CMB Angular Power Spectrum estimation

CMB Angular Power Spectrum estimation 

Simona Donzelli 

University of Milan 

PhD School XXI cicle 

2D IDAPP meeting 

Ferrara, may 3-4 2007 

Simona Donzelli CMB Angular Power Spectrum estimation

Outline 

The Cosmic Microwave Background (CMB) 

Bias of the power spectrum estimation 

1 The Cosmic Microwave Background (CMB) 

Main features 

The angular power spectrum 

2 Bias of the power spectrum estimation 

Instrumental effects 

Incomplete sky coverage 




Main features 



Figure: The timeline of the Universe 




Main features 



main features 

photons coming from the “last scattering surface” (zR ∼ 1000, i.e. t 

∼ 400.000 yrs after the Big Bang) 

black body spectrum at T = 2.725 ± 0.001 K 

homogeneity and isotropy 

anisotropy of the order ∆T/T = 10 −5 

linearly polarized at a 10% level 




Statistics of the CMB sky 

spherical harmonic expansion 

Main features 


temperature T(^n) = 

aℓmYℓm(^n) 

ℓm 

polarization Q(^n) ± iU(^n) = 

ℓm 

a∓2,ℓm ∓2Yℓm(^n) 

where a±2,ℓm = Eℓm ± iBℓm, (E, B grad and curl component) 

angular power spectrum 

〈a Y ℓm〉 = 0, 〈a Y ′ 

∗Y 

ℓmaℓ ′ ′ 

m ′〉 = δℓℓ ′δmm ′〈CYY ℓ 〉, where Y, Y ′ can be T, E or B 

if the anisotropies obey Gaussian statistics then the spectra C TT 

ℓ , C EE 

ℓ , C BB 

ℓ , C TE 

ℓ 

′ 

YY 

describe all the statistical properties, where Cℓ = 1 ℓ 2ℓ+1 m=−ℓ (aY ′ 

∗Y 

ℓmaℓ ′ m ′ + c.c) 




Statistics of the CMB sky 

spherical harmonic expansion 

Main features 


temperature T(^n) = 

aℓmYℓm(^n) 

ℓm 

polarization Q(^n) ± iU(^n) = 

ℓm 

a∓2,ℓm ∓2Yℓm(^n) 

where a±2,ℓm = Eℓm ± iBℓm, (E, B grad and curl component) 

angular power spectrum 

〈a Y ℓm〉 = 0, 〈a Y ′ 

∗Y 

ℓmaℓ ′ ′ 

m ′〉 = δℓℓ ′δmm ′〈CYY ℓ 〉, where Y, Y ′ can be T, E or B 

if the anisotropies obey Gaussian statistics then the spectra C TT 

ℓ , C EE 

ℓ , C BB 

ℓ , C TE 

ℓ 

′ 

YY 

describe all the statistical properties, where Cℓ = 1 ℓ 2ℓ+1 m=−ℓ (aY ′ 

∗Y 

ℓmaℓ ′ m ′ + c.c) 




Main features 


Power Spectrum & High Precision Cosmology 

what can you learn from the spectrum? 

constrains on CMB power spectrum ⇒ constrains on cosmological parameters 




Main features 



state of the art: temperature 

Figure: The WMAP 3yr power spectrum 




Main features 



state of the art: temperature & polarization 

Figure: Plots of signal for TT (black), TE (red), EE (green) for the best fit model. The 

dashed line for TE indicates areas of anticorrelation 





in a real experiment 

the pseudo power spectrum eCℓ 



the direct spherical harmonics transform eCℓ of an anisotropies map is biased 

by: 










by: 

instrumental effects 

beam function of the antenna 

observation and data analysis processing 

instrumental noise 

incomplete sky coverage 

mode-mode coupling of the spherical harmonics 

E − B polarizations mode mixing 










by: 

















by: 

















by: 

















by: 











Debiasing from the instrumental effects 

the model 

In a full-sky hypothesis: 

contributions 



〈eCℓ〉 = FℓB 2 ℓ〈Cℓ〉 + 〈eNℓ〉 

window function B 2 ℓ: beam and pixelization smoothing effects 

transfer function Fℓ: effects of the data analysis (filtering, scanning 

strategy, etc.) 

instrumental noise 〈Nℓ〉: average noise power spectrum 





the model 

In a full-sky hypothesis: 

contributions 



〈eCℓ〉 = FℓB 2 ℓ〈Cℓ〉 + 〈eNℓ〉 

window function B 2 ℓ: beam and pixelization smoothing effects 

transfer function Fℓ: effects of the data analysis (filtering, scanning 

strategy, etc.) 

