If the physical size of the sound horizon at the surface of last scattering is fixed (e.g., Jungman et al. 1995), then the angular scale associated with this depends on the geometry of the universe.

In other words, using your definitions, you measure $\theta$, and $S$ is fixed, so you are effectively measuring $D = S/\theta$. Getting from there to a unique set of cosmological parameters would be difficult - you can't separate all the relevant parameters with one piece of information, although how flat the universe is (i.e. how close $\Omega_M + \Lambda$ is to 1) does have the dominant effect, along with $H_0$.

But I think what you are getting at, is doesn't $S$ also depend on the cosmological parameters? Well, yes it does, a bit, but not in the same way that $D$ varies with those parameters. This is what avoids the circularity you suggest. A good discussion can be found in section 5 of Reid et al. (2002).

Fortunately, that isn't how the cosmogical parameters are determined at all (other than conceptually). They are found by modelling the full CMB angular power spectrum and fitting that to the data. The additional information that comes from the height and shape of the peaks, not just the first peak, provides enough information to break most of the parameter degeneracies. The sensitivity of the angular power spectrum to those parameters is also discussed in Jungman et al (1995).

If the physical size of the sound horizon at the surface of last scattering is fixed (e.g., Jungman et al. 1995), then the angular scale associated with this depends on the geometry of the universe.

In other words, using your definitions, you measure $\theta$, and $S$ is fixed, so you are effectively measuring $D = S/\theta$. Getting from there to a unique set of cosmological parameters would be difficult - you can't separate all the relevant parameters with one piece of information, although how flat the universe is (i.e. how close $\Omega_M + \Lambda$ is to 1) does have the dominant effect, along with $H_0$

But I think what you are getting at, is doesn't $S$ also depend on the cosmological parameters? Well, yes it does, but not in the same way that $D$ varies with those parameters. This is what avoids the circularity you suggest. A good discussion can be found in section 5 of Reid et al. (2002).

Fortunately, that isn't how the cosmogical parameters are determined at all (other than conceptually). They are found by modelling the full CMB angular power spectrum and fitting that to the data. i.e. Choose your cosmological parameters; generate a model angular power spectrum; does it match the observed CMB power spectrum, if not then iterate the parameters until it does. The additional information that comes from the height and shape of the peaks, not just the first peak, provides enough information to break most of the parameter degeneracies. The sensitivity of the angular power spectrum to those parameters is also discussed in Jungman et al (1995).

If the physical size of the sound horizon at the surface of last scattering is fixed (e.g., Jungman et al. 1995), then the angular scale associated with this depends on the geometry of the universe.

In other words, using your definitions, you measure $\theta$, and $S$ is fixed, so you are effectively measuring $D = S/\theta$. Getting from there to a unique set of cosmological parameters would be difficult - you can't separate all the relevant parameters with one piece of information, although how flat the universe is (i.e. how close $\Omega_M + \Lambda$ is to 1) does have the dominant effect, along with $H_0$.

Fortunately, that isn't how the cosmogical parameters are determined. They are found by modelling the full CMB angular power spectrum and fitting that to the data. The additional information that comes from the height and shape of the peaks, not just the first peak, provides enough information to break most of the parameter degeneracies. The sensitivity of the angular power spectrum to those parameters is also discussed in Jungman et al (1995).

If the physical size of the sound horizon at the surface of last scattering is fixed (e.g., Jungman et al. 1995), then the angular scale associated with this depends on the geometry of the universe.

In other words, using your definitions, you measure $\theta$, and $S$ is fixed, so you are effectively measuring $D = S/\theta$. Getting from there to a unique set of cosmological parameters would be difficult - you can't separate all the relevant parameters with one piece of information, although how flat the universe is (i.e. how close $\Omega_M + \Lambda$ is to 1) does have the dominant effect, along with $H_0$.

But I think what you are getting at, is doesn't $S$ also depend on the cosmological parameters? Well, yes it does, a bit, but not in the same way that $D$ varies with those parameters. This is what avoids the circularity you suggest. A good discussion can be found in section 5 of Reid et al. (2002).

Fortunately, that isn't how the cosmogical parameters are determined at all (other than conceptually). They are found by modelling the full CMB angular power spectrum and fitting that to the data. The additional information that comes from the height and shape of the peaks, not just the first peak, provides enough information to break most of the parameter degeneracies. The sensitivity of the angular power spectrum to those parameters is also discussed in Jungman et al (1995).