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arXiv:astro-ph/0310749v1 27 Oct 2003
‘Circles in the Sky’ in twisted cylinders
G.I. Gomero
∗
,
Instituto de F´ısica Te´orica,
Universidade Estadual Paulista,
Rua Pamplona 145
S˜ao Paulo, SP 0 1405–900, Brazil
February 2, 2008
Abstract
It is shown here how prior estimates on the local s hape of the universe can be used
to reduce, to a small region, the f ull parameter space for the search of circles in the
sky. This is the fir st step towards the development of efficient estrategies to look for
these match ed circles in order to detect a possible nontrivial topology of our Universe.
It is shown how to calculate the unique point, in the parameter space, representing
a pair of matched circles corresponding to a given isometry g (and its inverse). As
a consequence, (i) given some fine estimates of the covering group Γ of the spatial
section of our universe, it is possible to confine, in a very effective way, the region
of the parameter space in which to perf orm the searches for matched circles, and
reciprocally (ii) once identified such pairs of matched circles, one could determine with
greater precision the topology of our Universe and ou r location within it.
It has recently been suggested t hat the quadrupole and octopole moments of the CMB
anisotropies are almost aligned, i.e. each multipole has a preferred axis along which power is
suppressed and both axes almost coincide. In fact, the angle between the preferred directions
of these lowest multipo les is ∼10
◦
, while the probability of this occurrence f or two randomly
oriented axes is roughly 1/62. There is also at present almost no doubt that the extremely low
value of the CMB quadrupole is a real effect, i.e. it is not an illusion created by foregrounds
[1].
Traditionally, the low value of the quadrupole moment has been considered as indirect
evidence for a non–trivial topology o f the universe. Actually, it was the fitting to these low
values of the quadrupole a nd octopole moments of t he CMB anisotropy which motivated
the recent proposal that our Universe would be a Poincare’s dodecahedron [2]. On the other
hand, the observed alignement of the quadrupole and the octop ole moments has recently
been used as a hint for determining the direction along which might occur the shortest closed
geodesics characteristic of multiply connected spaces [3].
However, in most of the studies reported, the model topology used for the comparison
with da ta has been the T
1
topology, i.e. the torus topology with one scale of compactification
∗
german@ift.unesp.br
1
of the order of the horizon radius, and the ot her two much larger. This is the simplest
topology after the trivial one. Tests using S-statistics [4] and the c i rcles in the sky method
[5] performed in [3] yielded a null result for a non–trivial topology of our universe. However
it should be reminded that multiply connected universe models cannot be ruled o ut on these
grounds. In fact, S-statistics is a method sensitive only to translational isometries, while the
search for the circles in the s ky, which in principle is sensitive to detect any topology, was
performed in a three-parameter version able to detect translations only.
If the t opology of t he Universe is detectable in the sense of [6], then CMB anisotropy maps
might present matched circles, i.e. pairs of circles along of which the anisotropy patterns
match [5]. These circles are actually the intersections (in the universal covering space of the
spatial sections of spacetime) of the topological imag es of the sphere of la st scattering, and
hence are related by the isometries of the covering group Γ. Since matched circles will exist
in CMB anisotropy maps of any universe with a detectable topology, i.e. regardless of its
geometry and topology, it seems that the search for ‘circles in the sky’ might be performed
without any a priori information of what the geometry and topology of the universe is.
However, any pair of matched circles is described as a point in a six–dimensional parameter
space, which makes a full–parameter search computationally expensive.
1
Nevertheless, such a titanic search is currently being performed, and preliminary results
have shown the lack of antipodal, and approximately antipodal, matched circles with radii
larger than 25
◦
[7]. These results rule out the Poincare’s dodecahedron model [2], and it has
also been suggested that they rule out the possibility that we live in a small universe, since
for the majority of detectable topologies we should expect antipodal or almost antipodal
matched circles. In particular, it is argued that this claim is exact in all Euclidean manifolds
with the only exception of the Hantzche–Wendt manifold (G
6
in Wolf’s notation [8]).
