POWER ASYMMETRY IN COSMIC MICROWAVE BACKGROUND FLUCTUATIONS FROM FULL SKY TO SUB-DEGREE SCALES: IS THE UNIVERSE ISOTROPIC?

Previously (Eriksen et al. 2004b; Hansen et al. 2004b), we estimated the power spectrum on hemispheres centered on 164 different positions on the sphere. In that earlier analysis, we applied the Gabor transform approach (Hansen et al. 2002; Hansen & Górski 2003). In order to speed up the analysis without significant loss of precision, we will here apply the much faster MASTER algorithm (Hivon et al. 2002). This allows the analysis of many more positions on the sphere, as well as a large number of simulations for each WMAP year of observation and for each channel.

The power spectrum is estimated from the pseudo power spectrum by (Hivon et al. 2002)

$$\begin{equation} C_\ell =\sum _{\ell ^{\prime }}K^{-1}_{\ell \ell ^{\prime }}(\tilde{C}_{\ell ^{\prime }}-N_{\ell ^{\prime }})\ \ \ \ \tilde{C}_\ell =\sum _{m=-\ell }^{\ell }\frac{\tilde{a}_{\ell m}^X\tilde{a}_{\ell m}^Y}{2\ell +1}, \end{equation}$$

where $\tilde{a}_{\ell m}^X$ is the spherical harmonic transform of channel X with a given mask, $\tilde{C}_\ell$ is the corresponding pseudo power spectrum, and N_ℓ is the noise power spectrum. The coupling kernel $K_{\ell \ell ^{\prime }}$ depends on the mask applied to the data as detailed by Hivon et al. (2002). In previous works, we only used the auto-spectra, i.e., X = Y. Here we will also use the cross power spectra X ≠ Y as an additional check. In the previous papers, we estimated the power spectrum on hemispheres centered on 164 different positions on the sphere. Here we use the positions of the 3072 pixel centers in a HEALPix ⁹ N_side = 16 map.

Using the above approach, we obtain for each multipole bin b an N_side = 16 map, M_i (b), where the value of each pixel i corresponds to the ℓ(ℓ + 1)C_ℓ power on a hemisphere centered on that pixel. In harmonic space, we bin the spectrum in 2 multipoles per bin, such that ℓ_f = 2b + 2 and ℓ_l = 2b + 3 are the first and last multipoles of bin b. We have also performed tests on more localized spectra, i.e., spectra estimated on disks of various sizes:

• 1.  
  hemispheres (diameter 180°), 2 multipoles per bin, ℓ_f = 2b + 2 and ℓ_l = 2b + 3;
• 2.  
  90° diameter disks, 4 multipoles per bin, ℓ_f = 4b + 2 and ℓ_l = 4b + 5;
• 3.  
  45° diameter disks, 16 multipoles per bin, ℓ_f = 16b + 2 and ℓ_l = 16b + 17;
• 4.  
  225 diameter disks, 16 multipoles per bin, ℓ_f = 16b + 2 and ℓ_l = 16b + 17.

In the previous papers, we tested for asymmetry in different multipole ranges using the set of opposite hemispheres which had the largest power spectrum ratio for each range. The problem with this approach is, however, that the maximum asymmetry axis, as well as the significance, is different for different multipole ranges. We made some attempts to determine whether the asymmetry continues to higher multipoles, and if so, what is the highest multipole where the asymmetry is present. This turned out to be difficult because of the instability of significances and directions as a function of the multipole range. Whether we have a significant asymmetry or not depends on which multipole range we choose to look at. Also the axis of asymmetry is slightly different for different multipole ranges.

We will now present an approach which first will solve the problem of choosing which multipole ranges to look at, and second will tell us to which degree a complicated asymmetric model is preferred by the data rather than the isotropic model. The approach is inspired by a similar idea proposed by Land & Magueijo (2007). In order to understand the results from the model selection method, we will present a second, much simpler, asymmetry test, which is based on testing the alignment of the axes of asymmetry between independent multipole ranges.

