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Either way, my gut says there's some other source of contamination that was missed.

My gut says "statistical fluctuation".

Look at the maps in Figure 1. The largest cold spot happens to lie just to the right of center, and the largest hot spot happens to lie just to the right of that. The hot spot roughly lines up with one of the two hot spots in the quadrupole plot (Figure 14a) and the cold spot roughly lines up with one of the two cold spots in the quadrupole plot.

Furthermore, that hot spot and cold spot also roughly line up with hot and cold spots in the octopole plot. Since the alignment of the quadrupole and octopole moments is dominated by those two features in the CMB map, perhaps it's not surprising that they roughly line up.

What are the odds of a random fluctuation of that size, in the absence of any preferred direction in space? I haven't the foggiest idea. I suspect it's not negligible, but I could be wrong.

I'll go ask Max.

I asked Max whether the alignment between the quadrupole and octopole moments was being driven by one or two specific features in the map. He said it was entirely possible. I asked him what the likelihood was that such a fluctuation could arise randomly, and he said that he was working on exactly that general problem, and I should ask him again in a day or two.

He did say that the COBE collaboration calculated that there was a 1 in 1000 chance that the quadrupole amplitude would be as small as what is observed. Their new analysis gives a greater amplitude (chance about 1 in 20).

He also reminded me that the paper assumes Gaussian random distributions, and that if the distributions were non-Gaussian, it could significantly affect our expectations.

N.B.: Max just now updated the paper on his website to a new version. There are some new plots and new information. What I referred to before as Figure 14 is now Figure 15.