Escape from the clutches of the dark sector?

By Sean Carroll | October 17, 2005 11:56 am

Dark matter and dark energy make up 95% of the universe — or at least, we think so. Since these components are “dark,” we infer their existence only from their gravitational influences. Some of us have been foolhardy enough to imagine that these observations signal a breakdown of gravity as described by general relativity, rather than new stuff out there in the universe; but so far, the smart money is still on the existence of a dark sector that we have not yet directly detected.

There remains another possibility worth considering — that there is no dark stuff, and that gravity is perfectly well described by general relativity, but that we just aren’t using GR correctly. In other words, that the conventional theory can explain the observations perfectly well without dark matter or dark energy, we just have to be clever enough to figure out how. This would be the most radically conservative approach to the problem, in John Wheeler’s sense: we should push the smallest number of assumptions as far as they can possibly go.

Recently, separate attempts have been made to explain away “dark matter” and “dark energy” by this kind of strategy. In a paper that somehow got mentioned in the CERN Courier and on Slashdot, authors Cooperstock and Tieu have suggested that nonlinear effects in GR could explain flat rotation curves in spiral galaxies (one of the historically important pieces of evidence for dark matter). And in two papers, Kolb, Matarrese, Notari and Riotto and then just Kolb, Matarrese, and Riotto have suggested that nonlinear effects in GR could explain the acceleration of the universe (a key piece of evidence for dark energy). Are these people making sense? Are they crazy? Is this worth thinking about? Have they actually explained away the entire dark sector? (Answers: occasionally, possibly, yes, no.)

In both cases, the relevant technical issue is perturbation theory, specifically in the context of general relativity. Imagine that we have some equation (in particular, Einstein’s equation for the curvature of spacetime), and we’d like to solve it, but it’s just too complicated. But it could be that physically interesting solutions are somehow “close to” certain very special solutions that we can find exactly. That’s when perturbation theory is useful.

Call the solution we are looking for f(x), the special solution we know f0(x), and the small parameter that tells us how close we are to the special solution ε. For example, gravity is weak, so in GR the small paramter ε is typically something proportional to Newton’s constant G. Then for a wide variety of situations, the sought-after solution can be written as the special solution plus a series of corrections:

f(x) = f0(x) + ε f1(x) + ε2 f2(x) + …

So there are a series of functions that come into the answer, each of which is accompanied by a progressively larger power of ε. By only knowing the first one to start, we can often plug that solution into the equation we are trying to solve, and get an equation for the next function fi(x) that is much simpler than the full equation we are struggling to solve.

The point, of course, is that we don’t really need to get the whole infinite series of contributions. Since ε is by hypothesis small, every time we raise it to a higher power we get smaller and smaller numbers. Often you do more than well enough by just “going to first order” — calculating the εf1(x) term and forgetting about the rest. But it’s certainly possible to get into trouble — for example, there could be “non-perturbative effects” that this procedure simply can’t capture, or the perturbation series itself could be sick, for example if the function f2(x) were so huge itself that it overwhelmed the extra factor of ε it comes along with. We would then say that perturbation theory was breaking down.

In both of the attempts to do away with DM and DE, the authors are essentially claiming that this is what happens — conventional perturbation theory isn’t good enough for some reason. Let’s turn first to the attempt by Cooperstock and Tieu to do away with dark matter. To be honest, there are a bunch of problems with this paper. For example, equations (1) and (2) seem mutually inconsistent — they have chosen one coordinate system in which to express the spacetime metric, and another in which to express the spacetime velocity of the particles in the galaxy. Ordinarilly, you have to pick one coordinate system and stick to it. More importantly, Korzynski has analyzed their solution carefully and noticed that they have secretly included not only the mass of the stars, but a completely imaginary thin sheet of infinite density in the galactic plane. So the fact that the rotation curves don’t decay as they should is really no surprise.

But the real reason why most astronomers and physicsts didn’t take the paper seriously is that it violates everything we know about perturbation theory. In the galaxy, there are two parameters that are very small — the gravitational potential is about 10-6, and the velocity of the stars (compared to the speed of light) is about 10-3. So it would be surprising indeed if perturbation theory weren’t doing a really good job in this situation, even just including the first-order contribution. The real reason why nobody paid much attention to Cooperstock and Tieu is that they didn’t even seem to recognize that this was a problem, much less offer some proposed explanation as to why perturbation theory was breaking down. Extraordinary claims require extraordinary evidence, and we would need to be given a compelling reason to think that our perturbative intuition was failing before anyone would put a lot of effort into analyzing this paper.

