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arXiv:astro-ph/0205155v1 10 May 2002

Mon. Not. R. Astron. Soc. 000, 1–?? (2002)Printed 1 February 2008 (MN LATEX style file v1.4)

Topology of the universe from COBE-DMR; a wavelet

approach

G. Rocha1,2, L. Cay´ on3, R. Bowen4, A. Canavezes5,2, J. Silk4, A. J. Banday6

and K. M. G´ orski7,8

1Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, United Kingdom

2Centro de Astrof´ ısica da Universidade do Porto, R. das Estrelas s/n, 4150-762 Porto, Portugal

3Instituto de F´ ısica de Cantabria, Fac. Ciencias, Av. los Castros s/n, 39005 Santander, Spain

4Department of Physics, Denys Wilkinson Building, University of Oxford, Keble Road, Oxford OX1 3RH, United Kingdom

5Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, United Kingdom

6Max-Planck Institut fuer Astrophysik (MPA), Karl-Schwarzschild Str.1, D-85740, Garching, Germany

7European Southern Observatory (ESO), Karl-Schwarzschild Str.2, D-85740, Garching, Germany

8Warsaw University Observatory, Poland

1 February 2008

ABSTRACT

In this paper we pursue a new technique to search for evidence of a finite Uni-

verse, making use of a spherical mexican-hat wavelet decomposition of the microwave

background fluctuations. Using the information provided by the wavelet coefficients at

several scales we test whether compact orientable flat topologies are consistent with

the COBE-DMR data. We consider topological sizes ranging from half to twice the

horizon size. A scale-scale correlation test indicates that non-trivial topologies with

appropriate topological sizes are as consistent with the COBE-DMR data as an in-

finite universe. Among the finite models the data seems to prefer a Universe which

is about the size of the horizon for all but the hypertorus and the triple-twist torus.

For the latter the wavelet technique does not seem a good discriminator of scales for

the range of topological sizes considered here, while a hypertorus has a preferred size

which is 80% of the horizon. This analysis allows us to find a best fit topological size

for each model, although cosmic variance might limit our ability to distinguish some

of the topologies.

1 INTRODUCTION

It is commonly assumed in cosmology that the Universe is in-

finite. While this assumption greatly simplifies calculations,

there is nothing forbidding a geometry of space-time with

a finite topology. General Relativity specifies the local cur-

vature of space-time but the global geometry still remains

undefined.

The CMB contains a wealth of information about our

Universe. One expects that observation of CMB anisotropies

will yield information on local geometry parameters such as

the expansion rate of the Universe, its dark matter con-

tent as well as its nature, luminous matter content, the lo-

cal curvature, etc. It is also expected that this same source

might help us infer the global topology of the universe since

topological scales are on the order of the largest observ-

able scales. The COBE-DMR data has been used to place

constraints on some non-trivial topologies such as flat and

limited open topologies. (A summary of such constraints as

well as a brief account of several methods used is given in

Section 3). Meanwhile an improvement in the sensitivity of

CMB experiments has stimulated a renewed interest in the

search for the space-time topology.

Here we aim to constrain these finite topologies using

spherical Mexican Hat wavelets. In Section 2 we start by

giving a a brief account of the topological models investi-

gated in our analysis. In Section 3 we give a summary of

other methods used to search for the topology of the uni-

verse. In Section 4 we describe the wavelet-based technique

applied in the analysis presented here, and in particular,

how we calculate the correlation between the wavelets maps

and the subsequent statistical analysis. We show that these

wavelets allow one to discriminate between some topologies

while simultaneously constraining the best fit topological

size for each of the models, within the range of topologi-

cal sizes and scales studied in this work. In Section 5. we

present the methodology used as well as the results. Finally

the conclusions are given in Section 6.

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2 TOPOLOGY

Topology is the mathematical framework concerned with

studying continuity. Therefore a topologist does not distin-

guish between a circle, a square and a triangle, or between

a doughnut and a coffee cup for example. However topology

distinguishes a coffee cup from a bowl. Here we are interested

in the study of the imprints left on the CMB by multicon-

nected universes. A simply connected manifold is a surface

where any loop (closed path) can be continuously deformed

into a point; if not the manifold is said to be multiconnected.

Recent high resolution experiments, such as Boom-

erang, MAXIMA and DASI (Netterfield et al. 2001, Lee et

al. 2001, Pryke et al. 2001), seem to favour an approximately

flat observable universe. Consequently we restrict here our

attention to non-trivial topologies with a flat geometry, and

even if Λ models are favoured by those data we will not con-

sider them in the analysis presented here. For a comprehen-

sive review on topology see among others Lachieze-Rey &

Luminet 1995, Luminet & Roukema 1999, Levin 2001. The

description given here follows closely that of Levin, Scanna-

pieco & Silk 1998 and Scannapieco, Levin, & Silk 1999.

