Constraining topology in harmonic space

0.45 MB·English

by  M. Kunz

Constraining topology in harmonic space
M. Kunz,1,2,∗ N. Aghanim,3 L. Cayon,4 O. Forni,3 A. Riazuelo,5 and J.P. Uzan5
1D´epartement de Physique Th´eorique, Universit´e de Gen`eve,
24 quai Ernest Ansermet, CH-1211 Geneva 4, Switzerland
2Astronomy Centre, SciTech, University of Sussex, Brighton, BN1 9QJ, UK
3Institut d’Astrophysique Spatiale (IAS), Bˆatiment 121, F-91405,
Orsay (France); Universit´e Paris-Sud 11 et CNRS (UMR 8617)
4Department of Physics, Purdue University, 525 Northwestern Ave., West Lafayette, IN 47907-2036, USA
5Institut d’Astrophysique de Paris, Universit´e Pierre et Marie Curie,
Paris VI, 98 bis bd Arago, 75014 Paris, France
(Dated: February 5, 2008)
We consider several ways to test for topology directly in harmonic space by comparing the mea-
6 sured aℓm with the expected correlation matrices. Two tests are of a frequentist nature while we
computetheBayesianevidenceasthethirdtest. Usingcorrelationmatricesforcubicandslab-space
0
0 tori,westudyhowthesetestsbehaveasafunctionoftheminimalscaleprobedandasafunctionof
2 thesize oftheuniverse. Wealso applythemtodifferentfirst-yearWMAPCMB mapsand confirm
that the universe is compatible with being infinitely big for the cases considered. We argue that
n
there is an information theoretical limit (given by the Kullback-Leibler divergence) on the size of
a
thetopologies that can be detected.
J

8 PACSnumbers: 98.80.Es,04.20.Gz,98.70.Vc,98.80.Bp
2

2 I. INTRODUCTION analysis by Cornish et al of the first-year WMAP maps
v
basedona searchfor matchingcircles has notfound any
4
evidence for a non-trivial topology [10]. However, it is
6 GeneralRelativityhasbeenextremelysuccessfulinde-
difficult to quantify the probability of missing matching
1 scribing the large-scale features of our universe. But the
0 circles, and other groups have claimed a tentative detec-
global shape of space-time is a quantity that is not de-
1 tion of circles at scales not probed by Cornish et al (see
termined by the local equations of General Relativity.
5 e.g. [11,12]). Inthispaperwestudyadifferentapproach
An intriguing possibility is therefore that our universe is
0 which can in principle yield both an optimal test as well
/ much smaller than the size of the particle horizontoday.
h as a rigorous assessment of the fundamental detection
In the standard model, the universe is described by
p power of the CMB for a cosmic topology.
- a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) type Instead of working directly with the observed map of
o metric which is homogeneousandisotropic. If the topol-
CMB temperature fluctuations, we expand the map in
r
t ogyoftheuniverseisnottrivial,thenwearedealingwith terms of spherical harmonics,
s
aquotientspaceX/ΓwhereX isoneoftheusualsimply
a
v: cboonlince)catenddFΓLRisWasdpiasccreest(espahnedrificaxle,dE-puocilnidteafrneeorsyhmypmeer-- T(x)= aℓmYℓm(x), (1)
i ℓ,m
X try groupthat describes the topology. This construction X
does not affect local physics but changes the boundary
r where x are the pixels. Both the pixels and the expan-
a conditions (see eg. [1, 2] and references therein).
sion coefficients a are random variables. In the sim-
ℓm
This could potentially explain some of the anomalies plestmodelsoftheearlyuniverse,theyaretoagoodap-
found in the first-year WMAP data. For example, the proximation Gaussian random variables, an assumption
perturbations of the cosmic fluids need to be invariant that we will make throughout this paper. Their n-point
underΓ. Thereforethelargestwavelengthofthefluctua- correlation functions are then completely determined by
tions in the CMB cannotexceed the size of the universe, thetwo-pointcorrelationfunction. Thehomogeneityand
and so the suppression (and maybe the strange align- isotropy of the simply-connected FLRW universe addi-
ment) of the lowest CMB multipoles might be due to a tionally requires the two-point correlation of the a to
ℓm
non-trivial topology [3, 4, 5, 6, 7, 8]. Additionally, the be diagonal,
last scattering surface can wrap around the universe. In
this case we receive CMB photons, which originated at haℓma∗ℓ′m′i=Cℓδℓℓ′δmm′. (2)
the same physicallocationonthe lastscatteringsurface,
from different directions. Observationallythis would ap- The symmetry group Γ will introduce preferred direc-
pear as matched (correlated) circles in the CMB [9]. An tions, which will break global isotropy. This in turn in-
duces correlations between off-diagonal elements of the
two-point correlation matrix. In this paper we study
methods to find such off-diagonal correlations. Such
∗Electronicaddress: [email protected] a test is complementary to the matched-circle test of
2
[9,10],andiftheinitialfluctuationsareGaussianthenit selection. We then talk about the issues that we ne-
canusealltheinformationpresentintheCMBmapsand glected in this paper, and finish with conclusions.
