 Angular sound-horizon degeneracy

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.