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Questions such as whether we live in a spatially finite universe, and what its shape and size may be, are among the fundamental open problems that high precision modern cosmology needs to resolve. These questions go beyond the scope of general relativity (GR), since as a (local) metrical theory GR leaves the global topology of the universe undetermined. Despite our present-day inability to predict the topology of the universe, given the wealth of increasingly accurate astro-cosmological observations it is expected that we should be able to detect it. An overview of basic features of cosmic topology, the main methods for its detection, and observational constraints on detectability are briefly presented. Recent theoretical and observational results related to cosmic topology are also discussed.
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arXiv:astro-ph/0402324v1 13 Feb 2004
Cosmic Topology: a Brief Overview
M.J. Rebou¸cas
Centro Brasileiro de Pesquisas ısicas
Departamento de Relatividade e Part´ıculas
Rua Dr. Xavier Sigaud, 150 ,
22290-180 Rio de Janeiro RJ, Brazil
G.I. Gomero
Instituto de F´ısica Te´orica
Universidade Estadual Paulista
Rua Pamplona, 145, 01405-900 ao Paulo SP, Brazil
(Dated: February 2, 2008)
Questions such as whether we live in a spatially finite universe, and what its shape and size may be,
are among the fundamental open problems that high precision modern cosmology needs to resolve.
These questions go beyond the scope of general relativity (GR), since as a (local) metrical theory
GR leaves the global topology of the universe undetermined. Despite our present-day inability to
predict the topology of the universe, given the wealth of increasingly accurate astro-cosmological
observations it is expected that we should be able to detect it. An overview of basic features of
cosmic topology, the main methods for its detection, and observational constraints on detectability
are briefly presented. Recent theoretical and observational results related to cosmic topology are
also discussed.
I. INTRODUCTION
Is the space where we live finite or infinite? The popu-
lar ancient Greek finite-world response, widely accepted
in medieval Europe, is at a first sight open to a devastat-
ing objection: in being finite the world must have a lim-
iting boundary. But this is impossible, because a bound-
ary can only separate one part of the space from another:
why not redefine the universe to include that other part?
In this way a common-sense response to the above old
cosmological question is that the universe has to be infi-
nite otherwise something else would have to exist beyond
its limits. This answer seems to be obvious and needing
no further proof or explanation. However, in mathemat-
ics it is known that there are compact spaces (finite) with
no boundary. They are called closed spaces. Therefore,
our universe can well be spatially closed (topologically)
with nothing else beyond its ’spatial limits’. This may
be difficult to visualize because we are used to viewing
from ’outside’ objects which are embedded in our regular
3–dimensional space. But there is no need to exist any
region beyond the spatial extent of the universe.
Of course, one might still ask what is outside such a
closed universe. But the underlying assumption behind
this question is that the ultimate physical reality is an
infinite Euclidean space of some dimension, and nature
needs not to adhere to this theoretical embedding frame-
work. It is perfectly acceptable for our 3–space not to
be embedded in any higher-dimensional space with no
Electronic address: reboucas@cbpf.br
Electronic address: german@ift.unesp.br
physical grounds.
Whether the universe is spatially finite and what is its
size and shape are among the fundamental open prob-
lems that high precision modern cosmology seeks to re-
solve. These questions of topological nature have become
particularly topical, given the wealth of increasingly ac-
curate astro-cosmological observations, especially the re-
cent observations of the cosmic microwave background
radiation (CMBR) [1]. An important point in the search
for answers to these questions is that as a (local) metrical
theory general relativity (GR) leaves the global topology
of the universe undetermined. Despite this inability to
predict the topology of the universe we should be able to
devise strategies and methods to detect it by using data
from astro-cosmological observations.
The aim of the article is to give a brief review of the
main points on cosmic topology addressed in the talk
delivered by one of us (MJR) in the XXIV Brazilian Na-
tional Meeting on Particles and Fields, and discuss some
recent results in the field. The outline of our paper is as
follows. In section II we discuss how the cosmic topology
issue arises in the context of the standard cosmology, and
what are the main observational consequences of a non-
trivial topology for the spatial section of the universe. In
section III we review the two main statistical methods to
detect cosmic topology from the distribution of discrete
cosmic sources. In section IV we describe the search for
circles in the sky, an important method which has been
devised for the detection of cosmic topology from CMBR.
In section V we discuss the detectability of cosmic topol-
ogy and present examples on how one can decide whether
a given topology is detectable or not according to recent
observations. Finally, in section VI we briefly discuss
recent results on cosmic topology, and present some con-
2
cluding remarks.