There are 6 compact orientable flat topologies. As de-

scribed below, these topologies can be constructed using a

parallelepiped or a hexagonal prism as the finite fundamen-

tal domain whose opposite faces are glued together (ie. iden-

tified) in some particular way. These identifications can be

represented by tiling the space with copies of the fundamen-

tal domain. Four of these topologies are obtained from the

parallelepiped and two from the hexagon.

• Model 1: The simplest one is the hypertorus (3-torus)

which is obtained from a parallelepiped with pairs of op-

posite faces glued together. In other words this manifold

is built out of a parallelepiped by identifying x → x + h,

y → y + b and z → z + c.

The other three manifolds are variations of the hypertorus

involving identifications on opposite faces of a twisted par-

allelepiped.

• Model 2: One of these has opposite faces identified with

one pair rotated through the angle π (here called π twist

torus);

• Model 3: the second identifies opposite faces with one

rotated by π/2 (the π/2 twisted torus),

• Model 4 : the third one is obtained by proceeding with

the following identifications: (x,y,z) → (x+h,−y,−z) cor-

responding to translation along x and rotation around x by

π; next (x,y,z) → (−x,y + b,−(z + c)) corresponding to

translation along y and z followed by rotation around y by

π; and finally (x,y,z) → (−(x + h),−(y + b),z + c) trans-

lation along x,y,z followed by rotation around z by π (here

called the triple twist torus).

Two other topologies are built out of a hexagon by identi-

fying the three pairs of opposite sides while in the z direction,

• Model 5: the faces are rotated relatively to each other

by 2π/3 (2π/3 hexagon)

• Model 6: and by π/3 (here called π/3 hexagon).

For large scales, the temperature fluctuations in the

CMB are mainly due to the potential fluctuations on the

last scattering surface, the so called Sachs-Wolfe effect, SW

(Sachs & Wolfe 1967). This effect corresponds to a redshift-

ing of the photon as it climbs out of the potential well on

the surface of last scattering, and also to a time dilation

effect which allows us to see them at a different time from

the unperturbed photons (Coles & Lucchin 1995, Efstathiou

1990). Its expression is given by:

?∆T

where Φ is the gravitational potential and ∆η is the confor-

mal time between the decoupling time and today, ie, the

radius of the Last Scattering Surface (LSS) in comoving

units. The observable universe is defined by the diameter

of the LSS 2∆η. The angular power spectrum is defined as

Cl = ?|alm|2?, where the multipole moments alm are the

coefficients of the standard expression of temperature fluc-

tuations of the CMB on the celestial sphere, in terms of an

expansion in spherical harmonics:

T

(ˆ n)

?

sw=1

3Φ(∆ηˆ n) (1)

∆T

T

(α,φ) =

∞

?

l=2

l?

m=−l

almYlm(α,φ) (2)

where (α,φ) are the polar coordinates of a point on the

spherical surface. For flat (Ω = 1) scale invariant (n = 1)

models with no contribution from a cosmological constant

the SW effect gives rise to a flat power spectrum, i.e.,

l(l + 1)Cl =constant, where the quantity l(l + 1)Cl is the

power of the fluctuations per logarithmic interval in l. Its

shape is altered if one assumes a tilted initial power spec-

trum with a power law, P(k) = Aknor if one incorporates

a cosmological constant, or assumes other than flat models.

In the last two cases another source of anisotropy appears,

the so-called integrated Sachs-Wolfe effect, ISW, and is due

to the fact that potential fluctuations are no longer time

independent.

Now the gravitational potential can be decomposed into

eigenmodes:

Φ =

?+∞

−∞

In general, inflationary scenarios predict a Gaussian dis-

tribution of fluctuations independent of scale. This implies

that theˆΦ?kare obtained from a Gaussian distribution with

?ˆΦ∗

an ensemble average and Pφ is the predicted power spec-

trum. A Harrison-Zeldovich (ie scale invariant) spectrum

corresponds to Pφ= constant. For a compact manifold, the

identifications on the fundamental domain are expressed in

terms of boundary conditions on Φ (see appendix A). Hence

the continuous?k becomes a discretized spectrum of eigenval-

ues. In general the temperature fluctuations in a compact,

flat manifold become:

δT

T

−∞<kx,ky,kz<∞

with additional relations on theˆΦ?k. Once these eigenmodes

are known one can obtain the angular power spectrum Cl

(see Levin, Scannapieco & Silk 1998, for example). The set of

restricted discrete eigenvalues and relations between the co-

efficients Φkxkykzset by their specific identifications on the

fundamental domain, are extensively given in Levin, Scan-

napieco & Silk 1998 and Scannapieco, Levin, & Silk 1999.

For the sake of completeness we give in Appendix A, a list

of these solutions.

d3?kˆΦ?kexp(i∆η?k.ˆ n)(3)

?kˆΦ?

k′? =

2π

k3Pφ(k)δ3(?k −?k′), where the brackets indicate

∝

?

ˆΦkxkykzexp(i∆η?k.ˆ n) (4)

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Topology of the universe from COBE-DMR; a wavelet approach

3

Figure 1. From left to right in the first row: Simulations for a

Torus, π twist Torus; in the second row: π/2 twist Torus, triple

twist Torus; and in the third row: π/3 Hexagon, 2π/3 Hexagon,

for topological scale equal to half the horizon size (ie j=0.5). The

maps are in HEALPix pixelization with Nside=32 and COBE-

DMR resolution.

