Pixel-Pixel Covariance

• Pixel-Pixel Covariance
• • Definition
  • Ideal Pixel-Pixel Covariance
  • Reobserved Pixel-Pixel Covariance
  • Properties and Symmetries
  • Mathematical Details

Definition

The following definitions are a lightly-modified reproduction of Appendix A of Tegmark & Oliveira-Costa (2001).

The pixel-pixel covariance is best constructed in stages. We start by defining the covariance $\langle X_i Y_j\rangle$ between any two Stokes fields $X$ and $Y$ at pixels $i$ and $j$ for a local coordinate system where the $Q$ axis is pointed along the great circle connection pixel $i$ to pixel $j$. In this simplified case, the 6 unique covariances are

\[\begin{align} \covTT &\equiv \frac{1}{4\pi} \sum_\ell C_\ell^{TT} \covF{00} \label{eqn:theorycov:tt} \\ \covTQ &\equiv -\frac{1}{4\pi} \sum_\ell C_\ell^{TE} \covF{10} \label{eqn:theorycov:tq} \\ \covTU &\equiv -\frac{1}{4\pi} \sum_\ell C_\ell^{TB} \covF{10} \label{eqn:theorycov:tu} \\ \covQQ &\equiv \frac{1}{4\pi} \sum_\ell \left[ C_\ell^{EE} \covF{12} - C_\ell^{BB} \covF{22} \right] \label{eqn:theorycov:qq} \\ \covUU &\equiv \frac{1}{4\pi} \sum_\ell \left[ C_\ell^{BB} \covF{12} - C_\ell^{EE} \covF{22} \right] \label{eqn:theorycov:uu} \\ \covQU &\equiv \frac{1}{4\pi} \sum_\ell C_\ell^{EB} \left[ \covF{12} + \covF{22} \right] \label{eqn:theorycov:qu} \end{align}\]

for $z_{ij} = \cos(\sigma_{ij})$ and some fiducial spectrum $C_\ell$. The polarization weighting functions are simple functions of the $P_\ell$ and $P_\ell^2$ associated Legendre polynomials,

\[\begin{align} \covF{00} &\equiv (2\ell + 1) P_\ell(z_{ij}) \label{eqn:theorycov:F00} \\ \covF{10} &\equiv \chi_\ell \left[ \frac{z_{ij}}{1-z_{ij}^2} P_{\ell-1}(z_{ij}) - \left( \frac{1}{1-z_{ij}^2} + \frac{\ell-1}{2} \right)P_\ell(z_{ij}) \right] \label{eqn:theorycov:F10} \\ \covF{12} &\equiv \gamma_\ell \left[ \frac{\ell+2}{1-z_{ij}^2} z_{ij} P^2_{\ell-1}(z_{ij}) - \left( \frac{\ell-4}{1-z_{ij}^2} + \frac{\ell(\ell-1)}{2} \right) P^2_\ell(z_{ij}) \right] \label{eqn:theorycov:F12} \\ \covF{22} &\equiv 2\gamma_\ell \left[ \frac{\ell+2}{1-z_{ij}^2} P^2_{\ell-1}(z_{ij}) - \frac{\ell-1}{1-z_{ij}^2} z_{ij} P^2_\ell(z_{ij}) \right] \label{eqn:theorycov:F22} \end{align}\]

where^[1]

\[\begin{align} \chi_\ell &\equiv \frac{2\ell (2\ell + 1)} {\sqrt{(\ell-1)\ell(\ell+1)(\ell+2)}} \\ \gamma_\ell &\equiv \frac{2(2\ell + 1)} {(\ell-1)\ell(\ell+1)(\ell+2)} \end{align}\]

For a single pixel-pixel pair, the covariance terms form a symmetric $3 \times 3$ matrix $\mat M$

\[\begin{align} \mat M(z_{ij}) &= \begin{bmatrix} \covTT & \covTQ & \covTU \\ \covTQ & \covQQ & \covQU \\ \covTU & \covQU & \covUU \end{bmatrix} \end{align}\]

which is rotated from the local to a global Stokes coordinate system by application of the rotation matrix

\[\begin{align} \mat R(\alpha) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(2\alpha) & -\sin(2\alpha) \\ 0 & \sin(2\alpha) & \cos(2\alpha) \end{bmatrix} \end{align}\]

where $\alpha$ are bearing angles, such that the pixel-pixel covariance ${\mat C}_{ij}$ in global coordinates is defined by

\[\begin{align} {\mat C}_{ij} &= \mat R(\alpha_{ij}) \mat M(z_{ij}) \mat R(\alpha_{ji})^\top \label{eqn:theorycov:rotate} \end{align}\]

Pay attention to the fact that the left-hand $\mat R$ uses the bearing angle $\alpha_{ij}$ at pixel $i$ whereas the right-hand $\mat R$ uses the bearing angle $\alpha_{ji}$ at pixel $j$. Also note that these rotations move the covariances into the IAU polarization convention — to move into the HEALPix polarization convention, the rotations both need to be reversed, i.e. $\mat R \leftrightarrow \mat R^\top$.

The pixel-pixel covariance matrix $\mat C$ for some set of pixels is constructed by calculating the terms in ${\mat C}_{ij}$ for each pair of pixels and filling in the 9 values as the $(i,j)$-th entries of the covariance blocks in $\mat C$. By blocks, we refer to the fact that $\mat C$ can be block-decomposed as

\[\begin{align} \mat C &= \begin{bmatrix} \mat C^{TT} & \mat C^{TQ} & \mat C^{TU} \\ \mat C^{QT} & \mat C^{QQ} & \mat C^{QU} \\ \mat C^{UT} & \mat C^{UQ} & \mat C^{UU} \end{bmatrix} \end{align}\]

For example, if one holds $j$ constant and varies $i \in P$ across all pixels, then one would form 3 columns of $\mat C$, one in each of the block-columns.

