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

Generalized Spherical Harmonics for All-Sky Polarization Studies
P. B. Keegstra1 , C. L. Bennett2 , G. F. Smoot3 , K. M. Gorski4 , G. Hinshaw2 , L. Tenorio5
1 2

Hughes STX Corporation

Laboratory for Astronomy and Solar Physics, NASA/Goddard Space Flight Center Lawrence Berkeley Laboratory, University of California, Berkeley Theoretical Astrophysics Center, Denmark and Warsaw University Observatory, Poland Universidad Carlos III de Madrid, Spain

3 4

5

Abstract. When whole-sky linear p olarization is expressed in terms of Stokes parameters TQ and TU , as in analyzing p olarization results from the Differential Microwave Radiometers (DMR) on NASA's Cosmic Background Explorer (COBE ), coordinate transformations produce a mixing of TQ and TU . Consequently, it is inappropriate to expand TQ and TU in ordinary spherical harmonics. The prop er expansion expresses b oth TQ and TU simultaneously in terms of a particular order of generalized spherical harmonics. The approach describ ed here has b een implemented, and is b eing used to analyze the p olarization 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 prop erties at each p oint on the sphere are more complex than just scalars. The general form Tn,m (, ) = eim Pn,m ( ) has three indices , m, and n where - m and - n (Gel'fand et al. 1963). The forms appropriate for expanding complex Stokes parameters TQ and TU are (Sazhin & KorolЁ 1985) ev


TQ + iTU =
=2 m=-

D

,m T2,m

(, )

TQ - iTU =
=2 m=-

E 198

,m T-2,m

(, )

© Copyright 1997 Astronomical Society of the Pacific. All rights reserved.


Generalized Spherical Harmonics for All-Sky Polarization

199

Since TQ and TU are real and T-2,m = T2,-m , 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 Tm . The D ,m are complex expansion coefficients, analogues of the a ,m of ordinary spherical harmonic expansions of scalar quantities. Like them, the D ,m for a given transform among themselves in a coordinate transformation, but the absolute sum m D ,m D ,m is invariant. Following recent work by Zaldarriaga & Seljak (1997) and Kamionkowski et al. (1997), we can partition the 4 + 2 indep endent real parameters p er value of into those associated with even-parity solutions and odd-parity solutions, called E-like and B-like resp ectively by Zaldarriaga & Seljak. The formula appropriate for the phase convention used here is D D 2.
E ,m B ,m

= -(D = i(D

,m ,m

+(-1) - (-1)

+m +m

D D

, -m , -m

)/2 )/2

(1)

Prop erties and Computation · Generalized spherical harmonics start at - m . · P2,
-m

= 2, and for each

,

( ) = P2,m (180 - ).

· P2,m is real for m even, and pure imaginary for m odd. · All functions are zero at the p oles except P2,2 , which is nonzero at the North Pole ( = 0 ), and P2,-2 , nonzero at the South Pole ( = 180 ). · Functions are normalized such that for any value of , the integral over the sphere of the sum of squares for all m gives unity. Thus the "strength" of an individual function decreases as increases when contrasted with the usual normalization for ordinary spherical harmonics, where each m individually integrates to unity. · Function evaluation is by recursion. Recurrences on and then on m are used to reach each particular function. (Note that here refers to the colatitude, not the latitude.) ( + m +1)( - m +1)( + j +1)( - j +1) +1 mj Pj,m ( )+ P ( )+ (2 +1)( +1) ( +1) j,m ( + m)( - m)( + j )( - j ) -1 (2) Pj,m ( ) = cos Pj,m ( ) (2 +1) ( + m +1)( - m)Pj, ( ) - ( + m)( - m +1)Pj,m-1 ( ) = m cos - j 2i Pj,m ( ) sin

m+1

(3)


200

Keegstra et al.
z
2-1

V2 V1
1

180-2

1 2

y
2 1

x

Figure 1.

Geometry for definitions of 1 and 2 (Kosowsky 1996).

· The recursion is anchored by explicit formulas for the generalized quad2 rup ole P2,m ( ). (The recursion in at = 2 defines = 3, since the coefficient for = 1 vanishes.) 1 (cos - 1)2 4 i sin (cos - 1) 2 3 (cos2 - 1) 8 i sin (cos +1) 2 1 (cos +1)2 4

2 P2, 2 P2,

-2 -1

( ) = ( ) =

2 P2,0 ( ) = 2 P2,1 ( ) = 2 P2,2 ( ) =

(4)

3.

Sum Rules and Correlation Functions

Generalized spherical harmonics ob ey a sum rule analogous to a familiar one for ordinary spherical harmonics, but it includes an explicit phase factor which dep ends on the orientation of the two lines of sight. That phase factor dep ends on the angle which carries the reference direction for line of sight v1 into the reference direction for line of sight v2 . The geometry of v1 and v2 is illustrated in Figure 1. = 1 + 2 , which has the following geometric interpretation. The reference direction is rotated by 1 into the great circle from v1 to v2 , translated to v2 , and then rotated through 2 to bring it into alignment with the reference direction at v2 . ( is the angle b etween v1 and v2 ). cos sin 1 = = cos 1 cos 2 +sin 1 sin 2 cos(2 - 1 ) sin 2 sin(2 - 1 )/ sin


Generalized Spherical Harmonics for All-Sky Polarization cos 1 sin 2 cos 2 = = = (sin 2 cos 1 sin(2 - 1 ) - sin 1 cos 2 )/ sin sin 1 sin(1 - 2 )/ sin (sin 1 cos 2 sin(2 - 1 ) - sin 2 cos 1 )/ sin

201

(5)

With that definition of , the sum rule relating generalized spherical harmonics along two lines of sight to the angle b etween those lines of sight is P2,2 ( ) = e-
2i m =-

T2,m (1 ,1 )T2,m (2 ,2 ).

If this phase factor is included in the definition of the spherical average over all directions vi and vj separated by an angle C ( ) =< TQ (vi )TQ (vj )+ TU (vi )TU (vj ) >=
ij =

e-

2i

Z (vi )Z (vj )

where Z = TQ +iTU , then this allows us to define rotationally invariant analogues C P to the p ower sp ectra C : C ( ) = C P P2,2 ( ) = P2,2 ( )
m

D

,m

D

,m

.

Additionally, we can construct analogous sums of DEm and D Bm , which we , , denote as C E and C B resp ectively. These are the appropriate quantities to use for comparison to theoretical treatments of p olarization. The partitioning into C E and C B is p ertinent since Zaldarriaga & Seljak (1996) and Kamionkowski et al. (1996) b oth show that scalar p erturbations cannot produce a nonzero C B . l It is interesting to note that P2,2 (cos(180 )) = 0, which implies that correlations b etween physical p olarization signals vanish at the antip odes. Acknowledgments. The National Aeronautics and Space Administration (NASA)/Goddard Space Flight Center (GSFC) is resp onsible for the design, development, and op eration of the Cosmic Background Explorer (COBE ). Scientific guidance is provided by the COBE Science Working Group. GSFC is also resp onsible 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. References 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., & KorolЁ V. A. 1985, Sov. Astron. Lett., 11, 204 ev, Zaldarriaga, M., & Seljak, U. 1997, Phys. Rev. D, in press