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arXiv:gr-qc/0105002v3 7 Sep 2001
Detectability of cosmic topology in almost
flat universes
G.I. Gomero
1∗
, M.J. Rebou¸cas
1†
, R. Tavakol
1,2‡
1
Centro Brasileiro de Pesquisas F´ısicas,
Rua Dr. Xavier Sigaud 150
22290-180 Rio de Janeiro – RJ, Brazil
2
Astronomy Unit, Scho ol of Mathematical Sciences,
Queen Mary, University of Lo ndon,
Mile End Road, London E1 4NS, UK
February 7, 2008
Abstract
Recent observations suggest that the r atio of the total den sity to the critical density
of the universe, Ω
0
, is likely to be very close to one, with a significant proportion of
this energy being in the form of a dark component with negative pressure. Motivated
by this result, we study the question of observational detection of possible non-trivial
topologies in universes with Ω
0
∼ 1, which include a cosmological constant. Using a
number of indicators we find that as Ω
0
→ 1, increasing families of possible manifolds
(topologies ) become either undetectable or can be excluded observationally. Further-
more, given a non-zero lower bound on |Ω
0
− 1|, we can rule out families of topologies
(manifolds) as possible candidates for the shape of our universe. We demonstrate these
findings concretely by considering families of manifolds and employing bounds on cos-
mological parameters from recent observations. We find that given the present bounds
on cosmological parameters, there are families of both hyp erbolic and spherical mani-
folds that remain undetectable and families that can be excluded as the shape of our
universe. These results are of importance in future search strategies for th e detection
of the shape of ou r universe, given that there are an infinite number of th eoretically
possible topologies and that the future observations are expected to put a non-zero
lower bound on |Ω
0
− 1| which is more accurate and closer to zero.
∗
gomero@cbpf.br
†
reboucas@cbpf.br
‡
r.tavakol@qmw.ac.uk
1
1 Introduction
A number of recent observations suggest that the ratio of the baryonic (plus dark) matter
density to the critical density, Ω
m0
, is significantly less than unity, being of the order o f Ω
m0
∼
0.2−0.3 (see e.g. [1] and references therein). On the other hand, recent measurements of the
position of the first acoustic peak in the angular power spectrum of CMBR anisotropies,
by BOOMERANG-98 and MAXIMA-I experiments, seem to provide strong evidence that
the corresponding ratio for the total density to the critical density, Ω
0
, is close to one [2] –
[7]. Added to this is the evidence fr om the spectral and photometric observations of Type
Ia Supernovae [8, 9] which seem to suggest that the universe is undergoing an accelerated
expansion at the present epoch [10, 11]. This rather diverse set of observations has led to an
evolving consensus among cosmologists that the total density of the universe, Ω
0
, is likely to
be very close to unity, a significant part of which resides in the form of a dark component
which is smooth on cosmological scales and which possesses negative pressure. A candidate
that satisfies these criteria is the vacuum energy corresponding to a cosmological constant.
Somewhat parallel to these developments, and to some extent unrelated to them, a great
deal of work has also recently gone into studying the possibility that the universe may possess
compact spatial sections with a non-trivial topo logy (see, for example, [12] – [17]), including
the construction of different topolo gical indicators (see e.g. [18] – [33]). A fair number of
these studies have concentrated on cases where the densities corresponding to matter and
vacuum energy ar e substantially smaller than the critical density. This was reasonable
because until very recently observations used to point to a low density universe. In addition
an impo rt ant aim of most of these previous works has often been to produce examples where
the topology of the universe has strong observational signals and can therefore be detected
and even determined.
The main aims of this paper, which are complementary to these earlier works, are twofold.
Firstly, to study the question of detectability of a possible non-trivial compact topology in
locally homogeneous and isotropic universes whose total density parameter is taken to be
close to one (Ω
0
∼ 1). Secondly, to show how given a non-zero lower bound on |Ω
0
− 1|, we
can exclude certain families of compact manifolds as viable candidates for the shape of the
universe. We shall do this by employing two indicators, namely the ratios of the so called
injectivity radius ( r
inj
), a nd maximal inradius ( r
max
−
) to the depth χ
obs
of given catalogues.
We study almost flat models with both Ω
0
> 1 and Ω
0
< 1, and our results apply t o any
method of detection of topology based on observations of multiple images of either cosmic
objects or spots of microwave background radiation. We note that even though the idea that
non-trivial topologies become harder to detect as Ω
0
→ 1 is implicitly present in some other
works [22, 27 ], it has, however, often been passed over as an uninteresting limit concerning
non-trivial topologies, given the observations at the time. Here, in addition to studying
this limit in detail by considering concrete families of manifolds (topolog ies), rather than
individual examples as has o ften been done before, we treat it as the most relevant limit
concerning the geometry of the universe today in view of the recent observations [2] – [7].
2
The structure of the paper is as fo llows. In Section 2 we give an account of the cos-
mological models employed. Section 3 contains a brief account of our topological setting.
In Section 4 we discuss the question of detectability of cosmic topology using a number of
indicators. Section 5 contains a detailed discussion of the question of detection of cosmic
topology in universes with |Ω
0
− 1| ≪ 1, with the help of concrete examples, and finally
Section 6 contains summary of our main results and conclusions.
