Implementation of a Fourier Matched Filter in CMB Analyses. Application to ISW Studies

Aims: Implement a matched filter (MF) cross-correlation algorithm in multipole space and compare it to the standard Angular Cross Power Spectrum (ACPS) method. Apply both methods on a Integrated Sachs Wolfe (ISW) - Large Scale Structure (LSS) cross correlation scenario and study how sky masks influence the multipole range where signal arises and its comparison to theoretical predictions. Methods: The MF requires the inversion of a multipole covariance matrix that if $f_{sky} \lt 1$ is generally non-diagonal and singular. We use a SVD approach that focuses on those modes carrying most of the information. We compare the MF to the ACPS in ISW-LSS Monte Carlo simulations, paying attention on the effect that a limited sky coverage has on the cross-correlation results. Results: Within the linear data model for which the MF is defined, the MF performs comparatively better than the ACPS for smaller values of $f_{sky}$ and scale dependent (non-Poissonian) noise fields. In the context of ISW studies both methods are comparable, although the MF performs slightly more sensitively under more restrictive masks. A preliminary study predicts that most of the ISW--LSS cross correlation S/N ratio should be found in the very large scales (50% of the S/N at $l\lt 10$, 90% at $l\lt 40-50$), and this is confirmed by Monte Carlo simulations. The statistical significance of our cross-correlation statistics reaches its maximum when considering $l\in [2,l_{max}]$, with $l_{max} \in[5,40]$ for all values of $f_{sky}$ observed, despite of the smoothing and power aliasing that aggressive masks introduce in Fourier space. This $l$-confinement of the ISW-LSS cross correlation should enable a safe distinction from other secondary effects arising at smaller angular scales.