We can look at the asymmetry in the following way: the asymmetry is the result of a dipole component in the maps, M^b _i , which is common for several multipole bins b. The dipole for a given multipole bin b is given by

$$a_{1m}^b=\sum _iM_i^bY_{1m}^i.$$

We thus propose the following asymmetric model,

$$a_{1m}^b=a_{1m}^b(0)+A(b)a_{1m}(\theta,\phi),$$

where a^b _1m (0) is the random dipole expected in an isotropic model, and a_1m (θ, ϕ) is the common dipole component. The parameters (θ, ϕ) are assumed to be independent of the bin. We will allow variations of the asymmetry amplitude A with bin. To test this hypothesis, we will apply a simple χ² fit,

$$\begin{equation} \chi ^2=\mathbf {d}^\dagger \mathbf {C}^{-1}\mathbf {d}, \end{equation} \tag{ 1 }$$

where the elements of the data vector d are given by d_mb = a^b _1m (obs) − A(b)a_1m (θ, ϕ) for all m = [ − 1, 1] and all bins b. Here a^b _1m (obs) is the dipole of the map M^b _i for the data to be tested. The elements of the correlation matrix are simply $C_{mb,m^{\prime }b^{\prime }}=\langle a_{1m}^b(a_{1m^{\prime }}^{b^{\prime }})^*\rangle$. We will minimize this χ² with respect to the parameters (A(b), θ, ϕ). The resulting asymmetry direction is the best-fit common dipole component for all bins included in the analysis. The value of χ² at the minimum will be compared to χ² for the isotropic hypothesis,

$$\begin{equation} \chi ^2_0=\mathbf {d}^\dagger _0\mathbf {C}^{-1}\mathbf {d}_0, \end{equation} \tag{ 2 }$$

where the elements of d₀ are given only by the observed dipole d_0,mb = a^b _1m (obs). The χ² improvement Δχ² = χ² ₀ − χ² due to the additional parameters will be calibrated with a set of isotropic Gaussian simulations. These simulations will give us the expected improvement Δχ² from the additional parameters, with which the χ² improvement in the data can be compared. If the improvement Δχ² in the data turns out to be significantly better than in the simulations, this would indicate that the asymmetric model is preferred by the data.

We will test three different models for the asymmetry amplitude A(b).

• 1.  
  Constant amplitude. We will test (a) a three-parameter model (A₀, θ, ϕ) with constant amplitude A₀ for a given set of multipole ranges; (b) a four-parameter model where the asymmetry is assumed to have the constant amplitude A₀ starting at ℓ = 2 until ℓ_max, where the latter is the fourth free parameter; (c) a five-parameter model where both the lowest and highest multipoles in the asymmetric multipole range, ℓ_min and ℓ_max, are free parameters.
• 2.  
  Linearly decreasing amplitude. In this four-parameter model, we will assume the asymmetry to be maximal at ℓ = 2 and decrease linearly according to A = A₀(1 − αℓ), where α and A₀ are free parameters.
• 3.  
  Gaussian multipole dependence. We will test a model where the asymmetry peaks at a certain multipole ℓ₀ and falls off to both sides following a Gaussian, $A=A_0e^{-(\ell -\ell _0)^2/(2\sigma ^2)}$. We will test (a) a four-parameter model which is assumed to peak at ℓ₀ = 2 with σ as free parameter and a five-parameter model where both ℓ₀ and σ are free parameters.

Before presenting the results, we will briefly describe the procedure that we use to obtain these results. The simulations are made with a maximum multipole of L_max, so for model 1 above, the maximum and minimum multipoles (the free parameters ℓ_min and ℓ_max) will be sought within the multipoles available from the simulation in the range from L_min = 2 to L_max. Since we will be looking for asymmetry extending over large multipole ranges, we will only look for multipole ranges with a minimum number of Δℓ = ℓ_max − ℓ_min multipoles with a common dipole.

