I understand that we can measure the sound horizon from the anisotropies in the CMB. Large galaxy surveys today can construct two-point correlation functions, which reveal peaks at distances equal to today's sound horizon, I think because the BAOs at decoupling essentially created dark matter gravitational wells for structure to begin forming in (correct me if I'm wrong). I'm under the impression that the sound horizon is a fixed physical distance, so its measured comoving distance at any redshift should always be the same. Am I right or wrong in this assumption?
I read some information from a few different sources on this topic which confused me. On Wikipedia: "Peaks have been found in the correlation function (the probability that two galaxies will be a certain distance apart) at 100 h−1 Mpc, indicating that this is the size of the sound horizon today, and by comparing this to the sound horizon at the time of decoupling (using the CMB), we can confirm the accelerated expansion of the universe."
But if the sound horizon is always at the same physical scale, how can we use measurements today to determine the rate of expansion? Do the correlation functions give you proper separations, and from that with different cosmological models we can determine what $H_0$ or $H(z)$ value gives a comoving separation equal to the sound horizon? And is this the same as the BAO signal?