
These can be used to compute estimates of the angular power spectrum
as
The HEALPix package contains the Fortran90 facility synfast
which takes as input a power spectrum C_{l} and generates a realisation of
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 a_{00} is the
integral of the
over the sphere. To
obtain realisations of functions which have a_{00} distributed as a Gaussian
with zero mean and variance 1, set C_{0} to 1. The value of the
synthesised function at each pixel will
be Gaussian distributed with mean zero and variance .
As required, the integral of over the full
solid angle of the sphere has zero mean and variance .
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

, a dimensionless quantity measuring relative
fluctuations about the average CMB temperature.

The absolute quantity in or K.
Eric Hivon
20100618