II. NONTRIVIAL TOPOLOGY AND PHYSICAL
CONSEQUENCES
The isotropic expansion of the universe, the primor-
dial abundance of light elements and the nearly uniform
cosmic microwave background radiation constitute the
main observational pillars for the standard cosmologi-
cal model, which provides a very successful description
of the universe. Within the framework of standard cos-
mology, the universe is described by a space-time mani-
fold M4=R×Mendowed with the homogeneous and
isotropic Robertson-Walker (RW) metric
ds2=c2dt2+R2(t){2+f2(χ) [ 2+ sin2θ 2]},
(1)
where tis a cosmic time, f(χ) = (χ, sin χ, sinh χ) de-
pending on the sign of the constant spatial curvature
k= (0,1,1), and R(t) is the scale factor. The spa-
tial section Mis often taken to be one of the following
(simply-connected) spaces: Euclidean E3, spherical S3,
or hyperbolic space H3. This has led to a common mis-
conception that the Gaussian curvature kof Mis all
one needs to decide whether this 3–space is finite or not.
However, the 3-space Mmay equally well be one of the
possible quotient manifolds M=f
M/Γ, where Γ is a dis-
crete and fixed-point free group of isometries of the cor-
responding covering space f
M= (E3,S3,H3). Quotient
manifolds are multiply connected: compact in three in-
dependent directions with no boundary (closed), or com-
pact in two or at least one independent direction. The
action of Γ tessellates f
Minto identical cells or domains
which are copies of what is known as fundamental poly-
hedron (FP). In forming the quotient manifolds Mthe
essential point is that they are obtained from f
Mby iden-
tifying points which are equivalent under the action of the
discrete group Γ. Hence, each point on the quotient man-
ifold Mrepresents all the equivalent points on the cover-
ing manifold f
M. A simple example of quotient manifold
in two dimensions is the 2–torus T2=S1×S1=E2/Γ.
The covering space clearly is E2, and a FP is a rectangle
with opposite sides identified. This FP tiles the covering
space E2. The group Γ consists of discrete translations
associated with the side identifications.
In a multiply connected space any two points can al-
ways be joined by more than one geodesic. Since the ra-
diation emitted by cosmic sources follows geodesics, the
immediate observational consequence of a spatially closed
universe is that light from distant objects can reach a
given observer along more than one path the sky may
show multiple images of radiating sources [cosmic objects
or cosmic microwave background radiation from the last
scattering surface - (LSS)]. Clearly we are assuming here
that the radiation (light) must have sufficient time to
reach the observer at pM(say) from multiple direc-
tions, or put in another way, that the universe is suf-
ficiently small so that this repetitions can be observed.
In this case the observable horizon χhor exceeds at least
the smallest characteristic size of Mat p[55], and the
topology of the universe is in principle detectable.
A question that arises at this point is whether one can
use the topological multiple images of the same celestial
objects such as cluster of galaxies, for example, to de-
termine a nontrivial cosmic topology [56]. Besides the
pioneering work by Ellis [2], others including Sokolov
and Shvartsman [3], Fang and Sato [4], Starobinskii [5],
Gott [6] and Fagundes [7] and Fagundes and Wichoski [8],
used this feature in connection with closed flat and non-
flat universes. It has been recently shown that the topol-
ogy of a closed flat universe can be reconstructed with
the observation of a very small number of multiple im-
ages [9].
In practice, however, the identification of multiple im-
ages is a formidable observational task to carry out be-
cause it involves a number of problems, some of which
are:
Images are seen from different angles (directions),
which makes it very hard to recognize them as iden-
tical due to morphological effects;
High obscuration regions or some other object can
mask or even hide the images;
Two images of a given cosmic object at different
distances correspond to different periods of its life,
and so they are in different stages of their evolu-
tions, rendering problematic their identification as
multiple images.
These difficulties make clear that a direct search for
multiples images is not overly promising, at least with
available present-day technology. On the other hand,
they motivate new search strategies and methods to de-
termine (or just detect) the cosmic topology from obser-
vations. In the next section we shall discuss statistical
methods, which have been devised to determine a possi-
ble nontrivial topology of the universe from the distribu-
tion of discrete cosmic sources.
III. PAIR SEPARATIONS STATISTICAL
METHODS
On the one hand the most fundamental consequence of
a multiply connected spatial section Mfor the universe is
the existence of multiple images of cosmic sources, on the
other hand a number of observational problems render
the direct identification of these images practically im-
possible. In the statistical approaches we shall discuss in
this section instead of focusing on the direct recognition
of multiple images, one treats statistically the images of
a given cosmic source, and use (statistical) indicators or
signatures in the search for a sign of a nontrivial topology.
