= Angular sound-horizon degeneracy
{title2=$\theta_*=r_s(z_*)/D_M(z_*),\quad\ell_A=\pi/\theta_*$}
The angular <sound horizon> compares two lengths using the same distance convention. If $D_A$ is the <angular diameter distance> to last scattering at <redshift> $z_*$, define the transverse <comoving distance> $D_M=(1+z_*)D_A$. A comoving <sound horizon> $r_s$ has proper length $r_s/(1+z_*)$ at emission, so its angular size is $\theta_*=r_s/D_M$. Mixing a comoving ruler with $D_A$ would introduce an erroneous factor $1+z_*$. In a curved <FLRW metric>, $D_M$ is generally different from the radial <comoving distance>.
For a stipulated example, $r_s=144\,\mathrm{Mpc}$ and $D_M=14000\,\mathrm{Mpc}$ give $\theta_*=0.010286$ radians, or $0.589^\circ$, and an acoustic scale $\ell_A=305.4$. Multiplying both lengths by $1.1$ gives $158.4\,\mathrm{Mpc}$ and $15400\,\mathrm{Mpc}$ with exactly the same $\theta_*$ and $\ell_A$. This example establishes a ratio degeneracy, without claiming that any arbitrary rescaling corresponds to an allowed cosmological model. Equivalently,
$$
\delta\ln\theta_*=\delta\ln r_s-\delta\ln D_M.
$$
Other observables are needed to separate changes that preserve the ratio.
The approximate <adiabatic acoustic-peak spacing> is $\ell_A$. The first observed temperature peak is shifted by gravitational driving and <angular projection of density and Doppler acoustic sources>; it need not occur at $\ell_A$. In particular, neither the illustrative $305.4$ nor the <BOOMERANG acoustic-peak measurement> of $197\pm6$ should be substituted for the other.
The <cosmological baryon density> enters through the <baryon loading parameter> and <cosmological recombination>, changing relative acoustic amplitudes as well as the <sound horizon>. The <baryon-to-photon ratio> counts net <baryon number> per <photon>, whereas $\theta_*$ is an angle. Constraining <cosmological baryon density> requires the pattern of relative peak heights within a model, with other parameters fitted jointly. <Big Bang nucleosynthesis> provides a separate abundance test. <Baryon-density consistency between nucleosynthesis and the microwave background> is consequently a test of one common cosmic history, rather than two direct measurements of the same angle. A first peak by itself cannot determine every component of the <cosmological density parameter> or uniquely fix <spatial curvature of an FLRW universe>.
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