Limits of crystallographic methods for detecting space topology

Abstract

We investigate to what extent the cosmic crystallographic methods aimed to
detect the topology of the universe using catalogues of cosmic objects would be
damaged by various observational uncertainties. We find that the topological
signature is robust in the case of Euclidean spaces, but is very fragile in the
case of compact hyperbolic spaces. Comparing our results to the presently
accepted range of values for the curvature parameters, the best hopes for
detecting space topology rest on elliptical space models.

Full text

arXiv:astro-ph/0005515v1 25 May 2000
A&A manuscript no.
(will be inserted by hand later)
Your thesaurus codes are:
12(12.03.4; 12.12.1 )
Limits of Crystallographic Methods for
Detecting Space Topology
Roland Lehoucq
1
, Jean–Philippe Uzan
2,3
, and Jean–Pierre Luminet
4
1
CE-Saclay, DSM/DAPNIA/Service d’Astrophysique, F-91191 Gif sur Yvette cedex, France
email: roller@discovery.saclay.cea.fr
2
Laboratoire de Physique Th´eorique, CNRS–UMR 8627, Bˆat. 210, Universit´e Paris XI, F-91405
Orsay cedex, France
email: uzan@th.u-psud.fr
3
D´epartement de Physique Th´eorique, Universit´e de Gen`eve, 24 quai E. Ansermet, CH-1211
Geneva (Switzerland).
4
D´epartement d’Astrophysique Relativiste et de Cosmologie, Observatoire de Paris, CNRS–
UMR 8629, F-92195 Meudon, France
email: Jean-Pierre.Luminet@obspm.fr
the date of receipt and acceptance should be inserted later
Abstract. We investigate to what extent the cosmic crystallographic methods aimed to
detect the topo logy of the universe using catalogues of cosmic objects would be damaged
by various observational uncertainties. We find that the topological signature is robust
in the case of Euclidea n spaces, but is very fragile in the case of compact hyperbolic
spaces. Comparing our results to the presently a c c epted range of values for the curvature
parameters, the best hopes for detecting space topology rest on elliptical space models.
Key words: large scale structure – topolo gy
Preprint: LPT–O RSAY 0 0/51; UGVA-DPT 00/05–1082
1. Introduction
The search for the topology of the spatial sections of the universe has made tremendous
progress in the past years (see Lachi`eze–Rey and Luminet (1995) for an introduction to
the subject and early references, Luminet and Roukema (1999) and Uzan et al. (1999b)
for a review of the late developments). Methods using two–dimensional data sets, such as
the cosmic microwave background maps planned to be obtained by the MAP and Pla nck
Send offprint requests to: Roland Lehoucq

2 R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods
Surveyor satellite missions, a nd three–dimensional (3D) data sets, such as galaxy, cluster
and quasar surveys with redshifts, have been developped.
Following our previous works (Lehoucq et al. 1996, Lehoucq et al. 1999 and Uzan
et al. 1999a), we focus on the topological informa tion that can be extracted from a
3D catalogue of cosmic objects. The key point of all the topology–detecting methods
is based on the “topological lens effect”, i.e. on the fact that if the spatial section of
the universe has at least one characteristic size smaller than the spatial scale of the
catalogue, then one should observe different images of the same object. The original
crystallographic method (Lehoucq et al. 1 996) used a pair separatio n histogram (PSH)
depicting the number of pairs of catalogue’s objects having the same three– dimensional
separations in the universal covering space, with the idea that spikes should stand out
dramatically at characteristic lengths related to the size of the fundamental domain and
to the holonomies of space. However, following a remark by Weeks (1998 ), we proved
(Uzan et al. 19 99a and Lehoucq et al. 1999) that shar p spikes can emerge in the PSH
only if the holonomies of space are Clifford translations – a result independently derived
by Gomero et al. (1998). As a consequence, the PSH method does not apply to the
detection of topology in universes with hyperbolic spatial sections.
Since then, various generalisations of the crystallographic method were proposed. Fa-
gundes and Gaussman (1 998) sugge sted to map the differe nce s between the PSH of a
simulated catalogue in a compact hyperbolic universe and the PSH in the correspond-
ing simply–connected universe having the same distribution of objects and cosmological
parameters. They noticed sharp osc illations on the scale of the bin width, modulated by
a broad oscillation on the scale of the curvature radius. However in Uzan et al. (19 99b),
we calculated on one ha nd the differential PSH between a simulated distribution in
a compact hyperbolic Weeks space and a similar distribution in the simply–connected
hyperbolic space H
3
, on the other hand the differe ntial PSH between two different dis-
tributions (with the same total number of objects) in the simply–connected hyperbolic
space H
3
. Both curves (fig. 10 of Uzan et al. 1999b) exhibit the same pattern of sharp
and broad oscillations, which shows that the topological significance of such a pattern is
highly doubtful.
Fagundes and Gaussman (1999) next proposed a modified crystallographic method in
which the topological images in simulated catalogues are pulled back to the fundamental
domain before the set of 3D distances is calculated. The distribution of pa ir distances
is ex pected to be peaked around zero. The main drawback of this method is that the
nature of the s ignal s trongly depends on the topological type, on the orientation of the
fundamental domain and on the position of the observer inside the latter. Thus the
pullback method can be useful only if the exact topology is already known. On the other

