Why Does Dark Energy Make the Universe Accelerate?

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:

\frac{\ddot a}{a} = -\frac{4\pi G}{3}(\rho + 3p) .

(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):

H^2 \equiv \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3} \rho.

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.

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120 Responses to Why Does Dark Energy Make the Universe Accelerate?

  1. Matt says:

    Sean–

    I’ve run into this same problem when teaching GR to my own students. But after I tried a simple explanation of the sort you describe here, one of my more inquisitive students asked me why the story changes if we were to imagine flipping the sign of the cosmological constant. I didn’t know what to say except point back to the Friedmann equations (and the Raychaudhuri equation)!

    So I put this challenge to you — if the explanation is as straightforward as you describe it in “The Right Way,” then how does one intuitively explain why the universe goes from accelerated expansion to accelerated contraction if we change the sign of the cosmological constant, going from a dS-like solution to an AdS-like solution?

    Thanks!

  2. George says:

    Am I missing something, as it seems this means that on the whole the total energy of the universe is increasing? If the universe expands and the dark energy density stays constant throughout, then there is more total energy moment to moment?

  3. vmarko says:

    @ George:

    “Am I missing something, as it seems this means that on the whole the total energy of the universe is increasing?”

    Yes, you missed to read this:

    http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

    HTH, 🙂
    Marko

  4. George says:

    Thanks Marko, that certainly helps. I think it will take a bit to internalize this!

  5. Sean Carroll says:

    Bee– I didn’t use the pressure, I used the dependence of the density on the scale factor. You can derive that dependence from the equation of state, but you could equally well derive the equation of state from the scale-factor dependence. Neither is more fundamental.

    James– The expansion *rate* (the Hubble parameter) has always been decreasing. But recently its rate of decrease has become so small that the velocity of galaxies is increasing.

    Andy– I was speaking in an approximation where there is only constant dark energy, in which case the Hubble constant actually is constant.

    Matt– It’s more complicated when Lambda < 0, because there are no solutions with zero curvature and no other forms of energy density. But the basic physics is still the same. Ordinary matter dilutes away, eventually the vacuum energy dominates, the universe decelerates, then matter begins to dominate once again and you crunch.

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  7. Michael Carasik says:

    I very much appreciate “the right way,” as I had just read a “negative pressure” explanation on Friday which made no sense to me.

    What I continue not to understand is this: If space itself is expanding, why don’t our measurement units expand similarly so that the expansion is transparent to us?

  8. What about (hypothesized) particles as explanation: the ‘fat’ graviton, ‘chameleon’, …

    Fat Gravity Particle Gives Clues to Dark Energy
    Force-carrying “gravitons” with mass could help to explain the universe’s accelerating expansion
    scientificamerican.com/article.cfm?id=fat-gravity-particle-gives-clues-to-dark-energy

    Dark-energy particle spotted?
    Reported ‘chameleon’ particle would change its mass to match its environs.
    nature.com/news/2009/090529/full/news.2009.531.html

  9. kashyap vasavada says:

    How about this argument based on elementary thermodynamics. dq = du + pdv = 0 for an isolated system. If energy density of vacuum is constant positive , u has to increase with increasing volume. Vacuum cannot be diluted! So du > 0. Then for dv > 0, p has to be negative. For the case when Lambda is negative (Matt’s question) this will also work because in that case du 0. Anyone sees problem with this argument?

  10. kashyap vasavada says:

    How about this argument based on elementary thermodynamics. dq = du + pdv = 0 for an isolated system. If energy density of vacuum is constant positive (Lambda > 0) , u has to increase with increasing volume. Vacuum cannot be diluted! There is more vacuum. So du > 0. Then for dv > 0, p has to be negative. For the case when Lambda is negative (Matt’s question) this will also work because in that case du 0 Anyone sees problem with this argument?
    I had some problem in editing. Please ignore the previous comment.

  11. kashyap vasavada says:

    Sorry to take so much space. Again problem with the editor. In the 6th line in the above I should have du < 0.

  12. Confused says:

    Jason Dick “That answer lies in how the rate of expansion has changed over time. The expansion rate in our early universe was much, much higher than it is today, and has been decreasing steadily as matter has spread out. Early-on, when the expansion rate was very high, a photon would leave this far-away galaxy moving in our direction.”

    Sean Carroll “James– The expansion *rate* (the Hubble parameter) has always been decreasing. But recently its rate of decrease has become so small that the velocity of galaxies is increasing.”

    “… a long time ago, the Universe was actually expanding more slowly than it is today.” http://science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy/

    And you wonder why the lay person is confused! I can recall physicists explaining with absolute certainty that our moon formed from a ring of debris around the Earth that coalesced by gravitational attraction to form our satellite. But wait! No, an asteroid hit the Earth and a large chunk of our planet broke away. No, wait, our satellite was formed when… And all of this is pronounced with such conviction!
    After many years spent in abandoned mines almost a mile deep, there has been no trace, whatsoever, of anything that can be attributed to dark matter yet this is a forgone conclusion.