2 Cosmological Setting
Let us begin by assuming that the universe is modelled by a 4-manifold M which a llows
a (1 + 3) splitting, M = R × M, with a locally isotropic and homogeneous Friedmann-
Lemaˆıtre-Robertson-Walker (FLRW) metric
ds
2
= −c
2
dt
2
+ R
2
(t)
h
dχ
2
+ f
2
(χ)(dθ
2
+ sin
2
θdφ
2
)
i
, (2.1)
where t is a cosmic time, the function f (χ) is given by f (χ) = χ , sin χ , or sinh χ ,
depending on the sign of the constant spatial curvature (k = 0, ±1), and R(t) is the scale
factor. Furthermore, we shall assume throughout this article that the 3-space M is a multiply
connected compact quotient manifold of the form M =
f
M/Γ, where Γ is a discrete group
of isometries of
f
M acting freely on the covering space
f
M of M, where
f
M can take one
of the forms E
3
, S
3
or H
3
[correspo nding, respectively, t o flat (k = 0), elliptic (k > 0) and
hyperbolic (k < 0) spaces]. The group Γ is called the covering group of M, and is isomorphic
to its fundamental group π
1
(M).
For non-flat models (k 6= 0), the scale factor R(t) is identified with the curvature radius of
the spatial section of the universe at time t, and thus χ can be interpreted as the distance of
any point with coordinates (χ, θ, φ) to the origin of coordinates (in
f
M), in units of curvature
radius, which is a natural unit of length and suitable for measuring areas and volumes.
Throughout this paper we shall use this natural unit.
In the light of current observations, we assume the current matter content of the universe
to be well approximated by dust ( of density ρ
m
) plus a cosmological constant Λ. The
Friedmann equation is then given by
H
2
=
8πGρ
m
3
−
kc
2
R
2
+
Λ
3
, (2.2)
where H =
˙
R/R is the Hubble parameter and G is the Newton’s constant, which upon
introducing Ω
m
=
8πGρ
m
3H
2
and Ω
Λ
≡
8πGρ
Λ
3H
2
=
Λ
3H
2
and letting Ω = Ω
m
+ Ω
Λ
, it can be
rewritten as
R
2
=
kc
2
H
2
(Ω − 1)
. (2.3)
For universe models having compact spatial sections with non-trivial topology, which we
shall be concerned with in this article, it is clear that any attempt at the discovery of such a
topology through observations must start with the comparison of the curvature radius a nd
3
the horizon r adius (distance) d
hor
at the present time. To calculate the latter, we recall the
redshift-distance relation in the above FLRW settings can be written in the f orm
d(z) =
c
H
0
Z
z
0
h
(1 + x)
3
Ω
m0
+ Ω
Λ0
− (1 + x)
2
(Ω
0
− 1)
i
−1/2
dx , (2.4)
where the index 0 denotes evaluation at present time. The horizon ra dius d
hor
is then defined
as
d
hor
= lim
z→∞
d(z) ,
which, by using (2.3) evaluated at the present time, can be expressed in units of the curvature
radius in the form
χ
hor
≡
d
hor
R
0
=
q
|1 − Ω
0
|
Z
∞
0
h
(1 + x)
3
Ω
m0
+ Ω
Λ0
− (1 + x)
2
(Ω
0
− 1)
i
−1/2
dx . (2.5)
We note that in practice t here are basically t hree types of catalogues which can be used
in order to search f or repeated patterns in the universe and hence the nature of the cosmic
topology: namely, clusters of galaxies, containing clusters with redshifts of up to z ≈ 0.3;
active galactic nuclei (mainly QSO’s and quasars), with a redshift cut-off of z
max
≈ 4; and
the CMBR f r om the decoupling epoch with a redshift of z ≈ 3000. It is expected that future
catalogues will increase the first two cut-offs to z
max
≈ 0.7 [34] and z
max
≈ 6 (by 2005 [35])
respectively. Thus, instead of χ
hor
(for which z → ∞ ) it is observationally more appropriate
to consider the largest distance χ
obs
= χ(z
max
) explored by a given astronomical survey.
3 Topological prerequisites
Let us start by recalling that, contr ary to the hyperbolic and elliptic cases, there is no natural
unit of length for the Euclidean geometry, since in this case the curvature radius is infinite.
Also, it is unlikely that astrophysical observations can fix the density parameter Ω
0
to be
exactly equal to one. Therefore, in this section we shall confine ourselves to nearly flat
hyperbolic and spherical spaces and briefly consider some relevant facts about manifolds
with constant non-zero curvature, which will be used in the following sections.
Regarding the hyperbolic case, there is at present no complete classification of these
manifolds. A number of important results are, however, known a bout them, including the
two importa nt theorems of Mostow [36] (see also Prasad [37]) and Thurston [38]. According
to the former, geometrical quantities of orientable hyperbolic manifolds, such as their (finite)
volumes and the lengths of their closed geodesics, are topological invariant s. Moreover, it
turns out that there are only a finite number of compact hyperbolic manifolds with the
same volume. According to the latter, there is a countable infinity of sequences of com-
pact orientable hyperbolic manifolds, with the manifolds of each sequence being ordered in
terms o f their volumes [38]. Moreover there exists a hyperbolic 3-manifold with a minimum
volume [38], whose volume is shown to be greater than 0.28151 [39].
Hyperbolic manifolds can be easily constructed and studied with the publicly available
software package SnapPea [40] (see e.g. [41]). In this way, Hodgson and Weeks have compiled
4
a census of 11031 closed hyperbolic manifolds, ordered by increasing volumes (see [40, 4 2]).