instrumental noise 〈Nℓ〉: average noise power spectrum 





the model 

estimation of the contributions 



〈eCℓ〉 = FℓB 2 ℓ〈Cℓ〉 + 〈eNℓ〉 (full-sky) 

window function B 2 ℓ: study of the antenna beam pattern 

calibrations with Monte Carlo (MC) simulations: 

transfer function Fℓ = 〈eCℓs〉〈C th 

ℓ 〉 −1 (B 2 ℓ) −1 , where C th 

ℓ is an arbitrary 

theoretical p.s. and 〈eCℓs〉 is obtained from “observations” of N MC full-sky 

and noise-free theoretical sky realizations. 

instrumental noise 〈eNℓ〉MC: N MC simulations of the noise cointained in the 

data 

⇒ an accurate knowledge on the noise characteristics is required! 





the model 








ℓ 〉 −1 (B 2 ℓ) −1 , where C th 





data 






the model 








ℓ 〉 −1 (B 2 ℓ) −1 , where C th 





data 






the model 








ℓ 〉 −1 (B 2 ℓ) −1 , where C th 





data 






the model 








ℓ 〉 −1 (B 2 ℓ) −1 , where C th 





data 







A new approach: cross power spectrum (CrossSpect) 

the idea 

I consider the maps of two independent detectors A e B 

gC AB 

ℓ 

= 


1 

2ℓ + 1 

ℓ 

m=−ℓ 

a A B ∗ 

ℓm aℓm ⇒ g C AB 

ℓ = B A ℓ B B ℓ F AB 

ℓ 〈Cℓ AB 〉 + 〈 g N AB 

ℓ 〉 

window function B A ℓ e B B ℓ 

transfer function F AB 

ℓ = p FA ℓ FB ℓ 

instrumental noise → is not correlated: 〈n A ℓmn 

B ∗ 

ℓ ′ m 

′〉 = 0 







the idea 


gC AB 

ℓ 

= 


1 

2ℓ + 1 

ℓ 

m=−ℓ 

a A B ∗ 




ℓ 〉 





B ∗ 

ℓ ′ m 

′〉 = 0 







the idea 


gC AB 

ℓ 

= 


1 

2ℓ + 1 

ℓ 

m=−ℓ 

a A B ∗ 




ℓ 〉 





B ∗ 

ℓ ′ m 

′〉 = 0 







the idea 


gC AB 

ℓ 

= 


1 

2ℓ + 1 

ℓ 

m=−ℓ 

a A B ∗ 




ℓ 〉 





B ∗ 

ℓ ′ m 

′〉 = 0 







the idea 


gC AB 

ℓ 

= 


1 

2ℓ + 1 

ℓ 

m=−ℓ 

a A B ∗ 




ℓ 〉 





B ∗ 

ℓ ′ m 

′〉 = 0 

⇒ ADVANTAGE: 〈 g NAB ℓ 〉 = 0 ! 




CrossSpect: fullsky results 



Figure: PLANCK 70 GHz - white noise – auto vs cross spectrum 




CrossSpect: fullsky results 



Figure: PLANCK 70 GHz - white noise 


Outline 





1 The Cosmic Microwave Background (CMB) 

Main features 


2 Bias of the power spectrum estimation 




mode-mode coupling 

the problem 





T(^n) = 

almYlm(^n) 

lm 

Q(^n) ± iU(^n) = 

a∓2,lm ∓2Ylm(^n) 

where a±2,lm = Elm ± iBlm 

if I introduce a weight function w(^n) 

 

˜Tlm = d^n w T (^n) T(^n)Y ∗ 

lm (^n) 

˜Elm = 1 

 

d^n w 

2 

P (^n) ˆ Q(^n) ` 2Y ∗ 

lm (^n) + −2Y∗ lm (^n)´ 

−iU(^n) ` 2Y ∗ 

lm (^n) − −2Y∗ lm (^n)´˜ 

˜Blm = − 1 

 

d^n w 

2 

P (^n) ˆ U(^n) ` 2Y ∗ 

lm (^n) + −2Y∗ lm (^n)´ 

+iQ(^n) ` 2Y ∗ 

lm (^n) − −2Y∗ lm (^n)´˜ . 

lm 







the problem 

espanding w: w(^n) = 

ℓm wlmYℓm(^n) 

˜Tlm = 

l ′ m ′ l ′′ m ′′ 

w T 

l ′′ m ′′ Tl ′ m ′ 

 

d^n Yl ′ m ′ (^n)Yl ′′ m ′′ (^n)Y ∗ 

lm (^n) 