The topological identifications performed on the fun-

damental domain alter the CMB power spectrum in that:

1) the boundary conditions give rise to a discretised set of

eigenmodes for the power spectrum 2) the compact spaces

are anisotropic and all except for the hypertorus are inhomo-

geneous, giving rise to anisotropic Gaussian CMB fluctua-

tions, hence non-Gaussian fluctuations (Inoue 2001, Ferreira

& Magueijo 1997) the existence of a cutoff in perturbations

at scales larger then the topological scale in a given direc-

tion. Point 2) implies that the correlation function between

any two points on the LSS is no longer simply a function of

the angular separation. These finite universes will also ex-

hibit spatial correlations due to geometrical patterns formed

by repetition of topologically lensed cold and hot spots. In

our analysis we investigate these spatial correlations.

The simulations presented here are based on the re-

lations listed in Appendix A, and were produced only for

the SW effect, that is, taking into consideration the sources

of CMB anisotropies on large angular scales alone. To find

out whether the topology scale is an observable, in other

words, whether its effects can be seen on the CMB, one

compares the topological scale (in-radius) with ∆η. There-

fore the number of clones of the fundamental domain (the

parallelepiped for example) that can be observed in a com-

pact universe may be calculated by the number of copies that

can fit within the LSS. In Fig.1 we show simulated maps for

the 6 topologies for a topological scale equal to half of the

horizon size, j = 0.5 (j ∝ kx, see appendix A for details) In

Fig.2 we plot a Torus with topological scale of 0.1, equal to

and 10 times the horizon size (j = 0.1,1,10 respectively) as

well as an infinite universe for comparative purposes.

Figure 2. Torus for j=0.1,1,10 ie topological scale of 0.1, equal,

10 times the horizon size and an infinite universe.

3 SEARCHING FOR TOPOLOGY: OTHER

METHODS

The search for topology can generally be divided into two

methods, direct statistical methods and geometric methods.

Statistical methods can be used to analyze properties

ranging from the correlation function to the angular power,

Cl, spectra of topology simulations and compare them to

the known data. Geometric methods encompass the search

for circles in the sky, look for planes or axes of symmetry,

search for pattern formation via maps of correlations, for

example, a map of antipodal correlations of the CMB maps,

to name a few. The statistical analysis makes use of averages

over the sky and therefore might not be sensitive to the

inhomogeneity and anisotropy characteristic of these non-

trivial topologies.

Ever since data from COBE-DMR was released, calcu-

lations and statistical analyses have been performed to see

how consistent a topologically compact universe is with the

actual data. Initial work comparing Cl spectra from topo-

logical simulations of compact flat spaces, with COBE-DMR

data seemed to set a lower limit on the topology scale at

80% of the horizon radius (40% of the horizon diameter)

(Stevens, Scott & Silk 1993, Levin, Scannapieco & Silk 1998,

Scannapieco, Levin & Silk 1999), still possibly detectable

by some of the methods mentioned above, but eliminating

the intriguing possibility of a very small topological scale. A

likelihood analysis was performed to compare the Cl’s of the

model with the full COBE-DMR range of the angular power

spectrum. Though compact topologies do not give rise to

isotropic temperature fluctuations, therefore any likelihood

analysis is bound to be ambiguous. While observations of

the power spectrum on large angular scales are used to put

limits in the minimum scale of the topology, the cosmic vari-

ance does not allow us to differentiate between some of these

models (for example between a hexagonal prism and a hy-

pertorus). Therefore the angular power spectrum is a poor

measure of topology. Several other methods have been put

forward, in particular the geometric methods.

One of the first geometric searches for topology placed

the lower limit even higher (see de Oliveira-Costa & Smoot,

1995). They showed that the most probable hypertorus was

1.2 times larger than the horizon scale. A more anisotropic

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Rocha et al.

topology was also considered such as a hypertorus with 1 or 2

dimensions smaller than the present horizon (T1and T2re-

spectively)(de Oliveira-Costa, Smoot & Starobinsky 1996).

Their analysis is based on the fact that an anisotropic hy-

pertorus will exhibit a symmetry plane or a symmetry axis

in the pattern of CMB fluctuations. They developed a small-

ness statistic to measure the level of symmetry that would

result in these cases. Their analysis constrained T1and T2

models to be larger than half the radius of the LSS, in their

small dimensions. A circle method has been applied after-

wards giving pessimistic results.

