A perspective on the CMB acoustic peak

Abstract

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.

4 pages; submitted to ApJ Letters