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Reconstruction of full sky CMB E and B modes spectra removing E-to-B leakage from partial sky using deep learning

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Abstract

Incomplete sky analysis of cosmic microwave background (CMB) polarization spectra poses a major problem of leakage between E- and B-modes. We present a machine learning approach to remove this E-to-B leakage using a convolutional neural network (CNN) in presence of detector noise. The CNN predicts the full sky E- and B-modes spectra for multipoles \(2 \le \ell \le 384\) from the partial sky spectra for \(N_\textrm{side} = 256\). We use tensor-to-scalar ratio \(r=0.001\) to simulate the CMB polarization maps. We train our CNN using \(10^5\) full sky target spectra and an equal number of noise contaminated partial sky spectra obtained from the simulated maps. The CNN works well for two masks covering the sky area of \(\sim \)80% and \(\sim \)10%, respectively after training separately for each mask. For the assumed theoretical E- and B-modes spectra, predicted full sky E- and B-modes spectra agree well with the corresponding target spectra and their means agree with theoretical spectra. The CNN preserves the cosmic variances at each multipole, effectively removes correlations of the partial sky E- and B-modes spectra, and retains the entire statistical properties of the targets avoiding the problem of so-called E-to-B leakage for the chosen theoretical model.

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Notes

  1. http://cmb-bharat.in/.

  2. https://camb.readthedocs.io/en/latest/.

  3. https://github.com/healpy/healpy.

  4. https://healpix.sourceforge.io/.

  5. https://pla.esac.esa.int/#results.

  6. https://pla.esac.esa.int/#results.

  7. https://www.tensorflow.org/.

  8. https://colab.research.google.com/.

  9. https://colab.research.google.com/.

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Acknowledgements

We acknowledge the use of openly available packages CAMB, https://camb.readthedocs.io/en/latest/ HEALPix, https://healpix.sourceforge.io/healpy, https://github.com/healpy/healpy TensorFlow https://www.tensorflow.org/ and Google Colab. https://colab.research.google.com/. We thank Ujjal Purkayastha, Sarvesh Kumar Yadav and Albin Joseph for constructive discussions related to this work.

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India

    Srikanta Pal & Rajib Saha

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  1. Srikanta Pal
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  2. Rajib Saha
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Correspondence to Rajib Saha.

Appendices

Appendix A: Mixing kernel

Using Equation (25) in Equations (26) and (27), the partial sky harmonic coefficients of E- and B-modes are given by:

$$\begin{aligned} \tilde{a}_{E,\ell m}&= -\frac{1}{2}\int \sum _{\ell 'm'}\Bigl [a^{\textrm{imp}}_{2,\ell 'm'}Y_{2,\ell 'm'}(\hat{n})W(\hat{n})Y^{*}_{2,\ell m}(\hat{n}) \nonumber \\&\quad +a^{\textrm{imp}}_{-2,\ell 'm'}Y_{-2,\ell 'm'}(\hat{n})W(\hat{n})Y^{*}_{-2,\ell m}(\hat{n})\Bigr ]d\Omega , \end{aligned}$$
(A1)
$$\begin{aligned} \tilde{a}_{B,\ell m}&= \frac{i}{2}\int \sum _{\ell 'm'}\Bigl [a^{\textrm{imp}}_{2,\ell 'm'}Y_{2,\ell 'm'}(\hat{n})W(\hat{n})Y^{*}_{2,\ell m}(\hat{n}) \nonumber \\&\quad -a^{\textrm{imp}}_{-2,\ell 'm'}Y_{-2,\ell 'm'}(\hat{n})W(\hat{n})Y^{*}_{-2,\ell m}(\hat{n})\Bigr ]d\Omega . \end{aligned}$$
(A2)