Hence the statistical methods are not plagued by direct
3
recognition difficulties such as morphological effects, and
distinct stages of the evolution of cosmic sources.
The key point of these methods is that in a universe
with detectable nontrivial topology at least one of the
characteristic sizes of the space section Mis smaller than
a given survey depth χobs, so the sky should show multi-
ple images of sources, whose 3–D positions are correlated
by the isometries of the covering group Γ. These methods
rely on the fact that the correlations among the positions
of these images can be couched in terms of distance corre-
lations between the images, and use statistical indicators
to find out signs of a possible nontrivial topology of M.
In 1996 Lehoucq et al. [10] proposed the first statis-
tical method (often referred to as cosmic crystallogra-
phy), which looks for these correlations by using pair
separations histograms (PSH). To build a PSH we sim-
ply evaluate a suitable one-to-one function Fof the dis-
tance dbetween a pair of images in a catalogue C, and
define F(d) as the pair separation: s=F(d). Then we
depict the number of pairs whose separation lie within
certain sub-intervals Jipartitions of (0, smax], where
smax =F(2χmax), and χmax is the survey depth of C. A
PSH is just a normalized plot of this counting. In most
applications in the literature the separation is taken to be
simply the distance between the pair s=dor its square
s=d2,Jibeing, respectively, a partition of (0,2χmax ]
and (0,4χ2
max].
The PSH building procedure can be formalized as fol-
lows. Consider a catalogue Cwith ncosmic sources and
denote by η(s) the number of pairs of sources whose sep-
aration is s. Divide the interval (0, smax] in mequal
sub-intervals (bins) of length δs =smax/m, being
Ji= (siδs
2, si+δs
2],;i= 1,2,...,m ,
and centered at si= (i1
2)δs . The PSH is defined as
the following counting function:
Φ(si) = 2
n(n1)
1
δs X
sJi
η(s),(2)
which can be seen to be sub ject to the normalization
condition Pm
i=1 Φ(si)δs = 1 .An important advantage
of using normalized PSH’s is that one can compare his-
tograms built up from catalogues with different number
of sources.
An example of PSH obtained through simulation for a
universe with nontrivial topology is given in Fig. 1. Two
important features should be noticed: (i) the presence
of the very sharp peaks (called spikes); and (ii) the ex-
istence of a ’mean curve’ above which the spikes stands.
This curve corresponds to an expected pair separation
histogram (EPSH) Φexp (si), which is a typical PSH from
which the statistical noise has been withdrawn, that is
Φexp(si) = Φ(si)ρ(si) , where ρ(si) represents the sta-
tistical fluctuation that arises in the PSH Φ(si).
The primary expectation was that the distance cor-
relations would manifest as topological spikes in PSH’s,
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2 4 6 8
FIG. 1: Typical PSH for a flat universe with a 3–torus topol-
ogy. The horizontal axis gives the squared pair separation s2,
while the vertical axis provides a normalized number of pairs.
and that the spike spectrum of topological origin would
be a definite signature of the topology [10]. While
the first simulations carried out for specific flat man-
ifolds appeared to confirm this expectation [10], his-
tograms subsequently generated for specific hyperbolic
manifolds revealed that the corresponding PSH’s exhibit
no spikes [11, 12]. Concomitantly, a theoretical statis-
tical analysis of the distance correlations in PSH’s was
accomplished, and a proof was presented that the spikes
of topological origin in PSH’s are due to just one type
of isometry: the Clifford translations (CT) [13], which
are isometries gtΓ such that for all pf
Mthe dis-
tance d(p, gtp) is a constant (see also in this regard [11]).
Clearly the CT’s reduce to the regular translations in
the Euclidean spaces (for more details and simulations
see [14, 15, 16]). Since there is no CT translation in
hyperbolic geometry this result explains the absence of
spikes in the PSH’s of hyperbolic universes with non-
trivial detectable topology. On the other hand, it also
makes clear that distinct manifolds which admit the same
Clifford translations in their covering groups present the
same spike spectrum of topological origin. Therefore the
topological spikes are not sufficient for unambiguously
determine the topology of the universe.
In spite of these limitations, the most striking evi-
dence of multiply-connectedness in PSH’s is indeed the
presence of topological spikes, which result from trans-
lational isometries gtΓ . It was demonstrated [13, 14]
that the other isometries gmanifest as very tiny deforma-
tions of the expected pair separation histogram Φsc
exp(si)
corresponding to the underlying simply connected uni-
verse [17, 18]. Furthermore, in PSH’s of universes with
nontrivial topology the amplitude of the sign of non-
translational isometries was shown to be smaller than
the statistical noise [14], making clear that one cannot
use PSH to reveal these isometries.