soleadtooptimalconstraintsonthesizeoftheuniverse. The appendices look in more detail at how the cor-
Investigating the amount of information introduced into relation and the likelihood method differ, and how their
the two-point correlation matrix by a given topology al- underlyingstructurecanbeusedtodefine“optimal”esti-
lows us to decide from an information theoretical stand- mators. We alsodiscusshowselectinganextremumover
point whether the CMB will ever be able to constrain all orientations can be linked to extreme value distribu-
that topology. tions, which allows us to derive probability distribution
We will use the following notation: We often combine functions that can be fitted to the data for quantifying
the ℓ and m indices to a single index s ℓ(ℓ+1)+m confidence levels. We finally consider a distance func-
≡
andmixbothnotationsfrequently. Thenoisycorrelation tion on covariance matrices, motivated by the Bayesian
matrix given by the data is Ass′ ≡ asa∗s′. We will write evidencediscussion,andstudyitsapplicationtothecom-
the correlation matrix which defines a given topology as parison between different topologies.
ss′. This is the expectation value of the two-point cor-
B
relationfunctionfora thatdescribeauniversewiththat
s
topology. II. DETECTING CORRELATIONS
All the simulations in this paper are based on a flat
ΛCDMmodelwithΩ =0.7,aHubbleparameterofh= Aprioriitisverysimpletocheckwhethertherearesig-
Λ
0.67,aHarrison-Zel’dovichinitialpowerspectrum(n = nificantoff-diagonaltermspresentinthetwo-pointcorre-
S
1) and a baryon density of Ω h2 = 0.019, as described lationmatrix: Onejustlooksattermswithℓ=ℓ′ and/or
in [13, 14]. With this choice obf cosmological parameters m = m′. But the variance of the aℓm is too6large as we
6
we finda Hubble radius1/H 4.8Gpc while the radius canobserve only a single universe. When computing the
0
≈
of the particle horizon is Rh 15.6Gpc. We will denote Cℓ weaverageoveralldirectionsm. This averagingthen
a toroidal topology as T[X,Y≈,Z] where X, Y and Z are leads to a cosmic variance that behaves like 1/√ℓ. But
the sizes of the fundamental domains, in units of the now we have to consider each element of the correlation
Hubble radius. As an example, T[4,4,4] is a cubic torus matrix separately, leading to a cosmic variance of order
of size (19.3Gpc)3. The volume of such a torus is nearly 1 for each element. The matrix is therefore very noisy
halfthatofthe observableuniverse. Thediameter ofthe andwe need to “dig out” the topologicalsignalfrom the
particle horizon is about 6.5/H . But we should note noise. Furthermore, if we detect the presence of signifi-
0
that there are non-zero off-diagonal terms in ss′ even cantoff-diagonalcorrelations,westillneedtoverifythat
B
for universes that are slightly larger than the particle they are due to a non-trivial topology and not to some
horizon. other mechanism that breaks isotropy.
We have a range of correlation matrices at our dis- A natural approach to the problem is then to use the
posal so far. Two of them are cubic tori with sizes 2/H expectedcorrelationmatrixforagiventopologyasakind
0
(T[2,2,2])and4/H (T[4,4,4]). Forthesetwowehavethe offilter. Tothisendwecomputeacorrelationamplitude
0
correlation matrices up to ℓ = 60 (corresponding to λwhichdescribeshowclosetwomatricesare. Wedothis
max
s = 3720). We also have two families of slab spaces. by minimising
max
The first one, T[X,X,1], has one very small direction of
size 1/H0. The secondone, T[15,15,X],has two largedi- χ2[λ]= (Ass′ −λBss′)2 (3)
rectionsthatareeffectivelyinfinite. Bothgroupsinclude ss′
X
all tori with X = 1,2,...,15, and we know their corre-
lations matrices up to ℓ = 16 (or s = 288). The where Ass′ = asa∗s′ is the correlation matrix estimated
correlationmatricesanalmyasxedinthispapmearxdonotcontain fromthedataandBss′ theonewhichcontainsthe topol-
ogy that we want to test. For a good fit we expect to
theintegratedSachs-Wolfecontributions(cfdiscussionin
find λ 1 while for a bad fit λ 0.
