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Appendix A: WMAP cosmic microwave background maps considered

As we are interested in identifying trends in the data, we consider a suite of 11 different renditions of WMAP data: the Tegmark et al. (2003) reduced-foreground CMB Map (TOH1), the internal linear combination (ILC) WMAP maps from the 3rd year (ILC W3, Hinshaw et al. 2007), 5th year (ILC W5, Gold et al. 2009), 7th year (ILC W7, Gold et al. 2011), and the 9th year (ILC W9, Hinshaw et al. 2012), as well as sparsely inpainted versions of these maps. We also consider the sparsely inpainted WMAP ILC 5th year data reconstructed by Delabrouille et al. (Dela W5, 2009) using wavelets. These are summarised in Table 1 of Rassat et al. (2013).

Appendix B: Statistical anomalies and impact of sparse inpainting

Appendix B.1: Preferred axis for low multipoles: Axis of Evil

It was first noted by de Oliveira-Costa et al. (2004) that both the quadrupole and octopole of the CMB appeared planar (i.e. anomalously dominated by m =  ± ℓ modes) and were also aligned along a similar axis.

Land & Magueijo (2005a) suggest searching for a more general axis by considering the power in any mode m, instead of focussing on planar modes. This can be quantified by considering their statistic: (B.1)The expressions are given by and for m > 0 and (2ℓ + 1)Ĉ_ℓ =  ∑ _m | a_ℓm | ², where corresponds to the value of the a_ℓm coefficients when the map is rotated to have as the z-axis. The above statistic finds both a preferred axis and a preferred mode m.

Land & Magueijo (2005a) find that the preferred axes for ℓ = 2,...,5 for WMAP 1 data seemed aligned along a similar axis in the direction of (ℓ,b) ~ ( − 100°,60°), where the l varied from ≃[−90°, −160°] and b varied from ≃[48°,62°]. By considering the mean interangle θ (i.e. the mean value of all possible angles between two axes and for ℓ,ℓ′ = 2,...,5 and ℓ ≠ ℓ′), they find that only 0.1% of simulations had a lower average value than the one measured in WMAP1 data (~20°) and therefore rejected statistical isotropy at the 99.9% confidence level. The preferred axis has been dubbed the “AoE”.

Appendix B.2: Mirror parity

Another test is mirror parity in CMB data, i.e. parity with respect to reflections through a plane: , where is the normal vector to the plane. Since mirror parity is yet another statistic for which preferred axes can be found (i.e. the normal to the plane of reflection), it is complementary to the search for a preferred axis described in Sect. B.1.

In practice, mirror parity and preferred axes found using Eq. (B.1) are statistically independent (Land & Magueijo 2005c), so the coincidental presence of both increases the significance of the preferred axis anomaly. With all-sky data, one can estimate the S-map for a given multipole by(B.2)Positive (negative) values of correspond to even (odd) mirror parities in the direction. The same statistic can also be considered summed over all the low multipoles one wishes to consider (e.g. the multipoles that have similar preferred axes) as in Ben-David et al. (2012): The parity estimator is redefined as , so that ⟨S⟩ = 0. The most even and odd mirror-parity directions for a given map can be considered by estimating (Ben-David et al. 2012): (B.3)where μ(S) and σ(S) are the mean and standard deviation.

Others have also studied point parity with different statistics, e.g. Land & Magueijo (2005c), who did not find significant point parity in the first WMAP data release. Kim & Naselsky (2010a,b, 2011), however, find evidence of odd point parity in later WMAP renditions, and link this anomaly with the low level of correlations on the largest scales.

