Can someone explain angular power spectrum

[hadoque]

Hi
I'm trying to get my head around this plot of the cosmic microwave background radiation.
http://www.astro.ucla.edu/~wright/CMB-LCDM-w-WMAP.gif
I've been searching all around to find out what exactly a angular power spectrum is. I know what an ordinary power spectrum is.
I'm a second year university physics student, haven't read any cosmology.


Thanks for any help

 

[Chalnoth]

Note: Gah, I give up on the latex. My Latex code is right, but it's still showing up as an error on my machine. If you don't see it, sorry, but you can check the link at the bottom for some more detail.

Okay, so what is done is we take the map of the CMB sky, and perform a spherical harmonic transform on it. A spherical harmonic transform is basically the same general concept as a Fourier transform, but the spherical harmonics are mutually orthogonal on the surface of a sphere, like so:

$$f(\theta, \phi) = \sum_{lm}a_{lm}Y(\theta, \phi)_{l}^{m}$$

Here the "l" index denotes the number of oscillations, and the "m" index is a way of encoding the direction of oscillation on the sphere, and varies from "-l" to "l". For example, l=0 is zero oscillation: this is the monopole that sets the overall scale. l=1 is a dipole: one full oscillation over the sphere, and there are three possible directions (x, y, and z). Go to higher and higher l values, and you get more (and therefore smaller) oscillations and more possible directions for those oscillations. The way they are typically written, the spherical harmonic functions $$Y(\theta, \phi)_{l}^{m}$$ are complex functions, and the coefficients are therefore complex. This is a minor issue, though. In order to build the power spectrum, we average over directions. This is done as follows:

$$C_l = \frac{1}{2l+1}\sum_{m=-l}^l a_{lm}a^*_{lm}$$

Finally, in building that plot, you may notice that the vertical axis is not $$C_l$$, but is instead $$C_l l(l+1)/2\pi$$. It turns out if we had a power spectrum that was uniform in a logarithmic interval in l, then multiplying said function by $$l(l+1)/2\pi$$ would give us a constant. Thus this multiplication allows us to interpret the function more easily, because inflation predicts that the primordial power spectrum, the one initially generated by inflation, would be nearly a constant in this space.

If inflation is true, then, all of the features you see in a power spectrum written as above that deviate from a constant stem from the dynamics of the universe between inflation and the emission of the big bang (plus some very slight modification between us and the CMB). For example, the long damping tail at high l stems from the fact that the surface of emission of the CMB is not instantaneous: the phase transition from a plasma to a gas happened over time, and the resultant blurring of the signal damps the small-scale fluctuations. There's also the ratio between the even and odd peaks of the power spectrum. This comes about because of the differences in the physics between normal matter and dark matter: dark matter just falls into potential wells, while normal matter bounces. The failure of dark matter to bounce causes a reduction of the even-numbered peaks relative to normal matter.

Anyway, if you want an in-depth description of the whole process, take a look at Max Tegmark's page:
http://space.mit.edu/home/tegmark/cmb/pipeline.html

 

[hadoque]

Thanks a lot for that lengthy explanation. I can read the latex-code by holding the cursor over the red text, so that's no problem.

Thanks again.

/Johan