We made two sets of 1400 WMAP simulations (limited by the numbers of CPU hours and processors available) of a given band with a given mask. One set was used to construct the correlation matrix C, while on the other set, as well as on the WMAP data, the χ² minimization procedure was run in order to find the best-fit parameters. In order to obtain a converged covariance matrix with a limited number of simulations, we were forced to increase the size of the power spectrum bins to 20 multipoles (for the hemispheres and 90° disks), for which the covariance matrix is well approximated as a diagonal matrix. For the smaller disks, we keep the original power spectrum bins of 16 multipoles.

The values for ℓ_min, ℓ_max, and θ which minimize χ² were found using a three-dimensional grid, whereas the best-fit values of ϕ and A₀ were found analytically. The analytical expression for ϕ and A₀ is easily found by writing out a_1m as a function of θ and ϕ and setting the derivatives of χ² with respect to ϕ and A₀ equal to zero. For each simulation, the difference Δχ² = χ² ₀ − χ² between the minimum value of χ² and χ² ₀ for the isotropic hypothesis is recorded. This value shows the improvement in χ² for this given simulation, when using an anisotropic model.

We then have an array of χ² improvements for the given anisotropic model from all of the simulations, as well as for the data. We now check the χ² improvement of the data with respect to the simulations. If we quote a significance of 1%, it means that 1% of the simulations had a similar or larger improvement of χ² using the anisotropic model. In the results, we will also list the best-fit multipole range (ℓ_min, ℓ_max) as well as the amplitude A and direction (θ, ϕ) of the common dipole in the data.

In Figure 2, we show the preferred direction for 1400 simulated maps using various galactic cuts. The plot shows the density of best-fit directions as a function of galactic co-latitude. The KQ85 and KQ75 cuts show a uniform distribution of best-fit directions, whereas for the larger cuts there is a preference for the poles. Thus, we would expect that the preferred direction of asymmetry will be shifted away from the galactic plane for large sky cuts. Because of the large cut, power spectra estimated on hemispheres centered close to the poles will be similar in the polar area. This produces a dipolar structure with an axis pointing toward the poles.

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In Section 4 we will see, using the model selection method described above, that a model with a common dipole component in the range from ℓ = 2 to ℓ = 600 is preferred by the data with high significance. There should thus be a strong correlation between the distribution of power in different independent multipole ranges between ℓ = 2 and ℓ = 600. The purpose of the alignment test is to check if the asymmetry is distributed over the full multipole range and thus showing up as an alignment of the dipole of the power distribution of different independent multipole ranges.

The idea for the alignment test is simple. We construct the maps

$$M_i(b1,b2)=\sum _{b=b1}^{b2}M_i(b),$$

where we sum over (A) blocks of 20 multipoles and (B) blocks of 100 multipoles. Thus, for test A, we obtain a set of maps M_i (2, 21), M_i (22, 41), M_i (42, 61), etc., and for test B we obtain M_i (2, 101), M_i (102, 201), M_i (202, 301), etc. (because of different binning for 45° and 225 tests, the ranges in test B will be as given in Table 3). The numbers of multipoles in the blocks, 20 and 100, have been chosen as a compromise between two factors; on the one hand we wanted bin sizes to be large enough to reduce the statistical noise, on the other hand we wanted them to be small enough to have a sufficient number of bins for the alignment check. We also required a set of large bins and a larger set of small bins, and therefore found 20 and 100 to be suitable bin sizes satisfying the two criteria.