The purp ose of this letter is twofold. First, it is shown how to use prior estimates on the
local shape of the universe to reduce the region of the full parameter space in a way that the
search for matched circles might become practical. In fact, it is shown how to calculate the
unique point in the parameter space representing a pair of matched circles corresponding
to a given isometry g (and its inverse). As a consequence, given some fine estimates of t he
covering gro up Γ of the present spatial section of our Universe, we may be able to confine,
in a very effective way, the region of the parameter space in which to perform the searches
for circles in the sky. This is the first important step towards the development of efficient
estrategies to loo k for these matched circles. Moreover, o nce such pairs of matched circles had
been identified, it is a simple matter to use its locatio n in the para meter space to determine
with greater precision the topology of our Universe.
Second, it emerges from the calculations that we should not expect (nearly) antipodal
matched circles from the majority of detectable topologies. In particular, any Euclidean
topology, with the exception of the torus, might generate pairs of circles that are not even
nearly antipodal, provided t he observer lies out of the axis of rotat io n of the isometry that
gives rise to the pair of circles. This result might be g eneralized to the spherical case, for
which work is in progress.
The main motivation for this work is the suspicion that the alignement of the quadrupole
and the octopole moments of CMB anisotropies observed by the satellite WMAP, together
with the anomalous low value of the quadrupole moment, is the topological signature we
should exp ect from a generic topolo gy in a nearly flat universe, even if its size is slightly
1
These parameters are the ce nter of each circle as a point in the sphere of last scattering (four parameters),
the angular radius of both circles (one parameter ), and the relative phase between them (one parameter).
2
larger than the horizon radius. Moreover, as has been shown in [9], if topology is detectable
in a very nearly flat universe, the observable isometries will behave nearly as translations. If
we locally approximate a nearly flat constant curvature space M with Euclidean space, the
smallest isometries of the covering group of M will behave a s isometries in Euclidean space.
Since these isometries are not translations, they must behave as screw motions, thus an
appropriate model to get a feeling of what to expect observationally in a nearly flat universe
with detectable topology is a twisted cylinder.
Thus, let us begin by briefly describing the geometry of twisted cylinders. An isometry in
Euclidean 3-space can always be written as (A, a), where a is a vector and A is an orthogonal
transformation, and its action on Euclidean space is given by
(A, a) : x 7→ Ax + a , (1)
for any po int x. The generator of t he covering group of a twisted cylinder is a screw motion,
i.e. an isometry where its orthogonal part is a rotation and its translational part has a
component parallel to the axis of r otation [8]. Thus we can always choose the origin and
aligne the axis of r otation with the z–axis to write
A =
cos α − sin α 0
sin α cos α 0
0 0 1
(2)
for the orthogonal part, and
a = (0, 0, L) (3)
for the translational part of the generator g = (A, a).
This is what is usually done when studying the mathematics of Euclidean manifolds, since
it simplifies calculations. However, in cosmological applications this amounts to assume that
the observer lies on the axis of rotation, which is a very unnatural assumption. In order to
consider the arbitrariness of the position of the o bserver inside space, we parallel transport
the axis of rotation, along the po sitive x–axis, a distance ρ from the origin which remains
to be the observer’s position. Thus the generator of the twisted cylinder is now g = (A, b),
with tr anslational par t given by
b = ρ(1 − cos α)
b
e
x
− ρ sin α
b
e
y
+ L
b
e
z
. (4)
The pair of matched circles related by the generator g = (A, b) are the intersections of
the sphere of last scattering with its images under the isometries g and g
−1
respectively, and
the centers of these images are located at g0 = b and g
−1
0 = −A
−1
b. Thus the angular
positions of the centers of the matched circles are
n
1
=
b
|b|
and n
2
= −
A
−1
b
|b|
= −A
−1
n
1
. (5)
There are four parameters we can determine using (2) and (4). The angle σ between n
1
and the axis of rotation, the angle µ between the centers of the pair of matched circles, the
angular size ν of both matched circles, and the phase–shift φ. It turns out that only three
of them are independent, as should be expected since a screw motion has only three free
parameters (ρ, L, α) .
We easily obtain
cos µ = n
1
· n
2
= −
1 + tan
2
σ cos α
1 + tan
2
σ
(6)
3
for the angular separation between both directions n
1
and n
2
, while σ is given by
tan σ =
ρ
L
q
2(1 − cos α) . (7)
One can see from (6) and (7) tha t the matched circles will be antipodal only when the
observer is on the axis of rotation (ρ = 0), or the isometry is a translation (α = 0). In
particular, this shows that in a universe with any topology G
2
–G
6
, if a screw motion of the
covering group generates a pair of matched circles, they will not necessarily appear nearly
antipodal to an observer located off the axis of rotation. As an example consider an observer
in a G
4
universe located at a distance ρ = L/2 from the axis of rotation of the generator
screw motion (α = π/2) . From (6) and (7 ) it follows that µ ≈ 132
◦
.
