I do not read the astro-ph archive on a daily basis (as any astro-* or *-ph archive) but I use liferea to stay up to date with a number of blogs. This news aggregator shows a small window with the heading if a new entry appears in the logs that I told it to monitor. This morning, it showed an entry form Physics Comments with the title "Local pancake defeats axis of evil". My first reaktion was that this must be a hoax, there could not be a paper with that title.

But the paper is genuine. I read the four pages over lunch and it looks quite interesting: When you look at the WMAP power spectrum (or COBE for that matters) you realize that for very low l there is much less power than expected from the popular models. Actually, the plot starts with l=2 because l=0 is the 2.73K uniform background and l=1 is the dipole or vector that is attributed to the Doppler shift due the motion of us (the sun) relative to the cosmic rest frame.

What I did not know is that the l=2 and l=3 have a prefered direction and they actually agree (althogh not with the dipole direction, they are perpendicular to it). This fact was realised by Copi, Huterer, Starkman, and Schwarz as I am reminded). I am not entirely sure what this means on a technical level but it could be something like "when this direction is chosen as the z-direction, most power is concentrated in the m=0 component". This could be either due to a systematic error or a statistical coincidence but Vale cites that this is unlikely with 99.9% confidence.

This axis encoded in the l=2 and l=3 modes has been termed "axis of evil" by Land and "varying speed of light and I write a book and offend everybody at my old university" Magueijo. The in the new paper, Vale offers an explanation for this preferred direction:

His idea is that gravitational lensing can mix the modes and what appears to be l=2 and l=3 is actually the dipole that is mixed into these higher l by the lensing. To first order, this is effect is given by

A = T + grad(T).grad(Psi)

(Jacques is right, I need TeX formulas here!) where T is the true temperature field, A is the apprarant temperature field and Psi is a potential that summarizes the lensing. All these fields are function of psi and phi, the coordinates on the celestial sphere.

He then goes on an uses some spherical mass distribution of twice the mass of the Great attractor 30Mpc away from us to work out Psi and eventually A. The point is that the l=1 mode of A is two orders of magnitude stronger than l=2 and l=3 so small mixing could be sufficient.

What i would like to add here is how to obtain some analytical expressions: As always, we expand everything in spherical harmonics. Then

A_lm = T_lm + integral( Y_lm T_l'm' Psi_l"m" grad(Y_l'm').grad(Y_l"m") )

I am too lazy, but with the help of Mathworlds page on spherical harmonics and spherical coordinates you should be able to work out the derivatives and the integral analytically. By choosing coordinates aligned with the dipole, you can assume that in the correction term only the l'=1, m'=0 term contribute.

Finally, the integral of three Y's is given by an expression in Wigner 3j-symbols and those are non-zero only if the rules for addition of angular momentum hold. Everybody less lazy than myself should be able to work out which A_lm are affected by which modes of Psi_l"m" and it should be simple to invert this formula to find the modes of Psi if you assume all power in A comes from T_10. Especially Psi_l"m" should only infulence modes with l and m not too different from l" and m". By looking at the coefficients, maybe one is even able to see that only the dipole component of Psi has a stron infuence and this only on l=2 and l=3 and only for special values of m.

This would then be an explanation of this axis of evil.

Still, the big problem not only remains but gets worse: The observed power in l=2 and l=3 are too small to fit the model and they even get smaller if one subtracts the leaked dipole.

Using the trackback mechanism at the arxive, I found that at CosmoCoffee there is also a discussion of this paper going on. There, people point out that what counts is the motion of the lense relative to the background (Vale is replying to comments there as well).

It seems to me that this should be viewed as a two stage process: First there is the motion of the lense and this is what converts power in l=1 (due to the lense's motion) to l=2 and l=3. Then there is our motion but that affects only l=1 and not l=2 and l=3 anymore. Is that right? In the end, it's our motion that turns l=0 into l=1.

## Tuesday, September 06, 2005

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I DID NOT PUT THE SIGN THERE! I just filled in the fields with my date of birth and blogger.com computed it all by itself. How do I get rid of it? I should probably look at the template...

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