For each map M_i (b1, b2), the dipole is extracted (by a simple spherical harmonic transform on the map M_i (b1, b2)), and the direction of the dipole is stored in a vector $\vec{v}_i$ where i is a multipole range (b1, b2). In order to assess whether these directions for different multipole ranges are significantly more aligned in the WMAP data than in isotropic simulations, we define the mean angular distance $\bar{\theta }$ as

$$\bar{\theta }=\sum _{ij}\arccos (\vec{v}_i\cdot \vec{v}_j),$$

where the sum over i and j is over subranges (b1, b2) up to the maximum multipole for the given case. We will in the following quantify the alignment of the power distribution in the WMAP data by specifying the number of simulations with a lower mean angular distance $\bar{\theta }$ between the dipoles of the power distribution.

As a first test of the model selection approach, we used the constant amplitude model with ℓ_min and ℓ_max fixed at the ranges ℓ = 2–19 and ℓ = 2–41, allowing only the amplitude and direction of asymmetry to vary. In this case, we found the best-fit asymmetry axis to be (138°, 220°) and (108°, 227°) being within 10°–15° of the asymmetry axis that Hansen et al. (2004b) found using the method described in Section 3.1. Note that due to the limited multipole range, we use power spectrum bins of two multipoles in this case, ignoring correlations between multipoles. In Figure 3, we show the distribution of χ² improvements χ²(isotropic) − χ²(A₀, ϕ, θ), with the three parameters (A₀, ϕ, θ) obtained with 1400 simulated isotropic maps. The vertical lines in the plots show Δχ² for the data. For ℓ = 2–41, we see that only 0.4% of the simulations have a drop in χ² similar to the drop seen in the data, showing that the anisotropic model is actually preferred by the data. For the multipole range ℓ = 2–19, however, 30% of the simulations show a similar drop in χ², and this asymmetry is therefore not significant alone, taking into account the additional number of parameters required to describe it.

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We now allow first ℓ_max (fixing ℓ_min = 2) and then later also ℓ_min to be free parameters, i.e., we set the amplitude A to a constant value A₀ in a multipole range (ℓ_min, ℓ_max) and to zero for all other multipoles. Again we measured the drop in χ² by the addition of four (fixing ℓ_min = 2 and varying ℓ_max) and five parameters (varying both ℓ_min and ℓ_max) for the simulations and compared to the data.

In Table 1, we show the results for larger scales. We extracted power spectra up to L_max = 300 from WMAP simulations of different channels with different galactic cuts. As we are looking for asymmetric models extending over a large range in multipole space, we restrict our model search to models with at least 10 consecutive multipole bins (200 multipoles) with a common dipole component. The results for the four-parameter fits (the four free parameters are the constant amplitude A₀, the direction (θ, ϕ) and the maximum multipole of asymmetry, ℓ_max) are presented in the table.

The table shows that within the multipole range ℓ = 2–300, there is a dipole component with common amplitude and direction for the multipole range ℓ = 2–221. For the KQ85 cut, only 0.1%–0.3% of the simulations show a similarly strong fit (similarly large χ² improvement) for an asymmetric model. We see that the result is stable with the frequency channel. The significance is dropping with larger galactic cuts, but even with the extended KQ75 cut, only 1.5% of the simulations give a similarly strong fit to an asymmetric model. Even with the largest cut, the direction is remarkably consistent. Note, however, the expected shift away from the galactic plane, as discussed above. The dipole fitting procedure has thus revealed that the hemispherical power asymmetry extends to at least ℓ = 221.

The fact that the range ℓ = 2–221 is found to be the best-fit asymmetric range does not mean that the asymmetry cannot extend beyond ℓ = 221. Our model consists of an isotropic field, which in our analysis is considered noise, and an anisotropic dipole component, which is considered the signal. The isotropic "noise" component to the dipole will randomly change the direction and amplitude of each single power spectrum bin away from the asymmetric direction. By including larger multipole ranges, this noise component is reduced, and asymmetries extending over larger multipole ranges, even beyond ℓ = 300, can give a good fit.

