Harmonic Multipoles and the CMB Sky

Introduction

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:

Math

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

$\begin{displaymath}T(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^{+l} a_{l,m} Y_{l,m}(\theta,\phi) \end{displaymath}$

(1)

where the a_l,m are complex as are the $Y_{l,m}(\theta,\phi)$ . These are the normalized spherical harmonics which are defined as,

$\begin{displaymath}Y_{l,m}(\theta,\phi) = n_{l,m} P_{l,m}(\cos\theta) e^{im\phi} \end{displaymath}$

(2)

where the P_l,m are the (real) associated Legendre polynomials and n_l,m is a normalization factor,

$\begin{displaymath}n_{l,m} = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} \end{displaymath}$

(3)

It follows from the definition of the P_l,m that,

$\begin{displaymath}Y_{l,-m} = (-1)^m Y_{l,+m}^\ast \end{displaymath}$

(4)

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

$\begin{displaymath}a_{l,-m} = (-1)^m a_{l,+m}^\ast \end{displaymath}$

(5)

I think the orthogonality of the Y_l,m means that there is no other way to ensure that T is real over the complete sphere. Note that a_l,0 must be real. Adding the l,+m and l,-m components we have,

$\begin{eqnarray*}a_{l,+m}Y_{l,+m} + a_{l,-m}Y_{l,-m} & = & a_{l,+m}Y_{l,+m} + a_{l,+m}^\ast Y_{l,+m}^\ast \\ & = & 2 Re(a_{l,+m}Y_{l,+m}) \\ \end{eqnarray*}$

where Re means the real part of a complex quantity. (Recall $x^\ast y^\ast = (xy)^\ast$ and $x+x^\ast = 2Re(x)$ .) Expanding out we have,

$\begin{eqnarray*}a_{l,m} & = & n_{l,m} P_{l,m} a_{l,m} e^{im\phi} \\ & = & n_{... ...,m}^I\sin m\phi)+ i(a_{l,m}^R\sin m\phi + a_{l,m}^I\cos m\phi)) \end{eqnarray*}$

Where a_l,m^R and a_l,m^I are the real and imaginary coefficients of a_l,m respectively. Therefore,

$\begin{displaymath}a_{l,+m}Y_{l,+m} + a_{l,-m}Y_{l,-m} = 2 n_{l,+m} P_{l,+m}(a_{l,+m}^R\cos m\phi - a_{l,+m}^I\sin m\phi) \end{displaymath}$

and we can re-write the sum as,

$\begin{displaymath}T(\theta,\phi) = \sum_{l=0}^{\infty} \left( n_{l,0} P_{l,0}(... ...s\theta) (a_{l,m}^R\cos m\phi - a_{l,m}^I\sin m\phi) \right) \end{displaymath}$

(6)

avoiding any complex quantities.

Images and Animations

Multipole Components

It is interesting to plot each component of the series; ie. the 0<m<l real $T(\theta,\phi)$ patterns for each l. l=0, m=0 is the monopole, l=1, m=0,1 the dipole, l=2, m=0,1,2 the quadrupole etc.

Static Components

Here is a static image showing the multipole components up to l=3. The color scale is $\pm 0.9$ common across all the plots.

Animated Components

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

The CMB Sky

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 C_l so C_l = Var(|a_lm,m|) where |a_l,m| denotes the magnitude of the complex a_l,m. The phase angle of the complex a_l,m is uniformly distributed. Cosmological theories predict the form of the function C_l = f(l), which is ussually plotted in the form $\frac{l(l+1)C_l}{2\pi} = f(l)$ .

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 l. To see this animation click on the left image below.

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.

Addendum

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).