First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Beam Profiles and Window Functions

TL;DR

This paper tackles the challenge of accurately characterizing the WMAP beam profiles, which determine how the sky signal is smoothed by the instrument and thus shape the recovered CMB power spectrum through the window function $w_\ell$.It employs in-flight Jupiter maps in the nominal CMB observing mode, a Hermite expansion of the symmetric beam to derive transfer functions $B_\ell$ and window functions $w_\ell$, and a physical optics model to interpret beam distortions, all while carefully propagating uncertainties.Key results include per-band main-beam solid-angle uncertainties ($$\sim2$–$2.6\%$$), quantified window-function uncertainties (typically a few percent), and targeted corrections for sidelobes and a W-band pedestal, with Jupiter-based calibration achieving $1$–$3\%$ accuracy relative to the CMB dipole.The work provides coadded-beam parameters for map analyses, flux-to-temperature conversion factors, and public data products (Jupiter maps and window functions), establishing a robust framework for precise CMB analysis and informing beam modeling for future missions.Overall, the study delivers a comprehensive, uncertainty-aware beam model that underpins accurate CMB angular power-spectrum estimation from the first-year WMAP data.

Abstract

Knowledge of the beam profiles is of critical importance for interpreting data from cosmic microwave background experiments. In this paper, we present the characterization of the in-flight optical response of the WMAP satellite. The main beam intensities have been mapped to < -30 dB of their peak values by observing Jupiter with the satellite in the same observing mode as for CMB observations. The beam patterns closely follow the pre-launch expectations. The full width at half maximum is a function of frequency and ranges from 0.82 degrees at 23 GHz to 0.21 degrees at 94 GHz; however, the beams are not Gaussian. We present: (a) the beam patterns for all ten differential radiometers and show that the patterns are substantially independent of polarization in all but the 23 GHz channel; (b) the effective symmetrized beam patterns that result from WMAP's compound spin observing pattern; (c) the effective window functions for all radiometers and the formalism for propagating the window function uncertainty; and (d) the conversion factor from point source flux to antenna temperature. A summary of the systematic uncertainties, which currently dominate our knowledge of the beams, is also presented. The constancy of Jupiter's temperature within a frequency band is an essential check of the optical system. The tests enable us to report a calibration of Jupiter to 1-3% accuracy relative to the CMB dipole.

Figures (6)

• Figure 1: Jupiter maps of the A and B side focal planes bennett/etal:2003 in the reference frame of the observatory. The contour levels are at 0.9, 0.6, 0.3, 0.09, 0.06, 0.03 of the peak value. W1 and W4 are the "upper" W-band radiometers. In W band, the lobes at the 0.09 contour level ($\approx -10~$dB) and lower are due to surface deformations.
• Figure 2: Left: The symmetrized beams (normalized at unity) and noise levels (below) from two seasons of Jupiter observations. Both polarizations have been combined. The noise rises at small radii because there are fewer pixels over which to average. With four years of observations, the noise level will be reduced by a factor of two. Right: The K (black) and W3 (grey) symmetrized beam profiles with their associated Ruze patterns (§\ref{['sec:bu']}). The noise level is at 20 dBi in all bands as seen in the plot (missing data corresponds to negative values). The maximum optical gains are 47.1 and 59.3 dBi in K and W bands as indicated by the horizontal lines. Table 1 shows the gain budget. The dashed lines are the Ruze patterns assuming a Gaussian shaped distortion with the parameters in the text. The lighter shaded dotted lines that meet the dashed lines at $\theta=0$ are for a tophat shaped distortion. In W band, the tophat prediction, which has a prominent lobe at $\theta=2\fdg5$ clearly does not fit the data. Plots for W14 show the Ruze pattern to be above the beam profile for $\theta<1^{\circ}$ suggesting the magnitude of the deformations is not greater than those we use. However, some fraction of $\Omega_B$ could be at or near the noise level for $1\fdg5<\theta<2\fdg0$. The vertical straight lines indicate the cutoff radii, $\theta_{Rc}$, for the Gaussian distortion model.
• Figure 3: Left: A mosaic of the A-side W and V-band measured beams. One should focus on the main beams areas. Different noise levels in the constituent mosaics lead to apparent artifacts away from the beam centers. Right: Model of the A-side beams based on the physical optics calculations described in the text. The same surface is used for all beams. Most of the features in the measured beams are reproduced in the model indicating that the source of the distortions has been identified. The noise from the measurement has been added to model on the right to make the comparison more direct. The separation between different W-band beams is $1\fdg1$, less than the cutoff radius for the determining the W-band solid angles.
• Figure 4: The ten window functions, $w_\ell$, computed from the Hermite expansion. The window functions for the two polarizations in each feed are nearly indistinguishable at the resolution of the plot.
• Figure 5: The transfer functions and their statistical and systematic uncertainty. The y axis of each panel shows the fractional uncertainty. The green curve is the statistical uncertainty in the Hermite-based transfer function. The orange curve corresponds to twice the statistical uncertainty (it is mostly hidden by the black curve). The red curve is the fractional difference between the $b_l$ computed from the spherical harmonic decomposition of the time stream and the Hermite fit. The blue line is the fractional difference between the $b_l$ derived from the Juipter maps, after dividing by the $2.4^{\prime}$ pixelization window function, and the Hermite-based transfer function. The black curve is the adopted 1$\sigma$ uncertainty used in all analyses. It corresponds to the absolute value of the maximum deviation from zero of the red, blue, and orange curves. The uncertainties on the window function, $w_\ell$, are twice these, but average down when multiple channels are combined. The uncertainty at $\ell=1$ is small because we calibrate on the CMB dipole. The uncertainty in $\Omega_B$ is manifest at high $\ell$.