In brief, the only significant (measurable) sign of a
nontrivial topology in PSH are the spikes, but they can
4
be used merely to disclose (not to determine) a possi-
ble nontrivial topology of universes that admit Clifford
translations: any flat, some spherical, and no hyperbolic
universes.
The impossibility of using the PSH method for the de-
tection of the topology of hyperbolic universes motivated
the development of a new scheme called collecting cor-
related pairs method (CCP method) [19] to search for
cosmic topology.
In the CCP method it is used the basic feature of the
isometries, i.e., that they preserve the distances between
pairs of images. Thus, if (p, q) is a pair of arbitrary im-
ages (correlated or not) in a given catalogue C, then for
each gΓ such that the pair (gp, gq ) is also in Cwe
obviously have
d(p, q) = d(gp, gq).(3)
This means that for a given (arbitrary) pair (p, q ) of im-
ages in C, if there are nisometries gΓ such that both
images gp and gq are still in C, then the separation s(p, q)
will occur ntimes.
The easiest way to understand the CCP method is
by looking into its computer-aimed procedure steps, and
then examine the consequences of having a multiply con-
nected universe with detectable topology. To this end,
let Cbe a catalogue with nsources, so that one has
P=n(n1)/2 pairs of sources. The CCP procedure
consists on the following steps:
1. Compute the Pseparations s(p, q), where pand q
are two images in the catalogue C;
2. Order the Pseparations in a list {si}1iPsuch
that sisi+1 ;
3. Create a list of increments {i}1iP1, where
i=si+1 si;.
4. Def ine the CCP index as
R=N
P1,
where N=Card{i: i= 0}is the number of
times the increment is null.
If the smallest characteristic length of Mexceeds the
survey depth (rinj > χobs) the probability that two pairs
of images are separated by the same distance is zero, so
R 0. On the other hand, in a universe with detectable
nontrivial topology (χobs > rinj ) given gΓ, if pand q
as well as gp and gq are images in C, then: (i) the pairs
(p, q) and (gp, gq) are separated by the same distance;
and (ii) when Γ admits a translation gtthe pairs (p, gtp)
and (q, gtq) are also separated by the same distance. It
follows that when a nontrivial topology is detectable, and
a given catalogue Ccontains multiple images, then R>0,
so the CCP index is an indicator of a detectable nontriv-
ial topology of the spatial section Mof the universe. Note
that although R>0 can be used as a sign of multiply
connectedness, it gives no indication as to what the ac-
tual topology of Mis. Clearly whether one can find out
that Mis multiply connected (compact in at least one
direction) is undoubtedly a very important step, though.
In more realistic situations, uncertainties in the deter-
mination of positions and separations of images of cosmic
sources are dealt with through the following extension of
the CCP index:
Rǫ=Nǫ
P1,
where Nǫ=Card{i: iǫ}, and ǫ > 0 is a parameter
that quantifies the uncertainties in the determination of
the pairs separations.
Both PSH and CCP statistical methods rely on the
accurate knowledge of the three-dimensional positions of
the cosmic sources. The determination of these positions,
however, involves inevitable uncertainties, which basi-
cally arises from: (i) uncertainties in the determination
of the values of the cosmological density parameters m0
and Λ0; (ii) uncertainties in the determination of both
the red-shifts (due to spectroscopic limitations), and the
angular positions of cosmic objects (displacement, due
to gravitational lensing by large scale objects, e.g.); and
(iii) uncertainties due to the peculiar velocities of cosmic
sources, which introduce peculiar red-shift corrections.
Furthermore, in most studies related to these methods
the catalogues are taken to be complete, but real cata-
logues are incomplete: objects are missing due to selec-
tion rules, and also most surveys are not full sky coverage
surveys. Another very important point to be considered
regarding these statistical methods is that most of cos-
mic objects do not have very long lifetimes, so there may
not even exist images of a given source at large red-shift.
This poses the important problem of what is the suitable
source (candle) to be used in these methods.
Some of the above uncertainties, problems and limits of
the statistical methods have been discussed by Lehoucq
et al. [20], but the robustness of these methods still de-
serves further investigation. So, for example, a quanti-
tative study of the sensitivity of spikes and CCP index
with respect to the uncertainties in the positions of the
cosmic sources, which arise from unavoidable uncertain-
ties in values of the density parameters is being carried
out [21]. In [21] it is also determined the optimal values
of the bin size (in the PSH method) and the ǫparame-
ter (in the CCP method) so that the correlated pairs are
collected in a way that the topological sign is preserved.
For completeness we mention that Bernui [22] has
worked with a similar method which uses angular pair
separation histogram (APSH) in connection with CMBR.
