Galactic foreground residue biases in cosmic-microwave-background lensing-convergence reconstruction and delensing of B-mode maps

Open Access

Table 1.

The lensing response function, f_{ℓ₁Lℓ₂}, for different XY field pairs.

XY	f_l₁Ll₂^XY
TT	${\tilde{C}}_{ℓ_{1}}^{TT} F_{l_{2} L l_{1}}^{(0)} + {\tilde{C}}_{ℓ_{2}}^{TT} F_{l_{1} L l_{2}}^{(0)}$ $Mathematical equation: $ \tilde{C}_{\ell_1}^{TT}F^{(0)}_{l_2Ll_1} + \tilde{C}_{\ell_2}^{TT}F^{(0)}_{l_1Ll_2} $$
EE	${\tilde{C}}_{ℓ_{1}}^{EE} F_{l_{2} L l_{1}}^{(2)} + {\tilde{C}}_{ℓ_{2}}^{EE} F_{l_{1} L l_{2}}^{(2)}$ $Mathematical equation: $ \tilde{C}_{\ell_1}^{EE}F^{(2)}_{l_2Ll_1} + \tilde{C}_{\ell_2}^{EE}F^{(2)}_{l_1Ll_2} $$ , even
TE	${\tilde{C}}_{ℓ_{1}}^{TE} F_{l_{2} L l_{1}}^{(2)} + {\tilde{C}}_{ℓ_{2}}^{TE} F_{l_{1} L l_{2}}^{(0)}$ $Mathematical equation: $ \tilde{C}_{\ell_1}^{TE}F^{(2)}_{l_2Ll_1} + \tilde{C}_{\ell_2}^{TE}F^{(0)}_{l_1Ll_2} $$ , even
TB	$i {\tilde{C}}_{ℓ_{1}}^{TE} F_{l_{2} L l_{1}}^{(2)}$ $Mathematical equation: $ i \tilde{C}_{\ell_1}^{TE}F^{(2)}_{l_2Ll_1} $$ , odd
EB	$i {\tilde{C}}_{ℓ_{1}}^{EE} F_{l_{2} L l_{1}}^{(2)}$ $Mathematical equation: $ i \tilde{C}_{\ell_1}^{EE}F^{(2)}_{l_2Ll_1} $$ , odd

Notes. We considered the unlensed ${\tilde{C}}_{ℓ}^{BB} \sim 0$ $Mathematical equation: $ \tilde{C}_\ell^{BB} \sim 0 $$ . ‘Odd’ and ‘even’ denote whether the function is non-zero when l₁ + l₂ + L is odd or even, respectively.

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