THE PYTHAGOREAN KOSMOS WAS EXPLAINED IN A WAY BY PLATO


Dodecahedral space topology as an  explanation for weak wide-angle  temperature correlations in the  cosmic microwave background

The current ‘standard model’ of cosmology posits an infinite flat  universe forever expanding under the pressure of dark energy.
First-year data from the  Wilkinson Microwave Anisotropy Probe (WMAP) confirm this model to spectacular precision on all but
the largest scales1,2. Temperature correlations across the microwave sky match expectations on angular scales narrower than 60 grades
but, contrary to predictions, vanish on scales wider than 60 grades.
Several explanations have been proposed3,4. One natural  approach questions the underlying geometry of space—namely,
its curvature5 and topology6. In an infinite flat space, waves from  the Big Bang would fill the universe on all length scales. The
observed lack of temperature correlations on scales beyond 60 grades  means that the broadest waves are missing, perhaps because  space itself is not big enough to support them. Here we present a  simple geometrical model of a finite space—the Poincare´ dodecahedral  space—which accounts for WMAP’s observations with  no fine-tuning required. The predicted density is Q0 < 1.013 > 1,and the model also predicts temperature correlations in matching  circles on the sky7.
Temperature fluctuations on the microwave sky may be expressed  as a sum of spherical harmonics, just as music and other sounds may  be expressed as a sum of ordinary harmonics. A musical note is the  sum of a fundamental, a second harmonic, a third harmonic, and so  on.

The relative strengths of the harmonics—the note’s spectrum—determines the tone quality, distinguishing, say, a sustained middle  C played on a flute from the same note played on a clarinet.
Analogously, the temperature map on the microwave sky is the  sum of spherical harmonics. The relative strengths of the harmonics—the power spectrum—is a signature of the physics and geometry  of the Universe. Indeed, the power spectrum is the primary  tool researchers use to test their models’ predictions against  observed reality.
The infinite universe model gets into trouble at the low end of the  power spectrum (Fig. 1). The lowest harmonic—the dipole, with
wave number l = 1—is unobservable because the Doppler effect of  the Solar System’s motion through space creates a dipole 100 times   stronger, swamping out the underlying cosmological dipole.

The  first observable harmonic is the quadrupole, with wave number  l = 2.  WMAP  found a quadrupole only about one-seventh as strong  as would be expected in an infinite flat space. The probability that  this could happen by mere chance has been estimated at about 0.2%  (ref. 2). The octopole term, with wave number l = 3, is also weak at  72% of the expected value, but not nearly so dramatic or significant  as the quadrupole. For large values of l, ranging up to l = 900 and  corresponding to small-scale temperature fluctuations, the spectrum  tracks the infinite universe predictions exceedingly well.
Cosmologists thus face the challenge of finding a model that  accounts for the weak quadrupole while maintaining the success of  the infinite flat universe model on small scales (high l). The weak  wide-angle temperature correlations discussed in the introductory  paragraph correspond directly to the weak quadrupole.
Microwave background temperature fluctuations arise primarily   (but not exclusively) from density fluctuations in the early Universe,because photons travelling from denser regions do a little extra work  against gravity and therefore arrive cooler, while photons from less  dense regions do less work against gravity and arrive warmer. The  density fluctuations across space split into a sum of three-dimensional  harmonics—in effect, the vibrational overtones of space  itself—just as temperature fluctuations on the sky split into a sum  of two-dimensional spherical harmonics and a musical note splits  into a sum of one-dimensional harmonics. The low quadrupole  implies a cut-off on the wavelengths of the three-dimensional  harmonics. Such a cut-off presents an awkward problem in infinite flat space, because it defines a preferred length scale in an otherwise  scale-invariant space. A more natural explanation invokes a finite  universe, where the size of space itself imposes a cut-off on the  wavelengths (Fig. 2). Just as the vibrations of a bell cannot be larger  than the bell itself, the density fluctuations in space cannot be larger
than space itself.Whereas most potential spatial topologies fail to fit  the WMAP results, the Poincare´ dodecahedral space fits them very  well.

image

Figure 1 Comparison of the WMAP power spectrum to that of Poincare´ dodecahedral
space and an infinite flat universe. At the low end of the power spectrum, WMAP’s results
(black bars) match the Poincare´ dodecahedral space (light grey) better than they match
the expectations for an infinite flat universe (dark grey). Computed for Q m = 0.28 and
Q L = 0.734 with Poincare´ space data normalized to the l = 4 term.

