Foreground Contamination of CMB Maps By Tuhin Ghosh Under the Supervision of Prof. Tarun Souradeep.

Foreground Contamination of CMB Maps

ByTuhin Ghosh

Under the Supervision of

Prof. Tarun Souradeep

The Angular power spectrum of CMB anisotropy is considered a powerful tool for constraining cosmological parameters.

Fig. M. White 1997The Angular power spectrum of the CMB anisotropy dependssensitively on the present matter contained of the universe and the spectrum of primordial perturbations

lC

Angular Power Spectrum

2

l lmC a

0

( , )l

lm lml m l

Ta Y

T

( , )T T T

l=220

l=2

fig: WMAP 3rd YEAR DATA

The temperature anisotropy observed on the sky can be expanded as:

Where, T is mean temperature

Angular Power Spectrum is,

Where alm =expansion coefficient of temperature anisotropy

l is the spherical harmonics

List systematics

K band 23 GHz

Ka band 33 GHz

Q band 41 GHz

V band 61 GHzW band 94 GHz

CMB anisotropy signal

WMAP multi-frequency maps

CMB anisotropy signal is frequency independent

Foreground Properties

Variation of Foregrounds with latitude

From WMAP multifreq maps

Variation of Foregrounds with frequency

Antenna Temperature is related to the thermodynamic temperatures by the relation ,

where x=h/KT

Small fluctuations in antenna temperature can be converted to thermodynamic temperature fluctuation using,

TA is related to specific intensity for an extended source

S >> beam by the relation,

Antenna Temp. vs. Thermodynamic Temp.

Frequency dependence of the foreground emission:

Free-free emission has, (TA)free ~ f (for > 10GHz)

where f = -2.15

Synchrotron Spectrum is modeled by, (TA)Synch ~ s

where s ~ -3 at high galactic latitude and s ~ -2.5 close to galactic plane.

Dust Spectrum is given by,

with Tdust =18K and d~ 2.67 at mm wavelength.

CMB anisotropy signal is frequency independent

Mention template based cleaning for angular

Requires model

Model Independent Approaches toForeground Cleaning

WMAP: Cleaning in Pixel Space

T(p) =P 5

i=1 wi Ti (p)

P 5i=1 wi = 1

WMAP cleaning is done in the following way :

(1) Smooth each map with 10 beam resolution.

(2) Divide the sky in 12 disjoint regions.

(3) Internally combine all the channels such that:

And

e.g :- For the region outside the inner Galactic plane, the weights are wi =0.109, -0.684, -0.096, 1.921 and -0.250 for K, Ka, Q, V and W band respectively

Add Cl comparison From ILC

Reorder empasise (3)

nonumber

Improvement to this Method This method can be improved by taking the

assumption that the weights depend also on the angular scale (as well as galactic latitude).

As, the foregrounds dominates at large angular scale

whereas the detector noise dominates at small angular scale, so in angular scale weights are very useful.

Also, the angular scales resolved is not limited by lowest resolution channel but with high resolution channel.

Methodology

reword

CMB contributes equally in all 9 channels in terms of thermodynamic temperature.

It is assumed that foregrounds forms a positive covariance matrix. As variance differs drastically between clean and dirty regions. So,

we divide the sky into a set of regions of increasing cleanliness.

The maps at Planck frequencies di , where i=1,2,..,9 at related to the truesky b by the linear relation,

di = X i b+ X i f i +ni

di = X i b+ ci ci = X i f i + ni

W X = IP N

i=1 wi = 1:

which can be written as,

where

So, the true sky can be obtained from,~b= W di

~b= W X i b+ W ci

Basic Assumptions:

Imposes condition,

Split slide

Redo along lines of ..

(fig: Bond, 1996)

10 Ghz 30Ghz

60 Ghz 600 Ghz

ARCADE 2004

future: 3-90 GHz

10 GHz

30 GHz

Revise with ARCADEversion

We perform the cleaning method on harmonic space but over different portions of the sky. This method is using different weights for different multipoles also. The clean alm can be written as,

acleanlm =

P Ni=1 W i

lai

l mB i

l

Ccleanl = haclean

lm a¤cleanlm i

Ccleanl = h

P Ni=1 W i

lai

l mB i

l

P Nj =1 W j

laj

l m

B jl

i

So, the angular power spectrum is given by,

where, e = (1, 1, …,1) is a column vector of N ones.