It is also convenient to define the polarization-only block decompositions of the total covariance matrix as

\[\begin{align} \mat C^{\mathrm{Pol}} &\equiv \begin{bmatrix} \mat C^{QQ} & \mat C^{QU} \\ \mat C^{UQ} & \mat C^{UU} \end{bmatrix} \end{align}\]

Ideal Pixel-Pixel Covariance

Reobserved Pixel-Pixel Covariance

Properties and Symmetries

• The entire covariance matrix $\mat C$ is symmetric. Therefore the on-diagonal sub-blocks $\mat C^{TT}$, $\mat C^{QQ}$, and $\mat C^{UU}$ are individually symmetric as well, and the off-diagonal blocks must be related to one another as

  \[\begin{align*} \mat C^{TQ} &= \left(\mat C^{QT}\right)^\top & \mat C^{TU} &= \left(\mat C^{UT}\right)^\top & \mat C^{QU} &= \left(\mat C^{UQ}\right)^\top \end{align*}\]

  This directly implies the polarization-only matrix $\mat C^{\mathrm{Pol}}$ is symmetric as well.
• The covariance matrices $\mat C$, $\mat C^{TT}$, and $\mat C^{\mathrm{Pol}}$ are at least positive-semidefinite for appropriate non-zero fiducial input spectra. Positive-definiteness only occurs when there are at least as many non-zero harmonic modes ($C_\ell^{XY} \neq 0$ over all spectra $XY$) as there are diagonal elements in the matrix (the number of pixels in the map(s) that the covariance describes).
• The covariance matrices are linear in the fiducial spectra. For example, an $EE$-only covariance matrix $[\mat C]^{EE}$ (wherein all $C_\ell = 0$ except for $C_\ell^{EE}$ which is non-zero somewhere) and an $EB$-only covariance matrix $[\mat C]^{EB}$ can be summed to define

  \[\begin{align*} [\mat C]^{EEEB} = [\mat C]^{EE} + [\mat C]^{EB} \end{align*}\]

  which is equivalent to the covariance matrix which would have been produced if the fiducial spectrum had included the both of the $C_\ell^{EE}$ and $C_\ell^{EB}$ spectra from the start.
• Since the IAU and HEALPix polarization conventions differ by the direction of rotation in $\mat R(\alpha)$ which corresponds to a change in the sign of the $\sin$ terms ($\sij$ and $\sji$), the cosmologically-interesting case where $C_\ell^{EB} = 0$ and $C_\ell^{TB} = 0$ also simplifies such that the following are also true:

  \[\begin{align*} {\mat C}_{ij,\mathrm{Healpix}}^{TU} &= -{\mat C}_{ij,\mathrm{IAU}}^{TU} & {\mat C}_{ij,\mathrm{Healpix}}^{UT} &= -{\mat C}_{ij,\mathrm{IAU}}^{UT} \\ {\mat C}_{ij,\mathrm{Healpix}}^{QU} &= -{\mat C}_{ij,\mathrm{IAU}}^{QU} & {\mat C}_{ij,\mathrm{Healpix}}^{UQ} &= -{\mat C}_{ij,\mathrm{IAU}}^{UQ} \end{align*}\]

  and the remaining block components are unchanged.

Mathematical Details

It is often useful to have the fully-expanded expressions for each of the pixel-pixel covariance terms after applying the local-to-global coordinate system rotations. If we define short-cut notation for each of the terms in the rotation matrices as

\[\begin{align*} \mat R(\alpha_{ij}) &\equiv \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cij & -\sij \\ 0 & \sij & \cij \end{bmatrix} & \mat R(\alpha_{ji})^\top &\equiv \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cji & \sji \\ 0 & -\sji & \cji \end{bmatrix} \end{align*}\]

then expanding Eqn. $\ref{eqn:theorycov:rotate}$ explicitly (and grouping the terms by block-columns):

\[\begin{align*} \begin{bmatrix} {\mat C}_{ij}^{TT} \\ {\mat C}_{ij}^{QT} \\ {\mat C}_{ij}^{UT} \end{bmatrix} &= \begin{bmatrix} \covTT \\ \covTQ\cij - \covTU\sij \\ \covTQ\sij + \covTU\cij \end{bmatrix} \\ \begin{bmatrix} {\mat C}_{ij}^{TQ} \\ {\mat C}_{ij}^{QQ} \\ {\mat C}_{ij}^{UQ} \end{bmatrix} &= \begin{bmatrix} \covTQ\cji - \covTU\sji \\ \covQQ\cij\cji - \covQU(\cij\sji + \sij\cji) + \covUU\sij\sji \\ \covQQ\sij\cji + \covQU(\cij\cji - \sij\sji) - \covUU\cij\sji \end{bmatrix} \\ \begin{bmatrix} {\mat C}_{ij}^{TU} \\ {\mat C}_{ij}^{QU} \\ {\mat C}_{ij}^{UU} \end{bmatrix} &= \begin{bmatrix} \covTQ\sji + \covTU\cji \\ \covQQ\cij\sji + \covQU(\cij\cji - \sij\sji) - \covUU\sij\cji \\ \covQQ\sij\sji + \covQU(\cij\sji + \sij\cji) + \covUU\cij\cji \end{bmatrix} \end{align*}\]

Footnotes