Using Equation (28) in Equations (A1) and (A2), the partial sky harmonic coefficients of these polarization modes can be expressed as:

$$\begin{aligned} \tilde{a}_{E,\ell m}&= -\frac{1}{2}\sum _{\ell 'm'}[a^{\textrm{imp}}_{2,\ell 'm'}W^{(+2)}_{\ell m\ell 'm'}+a^{\textrm{imp}}_{-2,\ell 'm'}W^{(-2)}_{\ell m\ell 'm'}] , \end{aligned}$$
(A3)
$$\begin{aligned} \tilde{a}_{B,\ell m}&= \frac{i}{2}\sum _{\ell 'm'}[a^{\textrm{imp}}_{2,\ell 'm'}W^{(+2)}_{\ell m\ell 'm'}-a^{\textrm{imp}}_{-2,\ell 'm'}W^{(-2)}_{\ell m\ell 'm'}] . \end{aligned}$$
(A4)

We can simplify these Equations (A3) and (A4) as:

$$\begin{aligned} \tilde{a}_{E,\ell m}&= \sum _{\ell 'm'}\Biggl [\Biggl \{\frac{1}{2}(W^{(+2)}_{\ell m\ell 'm'}+W^{(-2)}_{\ell m\ell 'm'})\Biggr \} \nonumber \\&\quad \times \Biggl \{-\frac{1}{2}(a^{\textrm{imp}}_{2,\ell 'm'} +a^{\textrm{imp}}_{-2,\ell 'm'})\Biggr \}\Biggr ] \nonumber \\&\quad +\sum _{\ell 'm'}i\Biggl [\Biggl \{\frac{1}{2}(W^{(+2)}_{\ell m\ell 'm'}-W^{(-2)}_{\ell m\ell 'm'})\Biggr \} \nonumber \\&\quad \times \Biggl \{\frac{i}{2}(a^{\textrm{imp}}_{2,\ell 'm'} -a^{\textrm{imp}}_{-2,\ell 'm'})\Biggr \}\Biggr ] ,\nonumber \\&= \sum _{\ell 'm'}[K^{(+)}_{\ell m\ell 'm'}a^{\textrm{imp}}_{E,\ell 'm'}+iK^{(-)}_{\ell m\ell 'm'}a^{\textrm{imp}}_{B,\ell 'm'}] , \end{aligned}$$
(A5)
$$\begin{aligned} \tilde{a}_{B,\ell m}&= \sum _{\ell 'm'}\Biggl [\Biggl \{\frac{1}{2}(W^{(+2)}_{\ell m\ell 'm'}+W^{(-2)}_{\ell m\ell 'm'})\Biggr \} \nonumber \\&\quad \times \Biggl \{\frac{i}{2}(a^{\textrm{imp}}_{2,\ell 'm'}- a^{\textrm{imp}}_{-2,\ell 'm'})\Biggr \}\Biggr ] \nonumber \\&\quad -\sum _{\ell 'm'}i\Biggl [\Biggl \{\frac{1}{2}(W^{(+2)}_{\ell m\ell 'm'}-W^{(-2)}_{\ell m\ell 'm'})\Biggr \} \nonumber \\&\quad \times \Biggl \{-\frac{1}{2}(a^{\textrm{imp}}_{2,\ell 'm'}+a^{\textrm{imp}}_{-2,\ell 'm'})\Biggr \}\Biggr ], \nonumber \\&= \sum _{\ell 'm'}[K^{(+)}_{\ell m\ell 'm'}a^{\textrm{imp}}_{B,\ell 'm'}-iK^{(-)}_{\ell m\ell 'm'}a^{\textrm{imp}}_{E,\ell 'm'}] , \end{aligned}$$
(A6)

where the mixing kernels are defined as:

$$\begin{aligned} K^{(\pm )}_{\ell m\ell ' m'} = \frac{1}{2}[W^{(+2)}_{\ell m\ell ' m'}\pm W^{(-2)}_{\ell m\ell ' m'}] . \end{aligned}$$
(A7)

Appendix B: Mixing matrix

Ensemble average of the realizations of the power spectrum is given by:

$$\begin{aligned} \langle C_{l}\rangle = \frac{1}{2\ell +1}\sum _{m}\langle a_{\ell m}a^{*}_{\ell m}\rangle . \end{aligned}$$
(B1)

Moreover, since the ensemble average of the entire realizations of the power spectrum deliver the theoretical power spectrum, the ensemble average of the multiplication between harmonic coefficient and its complex conjugate is given by:

$$\begin{aligned} \langle a_{\ell m}a^{*}_{\ell 'm'}\rangle = \langle C_{l}\rangle \delta _{\ell \ell '}\delta _{mm'} , \end{aligned}$$
(B2)

where \(\delta \) defines the Kronecker delta function.

