A Method to Estimate the Primordial Power Spectrum from Cosmic Microwave Background Data

Abstract

A precise determination of the primordial spectrum of matter density fluctuations at superhorizon scales is essential to understanding large-scale structure in the universe. Attempts to constrain or obtain the primordial spectrum using data on cosmic microwave background (CMB) anisotropies have relied on statistical and correlation analyses that assume a power-law spectrum. We propose a method to derive P(k) directly from the CMB angular power spectrum that does not presuppose the need to know anything about its functional form. The method consists of a direct inversion of the Sachs-Wolfe formula. Using this new analysis technique and COBE data, we obtain an empirical P(k) that (1) supports a power-law parameterization and (2) has an amplitude and spectral index consistent with previous analyses of the same data. We obtain for the spectral index n = 1.52±0.4 when the second-year COBE data are used and n = 1.22±0.3 using the 4 year data set.