To close this section we refer the reader to refer-
ences [23, 24], which present variant statistical methods
(see also the review article [25]).
5
IV. CIRCLES IN THE SKY
The deepest surveys currently available are the CMBR
temperature anisotropy maps with zLS S 103. Thus,
given the current high quality and resolution of such
maps, the most promising searches for cosmic topology
through multiple images of radiating sources are based
on pattern repetitions of these CMBR anisotropies.
The last scattering surface (LSS) is a sphere of radius
χLSS on the universal covering manifold of the comoving
space at present time. If a nontrivial topology of space is
detectable, then this sphere intersects some of its topo-
logical images. Since the intersection of two spheres is
a circle, then CMBR temperature anisotropy maps will
have matched circles, i.e. pairs of equal radii circles (cen-
tered on different point on the LSS sphere) that have the
same pattern of temperature variations [26].
These matched circles will exist in CMBR anisotropy
maps of universes with any detectable nontrivial topol-
ogy, regardless of its geometry. Thus in principle the
search for ‘circles in the sky’ can be performed without
any a priori information (or assumption) on the geome-
try, and on the topology of the universe.
The mapping from the last scattering surface to the
night sky sphere is a conformal map. Since conformal
maps preserves angles, the identified circle at the LSS
would appear as identified circles on the night sky sphere.
A pair of matched circles is described as a point in a
six-dimensional parameter space. These parameters are
the centers of each circle, which are two points on the
unit sphere (four parameters), the angular radius of both
circles (one parameter), and the relative phase between
them (one parameter).
Pairs of matched circles may be hidden in the CMBR
maps if the universe has a detectable topology. There-
fore to observationally probe nontrivial topology on the
available largest scale, one needs a statistical approach to
scan all-sky CMBR maps in order to draw the correlated
circles out of them. To this end, let n1= (θ1, ϕ1) and
n2= (θ2, ϕ2) be the center of two circles C1and C2with
angular radius ν. The search for the matching circles
can be performed by computing the following correlation
function [26]:
S(α) = h2T1(±φ)T2(φ+α)i
hT1(±φ)2+T2(φ+α)2i,(4)
where T1and T2are the temperature anisotropies along
each circle, αis the relative phase between the two cir-
cles, and the mean is taken over the circle parameter
φ:h i =R2π
0. The plus (+) and minus () signs
in (4) correspond to circles correlated, respectively, by
non-orientable and orientable isometries.
For a pair of circles correlated by an isometry (per-
fectly matched) one has T1(±φ) = T2(φ+α) for some
α, which gives S(α) = 1, otherwise the circles are un-
correlated and so S(α)0. Thus a peaked correlation
function around some αwould mean that two matched
circles, with centers at n1and n2, and angular radius ν,
have been detected.
From the above discussion it is clear that a full search
for matched circles requires the computation of S(α),
for any permitted α, sweeping the parameter sub-space
(θ1, ϕ1, θ2, ϕ2, ν), and so it is indeed computationally
expensive. Nevertheless, such a search is currently in
progress, and preliminary results using the first year
WMAP data indicate the lack of antipodal, and nearly
antipodal, matched circles with radii larger than 25[27].
Here nearly antipodal means circles whose center are sep-
arated by more than 170.
According to these first results, the possibility that our
universe has a torus-type local shape is discarded, i.e. any
flat topology with translations smaller than the diameter
of the sphere of last scattering is ruled out. As a matter of
fact, as they stand these preliminary results exclude any
topology whose isometries produce antipodal images of
the observer, as for example the Poincar´e dodecahedron
model [28], or any other homogeneous spherical space
with detectable isometries.
Furthermore, since detectable topologies (isometries)
do not produce, in general, antipodal correlated circles,
a little more can be inferred from the lack or nearly an-
tipodal matched circles. Thus, in a flat universe, e.g., any
screw motion may generate pairs of circles that are not
even nearly antipodal, provided that the observer’s posi-
tion is far enough from the axis of rotation [29]. As a con-
sequence, our universe can still have a flat topology, other
than the 3-torus, but in this case the axis of rotation of
the screw motion corresponding to a pair of matched cir-
cles would pass far from our position. Similar results
also hold for spherical universes with non-translational
isometries generating pairs of matched circles. Indeed,
the universe could have the topology of, e.g., an inho-
mogeneous lens space L(p, q ), but with both equators of
minimal injectivity radius passing far from us [57]. These
points also make clear the crucial importance of the posi-
tion of the observer relative to the ’axis of rotation’ in the
matching circles search scheme for inhomogeneous spaces
(in this regard see also [30]).
