Combined Spherical Harmonics

Now that we hopefully have a handle on what the individual spherical harmonics (the $Y_l^m$ ) look like, we can think about combining them. To begin with, we will just look at the results of combining the orders ($m$ ) at fixed degree ($l$ ).

$$\sum_{m=-l}^{l} a_{lm} Y_l^m(\theta, \phi)$$

There is one small subtlety to this summation. The convention is that when $m \geq 0$ we use the real component of $Y_l^m$ , and when $m < 0$ the imaginary component. This gives a full set of both real and imaginary orders (the imaginary component of $m = 0$ is 0).

The one piece of new information we need to do the summation is the value of the $a_{lm}$ . In the most general case, these could be anything. Fortunately for the CMB we think that these are drawn from a Gaussian distribution.

$$a_{lm} \sim \mathcal{N}(0, C_l^{1/2})$$

We will not dig in to why we expect this to be so.¹ Instead we’ll look at the results for various $l$ , assuming $C_l = 1$ . The large figure shows the sum of all $m$ at the given $l$ . The small inlay shows the contribution of the individual $m$ that is selected.

Working with Data

Thus far we have working entirely from the theory side with little consideration for the data. We’ve seen how to pixelate maps, a set of basis functions that allow us to express those maps in frequency space, and here we’ve started to look at how those basis functions combine to generate maps assuming we know (or can generate) the coefficients. But, in reality, cosmologists start with the maps (images of the CMB) and need to infer the coefficients ($a_{lm}$ ) and hence the power in each harmonic ($C_l$ ). Doing this requires understanding and modeling a vast array of physical processes that affect the light of the CMB during its journey to earth. We’ll ignore all this for now² and just consider the statistics of this inference.

For short wavelength (large $l$ ), we will have a large number of $a_{lm}$ coefficients ($2l + 1$ of them) and will therefore be able to make an accurate estimate of $C_l$ . However, for small $l$ , the small number of coefficients makes any estimate of $C_l$ very uncertain. This uncertainty is not something we can improve by gathering more data or building more sensitive equipment. There simply aren’t many coefficients and so we don’t have enough data to make an accurate estimate of the variance of the normal that they are drawn from ($C_l$ ).

The maximum likelihood estimate of $C_l$ will be (Tristram and Ganga 2007),

$$\hat{C}_l = \frac{1}{2l + 1} \sum_{m=-l}^{l} a_{lm} a_{lm}^*$$

where $a_{lm}^*$ is the complex conjugate of $a_{lm}$ . This estimate has an uncertainty of (eq. 11),

$$\text{Std}(C_l) = \sqrt{\frac{2}{n - 1}}{C_l} = \sqrt{\frac{2}{2l}}{C_l} = \frac{C_l}{\sqrt{l}}$$

Which as you can see is largest for small values of $l$ . This is generally called “cosmic variance”, and can easily be seen in the Planck results.

References