R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods 3
hand, Gomero et al. (1999) introduced a mean PSH, aimed to reduce the statistical nois e
that may mask the to pological signature. They claimed that such a technique should
detect the contribution of non-translational isometries to the topolog ical signal, whatever
the curvature of space, but the applica bility of their method to real data has not be e n
demonstrated.
In order to improve the signal–to–noise ratio inherent to PSH’s , instead of reduc-
ing the statistical noise we can enhance the topolo gical signa l by co llec ting all distance
correla tio ns into a single index. Such was our purpose in Uzan et al. 1999a, where we
reformulated cosmic crystallography as a collecting-correlated-pairs method (CCP). The
CCP technique rests on the basic fact that in a multiply connected universe, equal dis-
tances appear more often than just by chance, whatever the curvature of space and the
nature of holonomies. We also showed that the extraction of a topological signal dra sti-
cally depends on the rather ac c urate knowledge of the cos mological parameters, namely
the density parameter, Ω
0
, and the cosmological constant parameter, Ω
Λ0
. This is due to
the fact that all cosmic crystallography methods require the determination of 3D separa-
tions betwe e n two any images: observations use redshifts for determining the coordinate
distance in the universal covering space (in addition to the angular positions on the c e -
lestial sphere), and the redshift–distance relation involves the cosmo logical parameters.
Conversely, the detection of a topological signa l would help to deter mine accurately the
curvature parameters (see also Ro ukema and Luminet 1999).
Howe ver it is to be recognized that, in the framework of cosmic crystallography,
the application of numerical simulations to real data rest on two idealized assumptions,
namely:
1. all objects are strictly comoving
2. the catalogues of observed objects (quasars, ga laxy clusters) are complete.
Although quoted in Lachi`eze–Rey and Luminet (1995), the quantitative effects of
such simplifications have not been fully discussed in the literature. In Lehoucq et al.
(1996), the authors qualitatively argued that the distortion due to peculiar velocities
was negligible, and they performed numerical simulations to study the influence o f the
angular resolution of the surveys. In Roukema (1996), the influence of the astrophysical
uncertainties (mainly of sp e c troscopic measurements and of peculiar velocities) on a
method trying to find quintuplets of quasars w ith the same geometry (and thus which
may be topological images) was evaluated. It was shown that, in such a ca se, the most
serious uncertainty comes from the radial peculiar velocities. In Uzan et al. (1999a ), we
discussed the effect of the p e c uliar velocities and of the errors in the spatial position
arising from the imprecisions on the values of the cosmological parameters. To finish, the
errors on the determination of the pos itio n and of the peculiar velocities were discussed

4 R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods
in Roukema (1996), and the way that the constraints on Ω
0
and Ω
Λ0
depend on the
redshifts of multiple topological images and on their r adial and tangential sepa rations
was calculated in Roukema and Luminet (1999).
The goal of the present article is to have a critical attitude on the methods we have
developped so far, by listing all the sources of observational uncertainties and by eval-
uating their effects on the theoretical efficiencies of the PSH method (Lehoucq et al.
1996) and of the CCP method (Uzan et al. 1999a ) respectively. In § 2, we discuss the
nature of uncer tainties and in § 3 the numerical methods used to estimate their effects
on the topological signal. We then quantify the magnitude of each effect in the Euclidean
and hyperbolic cases (§ 4) (we po stpone the case of elliptic spaces to a further study).
In conclusion (§ 5) we compare our results to the performances of current and future
observational programs aimed to detect the topology of the universe.
Notations and descriptions
We keep the notations of our pr e vious articles (Lehoucq et al. 1999, Uzan et al. 1999a ).
The local geometry of the universe is described by a Friedmann–Lemaˆıtre metric
ds
2
= −dt
2
+ a
2
(t)

dχ
2
+ f
2
(χ)