While for compact flat models the eigenmodes decom-

position has been done successfully, its analytical calculation

for compact hyperbolic models is actually impossible. Meth-

ods to constrain these spaces, include brute force numerical

calculation of the eigenmodes (Inoue 1999, Inoue, Tomita &

Sugyiama 2000, Cornish & Spergel 2000, Aurich 1999, Au-

rich & Marklof 1996) the method of images construction of

CMB maps (Bond, Pogosian & Souradeep 1998, 2000) and of

course geometric methods. Using the eigenmodes computed

numerically a comparison of the Cl’s with COBE-DMR data

alone seems to not rule out any of the CH models (Aurich

1999, Aurich & Steiner 2001, Inoue 1999, Cornish & Spergel

2000, Bond, Pogosyan & Souradeep 2000) Although a com-

parison of the full correlation function does not survive con-

sistency according to the method of images analysis, the

correlation function can be calculated using the method of

images without the explicit knowledge of eigenmodes and

eigenvalues (see Bond, Pogosyan & Souradeep, 2000). They

find anisotropic patterns and a long wavelength cutoff in

the power spectrum. The anisotropic patterns correspond

to spikes of positive correlation between a point on the LSS

and its image. The suppression of power for compact hy-

perbolic models is in part compensated by the ISW effect

and therefore is not so noticeable as in flat models. They

used a Bayesian analysis to compare the CH models with

the COBE-DMR data and found that CH models are in-

consistent with the data for most orientations. Only a small

set of orientations fit the data better than the standard infi-

nite models. Their analysis showed the inadequacy of using

analysis based on the C′

error bars on the C′

the models topologically connected, although there is some

controversy over the interpretation of these error bars (In-

oue 2001), and the finding of a spatially dependent variance

due to the global breaking of homogeneity. Some other work

explored the idea of a correct orientation for the CH model

and used the cold and hot spots of COBE-DMR to infer the

orientation of this manifold (Fagundes 1996, 2000).

As pointed out these direct methods are model depen-

dent in that they depend for example of the choice of mani-

fold, the location of the observer, the orientation of the man-

ifold, the local parameter values and finally of the statistics

used.

Another of the geometric methods for searching for a

finite topology is to look for circles in the sky (Cornish,

Spergel & Starkman 1998). These circles would be a matched

pair, not circles of identical temperature in the sky, but

circles of identical temperature fluctuations. Each topol-

ogy would have its own distinct pattern of matched circles,

and presumably if all the circle pairs could be identified,

the topology of the universe could be inferred from there.

ls alone, by pointing out the large

ls due to the large cosmic variance in

This method applies to all multiconnected topologies and

it is model independent. To get the correlated circles a sta-

tistical approach is used. Work was then done to include

the Doppler effects in the analysis (Roukema 2000). Using

COBE-DMR data this method has also been applied to con-

strain asymmetric flat 3-spaces (Roukema 2000). Intriguing

as this possibility is, it has its difficulties. Circles are ob-

scured in practice however by the ISW effect, which since

it occurs after the last scattering surface would not be cor-

related in the circles, and by the fact that the galactic cut

will interfere with many of the circles, making them more

difficult to detect (Levin 2001). Even if these obstacles were

surmounted, this method is useless if the topological scale is

larger than the horizon size, for if the surfaces of last scat-

tering of the observer do not intersect, there are no circles.

Cornish and Spergel applied this test to simulated maps to

conclude that the ISW interfered in the statistic giving rise

to a poor match for circle pairs. This effect affects mainly the

low order multipoles and therefore an improvement is to be

expected from future satellite missions. Bond Pogosian and

Souradeep found good matches even at COBE-DMR resolu-

tion, but found that at low values of the density parameter

Ω0 when the ISW effect contribution is larger the corre-

lations along circles get worse. Hence the COBE-DMR is

not useful for detecting circles in hyperbolic universes since

the ISW will contribute to the multipole range covered by

COBE-DMR. However this approach might still be useful

for flat spaces without a cosmological constant.

Another geometric method that has been used to search

for possible topology is to look for pattern formation. Peri-

odic boundary conditions imposed by the topological iden-

tifications would lead to the formation of distinct patterns

at individual modes on the last scattering surface. These

patterns are not readily apparent because the superposition

of many modes drowns out what would be obvious in one

mode. Looking for correlations is one of the best methods to

search for patterns. An antipodal correlation, as described in

Levin et al. 1998, ie, correlation between points on opposite

sides of the last scattering surface, would give a monopole,

that is, no correlation at all, in a simply connected uni-

verse since these points should be out of causal contact. In

a multi-connected universe though, they could in fact be

very close to each other (even may be the same point) and

show significant correlation. Hence this technique searchs

for ghost images and in doing so uncovers the symmetries

of space. These correlated maps satisfy a desirable property

since they are model independent. In theory it would then

be possible to extrapolate from the patterns and symmetries

in the anti-podal maps to the global topology. The resolu-

tion of the spots needed for the correlation calculation is far

higher than that obtained by COBE-DMR, somewhere on

the Silk damping scale, but this could be obtained by MAP

or Planck (Levin & Heard, 1999). This method runs into

the same problems as the circles in the sky method, partic-

ularly with regards to noise and disentangling correlations

on the last-scattering surface from the uncorrelated ISW and

foreground effects that obscure it. A real space statistic is

needed to overcome these obstacles, as proposed below.

