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Re-presentation of Spherical Harmonic Coefficients

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 ${\ell}$ by summing all the $m$ modes pertaining to that ${\ell}$:

T_{\ell}(\theta, \varphi) = \sum_{m=-{\ell}}^{\ell}a_{{{\ell}m}}Y_{{{\ell}m}}(\theta,\varphi).
\end{displaymath} (21)

The panels below are the composite maps (from ${\ell}=2$ to $10$) of NASA Wilkinson Microwave Anisotropy Probe (WMAP) 3-year Internal Linear Combination (ILC) map that is supposed to be close to the CMB signal. All the colour bars of the panels are $[-50,50] (\mu$K).
And blow are the foreground maps (sum of the synchrotron, free-free and dust templates) in WMAP V band (61 Gz channel) with all colour bars $[-200,200] (\mu$K):
Based on the a priori assumption that the CMB should be statistically independent of the foregrounds, consequently they should have no significant correlation. And comparing mode by mode from the above two panels, one might agree there is no correlation !?

I device a new representation of the spherical harmonic coefficients $a_{{{\ell}m}}$ for each ${\ell}$-mode, which I call "one dimensional Fourier representation" (1DFR). The 1DFR is a represention of the $a_{{{\ell}m}}$ in each ${\ell}$ by an inverse Fourier transform, as the $a_{{{\ell}m}}$ is now a function of a single variable $m$. Such a 1DFR for $a_{{{\ell}m}}$ from the spherical harmonic decomposition of the sky is written as

...^{\ell}\vert a_{{{\ell}m}}\vert \cos(m\varphi+\Phi_{{\ell}m}),
\end{displaymath} (22)

where for the negative $m$ we use the complex conjugate ( $a_{{\ell},-m}=a_{{{\ell}m}}^\ast$) to make $T_{\ell}(\varphi)$ real. For real $T$ on a sphere the statistics are registered only in $m\ge0$ modes, the $T_{\ell}(\varphi)$ curves assembled in this way contain the same amount of information as the spherical $T_{\ell}(\theta, \varphi)$.

For a Gaussian random field on a sphere, the real and imaginary parts of the spherical harmonic coefficients $a_{{{\ell}m}}$ in each ${\ell}$ are mutually independent and both Gaussian distributed with zero mean and variance $C_{{\ell}}/2$, where $C_{{\ell}}$ is its angular power spectrum. Thus if the $a_{{{\ell}m}}$ are a result of Gaussian process, the 1DFR $T_{\ell}(\varphi)$ 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 ${\ell}$. 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 $x$ axis is $\varphi$ and is plotted reversely to follow the conventional Galactic longitude coordinate $l$ and the $y$ axis is in unit of thermodynamic temperature ($\mu$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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next up previous
Next: Limited Number of Ensemble Up: Statistical Methods Previous: Mean Chi-square Statistics of
Chiang Lung-Yih 2013-10-02