dϑ
2
+ s in
2
ϑdϕ
2

, (1)
where a is the scale factor, t the cosmic time, χ the comoving radial distance, and
f(χ) = (sin χ, χ, sinhχ) according to the sign of the space curvature k = (+1, 0, −1). The
time evolution of a is obtained by solving the Friedmann and the conservation equations
H
2
= κ
ρ
3
−
k
a
2
+
Λ
3
, (2)
˙ρ = −3H(ρ + P ) (3)
where κ ≡ 8πG/c
4
, ρ and P are respectively the matter density and the pressure, Λ is
the cosmological constant and H ≡ ˙a/a is the Hubble parameter (with a dot referring to
a time derivative). We also use the standard para meters
Ω ≡
κρ
3H
2
and Ω
Λ
≡
Λ
3H
2
. (4)
This completely specifies the properties and the dynamics of the universal covering space.
In the following, we assume that we are in the matter do mina ted era, so that P = 0 and
ρ ∝ a
−3
.
The topology of the spatial sections is des c ribe d by the fundamental domain, a poly-
hedron whose faces are pairwise identified by the elements g of the holonomy group Γ
(see Lachi`eze–Rey and Luminet (1995) for the details ).
2. Observational uncertainties
Early methods used to put a lower bound on the size of the universe and based on the
direct recognition of multiple images of given objects – such as our Galaxy (Sokolov

R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods 5
and Schvartsman 1974), the Coma cluster (Gott 1980) – had to face a major drawback
due to the fact that the same object would be seen at different loo kback times. Thus,
evolution effects such as photometric or morphologic changes will, in most cases, render
the identification of objects impossible (see however Roukema and Edge, 1997). Statistical
techniques such as the PSH method (Lehoucq et al. 1999) and the CCP method (Uzan
et al. 1999a) are free from such biases.
The two main sources of uncertainties for all the statistical metho ds trying to detect
the topology of the universe in catalogues of cosmic objects are
(A) the error s in the positions of observed objects, which can be separated into:
(A1) the uncertainty in the determination of the redshifts due to spectroscopic impreci-
sion; such an effect is purely experimental a nd exists even if the objects are stric tly
comoving
(A2) the uncertainty in the position due to peculiar velocities of objects, which induce
peculiar redshift correctio ns
(A3) the uncertainty in the c osmological parameters, which induces an error in the
determination of the radial distance (via the redshift – distance relation)
(A4) the angular displacement due to gravitational lensing by large scale structure.
(B) the inco mpleteness of the catalogue, which has two main origins:
(B1) selection e ffects implying that some objects are missing from the catalog ue
(B2) the partial c overage of the cele stial sphere , due either to the presence of the galactic
plane, or to the fact that surveys are performed within solid angles much less than
4π.
Effects of peculiar velocities
A peculiar velocity has two effects on the determination of the p osition of a cosmic
object:
(i) An integrated effect, coming from the fact that if a galaxy has a proper velocity, then
its true position differs from its comoving one.
(ii) An instantaneous effect, due to the fact that the radial component of the proper
velocity will add to the cosmo logical redshift an extra term, ∆z
Dop
.
Assuming that the vector velocity v of a galaxy is constant, which is indeed a good
enough approximation to estimate the effects of peculiar velocities, a galax y at redshift
z has moved from its comoving position by a comoving distance
δℓ = vτ[z] (5)
where τ[z] is the loo k–back time, obtained by integrating the photon geodes ic equa tion
τ[z] =
1
H
0
Z
1
1
1+z
dx
p
Ω
Λ0
x
2
+ (1 − Ω
0
− Ω
Λ0
) + Ω
0
/x
. (6)

6 R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods
0 1 2 3 4 5
0
2
4
6
8
10
∆ z ( v
pos
||
/c)
−1
z
Ω
0
=1, Ω
Λ
=0
Ω
0
=0.3, Ω
Λ
=0.7
Ω
0
=0.3, Ω
Λ
=0
Fig. 1. Variation of ∆z
pos
in units of (v
k
/c) as a function of the redshift, assuming the
galaxy peculiar velocity is constant.
This reduces to the well known expression
τ[z] =
2
3H
0

