Signal: The Foundation of all our Algorithms

In the last post we referred to the signal metric, which "we really didn't understand." As subscribers to Olin College's philosophy of "do-learn," the next step was understanding the signal metric. It played a crucial role in the rest of the algorithms we developed, so this post will be devoted to an in-depth explanation of what it is.

In particular, we looked at a number known as the signal at every point in each sky. The signal is a measurement of how the ellipticities of each galaxies align with each other at a given point in a sky. Here's the basic idea behind it: all galaxies are elliptical to some extent. If the sky is free of dark matter, we'd expect that their major axes would all be pointing in random directions, as shown in Figure 1.

Figure 2: Elliptical galaxies in a sky with dark matter

Figure 1: Elliptical galaxies in a sky with no dark matter

However, in the presence of dark matter, the major axis of each galaxy will have a tendency to be tangential to the halo, as shown in Figure 2.

The signal is a unit-less measurement of how tangential all the galaxies are to a given point in the sky. It is an average of the tangential ellipticity of each galaxy relative to that given point. The ellipticity of a point (x,y) tangential to (x' y') is as follows (equations from here)

In pseudocode, this algorithm is as follows.

for each x point:
   for each y point:
      sig = average([e_tangential for every galaxy in this sky])

      if sig > max_sig:
          predicted_x = this x
          predicted_y = this y
          max_sig = sig
      
      add this signal to a matrix of (X, Y, Signal) #used for plotting
predicted_halo = (predicted_x, predicted_y)

To gain a better intuition about signals, take another look at Figures 1 and 2. In Figure 2, the point at the center of the figure has a large, positive signal. In Figure 1, there is no net alignment of the galaxies. Therefore the point at the center of the galaxy would have a signal number near 0.

Figure 3: Elliptical galaxies in a sky with no halos (again)

If it appears that galaxies are aligning themselves away from a point (as they are in Figure 3, then it is very unlikely that there's a halo at that point. The center of this figure would have a negative signal number. If a sky has a small area that looked like Figure 3, it is likely there is a halo(s) elsewhere in the sky stretching the galaxies beyond their natural ellipticity.

The larger the signal is at a point, the higher the probability is that a halo exists there. If we calculate the signal at every point in a sky, we get a pretty good estimate of where one of the halos is located. For efficiency reasons, we calculated the signal at every 100 points, not at every single point. Figure 4 contains an example sky plotted on the same grid as its signal. 

Figure 4: Signal for a sky with one halo.

Figure 5: Signal map of a sky with 3 halos

This sky only has one halo, marked by the star. The galaxies are represented by the black ellipses on the figure, and the color map represents the signal. As you can see, the strongest signal occurs right where the halo is. So this measure seems to work very well!

Here's the problem: we don't always only have one halo in the sky. A lot of times there's two or three halos. The signal metric doesn't account for this, since it chooses the single point with the highest signal. Take a look at Figure 5 - it clearly indicates the location of one halo, but without the blue and pink stars, it would tell you nothing about the other two halos' existence. Almost all the skies we examined looked like this, regardless of how many halos were actually in the sky. Since one halo is generally larger than the others, it overshadows the signal at the other halos.

We hoped to expand this method to work for predicting more than 1 halo. In addition we needed a solid way of predicting how many halos are in a given sky. We had a few ideas, but some turned out to be more successful than others...