Contamination cannot explain the lack of large-scale power in the cosmic microwave background radiation

Several anomalies appear to be present in the large-angle cosmic microwave background (CMB) anisotropy maps of the Wilkinson Microwave Anisotropy Probe. One of these is a lack of large-scale power. Because the data otherwise match standard models extremely well, it is natural to consider perturbations of the standard model as possible explanations. We show that, as long as the source of the perturbation is statistically independent of the source of the primary CMB anisotropy, no such model can explain this large-scale power deficit. On the contrary, any such perturbation always reduces the probability of obtaining any given low value of large-scale power. We rigorously prove this result when the lack of large-scale power is quantified with a quadratic statistic, such as the quadrupole moment. When a statistic based on the integrated square of the correlation function is used instead, we present strong numerical evidence in support of the result. The result applies to models in which the geometry of spacetime is perturbed (e.g., an ellipsoidal universe) as well as explanations involving local contaminants, undiagnosed foregrounds, or systematic errors. Because the large-scale power deficit is arguably the most significant of the observed anomalies, explanations that worsen this discrepancy should be regarded with great skepticism, even if they help in explaining other anomalies such as multipole alignments.

Figure 1

The two-point correlation function for the WMAP data. The solid curve shows the correlation function for the three-year WMAP internal linear combination data at resolution $N_{\mathrm{side}}=32$, computed with the SpICE software [32]. The dashed curves show the 95% confidence range in a set of simulations. The simulations virtually never produce correlation functions that are as close to zero at large angles as the real data.

Figure 2

Cumulative probability distributions for the quadrupole power $C_2$ in an ellipsoidal universe, in dimensionless $\Delta T/T$ units. In the left panel, data from the entire sky were used, and in the right panel the WMAP Kp0 cut was applied. The solid curve shows a standard LCDM model with no eccentricity. From bottom to top, the other three curves correspond to eccentricities $5\times {10}^{-3}$, $6.2\times {10}^{-3}$, $7.4\times {10}^{-3}$. The horizontal line shows the quadrupole found in the actual data.

Figure 3

Cumulative probability distributions for the statistic $S_{1/2}$. As in Fig. 2, the left panel is for full-sky data, while the right panel is for the Kp0 cut. Curves are as in the previous figure.

Figure 4

Cumulative probability distributions for the statistic $S_{1/2}$ for Bianchi VIIh (rotating universe) models. The values of the shear are 0 (solid lines), $2.4\times {10}^{-10}$ (dotted lines), $5\times {10}^{-10}$ (dashed lines), and $1\times {10}^{-9}$ (dot-dashed lines). As in the previous figures, no-cut probabilities are shown on the left, and Kp0-cut probabilities are shown on the right.

Figure 5

Eigenvalues of the second derivative matrix $\mathbf{H}$. Two independent calculations of the matrix were performed, each based on 80 000 simulations using the Kp0 cut. The matrices were truncated to include only multipoles $l=2$ through 8. The solid curve shows the eigenvalues computed from one matrix, sorted from largest to smallest. The dashed curve shows the quantities $\mathbf{v}·\mathbf{H}·\mathbf{v}$, where $\mathbf{v}$ are the eigenvectors computed from the first matrix and $\mathbf{H}$ is the second matrix. The difference between the two curves gives an indication of the numerical error in the Monte Carlo integration.

Figure 6

The sky pattern corresponding to the most negative eigenvalue of $\mathbf{H}$.

Figure 7

Cumulative probability distributions for models in which a fixed contaminant of the form shown in Fig. 6 is added to the sky. This pattern corresponds to the most negative eigenvalue of the matrix $\mathbf{H}$ and so might be expected to increase the probability of finding a low value of $S_{1/2}$. The curves shown in the figure are for varying amplitudes of the contaminant, with root-mean-square pixel values of 0 (solid lines), $2\mu \mathrm{K}$ (dotted lines), $4\mu \mathrm{K}$ (dashed lines), $8\mu \mathrm{K}$ (dot-dashed lines). No value of the amplitude causes the probability of getting values as low as those in the real data to increase noticeably.