1 −
1
(1 + z)
3/2

(7)
when Ω
0
= 1 and Ω
Λ0
= 0.
Now, δℓ can be expressed in terms of a peculiar redshift, ∆z
pos
, and of an angular
displacement, ∆θ
pos
, which depend on the galax y velocity as
∆z
pos
=
v
k
c
cτ[z]
χ
′
[z]
(8)
∆θ
pos
=
|v
⊥
|
c
cτ[z]
χ[z]
(9)
where a prime refers to derivative with re spect to z, v
k
is the component of the velocity v
with respect to the line–of–sight direction γ, and v
⊥
is the non–r adial peculiar velocity,
defined as
v
k
≡ v.γ and v
⊥
≡ v − v
k
γ. (10)
χ[z] is the observer area distance, given by
χ[z] =
c
a
0
H
0
Z
1
1
1+z
dx
p
Ω
Λ0
x
4
+ (1 − Ω
0
− Ω
Λ0
)x
2
+ Ω
0
x
. (11)
In figures 1 and 2, we respectively depict the variations of ∆z
pos
and ∆θ
pos
as a function
of the redshift.
Concerning the instantaneous e ffect ii), the redshift uncertainty ∆z
Dop
can be related
to the galaxy proper velocity as follows.
If we consider the trajectory of a photon x
µ
(s), s being the affine parameter along
the null geodesic, the relation between the emission (E) wavelength λ
E
and the reception
(R) wavelength λ
R
can be expressed a s
λ
R
λ
E
=

k
µ
u
obs
µ

E

k
µ
u
gal
µ

R
(12)

R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods 7
0 1 2 3 4 5
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
∆θ (v
⊥
/c)
−1
[rad]
z
Ω
0
=1, Ω
Λ
=0
Ω
0
=0.3, Ω
Λ
=0.7
Ω
0
=0.3, Ω
Λ
=0
Fig. 2. Variation of ∆θ
pos
as a function of the redshift. As long as z < 5, we have that
∆θ
pos
/1 rad < 0.5(v
⊥
/c). Note also that when z → 0 , ∆θ
pos
/1 rad → (v
⊥
/c).
where k
µ
≡ dx
µ
/ds is the tangent vector to the photon geodesic; u
obs
µ
and u
gal
µ
are
respectively the 4–velocity of the observer and of the gala xy. Neg lec ting the pe rturbations
of the metric and of the matter and focusing on the Do ppler effect, one can easily show
that
k
µ
u
µ
= −
1
a(t)
h
1 + γ
i
v
i
c
i
, (13)
where γ
i
is the direction in which the galaxy is observed and v
i
is its (Newtonian)
velocity. The observed (spectroscopic) redshift, z
obs
, and the cosmological redshift, z
cosm
,
respectively defined by
1 + z
obs
≡
λ
R
λ
E
, 1 + z
cosm
≡
a(t
E
)
a(t
R
)
(14)
are thus related by
1 + z
obs
= (1 + z
cosm
)
"
1 + γ
i
(v
gal
i
− v
obs
i
)
c
#
. (15)
Assuming that we can substract the Earth’s velocity, an error ∆z ≡ z
obs
− z
cosm
in the
determination of z
obs
can be interpreted in terms of a radial peculiar velo c ity given by
∆z
Dop
=
v
gal
k
c
(1 + z
cosm
). (16)
As expected, a galaxy receding from us, i.e. with γ
i
v
gal
i
> 0 will have an additional red-
shift, whereas a galaxy drawing nearer to us will induce a Doppler blueshift correction.
The considerations above are also us eful for discussing the effects of angular dis-
placement ∆θ
gl
due to gravitational lensing by large scale structure (Mellier 1999, van
Waerbeke et al. 2000). A typical va lue ∆θ
gl
∼ 1 ar csec corresponds to a peculiar velocity
a few km.s
−1
as long as the reds hift is less than 5 (as can be seen from Fig. 2). Thus
effect (A4) is much less important than effect (A2).