To conclude, ‘circles in the sky’ is a promising method
in the search for the topology of the universe, and may
provide more general and realistic constraints on the
shape and size of our universe in the near future. An im-
portant point in this regard is the lack of computational
less expensive search for matched circles, which can be
archived by restricting (in the light of observations) the
expected detectable isometries, confining therefore the
parameter space of realistic search for correlated circles
as indicated, for example, by Mota et al. [31].
V. DETECTABILITY OF COSMIC TOPOLOGY
In the previous sections we have assumed that the
topology of the universe is detectable, and focussed our
attention on strategies and methods to discover or even
6
determine a possible nontrivial topology of the universe.
In this section we shall examine the consequences of this
underlying detectability assumption in the light of the
current astro-cosmological observations which indicate
that our universe is nearly flat (Ω01) [32]. Although
this near flatness of the universe does not preclude a non-
trivial topology it may push the smallest characteristic
size of Mto a value larger than the observable horizon
χhor, making it difficult or even impossible to detect by
using multiple images of radiating sources (discrete cos-
mic objects or CMBR maps). The extent to which a
nontrivial topology may or may not be detected has been
examined in locally flat [33], spherical [34, 35, 36] or hy-
perbolic [36, 37, 38, 39] universes. The discussion below
is based upon our articles [34, 36, 37, 38], so we shall fo-
cus on nearly (but not exactly) flat universes (for a study
of detectability of flat topology see [33]).
The study of the detectability of a possible nontrivial
topology of the spatial sections Mrequires topological
typical scale which can be put into correspondence with
observation survey depths. A suitable characteristic size
of Mis the so-called injectivity radius rinj (x) at xM,
which is defined in terms of the length of the smallest
closed geodesics that pass through xas follows.
A closed geodesic that passes through a point xin a
multiply connected manifold Mis a segment of a geodesic
in the covering space f
Mthat joins two images of x. Since
any such pair of images are related by an isometry gΓ,
the length of the closed geodesic associated to any fixed
isometry g, and that passes through x, is given by the
corresponding distance function
δg(x)d(x, gx).(5)
The injectivity radius at xthen is defined by
rinj (x) = 1
2min
ge
Γ
{δg(x)},(6)
where e
Γ denotes the covering group without the iden-
tity map. Clearly, a sphere with radius r < rinj (x) and
centered at xlies inside a fundamental polyhedron of M.
For a specific survey depth χobs a topology is said to be
undetectable by an observer at a point xif χobs < rinj (x),
since in this case every image catalogued in the survey lies
inside the fundamental polyhedron of Mcentered at the
observer’s position x. In other words, there are no multi-
ple images in the survey of depth χobs , and therefore any
method for the search of cosmic topology based on their
existence will not work. If, otherwise, χobs > rinj (x),
then the topology is potentially detectable (or detectable
in principle).
In a globally homogeneous manifold, the distance func-
tion for any covering isometry gis constant. Therefore,
the injectivity radius is constant throughout the whole
space, and so if the topology is potentially detectable
(or undetectable) by an observer at x, it is detectable
(or undetectable) by any other observer at any other
point in the same space. However, in globally inhomoge-
neous manifolds the injectivity radius varies from point
to point, thus in general the detectability of cosmic topol-
ogy depends on both the observer’s position xand survey
depth. Nevertheless, for globally inhomogeneous mani-
folds one can define the global injectivity radius by
rinj = min
xM{rinj (x)},(7)
and state an ’absolute’ undetectability condition. Indeed,
for a specific survey depth χobs a topology is undetectable
by any observer (located at any point x) in the space
provided that rinj > χobs.
Incidentally, we note that for globally inhomogeneous
manifolds one can define the so-called injectivity profile
P(r) of a manifold as the probability density that a point
xMhas injectivity radius rinj(x) = r. The quantity
P(r)dr clearly provides the probability that rinj (x) lies
between rand r+dr, and so the injectivity profile curve
is essentially a histogram depicting how much of a man-
ifold’s volume has a given injectivity radius (for more
detail on this point see Weeks [39]). An important point
is that the injectivity profile for non-flat manifolds of
constant curvature is a topological invariant since these
manifolds are rigid.
In order to apply the above detectability of cosmic
topology condition in the context of standard cosmology,
we note that in non-flat RW metrics (1) , the scale factor
R(t) is identified with the curvature radius of the spatial
section of the universe at time t, and thus χcan be in-
terpreted as the distance of any point with coordinates
(χ, θ, φ) to the origin (in the covering space) in units of
curvature radius, which is a natural unit of length.
