Department of Statistics Unitmark
Dietrich College of Humanities and Social Sciences

A Non-parametric Analysis of the CMB Power Spectrum

Publication Date

September, 2001

Publication Type

Tech Report


Christopher J. Miller, Robert C. Nichol, Christopher Genovese and Larry Wasserman


We examine Cosmic Microwave Background (CMB) temperature power spectra from the BOOMERANG, MAXIMA, and DASI experiments. We non-parametrically estimate the true power spectrum with no model assumptions. This is a significant departure from previous research which used either cosmological models or some other parameterized form (e.g. parabolic fits). Our non-parametric 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 non-parametric 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 (\( 0 \le \ell \le 1100\)). Therefore, the current data prefer two peaks at the \(1\sigma\) level. If we assume that there are three peaks in the data, we find their locations to be within \(\ell_1\) = (118,300), \(\ell_2\) = (377,650), and \(\ell_3\) = (597,900). We find the ratio of the first peak-height to the second \((\frac{\Delta T_1}{\Delta T_2})^2 = (1.06, 4.27)\) and the second to the third \((\frac{\Delta T_2}{\Delta T_3})^2 = (0.41, 2.5)\).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 MAP and Planck CMB experiments, we could eliminate two-peak models at the 95% confidence limit. The non-parametric methodology discussed in this paper has many astrophysical applications.