Figure 1

An illustration of the low value of ${\mathcal{S}}_{1/2}^{\mathrm{cut}}$ and its origin. The upper left-hand panels show the pattern of ILC7 temperature fluctuations in $\ell =3$, 5 and 7 modes from top downwards; the larger panel underneath shows the sum, in which anticorrelations between the modes cause power near the poles to be small. The shaded regions bounded by solid and dashed lines represent a 20% azimuthal mask in the Galactic and $\ell =3$ angular momentum planes, respectively, (defined in text). The large uppermost panel on the right-hand side shows the values of ${\mathcal{S}}_{1/2}^{\mathrm{cut}}$ for a variable width Galactic azimuthal mask (solid line). The dashed line shows the same result using QML, rather than PCL, reconstruction techniques; the dotted line shows the QML result when the mask is applied in the $\ell =3$ angular momentum plane, demonstrating that full-sky power can be efficiently hidden even from the QML estimator. The two right-hand panels underneath the main plot show the corresponding estimated values ${\widehat{C}}_2$ and ${\widehat{C}}_3$. Plotted points show the results from using the WMAP team’s KQ85y7 and KQ75y7 masks for PCL (circles) and QML (crosses) estimators, respectively, exhibiting the QML estimator’s relative sensitivity to mask shape.Reuse & Permissions

Figure 2

The difference between the cut-sky estimates ${\widehat{C}}_{\ell }$ and the full-sky measured values ${\mathcal{C}}_{\ell }$ on the ILC7 map, expressed for ease of viewing as a fraction of the fiducial theoretical best-fit $C_{\ell }$’s published by the WMAP team [22]. The results from a PCL and QML estimator for a 20% azimuthal sky-cut are shown by dots and crosses, respectively. The shaded bands show the expected deviation of the cut-sky from the full-sky values. The larger, outer band represents the PCL deviation while the smaller, inner band (with dotted edges) represents the QML deviation. The low value of ${\mathcal{S}}_{1/2}^{\mathrm{cut}}$ arises because the PCL power spectrum reconstruction for $l\le 8$ typically falls below, never significantly above, the full-sky value.Reuse & Permissions

Figure 3

For various anisotropic theories we plot the biases (left panels) and change in the diagonal part of the variance (right panels) compared to the isotropic case. In this context nonzero “bias” can be desirable, for instance in the case illustrated in the uppermost panels, where it merely reflects that the full-sky power spectrum is contaminated by the presence of the galaxy. The solid lines show statistics for PCL reconstructions of a $f_{\mathrm{sky}}=20\%$ azimuthally masked sky; the dashed lines show the corresponding QML reconstructions. As expected, the QML reconstructions are generally more accurate (despite the anisotropy of the underlying theory); for more details see text. We scale the biases by the isotropic theory power spectrum and the change in the variances by the isotropic cosmic variance. The first three theories are simple existing models while the latter two are tests which are designed specially to reproduce the ${\mathcal{S}}_{1/2}^{\mathrm{cut}}$ anomaly. Note the different scales on each panel.Reuse & Permissions

Figure 4

The likelihoods for ${\mathcal{S}}_{1/2}^{\mathrm{cut}}$ compared between three theories: the isotropic concordance theory (solid line), the “designer” theory (dashed line) and “picture” (dash-dotted line) theory. The latter two are specifically designed to reproduce low ${\mathcal{S}}_{1/2}^{\mathrm{cut}}$. The improvement in the log likelihood over the concordance cases are, respectively, 4.2 and 3.7, which are very small improvements given the fine-tuning involved. Since these theories should produce the greatest possible gains in likelihood, the value of the observed ${\mathcal{S}}_{1/2}^{\mathrm{cut}}$ statistic is not a strong objection to the concordance theory.Reuse & Permissions

Figure 5

The weighting which different estimators give to full-sky modes of differing $m$ in estimating the $C_{\ell }$’s (here illustrated for $\ell =3$ with a 20% azimuthal mask). The full-sky estimator (thin solid line) by definition weights each $m$ equally; see Eq. (a7). Masking the sky then using the QML estimator (dashed line) comes close to reproducing this weighting, despite the loss of information associated with the first operation. The action of masking the sky then estimating $\ell =3$ power using the PCL technique (thick solid line) favors $|m|=2$ and $m=0$ while down-weighting $|m|=3$ modes, thus giving a less reliable estimate of the full-sky power. Similar trends are seen at other $\ell$.Reuse & Permissions

Figure 6

An example of how the QML estimator typically remains superior to the PCL estimator is given by comparing, for 200 000 random anisotropic theories, the bias on the cut-sky power spectrum estimates (here illustrated for $\ell =5$ with a 20% azimuthal mask). The width of the curves show that, for a given theory, the QML estimator (dashed curve) is typically significantly less biased than the PCL estimator (solid curve). Similar results hold at other $\ell$’s and for the extra variance induced by the anisotropy.Reuse & Permissions