Mathematical Sciences: New Methods for Inference from COBE Data

引力不稳定性理论通常假设原始密度涨落的频谱服从幂定律，其幅度来自于高斯分布，并且存在某种形式的暗物质。另一种理论依赖于拓扑缺陷，如单极、宇宙弦、纹理或其他动力学效应来产生今天的结构。不同的理论预测了CMB涨落的不同角谱。拓扑缺陷模型一般预测起伏幅度分布的非高斯统计特性。我们将开发方法和软件来寻找CMB温度分布的矩、多极系数和某些理论预测的“幂定律”的光谱参数的可信区间。这些估计也考验了最初波动的高斯性。我们的新方法将包括极小极大估计和“严格界”，并将严重依赖于大规模的非线性优化。我们将使用一个由35个Sun SPARC站点组成的网络来执行分布式模拟并并行解决大型优化问题。极小极大法和严格界方法都需要对CMB的先验约束，我们将从物理理论和结合不同空间尺度上的其他实验结果得出CMB的先验约束。宇宙背景探测器(COBE)团队近日宣布探测到宇宙微波背景(CMB)温度波动。CMB的波动追踪了物质和能量密度的原始变化，因为它们是在大爆炸后大约30万年的时候。宇宙学理论需要这些密度变化来解释观测到的当今大尺度结构，例如星系、星系团、超星系团和空洞。不同的理论需要不同的原始密度结构，一个主要的目标是使用COBE数据来区分这些理论。目前在天体物理学中使用的统计工具不允许在这些理论之间进行严格的区分：这些工具受到已知偏差的影响，这些偏差的大小在很大程度上是未知的。我们将开发和应用新的工具，使用来自宇宙学理论和其他实验的约束来限制可能的偏差，减少不确定性。

Gravitational instability theories usually assume that the spectrum of primordial density fluctuations follows a power law with amplitudes drawn from a Gaussian distribution, and the existence of some form of dark matter. Alternative theories depend upon topological defects, such as monopoles, cosmic strings, textures, or other dynamical effects to generate present day structures. Various theories predict different angular spectra for CMB fluctuations. Topological defect models generally predict non-Gaussian statistics for the fluctuation amplitude distribution. We will develop methodology and software to find confidence intervals for moments of the CMB temperature distribution, multipole coefficients, and spectral parameters of ``power laws'' predicted by some theories. These estimates also test the gaussianity of the initial fluctuations. Our new methodology will include minimax estimation and ``strict bounds,'' and will rely heavily on large-scale nonlinear optimization. We will employ a network of 35 Sun SPARCStations to perform distributed simulations and solve large optimization problems in parallel. Both the minimax and strict bounds approaches require a priori constraints on the CMB, which we will derive from physical theory and by incorporating the results from other experiments on different spatial scales. The Cosmic Background Explorer (COBE) team recently announced the detection of fluctuations in the temperature of the Cosmic Microwave Background (CMB). CMB fluctuations trace the primordial variations in the density of matter and energy as they were about 300,000 years after the Big Bang. These density variations are required by cosmological theories to account for observed present-day large-scale structure, e.g., galaxies, clusters of galaxies, superclusters and voids. Different theories require different structure in the primordial density, and a principal goal is to use COBE data to discriminate among these theories. The statistical tools currently used in the astrophysics comm unity do not permit rigorous discrimination among these theories: the tools are subject to known biases whose magnitudes are largely unknown. We will develop and apply new tools that use constraints derived from the cosmological theories and from other experiments to limit the possible bias and reduce the uncertainty.