CMB Anisotropy & Polarization in Multiply Connected Universes

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By: Ehsan Kourkchi

IUCAA & Sharif Univ. of Tech.

Supervisors: T. Souradeep & S. Rahvar

8th

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Outline

What is the CMB?

The Statistics of CMB

The Different Possible Topologies of the Flat Universe

The Simplest Toroidal Compact Universe

Calculation of Correlation Function Using Naive Sachs-Wolf effect

CMB Map Generating

Considering the Other Physical Sources in Correlation Function

Map Analyzing

WMAP: First year

results announced on Feb. 11,

2003 !

NASA/WMAP science team

Isotropy and Homogeneity

CMB can be treated as a Gaussian Random Field.

T

)ˆ()ˆ()ˆ,ˆ( 2121 nTnTnnC

. Mean

. Correlation

<…> is ensemble average, i.e. an average over all possible

realizations

Nji TTT .... N point Correlation

The Whole information could be found in two-point correlation function

TOPOLOGY

A Toroidal Universe

Pictures: Weeks et. al. 1999 Slide by: Amir Hajian

Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

Different Flat Topologies

Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

Different Flat Topologies

Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

Different Flat Topologies

Table: Riazuelo et al. arXiv:astro-ph/0311314 v1 13 Nov 2003

Different Flat Topologies

Slab Space

Slab Space With Flip

2d Torus

Imagine a cube which each parallel pair of its

faces has been identified

Then

Confine the Last Scattering Surface into

a 3d Torus

Calculation of Correlation Function

On large angular scales where topological effect becomes important, Sachs-Wolf effect is dominant and the relation between temperature of Last Scattering Surface and gravitational potential is:

Conformal time

Correlation using only Sachs-Wolf effect

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Considering homogeneity dictate that:

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Calculation of Correlation Function . . .

Fourier Transform

xxikekkPdxxC )(

2

1),( 3

3

3k

Harrison-Zeldovich Spectrum

*

Calculation of Correlation Function . . .

xxikekkPdxxC )(

2

1),( 3

3

Correlation in a compact toroidal

universe

Correlation Function in a Compact Toroidal Universe

xx’

RR

L

ni

n

n

ekPRC

2

)()(

Using FFT method one can easily find the two point correlation

function for each pair very fast

Using FFT method and generating map realization

RL

ni

n

n

ekPRC

2

)()(

1) First we need to generate correlation matrix for each two point. For the last scattering surface we use HEALPix pixelization.

2) Decompose the covariance matrix into two matrices.

3) Multiply the decomposed matrix into a random matrix to have a map realization.

TAAC

ii AMAP

Random matrix,< >

Correlation maps …

RL

The correlations between the pole of last scattering

surface and the other points of the sphere.

R/L = 1

The correlations between the pole of last scattering

surface and the other points of the sphere.

R/L = 1.5

Correlation function between two points on a surface R=L/2

Correlation function between two points on a circle vs. angle separation R=L

Corr

ela

tion

2

Correlation Function in a Compact Torus UniverseUsing all physical sources.

To considering all physical effect (not only naive Sachs-Wolf effect, we have such relation:

S is the source function which

contains all information since the CMB photons

emitted to this point we observe them

If we have statistical isotropy, the angular parts could be taken out and calculated easily to reduce the relation to:

Regarding to this condition the above integral should be taken over 1 dimensional k space and the process is fast enough.

But, to investigate the topological effects we can no longer do the previous method. The integral over 3 dimensional k space is also taking the huge time (e.g. its order of magnitude is something like the Universe age ) What to

do ! ?

Correlation Function in a Compact Torus UniverseUsing all physical sources.

Separation of the

Integral

More calculations ….

Adding topological constraints, only some special Ks contribute in the summation,

It is under progress …

Statistical analysis of different generated maps …

T1

T2

St = < (T1-T2)2 >1/2

Symmetrical Maps (smaps):

For each point of map it can be defined another temperature which is the square root of mean square of difference of each

point temperature and its image regarding to the plain which it normal

vector is the axis of symmetry connecting the main point and the center of the

sphere.

Symmetrical Maps (smaps):

The point which has lower temperature shows the axis around which the map is most symmetrical.

Doing some statistical analysis might enable us to get some particular limits on most probable volume of the compact space.

Oliveira-Costa & Smoot 1995Oliveira-Costa 2003

Statistical analysis of different generated maps …(Naive Sachs-Wolf effect)

Smap generated using a map with R/L=1

Some cool points show that there are some proffered axis in our universe.

Absolutely, having a torus topology make the Universe some symmetrical axis.

Smap

Smap generated using a map with R/L=1.5

Smap Analysis …

<St>min = S0

S0

Map Number

Smap

<St>min = S0

Probability of finding a map which its Smin is less than S0

S0

Different figures for different

R/L ratios.

Under Progress ….

generating different map realization containing all physical sources using

appropriate calculated correlation matrices, we will be able to predict the

properties of real toroidal compact spaces …

Hope to be done

Calculating correlation matrix using faster methods

Generating map realization using all physical sources using appropriate correlation matrices

Analyzing our generated maps for different compact spaces

Investigating of non-statistical isotropic maps using different methods (Bips, S-map, …) and put some constraint on the type and size of the possible compact fundamental domains

Using the source functions of polarization in compact spaces and do everything again