Gaussianity of the cosmic microwave background: smooth goodness-of-fit tests applied to interferometric data

Abstract

We adapt the smooth tests of goodness-of-fit developed by Rayner and Best to the study of the non-Gaussianity of interferometric observations of the cosmic microwave background (CMB). The interferometric measurements (visibilities) are transformed into signal-to-noise ratio eigenmodes, and then the method is applied directly in Fourier space. This transformation allows us to perform the analysis in different subsets of eigenmodes according to their signalto-noise ratio level. The method can also deal with non-uniform or incomplete coverage of the UV plane. We explore here two possibilities: we analyse either the real and imaginary parts of the complex visibilities (Gaussianly distributed under the Gaussianity hypothesis) or their phases (uniformly distributed under the Gaussianity hypothesis). The power of the method in discriminating between Gaussian and non-Gaussian distributions is studied by using several kinds of non-Gaussian simulations. On the one hand, we introduce a certain degree of non-Gaussianity directly into the Fourier space using the Edgeworth expansion, and afterwards the desired correlation is introduced. On the other hand, we consider interferometric observations of a map with topological defects (cosmic strings). To these previous non-Gaussian simulations we add different noise levels and quantify the required signal-to-noise ratio necessary to achieve a detection of these non-Gaussian features. Finally, we have also studied the ability of the method to constrain the so-called non-linear coupling constant f NL using χ 2 simulations. The whole method is illustrated here by application to simulated data from the Very Small Array interferometer.

Testing Gaussianity on Archeops data

“…For the Gaussian case, the mean should be equal to 1 and the dispersion equal to √ 2 (this is the null hypothesis, H 0 ). Following Aliaga et al (2005) and Rubiño-Martín et al (2006), the U 2 i are computed for a subset of signal-to-noise eigenmodes which are those associated with eigenvalues of the signal-to-noise matrix A satisfying (D A ) i ≥ (s/n) 2 c , where (s/n) c is a given signal-to-noise ratio cut. In Fig.…”

“…Aliaga et al 2003Aliaga et al , 2005. We used the five first statistics U 2 i which can be related to the k-order moments as follows,…”

“…This method has already been applied successfully to the MAXIMA (Cayón et al 2003b) and VSA experiments (Aliaga et al 2005;Rubiño-Martín et al 2006). The Archeops data will also be analysed with the morphological descriptors known as Minkowski functionals (Schmalzing & Górski 1998;Gott et al 1990).…”

Non-Gaussianity in the Very Small Array cosmic microwave background maps with smooth goodness-of-fit tests

“…In this section, we summarize some aspects of the smooth goodness‐of‐fit tests applied to CMB interferometers. For a more detailed description, see (Aliaga et al 2005, hereafter, A05).…”

Constraints on the non-linear coupling parameterfnlwith Archeops data

“…The Minkowski functionals have been applied to different recent experiments (Spergel et al 2007;de Troia et al 2007;Gott et al 2007;Curto et al 2007); other examples are the smooth tests of goodness-of-fit (Cayón et al 2003b;Aliaga et al 2005;Rubiño-Martín et al 2006;Curto et al 2007), wavelets (Barreiro et al 2000;Cayón et al 2001;Vielva et al 2004;Mukherjee & Wang 2004;Cruz et al 2005Cruz et al , 2007a, local estimators of the n-point correlations (Eriksen et al 2004(Eriksen et al , 2005, steerable filters (Wiaux et al 2006;Vielva et al 2007), and the CMB 1-pdf (Monteserín et al 2007) among others.…”