As examples, we show in Ta ble 1 the first 10 manifolds fr om this census with the lowest
volumes, together with their volumes, as well as the lengths l
M
of their smallest closed
geodesics and their inradii r
−
, which are discussed below. The first two entries (rows)
of Table 1 are known as Weeks’ and Thurston’s manifolds respectively, with the former
conjectured to be the smallest (minimum volume) closed hyperbolic manifold.
M V ol(M) l
M
r
−
m003(-3,1) 0.942707 0.5846 04 0.519162
m003(-2,3) 0.981369 0.5780 82 0.535437
m007(3,1) 1.014942 0.8314 43 0.527644
m003(-4,3) 1.263709 0.5750 79 0.550153
m004(6,1) 1.284485 0.4803 12 0.533500
m004(1,2) 1.398509 0.3661 31 0.548345
m009(4,1) 1.414061 0.7941 35 0.558355
m003(-3,4) 1.414061 0.3648 95 0.562005
m003(-4,1) 1.423612 0.3523 72 0.535631
m004(3,2) 1.440699 0.3615 22 0.556475
Table 1: First ten manifolds in the Hodgson-Weeks census of closed hyperbolic manifolds, together
with their corresponding volu mes, length l
M
of their smallest closed geodesics and their inradii r
−
.
Regarding the spherical manifolds, we recall that the isometry gro up of the 3-sphere is
O(4). However, since any isometry of S
3
that has no fixed points is orientation-preserving, to
construct multiply connected spherical manifolds it is sufficient to consider the subgroups
of SO(4), because for these subgroups the orientation is necessarily preserved. Thus any
spherical 3-manifold of po sitive constant curvature is o f the form S
3
/Γ, where Γ is a finite
subgroup of SO(4) acting f reely on the 3-sphere. The classification of spherical 3-dimensional
manifolds is well known and can be found, fo r instance in [44, 45, 46] (see also [47] for a
description in the context of cosmic topology). In the remainder of this section we shall focus
our attention o n the important subset of such spaces, referred to a s lens spaces, in order to
examine the question of detectability of cosmic topology of nearly flat spherical universes.
Briefly, there are an infinite number of 3-dimensional lens spaces that are globally homo-
geneous and also an infinite number that are only locally homogeneous. In both cases the
lens spaces are quotient spaces of the f orm S
3
/Z
p
where the covering groups are the cyclic
groups Z
p
(p ≥ 2). The cyclic groups Z
p
can a ct on S
3
in different ways parameterized by
an integer parameter, denoted by q, such that p and q are relatively prime integers such that
1 ≤ q < p/2 . These quotients are the lens spaces L(p, q). The action of Z
p
on S
3
gives rise
to globally homogeneous lens spaces only if q = 1; in all other cases the quotient S
3
/Z
p
gives
rise to lens spaces that are globally inhomogeneous.
One can give a very simple description of the actions of Z
p
on S
3
that give rise to lens
5
spaces. In fact, the 3-sphere can be described as the points of the bi-dimensional complex
space C
2
with modulus 1 , thus
S
3
=
n
(z
1
, z
2
) ∈ C
2
: |z
1
|
2
+ | z
2
|
2
= 1
o
,
and any action of Z
p
on S
3
giving rise to a lens space is generated by an isometry of the
form
α
(p,q)
: S
3
→ S
3
(z
1
, z
2
) 7→
e
2πi/p
z
1
, e
2πiq/p
z
2
, (3.6)
where p and q are given as above.
To close this section it is important to recall that since any covering group of spher-
ical space forms is of finite order, all of its elements are cyclic. As a consequence, any
3-dimensional spherical manifold is finitely covered by a lens space. Thus lens spaces play a
special role in studying the topolog y of spherical manifolds, which motivates their employ-
ment below in our study of the problem of detectability of the cosmic topology in nearly flat
spherical universes.
4 Detectability of cosmic topology
Unless there are fundamental laws that restrict the topology of the universe, its detection
and determination is ultimately expected to be an observational problem, at least at a
classical level. As a first step, according to eq. ( 2.5), one would expect the topology of
nearly fla t compact universe to be detectable from cosmological observations, provided the
bound χ
hor
≥ 1 (d
hor
≥ R
0
) holds, since the typical lengths of the simplest hyperbolic and
spherical manifolds are of the order of t heir curvature ra dii (see, for example, the values of
l
M
and r
−
in Table 1).
This is, however, a rough estimate and a more appropriate bound will crucially depend
upon the detailed shape of the universe, our position in it, and on the cosmological parame-
ters. This is particularly true in view of the fact that a crucial f eature of generic 3-manifolds is
their complicated shapes. Moreover, the fact that most 3-manifolds with non-zero curvature
are globally inhomogeneous introduces the possibility of observer dependence (or location
dependence) in the topological indicators, in the sense made precise below, which makes it
likely for these bounds to be location dependent. What is therefore called for are indicators
which are more sensitive (accurate) than χ
hor
and which at the same time take into account
the uncertainty in our location in such compact universes.
A natural way to characterize the shape of compact manifolds is through the size of their
closed geodesics. More precisely, for any x ∈ M, the distance function δ
g
(x) for a given
isometry g ∈ Γ is defined by
δ
g
(x) = d(x, gx) , (4.7)
6
which gives the length of the closed geodesic associated with the isometry g that passes
through x. So one can define the length of the smallest closed geodesic that passes through
x ∈ M as
ℓ(x) = min
g∈
e
Γ
{δ
g
(x)} ,
where
e
Γ denotes the covering group without the identity map, i.e.
e
Γ = Γ \ {id}.
