We consider structure formation and cosmic microwave background (CMB) anisotropies in a closed universe, both with and without a cosmological constant. The CMB angular power spectrum and the matter transfer function are presented, along with a discussion of their relative normalization. This represents the first full numerical evolution of density perturbations and anisotropies in a spherical geometry. We extend the likelihood function versus Ω from the CO BE 2 year data to Ω ≥ 1. For large ? the presence of a very steep rise in the spectrum toward low l allows us to put an upper limit of Ω ≤ 1.5 (95% CL) for primordial spectra with n ≤ 1. This compares favorably with existing limits on Ω. We show that there are a range of closed models that are consistent with observational constraints while being even older than the currently popular flat models with a cosmological constant. Future constraints from degree scale CMB data may soon probe this region of parameter space. A derivation of the perturbed Einstein, fluid, and Boltzmann equations for open and closed geometries is presented in an Appendix.
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机译：我们考虑有无宇宙常数的封闭宇宙中的结构形成和宇宙微波背景（CMB）各向异性。介绍了CMB角功率谱和物质传递函数，并讨论了它们的相对归一化。这代表了球形几何结构中密度扰动和各向异性的第一个完整的数值演化。我们将CO BE 2年数据中的似然函数与Ω的关系扩展到Ω≥1。频谱朝着低l处非常陡峭的上升，使我们可以将n≤1的原始频谱的上限设置为Ω≤1.5（95％CL）。这与Ω的现有限制相比是有利的。我们显示，有一系列与观测约束一致的封闭模型，但比具有宇宙学常数的当前流行的平面模型还要老。来自度数级CMB数据的未来约束可能很快会探查此参数空间区域。附录中介绍了开放和封闭几何形状的摄动爱因斯坦方程，流体方程和玻尔兹曼方程。
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