The theoretical and experimental CMB power spectrums are customarily presented in the context of spherical harmonic multipoles. In my own rather feeble attempts to understand intuitively what the maths actually means in terms of patterns and pictures I generated some nice images and animations which I will post here in case anyone else is interested. First the basic math:

Any scalar field on a sphere can be expressed as a series of spherical harmonic multipoles,

(1) |

where the ** a_{l,m}** are complex as are the
.
These are the normalized spherical harmonics which are defined as,

(2) |

where the ** P_{l,m}** are the (real) associated Legendre
polynomials and

(3) |

It follows from the definition of the ** P_{l,m}** that,

(4) |

where the star represents complex conjugation.
For real ** T** this also implies that,

(5) |

I think the orthogonality of the ** Y_{l,m}** means that there
is no other way to ensure that

where ** Re** means the real part of a complex quantity.
(Recall
and
.)
Expanding out we have,

Where ** a_{l,m}^{R}** and

and we can re-write the sum as,

(6) |

avoiding any complex quantities.

It is interesting to
plot each component of the series; ie. the **0< m<l** real
patterns for each

Click on the image below to see an animation showing the components
of each multipole up to ** l=24**.

It is normally assumed that the CMB sky is a *Gaussian
random field*.
This means that the pattern can be completely described
by the variance of the amplitudes of the multipole components
as a function of the ** l** number.
This variance is conventionally denoted by

Let's see what this looks like for a currently favoured model
of the universe by
sampling **| a_{l,m}|** with the
appropriate variance, and constructing each multipole

Finally let's stack the multipoles from the last animation to
make a cummulative image of the sky as we include higher and
higher ** l** numbers.
To see this animation click on the right image below.

Experiments like the upcoming MAP
will which have all sky coverage can potentially see the patterns like
that on the sphere above.
Current experiments like our own
DASI see only a smaller area of the sky like that shown in the zoomed
in region above, but as one can see for the higher ** l** numbers
this is enough.

For computing simulated CMB skys there are much smarter techniques than the brute force one I used here using Fourier transforms to make the complete sky in one shot. See for instance astro-ph/9703084 (see here for implemented code). The very fancy HEALPix package includes a sky generator.

I made these pages using MATLAB, Gifsicle, and latex2 html

This page prepared by Clem Pryke (pryke@aupc1.uchicago.edu).