The Right Way” explanation, trying to think about “pushing” a galaxy until it recedes from us in more than speed of light,
I can imagine that any amount of energy in spacetime will do.
The only thing that would change – is how long will we have to wait.
With a small amount of energy, we will just have to wait longer.
If that is the case – than it is to me – a clear cut demonstration of the difference between apparent and regular velocities.
It is very impressive and rewarding to see a top tier Scientist dedicating precious time to public outreach.
I hope you will find enough of a general interest in my question to merit an answer.
Yair
Yair,
Imagine a rubber sheet which is being stretched away indefinitely, and put two marbles on it. One marble (our galaxy) can in principle move with respect to the rubber directly beneath it. Its velocity with respect to this rubber cannot exceed the speed of light, and is the “regular” velocity. The other marble (distant galaxy) is further away on the rubber sheet, and it may move with respect to rubber directly beneath it, with velocity again only less than speed of light. That is also the “regular” velocity.
However, at any point in time, you can measure the distance between the two marbles/galaxies, along the straightest possible line on the rubber sheet. Now, even if the “regular” velocities of the two marbles are always kept zero, the distance between them will grow, since the rubber in between is being stretched. If you divide this distance with time, you get the “apparent” velocity. This velocity has nothing to do with the motion of matter through space (but only with the motion of the space itself), so there is no speed-of-light limit.
As for energy balance, you need to invest some energy to move each marble with respect to the piece of rubber it’s standing on, and it takes an ever-growing amount of energy to make the marble’s “regular” velocity closer to the speed of light. On the other hand, there is absolutely nothing needed to invest in order to make the “apparent” velocity as large as you like, because that velocity is the property of the motion of space itself, rather than matter through space.
In order to get this stretching motion of “space itself” appropriately consistent with observations, we need to plug a nonzero (positive) cosmological constant term into Einstein equations of general relativity. It is commonly called “dark energy”, but I still claim that this is a very unfortunate terminology. It is more than just “energy”, and it behaves in a way in which any ordinary energy would never behave.
Finally, note that in general relativity energy is not conserved — at least not in a way you would expect — regardless of the presence or absence of any CC, dark energy, etc. Therefore you should not even try to think about the kinetic energy balance between galaxies, because there is no balance and there is not supposed to be any.
HTH, 🙂
Marko
Dr Carroll,
If Dark Matter is keeping the Milky Way Galaxy together, why do we often hear that the acceleration of the Universe will effect us? I understand that the Universe is accelerating, but doesn’t Dark Matter keep the Galaxy together therfore not having an effect on “us”?
Is there a reason it’s described as energy rather than mass? Could you just as well give it in kg/m^3? Or does mass/energy equivalence break down here somehow?
It’s not obvious to me that constant curvature implies a constant proportional rate of expansion. I don’t think I have a good enough intuitive notion of what 4D curvature looks like to make inferences like that about it. Is there a way to explain how you measure curvature in spacetime that makes the relationship between curvature and constant expansion clear?
Does it work to imagine rockets (actually unpowered probes) heading in opposite directions, and then having one sending a beam of light back to the other? That gives you a triangle and you can look at the angle sum and compare it to pi. If there’s an angle deficit then that suggests the probes are accelerating apart. So if my way of measuring curvature is right, then I guess I do buy the claim that constant curvature across spacetime leads to a constant “growth rate of space”. But my reasoning might be completely wrong.
@Sean
Congratulations for your award winning book.A great read indeed!
For those of you who like me are constantly seeking to answer the questions ;what is Dark Matter?and What is Dark Energy? you can find my views to these questions in a paper I wrote http://dx.doi.org/10.4236/ijaa.2013.33028. Please excuse me for the missing 2pi in the expression r=2pi/k