In a globally homogeneous manifold, the distance function for any covering isometry g
is constant, as is the length o f the closed geodesic a ssociated with it. However, this is not
the case in a locally, but non-globally, homogeneous manifold, where in general the length
of the smallest closed geodesic in M is given by
1
l
M
= inf
x∈M
{ℓ(x)} = inf
(g,x)∈
e
Γ×M
{δ
g
(x)} , (4.8)
where inf
x∈M
{ℓ(x)} denotes the absolute (global) minimum of ℓ(x) , and l
M
clearly satisfies
l
M
≤ ℓ(x). The injectivity radius, which is the radius of the smallest sphere inscribable
in M (terminology used in the SnapPea package [40] and in refs. [14, 22]) is then given by
r
inj
= l
M
/2. One can similarly define the injectivity radius at any point x as r
inj
(x) = ℓ(x)/2.
Furthermore, the maximal inradius r
max
−
, can be defined as the radius of t he largest
sphere embeddable in M, and is given by
r
max
−
=
1
2
sup
x∈M
{ℓ(x)} ,
where sup
x∈M
{ℓ(x)} indicates the absolute (global) maximum of ℓ(x) . The maximal in-
radius r
max
−
is half of the largest distance any point in M can be from its closest image.
Clearly in the covering space
f
M, the maximal inradius r
max
−
is also the radius of the largest
sphere inscribable in any fundamental polyhedron (FP) of the set of all possible fundament al
polyhedra of M.
An indicator that has been utilized in most studies rega r ding searches for topological
multiple images, is the ratio of the injectivity radius r
inj
to χ
hor
(see, for example, [14]
and reference therein and also [21] for a similar measure). At first sight this seems to be a
very accurate indicator, since it defines the minimum scale required for multiple images to
be in principle observable in a given multiply connected universe. This can be made more
practical from an observational point of view by taking instead a different indicator, defined
as the ratio of the injectivity radius to the largest distance explored by some astronomical
survey χ
obs
[14], namely
T
inj
=
r
inj
χ
obs
. (4.9)
It turns out, however, that such individual indicators based on topological invariant s do
not encode all the information required to fix the topo lo gy uniquely. Thus, more than one
indicator is o ften necessary in practice. This is the case for the above indicator T
inj
, an
1
We are assuming that the topology of the spacelike section M of our universe is compact, so for a ny
possible manifold a closed geodesic of non-zero minimum length always exists.
7
important limitation of which arises from the fact that generic (globally inhomogeneous)
manifolds are likely to have complicated shapes, making it unlikely fo r the smallest closed
geodesic to pass precisely through our location in the universe. There is also the fact that in
the set of all compact manifo lds there is no lower bound to the length of the smallest closed
geodesics (or equivalently on r
inj
), even though each given manifold does have a lower bound
l
M
.
To partially remedy the first shortcoming above, one can consider, f or an observer O(x)
situated at x ∈ M, the analogous location dependent measure
T
inj
(x) =
r
inj
(x)
χ
obs
. (4.10)
However, there is still a major problem associated with this indicator, since our uncertainty
regarding our location in the universe makes r
inj
(x) uncertain.
As another indicator, we take the ratio of the length of the maximal inradius to χ
obs
,
namely
T
max
=
r
max
−
χ
obs
. (4.11)
A crucial feature of this indicator is that it provides a bound which holds for a ll observers,
regardless of their location and in this sense, therefore, circumvents the problem of our
ignorance regarding our location in the universe. We note also that assuming a fixed astro-
nomical survey for a given compact universe, T
inj
(x) is bounded by the values of r
inj
and
r
max
−
, thus
T
inj
≤ T
inj
(x) ≤ T
max
. (4.12)
It should be noted that one cannot guarantee that the inradius (given in the fourth column
of Table 1) is the maximal inradius for the corresponding hyperbolic manifolds. This is so
because the inradius is calculated for a specific Dirichlet domain, which depends on the
basepoint used for its construction. Thus, the same manifold can have different Dirichlet
domains a nd so different inradii. The maximal inradius r
max
−
of a given manifold M, however,
is unique [43] and corresponds to the largest sphere inscribable in any possible fundamental
polyhedra of M . The package SnapPea chooses the basepoint x at a local maximum of the
inradius, which may or may not be the global (absolute) maximum r
max
−
, but the relation
r
max
−
≥ r
−
(x) clearly holds for all compact hyperbolic (a nd elliptic) manifolds. We shall
show in the next section t hat the indicator T
max
will enable us to exclude certain families of
manifolds (topologies) as the shape of our universe.
Let us close this section by computing r
inj
and r
max
−
for the lens spaces which will be
used in the following sections. Recall that the distance between two points z = (z
1
, z
2
) and
w = (w
1
, w
2
) on S
3
is the angle between these points viewed from the origin of complex
bi-dimensional space C
2
,
cos (d(z, w)) = hz, wi = Re(z
1
w
∗
1
+ z
2
w
∗
2
) ,
8
where h , i indicates the inner product in C
2
, and Re takes the real part o f any complex
expression. Taking w = α
(p,q)
z, with α
(p,q)
as in (3.6), one has
cos (d(z, w)) = cos
2π
p
+
cos
2πq
p
− cos
2π
p
!
|z
2
|
2
. (4.13)
If q = 1, then d(z, w) = 2π/p is p osition independent, showing that L(p, 1) is globally
homogeneous, as anticipated in the previous section. In the general case of a lens space
L(p, q) one obtains
r
inj
=
π
p
and r
max
−
=
πq
p
, (4.14)
since the coefficient of |z
2
|
2
in (4.13) is always non- positive.
