Implementation of a Fourier matched filter in CMB analyses

Application to ISW studies

Max Planck Institut für Astrophysik, Karl Schwarzschild Str.1, 85741 Garching bei München, Germany e-mail: chm@mpa-garching.mpg.de

Received: 30 March 2008
Accepted: 11 July 2008

Abstract

We implement a matched filter (MF) cross-correlation algorithm in multipole space and compare it to the standard angular cross power spectrum (ACPS) method. Then we apply both methods to a integrated Sachs Wolfe (ISW) – large scale structure (LSS) cross correlation scenario. We study how sky masks influence the multipole range where the cross correlation signal arises and compare it to theoretical predictions. The MF requires the inversion of a multipole covariance matrix that, under partial sky coverage (), is generally non diagonal and singular. We chose a singular value decomposition (SVD) approach that enables identification of those modes carrying most of the information from those more likely to introduce numerical noise (that are dropped from the analysis). We compared the MF to the ACPS in ISW-LSS Monte Carlo simulations, focusing on the effect that a limited sky coverage has on the cross-correlation results. Within the data model where is Gaussian noise and is a known filter, we find that the MF performs better than the ACPS for lower 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 (lower values of f_sky). A preliminary analytical study of the ISW–LSS cross correlation signal-to-noise (S/N) ratio shows that most of it should be found on very large scales (50% of the S/N at , 90% at –50), and this is confirmed by Monte Carlo simulations. The statistical significance of our cross-correlation statistics reaches its maximum when considering , with for all values of f_sky observed, despite 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 on smaller (higher l-s) angular scales.