How to measure CMB anisotropies *In Practice*

[astrolollo]

Hi everyone
Given the definition of $C_{l}$, $C_{l}=\frac{1}{2l+1} \sum a_{l,m} Y_{l,m}$, I was wondering how it is possible to measure the $C_{l}$s in practice. How does one compute this quantity, having a map of the temperature anisotropies of the CMB?

 

[Orodruin]

The temperature of the CMB is a function on a sphere. As such, it can be expanded in terms of the spherical harmonics $Y_\ell^m$. The way of computing the expansion coefficients is the same as you would do in any expansion in terms of the eigenfunctions of a Sturm-Liouville operator - you simply take the inner product of the function you wish to expand (in this case the CMB temperature function) and the orthogonal basis functions (in this case the spherical harmonics), this corresponds to an integral over the domain of the functions (in this case a sphere).

The procedure is exactly equivalent to finding the expansion coefficients in a Fourier series.

 

[Chalnoth]

Hi everyone
Given the definition of $C_{l}$, $C_{l}=\frac{1}{2l+1} \sum a_{l,m} Y_{l,m}$, I was wondering how it is possible to measure the $C_{l}$s in practice. How does one compute this quantity, having a map of the temperature anisotropies of the CMB?

To expand a little bit upon what Orodruin said, the first step is to take a spherical harmonic transform:

$$a_{\ell m} = \int Y_{\ell m}(\theta, \phi) T(\theta, \phi) sin(\theta) d\theta d\phi$$

Here $T(\theta, \phi)$ is the CMB temperature in different directions on the sky. There are various mathematical tricks used to make this integral efficient to compute on a computer (a common library used for this is Healpix).

The power spectrum is then the variance of the amplitude at a given $\ell$. Each $Y_{\ell m}$ can be thought of as a wave on the surface of a sphere, with the wavelength given by $\ell$ and the direction given by $m$. Each $a_{\ell m}$, then, is the amplitude of the wave with wavelength $\ell$ and direction $m$. To get the power spectrum, we take the variance of the wave amplitudes of each direction at a given wavelength:

$$C_\ell = {1 \over 2\ell + 1} |a_{\ell m}|^2$$

Here the factor of $a/(2\ell + 1)$ comes from the fact that $\ell$ and $m$ are integers, with $\ell \ge 0$ and $-\ell \le m \le \ell$. For example, if $\ell = 2$, then $m = {-2, -1, 0, 1, 2}$.

There are lots of other complications that come in from the fact that there is stuff between us and the CMB, so that we don't have a perfectly-clean full-sky map. Various techniques are used to extract the CMB signal, but I think I'll stop here for now.

Does this answer your question? Did you have any other concerns?