The complex conjugate expressions of Equations (A5) and (A6) are given by:

$$\begin{aligned} \tilde{a}^{*}_{E,\ell m}&= \sum _{\ell 'm'}\Bigl [K^{(+)*}_{\ell m\ell 'm'}a^{\mathrm{imp*}}_{E,\ell 'm'}-iK^{(-)*}_{\ell m\ell 'm'}a^{\mathrm{imp*}}_{B,\ell 'm'}\Bigr ] , \end{aligned}$$
(B3)
$$\begin{aligned} \tilde{a}^{*}_{B,\ell m}&= \sum _{\ell 'm'}\Bigl [K^{(+)*}_{\ell m\ell 'm'}a^{\mathrm{imp*}}_{B,\ell 'm'}+iK^{(-)*}_{\ell m\ell 'm'}a^{\mathrm{imp*}}_{E,\ell 'm'}\Bigr ] . \end{aligned}$$
(B4)

The E-mode polarization has the opposite parity to the B-mode polarization. This provides:

$$\begin{aligned} \langle a_{E,\ell m}a^{*}_{B,\ell 'm'}\rangle = \langle a_{B,\ell m}a^{*}_{E,\ell 'm'}\rangle = 0 . \end{aligned}$$
(B5)

Therefore, using Equations (22)–(24) and (B5), we get:

$$\begin{aligned} \langle a^{\textrm{imp}}_{E,\ell m}a^{\mathrm{imp*}}_{B,\ell 'm'}\rangle = \langle a^{\textrm{imp}}_{B,\ell m}a^{\mathrm{imp*}}_{E,\ell 'm'}\rangle = 0 . \end{aligned}$$
(B6)

Taking the multiplication of Equation (A5) with Equation (B3) as well as the multiplication of Equation (A6) with Equation (B4) and also using Equation (B6), we can express the ensemble average of these multiplications between the partial sky harmonic mode and its complex conjugate. This expression corresponding to E-mode polarization is written as:

$$\begin{aligned}&\langle \tilde{a}_{E,\ell m}\tilde{a}^{*}_{E,\ell m}\rangle = \sum _{\ell 'm'}\sum _{\ell ''m''}[\{K^{(+)}_{\ell m\ell 'm'}K^{(+)*}_{\ell m\ell ''m''} \nonumber \\&\quad \times \langle a^{\textrm{imp}}_{E,\ell 'm'}a^{\mathrm{imp*}}_{E,\ell ''m''}\rangle \}+ \{K^{(-)}_{\ell m\ell 'm'}K^{(-)*}_{\ell m\ell ''m''}\nonumber \\&\quad \times \langle a^{\textrm{imp}}_{B,\ell 'm'}a^{\mathrm{imp*}}_{B,\ell ''m''}\rangle \}] . \end{aligned}$$
(B7)

Using Equations (20)–(24) and (B2), Equation (B7) can be simplified as:

$$\begin{aligned} \langle \tilde{a}_{E,\ell m}\tilde{a}^{*}_{E,\ell m}\rangle&= \sum _{\ell 'm'}\sum _{\ell ''m''}\Biggl [\Biggr \{K^{(+)}_{\ell m\ell 'm'}K^{(+)*}_{\ell m\ell ''m''} \nonumber \\&\times \Biggl (\langle C^{EE}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{EE}\rangle }{\mathcal {B}_{E}^{2}(\ell ')}\Biggr )\delta _{\ell '\ell ''}\delta _{m'm''}\Biggr \} \!\nonumber \\&+ \! \Biggl \{K^{(-)}_{\ell m\ell 'm'}K^{(-)*}_{\ell m\ell ''m''}\nonumber \\&\times \left( \langle C^{BB}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{BB}\rangle }{\mathcal {B}_{B}^{2}(\ell ')}\right) \delta _{\ell '\ell ''}\delta _{m'm''}\Biggr \}\Biggr ] \nonumber \\&= \sum _{\ell 'm'}\left[ |K^{(+)}_{\ell m\ell 'm'}|^{2}\Biggl (\langle C^{EE}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{EE}\rangle }{\mathcal {B}_{E}^{2}(\ell ')}\Biggr )\right. \nonumber \\&\left. + |K^{(-)}_{\ell m\ell 'm'}|^{2}\left( \langle C^{BB}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{BB}\rangle }{\mathcal {B}_{B}^{2}(\ell ')}\right) \right] . \end{aligned}$$
(B8)

Multiplying the both sides of Equation (B8) by \(\frac{1}{2\ell +1}\) after taking the sum over m index and then using Equation (B1), the more simplified form of Equation (B8) is expressed as:

$$\begin{aligned} \langle \tilde{C}^{EE}_{\ell }\rangle&= \sum _{\ell '}\sum _{mm'}\left[ \frac{|K^{(+)}_{\ell m\ell 'm'}|^{2}}{2\ell +1}\left( \langle C^{EE}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{EE}\rangle }{\mathcal {B}_{E}^{2}(\ell ')}\right) \right. \nonumber \\&\qquad \left. +\frac{|K^{(-)}_{\ell m\ell 'm'}|^{2}}{2\ell +1}\left( \langle C^{BB}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{BB}\rangle }{\mathcal {B}_{B}^{2}(\ell ')}\right) \right] , \nonumber \\&\quad = \sum _{\ell '}\left[ M^{(+)}_{\ell \ell '}\left( \langle C^{EE}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{EE}\rangle }{\mathcal {B}_{E}^{2}(\ell ')}\right) \nonumber \right. \\&\qquad \left. + M^{(-)}_{\ell \ell '}\left( \langle C^{BB}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{BB}\rangle }{\mathcal {B}_{B}^{2}(\ell ')}\right) \right] , \end{aligned}$$
(B9)

where

$$\begin{aligned} \langle \tilde{C}^{EE}_{\ell }\rangle = \frac{1}{2\ell +1}\sum _{m}\langle \tilde{a}_{E,\ell m}\tilde{a}^{*}_{E,\ell m}\rangle . \end{aligned}$$
(B10)

In Equation (B9), the mixing matrices are defined as:

$$\begin{aligned} M_{\ell \ell '}^{(\pm )} = \sum _{mm'}\frac{1}{2\ell +1}|K^{(\pm )}_{\ell m\ell ' m'}|^2 . \end{aligned}$$
(B11)

Equation (B9) denotes the relation between the full sky and the partial sky E-mode power spectra. Following the similar procedure, the partial sky power spectrum corresponding to the B-mode polarization, in terms of full sky spectrum, can be expressed by:

$$\begin{aligned} \langle \tilde{C}^{BB}_{\ell }\rangle&= \sum _{\ell '}\left[ M^{(+)}_{\ell \ell '}\left( \langle C^{BB}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{BB}\rangle }{\mathcal {B}_{B}^{2}(\ell ')}\right) \right. \nonumber \\&\qquad \left. +M^{(-)}_{\ell \ell '}\left( \langle C^{EE}_{\ell '}\rangle +\frac{\langle N_{\ell '}^{EE}\rangle }{\mathcal {B}_{E}^{2}(\ell ')}\right) \right] . \end{aligned}$$
(B12)

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Pal, S., Saha, R. Reconstruction of full sky CMB E and B modes spectra removing E-to-B leakage from partial sky using deep learning. J Astrophys Astron 44, 84 (2023). https://doi.org/10.1007/s12036-023-09974-4

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