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Angular power spectrum conventions

These $\hat{a}_{lm}$ can be used to compute estimates of the angular power spectrum $\hat{C}_l$ as
$\displaystyle \hat{C}_l$ $\textstyle \myequal $ $\displaystyle \frac{1}{2l +1}\sum_{m} \vert\hat{a}_{lm}\vert^2.$ (3)

The HEALPix package contains the Fortran90 facility synfast which takes as input a power spectrum Cl and generates a realisation of $f(\gamma_p)$ on the HEALPix grid. The convention for power spectrum input into synfast is straightforward: each Cl is just the expected variance of the alm at that l.

Example: The spherical harmonic coefficient a00 is the integral of the $f(\gamma)/\sqrt{4 \pi}$ over the sphere. To obtain realisations of functions which have a00 distributed as a Gaussian with zero mean and variance 1, set C0 to 1. The value of the synthesised function at each pixel will be Gaussian distributed with mean zero and variance $1/(4\pi)$. As required, the integral of $f(\gamma)$ over the full $4\pi$ solid angle of the sphere has zero mean and variance $4\pi$.
Note that this definition implies the standard result that the total power at the angular wavenumber l is (2l+1)Cl, because there are 2l+1 modes for each l.

This defines unambiguously how the Cl have to be defined given the units of the physical quantity f. In cosmic microwave background research, popular choices for simulated maps are

* $\Delta T/T $, a dimensionless quantity measuring relative fluctuations about the average CMB temperature.
* The absolute quantity $\Delta T$ in $\mu K$ or K.

Eric Hivon 2010-06-18
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