Fast directional correlation on the sphere with steerable filters

A fast algorithm is developed for the directional correlation of scalar band-limited signals and band-limited steerable filters on the sphere. The asymptotic complexity associated to it through simple quadrature is of order O(L^5), where 2L stands for the square-root of the number of sampling points on the sphere, also setting a band limit L for the signals and filters considered. The filter steerability allows to compute the directional correlation uniquely in terms of direct and inverse scalar spherical harmonics transforms, which drive the overall asymptotic complexity. The separation of variables technique for the scalar spherical harmonics transform produces an O(L^3) algorithm independently of the pixelization. On equi-angular pixelizations, a sampling theorem introduced by Driscoll and Healy implies the exactness of the algorithm. The equi-angular and HEALPix implementations are compared in terms of memory requirements, computation times, and numerical stability. The computation times for the scalar transform, and hence for the directional correlation, of maps of several megapixels on the sphere (L~10^3) are reduced from years to tens of seconds in both implementations on a single standard computer. These generic results for the scale-space signal processing on the sphere are specifically developed in the perspective of the wavelet analysis of the cosmic microwave background (CMB) temperature (T) and polarization (E and B) maps of the WMAP and Planck experiments. As an illustration, we consider the computation of the wavelet coefficients of a simulated temperature map of several megapixels with the second Gaussian derivative wavelet.Comment: Version accepted in APJ. 14 pages, 2 figures, Revtex4 (emulateapj). Changes include (a) a presentation of the algorithm as directly built on blocks of standard spherical harmonics transforms, (b) a comparison between the HEALPix and equi-angular implementation

Fast, exact CMB power spectrum estimation for a certain class of observational strategies

We describe a class of observational strategies for probing the anisotropies in the cosmic microwave background (CMB) where the instrument scans on rings which can be combined into an n-torus, the {\em ring torus}. This class has the remarkable property that it allows exact maximum likelihood power spectrum estimation in of order $N^2$ operations (if the size of the data set is $N$) under circumstances which would previously have made this analysis intractable: correlated receiver noise, arbitrary asymmetric beam shapes and far side lobes, non-uniform distribution of integration time on the sky and partial sky coverage. This ease of computation gives us an important theoretical tool for understanding the impact of instrumental effects on CMB observables and hence for the design and analysis of the CMB observations of the future. There are members of this class which closely approximate the MAP and Planck satellite missions. We present a numerical example where we apply our ring torus methods to a simulated data set from a CMB mission covering a 20 degree patch on the sky to compute the maximum likelihood estimate of the power spectrum $C_\ell$ with unprecedented efficiency.Comment: RevTeX, 14 pages, 5 figures. A full resolution version of Figure 1 and additional materials are at http://feynman.princeton.edu/~bwandelt/RT

How well-proportioned are lens and prism spaces?

The CMB anisotropies in spherical 3-spaces with a non-trivial topology are analysed with a focus on lens and prism shaped fundamental cells. The conjecture is tested that well proportioned spaces lead to a suppression of large-scale anisotropies according to the observed cosmic microwave background (CMB). The focus is put on lens spaces L(p,q) which are supposed to be oddly proportioned. However, there are inhomogeneous lens spaces whose shape of the Voronoi domain depends on the position of the observer within the manifold. Such manifolds possess no fixed measure of well-proportioned and allow a predestined test of the well-proportioned conjecture. Topologies having the same Voronoi domain are shown to possess distinct CMB statistics which thus provide a counter-example to the well-proportioned conjecture. The CMB properties are analysed in terms of cyclic subgroups Z_p, and new point of view for the superior behaviour of the Poincar\'e dodecahedron is found

The extended empirical process test for non-Gaussianity in the CMB, with an application to non-Gaussian inflationary models

In (Hansen et al. 2002) we presented a new approach for measuring non-Gaussianity of the Cosmic Microwave Background (CMB) anisotropy pattern, based on the multivariate empirical distribution function of the spherical harmonics a_lm of a CMB map. The present paper builds upon the same ideas and proposes several improvements and extensions. More precisely, we exploit the additional information on the random phases of the a_lm to provide further tests based on the empirical distribution function. Also we take advantage of the effect of rotations in improving the power of our procedures. The suggested tests are implemented on physically motivated models of non-Gaussian fields; Monte-Carlo simulations suggest that this approach may be very promising in the analysis of non-Gaussianity generated by non-standard models of inflation. We address also some experimentally meaningful situations, such as the presence of instrumental noise and a galactic cut in the map.Comment: 15 pages, 6 figures, submitted to Phys. Rev.

All-sky convolution for polarimetry experiments

We discuss all-sky convolution of the instrument beam with the sky signal in polarimetry experiments, such as the Planck mission which will map the temperature anisotropy and polarization of the cosmic microwave background (CMB). To account properly for stray light (from e.g. the galaxy, sun, and planets) in the far side-lobes of such an experiment, it is necessary to perform the beam convolution over the full sky. We discuss this process in multipole space for an arbitrary beam response, fully including the effects of beam asymmetry and cross-polarization. The form of the convolution in multipole space is such that the Wandelt-Gorski fast technique for all-sky convolution of scalar signals (e.g. temperature) can be applied with little modification. We further show that for the special case of a pure co-polarized, axisymmetric beam the effect of the convolution can be described by spin-weighted window functions. In the limits of a small angle beam and large Legendre multipoles, the spin-weight 2 window function for the linear polarization reduces to the usual scalar window function used in previous analyses of beam effects in CMB polarimetry experiments. While we focus on the example of polarimetry experiments in the context of CMB studies, we emphasise that the formalism we develop is applicable to anisotropic filtering of arbitrary tensor fields on the sphere.Comment: 8 pages, 1 figure; Minor changes to match version accepted by Phys. Rev.

The significance of the largest scale CMB fluctuations in WMAP

We investigate anomalies reported in the Cosmic Microwave Background maps from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite on very large angular scales and discuss possible interpretations. Three independent anomalies involve the quadrupole and octopole: 1. The cosmic quadrupole on its own is anomalous at the 1-in-20 level by being low (the cut-sky quadrupole measured by the WMAP team is more strikingly low, apparently due to a coincidence in the orientation of our Galaxy of no cosmological significance); 2. The cosmic octopole on its own is anomalous at the 1-in-20 level by being very planar; 3. The alignment between the quadrupole and octopole is anomalous at the 1-in-60 level. Although the a priori chance of all three occurring is 1 in 24000, the multitude of alternative anomalies one could have looked for dilutes the significance of such a posteriori statistics. The simplest small universe model where the universe has toroidal topology with one small dimension of order half the horizon scale, in the direction towards Virgo, could explain the three items above. However, we rule this model out using two topological tests: the S-statistic and the matched circle test.Comment: N.B. that our results do not rule out the recently proposed dodecahedron model of Luminet, Weeks, Riazuelo, Lehoucq & Uzan, which has a 36 degree twist between matched circles. 12 pages, 5 figs; more info at http://www.hep.upenn.edu/~angelica/topology.htm