An analysis of astronomical data suggests not only that the Universe is finite, but also that it has a specific, rather rigid topology. If confirmed, this is a major discovery about the nature of the Universe.
What shape is space? On page 593 of this issue, Luminet et al.1 suggest that the topology of the Universe may be a 'Poincaré dodecahedral space' — as illustrated on this week's cover. And this is no idle abstraction: Luminet et al. show that this topology, unlike many others, is supported by data from NASA's Wilkinson Microwave Anisotropy Probe (WMAP), published earlier this year2.
In thinking about the large-scale shape of the Universe, three interlinked questions must be confronted. First, what is its spatial curvature? There are three possible answers. Three-dimensional sections of space-time may be 'flat' — in such space sections, parallel lines stay the same distance apart and never meet (as in Euclidean space). Or the space sections may be 'negatively curved', such that parallel lines diverge from one another and never meet (the three-dimensional analogue of a Lobachevsky space). Finally, they may be 'positively curved', such that parallel lines converge and eventually intersect (the three-dimensional analogue of the surface of a sphere). The particular case that exists depends on how well the amount of matter in the Universe, coupled with the driving force of dark energy, balances the Universe's kinetic energy of expansion. This is usually expressed in terms of the normalized density parameter Ω0, which is unity for flat space sections; for positive spatial curvature, Ω0 is greater than one.
The second question is whether the Universe is 'open' or 'closed' — that is, is it spatially infinite, containing an infinite amount of matter, or is it spatially finite, containing a finite amount of matter? Positively curved space sections are necessarily closed, but the converse does not necessarily follow: both flat and negatively curved space sections can be finite if their connectivity is more complicated than in Euclidean space, meaning that their topology is quite unusual3,4 (for example, in a flat toroidal space, as you exit right you enter left, and space is finite). So the third issue is, what is the large-scale topology of the Universe?
It is worth noting that none of these features is determined by the Einstein gravitational field equations, which are differential equations that govern local, rather than global, properties of space-time5. Topology and curvature seem to be fixed by the initial conditions at the start of the Universe that have since determined its dynamical evolution. To investigate the topology and curvature of the Universe, we must use astronomical observations; from observed values of the energy densities and the expansion rate in the Universe, the curvature can be deduced using Einstein's field equations.
If the Universe is closed, and has a small enough diameter, we may be able to see right round it because photons can traverse the whole Universe — Luminet et al.1 illustrate this point well with the image of an insect crawling around the surface of a cylinder (Fig. 2 on page 593). If this is so, we may be able to identify multiple images of the same structure but in different directions in the sky4,6, or see an effect on the statistics of clustering of galaxies4. Furthermore, the existence of such a 'small universe' should be detectable through its influence on that Rosetta Stone of present-day cosmology, the cosmic microwave background (CMB) radiation7,8.
The CMB is the relic radiation of the Big Bang, and the very slight changes in its temperature across the sky record the density fluctuations that existed at a certain point in the early history of the Universe. Data on these anisotropies2,9 from WMAP and other sources suggest that the density parameter Ω0 has the value 1.02 ± 0.02. This is compatible with flat space sections. Further data might move the value of Ω0 towards or even below unity. But taking the present data at their face value, the conclusion is that the Universe is spatially closed and hence has finite space sections. This tentatively answers two of the three questions mentioned above.
But there is a further intriguing feature in the data. The so-called power spectrum of anisotropies (Fig. 1) shows a distinctive set of peaks when the anisotropy is compared between regions of sky separated by small angles. But on large angular scales (for regions typically more than 60° apart), there is a strange loss of power that does not fit with the expectations of standard cosmological models (in particular, there is less power in the quadrupole than expected). Some have proposed that this can be attributed to as yet undiscovered laws of physics at work in the early Universe10.
Luminet et al.1, however, suggest it is because we live in a universe with positively curved space sections and non-standard topology. The smaller-than-usual diameter of the Universe means that there is a maximum length scale that fits into it — and consequently there is a loss of power in the CMB spectrum on scales that are larger than this maximum7. The authors propose that the spatial sections of the Universe are dodecahedral sections of a space of positive curvature, fitted together to make finite three-dimensional spaces. This topology accounts for the WMAP data better than do standard models1.
Can this proposal be confirmed? Yes indeed. First, Luminet and colleagues' model suggests that Ω0 = 1.013, and future observations will produce data that should pin down the value of Ω0 to this level of accuracy (the current value from WMAP is accurate only to 2%). Second, a remarkable paper by Cornish et al.11 showed that, however complex the spatial topology, in a small universe there will be circles of identical temperature fluctuations in the CMB sky that could be identified from data on the CMB anisotropies. Luminet et al. determine what those circles would be in their model. Future analyses of WMAP data — and of data from its successor, the Planck satellite, to be launched in 2007 — should be able to verify whether the circles are there or not.
In many models of the Universe, it is assumed that spatial homogeneity extends outside our visual horizon for ever. However, in the case of chaotic inflation12,13 — a variant of the inflationary model for an exponential expansion that occurred in the wake of the Big Bang — the Universe is very inhomogeneous on scales much larger than we can observe, and we are in one expanding bubble in the middle of innumerable other, similar ones. But if Luminet et al. are correct, chaotic inflation is ruled out: there is only one expanding universe bubble, and we can see almost all the way round it.
In 1917, Einstein14 proposed that spatially closed universes are advantageous because that would remove the problem of boundary conditions at infinity15. A small universe in which we have seen most of what exists is even more advantageous6; indeed, strictly speaking, they are the only universes in which we can predict the astronomical future — the return of Halley's comet, for example — because only in them do we have access to all the data needed to make such predictions. The WMAP data, as interpreted by Luminet et al.1, suggest that we might indeed live in such a small closed universe.
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Hypersingular integral equations, Kähler manifolds and Thurston mirroring effect in ϵ(∞) Cantorian spacetime
Chaos, Solitons & Fractals (2007)
Chaos, Solitons & Fractals (2006)
Chaos, Solitons & Fractals (2006)
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Chaos, Solitons & Fractals (2005)