Ccleanl = CS

l +W lCRl W T

l

W le=eT W Tl = I

To preserve the CMB power spectrum,we can write the above equation as:

With the condition that,

W Tl =

C ¡ 1l e

eT C ¡ 1l e

Accordingtoabovediscussion, thewholeproblemisreduced to theconditionthat W lC clean

l W Tl should beminimized with thecondition that W le=eT W T

l =I .

This is standard Lagrangian multiplier problem solution of which is given by,

WMAP Results

This part is done by Rajib Saha, Pankaj Jain and Tarun Souradeep

1 DA 2 DA

1 DA

2 DA4 DA

WMAP multi-frequency maps

(K,Ka)+Q1+V2+W12= (C13,CA13)(K,Ka)+Q1+V2+W13= (C14,CA14)(K,Ka)+Q1+V2+W14= (C15,CA15)(K,Ka)+Q1+V2+W23= (C16,CA16)(K,Ka)+Q1+V2+W24= (C17,CA17)(K,Ka)+Q1+V2+W34= (C18,CA18)

(K,Ka)+Q2+V1+W12= (C19,CA19)(K,Ka)+Q2+V1+W13= (C20,CA20)(K,Ka)+Q2+V1+W14= (C21,CA21)(K,Ka)+Q2+V1+W23= (C22,CA22)(K,Ka)+Q2+V1+W24= (C23,CA23)(K,Ka)+Q2+V1+W34= (C24,CA24)

(K,Ka)+Q2+V2+W12= (C7,CA7)(K,Ka)+Q2+V2+W13= (C8,CA8) (K,Ka)+Q2+V2+W14= (C9,CA9)(K,Ka)+Q2+V2+W23= C10,CA10)(K,Ka)+Q2+V2+W24= (C11,CA11)(K,Ka)+Q2+V2+W34= (C12,CA12)

(K,Ka)+Q1+V1+W12= (C1,Ca1)(K,Ka)+Q1+V1+W13= (C2,CA2)(K,Ka)+Q1+V1+W14= (C3,CA3)(K,Ka)+Q1+V1+W23= (C4,CA4)(K,Ka)+Q1+V1+W24= (C5,CA5)(K,Ka)+Q1+V1+W34= (C6,CA6)

Construct 48 CLEAN Maps2 x 2 x 2 x 6

K/Ka , Q , V ,Wij

IIT Kanpur + IUCAA

Independent, self contained analysis of WMAP multi-frequency maps

Saha, Jain, Souradeep(WMAP1: Apj Lett 2006)

WMAP 2nd release : Eriksen et al. ApJ. 2006 (5 international groups)

WMAP: Angular power spectrum

Planck Mission

Planck Surveyor SatelliteEuropean Space Agency: Launch 2008

European space agency (ESA)

Scheduled Launch in 2007

Planck Satellite on display at Cannes, France (Feb. 1, 2007)

30GHz 44 GHz 70GHz

100 GHz 143 GHz 217GHz

353 GHz 545 GHz 857 GHz

Synchrotron Radiation

30GHz 44 GHz 70 GHz

100 GHz 143GHz 217 GHz

353 GHz 545 GHz 857 GHz

Dust emission

30 GHz 44 GHz 70 GHz

100 GHz 143 GHz 217 GHz

353 GHz 545 GHz 857 GHz

Free-free emission

30 GHz 44 GHz 70 GHz

100 GHz 143 GHz 217 GHz

353 GHz 545 GHz 857 GHz

CMB Maps at Planck Frequencies

In this project, weassumethat noise is uncorrelated frompixel to pixel.This is a simplifyingassumption, that would beimproved oncerealistic noisecovariancematricesaremadeavailableby thePlanck collaboration.

<¢Tnoise(n)¢Tnoise(n0) >=¾2±nn0 (1)Thevalueof ¾0 at every channel isknownfromPlanck andNobs isthemapofthee®ectivenumber of observationsprovidedwiththedata. So, thenoisemapscanbeeasily calculatedusingtherelation,

¾= ¾0 gpNobs

(2)

wheregisaGaussianrandomvariablewithunit varianceandzeromean.

