QuickPol

We provide utilities to compute beam matrices in the QuickPol formalism (Hivon et al. 2017). We introduce some additional steps here for computational efficiency. In this section, we use the indices $\ell, \ell', \ell''$ such that we don't need to change indices at the end in order to match Hivon. Define a scaled version of the scan spectrum

\[W_{\ell'}^{\nu_1,\nu_2,s_1,s_2,j_1,j_2} = \sum_{m^\prime=-\ell^\prime}^{\ell^\prime} \left(_{s_1+\nu_1}\tilde{\omega}^{(j_1)}_{\ell^\prime m^\prime}\right) \left(_{s_2+\nu_2}\tilde{\omega}^{(j_2)}_{\ell^\prime m^\prime}\right)^*\]

Define the matrix,

\[\begin{aligned} \mathbf{\Xi}^{\nu_1,\nu_2,s_1,s_2,j_1,j_2}_{\ell^{\prime\prime},\ell} &= (-1)^{s_1 + s_2 + \nu_1 + \nu_2} \sum_{\ell^{\prime}} \, \rho_{j_1,\nu_1} \rho_{j_2, \nu_2} W_{\ell'}^{\nu_1,\nu_2,s_1,s_2,j_1,j_2} \\ &\qquad\qquad \times \begin{pmatrix} \ell & \ell^{\prime} & \ell^{\prime\prime} \\ -s_1 & s_1+\nu_1 & -\nu_1 \end{pmatrix} \begin{pmatrix} \ell & \ell^{\prime} & \ell^{\prime\prime} \\ -s_2 & s_2+\nu_2 & -\nu_2 \end{pmatrix} \end{aligned}\]

This matrix does not depend on $u_1, u_2$. We can then write the beam matrix in terms of $\mathbf{\Xi}$,

\[\mathbf{B}_{\ell^{\prime\prime},\ell}^{\nu_1,\nu_2, u_1, u_2} \,= \sum_{j_1, j_2, s_1, s_2} \frac{2\ell + 1}{4\pi} \,_{u_1}\hat{b}^{(j_1)*}_{\ell, s_1} \,_{u_2}\hat{b}^{(j_2)*}_{\ell, s_2} \, \frac{k_{u_1} k_{u_2}}{k_{\nu_1} k_{\nu_2}} \, \mathbf{\Xi}^{\nu_1,\nu_2,s_1,s_2,j_1,j_2}_{\ell^{\prime\prime},\ell}\]

With this definition, the beam matrices $\mathbf{B}$ are sub-blocks of the linear operator relating the cross-spectrum to the beamed cross-spectrum (Hivon+17 eq. 38),

\[\tilde{C}^{\nu_1,\nu_2}_{\ell^{\prime\prime}} = \sum_{u_1,u_2}\left(\sum_{\ell} \mathbf{B}_{\ell^{\prime\prime},\ell}^{\nu_1,\nu_2, u_1, u_2} C_{\ell}^{u_1, u_2} \right).\]

Note that the inner sum is just a matrix-vector multiplication.

quickpolΞ!(𝚵::AA, ν₁, ν₂, s₁, s₂, ω₁, ω₂)

quickpolW(alm₁::Alm{Complex{T}}, alm₂::Alm{Complex{T}})

kᵤ([T=Float64], u)