Angular power spectrum conventions

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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 C_l and generates a realisation of $f(\gamma_p)$ on the HEALPix grid. The convention for power spectrum input into synfast is straightforward: each C_l is just the expected variance of the a_lm 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)C_l, because there are 2l+1 modes for each l.

This defines unambiguously how the C_l 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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