5 Detectability of topology of nearly flat universes
In this section we study how the bounds provided by recent cosmological observations can
constrain the set of detectable to pologies a nd how given a non-zero lower bound on |Ω
0
− 1|,
we can exclude certain families of manifolds as viable candidates for the shape of our universe.
To study the constraints on detectability as a function o f Ω
0
, we begin by considering
the horizon r adius (2.5). Figures 1a and 1b show the behaviour of χ
hor
as a function o f
Ω
m0
and Ω
Λ0
. Since in Ω
m0
– Ω
Λ0
plane (hereafter referred to as the parameter space) the
flat universes a re obviously characterized by the straight line Ω
m0
+ Ω
Λ0
= 1, these figures
clearly demonstrate that as |Ω
0
− 1| → 0 then χ
hor
→ 0, hence showing that, for a given
manifold M with non-zero curvature, there are values of |Ω
0
− 1| below which the topology
of the universe is undetectable (χ
hor
< 1) fo r any mix of Ω
m0
and Ω
Λ0
. A crucial feature of
this result is the ra pid way χ
hor
drops to zero in a narrow neighbourhood of the Ω
0
= 1 line.
From the observational point of view, this is significant since it shows that the detection of
the topology of the nearly flat models (Ω
0
∼ 1) becomes more and more difficult as Ω
0
→ 1,
a limiting value favoured by recent observa t io ns.
To show concretely how T
inj
and T
max
can be used to set constraints on the topology of
the universe, we recall two recent sets o f bounds on cosmological parameters [4, 5], namely
(i) Ω
0
= 1.08 ± 0.06, and Ω
Λ0
= 0 .66 ± 0.06, obtained by combining the data from CMBR
and galaxy clusters; and
(ii) Ω
0
= 1.04±0.05, and Ω
Λ0
= 0.68±0.05, obtained by combining the CMBR, supernovae
and large scale structure observations.
The bound (i) exclusively implies positive curvatures for the spatial 3-surfaces, while (ii)
allows negative curvatures as well. We note that even though the precise values of these
bounds are likely to be modified by future observations, t he closeness of Ω
0
to 1 is expected
to be confirmed. We have chosen these bounds here a s concrete examples of how recent
observatio ns may be employed in order to constrain the to pology of the universe, and clearly
similar procedures can be used for any modified future bounds on cosmological para meters.
9
Let us return t o the question of detectability of the topology in the neighbourhood Ω
0
∼ 1
and assume that a particular catalogue which covers the entire sky up to a redshift cut-off
z
max
. For a given universe [with fixed (Ω
m0
, Ω
Λ0
)] this assumption fixes the values of χ
obs
used in (4.9) – (4.1 1).
Suppose that the universe is compact and let r
inj
denote its injectivity radius. Now if
χ
obs
< r
inj
, then the topology of the universe is undetectable by any survey of depth up to
the corresponding z
max
. Note that this assertion holds regardless of our particular position
in the universe, as r
inj
≤ r
inj
(x), for all x. Also, since χ
hor
is the farthest distance at which
events are causally connected to us, one can obtain limits on undetectability by considering
χ
hor
rather than χ
obs
. For example, fo r a given universe [fixed (Ω
m0
, Ω
Λ0
)] any manifold
M whose r
inj
has a value that lies above the bird-like surface in Figure 1 (thus ensuring
T
inj
> 1) is undetectable. The crucial point here is that as we approach the line Ω
0
= 1, the
rapid decrease in the allowed values of χ
obs
will result in a rapid elimination of families of
detectable manifolds (topologies). This is impor tant given that recent observations seem to
restrict the cosmological parameters to small regions near the Ω
0
= 1 line.
As concrete examples o f constraints set by the bound (i), we consider universes that
possess lens space L(p, q) topologies. Using (4.14), and recalling that the topology cannot
be detected if T
inj
> 1 , one finds that for this family of universes t he t opology will be
undetectable if
p <
π
χ
obs
≤ p
∗
, (5.15)
where p
∗
is the smallest integer larger than π/χ
obs
. Using (2.5) and (5.15), we have compiled
in Table 2 the values of p
∗
for different sets of values of (Ω
0
, Ω
Λ0
) contained in bounds (i)
and four cata lo gues with distinct values of z
max
.
As concrete examples, we note that given Ω
0
= 1 .08 and Ω
Λ0
= 0 .66, then according to
Table 2, it would be impossible to detect any multiple images if the universe turns out to
be the projective 3-space L(2, 1) (for which r
inj
= 1.57080 ), or the lens space L(3, 1) (for
which r
inj
= 1.04712 ), as the ent ire observable universe would lie inside some fundamental
polyhedron of M. Similarly, if the universe turns out to be either of the lens spaces L(4, 1),
L(5, 1), L(5, 2), or L(6, 1), it would be impossible to detect its topology using catalogues
of quasars up to z
max
= 6. It is important t o note that from Table 2, one can see that
as Ω
0
approaches unity, p
∗
increases, which implies that an increasing subset of lens space
topologies are undetectable, in agreement with Figures 1. Note also that the greater the
depth z
max
of survey, the smaller is t he number of undetectable topologies, as expected.