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Topology of the universe from COBE-DMR; a wavelet approach

5

4 WAVELETS AS SCALE INFORMATION

PROVIDERS

We propose to apply a different method which makes use

of wavelets to detect these patterns on the CMB sky. It has

been shown that this tool is quite helpful in assessing the

contamination of CMB data by discrete radio sources, as

well as in detecting and characterizing non-Gaussian struc-

ture in maps of the CMB. These functions are widely used

in analysis and compression of data because they are com-

putationally efficient and have a better localization in both

space and frequency than the usual Fourier methods.

Wavelet deconvolution of an image provides information

at each location of the contribution of different scales. It is

our aim in this paper to distinguish topologies through the

imprint left on the CMB. The geometry of space will intro-

duce spatial correlations in the CMB and thereby generate

patterns in the observed maps. Full use of all the informa-

tion provided by wavelets would consist of interpreting the

localized values of wavelet coefficients at each scale. Our ap-

proach will however be a statistical one. The information

gathered at each scale and each location will be combined

into the scale-scale correlation between scales k and k′de-

fined as

C(k,k′) = (1/Np)

Np

?

i=1

[∆T/T(k,i)][∆T/T(k′,i)])

where Np represents the number of pixels. ∆T(k,i) is the

value of the wavelet coefficient at pixel i after convolution

with a wavelet of size k. That is, if pixel i corresponds to

direction?b, then

∆T/T(k,?b) =

?

d? x∆T/T(? x)Ψ(k,?b;? x),

where Ψ(k,?b;? x) = (1/k)ψ(|? x −?b|/k) and ψ is the so called

“mother” wavelet.

Planar Daubechies wavelets have been implemented in

the HEALPix pixelization⋆(G´ orski, Hivon & Wandelt 1999),

and applied to projections of simulated CMB sky maps of

flat models with a non-trivial global topology (Canavezes et

al. 2000). Their results are consistent with our own, shown

in section 5. Only two spherical wavelets have been up to

now implemented on the HEALPix pixelization. The spher-

ical Haar wavelet has been used in the quad-cube pixeliza-

tion (Tenorio et al. 1999) as well as in the HEALPix one

(Barreiro et al. 2000). The spherical Mexican Hat was im-

plemented on the HEALPix pixelization by Cay´ on et al.

2001 and Mart´ ınez-Gonz´ alez et al. 2001 (details about the

spherical Mexican Hat wavelet can be found in those pa-

pers). The scales that can be studied by the Haar wavelet

are powers of two times the pixel size. The scale-scale cor-

relation function therefore will not be calculated in evenly

discretized scales. On the contrary, the spherical Mexican

Hat allows us to determine wavelet coefficients at any scale,

although successive scales will not be independent. More-

over, Mart´ ınez-Gonz´ alez et al. 2001 have shown that this

wavelet is better suited to detect certain non-Gaussian fea-

tures than the spherical Haar.

The scale-scale correlation (defined as above or as the

⋆http://www.eso.org/science/healpix/

Figure 3. Simulation of a Torus with j=0.1 ie with a topological

scale 10% of the horizon size (left hand side), all the others cor-

respond to the simulated map convolved with the Mexican hat

wavelet for scales (from left to right): 290,470,830,arcminutes.

Figure 4. Simulation of a Torus with j=1 ie with a topological

scale of the order of the horizon size (left hand side), all the others

correspond to the simulated map convolved with the Mexican hat

wavelet for scales (from left to right): 290,470,830 arcminutes.

correlation of the squared values of ∆T/T) has been previ-

ously used to search for non-Gaussianity in the COBE-DMR

data (Barreiro et al. 2000, Cay´ on et al. 2000). COBE-DMR

values of this quantity are found to be consistent with Gaus-

sianity. Previous efforts have been dedicated to comparing

the power spectrum of the COBE-DMR data with predic-

tions from different topologies as mentioned in the previous

section. The power spectrum can be viewed as the correla-

tion of two equal scales. Power spectra for flat topologies

such those studied in this paper are presented in Scanna-

pieco et al. 1999.

5 RESULTS

The simulations presented here consider only the SW ef-

fect. The CMB temperature fluctuations are obtained using

Equation 4 of Section 2, for the restricted set of discrete

eigenvalue spectra. There are additional relations between

the coefficients of the eigenmodes (listed in Appendix A), set

by the specific identifications on the fundamental domain.

The simulations consider topologies with equal-sided funda-

mental domains, ΩΛ = 0 and a Harrison-Zeldovich Gaussian

power spectrum (see Section 2). We simulate a whole sky

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Rocha et al.

topology map in HEALPix pixelization which is then con-

volved with the COBE-DMR beam (Wright et al. 1994). The

COBE-DMR noise is then added to the simulated map and

a galactic mask (as defined in Banday et al. 1997) is applied.

