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Study of the Non-Gaussianity of the CMB

E. Martínez-González, A. Curto, R.B. Barreiro, P. Vielva, J.L. Sanz Grupo de Cosmología Observacional e Instrumentación

Instituto de Física de Cantabria (CSIC-UC), Santander (Spain)

2

nd

user’s coference, RES, Santander, 23 September 2009

(2)

Overview

• Introduction

• The statistical analysis based on wavelets

• Constraints on f

nl

using WMAP 5-year data

• Wavelets and the bispectrum

• Results and conclusions

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Introduction: local f nl

• A model that introduces the non-Gaussianity in the CMB through the primordial gravitational potential

Φ (x)

• At large scale, the dominant effect is the Sachs-Wolfe effect (see the n-pdf analysis by Vielva and Sanz 09)

• At small angular scales the transfer function should be taken into account

(

Φ Φ

)

+ Φ

=

Φ(x) L(x) fnl 2L(x) 2L(x)

3 ) (x T

T Φ

=

= 2

2 2

3 T

T T

f T T

T T

T L L

nl L

(4)

Data and methods

• DATA:

– Archeops baloom experiment

– Wilkinson Microwave Anisotropy Probe (WMAP)

• METHODS:

– Minkowski Functionals (for Archeops)

– Spherical Mexican Hat wavelet (for WMAP)

– N-point probability density function (Sachs-Wolfe regime)

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The wavelet

• The spherical Mexican hat wavelet (SMHW)

• Continuous wavelet transform

) 2 /(

2 2 2

2

2 2

1 2 ) ( 2 ) 1

;

( y R

S e

R y y

R

R N

+

=

Ψ θ π

Ψ

= ( ) ( ; , ) )

;

(b R dnf n n b R w

r r r

r r

) ,

; (x b R

S

r r Ψ

position

traslation

scale

) 2 / tan(

2 θ

= y

) , ( )

( )

;

( lm θ ϕ

lm

lm l R a Y b

R b

w r =

(6)

{ , , ...}

Wavelets

(7)

Statistics with the wavelet

Select a characteristic set of angular scales Rj

Compute the wavelet transform map w(bi,Rj) at these scales Rj

• Compute third order statistics by combining different scales

• Combine in a χ2 test all the estimators

= w b R w b R w b R db

q i j k

k j i ijk

r ) , ( ) , ( ) , 1 (

σ σ σ

=

rst ijk

rst rst

rst ijk ijk

ijk q C q q

q

,

1 ,

2 ( ) ( )

χ

=

rst ijk

f rst rst

rst ijk f

ijk ijk

nl q q nl C q q nl

f

,

1 ,

2( ) ( ) ( )

χ

data Gaussian model data fnl model

(8)

The data: WMAP

• Data from the WMAP (NASA) 5-year observations

– Frequencies: 41GHz (Q), 61 GHz (V), 94 GHz (W) – 8 radiometers: Q1, Q2, V1, V2, W1, W2, W3, W4

– Combine the data maps using as weights the inverse of the noise variance (Spergel et al. 2003)

– Considered maps: Q, V, W, V+W

– Resolution 6.9’ (Nside = 512 in HEALPix)

(9)

Results with the SMHW

• We consider 12 scales log spaced between 6.9’ and 500’

• 364 possible third order statistics with the wavelet coefficients

• We compare the data value with Gaussian simulations through a χ2 test

• We analyze the maps Q, V, W, V+W

47.9 47.8 48.7 49.2

0.35 376

364 354

W

0.27 377

364 348

V

0.63 378

364 384

Q

0.29 379

364 349

V+W

DOF

MAP χ data2 χ2 σ(χ2) P(χd2 χ2)

arXiv:0902.1523

(10)

Constraints on f nl

We calculate the expected values of the non-Gaussian models with fnl using simulations (Liguori et al. 2003,2007)

The minimum χ2(fnl) vs fnl provides the best-fitting value

We obtain the frequentist error bars with Gaussian simulations

Point sources contribution

7 6 ±

=

fnl

arXiv:0902.1523

30 65

W

30 23

V

33 11

Q

25 39

V+W

σ σ σ σ(fnl) Best fnl

MAP

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Analysis of the W band

The best-fitting fnl value is not compatible with zero at 95% CL for the W band but is compatible with the Q and V bands

• Considering the raw maps and the differences among different channels this deviation might be explained by systematics in the W band

0.34 -0.01

clean V1-V2

0.35 0.85

clean W1-W2+W3-W4

0.34 -0.52

clean W1+W2-W3-W4

44 59

clean W4

47 23

clean W3

45 91

clean W2

41 39

clean W1

0.29 1.02

clean V-W

0.29 -0.02

raw V-W

30 60

raw W

30 16

raw V

33 -3

raw Q

25 34

raw V+W

σ(fnl) Best fnl

Foreground Map

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Constraints on fnl

Rudjord et al. (2009) WMAP-5

Wavelets (needlet bispectrum)