The Kolb et al. work (which I’ve talked about before) is a slightly different story. These guys are appealing to second-order effects in cosmological perturbation theory to explain away dark energy. What they want to do is to point out that the real universe isn’t completely homogeneous and isotropic, it has fluctuations in it. The gravitational field of these fluctuations can be thought of as an effective source of energy and momentum, and should therefore contribute to the expansion history of the universe. Everyone agrees with this. The surprising part of the claim is that the second-order effects can be appreciably big, even though conventional perturbation theory would say they are small.

The first version of the claim, in the Kolb, Matarrese, Notari and Riotto paper, relied on perturbations that were super-Hubble-radius in size: larger than our currently observed universe. This really seemed surprising, as the situation outside our observable patch shouldn’t be able to affect us in any way. And indeed, the claim was more or less squashed in papers by Flanagan, Geshnizjani, Chung, and Afshordi, and Hirata and Seljak.

But like Rasputin, these guys are hard to kill off, and now they’re back with the paper by Kolb, Matarrese, and Riotto. (See also recent papers by Buchert.) Now they have ditched the super-Hubble idea, and are concentrating purely on second-order effects from small-scale perturbations. It’s a tricky problem, for various reasons. For example, you would like to average the effects of the perturbations over some region of space, and then use that to calculate the effect on the expansion of the universe. But it turns out to matter whether you first average and then evolve forward in time, or first evolve forward and then average. So, tricky. In addition, Ishibashi and Wald have written a careful paper that purports to show that a mechanism like this cannot possibly work, no matter what averaging procedure you use, although I haven’t looked carefully at that paper.

Still, the skepticism from most people stems from the simple fact that first-order perturbations are quite small, so second-order perturbations should be even smaller! Kolb et al. are experts, and they understand perfectly well that this is the issue; they’ve gone through some heroic calculations and are making the claim that perturbation theory is, indeed, breaking down. They themselves admit that this is far from sufficient to show that this effect makes the universe accelerate; but it certainly is necessary. A lot more work will need to be done before people have verified to their own satisfaction that the second-order terms really are anomalously large; it would be surprising, but interesting, and is worth understanding in any event.

As a final note to the conspiracy theorists out there, something that I like to emphasize: it would be great if any of these ideas were right. We’re not officers of the Establishment, trying to protect the unclothed Emperor of the Dark Sector from the taunts of cheeky truth-tellers. We’re all trying to figure out how the universe works, and any good ideas are more than welcome — so long as they make sense.

  • Dissident

    I typically get very excited and jump all over the place when one of these papers comes out (witness my recent – and much belated – reaction to the paper by Cooperstock and Tieu). After all, I do have a reputation as The Dissident to live up to, if only in my own mind…

    Alas, I must admit that were I to cast a vote here,

    I would now side with the “wrong” crowd. Still, it was a valiant attempt, and very much in line with what I’d like to see more people doing: thinking very carefully about all possible explanations within the existing framework before venturing out into cookoo-land.

  • tom

    Thanks for comprehensive roundup. I had heard about one of these papers but have not yet had the time to look into it. I totally agree that such a conservative solution would be great! :-)

  • Greg

    Thanks Sean. There are a surprising number of dark matter haters on slashdot, who reacted to Cooperstock and Tieu with much rejoicing. Perhaps they just like being contrarians.

  • Ben L

    So, here’s the reason I love Cosmic Variance. I got back to Chicago today, got to my office and at some point read the CERN Courier. I read the peice on Cooperstock and Tieu, and thought “Damn! That would be amazing if it’s true. I want someone who really understands GR to explain this to me. Maybe I should bug Sean to blog about it.” Then I went here.

    So, thanks Sean.

  • David Moles

    Nice explanation, Sean. Much more clear than the “just use GR!” vibe this reader (whose mathematical education stopped fifteen years ago with calculus in three dimensions) got from skimming Cooperstock and Tieu.

  • Plato

    The fundamentally new science of gravitational wave astronomy opens up a new window on the universe. Up until now, astronomy has relied on observations of electromagnetic wave signals (e.g. visible light, radio waves). The detection of gravitational waves offers a completely new perspective on the universe: they will enable us to “hear” the cosmic orchestra as well as to see it! GR17 will provide the scientific community with one of the earliest opportunities to discuss the first scientific results of this era.

    If you did not have a leading perspective in GR, to the conclusions of a cosmic scenario, as such with elliptical valuation might we ever discerned in global perspective (we know gravity waves are theoretical right?)[Taylor and Hulse]

    Then how indeed would any valuation on dark energy suffice us to say, that “Omega” would have a value in the expansitory situation at the edge of that same universe? Would also lead us to the “edge” on questions about the blackhole horizon?