After monopole and dipole removal the map is renormalized

to the COBE-DMR value of C10. The simulated map is then

convolved with the Mexican Hat wavelet coefficients for a set

of scales. Due to the high computational cost of each topol-

ogy simulation we present statistical results for only 100

simulations in each case. The error we introduce is studied

in the case of model 2, for j = 1.5 for which we ran 500 sim-

ulations (see comments later in this Section). In Fig.3 and

Fig.4 we plot the wavelet maps, ie the topology maps con-

volved with a Mexican Hat wavelet, at different scales. We

then compute the scale-scale correlations between successive

scales of such maps. In Figs. 5 to 10 we plot these scale-scale

correlations of the 6 topologies for 3 of the 5 different topo-

logical scales considered j=0.5,0.8,1.0,1.5,2.0 (ie. topological

scale 0.5,0.8,1,1.5,2 times the horizon size respectively), for

Gaussian simulations of an infinite universe and its values

for the COBE-DMR data.

For most of the topologies the shape of the curve

changes with topological scale, with its peak moving to-

wards larger angular scales with increasing topological scale.

The peak possibly arises due to a cutoff of power at large

scales (corresponding to the low l cutoff in the Cl). This

feature however allows the wavelet analysis to distinguish

characteristic scales for most of the models. For small uni-

verses these curves differ substantially from the standard

infinite model curve, hence allowing us to discriminate be-

tween them. As the universe becomes bigger the shape of

the correlation curves tend to that of an infinite Gaussian

universe as should be expected. Nevertheless the mean value

of the correlations, hence the cosmic variances, for a large

universe does not tend to that of an infinite universe, rather

being substantially larger. The cosmic variances are smaller

for a small universe as expected. This might indicate that we

did not consider a large enough upper limit on the topolo-

gial sizes of the universes considered. It is to be expected

that as the size of the universe increases the mean value and

hence the error bars of the scale-scale correlations decrease

and tends to that of an infinite universe. This might be-

come apparent for sizes larger then twice the radius of the

LSS which is the upper limit of the range of sizes considered

here.

The triple twist torus seems to behave in a quite dis-

tinct way when compared with the other models. The scale-

scale correlation peaks towards the highest angular scale for

all topological sizes with roughly the same shape but with

similar uncertainties. Therefore this analysis does not allow

one to differentiate topological scales for this model as eas-

ily. This behaviour might result from the limited range of

scales studied. The wavelet analysis has only been applied

for scales up to 1800 arcminutes. It might happen that the

triple twist torus exhibits a correlation curve shape similar

to that of the other models peaking towards scales larger

then our upper limit. This would explain why we do not see

such a peak. For this particular model one might need to go

to larger topological sizes to get similar results to the other

models, although this seems to be in variance with the anal-

ysis based on the power spectrum which indicates that all

6 compact, orientable and flat topologies are cut off at the

same wavelength as the hypertorus (Levin, Scannapieco &

Silk 1998).

To estimate the error introduced in our analysis by the

use of 100 simulations, we plot in Fig. 11 the scale-scale

correlation curve and its 1σ error bar for 100 and 500 simu-

lations. This plot shows that a larger number of simulations

slightly increases the mean value of the scale-scale correla-

tion and increases the variance, although these differences

are small when compared with the error bar size and there-

fore for intercomparative purposes a number of 100 simula-

tions is sufficient to draw the relevant conclusions.

We then compare these scale-scale correlations for our

topology simulations with the values for COBE-DMR as well

as with the scale-scale correlations for a Gaussian model

(ie, infinite universe with a trivial topology). The data is

compared with the simulations by performing a χ2statistic:

χ2

k,k′(i,j)) = (Ci,j(k,k′) − Ccobe(k,k′))2/σ2

and χ2(i,j) =

?

scale-scale correlation averaged over the 100 simulations for

topological model i with topological size j, between scales

(k,k′); Ccobe(k,k′) is the observed COBE-DMR value; and

σi,j(k,k′) is the standard deviation for model i with topolog-

ical size j, for scales (k,k′) computed from the 100 simulated

CMB maps.

The presence of foregrounds in the COBE-DMR maps

might substantially affect the calculation of these correla-

tions. To tackle this issue we compute the scale-scale cor-

relation for the COBE-DMR coadded map after foreground

(ie. COBE-DIRBE) correction. These correlations for the

COBE-DMR both before and after foreground correction

are plotted in Fig.12, along with the correlations for a torus

(model 1) and an infinite universe. From Fig.12 we conclude

that the differences between the two COBE-DMR curves

are too small to be noticeable in our statistical analysis. We

do not expect the foregrounds correction to affect our final

results.

In order to quantify these results, we compare the mean

scale-scale correlation and the 1σ scatter with the values

of the scale-scale correlation for each simulation via the χ2

values. We get the distribution of the χ2for each model

and each topological size. We then compute the probability

of getting a χ2value larger than or equal to that of the

COBE-DMR value, and the results are displayed in Table

1. For all but the triple twist torus, the smallest universe

with j = 0.5 is excluded. For the triple twist torus (model

4), however, this is the favoured case with a probability of

70%. The hypertorus (model 1) with a universe size 80% of

the horizon size has a probability of 47% while the π and

the π/2 twist torus (model 2 and 3 respectively) with a size

equal to the horizon size has a probability of 53% and 50%

respectively. Finally, the 2π/3 and the π/3 twist hexagon

(models 5 and 6 respectively) with size equal to the horizon

size have probabilities of 36% and 38% respectively.

