Published online 20 February 2008 | Nature | doi:10.1038/news.2008.610


Cosmic coincidence spotted

An absurdly large number could hold the key to universal mysteries.

This unimaginably large number keeps popping up in descriptions of the Universe.

The secret of the Universe is not 42, according to a new theory, but the unimaginably larger number 10122. Scott Funkhouser of the Military College of South Carolina (called The Citadel) in Charleston has shown how this number — which is bigger than the number of particles in the Universe — keeps popping up when several of the physical constants and parameters of the Universe are combined1. This ‘coincidence’, he says, is surely significant, hinting at some common principle at work behind the scenes.

The number first turned up when, more than a decade ago, physicists discovered that the expanding Universe is accelerating. This implies that there is a force that opposes gravity on very large scales, which physicists call dark energy. It is quantified by a parameter called the cosmological constant.

One interpretation of dark energy is that it results from the energy of empty space, called vacuum energy. The laws of quantum physics imply that empty space is not empty at all, but filled with particles popping in and out of existence. This particle ‘fizz’ should push objects apart, just as dark energy seems to require. But the theoretical value of this energy is immense — so huge that it should blow atoms apart, rather than just causing the Universe to accelerate.

Physicists think that some unknown force nearly perfectly cancels out the vacuum energy, leaving only the amount seen as dark energy to push things apart. This cancellation is imperfect to an absurdly fine margin: the unknwon 'energy' differs from the vacuum energy by just one part in 10122. It seems incredible that any physical mechanism could be so finely poised as to reduce the vacuum energy to within a whisker of zero, but it seems to be so.

Five more times

Now Funkhouser says that this is not the only appearance of this vast number among the parameters of the Universe. He lists five other instances in which the ratios between various cosmic quantities turn out to be equal to 10122, give or take a factor of ten (which matters very little at such huge scales, and could be due to errors in our understanding of the numbers involved).

For example, the ratio of the mass of the observable Universe to that of the smallest possible ‘quantum’ of mass is about 6x10121. And the number of ways in which the particles of the current Universe can be arranged throughout space (a measure of entropy) is 2.5x10122.

This isn’t just numerological number juggling. “If you take the basic parameters of the Universe there are only so many ways you can put them together to make ‘pure numbers’ with no units,” Funkhouser says — and less still ones that have any obvious physical interpretation. So the fact that even a handful of these give ratios that are so huge and yet so similar seems significant. “It is unlikely for chance alone to be responsible for generating so many pure numbers from just several fundamental parameters,” says Funkhouser.

Logic behind the scenes

“For the same basic set of parameters to produce two large-number coincidence problems is essentially preposterous.”

Scott Funkhouser

A similar ‘large-number coincidence’ was noted in the 1930s by the astronomer Arthur Eddington and the physicist Paul Dirac. They saw that several other combinations, such as the ratio of the electrostatic attraction between an electron and a proton to the gravitational attraction, are equal to about 1040.

Like Dirac, Funkhouser thinks that these large-number coincidences can’t be accidental. All the instances of 10122, he argues, must stem from the same basic reason. It’s rather like noticing that the recurrence of spring and neap tides coincides with the phases of the Moon: both are due to the motions of the Moon and Earth.

At the root of the issue, Funkhouser says, is the fact that the current density of matter in the Universe is about the same as the observed vacuum energy density — a puzzling fact that he calls the ‘cosmic coincidence’. The vacuum energy density is thought to be constant, but the mass density changes as the Universe expands. Why they happen to be equal right now — a balance that helps us to detect dark energy amidst matter — is not known. Some have suggested explanations based on the anthropic principle: basically, it’s only under these conditions that life becomes possible, so if things were otherwise, we wouldn’t be here to see it.

Given this single coincidence, Funkhouser shows, all the 10122 ratios inevitably follow from the standard laws of physics and cosmology. “The major parameters of the Universe are the cosmological constant and the total mass and radius,” he says. “Due to the cosmic coincidence, they are related.” He has shown previously that similar reasoning accounts for the Eddington-Dirac large-number coincidence2.

Too many coincidences

The existence of two large-number coincidences (1040 and 10122) now presents a puzzle in itself.


“It is remarkable enough that the parameters of nature should somehow produce one large-number coincidence,” says Funkhouser. “For the same basic set of parameters to produce two large-number coincidence problems is essentially preposterous — unless the two problems are related.” But how?

Funkhouser notes that 10122 is about equal to the cube of 1040. Is there some reason why the two sets of large numbers should be linked in that way? It would follow, Funkhouser says, if there happens to be a certain mathematical relationship linking the mass of a nucleon (a proton or neutron) with the speed of light, the gravitational constant, Planck’s constant and the cosmological constant.

That sounds pretty exotic, but in fact such a link was proposed in 1967 by the Russian physicist Yakov Zel’dovich based on an entirely separate argument. And he’s not the only one to suggest this.

“The interesting thing is that the relationship has been proposed before by several different authors, each with a different explanation,” says Funkhouser. And in a separate paper3, he’s come up with yet another justification for it. “I have shown that a simple model for the origin of our Universe involving ten spatial dimensions leads naturally to this relation,” he says. It follows if seven of the dimensions shrank while the other three “puffed out” to form the reality we observe today. 

  • References

    1. Funkhouser, S. Proc. R. Soc. A advance publication doi:10.1098/rspa.2007.0370 (2008).
    2. Funkhouser, S. Proc. R. Soc. A 462, 2076 (2006).
    3. Funkhouser, S. Int. J. Theor. Phys. (in the press).
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