ISOTROPY IN THE TWO-POINT ANGULAR CORRELATION FUNCTION OF THE COSMIC MICROWAVE BACKGROUND

With the launch of the Wilkinson Microwave Anisotropy Probe (WMAP) in 2003, the CMB has been measured in highly detailed full-sky maps (Bennett et al. 2003; Spergel et al. 2007; Hinshaw et al. 2009; Jarosik et al. 2011), which have been examined in great detail over the past few years (Cruz et al. 2005; de Oliveira-Costa et al. 2004; Land & Magueijo 2005; Hoftuft et al. 2009; Hansen et al. 2009; Kim & Naselsky 2010; Bennett et al. 2011). In particular, the angular two-point correlation function C(θ) has been studied; it is defined as the average product between the temperature of two points an angle θ apart

$$\begin{equation} C(\theta)=\overline{T(\hat{\Omega }_{1}),T(\hat{\Omega }_{2})} |_{\hat{\Omega }_{1} \cdot \hat{\Omega }_{2} = \cos (\theta)}. \end{equation} \tag{ 1 }$$

Here, $T(\hat{\Omega })$ is the fluctuation around the mean of the temperature in direction $\hat{\Omega }$ on the sky. Several anomalies have been claimed in the angular correlation function, especially the missing power on large angular scales (for a review, see Copi et al. 2010). Specifically, the angular correlation function is very nearly zero at scales above 60°; such a low correlation has a significance of ⩾3.2σ in the standard Gaussian random, statistically isotropic cosmology. This result, first observed in COBE data, has been strengthened with WMAP first-year data (Hinshaw et al. 1996), as well as later WMAP observations (Copi et al. 2007, 2009; Sarkar et al. 2011), though questions have been raised regarding its significance (Pontzen & Peiris 2010; Efstathiou et al. 2010; Ma et al. 2011)

In this brief report, we explore the directional contributions to the angular two-point correlation function. Directional information is lost in the original definition of C(θ), as Equation (1) assumes statistical isotropy and averages over all directions on the sky. Our goal is to provide a generalization of Equation (1), and ascertain if there are specific directions that result in the unusually low correlation.

The angular two-point correlation function is defined by Equation (1), where the average is taken over all pairs of points separated by an angle θ on the sky.

We now propose a new statistical quantity, a fixed-vertex correlation function:

$$\begin{equation} C(\hat{\Omega },\theta)=\overline{T(\hat{\Omega }), T(\hat{\Omega }_{2})} |_{\hat{\Omega } \cdot \hat{\Omega }_{2} = \cos (\theta)}, \end{equation} \tag{ 2 }$$

where $\hat{\Omega }$ is a direction in the sky, and the average is taken over a ring of pixels separated by θ from the direction $\hat{\Omega }$. In particular, notice that $C(\theta)=({1}/{N})\sum _{i=1}^N C(\hat{\Omega }_i, \theta)$.¹ Our new statistic preserves directional information that is normally lost when averaging over the entire sky²; as a result, it may be used to test isotropy. Maps of $C(\hat{\Omega },\theta)$ for θ = 60° and θ = 120° are shown in Figure 1.

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In order to quantify the low power in the angular two-point function between approximately 60° and 180°, the statistic (Spergel et al. 2003)

$$\begin{equation} S_{1/2} = \int _{-1}^{1/2}C(\theta)^2 d(\cos \theta) \end{equation} \tag{ 4 }$$

has been widely used. In order to examine possible anisotropies in C(θ), we attempt to separate S_1/2 into contributions from specific directions $\hat{\Omega }$; we define

$$\begin{eqnarray} S_{1/2}(\hat{\Omega }) &\equiv & \frac{1}{N} \int _{-1}^{1/2}C(\hat{\Omega },\theta)C(\theta)d(\cos \theta), \end{eqnarray} \tag{ 5 }$$

so that the sum of $S_{1/2}(\hat{\Omega }, \theta)$ over all pixels is S_1/2.³

Note that the quantity $S_{1/2}(\hat{\Omega })$ is non-local, in that it receives contributions from not only the direction $\hat{\Omega }$, but from the whole sky, due to the term C(θ) in Equation (5). So, for example, excluding a pixel $\hat{\Omega }$ will not change the value of S_1/2 simply by $-S_{1/2}(\hat{\Omega })$, because there is also an indirect effect of the changed global C(θ) as well as the changed monopole that is being subtracted. These two are approximately constant if only a small area of the sky is affected, so analysis of $S_{1/2}(\hat{\Omega })$ is suitable for examination of the effect of small regions of the sky on S_1/2. However, these indirect effects must be taken into consideration while examining much larger regions.

3.1. Random Map Analysis

We first examine plots of S_1/2 in random maps to obtain an idea of what to expect. We generate 80,000 synthetic Gaussian random statistically isotropic maps, based on the underlying best-fit ΛCDM cosmological model (Larson et al. 2011).

A strong relationship between contributions to power in S_1/2 and the regions of the quadrupole and octupole (QO regions for simplicity) is consistently seen in random maps. An example may be seen in the right panels of Figures 3 and 4 with the full-sky Internal Linear Combination (ILC) map.