    Something had to be created out of “nothing” right? Or was there something always there, that we never wanted to take to a limit in terms of theoretical developement?

    The circle/sphere being very large ( our cosmos), had to signal somekind of collapse, or could we ever find this association in what happens to the blackhole circle valuation in energy detrmination, would also define some edge for the universe?

    Tegmark said no to the “soccer ball”, maybe he might say no to the circles too?

  • citrine

    How will the non-linear effects modify the (interpretation of the?) metric components?

  • Brad Holden

    How funny, today’s CosmoClub was on this very subject. We had a speaker from the University of Florida, named Ethan Siegel, who talked about doing a better calculation like that of Kolb et al.

    The gist of it was that you need to calculate the scale factor evolution to arbitary order in density contrast but most other terms can be done to linear order. The original Seljak result was wrong, according to our speaker, but not very wrong. He had interesting things to say about super-horizon perturbations as well, namely that most inflationary models would require a fair bit of fine-tuning to produce the observations.

    The paper is published and, of course, on astro-ph.

  • agm

    for example, there could be “non-perturbative effects” that this procedure simply can’t capture, or the perturbation series itself could be sick, for example if the function f2(x) were so huge itself that it overwhelmed the extra factor of ε it comes along with.

    Still, the skepticism from most people stems from the simple fact that first-order perturbations are quite small, so second-order perturbations should be even smaller!

    I agree that in a given application of perturbation theory, a second order effect is likely to be small, if perturbation is valid way of looking at something. And since I know crap all about GR, I’m gonna take your word for it. However, you’re logic is truly dizzying. You are saying that it’s unlikely that Kolb et al are correct because their results are inconsistent with perturbation theory, after explaining that perturbation theory doesn’t apply when perturbative effects are hair on the back of the non-perturbative effects doing the heavy lifting.

  • Kaleberg

    And it might just turn out that these second order effects that play hob with applying perturbation theory ARE dark matter. Why do I keep thinking of that GR paper, was it by Wisdom, about how one can flap one’s wings in empty space and push oneself forward? It seems that it isn’t just quantum theory that says that there is something in a vacuum.

    Then again, I’m not a physicist. I sell vacuum cleaners.

  • Shantanu

    Thanks a lot Sean for the detailed description. This was helpful for
    non-experts like me. On a related note I am very curious to know
    your take on the speed of gravity/speed of light controversy from two years ago.
    In the last two weeks itself there have been many more papers on this.
    What is your take on it? If Kopeikin’s claims are correct, then does that
    mean that some brane/string based models which predict speed of gravity
    to be different from speed of light are ruled out? Note same question also
    holds for Mark, Risa, Clifford and Joanne and (other experts who read this blog).
    (If you have discussed this before, could you point me to the link?)

  • Sean

    Shantanu, I am not an expert. But as it says in the link you included, a consensus of people who are experts agree that this is not a measurement of something we would call “the speed of gravity.” I would tend to believe them.

  • Dissident

    Oh oh… here we go again:

    Off to jump all over the place.

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  • Arun

    Another paper has appeared:

    Presence of exotic matter in the Cooperstock and Tieu galaxy model

    Authors: D. Vogt, P. S. Letelier
    Comments: 4 pages

    We analyze the presence of an additional singular thin disk in the recent General Relativistic model of galactic gravitational field proposed by Cooperstock and Tieu. The physical variables of the disk’s energy-momentum tensor are calculated. We show that the disk is made of exotic matter, either cosmic strings or struts with negative energy density.

  • Thomas Larsson

    Another paper has appeared:


    Let me quote the entire section 4, Conclusion:

    “Dark matter does not exist.”

    Do the experts have an opinion about this paper?

  • Dissident

    Far from being an expert, I’d still expect those who claim such lofty status to focus on assumption #2: “in the limit of no rotation halo spacetime is Minkowski spacetime”. Why this should be more “likely” than assumption #4 (perfect fluid) is not obvious, at least to me.

    But for the time being, I’ll be doing my usual excited-jumping-around-routine. Thanks for pointing out this paper – I had (again!) completely missed it!

  • Count Iblis
  • Arun

    Thomas, how does the conclusion – dark matter is not a perfect fluid lead to the conclusion that dark matter doesn’t exist?

  • Dissident

    …and here’s the latest from Cooperstock & Tieu: no “thin sheet of infinite density”.

    Eagerly awaiting the response of the Dark Forces…

  • Scott

    Eagerly awaiting the response of the Dark Forces

    or simply an expert in GR. It seemed to refute the non-physical argument, but I had trouble following all the math, as the only background in GR is what I taught myself from Sean’s book this summer not to mention the fact that I am drained from lack of sleep and finals.

  • Scott

    background i have in

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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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