˜Elm = 1 

2 

l ′ m ′ l ′′ m ′′ 

w P 

l ′′ m ′′ 

» 

El ′ m ′ d^n Yl ′′ m ′′ (^n) ` 2Yl ′ m ′ (^n) 2Y ∗ 

lm (^n) + −2Yl ′ m ′ (^n) −2Y∗ lm (^n)´ 

 

+iBlm d^n Yl ′′ m ′′ (^n) ` 2Yl ′ m ′ (^n) 2Y ∗ 

lm (^n) − −2Yl ′ m ′ (^n) −2Y∗ lm (^n)´– 

˜Blm = 1 

2 

l ′ m ′ l ′′ m ′′ 

w P 

l ′′ m ′′ 

» 

Bl ′ m ′ d^n Yl ′′ m ′′ (^n) ` 2Yl ′ m ′ (^n) 2Y ∗ 

lm (^n) + −2Yl ′ m ′ (^n) −2Y∗ lm (^n)´ 

 

−iElm d^n Yl ′′ m ′′ (^n) ` 2Yl ′ m ′ (^n) 2Y ∗ 

lm (^n) − −2Yl ′ m ′ (^n) −2Y∗ lm (^n)´– . 



the coupling term 





we have 

d^n NY ∗ 

lm (^n) N ′ Yl ′ m ′ (^n) N ′′ Yl ′′ m ′′ (^n) = 

(−1) N+m 

» ′ ′′ – 1/2 „ 

(2l + 1)(2l + 1)(2l + 1) 

4π 

l 

′ 

l l ′′ 

−N N ′ 

N ′′ 

« „ 

l 

−m 

′ 

l 

m ′ 

we can compute 

0 

B 

@ 

˜c TT 

l 

˜c EE 

l 

˜c BB 

l 

˜c TE 

l 

˜c TB 

l 

˜c EB 

l 

1 0 

C B 

C B 

C B 

C = B 

C B 

A @ 

M TT,TT 

ll ′ 

M EE,EE 

ll ′ M EE,BB 

ll ′ 

M BB,EE 

ll ′ M BB,BB 

ll ′ 

M TE,TE 

ll ′ 

M TB,TB 

ll ′ 

M EB,EB 

ll ′ 

1 0 

C B 

C B 

C B 

C B 

C B 

C B 

A @ 

Simona Donzelli CMB Angular Power Spectrum estimation 

l ′′ 

m ′′ 

« 

c TT 

l ′ 

c EE 

l ′ 

c BB 

l ′ 

c TE 

l ′ 

c TB 

l ′ 

c EB 

l ′ 

1 

C . 

C 

A




mode-mode coupling kernel 

M TT,TT 

ll ′ = (2l ′ + 1) 

4π 

M TE,TE 

ll ′ = MTB,TB 

ll ′ 

= 

(2l ′ + 1) 

8π 

M EE,EE 

ll ′ = MBB,BB 

ll ′ 

= 

(2l ′ + 1) 

16π 

M EE,BB 

ll ′ = MBB,EE 

ll ′ 

= 

(2l ′ + 1) 

16π 

 

l ′′ 

 

l ′′ 

W TT 

l ′′ 

„ ′ 

l l l ′′ 

0 0 0 

W TP 

l ′′ 

„ ′ 

l l l ′′ 

0 0 0 



« 2 

 

l ′′ 

W PP 

l ′′ 

»„ ′ 

l l l ′′ 

−2 2 0 

»„ ′ 

l l l 

× 

′′ « „ ′ 

l l 

+ 

−2 2 0 

« »„ 

l 

′ 

l l ′′ 

−2 2 0 

l ′′ 

2 −2 0 

 

l ′′ 

W PP 

l ′′ 

»„ 

l 

′ 

l l ′′ 

»„ 

× 

−2 

′ 

l l l 

2 0 

′′ « „ 

− 

−2 2 0 

l 

′ 

l 

 

= 

l ′′ 

2 −2 0 

« „ ′ 

l l l 

+ 

′′ 

2 −2 0 

«– 

« „ ′ 

l l l 

− 

′′ 

2 −2 0 

«– 

« „ ′ 

l l l 

+ 

′′ 

2 −2 0 

«– 

«– 

where W ab 

l w 

m 

a 

lmwb∗ lm 


«–



Kernel Mℓℓ ′ computation 

Wigner 3-j symbol 

it’s not null only if (triangle condition): 



„ « 

l1 l2 l3 

having 

m1 m2 m3 

|l1 − l2| ≤ l3 ≤ l1 + l2 e m1 + m2 + m3 = 0 

simmetry property: 