From Table 1, we also see that the best-fit asymmetric multipole range is slightly larger for the larger galactic cuts. For KQ85, ℓ = 2–221 is the best-fit asymmetric range, whereas for KQ75 ℓ = 2–281 gives a better fit, and for larger cuts ℓ = 2–300 is preferred. As discussed above, the fact that a smaller range ℓ = 2–221 gives the best-fit for KQ85 does not mean that the larger range ℓ = 2–281 is a bad fit for this mask. In the table, we have included in the parenthesis the significance for ℓ = 2–281 for KQ85. Clearly ℓ = 2–281 is also significantly asymmetric for KQ85, but because of the random noise component, ℓ = 2–221 gives a slightly better fit.

Note that we have not included the results of the five-parameter fits. The reason for this is that the five-parameter fits in all these cases show identical best-fit model parameters (that is, they show that the best-fit value for the fifth parameter ℓ_min equals ℓ_min = 2 for all cases) to the four-parameter fits.

To investigate properly the maximum multipole for asymmetry, we run a set of simulations with L_max = 500 and L_max = 800. In order to reduce the CPU time for the L_max = 800 case, we now use disks of 90°, 45°, and 225 diameter. In this way, we also obtain more localized spectra. For the 45° and 225 disks, we had to use multipole bins of 16 multipoles in the power spectrum estimation, and therefore also in the dipole fitting, instead of 20 used in the previous analysis. For the 225 disks, the variance of the power spectrum estimate close to the galactic plane was so large that the map M^b _i needed to be normalized by its standard deviation (obtained from simulations) before a dipole fit could be performed. As a result, the values for the amplitude obtained in this case are different from the amplitudes obtained for other disk sizes.

The results are shown in Table 2. The hemisphere results with L_max = 500 show that the asymmetry extends to ℓ = 481 with a similar direction of asymmetry as for lower multipoles, but now with a lower significance (p = 2.3%). For the more localized spectra, however, the asymmetry is found highly significant (p = 0.4% for the most localized spectra) for the range ℓ = 2–600 for all disk sizes, but no evidence is found for an asymmetry extending beyond ℓ = 600.

Table 2. Significances (in %) and Parameters (ℓ_max, θ, ϕ, A₀) of the Best-fit Asymmetric Four-parameter Model

Diameter	Mask	ℓ-range	ℓmax	θ (deg)	ϕ (deg)	p (%)	A0 (μK2/2π)
180°	Kq85	[2, 500]	481	102 ± 12	235 ± 12	2.3	70 ± 20
180°	Kq75	[2, 500]	481	104 ± 16	224 ± 16	14	50 ± 20
90°	Kq85	[2, 800]	601	105 ± 11	225 ± 11	2.6	130 ± 30
45°	Kq85	[2, 800]	591	102 ± 9	223 ± 10	0.6	170 ± 40
225	Kq85	[2, 800]	591	107 ± 11	216 ± 10	0.4	(1.4 a ± 0.4)×10−4

Notes. Approximate Fisher matrix errors for θ, ϕ, and A₀ are also given. Please refer to the text for details about the asymmetric models and their parameters. The significances specify the percentage of simulated maps with a larger drop in χ² for the asymmetric model (considering only models with Δℓ>400) than found in the WMAP data. The results are based on 1400 simulations with the co-added V+W channels. ^aThe power maps obtained from 225 disks are normalized (as described in the text), and the amplitude A₀ is therefore unitless in this case.

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Finally, we will make a consistency check by performing the five-parameter dipole fit in individual subranges of 100 multipoles from ℓ = 2 to ℓ = 800 using 225 disks. Note that the size of these subranges is not exactly 100, since each power spectrum bin has 16 multipoles. We show the results in Table 3. We see clearly that the best-fit direction in each subrange up to ℓ = 600 is consistent with the best-fit direction, (θ = 107, ϕ = 216), for the full range ℓ = 2–591 for this disk size. The two bins above ℓ = 600, however, show a very different dipole direction. We see that the asymmetry can be seen as an alignment of the power distribution dipoles between multipoles from ℓ = 2 up to ℓ = 600. We will now study this alignment in more detail.