section VIIIC). We c≈an easily solve dχ2/dλ=≈0 and find that
This paper is organised as follows: We start out by
matchingthemeasuredcorrelationstoagivencorrelation λ= ss′Ass′Bss′ (4)
mbeattwreixe.nWdieffethreenntschoorwreltahtaiotnamsimatirliacrespocwanerbteoadcihsiteinvegduibshy P ss′(Bss′)2
using the likelihood. In general we do not know the rel- minimises Eq. (3). As Pwe know that we will have to
ative orientation of the map and the correlation matrix, compare our method against maps from an infinite uni-
andwediscusshowtodealwiththisissuenext. Wethen versewiththesamepowerspectrum,wedonotsumover
present a first set of results from this analysis, before the diagonal s = s′ (which corresponds to ℓ = ℓ′ and
embarking on a simplified analysis of the WMAP CMB m=m′)toimprovethesignaltonoise. Thiscorresponds
data and toroidal topologies. to replacing the correlation matrix through
B → B−D
So far the methods were all of a frequentist nature. where isadiagonalmatrixwiththepowerspectrumon
D
Using the likelihoodwe canalsostudy the evidence fora the diagonal. If the power spectrum is constant so that
giventopology,whichis the Bayesianapproachtomodel = C 1 then the eigenvectors of the new correlation
D ×
3
matrix arethe same as thoseofthe originalone,andthe The summation depends again if we keep the diagonal
eigenvalues are replaced by ǫ(i) ǫ(i) C. In this case elementsornot. Forawhitenedmap,theresultsimplifies
→ −
they will no longer be positive. to σ2 =2/U. In a finite universe,
∞
WecouldalsointroduceacovariancematrixinEq.(3).
2
In the presence of noise this may be useful. In this σ2 = tr( ∗ ∗), (10)
study we will assume throughout an idealised noise-free B U2 BB BB
and full-sky experiment for simplicity. At any rate the
howevernowweneedtobemorecarefulifwediscardthe
WMAP data will be cosmic variance dominated at the
diagonal elements, as then
low ℓ that we consider here, see section VIIIA. Ne-
glecting the noise contribution, the covariance matrix is 2
Cqq′ = q q′ where q = s,s′ . But as the correlation σB2 = U2 (Bs1s2Bs∗2s3Bs3s4Bs∗4s1) (11)
matricehsBarBe ailready expec{tatio}n values, we end up with s16=sX2,s36=s4
amatrixthathasasinglenon-zeroeigenvalueǫ= 2.
qBq Table I shows the expectation values of variances for a
If we invert this singular matrix with the singular value
selectionoftopologies,computedwiththeseformulas. It
P
decomposition (SVD) method (setting the inverse of the
may be surprising that the variance of λ for an infinite
zero eigenvalues to zero) and minimise the resulting ex-
universe depends on the test-topology. However, Eq. (4)
pression for the χ2, we find again Eq. (4).
depends on even if the a do not. The variance is a
ℓm
It is straightforwardto compute the expectation value B
measureofhowdifferent isfromthediagonal“correla-
andvarianceoftheλfunctionfortwoimportantcases. In B
tion matrix” of an infinite universe, Eq. (5). The larger
thefirstcasetheuniverseisinfinite,sothatthespherical
the difference, the smaller the variance of λ, as the ran-
harmonics a are characterised by the usual two-point
ℓm dom off-diagonal correlations present in the a are less
ℓm
function,
likely to match those of the test-matrix . The value
B
haℓma∗ℓ′m′i∞ =Cℓδℓℓ′δmm′. (5) owfilℓlmdaisxcuinsstlhaetetrahbolewwitaisncflhuoesnecnesbtahseicmalelyasaurrbeimtreanrtilsy.,Wwee
In the second case the universe has indeed the topology have also introduceda “signalto noise ratio”S/N which
described by the correlation matrix against which we isthedifferenceoftheexpectationvalues,dividedbythe
B
test the a . In this case the two point function of the errors added in quadrature,
ℓm
spherical harmonics is given by
X X
∞ B
haℓma∗ℓ′m′iB =Bss′ (6) S/N(B,X)= |σh(Xi)2∞−+hσ(iX|)2B. (12)
In both cases the spherical harmonics obey a Gaussian Here X is the estimator pused. This gives only a rough
statistics and the higher n-point functions are uniquely
indication of the true statistical significance with which
determinedbythetwo-pointfunctionviaWickstheorem.