(−1) l1 +l2 +l „ 

3 

l1 

m1 

l2 

m2 

l3 

m3 

« „ 

l2 l1 l3 

= 

m2 m1 m3 


«



Kernel Mℓℓ ′ computation 

the code 



for l1 = 2, lmax 

for l2 = 2, l1 

for l3 = |l1 − l2|, l1 + l2 

if l1 + l2 + l3 =even 

M TT 

l1 ,l2 + = W TT 

„ 

l1 

l3 0 

l2 

0 

l3 

0 

« 2 

M TP 

l1 ,l2 + = W TP 

„ 

l1 

l 2 

3 0 

l2 

0 

l3 

0 

« „ 

l1 

−2 

l2 

2 

l3 

0 

M PT 

l1 ,l2 + = W PT 

„ 

l1 

l 2 

3 0 

l2 

0 

l3 

0 

« „ 

l1 

−2 

l2 

2 

l3 

0 

M PP 

l1 ,l2 + = W PP 

„ 

l 4 

3 

l1 

−2 

l2 

2 

l3 

0 

« 2 

if l1 + l2 + l3 =odd 

M PPleak 

l 1 ,l 2 

+ = W PP 

„ 

l1 l2 l3 

l 4 

3 −2 2 0 

« 2 


« 

«



mode-mode coupling kernel 

an example of kernel 



Figure: Binned kernel and its inverse (absolute values, with green color indicating the 

negative elements) for an elliptically top hat sky window 


Decoupling 





computation of the power spectrum 

〈eCℓ〉 = 

ℓ ′ 

Mℓℓ ′ Fℓ ′ B 2 

ℓ 〈Cℓ ′ 〉 , 

I consider Cℓ ≡ ℓ(ℓ + 1)Cℓ/2π. 

for a set of bins nbins, ℓ (b) 

min < ℓ(b) < ℓ (b+1) 

min , we can define the operator 

⎧ 

⎨ 1 ℓ(ℓ+1) 

2π 

Pbℓ = ℓ 

⎩ 

(b+1) 

min 

−ℓ(b) 

if 2 ≤ ℓ 

min 

(b) 

min ≤ ℓ < ℓ(b+1) 

min 

0 otherwise 

and the binned power spectrum is Cb = PbℓCℓ. The reciprocal operator is: 

 

2π se 2 ≤ ℓ 

Qℓb = ℓ(ℓ+1) 

(b) 

min ≤ ℓ < ℓ(b+1) 

min 

0 otherwise. 

Inverting we obtain: 

where 

〈Cb〉 = K −1 

bb ′ ` 

Pb ′ ℓ 〈eCℓ〉 ´ , 

Kbb ′ = PbℓMℓℓ ′ Fℓ ′ B 2 

ℓ ′ Qℓb ′ . 




CrossSpect: cut-sky results 



Figure: input vs output. Error bars evaluated from MC simulations 




conclusions and future 

conclusions 



the CrossSpect code is able to debiase the power spectrum from the 

main “observational” effects without requiring noise estimation 

actually it’s one of the code candidated for the future PLANCK power 

spectrum estimation: testing ongoing on simulations with increasing 

numbers on biasing effects 

possible developing 

take into account the correlated component of the noise 

estimate the error bars analytically 





conclusions 















conclusions 















conclusions 















conclusions 














CrossSpect: the core of the code 



tt[l] + = 2(alm1T (l, m).re alm2T (l, m).re + alm1T (l, m).im alm2T (l, m).im); 

gg[l] + = 2(alm1 G(l, m).re alm2 G(l, m).re + alm1 G(l, m).im alm2 G(l, m).im); 

cc[l] + = 2(alm1 C(l, m).re alm2 C(l, m).re + alm1 C(l, m).im alm2 C(l, m).im); 

tg[l] + = (alm1T (l, m).re alm2 G(l, m).re + alm1T (l, m).im alm2 G(l, m).im 

+alm1 G(l, m).re alm2T (l, m).re + alm1 G(l, m).im alm2T (l, m).im); 

tc[l] + = (alm1T (l, m).re alm2 C(l, m).re + alm1T (l, m).im alm2 C(l, m).im 

+alm1 C(l, m).re alm2T (l, m).re + alm1 C(l, m).im alm2T (l, m).im); 

gc[l] + = (alm1 G(l, m).re alm2 C(l, m).re + alm1 C(l, m).re alm2 G(l, m).re 

+alm1 G(l, m).im alm2 C(l, m).im + alm1 C(l, m).im alm2 G(l, m).im); 




CrossSpect: primi risultati 



Figure: 70 GHz - white noise – distribuzione C MC 

ℓ

CMB Angular Power Spectrum estimation

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