Table 3. Significances (in %) and Parameters (ℓ_min, ℓ_max, θ, ϕ, A₀) of the Best-fit Asymmetric Five-parameter Model

Diameter	Mask	ℓ-range	ℓmin	ℓmax	θ (deg)	ϕ (deg)	p (%)	Aa 0 ×10−4
Δℓ>40
225	Kq85	[2, 95]	2	63	110 ± 15	226 ± 16	8.4	0.38 ± 0.14
225	Kq85	[96, 191]	96	191	100 ± 16	200 ± 16	19	0.27 ± 0.10
225	kq85	[192, 303]	208	281	100 ± 35	238 ± 34	99	0.13 ± 0.11
225	kq85	[304, 399]	352	399	83 ± 24	182 ± 24	77	0.23 ± 0.13
225	kq85	[400, 495]	432	479	113 ± 19	224 ± 19	36	0.32 ± 0.14
225	kq85	[496, 591]	496	591	112 ± 20	210 ± 20	48	0.21 ± 0.10
225	kq85	[592, 703]	608	687	45 ± 20	111 ± 24	43	0.26 ± 0.11
225	kq85	[704, 799]	736	799	35 ± 26	47 ± 36	63	0.25 ± 0.13

Notes. Approximate Fisher matrix errors for θ, ϕ, and A₀ are also given. Please refer to the text for details about the asymmetric models and their parameters. The significances specify the percentage of simulated maps with a larger drop in χ² for the asymmetric model than found in the WMAP data. The results are based on 1400 simulations for the co-added V+W channels. ^aThe power maps obtained from 225 disks are normalized (as described in the text), and the amplitude A₀ is therefore unitless in this case.

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Before looking at significances, we will illustrate the direction of the dipoles of individual 100 multipole blocks with some figures. In Figure 4, we show the distribution of power in the WMAP V+W band data using hemispheres (the KQ85 cut was used in the power spectrum estimation). Each map shows the distribution of power, M_i (b1, b2), for a given 100 multipole range. We have already seen by eyes that there is a clear dipolar distribution, and that the direction of the dipole is very similar in each case. In Figure 5, we show the position of the dipole for subranges of 100 multipoles. The color of the disk indicates the multipole range (see Table 3 for the exact ranges used). The results in this plot are taken from power spectrum estimates on disks with diameter 225 using the KQ85 galactic cut. We see that all the individual multipole ranges have dipoles pointing in a direction close to the best-fit dipole for the full range ℓ = 2–600, indicated by the white hexagon.

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The WMAP team pointed out that ℓ = 22 is a strong negative outlier in the full sky power spectrum, and ℓ = 40 is a strong positive outlier. Hansen et al. (2004b) noted that these outliers in the full sky spectrum seem to be associated with the asymmetry; the high outlier at ℓ = 40 was associated with the high power in the hemisphere of maximum asymmetry, and the low outlier at ℓ = 22 was associated with the low power in the opposite hemisphere. In Figure 6, we show the distribution of power in these two bins, as well as for the first bin, ℓ = 2–3, and the bin ℓ = 28–29, which is a particularly asymmetric bin. The conclusions of Hansen et al. (2004b) still hold. One can see the same dipolar distribution of power, which is also seen in the maps of the six subranges within ℓ = 2–600 in Figure 4.

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In Table 4, we show the results for the alignment test using blocks of 20 multipoles. We limit the maximum multipole to 300, because noise is getting important after ℓ = 300, increasing the variance of the directions of blocks with only 20 multipoles. Later, for blocks of 100 multipoles, we will also consider higher multipoles, as the noise is reduced in each block when averaging over 100 instead of 20 multipoles. In the table, we have considered the alignment test for the V band, using different galactic masks. The numbers indicate the percentage of simulations with a lower mean angle $\bar{\theta }$ between dipole directions. For ℓ = 2–300, there is a significant (∼1% level) alignment between dipole directions for the 20 multipole blocks. Note in particular that for the extended KQ75 cut, none of the 1400 simulations are as strongly aligned as the WMAP data. Thus, the asymmetry is strong even with a large galactic cut. We also see that there is a significant (2%–4% level) alignment when considering only the first five 20 multipole blocks on large scales, ℓ = 2–100. Considering only the blocks on small scales, ℓ = 200–300, excluding the first 100 multipoles, there is no significant alignment present, however.

