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. Coel says:

    How much is wrong with this way of explaining it?:
    The basic equations of the universe encapsulate conservation of energy.
    The density of dark energy is constant, so expansion creates more of it.
    Thus the (negative) gravitational potential energy increases.
    Thus, by conservation of energy, kinetic energy must increase.
    Hence the expansion rate increases.

  2. Pingback: Why Does Dark Energy Make the Universe Accelerate? | Gordon's shares

  3. Vinicius says:

    Hi, very nice post.

    I think the preassure explanation its there because of the w=-1 dispersion relation.

    I already did a cosmology course and still can ge my head around the faster than light exapansion. Espacially because my professor told me that the equations governing exapnsion could be interpreted as everything moving away from everything else, kind like if the universe started with a constant speed (in this case, acceleration).

    Could you comment on that ? How come a galaxy, in the distant future, might move away from us faster than speed of light ?

    And one more questin : If the space has a propriety which is an amount of energy, as univese expands, more energy comes in because there´s more space. So, total energy in the universe is increasing as time passes ?


  4. Jake Maier says:

    What is meant by “the universe is doubling in size?”
    Is it the distance to the end of the visible universe that is doubling, or the area, or volume?
    Volume was my first guess, but if the volume doubles per fixed time the distancing galaxies would need to decelerate or is it the radius of the visible universe that is doubling, ? What am I missing.
    Being a forester who often is satisfied with a result of 2+2~3 I readily admit that some of this is not easy to grasp for me.

  5. Travis says:


    The explanation given in “The Right Way” doesn’t make sense because the normal effect of energy on the curvature of spacetime is to cause objects to get closer to each other. It seems to me that in order to explain why dark energy causes expansion, you need to explain why the effect on the curvature of spacetime due to dark energy causes the opposite of what is normally observed. The logic used in “The Right Way” would seem to imply that the mass of the sun should cause the sun to expand, albeit at an ever decreasing rate due to the energy density decreasing as the expansion occurs.

  6. Jason Dick says:

    Very much like this explanation 🙂 I’ve used a very similar explanation myself, but I hadn’t figured out how to do it adequately without any math at all.

    To Vinicius:
    Actually, most of the galaxies in the observable universe, by the usual definition, are currently and always have been receding from us faster than light (these galaxies are pretty far away, but there are a lot more galaxies far away than close by). The short, glib answer to this is that in General Relativity, there is no unique definition for the velocity of a far-away object. There are, instead, multiple possible definitions. Because of this, it doesn’t make sense to talk about the speed of light as being a limiting factor for the speeds of far-away objects. Instead, General Relativity only states that no object can outrun a light ray (in vacuum). A far-away galaxy with a recession velocity greater than that of light doesn’t break this rule, because it’s not outrunning any light rays traveling past it.

    That might potentially explain why it isn’t a violation of relativity for a galaxy’s recession velocity to climb faster than that of light. But it doesn’t explain why we see so many galaxies that have always had recession velocities that high.

    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. But the expansion would create more space between us and that photon than the photon could travel, so that even though it was traveling towards us at the speed of light, it had still further to travel to get here. Then, later, as the expansion slowed, the photon started to make headway, eventually reaching us. The galaxy, however, only continued to get further away due to the expansion. That photon that left the galaxy long ago managed to get close enough for it to pass matter whose recession velocity became lower than the speed of light, but the far-away galaxy never did.

    Does that help?

  7. Sean Carroll says:

    Jake– “Doubling in size” means that the distance between galaxies doubles in size. So the volume would go up by a factor of 8.

  8. vmarko says:

    I’d say that the main source of confusion comes from the terminology — the term “dark energy” is not exactly a suitable alternative name for the cosmological constant.

    While formally one can put the CC on the other side of Einstein equations and consider it as a part of an effective stress-energy tensor, the CC doesn’t actually behave as one would intuitively expect from “energy” to behave. So trying to understand the CC as a form of energy is bound to introduce confusion and break our intuitive idea of energy as something that produces attractive gravitational forces.

