Was Friedmann wrong?

Yesterday we wondered out loud whether cosmological evidence for dark matter might actually be pointing to something more profound: a deviation of the behavior of gravity from that predicted by Einstein’s general relativity (GR). Now let’s ask the same question in the context of dark energy and the acceleration of the universe.

We have (at least) two problems to face. First, if you do a back-of-the-envelope estimate of what the vacuum energy (the energy density inherent in empty space, or equivalently Einstein’s cosmological constant) should be, you get an answer that exceeds the observationally allowed value by the ridiculous factor of 120 orders of magnitude (10120). That’s bad, as far as agreement between theory and experiment is concerned. But nevertheless the universe is accelerating, which could be explained by a tiny amount of vacuum energy — about 10-8 ergs per cubic centimeter, if you care. Or perhaps by something else.

For example, maybe general relativity works for ordinary bound systems like stars and galaxies, but breaks down for cosmology, in particular for the expansion rate of the universe. In GR the expansion rate is described by the Friedmann equation, which sets the expansion rate proportional to the square root of the energy density. Ordinarily the density drops as the universe expands, and the expansion rate follows suit; if the density is constant (as with vacuum energy), the expansion rate can stay constant. (We would then interpret that as “accelerating”, oddly enough. A distant galaxy has a recession velocity v=Hd, where d is the distance and the Hubble constant H tells us the expansion rate. If H is constant and d is increasing, we would measure v to be increasing.)

So maybe Friedmann was somehow wrong. For example, maybe we can solve the problem of the mismatch between theory and experiment by saying that the vacuum energy somehow doesn’t make the universe accelerate like ordinary energy does. There are different ways to make this happen, none of which strike you as perfectly compelling. Arkani-Hamed, Dimopoulos, Dvali, and Gabadadze have proposed a “filter” by which smoothly-distributed energy doesn’t affect spacetime in the same way as localized energy; it’s somewhat ad hoc, but definitely interesting. With Laura Mersini I suggested that the pressure of a substance, as well as the energy density, contributes to the expansion rate, in just such a way as to make vacuum energy (which comes with a negative pressure) cancel exactly and leave spacetime unaffected. We were inspired by models with extra dimensions of space, but the particular models in question don’t quite work, so I’ve since spent a lot of time looking for models that are strictly four-dimensional. No luck yet.

Solving the cosmological constant problem is hard, and a popular strategy is to ignore it and try to account for dark energy separately. That is, imagine that the vacuum energy is set to zero by some mysterious mechanism, and something else is responsible for the acceleration of the universe. There are different approaches to this possibility as well. One is to be largely phenomenological, and just write down alternatives to the Friedmann equation to see what might work. This approach (inspired again by extra dimensions, but basically phenomenological) has been pursued by Dvali and Turner, and by Freese and Lewis. One issue here is that ultimately it wouldn’t be possible to distinguish experimentally between a modified Friedmann equation and some model of dark energy; both would only show up in the behavior of the expanding universe.

So the complementary approach is to come up with a whole new theory of gravity that can make the universe accelerate, and see if what that theory has to say about other tests of gravity. I’ve been thinking about this recently, having spent the weekend talking to Mark Trodden. He and I have written a paper with Duvvuri and Turner that explores an especially simple approach to doing this. We just suggest that the curvature of spacetime somehow resists going to zero, and can bounce back to infinity when it becomes small. (There are actually a lot of equations involved, but that’s the basic philosophy.) Then we can compare our model to experiments. Sadly, it fails. The simplest way to see this was pointed out by Chiba, who rewrote our theory in a way that resembles other models, and showed that ordinary tests of GR in the solar system are incompatible with our suggestion. Basically, the orbit of Mars would look different. But that’s okay; it just goes to show that GR is actually quite robust, and even an attempt to change it exclusively in cosmology ends up affecting all sorts of other things.

A more elaborate approach has been suggested by Dvali, Gabadadze, and Porrati (see a recent review by Dvali for more details). They again use an extra dimension of space, imagining that our world is a three-dimensional “brane” embedded in a four-dimensional space (plus one time dimension, as usual). They have invented a scheme whereby gravity can behave differently on and off the brane, become much weaker in the four-dimensional “bulk.” But at very large scales the bulk begins to affect our universe. They claim that, if we choose parameters appropriately (there is always a great deal of unnatural fine-tuning involved in these scenarios), we can straightforwardly explain the accelerated expansion of the universe. Even better, they are not yet (as far as we know) ruled bout by solar-system tests, but improved measurements might be able to detect new affects from the extra dimensions in the orbit of the moon (see also this paper by Lue and Starkman). This theory is not very well understood as yet, and deserves a lot more work to be fully explicated; but it’s interesting and promising, so we’ll have to see what happens as people think about it further. (There is a nice popular-level exposition by Dvali in Scientific American, although you have to pay for the full article online.)

So we don’t have any extremely compelling alternatives to Einstein’s theory, but there are a lot of possibilities and we have our work cut out for us. The good news is that observed phenomena (the dynamics of galaxies and clusters, the cosmic microwave background, the accelerating universe) are pushing us to think of profound new scenarios for gravity and cosmology.

By the way (as it were), John Scalzi links here and suggests that GR is bound to give way pretty soon. I hope that’s not the impression I’m actually giving; I personally think it’s an incredible long shot, but one worth pursuing. Probably the “standard” story of cold dark matter and vacuum energy are exactly right; this is a robust model that makes many more predictions than there are free parameters, and it’s well-motivated (although far from completely understood) in fundamental physics. In contrast, when we start modifying gravity, we’re just flailing around, hoping some good will come of it. But that’s how science works; the flailing always looks a little silly until it smacks into a bit of miraculous insight, which we then clean up and proclaim to be genius. Right now there’s a lot of talk of modifying gravity, because it’s well worth considering; but if you’re going to bet, Einstein gets much better odds even than Smarty Jones.

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