## A Non-parametric Analysis of the CMB Power Spectrum

September, 2001

Tech Report

### Author(s)

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

### Abstract

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.