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Paper text:
soon after the release of the first - year wmap data  @xcite , tegmark _ et al . _
@xcite noticed that the cmb quadrupole and octupole aligned with each other unusually well , at roughly the @xmath0 level .
multipole vectors  discovered by maxwell  @xcite in the @xmath1 century , widely forgotten , then reintroduced by copi _
et al . _
@xcite  provide a useful tool for analyzing the alignment in greater detail . while exact confidence levels vary depending on what one measures , all researchers agree that the quadrupole - octupole alignment is unusual at roughly the @xmath2 level or better  @xcite .
the combination of the 1-in-100 alignment with the 1-in-600 overall weakness of the low-@xmath3 modes motivates one to seek a physical explanation .
non - trivial spatial topology can explain the weakness of the low-@xmath3 modes .
might it also explain the quadrupole - octupole alignment ? the present paper simulates the cmb in a poincar dodecahedral space  @xcite and checks the quadrupole - octupole alignment .
absolutely no correlation is found .
we use the late jesper gundermann s simulation  @xcite of the cmb in the poincar dodecahedral space , with modes through @xmath4 . this simplified simulation , while neglecting the doppler contribution and the sound speed , nevertheless produces a low-@xmath3 power spectrum essentially identical to the spectra produced by more refined simulations .
thus we may be quite confident that if the dodecahedral topology imposed a nontrivial quadrupole - octupole alignment , this simulation would capture it . as we will see in section  [ sectionmeasuring ] , however , absolutely no such correlation is found .
even if one were to add a doppler term and sound speed to the simulation , the distribution in figure  [ figurequadocthistogram ] would change by at most a tiny amount , not nearly enough to introduce a nontrivial quadrupole - octupole correlation .
for each simulated cmb sky , we use the polynomial method  @xcite to compute the two quadrupole vectors @xmath5 and the three octupole vectors @xmath6 . following  @xcite
, we take the cross product @xmath7 , which we normalize to obtain a unit vector @xmath8 orthogonal to the plane of the quadrupole .
similarly , we take the cross product of each of the three possible pairs of octupole vectors @xmath9 which we normalize to obtain unit vectors @xmath10 orthogonal to each of the three octupole planes .
the three dot products @xmath11 then measure the extent to which the quadrupole plane does or does not align with each of the three octupole planes
.    in a simply connected universe one expects no correlation between the quadrupole vector @xmath12 and each octupole vector @xmath13 .
as @xmath12 and @xmath13 ( for some fixed @xmath14 ) wander randomly over the 2-sphere , their dot product follows a flat distribution on the interval @xmath15 $ ] ( this is a consequence of the wonderful fact that radial projection of a sphere onto a circumscribed cylinder via @xmath16 preserves area ) .
hence each @xmath17 , being the absolute value of the dot product , follows a flat distribution on @xmath18 $ ] .    in the real universe
the quadrupole aligns surprisingly well with the octupole , giving dot products @xmath19 for the dq - corrected tegmark ( dqt ) cleaning @xcite of the first - year wmap data or @xmath20 for the lagrange internal linear combination ( lilc ) cleaning @xcite of the same data .
the question of whether a multiconnected spatial topology might explain the observed quadrupole - octupole alignment may be rephrased more precisely as : does a given topology predict a flat distribution for each @xmath21 or does it predict a distribution skewed towards the high end ? for the poincar dodecahedral space , our simulations ( recall section  [ sectionsimulating ] ) yield a flat distribution ( figure  [ figurequadocthistogram ] ) , implying that the dodecahedral topology does nothing to explain the quadrupole - octupole alignment .    to be fully rigorous
we should point out that even though the individual dot products @xmath21 follow the same flat distribution in the dodecahedral topology that they do in the simply connected model , it s nevertheless conceivable that their sum @xmath22 might follow a slightly different distribution in the two cases , depending on the internal correlations among the three @xmath21 in the dodecahedral case . in practice , however , our simulations find the observed sum to be unusual at roughly the @xmath2 level regardless of whether we compare to the dodecahedral topology or a simply connected space .
the poincar dodecahedral space topology , while explaining the weakness of the low-@xmath3 modes , completely fails to explain the quadrupole - octupole alignment .
while this negative result leaves one feeling less optimistic , good scientific practice demands that one analyze a few other plausible topologies before reaching any firm conclusion about whether topology might play a role .
one must also keep an open mind about what observations may or may not be due to random chance alone .
the quadrupole - octupole alignment might be due to chance , while the weakness of the low-@xmath3 modes has a physical explanation . or perhaps exactly the reverse is true . at this point
the mystery remains open .
i dedicate this article to my friend and collaborator jesper gundermann , whose untimely death on 10 june 2006 saddened all who knew him .
his energetic enthusiasm and deep love of science brought joy to those of us lucky enough to work with him .
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the cmb quadrupole and octupole , as well as being weaker than expected , align suspiciously well with each other . 
 non - trivial spatial topology can explain the weakness . 
 might it also explain the alignment ? 
 the answer , at least in the case of the poincar dodecahedral space , is a resounding no .