Escape from the clutches of the dark sector?

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 f_i(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.