But even when z
max
→ ∞ there is still a subset of lens space topologies that remains
undetectable. The key point here is that proceeding in a similar way, one can translate
bounds on cosmological parameters to constraints on allowed topologies (here taken to be
lens topo logies).
So far we have considered the question of detectability of families of manifolds (topologies)
in universes with Ω
0
→ 1 . Alternatively, we may ask what is the region of the parameter
space for which a given topology is undetectable. To this end, we note that for a given
10
z
max
Ω
0
Ω
Λ0
χ
obs
p
∗
1.02 0.60 0.40885 8
∞ 1.08 0.66 0.82728 4
1.14 0.72 1.10777 3
1.02 0.60 0.40088 8
3000 1.08 0.66 0.81135 4
1.14 0.72 1.08670 3
1.02 0.60 0.24375 13
6 1.08 0.66 0.49596 7
1.14 0.72 0.66796 5
1.02 0.60 0.10278 31
1 1.08 0.66 0.20879 16
1.14 0.72 0.28073 12
Table 2: For each value of z
max
the first and third rows correspond to the smallest and largest
values of χ
obs
in the p arameter space given by bounds (i), while the second entry corresponds to
the central value. From this table one can see that the projective sp ace L(2, 1), and the lens space
L(3, 1) are undetectable in a universe with Ω
0
= 1.08, and Ω
Λ0
= 0.66; while for the same values of
the density parameters the lens spaces L(4, 1), L(5, 1), L(5, 2), and L(6, 1) are u ndetectable, using
catalogues of d ep th up to z
max
= 6.
topology (fixed r
inj
) and for a given catalogue cut-o ff z
max
, one can solve the equation
χ
obs
= r
inj
, (5.16)
which is an implicit function of Ω
m0
and Ω
Λ0
. Equivalently using (2.5), equation (5.16) can
be rewritten as the following implicit function of ε
0
≡ 1 − Ω
0
and ε
Λ
≡ 1 − Ω
Λ0
:
χ
obs
≡
q
|ε
0
|
Z
z
max
0
h
(ε
Λ
− ε
0
)(1 + x)
3
+ 1 − ε
Λ
+ ε
0
(1 + x)
2
i
−1/2
dx = r
inj
. (5.17)
Now, note that any point o f the parameter space for which χ
obs
< r
inj
, will lie below the
graph of the solution of eq. (5.17) in the plane (ε
Λ
, |ε
0
|). Thus the points below the solution
curves correspond to universe models for which any topology with injectivity radius larger
than that g iven by the fixed r
inj
are undetectable.
As concrete examples of the constraints set by the bound (ii), we consider the subint erval
of this bound with Ω
0
≤ 1 , correspo nding to nearly flat hyperbolic universes. We will take
r
inj
as the largest value of χ
obs
in this region of parameter space, which can be easily
calculated to be χ
obs
= 0.20125 for z
max
= 6 , and χ
obs
= 0.34211 for z
max
= 3000. We
solved equation (5.17) with these two specific extreme values of χ
obs
as the values for r
inj
,
and the results are shown in Fig. 2 as plots of ε
0
against ε
Λ
. As can be seen the allowed
hyperbolic region of the parameter space, given by ε
0
∈ (0, 0.01] and ε
Λ
∈ [0.27 , 0.37], lies
below both curves, showing that using quasars up to z
max
= 6 , nearly flat FLRW hyp erbolic
11
universes, with the density parameters in this region, will have undetectable topologies, if
their corresponding injectivity radii satisfy r
inj
≥ 0.20125 . Similarly, using CMBR, the
topology of nearly flat FLRW hyperbolic universes with r
inj
≥ 0.342 11 and the density
parameter in t he hyberbolic range of the bound (ii) are undetectable.
In Table 3 we have summarized the restrictions on detectability impo sed by the hyperbolic
subinterval of the bounds (ii) on the first seven manifolds of Hodgson-Weeks census of closed
orientable hyperbolic manifolds (there is no restrictions for the last three in the set of the
first ten smallest manifolds). In Table 3, U denotes that the t opology is undetectable by any
survey of depth up to the redshifts z
max
= 6 (quasars) or z
max
= 3000 (CMBR) respectively.
Thus using quasars, the topology of the five known smallest hyperbolic manifolds, as well
as m009(4,1), are undetectable within the hyperbolic region of the parameter space given
by (ii), while only topologies m007(3,1) and m009(4,1) remain undetectable even if CMBR
observatio ns are used.
M r
inj
quasars cmbr
m003(-3,1) 0.292302 U —
m003(-2,3) 0.289041 U —
m007(3,1) 0.415721 U U
m003(-4,3) 0.287539 U —
m004(6,1) 0.240156 U —
m004(1,2) 0.183065 — —
m009(4,1) 0.397067 U U
Table 3: Restrictions on detectability imposed by the hyperbolic interval of the recent bounds
(ii) for the first seven manifolds of Hodgson-Weeks census. The capital U stands for undetectable
using catalogues of quasars (u p to z
max
= 6) or CMBR (up to z
max
= 3000).