8 R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods
Effects of catalogue incomple teness
Objects can be missing from a catalogue due to strong evolution effects (this is par-
ticularly the case for quasars), and to absorption of light by intergalactic gas or dust in
some sky directions. Another well known selection effect is the Malmquist bias: statisti-
cal samples of astronomical objects which ar e limited by apparent magnitude have mean
absolute magnitudes which are different than those of distance–limited samples. An ap-
parent magnitude–limited sample contains, if luminosity function has a finite width, some
very luminous objects which, in spite of their large distances, can jump the a pparent-
magnitude limit of the catalo gue. While one looks further and further out one finds more
and more luminous objects. The consequence for an apparent–magnitude catalog (which
is the case with galaxy catalogues) is that dwarf galaxies fade out quickly with distance,
and fina lly at the largest distance s the extremely luminous and rare galaxies are the only
ones which can enter the catalogue.
In addition to effects (B1) and (B2), some noise spikes may appear due to gravita-
tional clustering of objects. The large scale distribution of g alaxies shows a variety of
landscapes co ntaining voids, walls, fila ments and clusters in a complex 3D sponge-like
pattern. For instance, since there are many galaxies in c lusters, the dista nce s associated
to cluster-cluster separations may appear as fake spikes in the PSH for galaxies. Typical
cluster – cluster separations are 130h
−1
Mpc (Guzzo et al. 1999). Such effects currently
occur in N–body simulations that show clustering (Park and Gott 1991). The trouble
can be relieved if galaxy clusters instead of gala xies are used as typical objects for prob-
ing the topo logy, although in that case sup e rcluster-supercluster separations might also
introduce noise spikes (at a lower level).
3. Numerical implementation
In the following, we perform numerical simulations to evaluate separately the magnitudes
of the effects lis ted above. As usual, we start from a random distribution of cosmic objects
in the fundamental domain. In a first step we generate a complete catalogue of comoving
objects by unfolding the distribution in the universal covering space. We refer to this
catalogue as the ideal catalogue since it would correspond to ideal obs e rvations. In a
second step we introduce the various errors (A
i
) − (B
i
) in order to build more realistic
catalogues which depart fr om the ideal one.

R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods 9
3.1. Uncertainties on positions
To study numerically the errors on the positions, we first assume that the cosmological
parameters are known with good enough accuracy, so that we do not discuss (A3). Then
we perform the following calculations.
1. To evaluate the effect of the observational imprecision, we give to each o bject of the
ideal catalogue (built on the assumption tha t the objects are strictly comoving) a
redshift error to be added to its ideal redshift. We assume that the distribution of the
redshift error is Gaussian, with mean value ¯z = 0 and dispersion ∆z.
Note that, since ∆z is absolute, the relative e rror will be less important when we deal
with catalogues of higher redshift objects.
2. To discuss (A2), we associate a peculiar velocity to each point of the catalogue before
unfolding. Hence, there will be correlatio ns between the velocities of two topological
images. It follows that, assuming that the time evolution of the peculiar velocity is
small, the obs e rved velocities of two topological images of the same object can have
different directions but simila r magnitudes (see e.g. Roukema and Bajtlik 1999). As
in the pr e vious case, we assume that the velo city distribution is Gaussian. Moreover,
velocities at different points of space are assumed to be uncor related both in magni-
tude and direction, which is indeed not strictly the case in real data since large sc ale
streaming motions of ga laxies have been observed (Strauss and Willick 1995).
More precisely, to generate the catalog of topological images taking into account the
peculiar velocities, we proceed as follows:
1. We generate a random collection of points M
i
(named original sources ) uniformly dis-
tributed inside the fundamental domain. Each point is assigned a velocity v
i
according
to a Gaussian distribution with mean ¯v and dispersion ∆v.
2. We unfold this catalogue of original sources to obtain the se t of points M
k,i
, imag e s
of M
i
by the holonomies g
k
∈ Γ.
3. Each image M
k,i
has a redshift z
k,i
corresponding to a look–back time τ
k,i
≡ τ(z
k,i
)
calculated with formula (6).
4. The final catalogue accounting for the p e c uliar velocities of the original sources is
obtained by applying the g
k
to M
i
+ v
i
× τ
k,i
for all c ouples (k, i).
Since ∆z can be interpreted either as an uncertainty on the redshift determination or
as due to a r adial peculiar velocity, both errors (A1) and (A2) can be investigated with
the same calculations. The interpretation of ∆z in terms of a radial peculiar velocity is
useful to compare the order s of magnitude of (A1) and (A2). In real data, the error due
to (A2) is expected to be greater than the error due to (A1).

