# How to relate CMB acoustic peak to BAO in galaxy surveys to better than a factor 2?

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 $$\ldots$$.