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Astronomical Data Analysis Software and Systems VI
ASP Conference Series, Vol. 125, 1997
Editors: Gareth Hunt and H. E. Payne

Generalized Spherical Harmonics for All-Sky Polarization Studies

P. B. Keegstra1, C. L. Bennett2, G. F. Smoot3, K. M. Gorski4, G. Hinshaw2, L. Tenorio5

1Hughes STX Corporation
2Laboratory for Astronomy and Solar Physics, NASA/Goddard Space Flight Center
3Lawrence Berkeley Laboratory, University of California, Berkeley
4Theoretical Astrophysics Center, Denmark and Warsaw University Observatory, Poland
5Universidad Carlos III de Madrid, Spain



When whole-sky linear polarization is expressed in terms of Stokes parameters and , as in analyzing polarization results from the Differential Microwave Radiometers (DMR) on NASA's Cosmic Background Explorer ( COBE ), coordinate transformations produce a mixing of and . Consequently, it is inappropriate to expand and in ordinary spherical harmonics. The proper expansion expresses both and simultaneously in terms of a particular order of generalized spherical harmonics. The approach described here has been implemented, and is being used to analyze the polarization signals from the DMR data.


1. Definition and Motivation

Generalized spherical harmonics are an extension of ordinary spherical harmonics, intended for expansion of functions whose transformation properties at each point on the sphere are more complex than just scalars. The general form

has three indices , m, and n where and (Gel'fand et al. 1963). The forms appropriate for expanding complex Stokes parameters and are (Sazhin & Korolëv 1985)

Since and are real and , the two expansions are degenerate, and we may restrict our consideration to the first form. Thus, for the DMR case we need only consider generalized spherical harmonics with n=2, which we will henceforth refer to as .

The are complex expansion coefficients, analogues of the of ordinary spherical harmonic expansions of scalar quantities. Like them, the for a given transform among themselves in a coordinate transformation, but the absolute sum is invariant.

Following recent work by Zaldarriaga & Seljak (1997) and Kamionkowski et al. (1997), we can partition the independent real parameters per value of into those associated with even-parity solutions and odd-parity solutions, called E-like and B-like respectively by Zaldarriaga & Seljak. The formula appropriate for the phase convention used here is

2. Properties and Computation

3. Sum Rules and Correlation Functions

Figure: Geometry for definitions of and (Kosowsky 1996). Original PostScript figure (48kB).

Generalized spherical harmonics obey a sum rule analogous to a familiar one for ordinary spherical harmonics, but it includes an explicit phase factor which depends on the orientation of the two lines of sight. That phase factor depends on the angle which carries the reference direction for line of sight into the reference direction for line of sight . The geometry of and is illustrated in Figure 1. , which has the following geometric interpretation. The reference direction is rotated by into the great circle from to , translated to , and then rotated through to bring it into alignment with the reference direction at . ( is the angle between and ).

With that definition of , the sum rule relating generalized spherical harmonics along two lines of sight to the angle between those lines of sight is

If this phase factor is included in the definition of the spherical average over all directions and separated by an angle

where , then this allows us to define rotationally invariant analogues to the power spectra :

Additionally, we can construct analogous sums of and , which we denote as and respectively. These are the appropriate quantities to use for comparison to theoretical treatments of polarization. The partitioning into and is pertinent since Zaldarriaga & Seljak (1996) and Kamionkowski et al. (1996) both show that scalar perturbations cannot produce a nonzero .

It is interesting to note that , which implies that correlations between physical polarization signals vanish at the antipodes.


The National Aeronautics and Space Administration (NASA)/Goddard Space Flight Center (GSFC) is responsible for the design, development, and operation of the Cosmic Background Explorer ( COBE ). Scientific guidance is provided by the COBE Science Working Group. GSFC is also responsible for the development of analysis software and for the production of the mission data sets.

Fruitful discussions with M. Jacobsen, University of Maryland Department of Mathematics, and B. Summey, Hughes STX, are gratefully acknowledged.


Gel'fand, I. M., Minlos, R. A., & Shapiro, Z. Y. 1963, Representations of the Rotation and Lorentz Groups and their Applications (New York: Pergamon Press)

Kamionkowski, M., Kosowsky, A., & Stebbins, A. 1997, Phys. Rev. Lett., 78, 2038

Kosowsky, A. 1996, Ann. Phys., 246, 49

Sazhin, M. V., & Korolëv, V. A. 1985, Sov. Astron. Lett., 11, 204

Zaldarriaga, M., & Seljak, U. 1997, Phys. Rev. D, in press

© Copyright 1997 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA

Next: Image Reconstruction with Few Strip-Integrated Projections: Enhancements by Application of Versions of the CLEAN Algorithm
Previous: Titan Image Processing
Up: Algorithms
Table of Contents - Index - PS reprint