Hitherto we have used T
inj
together with observational bounds on cosmological param-
eters in order to set bounds o n detectability. We shall now employ the indicator T
max
as
a way of excluding certain families of manifolds (topologies) for the universe. Recalling
that for a hyperbolic or spherical compact manifold M, the maximal inradius r
max
−
is the
radius of the largest sphere embeddable in M, then any catalogue of depth z
max
, such that
χ
obs
> r
max
−
, may contain multiple images of cosmic sources or CMBR spots. Thus if we
can be confident that multiple images do not exist up to a certain depth z
max
, then one can
claim that T
max
> 1 (r
max
−
> χ
obs
), a nd therefore any topology not satisfying this inequality
(i.e. those for which r
max
−
< χ
obs
) can be excluded by such observat io ns.
2
As an illustration f or the case of spherical manifolds, suppose again that our universe
2
Obviously it is possible that even exploring a region of the universe deeper than a ball of radius r
max
−
with catalogues of cosmic sources, we do not observe any multiple images. This may be the case when the
selection rules used to build the catalogues do not allow the recording of enough multiple images to have a
detectable topological signal (see [25] for a more detailed discussion on this point).
12
has a lens space topology. If using a catalogue of cosmic objects with redshift cut-off up to
z
max
yields no multiple images, then any lens space satisfying T
max
< 1 (r
max
−
< χ
obs
) can
be discarded as a model for the shape of our universe, while those with T
max
> 1 would be
either detectable or undetectable. Using (4.14) one can see that the discarded lens spaces
L(p, q) are those satisfying
p ≥ p
∗
q
, (5.18)
where p
∗
q
is the smallest integer larger than qπ/χ
obs
. As concrete examples, assume that a
catalogue of cluster of ga la xies with redshift cut-o ff z
max
= 1 was constructed and no multiple
images exist. Using the values of χ
obs
from Table 2, corresponding to the extremes a nd t he
central values of Ω
0
and Ω
Λ0
of the bounds (i), we can obtain values of p
∗
q
corresponding
to q ranging from 1 to 7, as shown in Table 4. Equation (5.18) together with this table
Ω
0
Ω
Λ0
q → 1 2 3 4 5 6 7
1.14 0.72 p
∗
q
→ 12 23 34 45 56 68 79
1.08 0.66 p
∗
q
→ 16 31 46 61 76 91 106
1.02 0.60 p
∗
q
→ 31 62 93 123 154 185 216
Table 4: Len s spaces L(p, q) with p ≥ p
∗
q
are discarded on the basis of observations if there are
no multiple images up to z
max
= 1 in universes with three different set of (Ω
0
, Ω
Λ0
) in the bound
(i). For a given universe [fixed (Ω
0
, Ω
Λ0
)] and for each value of q the corresponding value of p
∗
q
is
given, thus for example, any lens space of the family L(p, 3) with p ≥ 93 could not be the shape of
a universe with Ω
0
= 1.02 and Ω
Λ0
= 0.60 .
make it clear that, for example, any lens space belonging t o the family L(p, 4), with p ≥ 61,
cannot be the shape of an elliptic universe with the central values of the density parameters.
Furthermore, it is clear from Table 4 that as Ω
0
approaches unity, p
∗
q
increases for any fixed
q, which implies that a decreasing subset of lens space topolog ies are excluded.
Alternatively, we can ask how fixing the cosmological parameters results in the exclusion
of a given topology as a possible shape of our universe. To this end, given a fixed value of
r
max
−
, one can solve the equation
χ(z
max
) = r
max
−
, (5.19)
which is analogous to (5.16).
We recall that if one can be certain that there is no multiple images up to a certain depth
z
max
then one can claim that r
max
−
> χ
obs
, and therefore any topology for which r
max
−
< χ
obs
can be excluded by such observations. Thus, for a fixed catalogue depth z
max
, the points
above the graph of the solution of (5.19) in the plane ε
Λ
– ε
0
correspond to cosmological
models for which any topology with r
max
−
< χ
obs
can be excluded.
As an interesting example of application of the indicator T
max
, we consider the hyperbolic
manifolds and recall the important recent mathematical result according to which any closed
orientable hyperbolic 3–manifold contains a ball of radius r
0
= 0.24746 [39]. It therefore
follows that the absolute lower bound to r
max
−
for any compact hyperbo lic manifold is given
13
by r
0
. We solved Eq. (5 .1 9) for r
max
−
= r
0
and with catalogue depths z
max
= 6 and 3000 , a nd
the results are shown in F ig . 3. This fig ure shows that the allowed observatio na l hyperbolic
range of the cosmological parameters given by bounds (ii) intersects the lowest solution curve,
corresponding to the catalo gue with z
max
= 3000. Thus, g iven the current observa t io na l
bounds (ii), if no repeated patterns exist up to z
max
= 3000, there are families of hyperbolic
manifolds t hat can be excluded as the shape of our universe.
3
As concrete examples note
that in such cases the topolo gy of the first ten closed orientable hyperbolic manifolds with
smallest volumes given in Table 1 would not be excluded, as they all po ssess values of
r
max
−
≥ 0.519162, and therefore the corresponding solution curves would lie substantially
above the hyperbolic range of the observational bounds (ii).
Interestingly Fig. 3 also shows that if only quasars are used (redshift up to z
max
= 6)
and no multiple images exist, then no hyperbolic manifolds can be excluded as the shape of
our universe, as the corresponding solution curve lies a bove the current bounds (ii) on the
cosmological parameters.
The above considerations provide examples of the importance of considering families of
manifolds (topologies), rather than arbitra ry individual examples, in looking for the shape
of our universe in the infinite set of distinct possible topologies.
6 Final Remarks
We have made a detailed study of the question of detectability o f the cosmic topology in
nearly flat universes (Ω
0
∼ 1) , which a r e favoured by recent cosmological observations.
