A precise determination of the primordial spectrum of matter densityfluctuations at super-horizon scales is essential in understanding large scalestructure in the universe. Attempts to constrain or obtain the primordialspectrum using data on cosmic microwave background (CMB) anisotropies hasrelied on statistical and correlation analyses that assume a power-lawspectrum. We propose a method to derive $P(k)$ directly from the CMB angularpower spectrum which does not presuppose the need to know anything about itsfunctional form. The method consists of a direct inversion of the Sachs-Wolfeformula. Using this new analysis technique and COBE data we obtain an empirical$P(k)$ which 1) supports a power-law parameterization and 2) has an amplitudeand spectral index consistent with previous analyses of the same data. Weobtained for the spectral index, $n=1.52 \pm 0.4$ when the 2nd year COBE datais used and $n=1.22 \pm 0.3$ using the 4 year data set.
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机译：精确确定超水平尺度上物质密度波动的原始谱对于理解宇宙中的大尺度结构至关重要。利用宇宙微波背景（CMB）各向异性数据限制或获得原始光谱的尝试依赖于假设功率谱的统计和相关分析。我们提出了一种直接从CMB角功率谱中导出$ P（k）$的方法，该方法不以需要了解有关其功能形式的任何知识为前提。该方法包括Sachs-Wolfeformula的直接反演。使用这种新的分析技术和COBE数据，我们获得了一个经验$ P（k）$，其中1）支持幂律参数化，并且2）具有与先前对相同数据的分析一致的幅度和光谱指数。当使用第二年的COBE数据时，我们获得了光谱指数$ n = 1.52 \ pm 0.4 $，使用四年数据集获得了$ n = 1.22 \ pm 0.3 $。
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