Astronomy & Astrophysics

All Figures

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg01.ps}\end{figure}$ Figure 1: Dependence of the angular size ${\alpha }$ of matched circle radii on the total density parameter $\Omega _{{\rm tot}}$ and the non-relativistic matter density parameter $\Omega _{{\rm m}}$ for the Poincaré dodecahedral hypothesis. Clearly only slight changes in $\Omega _{{\rm tot}}$ are needed for the circle radius to change significantly. The values $\Omega_{{\rm tot}}=1.013, \alpha=35\mbox{$^\circ$ }$ expected by Luminet et al. (2003) are shown by a star. The word "approx'' is used in the label to remind the reader that this is not an exact calculation, since Euclidean geometry is assumed.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg02.ps}\end{figure}$ Figure 2: An inversion of Fig. 1, focussing on the angles found to be of interest in the WMAP ILC map. Independently of $\Omega _{{\rm m}}$ , the slope in the range $5\mbox{$^\circ$ }< \alpha < 15\mbox{$^\circ$ }$ is $\mbox{d} \Omega_{{\rm tot}}/\mbox{d}\alpha \approx 6 \times 10^{-5}~\mbox{deg}^{-1}$ .
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg03.ps}\end{figure}$ Figure 3: Dependence of $\Omega _{{\rm m}}$ on ${\alpha }$ , similarly to Fig. 2. The slope in the range $5\mbox{$^\circ$ }< \alpha < 15\mbox{$^\circ$ }$ is $\mbox{d} \Omega_{{\rm m}}/\mbox{d}\alpha \approx 2.2 \times 10^{-3}~\mbox{deg}^{-1}$ , with only weak dependence on $\Omega _{{\rm tot}}$ over this interval.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg04.ps}\end{figure}$ Figure 4: Values of the correlation statistic S(Eq. (9)) as a function of matched circle angular radius ${\alpha }$ for each pair of six matched circles implied by an orientation of the fundamental dodecahedron, where the rotation of the Clifford translation (screw motion) mapping from a circle to its image is $-36\mbox{$^\circ$ }$ (left-handed). The values of ${\alpha }$ are discrete, but a small random offset in ${\alpha }$ filling the $0.5\mbox{$^\circ$ }$ interval between discrete values is used for to better display individual points. The intervals in $l^{{\rm II}}, b^{{\rm II}}$ and $\theta$ are $2\mbox{$^\circ$ }$ . Points with excess correlation appear at ${\sim }$ $10\mbox{$^\circ$ }$ .
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg05.ps}\end{figure}$ Figure 5: As for Fig. 4, for a right-handed rotation. If a signal is present, it should be present for a left-handed rotation or a right-handed rotation, but not for both.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg06.ps}\end{figure}$ Figure 6: As for Fig. 4, for a zero rotation. This is unphysical, so only noise signals should be present.
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In the text

$\begin{figure} \includegraphics[width=8cm,clip]{0337fg07.ps}\end{figure}$ Figure 7: As for Fig. 4, showing values of the difference statistic $\sigma$ (Eq. (10)) as a function of matched circle angular radius ${\alpha }$ for each pair of six matched circles implied by an orientation of the fundamental dodecahedron, again for a left-handed rotation. A cluster of points with smaller differences $\sigma$ than the general distribution is visible at ${\sim }$ $10\mbox{$^\circ$ }$ .
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg08.ps}\end{figure}$ Figure 8: As for Fig. 7, for a right-handed rotation.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg09.ps}\end{figure}$ Figure 9: As for Fig. 7, for a zero rotation.
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{0337fg10.ps}\end{figure}$ Figure 10: Values of the correlation statistic S calculated for the six individual pairs of circles for dodecahedrons with orientations in the apparently favoured range (Eq. (16)), shown as the mode at each circle radius ${\alpha }$ . This is for a left-handed screw motion. All six pairs have high correlations at around ${\sim }$ $10\mbox{$^\circ$ }$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{0337fg11.ps}\end{figure}$ Figure 11: As for Fig. 10, for a right-handed screw motion. Although correlations of some circle pairs are high, there is no case where all six are high.
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{0337fg12.ps}\end{figure}$ Figure 12: As for Fig. 10, for a zero (unphysical) screw motion. Although correlations of some circle pairs are high, there is no case where all six are high.
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In the text

$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg13.ps}\end{figure}$ Figure 13: Temperature fluctuations in the WMAP ILC data around an $11\mbox{$^\circ$ }$ radius circle centred at position i=1 in Table 1 (continuous curve) and its opposite circle (dot-dashed circle), matched by a screw motion with a left-handed rotation of $36\mbox{$^\circ$ }$ shown against the distance around each circle for $(\Omega _{{\rm m}}=0.3,\Omega _\Lambda =0.7)$ . These distances are a slight underestimate of the true distances for the hypothesis, since Euclidean calculation is used for a "slightly'' positively curved space. The two curves shown are $\delta T/T$ In the general analysis in this paper and in this and following plots, point pairs where either point lies at a galactic latitudes with $\vert b^{{{\rm II}}}\vert < 2\mbox{$^\circ$ }$ and or within $20\mbox{$^\circ$ }$ of the galactic centre are excluded.
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In the text