Simulated Noise Maps

Planck noise level after 14 months of its survey:

First we try out the method using only a very small subset of the numerous combinations allowed by the Planck mission. This is just to test that how this method works and how well we can recover the input power spectrum. The combinations for the test case is shown below:

Test Case

Low frequency channel High frequency channel

Auto Power Spectrum of 20 Maps Cross Power Spectrum of 12 Maps

Final Cl comparison

Conclusion from test case From this, we know that auto-correlated maps shows

some excess power at large l. But this can be removed by using the cross-correlation of two maps which has no channels common and has statistically independent noise. Hence, cross-correlation works better than auto correlation.

In this case there remains some residual along the center of galactic latitude which are supposed to affect large l values. For l < 1000, this is not affecting our result. But for more accurate estimation and for large l value, we have to mask it out with Kp2 mask and estimate the full sky power spectrum from cut sky power spectrum.

Also, we have to correct it for low l bias coming from the no. of channels combined. To remove this bias, we make many realizations of same sky and final Cl is corrected.

We first divide the Planck frequencies in the group of 4 and perform the following steps:-

The full sky is subdivided into 6 sky parts according to the level of contamination in 4 maps. We then repeat the following cleaning procedure six times, once corresponding to each region j = 1, 2, 3, 4, 5, 6.

Obtain the power spectrum matrix of the ith region and obtain the weight factors Wl.

Get the spherical harmonics of four maps and out of compute the all sky clean map using the weights obtained above.

Replace the ith region with the corresponding clean region. And before replacement it is smoothed by the resolution of the last input map.

At the end of 6 iterations, we get the clean map of CMB.

The above steps are repeated to get the clean map out of 4 channels corresponding to the high frequency end.

Steps for Planck Cleaning

W Tl =

C ¡ 1l e

eT C ¡ 1l e

acleanlm =

P Ni=1 W i

lai

l mB i

l

Partitioning the Sky Take the beam out of each 4 frequency channel (30 GHz, 44GHz,

70GHz, 100GHz). Make the 3 difference maps out of 4 frequency channel (44-30GHz, 70-

44GHz, 100-70GHz). Now form the junk map out of three difference maps by comparing

pixel wise absolute maximum value and each pixel is assigned a maximum value among the corresponding pixels of three difference assemblies.

Downgrade the junk map to Nside=64 by using the HEALPIX subroutine udgrade.

Then we divide the sky in 6 different regions in almost logarithm scale with

0 T<0.03mK, 0.03 T<0.1mK, 0.1 T<0.3mK, 0.3 T<1mK, 1 T<3mK and T 3. The resulting 6 partitions is then converted back to Nside =512 using

HEALPIX subroutine udgrade and then smooth with Gaussian beam of FWHM of 2 degree.

Apply and cutoff at 0.25 and write a small code that check that each pixel is not there in two partitions.

Now the 6 partitions are ready for the remaining analysis.

Low Frequency Cleaning:

Less words few junk maps

Partition: Low frequency channel

Partition: High frequency channel

Results

30 GHz 44 GHz

70 GHz100 GHz

Result of Low frequency Cleaning

100 GHz143 GHz

217 GHz 353 GHz

Result of High frequency Cleaning

Comparison of Input and Output Map

Difference Map

Input Map Low frequency clean map

Variation of Wl with multipole l

Power Spectrum Estimation

Different Schemes for Optimal Power Spectrum Planck has a large number of frequency channel each with a significant

number of independent channels. This allows us to envisage many different schemes for implementing the method. There is a compromise of two basic considerations in devising a optimal scheme:

When there are a large number of independent channels at a given frequency, the noise in the channel map are reduced if co add maps into a smaller subset of independent maps. This would reduce the role of noise in determining the weights for the obtaining cleaned maps. This also reduces the noise level in each of the cleaned maps.

However, merging channels reduces the number of independent cross-power spectra available to estimate the final power spectrum. Although cross correlating removes the noise bias, noise in cleaned maps for a cross spectra does feed into the variance. Larger the number of independent cross-spectra, smaller will be the variance .