Next, to compute the angular size of these circles, let R
LSS
be the radius of the sphere
of last scattering. Simple g eometry shows that, since |b| is the distance between the two
centers of the spheres whose intersections generate one of the matched circles, the angular
size of this intersection is
cos ν =
|b|
2R
LSS
=
L
2R
LSS
cos σ
. (8)
The computation of the phase–shift between the matched circles (the last parameter
we wish to constrain) is more involved. First we need to have an operational definition of
this quantity. This is simply accomplished if we realize that there is a great circle that
passes through the centers, n
1
and n
2
, of the matched circles. Orient this great circle such
that it passes first through n
2
, and then through n
1
along the shortest path, and let v
2
,
u
2
, u
1
and v
1
be the intersections of the great circle with the matched ones following this
orientation. If there were no phase–shift, then we would have g(R
LSS
u
2
) = R
LSS
u
1
and
g(R
LSS
v
2
) = R
LSS
v
1
. Hence we define the phase shift as the rotation angle, aro und the
normal of the sphere at n
1
, that takes u
1
to
b
u
2
= g(R
LSS
u
2
)/R
LSS
, positive if the shift is
counterclockwise, and negative otherwise.
In order to use this operational definition to compute the phase–shift, recall first that
the great circle passing through n
2
and n
1
, with the required orientation, is given by
n(t) =
1
sin µ
[ n
2
sin(µ − t) + n
1
sin t ] , (9)
where t is the angular distance between n(t) and n
2
. Thus we have
u
1
=
1
sin µ
[ n
2
sin ν + n
1
sin(µ − ν) ] ,
u
2
=
1
sin µ
[ n
2
sin(µ − ν) + n
1
sin ν ] , and (10)
b
u
2
=
1
sin µ
[ An
2
sin(µ − ν) + An
1
sin ν ] +
b
R
LSS
,
Writing the positions of u
1
and
b
u
2
with respect to their projections to the axis n
1
, as
w
1
= u
1
− n
1
cos ν and w
2
=
b
u
2
− n
1
cos ν, enables us to express easily the phase–shift as
cos φ =
w
1
· w
2
sin
2
ν
, (11)
since |w
1
| = |w
2
| = sin ν. After a somewhat lengthy calculation one arrives at
cos φ =
2(1 + cos α)
1 − cos µ
− 1 (12)
4
for the phase–shift. It is easily seen that when the observer is on the axis o f rotation (ρ = 0),
the shift equals α;
2
while when the isometry is a translation (α = 0), the shift vanishes. In
general, however, the shift depends on the three parameters (ρ, L, α), but only throug h the
values of µ and α, thus it is not an independent parameter. In fact, for a given pair (σ, µ),
one can easily compute φ, since α is readily o bta ined from (6).
Summarizing, given estimates of the parameters (ρ, L, α), and having determined an
estimate of the axis of rotation of the screw motion, one can perform searches for pairs of
circles both with centers at an angular distance σ of this axis and separation µ between
them, given by (7) and (6) respectively. The phase–shift between the circles is fixed by t hese
two parameters. Moreover, we can also limit the search of matched circles to only those
with ang ular size ν given by (8). We have, thus, constrained three out of the six parameters
needed to locate a pair of matched circles. The three missing parameters are the position
(θ, ϕ) of the axis of rotation o f the screw motion and the azimuthal angle λ of the center of
one of the matched circles.
Under the hypothesis that the alignement of the quadrupole and the octopole moments
is due to the topology of space, the position of the a xis of rotation might be estimated fro m
this alignement. Work is in progress in this direction. Finally, only the a ng le λ remains
totally unconstrained.
Interestingly, a consequent precise identification of a pair of matched circles will allow,
reciprocally, to determine with greater precision the same topolog ical parameters with which
we started, together with our position and o r ientation in the Universe.
Acknowledgments
I would like to thank FAPESP for the gr ant under which this work was carried out (con-
tract 02/12328–6). I also thank B. Mota, A. Bernui and W. Hip´olito–Ricaldi for useful
conversations.
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We infere from this that, in the general case, both angles α and φ have the same sign.
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