    The other culprit in the confusion is that people are often told “in General Relativity the energy is the source of the gravitational field”. This is only half-true — it’s not just the energy, but also the momentum (or technically called “pressure”, “shear stress”, etc.) is a source of gravity as well. And given that in Newtonian gravity all momenta are small compared to rest masses, most people have no intuition on what momentum does to the gravitational field. With no intuition, they most often tend to simply ignore it as a source of gravity, thereby coming to wrong conclusions about the behavior of “dark energy”.

    The point is that the “dark energy” also carries along its own “dark momentum” (usually called pressure), which is three times as strong and contributes a repulsive gravitational effect. So one attractive energy plus three repulsive pressures give a net repulsive force of gravity. People who haven’t studied the details of GR have a hard time wrapping their heads around this.

    So IMHO, trying to provide an intuitive explanation for repulsive gravitational effects of dark energy is bound to fail — not because we are bad at explaining things, but because the audience (uninitiated in GR) has a hard time figuring out that energy is not the only source of gravity.

    HTH, 🙂

  9. If the distance between galaxies double in size, does the size of each individual galaxy double in size in a corresponding manner: that is, does the distance between the stars in a galaxy double in size? If not, why is the expansion selective according to scale?

  10. Vinicius says:

    Thanks for the answer Jason Dick.

    So you are saiyng that the velocity measurement on the galaxies gives us results faster than light but in their reference frame they are moving at lower-than-light speeds ?

  11. Sean Carroll says:

    Philip– Galaxies (or planets, or atoms) don’t expand, because they are held together by local forces. (Gravity, in the case of galaxies.) Only freely-moving objects get pulled along with the expansion of space.

    Vinicius– What we call the “velocity” of a distant galaxy is just a fake. In GR, velocities are only well-defined when objects are moving right past each other. Distant objects, with lots of curved spacetime in between them, have an “apparent velocity” at best.

  12. Marc Sher says:

    Sean—I agree that dropping “negative pressure” is a good idea. But there is still a source of confusion for the layperson. The first Friedman equation (typo warning – the a-dot/a term should be squared), is the same for either sign of a-dot. It gives either exponential expansion or exponential contraction for constant energy density. Most laypeople think of gravity (i.e. spacetime curvature) as always causing a contraction, and the hard part for me is explaining why more stuff makes the expansion faster….

  13. Serge says:

    Such a simple explanation, even I can understand it (I think).

    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. I can see how that could become a pet peeve.

    Perhaps the term dark energy is also misapplied, since it is not really energy in any conventional sense. It appears to be a property of spacetime and its only interaction with matter is through its effect on spacetime. But, I guess, the same can be said about gravity — not an energy in a conventional sense and same type of interaction with matter, through its effect on spacetime (gravity warps spacetime, dark energy expands it).

    But if they are both properties of spacetime, does it make sense to try and unify them with the other forces of nature? Of course, who is to say the other forces of nature are not themselves properties of spacetime?

    Alright, I admit it, I don’t understand anything after all.

  14. As a layperson, I will say that I find the second (right) explanation much more intelligible. But I thought dark energy was called ‘dark’ because we had no idea what it was. That we could see the effects, but not the cause or necessarily the mechanism. Has this changed?

  15. rob says:

    first paragraph…. change idea of gravity …make its apparent attraction into a differential of forces…
    radiation do the pushing …masses absorb …creating a differential across and between masses.
    La Place.s … Shadow Gravity…
    Newton.s universal gravitation law …the…product of two masses divided by square of distance … is just an averaging out…to set the modelbin it.s simplest form
    now big G can be seen as a tiny fraction of two thirds the cylinder ..between the masses .
    as radiation arrives from all points in space it sets up a cylidrical volume lf lower energy… inside are two cones of lowest energy. which is the absorbed volume.
    this leaves two thirds as the Differential . and is the value of energy causing the force / acceleration.