 

The Poincare´ dodecahedral space is a dodecahedral block of space  with opposite faces abstractly glued together, so objects passing out  of the dodecahedron across any face return from the opposite face.

image

Figure 2 Wavelengths of density fluctuations are limited by the size of a finite  ‘wraparound’ universe. a, A two-dimensional creature living on the surface of a cylinder  travels due east, eventually going all the way around the cylinder and returning to her
starting point. b, If we cut the cylinder open and flatten it into a square, the creature’s path  goes out of the square’s right side and returns from the left side. c, A flat torus is like a  cylinder, only now the top and bottom sides connect as well as the left and right. d, Waves  in a torus universe may have wavelengths no longer than the width of the square itself. To  construct a multi connected three-dimensional space, start with a solid polyhedron (for example, a cube) and identify its faces in pairs, so that any object leaving the polyhedron  through one face returns from the matching face. Such a multi connected space supports
standing waves whose exact shape depends on both the geometry of the polyhedron and  how the faces are identified. Nevertheless, the same principle applies, that the  wavelength cannot exceed the size of the polyhedron itself. In particular, the inhabitants of  such a space will observe a cut-off in the wavelengths of density fluctuations.

Light travels across the faces in the same way, so if we sit inside the   dodecahedron and look outward across a face, our line of sight reenters  the dodecahedron from the opposite face. We have the  illusion of looking into an adjacent copy of the dodecahedron. If  we take the original dodecahedral block of space not as a Euclidean  dodecahedron (with edge angles ,117grades ) but as a spherical dodecahedron  (with edge angles exactly 120grades), then adjacent images of the  dodecahedron fit together snugly to tile the hyper sphere (Fig. 3b),analogously to the way adjacent images of spherical pentagons (with  perfect 120 grades angles) fit snugly to tile an ordinary sphere (Fig. 3a).

image

Figure 3 Spherical pentagons and dodecahedra fit snugly, unlike their Euclidean  counterparts. a, 12 spherical pentagons tile the surface of an ordinary sphere. They fit  together snugly because their corner angles are exactly 120 grades. Note that each spherical  pentagon is just a pentagonal piece of a sphere. b, 120 spherical dodecahedra tile the  surface of a hyper sphere. A hyper sphere is the three-dimensional surface of a four dimensional  ball. Note that each spherical dodecahedron is just a dodecahedral piece of a  hyper sphere. The spherical dodecahedra fit together snugly because their edge angles  are exactly 120 grades . In the construction of the Poincare´ dodecahedral space, the
dodecahedron’s 30 edges come together in ten groups of three edges each, forcing the
dihedral angles to be 120 grades and requiring a spherical dodecahedron rather than a  euclidean one. Software for visualizing spherical dodecahedra and the Poincare´ dodecahedral space is available at khttp://www.geometrygames.org/CurvedSpacesl.

The power spectrum of the Poincare´ dodecahedral space depends  strongly on the assumed mass-energy density parameter Q0 (Fig. 4).

image

Figure 4 Values of the mass-energy density parameter Q 0 for which the Poincare´  dodecahedral space agrees with WMAP’s results. The Poincare´ dodecahedral space  quadrupole (trace 2) and octopole (trace 4) fit the WMAP quadrupole (trace 1) and
octopole (trace 3) when 1.012 < Q 0 < 1.014. Larger values of Q 0 predict an  unrealistically weak octopole. To obtain these predicted values, we first computed the  eigenmodes of the Poincare´ dodecahedral space using the ‘ghost method’ of ref. 10 with
two of the matrix generators computed in Appendix B of ref. 11, and then applied the  method of ref. 12, usingQ m = 0.28 andQ L = Q0 = 0.28, to obtain a power spectrum  and to simulate sky maps. Numerical limitations restricted our set of three-dimensional
eigenmodes to wave numbers k < 30, which in turn restricted the reliable portion of the  power spectrum to l = 2, 3, 4. We set the overall normalization factor to match the WMAP  data at l = 4 and then examined the predictions for l = 2, 3.
The octopole term (l =3) matches WMAP’s octopole best when   1.010 < Q 0 < 1.014. Encouragingly, in the subinterval
1.012 < Q 0 < 1.014 the quadrupole (l = 2) also matches the  WMAP value. More encouragingly still, this subinterval agrees  well with observations, falling comfortably within WMAP’s best fit   range of Q0 = 1.02 ^ 0.02 (ref. 1).
The excellent agreement with WMAP’s results is all the more  striking because the Poincare´ dodecahedral space offers no free  parameters in its construction. The Poincare´ space is rigid, meaning  that geometrical considerations require a completely regular dodecahedron.
By contrast, a 3-torus, which is nominally made by gluing  opposite faces of a cube but may be freely deformed to any  parallel piped, has six degrees of freedom in its geometrical construction.
Furthermore, the Poincare´ space is globally homogeneous,meaning that its geometry—and therefore its power  spectrum—looks statistically the same to all observers within it.
By contrast, a typical finite space looks different to observers sitting at different locations.
Confirmation of a positively curved universe (Q0> 1) would require revisions to current theories of inflation, but it is not certain
how severe those changes would be. Some researchers argue that  positive curvature would not disrupt the overall mechanism and
effects of inflation, but only limit the factor by which space expands  during the inflationary epoch to about a factor of ten8. Others claim  that such models require fine-tuning and are less natural than the  infinite flat space model9.
Having accounted for the weak observed quadrupole, the Poincare  ´ dodecahedral space will face two more experimental tests in the  next few years. (1) The Cornish–Spergel–Starkman circles-in-thesky method7 predicts temperature correlations along matching  circles in small multi connected spaces such as this one.