10 R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods
Table 1. The multiplication factor, i.e. the ratio between the total number of entries
in the catalogue and the number of “original” o bjects randomly distributed in the fun-
damental domain, is given as a function of the solid angle of the survey. We have als o
indicated the corre sponding sky coverage.
θ multiplication factor sky coverage (q%)
160
o
6.75 41.3
80
o
1.92 11.7
70
o
1.45 9.0
60
o
1.06 6.7
3.2. Catalogue incompleteness
Starting from an ideal catalogue, we simulate two kinds of incomplete catalo gues:
1. To acco unt for effect (B1), we randomly throw out p% of the objects from the ideal
catalogue. In our various runs we vary p while keeping approximately consta nt the
number of catalogue objects, which means that we must increase the number of
“original” objects in the fundamental domain when p is increasing.
2. To simulate effect (B2), we generate a catalogue limited in solid angle by selecting
only the objects that lie in a beam of aperture θ. The sky coverage is rela ted to θ via
q% =
1
2
(1 − cos(θ/2)) . (17)
Again, we keep constant the number of objects in the c atalogue when we vary θ .
Table 1 below gives typical number s.
As a matter o f fact, mo re than the aperture ang le, the depth of the survey will be
critical for the multiplication factor, since within a b e am of given angle, if the redshift
cut-off is great enough to encompass a distance (in the universal cove ring space) N times
greater than the size of the fundamental domain, at least N topological images will be
exp ected in the beam.
4. General Results
4.1. Euclidean spaces
We first consider the Euclidean case and we apply the PSH crystallographic method
as describ ed in Lehoucq et al. (1996), restricting to a typical situation where Ω
0
=
0.3 and Ω
Λ0
= 0.7. We choose the topology of the universe to be a cubic 3–tor us T
1
(see e.g. Lachi`eze–Rey and Luminet (1995) for description), with identification length
L = 3, 000 Mpc for a Hubble parameter H
0
= 75 km.s
−1
Mpc
−1
. In the various runs, the
number of objects in the catalogue is kept constant at 8500 (this is the order of magnitude

R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods 11
1 1.5 2 2.5 3 3.5 4 4.5 5
0
20
40
60
80
100
120
∆ z
l
× 10
3
z
cut
Fig. 3. Plot of ∆z
l
as a function of the depth of the catalogue, for a cubic hype rtorus of
size 3, 000 Mpc which corresponds to 45% of the Hubble radius (4, 950 h
−1
Mpc) when
Ω
0
= 0.3, Ω
Λ0
= 0.7 and h = 0.75, us ing the PSH method.
Table 2. Values of the critical percentage of rejection above which the PSH spikes
disappear, as a function of the redshift cut–off.
z
cut
1 2 ≥ 3
p
l
[%] 70 80 > 90
of the number of objects in current quasar cata logues). We examine separa tely the effects
of errors in position due to redshift uncertainty ∆z and peculiar velocities ∆v, and the
effects of catalogue incompleteness due to selection effects and partial sky coverage. Each
of these effects will contribute to spoil the sharpness of the topological signal. For a given
depth of the catalogue, namely a redshift cut-off z
cut
, we perform the runs to look for
the critical value of the error at which the topological signal fades o ut.
Figure 3 gives the critical redshift error ∆z
l
above which the topological spikes dis-
appear.
The effect of peculiar velocities is very weak, since we find that the peculiar velocity
must excee d ∆v
l
= 10, 000 km.s
−1
when z
cut
= 1 and ∆v
l
= 40, 000 km.s
−1
when z
cut
= 5
in order to make the topolo gical signal disappear. As already p ointed out in Lachi`eze–
Rey and Luminet (1995), the observed peculiar velocities of galaxies have typical values
much less that ∆v
l
.
The effects of catalogue incompleteness are summarized in tables 2 and 3. The to po -
logical signal would be destroyed only for a very large rejection percentage or a small
aperture angle.

12 R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods
Table 3. Values of the aperture angle below which the PSH spikes disappear, as a
function of the redshift cut–off.
z
cut
2 3 4 5
θ
l
110
o
80
o
70
o
60
o
4.2. Hyperbolic spaces
We now turn to universes with hyperbolic spatial sections and apply the CCP method as
described in Uzan et al. (1 999a). The obtention of a topological signal (the so–called CCP
index) strongly depends on the correct determination of the cosmological parameters. In
Uzan et al. (1999a) we discussed the problems arising fro m the spanning of the cosmolog-
ical parameters space with a required “accuracy bin” ε. Latest independent constraints
on these cosmolog ical parameters from the cosmic microwave background (de Bernardis
et al. 2000), the study of supernovae (Efstathiou et al. 1999), of la rge scale structure a t
z = 2 (Roukema and Mamon, 2000) and of gravitational lensing (Mellier 1999), make
us hope that we can r e strict further the parameters space to apply efficiently the CCP
method. In the following, we assume that the cosmolog ical parameters are known with
the required accura c y.
We choose the topolo gy of the universe to be described by a Weeks manifold (Weeks,
1985) (see also Le houcq et al. (1999) for the numerical implementation of this topology),
assuming the cosmological parameters given by Ω
0
= 0.3 and Ω
Λ0
= 0. In such a case the
number of copies of the fundamental domain within the horizon is about 190. However
our simulated catalogues are much smaller than the horizon volume. We fix the number
of objects to 1300, which is a compromise between a realistic catalogue population and
a reasona ble computing time.
Again we perform the tests by varying the errors ∆z and ∆v around their average
values ¯z = 0 and ¯v = 0 until when the CCP index falls down to noise level.
Contrarily to the Euclidean case, both slight changes in redshift and in peculiar
velocity induce errors in position which dramatically eradicate the topological signal: an
error of only ∼ 50 kms
−1
in velocity and an er ror in redshift of the order of the bin
accuracy ε ∼ 10
−6
drown the CCP index into noise.
The incompleteness effects are less dramatic, as shown in figures 4 and 5. Again, for
a given re ds hift cut–off, we per formed the runs by varying p and θ in order to find the
critical values at which the topological signal disa ppears.