In Table 5, we show the results from the alignment test of 100 multipole blocks, using more localized spectra. The V+W map with the KQ85 galactic cut was used in this analysis. We clearly see the result of the strong alignment, which was already obvious in Table 3. Using more localized spectra, the alignment appears even stronger than for hemisphere spectra. Note that for the range ℓ = 2–600, none of the simulations show a similarly strong alignment for any of the disk sizes. In particular, for the 225 (diameter) disk results, none of the 9800 simulations have a similarly strong alignment (which is close to a 4σ detection of asymmetry). Note also that the alignment is highly significant, also for separate multipole ranges at small and large scales, for instance, ℓ = 2–300 and ℓ = 300–600.

Table 5. Calculated Mean Angle $\bar{\theta }$ Between the Dipole Directions for all Blocks of 100 Multipoles Within the Given Multipole Range

Disk Size	180°	90°	45°	225	90°(KQ85N)
ℓ = 2–800		6.1	8.3	9.2	53
ℓ = 2–700		0.5	0.5	0.4	15
ℓ = 2–600		0	0	0	1.1
ℓ = 2–500	0.1	0	0	0.04	3.9
ℓ = 2–400	0.1	0	0.1	0.3	4.6
ℓ = 2–300	0.5	0.6	0.7	1.8	11
ℓ = 2–200	14	13	9.2	18	41
ℓ = 200–600		0	0	0	2.7
ℓ = 300–600		0.1	0.1	0.1	7.3
ℓ = 400–600		3.6	0.4	0.5	6.6

Notes. The numbers given in the table are the percentages of simulated maps with a lower mean angle $\bar{\theta }$. The results are based on 1400 simulations, except for the results for 225 disks, which is based on 9800 simulations. Zero entries mean that none of the simulated maps had a similarly low mean angle. The combined V+W map was used for obtaining all results in this table. The KQ85 mask was used, when another mask is not specified. Refer to the text for details of how $\bar{\theta}$ is calculated.

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In Figure 7, we show the spectra in the best-fit dipole direction for ℓ = 2–600 for various disk sizes from hemispheres to 45° (diameter) disks. Also in these plots, we see that the difference between the spectra in the opposite directions becomes larger with more localized spectra. In particular, the first part of the spectrum ℓ = 2–100, as well as the amplitudes of the first two peaks, is clearly different. We have investigated whether also the positions, and not only the amplitudes, of the first two peaks may have a similarly asymmetric distribution on the sky. For the localized spectra obtained on 90° (diameter) disks, using the V+W map with the KQ85 cut, we made a fit to the first two peaks. From this we constructed a map, M_i , for each peak with the peak position as a function of (θ, ϕ). No dipole structure similar to what was found for the power spectrum amplitude is seen in these maps.

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In this section, we will perform several tests in order to investigate whether foreground residuals or instrumental systematic effects may cause the observed asymmetry. In particular, we will look at the cross-spectra instead of the auto-spectra, we will look at the WMAP data year by year, and finally we will study in detail whether there are still clear signs of the asymmetry outside the |b|>30 cut.

The cross-spectra based on a_ℓm obtained from different channels and years of observations are less prone to systematic errors, and in particular to uncertainties in the noise model (Hinshaw et al. 2003). We have obtained the hemisphere spectra based on the spectrum obtained as a mean of all 780 possible combinations of the channels Q, V, and W, as well as the five years of observation. Making simulations with all 780 cross-spectra turned out to require too much CPU time, and we were therefore not able to perform a full statistical test using the cross-spectra. Using the WMAP data we found that the direction of the dipole, using the maps M^b _i based on cross-spectra, is consistent with the dipole based on the auto-spectrum.