In making use of the χ2statistic, we assume that the

error bars follow a Gaussian distribution. To assess the effect

of non-Gaussianity on these simulations of the distribution

of the scale-scale correlations, we plot in Fig.13 some his-

tograms for a scale of 500 arcminutes for models π and triple

twist torus, to conclude that indeed these are not Gaussian-

distributed. This might slightly bias the scale-scale correla-

tion to larger values. Hence assuming symmetric error bars

i,j(k,k′), (5)

k,k′χ2

k,k′(i,j), where Ci,j(k,k′) is the

c ? 2002 RAS, MNRAS 000, 1–??

Page 7

Topology of the universe from COBE-DMR; a wavelet approach

7

results in a slightly worse fit for the larger universes consid-

ered here.

The best fit topological scale varies with the model but

in the majority a value of j = 1.0 is favoured, ie a compact

flat universe with a topological scale approximately equal to

the horizon size. Although the data seems to prefer a triple

twist torus with a topological size of j = 0.5, i.e. a universe

half the horizon size, it is known that the wavelet analy-

sis is not a good scale discriminator for this model. This

result probably just reflects the fact that in the χ2anal-

ysis comparable 1σ error bars for all sizes are used while

the average value for a universe with half the horizon size

is closer to the COBE-DMR values. These error bars are

comparable to the larger uncertainties for the other models

and this would explain the smaller χ2value obtained. Apart

from this model, the data seems to prefer a topological size

equal to the horizon size for all but the hypertorus model.

For the hypertorus, a topological scale of j = 0.8 ie with

a universe size 80% of the horizon size is preferred, with a

47% probability. Therefore our analysis seems to be in agree-

ment with previous power spectrum analysis (Stevens, Scott

& Silk 1993, Scannapieco, Levin & Silk 1999), although our

favoured topological sizes are curiously close to the mini-

mum value allowed by the power spectrum analysis alone.

The most intriguing result is that for a triple twist torus:

according to the Cl analysis, a model as small as half the

radius of the LSS is excluded while here it is favoured. As

mentioned above, this is mainly due to the large cosmic vari-

ance and the similarity of the shape of the correlation curve

for all topological sizes. Actually the infinite universe corre-

lation curve is compatible with this model and the χ2anal-

ysis favours the finite universe mainly due to a larger cosmic

variance. With the finite universes displaying a larger cos-

mic variance, it is not surprising that we get a better χ2fit

for these models. Therefore the cosmic variance presents a

serious problem when we try to compare these models with

an infinite universe, but still allows us to distinguish some

of these non-trivial models. These results were also obtained

for a limited range of topological sizes.

Another point of interest is to investigate how a mis-

match of the assumed power spectrum compared to the ac-

tual one can lead to incorrect conclusions about the pre-

ferred topology. One should expect that changing the am-

plitude and shape of the power spectrum would change the

amplitude and shape of the scale-scale correlations, hence

resulting in a different χ2fit.

Cay´ on et al. found that the COBE-DMR data was con-

sistent with Gaussian universes. Here we find that finite

models with the appropriate topological size are as com-

patible with COBE-DMR data as an infinite universe (see

table 4).

6 CONCLUSIONS

In this paper, we have used spherical Mexican Hat wavelets

to study the scale-scale correlations exhibited by non-trivial

topologies. Wavelet deconvolution of an image provides in-

formation about the contribution of different scales at each

location in the map. Here we perform a statistical analy-

sis by combining the information gathered at each location

and each scale into the scale-scale correlation between two

Modeljχ2

Prob(%)

1

1

1

1

1

0.5

0.8

1.0

1.5

2.0

321.11

7.13

14.87

15.16

26.29

0

47

20

19

15

2

2

2

2

2

0.5

0.8

1.0

1.5

2.0

1017.09

41.05

7.32

17.79

12.41

0

6

53

27

14

3

3

3

3

3

0.5

0.8

1.0

1.5

2.0

188.46

34.76

8.51

15.58

21.93

2

12

50

27

16

4

4

4

4

4

0.5

0.8

1.0

1.5

2.0

4.14

7.62

14.55

24.80

16.56

70

39

21

16

21

5

5

5

5

5

0.5

0.8

1.0

1.5

2.0

42.68

50.28

14.06

16.04

21.70

06

6

36

24

12

6

6

6

6

6

0.5

0.8

1.0

1.5

2.0

116.78

10.01

9.60

16.32

30.97

1

38

38

21

13

Gaussian- 10.83 45

Table 1. COBE-DMR χ2values listed on third column. The last

column displays the probability of obtaining such value or higher

for each model and each topological size.

successive scales as defined in Section 4. We compute the

scale-scale correlations for 100 simulations of flat compact

topologies. These values were compared to the correspond-

ing quantities for an infinite topology and the values ob-

tained for COBE-DMR data. To quantify and extract con-

straints we perform a χ2analysis of the COBE-DMR data.