Appendix C: Validation of sparse inpainting to study large-scale anomalies

One can test for preferred axes directly on different renditions of WMAP data, for example, on ILC maps. However, these may be contaminated on large scales due to Galactic foregrounds (Hinshaw et al. 2012). Another approach is to use a different basis set than spherical harmonics, i.e. use a basis that is orthonormal on a cut sky (e.g. Rossmanith et al. 2012). Alternatively, one can use sparse inpainting techniques to reconstruct full-sky maps (Plaszczynski et al. 2012; Dupé et al. 2011; Rassat et al. 2013) or other inpainting methods, such as diffuse inpainting (Zacchei et al. 2011) or inpainting using constrained Gaussian realisations (Bucher & Louis 2012; Kim et al. 2012). Any inpainting technique should be tested for potential biases, specifically for the masks and statistical tests one is interested in. Here we use the sparse inpainting techniques first described in Abrial et al. (2008) and Starck et al. (2010) and refined in Starck et al. (2013b) to reconstruct regions of missing data. The advantage of this method is that is does not assume the “true” map is either Gaussian or isotropic, yet it allows it to be (see for e.g. Starck et al. 2013a).

Rassat et al. (2013) show sparse inpainting is a bias-free reconstruction method for the low quadrupole, quadrupole/octopole alignment, and octopole planarity tests. Here, we test whether sparse inpainting is a bias-free method for both the AoE and mirror parity. We calculate 1000 Gaussian random field realisations of WMAP7 best-fit cosmology using the WMAP7 temperature analysis mask (f_sky = 0.78).

Appendix C.1: Recovering the mean interangle (θ) with a realistic  Galactic mask

We compare the mean interangle θ for the statistic given in Eq. (B.1) for ℓ = 2−5, for each map before and after inpainting, using nside = 128 for the CMB maps. We find θ ~ 57° ± 9° in our simulations before and after inpainting, i.e. what is expected in the case of isotropic axes and a Gaussian random field (Land & Magueijo 2005a). After inpainting with the WMAP7 mask, we find that (θ_true − θ_inp) ~  − 0.55° ± 10.7°, showing there is no significant bias in the estimation of the mean interangle after sparse inpainting is applied. While the bias is small, the error bar on the mean interangle after sparse inpainting is not negligible (10°). Following Starck et al. (2013b), we test the statistic on an optimistic Planck-like mask with f_sky = 0.93 and find (θ_true − θ_inp) ~ 0.17° ± 7.3°, showing we can expect better reconstructions with Planck data and smaller masks.

Appendix C.2: Recovering mirror parity statistics (S ±) with a realistic  Galactic mask

To test for possible biases in the S_± statistics after sparse inpainting, we calculate S₊ and S₋ for each CMB simulation before and after inpainting, setting nside = 8 for the CMB maps, and nside = 64 for the parity maps (calculated using Eq. (B.2)), as in Ben-David et al. (2012). As in Fig. 6 of Ben-David et al. (2012), we find that the distributions of S₊ and S₋ populations do not change before and after inpainting (“True” and “After Inp.” in Table C.1). We do not find any significant bias in the S_± measurements (“Bias” column in Table C.1).

Following Starck et al. (2013b), we also test the statistic on an optimistic Planck-like mask with f_sky = 0.93 and find

ΔS₊ =  −0.0021 ± 0.081 and ΔS₋ = 0.00091 ± 0.10, showing we can expect significantly better reconstructions with future Planck data and smaller masks.

Appendix D: Recovering the primordial CMB

Since statistical isotropy is predicted for the early Universe, analyses should focus on the primordial CMB, i.e. one from which secondary low-redshift cosmological signals have been removed. The observed temperature anisotropies in the CMB, δ_OBS, can be described as the sum of several components: (D.1)where δ_prim are the primordial temperature anisotropies, and the total secondary temperature anisotropies due to the late-time Integrated Sachs Wolfe (ISW) effect (see Sect. 2.1 of Rassat et al. 2013).

In practice, the temperature ISW field can be reconstructed in spherical harmonics, , from the LSS field g_ℓm (Boughn et al. 1998; Cabré et al. 2007; Giannantonio et al. 2008): (D.2)where g_ℓm represent the spherical harmonic coefficients of a galaxy overdensity field g(θ,φ).

The spectra C_gg and C_gT are the galaxy (g) and CMB (T) auto- and cross-correlations measured from the data or their theoretical values (see Sect. 2.2 of Rassat et al. 2013).

© ESO, 2013