Similarly, we have studied the direction of the dipole for each single year of observation. We found that these directions are consistent with the directions obtained with the co-added maps with all years included. There is thus no sign of systematic errors in specific years causing the asymmetry. We have also considered the difference between the power distribution maps, M_i (b1, b2), obtained with different channels. Foreground residuals causing the asymmetry would show up in these differences between channels. The dipole directions of the two difference maps Q–V and V–W are not similar to the direction of asymmetry. There is thus no sign of a frequency-dependent foreground with the dipolar power distribution that we have detected in the individual bands. We further ran simulations to which we added foreground templates at the level expected to be found in the WMAP data. Although the dipole amplitudes were stronger in these simulations, no preferred direction was identified. We further checked whether there are more strong outliers in the 225 disk spectra near the galactic plane than close to the poles, but the outliers were found to be distributed isotropically on the sphere.

As discussed above, the asymmetry is no longer significant for the |b|>30 cut, but the fact that the direction of asymmetry is still (within the error bars) consistent with the direction of asymmetry found for smaller galactic cuts is a strong argument against foreground residuals causing the asymmetry. Still, in order to make sure that the drop in significance for the 60° cut is not due to the fact that the large mask is excluding some galactic residual causing the asymmetry, we made some further tests. First we made a mask that was equal to the regular KQ85 cut for the southern galactic hemisphere, but had an extended 30° galactic cut in the northern galactic hemisphere (this is the KQ85N mask). Then we made a similar mask extending only in the southern hemisphere (this is the KQ85S mask).

Using the southern KQ85S mask (for ℓ = 2–300), the significance of the asymmetry is still high (p = 1.2%) for the range ℓ = 2–261 (the direction of asymmetry is consistent with the results above). For the northern KQ85N mask, the significance has dropped to 12%, but the direction of asymmetry is still consistent. Using the more localized spectra estimated on 90° disks, we see from Table 5 that the alignment between 100 multipole blocks is significant at the 1.1% level for ℓ = 2–600, using the KQ85N cut. For ℓ = 2–400, the alignment is still significant at the 2σ level. This result combined with the fact that the direction of asymmetry has changed little with the large |b|>30 cut, as well as the consistency of results using different frequency bands, shows that an explanation of the asymmetry in terms of foreground residuals is difficult to find consistent with the results presented in this paper.

We have so far discussed a model where the common dipole has the same amplitude A₀ over the full multipole range in the best-fit model. We have also tested a model with A decreasing linearly with multipole, as well as a Gaussian shape of the amplitude. Both of these models are described in detail in Section 3.2. In Table 6, we present the results. The first part of the table shows the result with the linear fit, where α is the parameter describing how fast A decreases with multipole. We see that a model with decreasing amplitude is preferred by the data, in fact α = 0 is excluded at the 3σ level. In Figure 8, we show the multipole dependence of the asymmetry. The asymmetry decreases and vanishes close to ℓ = 600, consistent with the results in the previous sections.

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In the lower part of the table, the results with the Gaussian model are shown. We show the results for the four-parameter model, where the peak of the Gaussian is forced to ℓ_peak = 2, with the width σ allowed to vary, as well as for the five-parameter model with ℓ_peak as an additional free parameter. With the exception of the 45° disk results, the five-parameter model finds the same best-fit model with ℓ_peak = 2 as the four-parameter model. In Figure 8, we have plotted the best-fit Gaussian model for 90° disks on top of the linear model. We see that the two models show a consistent decrease in the amplitude of the asymmetry. We conclude that the asymmetry is larger for smaller multipoles and decreasing continuously toward ℓ = 600, where it disappears.