$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg14.ps}\end{figure}$ Figure 14: As for Fig. 13, for position i=2 in Table 1. This is the only one of the six circle pairs for which one of the Bennett et al. (2003) point sources lies within $0.5\mbox{$^\circ$ }$ . This source is GB6 J1635+3808 at $(l^{{\rm II}}=61.1\mbox{$^\circ$ }, b^{{\rm II}}=42.3\mbox{$^\circ$ })$ and is shown by a pair of overlapping circles at (arbitrarily) $\delta T/T = 0$ . Its sky position is within $0.1\mbox{$^\circ$ }$ of the circle shown by the solid curve, suggesting that it is a contaminant present in the solid curve (Sect. 4.5).
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In the text

$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg15.ps}\end{figure}$ Figure 15: As for Fig. 13, for position i=2 in Table 1.
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In the text

$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg16.ps}\end{figure}$ Figure 16: As for Fig. 13, for position i=2 in Table 1. The exclusion due to the Galaxy mentioned in Fig. 13 excludes many points in this plot.
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In the text

$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg17.ps}\end{figure}$ Figure 17: As for Fig. 13, for position i=2 in Table 1.
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In the text

$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg18.ps}\end{figure}$ Figure 18: As for Fig. 13, for position i=2 in Table 1.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg02.ps}\end{figure}$	Figure 2: An inversion of Fig. 1, focussing on the angles found to be of interest in the WMAP ILC map. Independently of $\Omega _{{\rm m}}$ , the slope in the range $5\mbox{$^\circ$ }< \alpha < 15\mbox{$^\circ$ }$ is $\mbox{d} \Omega_{{\rm tot}}/\mbox{d}\alpha \approx 6 \times 10^{-5}~\mbox{deg}^{-1}$ .
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$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg03.ps}\end{figure}$	Figure 3: Dependence of $\Omega _{{\rm m}}$ on ${\alpha }$ , similarly to Fig. 2. The slope in the range $5\mbox{$^\circ$ }< \alpha < 15\mbox{$^\circ$ }$ is $\mbox{d} \Omega_{{\rm m}}/\mbox{d}\alpha \approx 2.2 \times 10^{-3}~\mbox{deg}^{-1}$ , with only weak dependence on $\Omega _{{\rm tot}}$ over this interval.
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$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg05.ps}\end{figure}$	Figure 5: As for Fig. 4, for a right-handed rotation. If a signal is present, it should be present for a left-handed rotation or a right-handed rotation, but not for both.
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$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg06.ps}\end{figure}$	Figure 6: As for Fig. 4, for a zero rotation. This is unphysical, so only noise signals should be present.
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$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg08.ps}\end{figure}$	Figure 8: As for Fig. 7, for a right-handed rotation.
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$\begin{figure} \par\includegraphics[width=8cm,clip]{0337fg09.ps}\end{figure}$	Figure 9: As for Fig. 7, for a zero rotation.
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$\begin{figure} \par\includegraphics[width=7.7cm,clip]{0337fg10.ps}\end{figure}$	Figure 10: Values of the correlation statistic S calculated for the six individual pairs of circles for dodecahedrons with orientations in the apparently favoured range (Eq. (16)), shown as the mode at each circle radius ${\alpha }$ . This is for a left-handed screw motion. All six pairs have high correlations at around ${\sim }$ $10\mbox{$^\circ$ }$ .
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$\begin{figure} \par\includegraphics[width=7.7cm,clip]{0337fg11.ps}\end{figure}$	Figure 11: As for Fig. 10, for a right-handed screw motion. Although correlations of some circle pairs are high, there is no case where all six are high.
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$\begin{figure} \par\includegraphics[width=7.7cm,clip]{0337fg12.ps}\end{figure}$	Figure 12: As for Fig. 10, for a zero (unphysical) screw motion. Although correlations of some circle pairs are high, there is no case where all six are high.
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$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg15.ps}\end{figure}$	Figure 15: As for Fig. 13, for position i=2 in Table 1.
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$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg16.ps}\end{figure}$	Figure 16: As for Fig. 13, for position i=2 in Table 1. The exclusion due to the Galaxy mentioned in Fig. 13 excludes many points in this plot.
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$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg17.ps}\end{figure}$	Figure 17: As for Fig. 13, for position i=2 in Table 1.
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$\begin{figure} \par\includegraphics[angle=360,width=12.6cm,clip]{0337fg18.ps}\end{figure}$	Figure 18: As for Fig. 13, for position i=2 in Table 1.
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