Dodecahedral space topology as an explanation for weak wideangle temperature correlations in the CMB

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Dodecahedral space topology as an explanation for
weak wide-angle temperature correlations in the
CMB

• By Jean-Pierre Luminet, et. al.
• NATURE VOL 425 9 OCTOBER 2003

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Outline

• Cosmic microwave background
• Results from WMAP
• Poincare dodecahedral space

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• rapid expansion after Big Bang (faster than
  light)
• acoustic wave
• about 380,000 years after Big Bang, the radiation
  (photons) decoupled from the matter (z1000)
• 2800 K
• now 2.725 K stretched and red-shifted to a
  distance of 15G lys.
• almost exactly a black body spectrum

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CMB anisotropy by WMAP This map shows a range of
about 0.0005 K from the coldest (blue) to the
hottest (red) parts of the sky.
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Cosmic Microwave Background Anisotropies

• At the most abstract level, there are only two
  factors relevant to the formation of
  anisotropies
• gravitational interactions
• SW (Sachs-Wolfe) effect
• The combination of intrinsic temperature
  fluctuations and gravitational redshift at or
  before last scattering
• ISW effect after last scattering
• Compton scattering (such as SZ effect).

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The relative strengths of the harmonics The
angular power spectrum

• An angular spectrum just refers to the amplitude
  of regular variations of some quantity
  (temperature or intensity) with angle.
• The CMB anisotropy field ?T/T can naturally be
  described in terms of spherical harmonics as in
  equation
• The lack of any preferred direction requires that
  there is no dependence on m and so the angular
  power spectrum can be expressed purely in terms
  of l with (2l1) statistically independent
  sub-components contributing to each power
  spectrum estimator
• The angular power spectrum is usually plotted out
  in a log-linear form in terms of Cl l(l1), as
  this shows the relative contributions to
  anisotropy from each angular frequency range and
  so a flat response here will correspond to a
  scale-invariant spectrum of the kind expected in
  inflation theories.

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The CMB angular power spectrum

• The CMB anisotropy field ?T/T can naturally be
  described in terms of spherical harmonics as in
  equation.
• The angular power spectrum of the anisotropy of
  the CMB contains information about the formation
  of the Universe and its current contents.
• The angular power spectrum in next page is a plot
  of how much the temperature varies from point to
  point on the sky (the y-axis variable) vs.

l1, dipole pattern. This indicates that the
Solar System is moving at 369 km/sec relative to
the observable Universe
Quadrupole
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Too little power detected at largest
scale Weak quadrupole
Where angular scale 180o/l
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• Each peak in the angular power spectrum
  corresponds to oscillations with just the right
  wavelength that they hit these extrema (turning
  points) at the time of re-combination, when the
  Universe became transparent.
• 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 60o means that the
  broadest waves are missing.
• Perhaps because space it self is not big enough
  to support them. Just as the vibrations of a bell
  cannot be larger than the bell it self, the
  density fluctuations in space cannot be larger
  than space itself.

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12 spherical pentagons tile the surface of an
ordinary sphere. They fit together snugly because
their corner angles are exactly 120o. Note that
each spherical pentagon is just a pentagonal
piece of a sphere.
120 spherical dodecahedra tile the surface of a
hypersphere. A hypersphere is the 3-D surface of
a 4-D ball. Note that each spherical dodecahedron
is just a dodecahedral piece of a hypersphere.
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Properties of Poincare dodecahedral space

• 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. 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.
• The Poincare space is rigid, meaning that
  geometrical considerations require a completely
  regular dodecahedron.
• Furthermore, the Poincare space is globally
  homogeneous, meaning that its geometryand
  therefore its power spectrumlooks statistically
  the same to all observers within it.

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Infinite flat universe
Poincare space
WMAP
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The Poincare dodecahedral space quadrupole (trace
2) and octopole (trace 4) fit the WMAP quadrupole
(trace 1) and octopole (trace 3) when 1.012
lt?0lt1.014. Larger values of ?0 predict an
unrealistically weak octopole. We set the overall
normalization factor to match the WMAP data at l
4 and then examined the predictions for l 2,
3.
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Two experimental tests in the next few years

• The predicted density is 1.013gt1.
• The upcoming Planck Surveyor data should
  determine O0 to within 1. Finding O0lt1.01 would
  refute the Poincare space as a cosmological
  model, while O0 would provide strong evidence in
  its favor.
• The model also predicts temperature correlations
  in matching circles on the sky six pairs of
  Cornish-Spergel-Starkman circles of angular
  radius about 35o, centred on different points on
  the sky.

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Cornish-Spergel-Stark Circles in Finite Universe
The signature is not constant temperature along
each circle, but identical temperatures at
identified points lying along pairs of circles.
pairs of circles
The self-intersection of the SLS is easiest
to visualize from the perspective of the
universal covering space. Fixing our attention on
a constant time spatial hypersurface, there will
be copies or clones of ourselves dotted about the
universe at positions dictated by the fundamental
group. Surrounding each clone will be a copy of
the sphere of last scatter. If any of our clones
are situated a distance less than Dsls away from
us, then our sphere of last scatter will
intersect the clones. Of course, there is no
physical distinction between us and our clones,
so the intersections seen in the covering space
are, in fact, self-intersections of the sphere of
last scatter.
The sphere of last scatter viewed in the
universal cover. The dark sphere marks the
primary copy and the four lighter spheres are
intersecting clones. The intersection that leads
to the matched circle pair.
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Other models New Physics?

• Loss of power at large angles could be
  explained by lowering the primordial power
  spectrum of inflation fluctuations (P(l)) at
  small wave numbers. For example
• from a break in the inflation potential Sharp
  features in the inflation potential can suppress
  power at low l.
• or an oscillating scalar field which couples to
  the inflation.

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References

• Jean-Pierre Luminet et al. Dodecahedral space
  topology as an explanation for weak wide-angle
  temperature correlations in the cosmic microwave
  background. Nature Vol 425, 593-595 (2003)
• C. L. Bennett et al. First year Wilkinson
  Microwave Anisotropy Probe (WMAP) observations
  Preliminary maps and basic results. Astrophys. J.
  Suppl. 148, 1-27 (2003).
• J. M. Cline et al. Does the small CMB quadrupole
  moment suggest new physics? J. Cosmol. Astropart.
  Phys. 07, 002(2003).
• L. Page et al. First-Year Wilkinson Microwave
  Anisotropy Probe (WMAP) observations
  Interpretation of the TT and TE angular power
  spectrum peaks. Astrophys. J. Suppl. 148, 233-241
  (2003).
• Cornish et al. Circles in the sky finding
  topology with the microwave background radiation.
  Class. Quantum Grav. 15, 2657-2670 (1998).