When  Q0 = 1.013 the horizon radius is about 0.38 in units of the curvature  radius, while the dodecahedron’s in radius and out radius are 0.31  and 0.39, respectively, in the same units. In this case the horizon  sphere self-intersects in six pairs of circles of angular radius about  35grades, making the dodecahedral space a good candidate for circle  detection if technical problems (galactic foreground removal, integrated  Sachs–Wolfe effect, Doppler effect of plasma motion) can be  overcome. Indeed, the Poincare´ dodecahedral space makes circle  searching easier than in the general case, because the six pairs of  matching circles must a priori lie in a symmetrical pattern like the  faces of a dodecahedron, thus allowing the searcher to slightly relax  the noise tolerances without increasing the danger of a false positive.
(2) The Poincare´ dodecahedral space predicts Q0 = 1.013>  1. The  upcoming Planck Surveyor data (or possibly even the existing
WMAP data in conjunction with other data sets) should determine  Q0 to within 1%. FindingQ0 < 1.01 would refute the Poincare´ space  as a cosmological model, while Q0 > 1.01 would provide strong  evidence in its favour.

Since antiquity, humans have wondered whether our Universe is  finite or infinite. Now, after more than two millennia of speculation,observational data might finally settle this ancient question.

Jean-Pierre Luminet1, Jeffrey R. Weeks2, Alain Riazuelo3,Roland Lehoucq1,3 & Jean-Philippe Uzan4

1Observatoire de Paris, 92195 Meudon Cedex, France
215 Farmer Street, Canton, New York 13617-1120, USA
3CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France
4Laboratoire de Physique The´orique, Universite´ Paris XI, 91405 Orsay Cedex,France
………………………………………………………………………………………………………………………………………………………..

Received 23 June; accepted 28 July 2003; doi:10.1038/nature01944.
1. Bennett, C. L. et al. First year Wilkinson Microwave Anisotropy Probe (WMAP 1) observations:
Preliminary maps and basic results. Astrophys. J. Suppl. 148, 1–27 (2003).
2. Spergel, D. N. et al. First year Wilkinson Microwave Anisotropy Probe (WMAP 1) observations:
Determination of cosmological parameters. Astrophys. J. Suppl. 148, 175–194 (2003).
3. Contaldi, C. R., Peloso, M., Kofman, L. & Linde, A. Suppressing the lower multipoles in the CMB
anisotropies. J. Cosmol. Astropart. Phys. 07, 002 (2003).
4. Cline, J. M., Crotty, P. & Lesgourgues, J. Does the small CMB quadrupole moment suggest new
physics? J. Cosmol. Astropart. Phys. (in the press); preprint at hhttp://arXiv.org/astro-ph/0304558l
(2003).
5. Efstathiou, G. Is the low CMB quadrupole a signature of spatial curvature? Mon. Not. R. Astron. Soc.
343, L95–L98 (2003).
6. Tegmark, M., de Oliveira-Costa, A. & Hamilton, A. A high resolution foreground cleaned CMB map
from WMAP. Phys. Rev. D (in the press); preprint at hhttp://arXiv.org/astro-ph/0302496l (2003).
7. Cornish, N., Spergel, D. & Starkman, G. Circles in the sky: Finding topology with the microwave
background radiation. Class. Quant. Grav. 15, 2657–2670 (1998).
8. Uzan, J.-P., Kirchner, U. & Ellis, G. F. R. WMAP data and the curvature of space. Mon. Not. R. Astron.
Soc. (in the press).
9. Linde, A. Can we have inflation with Q . 1? J. Cosmol. Astropart. Phys. 05, 002 (2003).
10. Lehoucq, R., Weeks, J., Uzan, J.-P., Gausmann, E. & Luminet, J.-P. Eigenmodes of 3-dimensional
spherical spaces and their application to cosmology. Class. Quant. Grav. 19, 4683–4708 (2002).
11. Gausmann, E., Lehoucq, R., Luminet, J.-P., Uzan, J.-P. & Weeks, J. Topological lensing in spherical
spaces. Class. Quant. Grav. 18, 5155–5186 (2001).
12. Riazuelo, A., Uzan, J.-P., Lehoucq, R. &Weeks, J. Simulating cosmic microwave background maps in
multi-connected spaces. Phys. Rev. D (in the press); preprint at hhttp://arXiv.org/astro-ph/0212223l
(2002).
Acknowledgements J.R.W. thanks the MacArthur Foundation for support.
Competing interests statement The authors declare that they have no competing financial
interests.
Correspondence and requests for materials should be addressed to J.R.W. (weeks@northnet.org).

SOURCE  http://www.obspm.fr/

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