R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods 13
0 10 20 30 40 50 60 70
1
1.2
1.4
1.6
z
cut
=5
0 10 20 30 40 50 60 70
1
1.2
1.4
1.6
normalised CCP index
z
cut
=4
0 10 20 30 40 50 60 70
1
1.1
1.2
1.3
p [%]
z
cut
=3
Fig. 4. Plot of the CCP–index (normalized to background noise) as a function of the
rejection percentage of objects p
l
, for various values of of the depth of the catalogue. The
topological signal disappears when the rejection percentage is grea ter than p
l
.
140 160 180 200 220 240 260 280 300 320 340 360
1.2
1.3
1.4
1.5
z
cut
=5
140 160 180 200 220 240 260 280 300 320 340 360
1.1
1.2
1.3
1.4
normalised CCP index
z
cut
=4
140 160 180 200 220 240 260 280 300 320 340 360
1
1.05
1.1
1.15
1.2
1.25
θ [deg]
z
cut
=3
Fig. 5. Plot of the CCP–index as a function of θ
l
for various values of the depth of the
catalogue. The topological signal disappears when the aperture angle falls down below
θ
l
.
5. Conclusions and perspectives
Our numerical results have now to be compared with the precisions of present exp eri-
mental 3D data, and to the performances of observational programs started or expected
to be achieve d in the next decade.
At present day, a typical precision prac tical for the spectroscopic uncertainty is ∆z ∼
0.001 for an object such as a quasar. In clusters, spectros c opic redshifts can be found
very precisely for individual galaxies.

14 R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods
Concerning peculiar velocities, the typical dispersion velocity is 1000 km.s
−1
in rich
clusters. The X-ray velocity of the peak of the X-ray distribution would also provide a
way to estimate the true cluster redshift, including the peculiar velocity of the cluster
as a whole. For a quasar, a c onservative upper limit to the peculiar velocity, assuming
the quasar to be at the centre of a galaxy, can be taken as ∆z ∼ 0.002. So, from an
exp erimental point of view, the uncertainties on the redshifts will be dominated by effect
(A2), namely the peculiar velocities, rather than by the spectroscopic imprecision.
The main limitation of present 3D sa mples is the small volume of existing redshift
data. Future surveys will significantly improve both the redshift cut–off and the sky
coverage. For instance, the Sloan Digital Sky Survey (SDSS) (Loveday 1998) will map in
detail o ne-quarter of the entire sky, determining the po sitions and absolute brightnesses
of more than 100 million celestial objects. It will also measure dista nce s to more than a
million galaxies and quasar s. More precisely, SDSS will map a contiguous π steradians
area in the north Galactic cap, up to a limiting magnitude 23 for two thirds of the
observing time, together with three southern stripes centred at RA α = 5
◦
and with
central declinations of δ = +15
◦
, 0
◦
and −10
◦
for the remaining one third of the time.
Concerning the distance determinations, 10
6
galaxies and 10
5
quasars will be observed
sp e c troscopically with a res olution ∆z/z ∼ 5.10
−4
. The main galaxy sample will consist
of ∼ 900, 000 galaxies up to magnitude 18, with a median redshift z ∼ 0.1. A second
galaxy sa mple will consist in ∼ 100, 000 luminous red galaxies to magnitude 19.5 with a
median redshift z ∼ 0.5. Precision on redshifts can be estimated for the reddest galaxies
to ∆z ∼ 0.02. A sample of ∼ 100, 000 quasars will be obse rved, an order of mag nitude
larger than any existing quasar catalogue. The complete survey data will become public
by 2005. We can also mention the ESO–VLT Virmos Deep Survey Project (Lef`evre 2000),
a co mprehensive imaging and r e dshift survey of the deep universe based on more than
150 000 redshifts.
In the field of X–ray obs e rvations, the XMM satellite (Arnaud 1996) will provide deep
insight on X–ray galaxy clusters and active galac tic nuclei. Also, the XEUS project under
study by ESA (Parmar et al. 1999) will be a long–term X–ray observatory at 1 keV, with
a limiting sensitivity around 250 times better than XMM, allowing XEUS to study the
properties of galaxy groups at z = 2 a nd active galactic nuclei at z
<
∼ 3.
In the present paper we have investigated how the various observational uncertainties
will s poil the topologic al signa l expected to arise in the ideal situation when crystallo-
graphic methods are applied to complete c atalogues of perfectly comoving objets with
zero proper velocities and whose 3 D– positions are known with infinite accuracy. By nu-
merical simulations we have introduced random errors fo r each possible uncertainty, and
we varied the parameters to determine the limits at which the topological signal va nis hes.