We are hunting for the best scheme to get the power

spectrum.

Scheme:1

Possible Independent Combination is 1x1x3x2=6


High Frequency Combinations

Cross correlate each low frequency channel with the high frequency one.

Then we get 6x54=324 cross-correlate maps.

Result of Scheme I

Scheme:II

Low frequency combination High frequency combination



Possible Independent Cross-correlates Combined Maps is 12x1x2x1=24


Specialty : Two set of Cl

Quite feasible

No. of total Ch. 4 6 12 8 No. of total Ch. 8 12 12 12

Scheme:III

Get 2 set of Cl. The merging of more no. of differential assemblies results in increase in variance at large l due to lower number of cross correlation.






Expected not good result out of it

Scheme:IV






Quite Feasible

Scheme:V

There are huge no. of combination are possible for this case. In this case, the variance will gone down but in the cost of huge computational time and huge memory.So, this type of combination is also not computationally favorable. As, the size of each clean map is 12.5 Mb. So, the total space required for this cleaning is 12.5x13824 = 173 Gb for one realization. Say for 100 realization, we need 173Gbx100 = 2 Tb. For one clean map to made computer takes 10 minutes in Iucaa Cluster. Hence, for generating 13824 clean maps it will take 10x13824= 96 days on a single processor.


Possible Independent Combination is

4x6x12x8=2304


Quite Tough

Discussion The final goal would be select the most optimal one scheme

based on simulation. By choosing each differential assemblies as independent, we get large number of independent cross spectra. Large no. of cross spectra means it reduces the variance. But it is computationally challenging.

On the other hand, as the noise level in different frequency channels are different, hence it may make more sense to combine them in larger chunks (proportional to their noise level). However, co adding channels reduces the number of independent cross-power spectra available to estimate the final power spectrum.

In this project, we worked with full sky. We didn't mask out any portion of the sky because we found that it is not important for l < 1000. But as Planck will measure up to lmax ~ 4000. For that case we have to calculate actual full sky Cl from the cut sky Cl.

Conclusion In this project, using simulated data from the Planck

mission we establish that the model independent method of foreground cleaning gives the estimation of angular power spectrum solely based upon the Planck data.

The wider frequency coverage of the Planck mission and the larger number of independent detector channels with lower noise markedly improves the potential of this method compared to WMAP.

Moreover, the larger choice of channels combinations allow for many different implementation schemes for the basic method. Within the time limitations of project (and computational resources), here we have explored only one of the schemes.

Thank You

Motivation

For generation of foreground templates at Planck frequencies, we use Planck reference sky model (PSM), prepared by the members of Working Group 2 of the Planck Science collaboration. It generates the different foreground components like synchrotron emission, free-free emission and dust emission at Planck frequencies.

The input CMB Map is generated by using a software HEALPix (Hierarchical Equal Area and iso-Latitude Pixelization) from the

realization of best fit -CDM model and add pixel by pixel foreground component, correct beam and the noise,

where M is the map at given frequency, n is the noise in temperature, B is the FWHM of the beam .

M = B(Tcmb +Tsynchrotron +Tdust + Tf r ee¡ f ree) +n

To estimate the angular power spectrum out of 20 clean maps, we adopted the following steps:-

As the clean maps are smoothed with the highest frequency beam of the 4 channel combination. To get rid of the beam effect, basically alm is to be divided by the beam Bl of the highest beam.

The second step is to divide the final alm obtained in the above steps with the pixel window. In the software HEALPix, it always require convolve the map with the pixel window Pl of that resolution. So, basically combining the above two steps we have to divide the alm by Bl x Pl. This will remove both the effects.

To get the mean power spectrum, we can average all the 20 maps and compare with the input power spectrum. But due to noise which dominates at large l, we get excess power remains at large multipoles from the auto-correlation of the maps.

To get rid of noise bias, we cross correlate cleaned maps that do not have a common detector channel and have statistically independent noise. This way of cross correlation removes the residual noise. In other words, it remove the excess power at large l arising from the noise bias.

Power spectrum estimation

Foreground Contamination of CMB Maps By Tuhin Ghosh Under the Supervision of Prof. Tarun Souradeep.

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