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CMB and the Topology of the Universe. Dmitri Pogosyan, UAlberta. With: Tarun Souradeep Dick Bond and: Carlo Contaldi Adam Hincks. Simon White, Garching, last week: Open questions in Cosmology: Is our Universe is finite or infinite? Is its space curved ? ….

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Cmb and the topology of the universe
CMB and the Topology of the Universe

Dmitri Pogosyan, UAlberta


Tarun Souradeep

Dick Bond


Carlo Contaldi

Adam Hincks

Simon White, Garching, last week:

Open questions in Cosmology:

Is our Universe is finite or infinite?

Is its space curved ?


COSMO-05, Bonn 2005

Preferred parameter region tot 1 02 0 02
Preferred parameter region, tot=1.02 ± 0.02



COSMO-05, Bonn 2005

How Big is the Observable Universe ?

Relative to the local curvature & topological scales

COSMO-05, Bonn 2005

Absence of large scale power

COSMO-05, Bonn 2005

NASA/WMAP science team

Correlation function of isotropic cmb
Correlation function of isotropic CMB

Oscillations in the angular spectrum

come from this break

COSMO-05, Bonn 2005

Perturbations in compact space
Perturbations in Compact space

  • Spectrum has the lowest eigenvalue.

  • Spectrum is discrete, hence statistics in general is anisotropic, especially at large scales.

  • Statistical properties can be inhomogeneous.

  • However, perturbations are Gaussian, and CMB temperature fluctuations are fully described by pixel-pixel correlation matrix CT(p,p’)

  • We implemented general method of images to compute CT(p,p’) for arbitrary compact topology (Bond,Pogosyan, Souradeep, 1998,2002)

COSMO-05, Bonn 2005

Pixel pixel correlation with compact topology using method of images bps phys rev d 2000
Pixel-pixel correlation with compact topology (using method of images, BPS, Phys Rev D. 2000)

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Example of strong correlation on last scattering surface
Example of strong correlation on last scattering surface

These two points on LSS are identical

And so are these

COSMO-05, Bonn 2005

Correlated circles after cornish spergel starkman et al 1998 2004
Correlated Circles(after Cornish, Spergel, Starkman et al, 1998,2004)

(Figure: Bond, Pogosyan & Souradeep 1998, 2002)

COSMO-05, Bonn 2005

Temperature along the correlated circles pure lss signal
Temperature along the correlated circles(pure LSS signal)

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Positive curvature multiconnected universe
Positive curvature multiconnected universe ?



a Poincare dodecahedron “Soccer ball cosmos” ?

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Isotropic cpp
Isotropic Cpp’


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Surface term to dt t
Surface term to DT/T


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Complete surface and integrated large angle t t
Complete, surface and integrated large-angle ΔT/T


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Variation of the pixel variance
Variation of the pixel variance


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Constraining the models from maps
Constraining the models from maps

  • Complete topological information is retained when comparison with data is done on map level

  • Low res (Nside=16) maps contain most information, although special techniques as circle searching may benefit from finer pixalization. The cost – additional small scale effects which mask topological correlations.

  • Main signal comes from effects, localized in space, e.q on LSS. But even integrated along the line of sight contributions retain signature of compact topology.

  • Orientation of the space (and, possibly, position of observer) are additional parameters to consider. What is the prior for them ?

COSMO-05, Bonn 2005

Likelehood comparison of compact closed versus flat universe with 0 7
Likelehood comparison ofcompact closed versus flat Universe with =0.7

COSMO-05, Bonn 2005

WMAP: Angular power spectrum

NASA/WMAP science team

COSMO-05, Bonn 2005

Single index n:

(l,m) -> n


COSMO-05, Bonn 2005

Inadequacy of isotropized c l s a lm cross correlation
Inadequacy of isotropized Cl’s: alm cross correlation

Very small space

Just a bit smaller than LSS

2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8 9 10

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Compression to isotropic cls is lossy inhanced cosmic variance of cl s
Compression to isotropic Cls is lossy Inhanced cosmic variance of Cl’s

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Bipolar Power spectrum (BiPS) : variance of Cl’sA Generic Measure of Statistical Anisotropy

Bipolar multipole index

A weighted average of the correlation function over all rotations

Except for


(This slide is provided as a free advertisement for T. Souradeep talk, Wednesday)

COSMO-05, Bonn 2005

Conclusions variance of Cl’s

  • A fundamental question – whether our Universe is finite or infinite –is still open.

  • The region near Otot=1 is rich with possibilities, with negatively or positively curved or flat spaces giving rise to distinct topological choices.

  • Modern CMB data shows that small Universes with V < VLS are failing to describe the temperature maps. Reason – complex correlation are not really observed (in line with circle finding results).

  • This is despite the fact that it is not too difficult to fit the low l suppression of isotropized angular power spectrum.

  • Integrated along the line of sight contribution to temperature anisotropy masks and modifies topology signature. It must be taken into account for any accurate quantitative restrictions on compact spaces.

  • Full likelihood analysis assumes knowledge of the models. Model independent search for statistical anisotropy calls for specialized techniques – see talk on BiPS by Tarun Souradeep on Wednesday.

COSMO-05, Bonn 2005