Recent studies on the large angular scale properties such as alignment of the quadrupole and octupole in the CMB temperature anisotropy are based on
the composite map constructed for each by summing all the
modes pertaining to that :
I device a new representation of the spherical harmonic coefficients
for each -mode, which I call "one dimensional Fourier representation" (1DFR). The 1DFR is a represention of the in each by an inverse
Fourier transform, as the is now a function of a single
variable . Such a 1DFR for from the spherical harmonic
decomposition of the sky is written as
For a Gaussian random field on a sphere, the real and imaginary parts of the spherical harmonic coefficients in each are mutually independent and both Gaussian distributed with zero mean and variance , where is its angular power spectrum. Thus if the are a result of Gaussian process, the 1DFR curves shall possess all the usual Gaussian random properties such as two-point correlation, peak statistics, and Minkowski functionals...etc..
The advantage of 1DFR is that it offers different perspective and insight of the signal properties in each . For cross correlations between maps, for example, it is difficult to do in standard composite maps as seen above. I have employed 1DFR to test the peak statistics, cross correlation with the foregrounds.
Below is the 1DFR curves for WMAP 3-year ILC map (blue curves, which can be compared with the composite maps above) and those of the foreground maps at Q channel (orange curves), V channel (green curves) and W channel (red curves). The axis is and is plotted reversely to follow the conventional Galactic longitude coordinate and the axis is in unit of thermodynamic temperature (K). Notice strong anti-cross correlation for the quadrupole in 1DFR curves whereas in composite maps it is obscured.
For details of this idea see my papers :
The Auto and cross correlation of phases of the whole sky CMB and foreground maps from the 1-year Wilkinson Microwave Anisotropy Probe (WMAP) data, IJMPD, 15, 1283 (2006)
Testing Gaussian random hypothesis with the cosmic microwave background temperature anisotropies in the three-year WMAP data, ApJ, 664, 8 (2007)
The One-dimensional Fourier Representation and Large Angular Scale Foreground Contamination in the 3-year Wilkinson Microwave Anisotropy Probe data, JCAP, 7, 21 (2007)
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