To illustrate now the above condition for detectability
(undetectability) of cosmic topology, in the light of recent
observations [1, 32] we assume that the matter content
of the universe is well approximated by dust of density
ρmplus a cosmological constant Λ. In this cosmological
setting the curvature radius R0of the spatial section is
related to the total density parameter 0through the
equation
R2
0=kc2
H2
0(Ω01) ,(8)
where H0is the Hubble constant, kis the normalized
spatial curvature of the RW metric (1), and where here
and in what follow the subscript 0 denotes evaluation
at present time t0. Furthermore, in this context the
redshift-distance relation in units of the curvature radius,
R0=R(t0), reduces to
χ(z) = p|10|Z1+z
1
dx
px3m0+x2(1 0) + Λ0
,
(9)
where m0and Λ0 are, respectively, the matter and
the cosmological density parameters, and 0m0+
Λ0. For simplicity, on the left hand side of (9) and in
many places in the remainder of this article, we have left
7
implicit the dependence of the function χon the density
components.
A first qualitative estimate of the constraints on de-
tectability of cosmic topology from nearflatness can be
obtained from the function χ(Ωm0,Λ0, z ) given by (9)
for a fixed survey depth z. Figure 2 clearly demonstrates
the rapid way χdrops to zero in a narrow neighbourhood
of the 0= 1 line. This can be understood intuitively
from (8), since the natural unit of length (the curvature
radius R0) goes to infinity as 01, and therefore the
depth χ(for any fixed z) of the observable universe be-
comes smaller in this limit. From the observational point
of view, this shows that the detection of the topology of
the nearly flat universes becomes more and more difficult
as 01, a limiting value favoured by recent observa-
tions. As a consequence, by using any method which
relies on observations of repeated patterns the topology
of an increasing number of nearly flat universes becomes
undetectable in the light of the recent observations, which
indicate that 01.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
1
2
3
4
χ(z)
m
Λ
FIG. 2: The behaviour of χ(Ωm0,Λ0, z ) for a fixed z= 1100
as a function of the density parameters Λ0 and m0.
From the above discussion it is clear that cosmic topol-
ogy may be undetectable for a given survey up to a
depth zmax, but detectable if one uses a deeper survey.
At present the deepest survey available corresponds to
zmax =zLSS 103, with associated depth χ(zLSS ). So
the most promising searches for cosmic topology through
multiple images of radiating sources are based on CMBR.
To quantitatively illustrate the above features of the
detectability problem, we shall examine the detectabil-
ity of cosmic topology of the first ten smallest (volume)
hyperbolic universes.
To this end we shall take the following interval of the
density parameters values consistent with current ob-
servations: 0[0.99,1) and Λ0 [0.63,0.73]. In
this hyperbolic sub-interval one can calculate the largest
value of χobs(Ωm0,Λ0 , z) for the last scattering sur-
face (z= 1100), and compare with the injectivity radii
rinj to decide upon detectability. From (9) one obtains
χmax
obs = 0.337 .
Table I summarizes our results which have been refined
upon and reconfirmed by Weeks [39]. It makes explicit
that there are undetectable topologies even if one uses
Manifold rinj cmbr
m003(-3,1) 0.292
m003(-2,3) 0.289
m007(3,1) 0.416 U
m003(-4,3) 0.287
m004(6,1) 0.240
m004(1,2) 0.183
m009(4,1) 0.397 U
m003(-3,4) 0.182
m003(-4,1) 0.176
m004(3,2) 0.181
TABLE I: Restrictions on detectability of cosmic topology
for 0= 0.99 with Λ0 [0.63,0.73] for the first ten smallest
known hyperbolic manifolds. Here Ustands for undetectable
topology with CBMR (zmax = 1100), while the dash denotes
detectable in principle.
CMBR.
We note that similar results hold for spherical universes
with values of the density parameters within the current
observational bounds (for details see [34, 35, 36]). This
makes apparent that there exist nearly flat hyperbolic
and spherical universes with undetectable topologies for
01 favoured by recent observations.
The most important outcome of the results discussed
in this section is that, as indicated by recent obser-
vations (and suggested by inflationary scenarios) 0is
close (or very close) to one, then there are both spher-
ical and hyperbolic universe whose topologies are unde-
tectable. This motivates the development of new strate-
gies and/or methods in the search for the topology of
nearly flat universes, perhaps based on the local physi-
cal effect of a possible nontrivial topology. In this regard
see [40, 41, 42, 43, 44, 45], for example.
VI. RECENT RESULTS AND CONCLUDING
REMARKS
In this section we shall briefly discuss some recent re-
sults and advances in the search for the shape of the
universe, which have not been treated in the previous
sections. We also point out some problems, which we
understand as important to be satisfactorily dealt with
in order to make further progress in cosmic topology.