-80 < fnl < 80

Pietrobon et al. (2009) WMAP-5

Wavelets (needlets) -50 < fnl< 110

Curto et al. (2009) WMAP-5

Wavelets (SMHW) -18 < fnl< +80

Smith et al. (2009) WMAP-5

Bispectrum -4 < fnl< +80

Slosar et al. (2008) SDSS and others

LSS -65 < fnl< +70

Gott et al. (2007) WMAP-3

Minkowski -101 < fnl < +107

Yadav & Wandelt (2008) WMAP-3

Bispectrum +27 < fnl< +147

Curto et al. (2008) Archeops

Minkowski -920< fnl < +1075

De Troia et al. (2007) BOOMERANG

Minkowski -800< fnl < +1050

Komatsu et al. (2008) WMAP-5

Minkowski -178 < fnl < +64

Cabella et al. (2005) WMAP-1

Local curvature &

wavelets -180 < fnl < +170

Creminelli et al (2003) WMAP-3

Bispectrum -36 < fnl< +100

Komatsu et al.(2008) WMAP-5

Bispectrum -9 < fnl< +111

Curto et al. (2008) WMAP-5

Wavelets (SMHW) -8 < fnl< +111

Paper Experiment

Method Constraints (95%CL)

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North and South f nl

• The localization property of the wavelets allows a local analysis of f

nl

• We have constrained f

nl

for the Galactic North and South regions

• f

nl

= 46 ± 37 for the North (68% CL)

• f

nl

= 35 ± 38 for the South (68% CL)

• North and South values are compatible

(14)

The extended masks

•Consider several angular scales between 6.9’ and 500’

•Each angular scale needs a extended mask

arXiv:0807.0231

(15)

Conclusions

• Cubic statistics based on wavelets are very useful in constraining f

nl

:

– Computationally fast

– Provide an independent check of the results with respect to systematics.

– Allow for localized analysis of sky patches.

• WMAP data compatible with Gaussian simulations according to the SMHW

• f

nl

compatible with zero –18 < f

nl

< +80 (95% CL)

• North and South f

nl

compatible

• The SMHW method is optimal (as efficient as the

bispectrum) to constrain f

nl

when combining large and small scales, as least in the ideal case.

• Work in progress to make the method optimal also in

realistic situations.

(16)

Andrés Curto

Why do we need high- performance computing?

• Our statistical analyses of CMB maps require statistical estimators (usually evaluated with simulations)

• We need to average over many simulations in order to reduce the random errors of the estimators

– Mean values: ~100 simulations

– Variances: ~10.000 simulations (~500000 for the N-pdf)

• We usually follow two steps:

– Generate Gaussian and non-Gaussian simulations with the properties of the instrument

– Analyze these simulations with the corresponding estimator

(17)

Andrés Curto

How have we used the computational resources?

• SOFTWARE

– IBM XL FORTRAN compilers (available in Altamira)

– HEALPix: Hierarchical Equal Area isoLatitude Pixelization of a sphere (http://healpix.jpl.nasa.gov/)

– Minkowski functionals software (developed by the Observational Cosmology and Instrumentation Group at IFCA)

– Spherical Mexican hat wavelet on the sphere software (developed by the Observational Cosmology and Instrumentation Group at IFCA)

(18)

Andrés Curto

How have we used the computational resources?

HARDWARE

– From 100 to 200 CPU’s at Altamira node running simple jobs (1 job per CPU)

– 1.000 GB of permanent disc and 500 GB of scratch disc

– Total time: 104.254 hours

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Publications

“Constraints on the non-linear coupling parameter fnl with Archeops data”, Curto A., Macías-Pérez J. F., Martínez-González E., Barreiro R. B., Santos D., Hansen F. K., Liguori M., Matarrese S., 2008, Astronomy and Astrophysics, 486, 383

“WMAP 5-year constraints on fnl with wavelets”, Curto A., Martínez-González E., Mukherjee P., Barreiro R. B., Hansen F. K., Liguori, M. & Matarrese S. 2008, Monthly Notices of the Royal Astronomical Society, 393, 615

“Analysis of non-Gaussian cosmic microwave background maps based on the N- pdf. Application to WMAP data”, P. Vielva, J.L. Sanz, Monthly Notices of The Royal Astronomical Society, 2009, 397, 837

“Improved constraints on primordial non-Gaussianity for the WMAP 5-year data”

Curto A., Martínez-González E., Barreiro R. B., 2009, Monthly Notices of the Royal Astronomical Society, to be published. arXiv:0902.1523

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FUTURE: Planck satellite

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