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). - Paper 5
-
A paper applying the multipole vector formalism to the 9 year
WMAP data and the Planck first data release. The alignments
persist throughout the WMAP data (despite refinements in their
analysis including the identification of systematics) and in
the Planck data (despite being a completely different
detection technique from WMAP). All together this shows that
the alignments are present in the data and cannot easily be
attributed to systematics.
"Large-scale alignments from WMAP and Planck", C.J. Copi, D. Huterer, D.J. Schwarz, and G.D. Starkman, MNRAS, 449, 3458 (2015).
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
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
Copies of the figures from paper 2. See the papers for details.
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.
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.
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.
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.
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.
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 and Python is provided.
The code can be tracked, cloned, etc from GitHub at https://github.com/cwru-pat/cmb-multipole_vectors.
Alternatively, the latest version can be found 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.
When this code is used please include an acknowledgment in your publications of the form:
Some of the results in this paper hasve been derived using the multipole vector code available at http://www.phys.cwru.edu/projects/mpvectors/, which was developed with support from the US Department of Energy.
Also include a reference to paper1: C.J. Copi, D. Huterer, and G.D. Starkman, Phys. Rev. D., 70, 043515 (2004).
Current version: 1.20. Removed redundant equations and included Python implementation.
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.
Acknowledgment
This work was supported by a grant from the US Department of Energy to the Particle Astrophysics Theory Group at CWRU.