One of the intriguing results from the analysis of
WMAP data is the considerably low value of the CMBR
quadrupole and octopole moments, compared with that
predicted by the infinite flat ΛCDM model. Another
noteworthy feature is that, according to WMAP data
analysis by Tegmark et al., both the quadrupole and the
octopole moments have a common preferred spatial axis
along which the power is suppressed [58].
This alignment of the low multipole moments has been
suggested as an indication of a direction along which a
possible shortest closed geodesics (characteristic of multi-
ply connected spaces) of the universe may be [50]. Moti-
8
vated by this as well as the above anomalies, test using S-
statistics [51] and matched circles furnished no evidence
of a nontrivial topology with diametrically opposed pairs
of correlated circles [50]. It should be noticed, however,
that these results do no rule out most multiply connected
universe models because S-statistics is a method sensi-
tive only to Euclidean translations, while the search for
circles in the sky, which is, in principle, appropriate to
detect any topology, was performed in a limited three-
parameter version, which again is only suitable to detect
translations.
At a theoretical level, although strongly motivated by
high precision data from WMAP, it has been shown that
if a very nearly flat universe has a detectable nontriv-
ial topology, then it will exhibit the generic local shape
of (topologically) R2×S1[59], irrespective of its global
shape [31]. In this case, from WMAP and SDSS the data
analysis, which indicates that 01 [32], one has that
if the universe has a detectable topology, it is very likely
that it has a preferred direction, which in turn is in agree-
ment with the observed alignement of the quadrupole and
octopole moments of the CMBR anisotropies. In this
context, it is relevant to check whether a similar align-
ment of higher order multipole ( > 3) takes place in
order to reinforce a possible nontrivial local shape of our
3–space. In this connection it is worth mentioning that
Hajian and Souradeep [52, 53] have recently suggested
a set of indicators κ(= 1,2,3, ...) which for non-zero
values indicate and quantify statistical anisotropy in a
CMBR map. Although κcan be potentially used to dis-
criminate between different cosmic topology candidates,
they give no information about the directions along which
the isotropy may be violated, and therefore other indi-
cators should be devised to extract anisotropy directions
from CBMR maps.
In ref. [31] it has also been shown that in a very nearly
flat universe with detectable nontrivial topology, the ob-
servable (detectable) isometries will behave nearly like
translations. Perhaps if one use Euclidean space to lo-
cally approximate a nearly flat universe with detectable
topology, the detectable isometries can be approximated
by Euclidean isometries, and since these isometries are
not translations, they have to be screw motions. As
a consequence, an approximate local shape of a nearly
flat universe with detectable topology would look like a
twisted cylinder, i.e. a flat manifold whose covering group
is generated by a screw motion. Work toward a proof of
this conjecture is being carried out by our research group.
Before closing this overview we mention that the study
of the topological signature (possibly) encoded in CMBR
maps as well as to what extent the cosmic topology
CMBR detection methods are robust against distinct
observational effects such as, e.g., Suchs-Wolfe and the
thickness of the LSS effects, will benefit greatly from ac-
curate simulations of these maps in the context of the
FLRW models with multiply connected spatial sections.
A first step in this direction has been achieved by Ri-
azuelo et al. [54], with special emphasis on the effect
of the topology in the suppression of the low multipole
moments. Along this line it is worth studying through
computer-aided simulations the effect of a nontrivial cos-
mic topology on the nearly alignments of the quadrupole
and the octopole moments (spatial axis along which the
power is suppressed).
To conclude, cosmic topology is at present a very ac-
tive research area with a number of important problems,
ranging from how the characterization of the local shape
of the universe may observationally be encoded in CMBR
maps, to the development of more efficient computation-
ally searches for matching circles, taking into account
possible restrictions on the detectable isometries, and
thereby confining the parameter space which realistic
‘circles in the sky’ searches need to concentrate on. It
is also of considerable interest the search for the statisti-
cal anisotropy one can expect from a universe with non-
trivial space topology. Finally, it is important not to for-
get that there are almost flat (spherical and hyperbolic)
universes, whose spatial topologies are undetectable in
the light of current observations with the available meth-
ods, and our universe can well have one of such topolo-
gies. In this case we have to devise new methods and
strategies to detect the topology of the universe.
Acknowledgments
We thank CNPq and FAPESP (contract 02/12328-6)
for the grants under which this work was carried out. We
also thank A.A.F. Teixeira and B. Mota for the reading
of the manuscript and indication of relevant misprints
and omissions.
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possibly be used in the search for multiple images in the
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an important step in cosmic topology to the extent that
for the first time a possible nontrivial cosmic topology
was tested against accurate CMBR data.
[59] Or more rarely R×T2.
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