Introduction

This site contains information about the multipole vector decomposition initially developed by Copi, Huterer, and Starkman as another way of decomposing functions on a sphere. It has been applied to the CMB as a probe of the hypothesis that the primordial perturbations are statistically isotropic and Gaussian (at least on large scales).

Contained here are publications, images, and code associated with our work on these multipole vectors. All materials here are copyrighted by the authors (unless explicitly noted otherwise). The information here is free for use as long as proper credit is given to the authors.

Publications based on this work

Paper 1
Our initial paper describing the technique for performing a multipole vector decomposition and its application to the WMAP full-sky map is available on astro-ph and has been published in Physical Review D.
"Multipole Vectors--a new representation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2<=l<=8" C.J. Copi, D. Huterer, and G.D. Starkman, Phys. Rev. D., 70, 043515 (2004).
Paper 2
A paper focusing on the low l multipoles (l=2 and 3), the planarity of l=3 and the alignment of the l=2 and l=3 planes. We find the alignment to be inconsistent with statistical fluctuations around the standard inflationary expectations. This paper can be found on astro-ph and has been published in Physical Review Letters.
"Is the low-l microwave background cosmic?", D.J. Schwarz, G.D. Starkman, D. Huterer, and C.J. Copi, Phys. Rev. Lett., 93, 221301 (2004).
Paper 3
A paper collecting much of the work involving multipole vectors performed by us and by others. It provides a review of the multipole vector formalism and extensions to it. We focus mainly on the lowest cosmological multipoles (l=2 and 3) and show that alignment between the planes they define and the ecliptic persists at > 99.9% C.L. We explore many other alignments and extend the analysis in ways to address some systematics. This paper can be found on astro-ph and has been published in Monthly Notices of the Royal Astronomical Society.
You can download the paper with high resolution (and thus large) figures as either hi-res postscript (compressed, approximately 980kB) or hi-res pdf (approximately 850kB).
"On the large-angle anomalies of the microwave sky" C.J. Copi, D. Huterer, D.J. Schwarz, and G.D. Starkman, MNRAS, 367, 79 (2006).
Paper 4
A paper applying the multipole vector formalism to the 3 year WMAP data release confirming most of the large scale anomalies discovered in the earlier papers. We have also performed a preliminary evaluation of the "angular correlation function" including a careful discussion of what this means, assumptions built into various definitions, and the care one must take in light of the large scale anomalies. This paper can be found on astro-ph and has been published in Physical Review D.

"The Uncorrelated Universe: Statistical Anisotropy and the Vanishing Angular Correlation Function in WMAP Years 1-3", C.J. Copi, D. Huterer, D.J. Schwarz, and G.D. Starkman, Phys. Rev. D., 75, 023507 (2007).

Images of multipole vectors

The following images are provided as a convenience. These are (potentially) higher resolution images than can be found in the publications listed above. The images are provided for personal use only. Permission to republish or reprint these images must be obtained from the creator of the image and the journal in which they appeared.

Multipole Vectors

CMB Multipole Vectors An image of the sky as decomposed into the L=2 to 8 multipole moments based on the first year WMAP results (as cleaned by Tegmark et al.). Shown are the multipole moment shaded using the standard CMB color gradient. Included are the vectors calculated for these multipoles. The vectors are in cyan and magenta. They are only defined up to a pairwise sign (thus they are ''headless''). Click on the image to get a larger version. A version of this image was included in our first paper.

This figure was generated using the freely available geomview package. The files used to generate this figure are available as a tar.gz archive or a zip archive. With geomview you can then rotate the multipoles to better explore their 3-d structure. (I have been unable to find a java applet that allows viewing of oogl files that use CMESH, hence this option is not available online at the moment.)

Alignment of low multipoles

Dipole direction,
	    equinoxes, and l=2,3 oriented area vectors Quadrupole and octupole Quadrupole and
	    octupole, closeup, Tegmark Quadrupole and
	    octupole, closeup, ILC l=2 only l=3 only l=2, 3, 4

Copies of the figures from paper 2. See the papers for details.

Mollweide, Quad+Oct, ILC, Ecliptic Mollweide, Quad+Oct, ILC, Galactic Here are different views of the quadrupole (with kinematic corrections) and the octopole for the ILC map using the Mollweide projection and a standard CMB color scale. We show both Galactic and Ecliptic coordinates. The Ecliptic plane is shown as the solid line. Notice how the lobes are separated by the Ecliptic plane.

Mollweide, Quad+Oct, Tegmark, Ecliptic Mollweide, Quad+Oct, Tegmark, Galactic Same as above but for the Tegmark map. The separation is still visible, though, there are some observable differences in the lobe shapes.

Copies of some of the figures from paper 3 showing the peculiar properties of the quadrupole and octopole. See the paper for details.

