Multipole ${\it l}$ value for the second acoustic peak in the CMB power spectrum So according to the Planck paper https://arxiv.org/pdf/1502.01582.pdf page 20, the first peak is  ${\it l}$=220, but the second peak is ${\it l}$=537. Not quite the double of the first peak, why? If the fundamental wave vs harmonics analogy were to hold with the sound wave, shouldn't it be doubled? Is it because of the non exact relation between l and angular size? Is the /angular size/ of the second peak 1/2 of that of the first peak?
 A: There are a few reasons why the CMB angular power spectrum is more complicated than the spectrum for "waves on a string." To give a few examples:

*

*First, the waves exist in three dimensional space, but we only observe angular fluctuations on our sky. There is a projection that occurs while passing from three dimensional modes with a "wavenumber" $k$, to two dimensional modes with an "angular quantum number" $\ell$. Angular mode $\ell$ will therefore contain information from a range of different 3d modes with different values of $k$.


*Second, the primordial plasma had a mix of different components, that were not all in equilibrium. In particular, there was the relativistic primordial plasma -- consisting mostly of electrons, protons, and photons -- as well as dark matter. The primordial plasma will oscillate due to sound waves, creating the peak-like structure in the CMB. However, the dark matter does not oscillate, and only falls into local gravitational wells. This will tend to enhance the amplitude of plasma oscillations that "fall into" the wells, and decrease the amplitude of plasma oscillations that "climb out of" the wells (the relative heights of the second and third peaks is one piece of evidence for dark matter).


*The dispersion relationship for the plasma (which controls the wavelengths of each mode) will likely not be as simple as what you would get for a wave on a string, because of underlying interactions in the plasma.
I am not sure what precisely explains the specific point you brought up -- maybe someone else will be able to give a more detailed answer -- but this is just to say that there is more going on in the CMB than in the toy "waves on a string" problem, so you shouldn't expect the peaks in the CMB to follow a pattern derived for waves on a string.