  16. Sean Carroll says:

    Marc– Typo fixed, thanks. I think the confusion you point out is a good one, actually, because it helps understand what general relativity really says. It’s not just Newtonian gravity with some extra words about spacetime. It’s a dramatically different perspective, in which gravity is the curvature of spacetime. And spacetime curvature can take many different forms, depending on the distribution of energy. If you have a lump of positive energy, outside that lump spacetime will curve in such a way as to cause what looks like an attractive force. If you have a smooth distribution of constant density, the universe will expand or contract at a constant rate. What matters is that there is a balance between stuff and curvature.

    SelfAwarePatterns– Dark energy is called “dark” because it doesn’t interact with light, that’s all. In a strict sense we are not sure what it is, but there is certainly a simple and obvious candidate that fits all the data — the cosmological constant, which is what I was describing above.

  17. Truck Captain Stumpy says:

    So… let me get this straight….
    The expansion comes in much the same way the cooling does when a firefighter uses a fog pattern of water instead of a straight stream… meaning the surface area grows exponentially with the continued growth but the actual growth is really steady (much like the water is). If that is it, then I think I can really understand it.

    Or do I have something wrong?

  18. sagredo says:

    If there is no pressure, then what is the first law of thermodynamics for de Sitter space?

  19. Sagredo says:

    In the case of the accelerating expansion
    dU=єdV, where є is the energy density, є=const.
    But work would have to be done to cause the region to expand,
    dW=-pdV. Then p=-є.
    It is the negative pressure which is the driving force behind the accelerating expansion. Should we ban the negative pressure?

  20. AI says:

    Ok, but since the concept of scale is relative could the expansion of the Universe be reinterpreted as contraction of everything in it? Here the amount of space would stay the same but the dimensions of everything else – galaxies, stars, us – would shrink with time. Would Einstein equations still apply to this case? Could the energy momentum tensor be somehow modified for them to apply? Or are there some fundamental reasons why this picture is ruled out?

  21. See also:

    Why all these prejudices against a constant?
    Eugenio Bianchi, Carlo Rovelli

    The expansion of the observed universe appears to be accelerating. A simple explanation of this phenomenon is provided by the non-vanishing of the cosmological constant in the Einstein equations. Arguments are commonly presented to the effect that this simple explanation is not viable or not sufficient, and therefore we are facing the “great mystery” of the “nature of a dark energy”. We argue that these arguments are unconvincing, or ill-founded.


  22. Bee says:

    You need either both equations, or you need one and stress-energy conservation. Either way you turn it, the pressure will come in. You’ve implicitly used stress-energy conservation by starting on the assumption that you’re dealing with a source that is constant over time.

    That having been said, I like your explanation. The explanation that I myself hate most is saying that dark energy acts like ‘antigravitation’. Because I once made the effort of trying to take this idea seriously and now I’m like the only person on the planet who knows that antigravitation does not act like dark energy. (If you think about it, it’s trivial. Whatever antigravitating matter you have, it’s still matter. It’ll never give you the right w value, regardless of whether its energy density is positive or negative.)

  23. James Cross says:

    “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.”

    I thought observations have shown about 4 billion years the expansion began to speed up. Can it the rate be constant and fixed if it is changing?

  24. platohagel says:

    I am a layman, so how would it be better explained to me that you might help me understand this challenge?

    R for a simple model involving matter in the form of a perfect fluid:

    and then to have,

    the only way to make the expansion of such a universe actually accelerate is to fill it with some sort of stuff that has

    This is a interesting view to me, and at the same time I am asking what do you have in terms of the naturalness of what exists in the universe to explain this?

    I am drawn to spherical cows in terms of explanations and to this, being only an approximate, I wonder what evidence exists as to contribute to that expansion? So you take “a region in space” and examine it? The collapse of a supernova, knowing it had a pre-existent state, what is geometrically happening through that evolution?

    I am following of course with interest.


  25. Andy Odell says:

    Sorry, but I think there is a fundamental mistake in this. 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? A varying H implies that the energy density also varies with time; not a problem, energy is conserved, but not energy density.

    Most beginning astronomy students make this same mistake (though much more naively) – that ‘velocity proportional to distance’ applies to a given object – as its distance increases, so does its velocity. But this is incorrect.