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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
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.
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
, and
for each 
, 
.
.
is real for m even, and pure imaginary
for m odd.
, which is
nonzero at the North Pole (
= 0°), and 
,
nonzero at the South Pole (
= 180°).
, 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.
and then on m are used
to reach each particular function.
(Note that 
here refers
to the colatitude, not the latitude.)
. (The recursion in 
at 
defines 
, since the coefficient
for 
vanishes.)
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
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