2. Capítulo II ESTADO DEL ARTE
2.3 MARCO TEORICO
2.3.3 Realidad aumentada
Chiang et al. (2003); Chiang et al. (2006) and Coles et al. (2004) all attribute the non- Gaussianity they detect to residual foreground emission in the cleaned maps. Eriksen et al. (2005) describes simulated LILC data which characterise the foreground residuals one can expect in the real ILCs. This is naturally limited by our knowledge of these foregrounds, and we expect the real ILC map to have more residuals than the simulations, particularly along the plane, but they do indicate at least an approximate level and morphology of the residuals.
We can apply our phase analysis to these simulations to determine both the average behaviour and the likelihood of detecting these residuals on a given sky. As in the case of non-trivial topological models, we can see if they can guide us to the most sensitive test for the predicted morphology. They may also be able to inform our judgement of whether the non-Gaussian signals detected in the data are consistent with what we expect from foreground residuals.
Figure 6.12 shows the average super maps for 1000 simulations where the LILC is either in the Galactic reference frame or rotated such that the Galactic plane residuals are perpendicular to the equator or likewise diagonal. (These choices are motivated by the assumption that the Galactic plane itself would be the dominant structure in any such residuals, which may or may not be correct.) These maps show strong phase correlations for
150 6. Phase analysis
Figure 6.12 “Super” maps showing average significance of Kuiper’s statistic for GRF simu- lations compared to LILC simulations at several orientations. (a) gives those for Gaussian random field (GRF) simulations, while the subsequent plots give those for LILC simula- tions (b) in the Galactic reference frame and in reference frames where the Galactic plane runs (c) vertically (perpendicular to the coordinate equator) and (d) diagonally.
∆`even and ∆m = 0 as expected when the planar residuals are aligned with the coordinate equator. The correlations then mix into other modes depending on the orientation of the residuals. At all three orientations, there is some non-Gaussian signal at all gradients, though this is not detectable for an individual realisation as it remains well under the variance. The strongest correlation on average appears not to be in the Galactic frame but the perpendicular frame (c). Note that the return mapping statistic, not shown, does not appear to detect these residuals at any orientation or gradient.
We have generated such super maps for several different ranges of scales, namely 2 ≤
` ≤ 64, 2 ≤ ` ≤ 16, 16 ≤ ` ≤ 32, and 32 ≤ ` ≤ 64, and find that the phase correlations come from the smaller scales. In particular, the super maps for`≤16 and 16≤` ≤32 do not show a non-Gaussian signal on average. The signal is therefore likely due to residual point sources along the plane. This makes it unclear, then, whether the non-Gaussianities detected by Coles et al. (2004), for example (and which we confirm), at large angular scales, can be attributed to foreground residuals. Obviously, there remain residuals not mapped by the LILC simulations, but based on our current knowledge of the morphology of the Galactic emission, they are unlikely to explain that detected non-Gaussianity.
What is not clear from those averaged super maps is that the variance among reali- sations is quite large, and these signals cannot necessarily be seen for an individual sky. If we look at the pair which gives the strongest signal in the orientation which is likewise strongest, which is ∆m = 2 and ∆` = 2 in the perpendicular orientation, according to Figure 6.12, then we find that even this test only gives a strong (>99%) detection in∼30 out of 1000 simulations. An individual ILC simulation is usually not distinguishable from an individual GRF simulation. Though if we average enough of them, we start to see the non-Gaussian signal, it is simply too weak to expect any individual case to show a clear signal.
6.3 Application 151
Figure 6.13 Average K-statistic map for LILC simulations, arbitrary units.
But with the “super” maps as our guide, we can construct statistical maps as described in §6.2.3 to see if the additional morphological information will help. We use only the ranges −4≤∆m≤4 and 0 ≤∆` ≤4 and the full range of harmonics up to ` ≤64. The results are not as informative for the LILCs as for the compact models. Figure 6.13 shows that there is no clear morphology to the non-Gaussian signal. In this case, we cannot improve on the technique used in Coles et al. (2004) of examining the distribution over all orientations and comparing to the distribution for the null hypothesis, using for example a simple χ2 misfit statistic. But again, the variation among different realisations is great,
and of the 1000 simulations tested, only ∼ 80 of the LILC simulations are anomalous at the >95% level by this measure, and only 35 at the >99% level.
We must conclude from the above that this method is therefore not an effective tool for analysis of foreground residuals. The simulations indicate that the strongest non-Gaussian signal arising from known foregrounds comes from smaller scales, confirming that what Chiang et al. (2003) detect is likely such residuals but that the detection of Coles et al. (2004) at larger angular scales may not be.
6.3.5
WMAP data
Armed with the above information on the non-Gaussian signatures due to compact topolo- gies or foreground residuals, we turn to the WMAP data itself in the form of the LILC map (Eriksen et al., 2004a). As mentioned, incomplete sky coverage introduces correla- tions between modes which will confuse any true non-Gaussian signals. The internal linear combination maps are the best full sky, foreground cleaned maps of the CMB, despite the known foreground residuals that remain. We shall perform analyses like those described above to see if the phase statistics can tell us about the foreground residuals in the data or the possibility that the universe is finite.
First, we can look for the likely foreground residuals using the perpendicular orientation found in § 6.3.4. The left side of Fig. 6.14 shows the “super” maps for different ranges of
`. But as noted, the signal that appears when averaging over an ensemble of simulations is not visible in a single realisation, so it is unsurprising that there is nothing visible in these figures.
152 6. Phase analysis
K-statistic R-statistic
Figure 6.14 “Super” maps from the LILC map for different ranges of` as shown, on the left the R-statistic and on the right the K-statistic (perpendicular orientation).
Nor do we see any indication of the compact topology signature shown in the R-statistic shown at right in Fig. 6.14. The statistical maps are shown in Fig. 6.15, and again, there is little obvious structure though we have attempted to isolate the possible signals following the analysis above of cubic torus models and the LILC simulations. As noted in § 6.3.1, however, a universe with a cell size of only three Hubble radii is only detectable in roughly half of realisations by this method. Therefore, the implied lower limit this method can place on such a compact topology is closer to two Hubble radii. Lastly, for completeness, we look directly at two phase gradients in Fig. 6.16 and see no indication of non-Gaussianity. Further work is needed to characterise the various phase statistic distributions and to refine these tests to detect the known galactic contamination in these maps. Indeed, point sources are visible by eye along the plane, though this analysis has been at larger angular scales. And though these preliminary results do not show much of interest, as described above, there are many different ways to slice the data in order to search for the correlations.