Study of the Non-Gaussianity of the CMB

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

Overview

• Introduction

• The statistical analysis based on wavelets

• Constraints on f

_nl

using WMAP 5-year data

• Wavelets and the bispectrum

• Results and conclusions

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) f_nl ²_L(x) ²_L(x)

3 ) (x T

T −Φ

∆ =











 ∆

∆ −

∆ −

∆ = ²

2 2

3 T

T T

f T T

T T

T _{L L}

nl L

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)

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 =}

∑

{ , , ...}

⊗

Wavelets

Statistics with the wavelet

• Select a characteristic set of angular scales R_j

• Compute the wavelet transform map w(b_i,R_j) at these scales R_j

• 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

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’ (N_side = 512 in HEALPix)

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⁽χ_d² ≤χ²⁾

arXiv:0902.1523

Constraints on f _nl

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

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

• We obtain the frequentist error bars with Gaussian simulations

Point sources contribution

7 6 ±

=

∆f_nl

arXiv:0902.1523

30 65

W

30 23

V

33 11

Q

25 39

V+W

σ σ σ σ(f_nl) Best f_nl

MAP

Analysis of the W band

• The best-fitting f_nl 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(f_nl) Best f_nl

Foreground Map

Constraints on f_nl

Rudjord et al. (2009) WMAP-5

Wavelets (needlet bispectrum)

-80 < f_nl < 80

Pietrobon et al. (2009) WMAP-5

Wavelets (needlets) -50 < f_nl< 110

Curto et al. (2009) WMAP-5

Wavelets (SMHW) -18 < f_nl< +80

Smith et al. (2009) WMAP-5

Bispectrum -4 < f_nl< +80

Slosar et al. (2008) SDSS and others

LSS -65 < f_nl< +70

Gott et al. (2007) WMAP-3

Minkowski -101 < f_nl < +107

Yadav & Wandelt (2008) WMAP-3

Bispectrum +27 < f_nl< +147

Curto et al. (2008) Archeops

Minkowski -920< f_nl < +1075

De Troia et al. (2007) BOOMERANG

Minkowski -800< f_nl < +1050

Komatsu et al. (2008) WMAP-5

Minkowski -178 < f_nl < +64

Cabella et al. (2005) WMAP-1

Local curvature &

wavelets -180 < f_nl < +170

Creminelli et al (2003) WMAP-3

Bispectrum -36 < f_nl< +100

Komatsu et al.(2008) WMAP-5

Bispectrum -9 < f_nl< +111

Curto et al. (2008) WMAP-5

Wavelets (SMHW) -8 < f_nl< +111

Paper Experiment

Method Constraints (95%CL)

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

The extended masks

•Consider several angular scales between 6.9’ and 500’

•Each angular scale needs a extended mask

arXiv:0807.0231

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.

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

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)

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

Publications

• “Constraints on the non-linear coupling parameter f_nl 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

FUTURE: Planck satellite