How to use

• Make sure you have healpy, pysm, and pymaster installed
• Run the notebooks!

01_ILC in pixel space

Internal Linear Combination of simulated CMB maps and foregrounds in pixel space Linear combination of component maps:

$$ d_i(p) = a_i s_{CMB}(p) +f_i(p) +n_i(p) $$

map, cmb, foregrounds, noise respectively

Assumes:

• CMB spectral energy distribution is a perfect blackbody
• CMB is uncorrelated to other sky components

Affected by errors dramatically. The ILC of the CMB is obtained by linearly combining input maps with frequency dependent weights.

The weights:

• preserve CMB blackbody spectrum
• minimise variance

$$ s_{ILC} (p) = \sum_i w_i . d_i(p) $$

with $\sum_i w_i .a_i =1$

$$ s_{ILC} (p) = s_{CMB}(p) + \sum_i w_i . (f_i(p) + n_i(p)) $$

$$ \Bigl \langle s^2_{ILC} \Bigr \rangle = \Bigl \langle s^2_{CMB} \Bigr \rangle + \sum_{i,i^{\prime}} w_i (\langle f_i f_i ^{\prime}\rangle + \langle n_i n_i ^{\prime} \rangle ) w_i^{\prime} $$

ILC method

• Compute the covariance $C_{i i^{\prime}} = \langle d_i (p)d_{i^{\prime}}(p)\rangle$
• Compute the inverse of the covariance
• The CMB spectral energy distribution

$$ A_{CMB} = [a_0, a_1,...,a_8]=[1,1,...,1] $$

• ILC weights can be computed as:

$$ w = \frac{A^T_{CMB} C^{-1}}{A^T_{CMB} C^{-1}A_{CMB}} $$

• Combine the input maps with the weights:

$$ \hat{S}_{ILC}(p) = \sum_i w_i . d_i(p) $$

02_ILC in harmonic space

Internal Linear Combination of simulated CMB maps and foregrounds in harmonic space Performing ILC in harmonic space (as opposed to pixel space) allows for better control over spatial scales using spherical harmonic decomposition.

• Transform Frequency Maps to Harmonic Space:

Each map $M_i(\hat{n})$ (where $i$ represents a frequency channel and $\hat{n}$ represents a sky direction) is decomposed into spherical harmonics:

$$ M_i(\hat{n}) = \sum_{l,m} a^i_{lm}Y_{lm}(\hat{n}) $$

where $a^i_{lm}$ are the harmonic coefficients

• Compute Covariance Matrix

covariance matrix of the harmonic coefficients:

$$C^{ij}_l = \langle a^i_{lm} a^{j*}_{lm} \rangle$$

where $i$, $j$ are the frequency channels

• Compute weights that minimize variance of the combined map while preserving the CMB signal:

$$\omega_i(l) = \frac{\sum_j (C_l^{-1})_{ij} b_{j}}{\sum_{ij}b_i (C_l^{-1})_{ij} b_{j}}$$

where $b_i$ represents the CMB transfer function. $b_i = [1,...,1]$

• Combine maps

Construct the harmonic coefficients of the cleaned map.

$$a^{ILC}_{lm} = \sum_i w_i(l)a^i_{lm}$$

• Reconstruct the map
• Compute the power spectrum

03_HILC after applying a mask

Masking the galactic plane can help reduce the main foregrounds and make extracting the CMB signal a lot easier. A cosine smoothed mask can be used to avoid any sharp edges. The coupled cross power spectrum of the masked maps aka the "pseudo power spectra" is given as

$$\tilde{C}_{\ell}^{ij} =\frac{1}{2 \ell + 1} \sum_{m=-\ell}^{\ell} \left| \tilde{a}_{\ell m} \right|^2$$

The mode coupling matrix $M_{\ell \ell'}$

To get the decoupled power spectra $\hat{C}{\ell}^{ij}$ we needed to invert the mode coupling matrix and convolve it with the pseudo $C{\ell}$:

$$\hat{C}_l^{ij} = M_{ll'}^{-1} \tilde{C}_l^{ij}$$