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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 $z_{\ast }$, define the transverse comoving distance $D_M=(1+z_{\ast })D_A$. A comoving sound horizon $r_s$ has proper length $r_s/(1+z_{\ast })$ at emission, so its angular size is ${\theta }_{\ast }=r_s/D_M$. Mixing a comoving ruler with $D_A$ would introduce an erroneous factor $1+z_{\ast }$. 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 ${\theta }_{\ast }=0.010286$ 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 ${\theta }_{\ast }$ and ${\ell }_A$. This example establishes a ratio degeneracy, without claiming that any arbitrary rescaling corresponds to an allowed cosmological model. Equivalently,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 $197\pm 6$ 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 ${\theta }_{\ast }$ 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.

P. J. E. Peebles made the connection between microscopic processes and cosmological observables quantitative. His 1968 paper treated cosmological recombination as a kinetic problem. Capturing an Electron directly into the hydrogen ground state releases a photon that can ionize another atom. Net neutralization depends on escape of resonance radiation through cosmological redshift and on two-photon spontaneous emission from the $2s$ state. An equilibrium Saha equation alone does not give that history.^[ref]

This matters for an acoustic prediction because the hydrogen ionization fraction fixes the supply of free Electrons for Thomson scattering. In physical units the scattering rate per photon is ${\Gamma }_T=n_e{\sigma }_{T}c$. The cosmological visibility function describes the distribution of last-scattering times. Thus atomic physics helps determine when the photon-baryon acoustic oscillator is read out, as well as how rapidly tight coupling fails.

In 1970 P. J. E. Peebles and J. T. Yu followed adiabatic cosmological perturbations by numerically integrating the photon collision equation through cosmological recombination. Their calculation connected acoustic evolution and dissipation to residual matter structure and microwave brightness fluctuations. It did not assume that a perfect photon-baryon fluid remained valid throughout last scattering. Their illustrative cosmological models and normalization were historical choices, rather than the present Lambda-CDM model; their predicted brightness variance versus angular resolution should not be relabelled as a modern measured $C_{\ell }$ spectrum.^[ref]

The continuing idea is a forward prediction: specify initial conditions, evolve the coupled matter and radiation, project onto the sky, and compare the result with an instrument's readout. Coherent initial phases are needed for acoustic variance peaks explains why a collection of oscillating modes does not automatically produce a peaked power spectrum. The BOOMERANG acoustic-peak measurement provides a later observational test of this physical programme.