First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Angular Power Spectrum

TL;DR

This paper presents a robust measurement of the CMB angular power spectrum from the WMAP first-year data by combining 28 cross-power spectra across eight high-frequency differencing assemblies, while carefully modeling and marginalizing foregrounds, point sources, beam, and noise uncertainties. It employs two complementary estimation approaches (optimal cross-spectrum combination and a co-added sky map) and demonstrates consistency between methods, yielding a spectrum that is cosmic-variance limited up to $l\approx 350$ and shows clear acoustic peaks at $l\approx 220$ and $\approx 546$. The analysis provides strong support for adiabatic initial conditions and the standard $\Lambda$CDM cosmology, and makes available detailed data products and a likelihood routine to facilitate cosmological parameter inference. Overall, the work establishes a definitive, foreground-corrected CMB power spectrum from the first-year WMAP data and demonstrates rigorous handling of instrument systematics and foreground uncertainties.

Abstract

We present the angular power spectrum derived from the first-year Wilkinson Microwave Anisotropy Probe (WMAP) sky maps. We study a variety of power spectrum estimation methods and data combinations and demonstrate that the results are robust. The data are modestly contaminated by diffuse Galactic foreground emission, but we show that a simple Galactic template model is sufficient to remove the signal. Point sources produce a modest contamination in the low frequency data. After masking ~700 known bright sources from the maps, we estimate residual sources contribute ~3500 uK^2 at 41 GHz, and ~130 uK^2 at 94 GHz, to the power spectrum l*(l+1)*C_l/(2*pi) at l=1000. Systematic errors are negligible compared to the (modest) level of foreground emission. Our best estimate of the power spectrum is derived from 28 cross-power spectra of statistically independent channels. The final spectrum is essentially independent of the noise properties of an individual radiometer. The resulting spectrum provides a definitive measurement of the CMB power spectrum, with uncertainties limited by cosmic variance, up to l~350. The spectrum clearly exhibits a first acoustic peak at l=220 and a second acoustic peak at l~540 and it provides strong support for adiabatic initial conditions. Kogut et al. (2003) analyze the C_l^TE power spectrum, and present evidence for a relatively high optical depth, and an early period of cosmic reionization. Among other things, this implies that the temperature power spectrum has been suppressed by \~30% on degree angular scales, due to secondary scattering.

Figures (14)

• Figure 1: The effective noise as a function of $l$ for the 8 differencing assemblies used in the combined power spectrum analysis. These spectra were computed from end-end simulations of noise maps, as discussed in §\ref{['sec:inst_noise']}. For illustration, the spectra shown here were computed using the quadratic estimator with uniform pixel weights. The actual noise model uses three separate weighting schemes in three separate $l$ ranges, and thus has discontinuities where the effective noise level changes. The weights are defined in Appendix \ref{['sec:weights']}.
• Figure 2: The full set of individual cross-power spectra for $l<500$, computed from the 8 high frequency differencing assemblies Q1 through W4, 28 spectra in all. The spectra were evaluated from the uncorrected sky maps (no Galaxy model subtracted) using the Kp2 sky cut with uniform weighting. The data are plotted in color by effective frequency $\sqrt{\nu_i\nu_j}$ with red corresponding to 41 GHz and blue to 94 GHz. The top panel shows a very robust measurement of the first acoustic peak with a maximum near $l \sim 220$. There is also a clear indication of the rise to a second peak at $l \sim 540$ as discussed in §\ref{['sec:discuss']}. The bottom panel shows the ratio of each channel to the combined spectrum presented in §\ref{['sec:combined']}. This clearly shows the residual foreground emission due to diffuse Galactic radio emission at low $l$ and to point sources at higher $l$. The level of contamination, which is strongest at Q band, is consistent with the expected level of foreground emission. See Figure \ref{['fig:cross_power_corr']} for the spectra after foreground subtraction.
• Figure 3: The same set of cross-power spectra as shown in Figure \ref{['fig:cross_power_raw']}. Here, the foreground model discussed in §\ref{['sec:foregrounds']} has been subtracted from each channel. The bottom panel shows the ratio of each of the 28 cross-power spectra to the combined spectrum presented in §\ref{['sec:combined']}. Aside from a $\sim$10% discrepancy in the Q band data at $l < 20$, the data are consistent with each other to the sensitivity limits of the individual spectra. Because the WMAP data are not sensitivity limited at low $l$, we use only V and W band data in the final combined spectrum for $l < 100$ (see §\ref{['sec:combined']}) to minimize residual Galactic contamination.
• Figure 4: The same set of cross-power spectra as shown in Figure \ref{['fig:cross_power_corr']}, but showing the low $l$ spectrum unbinned. The agreement between the individual spectra is striking. The low value of the quadrupole moment, $C_2$, that was first seen by COBE-DMR is also seen in the WMAP data. The steep, nearly linear rise in the spectrum from $l=2$ to 5 translates to a near absence of power in the angular correlation function at separations larger than $\sim$60$^{\circ}$spergel/etal:2003bennett/etal:2003b. This was also seen in the COBE-DMR data, but it is now clear that this is not due to Galaxy modeling errors. The bottom panel shows the fractional rms among the 28 WMAP cross-power spectra in black, while the red curve shows the same statistic averaged over an ensemble of 1000 Monte Carlo realizations. Based on this we estimate the measurement error on the combined spectrum to be $<$2-5% for $l < 100$.
• Figure 5: The auto-power spectrum of the combined Q+V+W map evaluated with the three weighting schemes defined in Appendix \ref{['sec:weights']}. In each case, the spectrum was computed over the entire $l$ range, and black shows uniform weights, red shows inverse-noise weights, and green shows transitional weights. The agreement is excellent.