Most studies so far have concentrat ed on individual manifolds. Given t he infinite number
of theoretically possible topologies, we have instead concentrated on how to employ current
observatio ns and a number of indicators in order to find families of possible undetectable
manifold (topo lo gies) as well as families of manifolds (topologies) that can be excluded.
We have found that as Ω
0
→ 1, increasing families of possible manifolds (topologies)
become undetectable observationa lly. In this sense the topology of the universe can be said
to become more difficult to detect through observations of multiple images of either cosmic
objects or spots of microwave backgr ound radiation. We have also found that for any given
manifold M with non-zero curvature, there are values of |Ω
0
− 1| below which its topo lo gy
is undetectable (using pattern repetition) for any mix of Ω
m0
and Ω
Λ0
.
We have made a detailed study of the constraints that the most recent estimates of
the cosmological parameters place on the detectable and allowed topo lo gies. Considering
concrete examples of both spherical and hyperbolic manifolds, we find that, given the present
observaio nal bounds on cosmological parameters, there are families of both hyperbolic and
3
Clearly we are assuming here that the universe is multiply connected, and no multiple images exist.
These multiply connected universes are therefore indistinguishable (using pattern rep e tition) from simply-
connected universes with the same covering space, e qual radius, and identical distribution of cosmic sources.
In these cases the scale of multiply-connectedness is greater than the observations depth χ
obs
(z
max
) a nd no
sign of multiply-connectedness will arise.
14
spherical manifolds that remain undetectable and families that can be excluded as the shape
of our universe. We also demonstrate the import ance of considering families o f po ssible
manifolds (topologies), rather than arbitrary individual examples, in search strategies for
the detection of the shape of our universe.
Finally we note that even though the precise values of the recent bounds on the density
parameter used in this paper are likely to be modified by future observations, the closeness
Ω
0
∼ 1 is expected to be confirmed. We have chosen these bounds in this paper as concrete
examples o f how recent observations may be employed in order to constrain the topology of
the universe, and clearly similar procedures can be used for any modified future bounds on
cosmological parameters.
Acknowledgments
We are grateful to Jeff Weeks for his expert a dvice on topology and on the SnapPea program.
Our knowledge of hyperbo lic manifold was greatly enhanced by his very kind and useful
correspondence and comments. We also thank Dr A. Przeworski for helpful correspondence.
Finally we thank FAPERJ and CNPq for the grants under which this work was carried out.
Figure captions
Figure 1. The behaviour of the horizon radius χ
hor
in units of curvature radius, fo r FLRW
models with dust and cosmological constant, g iven by Eq. (2.5), as a function of the
cosmological density parameters Ω
Λ
and Ω
m
. These figures (1a & 1b) show clearly the
rapid way χ
hor
(as well as χ
obs
) falls off to zero for nearly flat (hyperbolic or elliptic)
universes, as Ω
0
= Ω
m0
+ Ω
Λ0
→ 1. In both figures the vertical axes represent χ
hor
,
while the horizontal axes are given respectively (and anti–clockwise), by Ω
Λ
and Ω
m
(in Fig. 1a) and Ω
m
and Ω
Λ
(in Fig. 1b).
Figure 2. The solutions of Eq. (5.17) as plots of ǫ
0
= 1 − Ω
Λ0
(vertical axis) versus ǫ
Λ
=
1−Ω
Λ0
(horizontal axis), with r
inj
taken as the largest values o f the χ
obs
in the range of
cosmological para meters given by the bounds (ii). The upper and t he lower solutions
curves correspond to the catalogues with z
max
= 3000 and z
max
= 6, respectively.
Note that the allowed hyperbolic region of the parameter space given by (ε
0
∈ (0, 0.01]
& ε
Λ
∈ [0.27, 0.3 7]), and represented by the dashed rectangular box, lies below both
curves, showing that using quasars up to z
max
= 6 nearly flat FLRW hyperbolic
universes, with the density parameters in this region, will have undetectable topologies,
if their corresponding inradii satisfy r
inj
≥ 0.20125 . Similarly, using CMBR, with
z
max
= 3000 , t he topology of nearly flat hyperbolic FLRW universes with r
inj
≥
0.34211 will be undetectable.
15
Figure 3. The solutions of Eq. (5.19) fo r r
max
−
= 0.24746, as plots of ǫ
0
(vertical axis) versus
ǫ
Λ
(horizontal axis),for the catalogues with z
max
= 6 and z
max
= 3000. As can be
seen, the allowed observatio nal hyperbolic range of the cosmological parameters given
by bounds (ii), and represented by the dashed rectangular box, intersects the lowest
solution curve, corresponding to the catalogue with z
max
= 3000, thus showing that
given the current observational bounds, there are families of hyperbolic manifolds that
remain undetectable a nd families that can be excluded as the shape of our universe,
if there are no repeated patterns up to z
max
= 3000. On the other hand if quasars
(z
max
= 6) are used, then no hyperbolic manifolds can be excluded as the shape of
our universe, if there are no repeated patterns up to that depth, as the corresponding
solution curve lies above the dashed rectangle.
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18
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
Figure 1a
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
5
Figure 1b
0.009
0.01
0.011
0.012
0.013
0.014
0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4
Figure 2
z_max = 3000
z_max = 6
0
0.005
0.01
0.015
0.02
0.025
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Figure 3
z_max = 6
z_max = 3000
r_max = 0.24746
