Question for a friendly cosmologist.

Let's define the sound horizon as $s = \int_0^{t_{ls}} c_s (1+z) dt$ where $c_s$ is speed of sound in primordial plasma (roughly $c/\sqrt{3}$ but you can include precision if you like) and $t_{ls}$ is the time at last scattering, which I take to be 380,000 years. As I understand it, this integral gives $s = 150$ Mpc. It means that this is the distance scale on which galaxies are more likely to be separated now. Back at last scattering, which I think means $z = 1092$, this implies a proper distance $d_{BAO} = 150/(1+z) = 0.137$ Mpc.

Next let's consider the first acoustic peak in CMB. The peak is at $l=200$ which I think corresponds to 1 degree. (If you would like to make this more precise, please go ahead). The angular diameter distance to the surface of last scattering is, I think, $d_A = (14 \;{\rm Gpc})/(1+z) = 12.8$ Mpc. Taking this and multiplying by 1 degree, I get a proper distance $d_{CMB} = 0.22$ Mpc.

My question is how to relate $d_{BAO}$ to $d_{CMB}$. The latter is some sort of correlation length and I guess a precise answer involves a lengthy analysis and numerical methods etc. However, is there a rough picture which gives a good sense of this---say to within better than a factor 2? I can't help noticing that $0.22/0.137$ is quite close to $\pi/2$ for example $\ldots$.


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