I How is the sound horizon at recombination and BAO measured in the CMB?

[jordy1113]

So I'm trying to understand the sound horizon measured at recombination and the sound horizon measured with BAO. Here is what I've gathered (PLEASE tell me if I'm wrong and if you could please explain, I've been trying to read but can't find a clear explanation):

~r(z*) is measured using CMB (could someone explain to me exactly how this is measured) at z*=1100 (recombination epoch)
~ r using BAO can be measured from the drag epoch to very very small redshifts
~ both sound horizons are basically the same (around 140Mpc, yes I know they're not exactly the same but they vary only a little)

Again if someone could explain how these are measure (like why choose angular size of 1 degree for r(z*)) I really need to clear up my confusion :( Thank you to everyone in advance

 

[Bandersnatch]

The experts seem to have vacated the premises, so you're stuck with me. My understanding is incomplete, though.

With that disclaimer - You get 1 degree as your base scale from theoretical considerations. You calculate the physical size of the sound horizon at recombination (r, about 0.14 Mpc, or 0.45 Mly, which is 150 Mpc in comoving terms). This gives you an idea of how big the largest 'bubble' leaving imprint on the CMB should be. You then calculate the proper distance from the observer to the CMB at recombination (R, about 45 Mly), so that you can relate the physical size to its observed size. Using small angle approximation, you get the angular size by dividing r/R, which netts about 1 degree. As the universe expands, this angular size remains constant.

You can then measure the size of the temperature fluctuations in the CMB to see how they compare to theory. E.g. a deviation here would suggest the geometry of the universe being different than the assumed flat one.

I have no hands-on experience with actually measuring these, nor have I seen it done.
I suspect it's done with some sort of statistical analysis across the whole sky, rather than taking a ruler to any single fluctuation, considering how messy it all is.