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8 th B A H M A N 1384. * * *. CMB Anisotropy & Polarization in Multiply Connected Universes. By: Ehsan Kourkchi IUCAA & Sharif Univ. of Tech. Supervisors: T. Souradeep & S. Rahvar. Saturday Jan. 28, 2006. Outline. What is the CMB? The Statistics of CMB - PowerPoint PPT Presentation
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CMB Anisotropy & Polarization in Multiply Connected Universes
* * *
By: Ehsan Kourkchi
IUCAA & Sharif Univ. of Tech.
Supervisors: T. Souradeep & S. Rahvar
8th
B
A
H
M
A
N
1384
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
*
Considering homogeneity dictate that:
*
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