A Nonparametric Analysis of the Cosmic Microwave Background Power Spectrum
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Abstract
We examine cosmic microwave background (CMB) temperature power spectra from the BOOMERANG, MAXIMA, and DASI experiments. We nonparametrically estimate the true power spectrum with no model assumptions. This is a significant departure from previous research that used either cosmological models or some other parameterized form (e.g., parabolic fits). Our nonparametric estimate is practically indistinguishable from the best-fit cosmological model, thus lending independent support to the underlying physics that governs these models. We also generate a confidence set for the nonparametric fit and extract confidence intervals for the numbers, locations, and heights of peaks and the successive peak-to-peak height ratios. At the 95%, 68%, and 40% confidence levels, we find functions that fit the data with one, two, and three peaks, respectively (0 ≤ l ≤ 1100). Therefore, the current data prefer two peaks at the 1 σ level. However, we also rule out a constant temperature function at the greater than 8 σ level. If we assume that there are three peaks in the data, we find their locations to be within l1 = (118, 300), l2 = (377, 650), and l3 = (597, 900). We find the ratio of the first peak height to the second (ΔT1/ΔT2)2 = (1.06, 4.27) and the second to the third (ΔT2/ΔT3)2 = (0.41, 2.5). All measurements are for 95% confidence. If the standard errors on the temperature measurements were reduced to a third of what they are currently, as we expect to be achieved by the Microwave Anisotropy Probe and Planck CMB experiments, we could eliminate two-peak models at the 95% confidence limit. The nonparametric methodology discussed in this Letter has many astrophysical applications.