The χ2values obtained for the preferred topological sizes

when compared with an infinite universe might occur due to

the large error bars displayed by the scale-scale correlations

of these models and not because these spaces do actually fit

better the data. If correct this would mean that we are cos-

mic variance limited. The wavelets technique does not seem

to be a good scale discriminator for a triple twist torus for

the range of topological sizes considered here, and one might

have to go to larger universe sizes to get useful information

on this particular model. Apart from this model, the data

seems to prefer a topological size equal to the horizon size for

all but the hypertorus model. For the hypertorus, a topo-

logical scale of j = 0.8 ie with a universe size 80% of the

horizon size is preferred, with a 47% probability. The π and

the π/2 twist torus with a size equal to the horizon size is

preferred with a probability of 53% and 50% respectively.

c ? 2002 RAS, MNRAS 000, 1–??

Page 8

8

Rocha et al.

500 10001500

0

j=0.5

j=1.0

j=2.0

Gaussian

COBE

scale ( Model 1: Torus )

Figure 5. Mean ± 1σ scale-scale correlations for Model 1

Finally the 2π/3 and the π/3 twist hexagon with size equal

to the horizon size is favoured with a probability of 36%

and 38% respectively. These results need to be interpreted

in view of the topological size and scales range used in this

analysis.

One expects that methods based on a pattern forma-

tion technique will help us to further constrain these mod-

els, in particular with forthcoming satellite experiments such

as MAP and Planck. In ongoing work we are including the

small angle sources of CMB temperature fluctuations in sim-

ulating the non-trivial topologies to allow a similar analysis

of Planck or MAP data, among others.

7 ACKOWLEDGMENTS

We would like to thank Janna Levin and Evan Scanna-

pieco for fundamental comments on the simulations of the

non-trivial topologies and Lance Miller, Proty Wu, Pedro

Ferreira, Dmitri Novikov and Pedro Avelino for useful dis-

cussions. We would also like to aknowledge the use of the

HEALPix pixelization. GR would like to acknowledge a Lev-

erhulme fellowship at the University of Cambridge. GR and

LC thank the Dept. of Physics of the University of Oxford

for support and hospitality during the progression of this

work.

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8 APPENDIX A

In a compact topology the gravitational potential is re-

stricted to a discrete spectra of eigenmodes:

φ(? x) =

?

?k

φ?kei?k.? x

(6)

with additional relations imposed on the φ?k. For example

consider the case of the hypertorus. The identifications on

the cube are expressed in terms of three boundary condi-

tions:

φ(x,y,z) = φ(x + h,y,z) = φ(x,y + b,z) = φ(x,y,z + c)(7)

The first boundary condition gives rise to the following re-

lation:

e−ikxx= e−ikx(x+h)

(8)

which imply that kx =

two other boundary conditions restrict the set of discrete

eigenvalue spectrum to:

2π

hj. This in conjunction with the

?k = 2π(j

h,wb,nc), (9)

with j,w,n running over all integers. There is a minimum

eigenvalue and hence a maximum wavelength fitting inside

the fundamental domain given by the paralelepiped: kmin =

2πmin(1

c) and λmax = max(h,b,c). For a π twist torus

the set of restricted eigenvalue spectrum is

h,1

b,1

?k = (2πj

h,2πw

b,πnc)(10)

with the additional relation on the coefficients of the eigen-

modes

Φjwn = Φ−j−wneiπn,

for a π/2 twist torus these are

(11)

?k = (2πj

h,2πw

h,πn2c) (12)

with

Φjwn

=Φw−jneinπ/2

Φ−w−jneinπ

Φ−wjnei3nπ/2,

(13)

=

=

while the triple twist torus’s discrete spectrum is

?k = (πj

h,πwb,πnc), (14)

with

Φjwn

=Φj−w−neiπj

Φ−jw−neiπ(w+n)

Φ−j−wneiπ(j+w+n).

(15)

=

=

The last two compact flat spaces have an hexagonal funda-

mental domain. The potential can be written as:

Φ=

?

n2n3nz

exp[i2π

Φn2n3nzeikzz

(16)

×

h[n2(−x +

1

√3y) + n3(x +

1

√3y)]]

For the 2π/3 hexagon one has

?k = (2πj

h,2πw

h,2πnz

3c), (17)

with

Φn2,n3,nz

=Φn3,−(n2+n3),nzei2πnz/3

Φ−(n2+n3),n3,nzei4πnz/3.=

While for the π/3 hexagon the spectrum becomes

?k = (2πj

h,2πw

h,πnz

3c),(18)

with

Φn2,n3,nz

=Φ(n2+n3),−(n2−n3)/√3,nzeiπnz/3

Φn3,−(n2−n3),nze2iπnz/3

Φ−n2,(n2−n3/√3,nzeiπnz

Φ−(n2+n3),n3,nzei4πnz/3.

(19)

=

=

=

c ? 2002 RAS, MNRAS 000, 1–??