Measuring the Geometry of the Universe in the Presence of Isocurvature Modes

The cosmic microwave background (CMB) anisotropy constrains the geometry of the Universe because the positions of the acoustic peaks of the angular power spectrum depend strongly on the curvature of three-dimensional space. In this Letter we exploit current observations to determine the geometry in the presence of isocurvature modes. Most previous analyses assumed that the primordial perturbations were adiabatic. A priori one might expect that allowing isocurvature modes would substantially degrade constraints on the curvature. We find, however, that with additional data sets, the geometry remains well constrained. When the most general isocurvature perturbation is allowed, the CMB alone can only poorly constrain the geometry to ${\Omega }_0=1.6\pm 0.3$. Including large-scale structure data, one obtains ${\Omega }_0=1.07\pm 0.03$, and $1.06\pm 0.02$ when supplemented by supernova data and the determination of $H_0$.