A Maximum Likelihood Analysis of the Low CMB Multipoles from WMAP

TL;DR

This paper demonstrates that a quadratic maximum likelihood (QML) estimator provides more reliable estimates of the low-$\ell$ CMB multipoles than the pseudo-$C_\ell$ (PCL) estimator when applied to Galaxy-cut maps. By analyzing WMAP-derived Galaxy-subtracted maps with modest sky cuts ($Kp2$, $Kp0$, and $Kp0+$), the authors find quadrupole amplitudes in the range $176$–$250\, (\mu K)^2$ and octopole amplitudes in $794$–$1183\, (\mu K)^2$, generally larger than the WMAP PCL values but stable across masks. The QML-based statistics, including the $S$-statistic for large-angle correlations, suggest that the discrepancy with the concordance $\Lambda$CDM model is only at the level of a few percent, in line with prior work by TdOH03 and E03b. Overall, the results argue that no exotic new physics is required to explain the low-$\ell$ CMB amplitude, given realistic foreground subtraction uncertainties and estimator performance.

Abstract

The amplitudes of the quadrupole and octopole measured from the Wilkinson Microwave Anisotropy Probe (WMAP) appear to be lower than expected according to the concordance Lambda CDM cosmology. However, the pseudo-Cl estimator used by the WMAP team is non-optimal. In this paper, we discuss the effects of Galactic cuts on pseudo-Cl and quadratic maximum likelihood estimators. An application of a quadratic maximum likelihood estimator to Galaxy subtracted maps produced by the WMAP team and Tegmark, de Oliveira-Costa and Hamilton (2003) shows that the amplitudes of the low multipoles are stable to different Galactic cuts. In particular, the quadrupole and octopole amplitudes are found to lie in the ranges 176 - 250 (micro K)**2 794 - 1183 (micro K)**2 (and more likely to be at the upper ends of these ranges) rather than the values of 123 (micro K)**2 and 611 (micro K)**2 found by the WMAP team. These results indicate that the discrepancy with the concordance Lambda CDM model at low multipoles is not particularly significant and is in the region of a few percent. This conclusion is consistent with an analysis of the low amplitude of the angular correlation function computed from quadratic maximum likelihood power spectrum estimates.

Figures (8)

• Figure 1: The upper figure shows the WMAP-ILC map of Bennett et al. (2003b) which is smoothed with a Gaussian beam of $1^\circ$ FWHM. The lower figure shows the Wiener filtered component separated map of TdOH03.
• Figure 2: Comparison of the PCL and QML estimates of the quadrupole and octopole from simulations with the Kp2 mask. The abscissae list the input values of the quadrupole and octopole used to generate the simulated skies. The ordinates give the output values from the PCL estimator (Figures 2a and 2c) and QML estimator (Figures 2b and 2d) after the application of the Kp2 mask.
• Figure 3: As Figure 2, but for simulations with the Kp0 mask.
• Figure 4: Temperature differences of the low resolution WMAP-ILC and TdOH03 maps. Figure 4a shows the whole sky, for which the rms temperature difference is $8.9 \mu K$. The Kp2 and Kp0 masks are shown in figures 4b and 4c. Figure 4d shows the Kp0+ mask defined in the text, which consists of the Kp0 mask and those pixels for which the temperature difference between the two maps differs by more than $\Delta T/T=3 \times 10^{-6}$ ($8.2 \; \mu K$).
• Figure 5: The low CMB multipoles computed from the WMAP-ILC map. The filled circles show the QML estimates of $\Delta T^2_\ell$ and the solid lines show the PCL estimates. The error bars on the points are computed from the diagonal components of the Fisher matrix $F_{\ell \ell^\prime}$, which assumes the fiducial $\Lambda$CDM model. The panels to the right show three rows of the Fisher matrix. Figure 5a shows the results if no Galactic mask is imposed. The Kp2 Galactic mask is used for Figure 5b, the Kp0 mask for Figure 5c, and the Kp0+ mask for Figure 5d. The dotted lines show the power spectrum of the fiducial $\Lambda$CDM model.