Mollweide projection of Tegmark Quadrupole plus Octopole Mollweide projection of Tegmark Octopole Mollweide projection of Tegmark Quadrupole The Quadrupole, Octopole, and Quadrupole + Octopole for the Tegmark map. In each of the plots the solid line is the ecliptic plane and the dashed line is the supergalactic plane. The directions of the equinoxes (EQX), dipole due to our motion through the Universe, north and south ecliptic poles (NEP and SEP) and north and south supergalactic poles (NSGP and SSGP) are shown. The quadrupole multipole vectors are plotted as the solid red symbols for each map, ILC (circles), TOH (diamonds), and LILC (squares). The octopole multipole vectors are plotted as the solid magenta symbols for each map. The open symbols of the same shapes and color are for the normal vector for each map. The dotted line is the great circle connecting each pair of multipole vectors for this map. For the quadrupole the minimum and maximum temperature locations are shown as the white stars. The light gray stars are particular sums of the multipole vectors which are very close to the temperature minima and maxima of the octopole. The solid black star shows the direction of the vector that appears in the trace of the octopole of the TOH map. The solid magenta star is the direction to the maximum angular momentum dispersion for the octopole, again for the TOH map. (The direction that maximizes the angular momentum dispersion of any of the maps coincides with the respective normal vector.) Notice how the ecliptic plane carefully threads its way between two extrema separating the weaker power in the northern ecliptic hemisphere from the stronger power in the southern ecliptic hemisphere as discussed in paper 3.

Motion of the Octopole vectors as foregrounds are added Motion of the Quadrupole vectors as foregrounds are added The trajectories of the Quadrupole and Octopole multipole vectors for the TOH map as the V band synchrotron foreground map is slowly added. Here c is the fraction of foreground power added to the sky map.

Alignment of low multipoles in 3 year WMAP data

Copies of some of the figures from paper 4 showing the peculiar properties of the quadrupole and octopole as they persist in the data. See the paper for details.

Mollweide projection of ILC123 Quadrupole plus Octopole Mollweide projection of ILC123 Octopole Mollweide projection of ILC123 Quadrupole The Quadrupole, Octopole, and Quadrupole + Octopole for the ILC123 map. These are quite similar to the plots from paper 3 but updated for the WMAP 3 year data release. In each of the plots the solid line is the ecliptic plane and the dashed line is the supergalactic plane. The directions of the equinoxes (EQX), dipole due to our motion through the Universe, north and south ecliptic poles (NEP and SEP) and north and south supergalactic poles (NSGP and SSGP) are shown. The quadrupole multipole vectors are plotted as the solid red symbols for each map, ILC1 (circles), TOH1 (diamonds), and LILC1 (squares) from the first year data release and additionally the ILC123 (triangles). The octopole multipole vectors are plotted as the solid magenta symbols for each map. The open symbols of the same shapes and color are for the normal vector for each map. The dotted line is the great circle connecting each pair of multipole vectors for this map. For the quadrupole the minimum and maximum temperature locations are shown as the white stars. The solid magenta star is the direction to the maximum angular momentum dispersion for the octopole, again for the ILC123 map. (The direction that maximizes the angular momentum dispersion of any of the maps coincides with the respective normal vector.) Notice how the ecliptic plane carefully threads its way between two extrema separating the weaker power in the northern ecliptic hemisphere from the stronger power in the southern ecliptic hemisphere as discussed in paper 3 and confirmed in paper 4.

Mollweide projection of ILC123 Quadrupole with vectors
showing how the V band, W band, and ILC quadrupole has changed. New systematic corrections in the WMAP analysis have moved the quadrupole vectors more than had been expected based solely on the pixel noise in the observation. Graphically the movement of these vectors is shown at the right. The filled triangles show the quadrupole vectors of the difference between the WMAP1 and WMAP123 maps for the V band (white), W band (black), and the ILC (gray). The filled triangles show the normal vector for these maps. Notice how the vectors are near the ecliptic plane, one being very close to the equinox, and the normal is near the ecliptic pole. These new systematics are related to the geometry of the solar system.

Downloadable Code

The algorithm for constructing multipole vectors from a standard spherical harmonic decomposition is described in our first paper. In particular in Appendix A. A reference implementation that solves equation (A3) from that paper in C is provided here as a tar.gz archive and a zip archive. This implementation requires the freely available Gnu Scientific Library. See the source code for more information about its use, license, and disclaimers.

Current version: 1.12. Fixed a memory leak in the previous version.

Latest code as a tarball: mpd_decomp_latest.tar.gz
Latest code as a zip file: mpd_decomp_latest.zip

Site Info

This site is provided as a convenient reference for information about the multipole vectors. The webpage design is based on the Gila design made freely available by haran on the open source web design site. The elegance of the layout should be attributed to haran, the flaws to me.

Contact Info

Comments or questions can be emailed directly to the authors or to the general address mpvectors@phys.cwru.edu.