R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods 15
Our numerical calculations of the spoiling effects due to the various uncertainties
(A
i
) − (B
i
) clearly show that the crystallogra phic methods are stable (in the sens e that
the topological signal is robust when data depart from the ideal ones) in the Euclidean
case, but highly unstable in the hyperbolic case. Indeed in a small multi–connected flat
space, realistic values of peculiar velocities o f objects, errors in redshift determinations
and partial sky coverage will not make the PSH method to fail. This can be understood
by the fact that the topological images of a given object are related together by Clifford
translations which enhance the topological signal. On the contrary, in a c ompact hyper-
bolic model, holonomies ar e not Clifford translations. The topo logical signal, built as a
CCP index, is destroyed as soon as small er rors are introduced in the position of objects,
due either to peculiar velocities or to redshift measure ment imprecision.
The same kind of critical analysis should be made with the 2D topology–detecting
methods based on the analysis of CMB data. For instance, in the pairs of matched circles
method (Cornish et al. 1998), it would be necessary to investigate how deviations to the
ideal situation, such as a non zero thickness of the last scattering surface or peculiar
motions of the emitting primordial plasma regions, would alter the pattern of perfectly
matched circles.
To conclude, let us comment on the future of observational cosmic to po logy, at
the light of observational constraints on the curvature parameters recently provided by
BOOMERanG and MAXIMA balloon measurements (de Bernardis et al. 2000, Hanany et
al. 2000). Under specific assumptions such as a cold dark matter model and a primordial
density fluctuation power s pectrum, the range o f values allowed for the energy-density
parameter Ω = Ω
0
+ Ω
Λ0
is restricted to 0.88 < Ω < 1.12 with 95% c onfidence. This
means that the curvature radius of space is as least as great as the radius of the observ-
able universe (delimited by the last scattering s urface). Such results, if accepted, leave
open all three cases of space curvature as well as mos t of multi–connected topologies.
A strictly flat space is quite improbable. Even inflationary scenarios predict a value
of Ω asymptoca lly close to 1, but not strictly equal. From a topolog ic al point of view, a
strictly Euclidean space would be interesting since we have shown that the PSH method
is robust enough to provide a to pological signal even when realistic uncertainties on the
data are taken into account.
For compact hype rbolic spaces, if we accept the recent observational constraints
0.88 < Ω
0
+ Ω
Λ0
< 1, a number of spaceforms such as the Weeks or the Thurston
manifolds still have topological lengths smaller than the horizon size (Weeks: SnapPea).
Howe ver the topological lens effects would be weaker in spaceforms with Ω close to 1
than in spaceforms with Ω ∼ 0.3 (see figs. 2 and 3 of Lehoucq et al. 1999). Further more,
only the CCP method can be applied for detecting the topolog y, and the present article

16 R. Lehoucq, J-P. Uzan & J-P. Luminet: Limits of Crystallographic Methods
shows that the topological signal will fall to noise level as soon as uncertainties smaller
than the experimental ones are taken into ac c ount in the simulations.
Eventually, elliptical spaceforms appear to be the most interesting case. On a theo-
retical point of view, as far as we know, no inflationary model is able to drive the density
parameter to a value greater tha n 1, so that if space happened to be really elliptical, new
models should be built in order to explain, e.g., the primordial fluctuations s pectrum or
the horizon problem. From a topological point of view, since the volumes of (all closed)
spaceforms are not bounded below (the order of the holonomy group can be a rbitrarily
large), one can always find an elliptical space which fits into the Hubble radius even
if 1 < Ω
0
+ Ω
Λ0
< 1.12. On the other hand, the holonomies of such spaces are Clifford
translations, so that we can hope to apply the robust PSH method to detect a to pological
signal. This w ill be the purp ose of our subsequent paper.
Acknowledgements: It is a pleasure to thank N. Aghanim, R. Juszk iew icz , Y. Mellier
and B. Roukema for discussions.
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