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 (ℓ peak). Numerical and analytic calculations show that ℓ peak is approximately a function of Ω K /Ω M where Ω K and Ω M are the curvature (Ω K > 0 implies an open geometry) and mass density today in units of critical density. Assuming Ω K /Ω M ≪ 1, one obtains ℓ peak ≈ 11 √ 3 9(√ a * +aeq− √ aeq) 2 + ΩK ΩM where a * and a eq are the scale factor at decoupling and radiation-matter equality. The derivation of ℓ peak gives another perspective on the widely-recognized Ω M-Ω Λ degeneracy in flat models. This formula for near-flat cosmogonies together with current angular spectrum data yields familiar parameter constraints. 1. introduction It has long been recognized that the first peak in the CMB angular spectrum provides information about the cosmic variance limited through ℓ ≈ 600, should make the definitive measurement in the near future (Page 2000). As new data resolve higher order peaks, attention shifts to the physics at angular scales beyond that of the first maximum. The object of the present analysis is to clarify the physics derived from the position of the first peak, hereafter called " the peak index " or ℓ peak. The size of the sound horizon r s * at an angular diameter distance D a * to decoupling determines the peak index. 3 This is widely recognized and serves as a starting point. In Sections 2, 3, and 4, numerical and analytic calculations yield the peak index as a function of Ω M and Ω K. All results are checked with CMBFAST (Seljak & Zaldarriaga 1996). In Section 5, a simple formula for ℓ peak , applicable to low Ω K /Ω M universes, is developed alongside geometric and classical interpretations of the formal results. It is found that the peak index approximates a function of Ω K /Ω M. Although our analysis grounds itself in the familiar concepts of D a * and r s * , the results are not widely recognized. While the physical effects responsible for ℓ peak are understood, the interplay that gives the Ω K /Ω M dependence is not intuitive and holds a number of surprises. We end the investigation by …