Motivated by this result, we seek to quantify these contributions by repeating our analysis of $S_{1/2}(\hat{\Omega })$ in only the regions of the sky containing the quadrupole and octupole. We compute $S_{1/2}(\hat{\Omega })$ in a "sky" containing only the quadrupole and octupole of each map, and define the QO region to correspond to pixels $\hat{\Omega }_{{\rm QO}}$ where $|S_{1/2_{{\rm QO}}}(\hat{\Omega }_{{\rm QO}})| \ge x$; x is selected so that $\sum S_{1/2}(\hat{\Omega }_{{\rm QO}})/\sum S_{1/2}(\hat{\Omega }) \approx 0.9$. We define the QO region of each individual synthetic map separately. The contribution of S_1/2 from the QO region is thus defined as

$$\begin{equation} S_{1/2}^{\rm QO}\equiv \sum _i S_{1/2}\big(\hat{\Omega }_i^{\rm QO}\big). \end{equation} \tag{ 7 }$$

There is an extremely strong, almost linear relationship between S^QO_1/2 and total S_1/2, plotted in Figure 2. In contrast, the total S_1/2 and the contributions from non-QO regions of the sky are essentially uncorrelated, except at very high (⩾10⁵ (μK)⁴) values of S_1/2, with which we are not concerned. Note that by definition, $S_{1/2} \equiv S_{1/2}^{{\rm QO}} + S_{1/2}^{\rm non{\rm -}QO}$.

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This result makes intuitive sense, as the statistic $S_{1/2}(\hat{\Omega })$ is based off $C(\hat{\Omega }_i,\theta)$, which is higher when the central point $\hat{\Omega }$ is in a high-temperature region such as the QO region. More qualitatively, $S_{1/2}(\hat{\Omega })$ is a measure of correlation on the largest scales in the sky—which are, of course, the quadrupole and octupole.

3.2. WMAP Map Analysis

We now examine WMAP's seven-year maps in our analysis (Jarosik et al. 2011). We use the full sky and cut-sky ILC maps.⁴ The maps are first degraded to N_side = 64 (corresponding to pixel scale ≃ 1°); next, the KQ75 mask is applied, and finally all resulting pixels that are more than 10% masked (i.e., with mask value of less than 0.9) are excluded from the analysis. Finally, the dipole was removed.

From inspection of the full-sky ILC $S_{1/2}(\hat{\Omega })$ (Figure 4, right panel), there is an obvious correlation between the magnitude of $S_{1/2}(\hat{\Omega })$ contributions and the regions where the quadrupole and octupole have power, as expected from our analysis of random maps. Quantifying this result using our previous definition of the QO region, we find that S^QO_1/2 = 8837 (μK)⁴, while $S_{1/2} ^{\rm non{\rm -}QO}$ = −349 (μK)⁴. ⁵ The values observed in the full-sky map are very typical, compared to our random map results.

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Moving on to the cut-sky map, there is much less power on large angular scales, as expected; only 0.06% of random maps have a similarly low value of S_1/2. And as the empirical relationship predicts, S^QO_1/2 is similarly low, at the 0.046% level. In comparison, the non-QO contribution, $S_{1/2}^{\rm non{\rm -}QO}$ is quite typical (15.6% confidence level.)⁶ Thus, the strong correlation that we observed in random maps continues to hold for the anomalous cut-sky map, as shown in Figure 2. What is unlikely about our sky is simply for S^QO_1/2 to exhibit such a low value in the first place.

It is known that the quadrupole in the CMB sky is lower than expected from ΛCDM results, though not anomalously so (Komatsu et al. 2011). It makes sense to consider what happens to our results above if we use the ΛCDM theoretical power spectrum for Monte Carlo simulations as before, but substituting the low WMAP quadrupole (C₂ = 210 (μK)²), in order to test if our result is still valid. As expected, the significance of S_1/2 relative to these new synthetic maps is now lower (0.70% compared to 0.06% before), and a similar effect is seen in our new statistics: S^QO_1/2 has a significance of 0.58% (previously 0.046%), and $S_{1/2}^{\rm non{\rm -}QO}$ is again typical—27.4% C.L., compared to 15.6% before. Therefore, comparison to Gaussian random skies conditioned to have low C₂ gives results that are somewhat less significant than before, which is expected since we already knew that part of the low-S_1/2 problem originated from low quadrupole. However, the strong correlation observed in random maps continues to hold. In addition, the various statistics are still low with ⩽1% probability, even compared to skies with the low quadrupole, and the lowness is once again due to lack of power from the quadrupole and octupole regions.

In this Letter, we first introduced the fixed-vertex correlation function $C(\theta, \hat{\Omega })$, which has the potential to be a useful tool for studying the statistical isotropy of the correlation function. We then observe that the value of S_1/2 in random skies is correlated with an expected anisotropy in the quadrupole and octupole regions of the sky. Our sky, known to have a low S_1/2, also receives low contributions from its quadrupole and octupole regions. We have a bit of a chicken and egg problem here, as we cannot tell if the low S^QO_1/2 caused a low S_1/2, or vice versa, or if both were caused by an underlying factor. It is worth considering that this result and the one of Pontzen & Peiris (2010) may stem from the same root cause.

It is also interesting that random Monte Carlo skies (which are generated from the assumption of statistical isotropy) and the reconstructed full-sky ILC (which assumes statistical isotropy in the reconstruction process) have similar values of S_1/2 and S^QO_1/2, while the cut-sky temperature map is anomalous. These findings may provide a hint for a future understanding of the origin of large-angle missing power in the CMB.

We thank Dragan Huterer for initial guidance in the project and many useful suggestions. We thank Glenn Starkman for introducing the idea of $C(\hat{\Omega },\theta)$. We thank Hiranya Peiris for a useful communication. We thank an anonymous reviewer for useful suggestions and comments. We acknowledge use of the HEALPix package (Górski et al. 2005) and the Legacy Archive for Microwave Background Data Analysis (LAMBDA). This work was partly supported by NSF under contract AST-0807564, NASA under contract NNX09AC89G, and the University of Michigan.