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.