A perspective on the CMB acoustic peak

CMB angular spectrum measurements suggest a flat universe. This paper clarifies the relation between geometry and the spherical harmonic index of the first acoustic peak ($ell_{peak}$). Numerical and analytic calculations show that $ell_{peak}$ is approximately a function of $Omega_K/Omega_M$ where $Omega_K$ and $Omega_M$ are the curvature ($Omega_K > 0$ implies an open geometry) and mass density today in units of critical density. Assuming $Omega_K/Omega_M ll 1$, one obtains a simple formula for $ell_{peak}$, the derivation of which gives another perspective on the widely-recognized $Omega_M$-$Omega_Lambda$ degeneracy in flat models. This formula for near-flat cosmogonies together with current angular spectrum data yields familiar parameter constraints.