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  14. Hal Swyers says:

    I kept this even more simplistic, but thought that was the intent of the original challenge

    http://thefurloff.com/2013/11/17/the-universe-and-pressure/

  15. Victor says:

    How about this explanation:

    Suppose that you live in a small apartment in which your many belongings make it look really messy. If you move to a larger place, say a mansion, with the same belongings, your house should logically look less messy. But mysteriously this very expectation is violated in our universe; a fact which has baffled very smart physicists for so long. Measurements prove that we are constantly, here exponentially, moving to a larger world as time passes. Yet the calculations of space-time dynamic based on Friedmann equations show that to keep this rate of expansion, our new mansion should look as messy, here better to say as massy, as before. Put it other way, there should be some phantom belongings in our apartment, alternatively phantom galaxies in our universe, often called dark matter by cosmologists, not yet seen and measured, that stay with us no matter what and fill up this continuously, exponentially, expanding space-time.

    What more puzzles me is what could be or occur on the boundary of our universe spacetime, what is beyond the farthest away galaxies? If this huge bubble is expanding so what is there outside of it?!

  16. Bee says:

    Sean: How do you know the ‘dependence of density on the scale factor’ without knowing the equation of state for which you need the pressure? You’ve basically postulated it (as I said, you assumed it’s constant), but we both know you’re not allowed to do this in GR. You can pick an initial value, the rest is dictated by the theory. And for that you need the pressure. You’ve just avoided that by picking a time-dependence that you like and avoid telling us that it’ll only solve the equations of motion if you have a negative pressure.

  17. “So the term acceleration is really a misnomer when talking about expansion of the universe, since the expansion occurs at a constant rate and the “acceleration” is only apparent. “

    No, it is not a misnomer. By “constant” Sean means that (in the limit where the cosmological constant is the only important thing) the expansion is exponential.

  18. From the point of view of the Friedmann equations, introducing a positive cosmological term is equivalent to adding some negative pressure, as we can see
    in this Wikipedia page about FLRW. In FLRW, just by looking at the geometry of the universe, there is no way to tell whether a cosmological term is present, or when there is more negative pressure. I think this is why people say that expansion is caused by negative pressure, and I see nothing wrong with this.

  19. “Sorry, but I think there is a fundamental mistake in this.”

    No, Sean is correct.

    “Hubble’s constant is constant in space (we would measure the same value no matter which galaxy we live in), but not necessarily in time. In fact, the inverse of H (called the Hubble time, the time it would take each galaxy to get to its current distance at current velocity), is sometimes called the age of the universe; this implies that if H were constant, then the age of the universe presumably wouldn’t change with time?”

    Right. I admit that there is some confusion, since the “constant” in the Hubble constant means it is only a constant in space at a given time (the actual origin is that is a constant in an equation like m and b constants in y = mx + b) while the cosmological constant is constant in time.

    “A varying H implies that the energy density also varies with time; not a problem, energy is conserved, but not energy density.”

    Energy is not conserved in GR or cosmology, at least not in the sense in which this is normally understood.

    Without the cosmological constant, in general H is not constant in time. So, this fact does not imply the cosmological constant.

  20. N. says:

    Great post, thank you Sean.

    How does all this look in the eyes of MOND?

    Or is that horse dead already?

  21. supermuf says:

    The tricky part is explaining why “expanding at a fixed rate” means “accelerating.”

    The point is that the new expanded space also expands, that’s the reason for the exponential rate.

  22. meh says:

    I think vmarko hit the nail on the head. Most people do not ever learn the intricacies of General Relativity. Though I squirm as I type that because it’s not like the fundamental concept of dark energy is too complicated to teach to a high school physics class. That being said, I think every high school student should take calculus by their senior year; and that certainly doesn’t happen.

    I understand, and prefer, the negative pressure explanation because it helps me visualize what is happening and implies the deformation of space. In my experience, the people that have difficulty understanding dark energy usually don’t want to understand it. Tends to be the same people that think scientists are involved in a conspiracy that involves tricking governments into paying for their Bugatti Veyron, mansions, and stripper laden pool parties…you know, crazies…who must be silenced if our plans are to succeed.

  23. kashyap vasavada says:

    @vmarko: I am aware of the fact that many people believe energy is not conserved in GR. But don’t you use energy conservation in thermodyamaics to get the second equation. If there are alternate derivations directly from GR (without using thermodynamics) doesn’t it lead to some conflict somewhere?

  24. meh says:

    N.,

    MOND :: Newton’s gravity

    TeVeS :: Einstein’s gravity

    I think those are both generally restricted to dark matter.

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