a universe with the given topology can be distinguished
Let us first define the auto-correlation U =
fromaninfiniteuniverse. Asthedistributionofλandχ2
secstsi′o|nBsesx′|c2l.udWeetrheemdiniadgothnealretaedrmersthsa=tssu′chexscuemptsiwnhtehries are not exactly Gaussian, S/N is not exactly measured
in units of standard deviations. However, it is sufficient
P
specifically mentioned. For an infinite universe, we no-
to compare the different methods and to illustrate how
tice that if we sum only over the non-diagonal elements
welldifferenttopologiescanbedetected. Forprecisesta-
s 6= s′ then, since hasa∗s′i∞ = Csδss′ the expectation tistical results we fit the full distribution, see appendix
value of lambda is zero, λ =0. Else,
h i∞ B.
1
hλi∞ = U CsBss. (7) topology ℓmax λ∞ λB S/N [σ]
Xs T[2,2,2] 60 0±0.017 1±0.102 9.7
If the map was whitened (see below), then λ = T[4,4,4] 60 0±0.046 1±0.082 10.6
∞
h i
tr( )/U. T[2,2,2] 16 0±0.03 1±0.34 2.9
B
For a finite universe, T[4,4,4] 16 0±0.09 1±0.22 4.2
T[6,6,1] 16 0±0.08 1±0.33 2.9
λ =1 (8)
B
h i T[15,15,6] 16 0±0.51 1±0.59 1.3
independently if we sum over the diagonal elements or
not, as we just recover the auto-correlation in the nu- TABLEI:Comparisonofthemeanandstandarddeviationof
merator. Of course the auto-correlation value depends λ for different topologies and different ℓmax, normalised with
thetruepowerspectrum. TheS/NvalueisgivenbyEq.(12).
on the summation convention.
Forthe variance,inthe caseofaninfinite universe,we
find
The power spectrum C depends of course on the cos-
ℓ
2
σ∞2 ≡hλ2i∞−hλi2∞ = U2 CsCs′|Bss′|2. (9) mlemolowgiecanloprmaraalmiseettehres.cTororemlaitniiomnismeatthriiscepsoetietnhteiralbpyrtohbe-
ss′
X
4
diagonal C a a∗ or by the usual orientation- aver- diagonal,ie.foruniverseswithaverylargecompactifica-
s ≡ h s si
aged power spectrum tion scale. A possibility is to decompose the correlation
matrixintoa sumovereigenvaluesandeigenvectors. We
C = 1 a 2, (13) canthenonlyretainthemostimportanteigenvectors. As
ℓ ℓm
2ℓ+1 | | the correlationmatrix is alsoa covariancematrix,this is
m
X somewhat analogous to principal component analysis or
via the prescription the Karhunen-Loevetransform. Fora correlationmatrix
we will write the decomposition as
B
ss′
Bss′ → √BCsCs′. (14) Bss′ = ǫ(i)vs(i)vs(i′)∗ = bs(i)bs(i′)∗. (15)
i i
X X
This is often called “whitening”, and it serves to enforce
The ǫ(i) are the eigenvalues of the matrix and they
the same (white noise)power spectrum in both the tem- B
arerealandpositiveasthe matrixishermitianandposi-
plate and the model being tested. After applying this
tive. This allows us to define effective sphericalharmon-
normalisation the power spectrum is just C = 1. We
s ics b(i) √ǫ(i)v(i), which have, for example, the same
apply the same normalisationto the aℓm. As we willnot s ≡ s
properties under rotation as the usual a .
in general know their “true” input power spectrum, we ℓm
use the one recovered from the a themselves. As can
ℓm
be seen in table II, the division by the recovered power
III. USING THE LIKELIHOOD
spectrum greatly reduces the variance of λ and so im-
proves the detection power for the different topologies.
Instead of considering the correlation between the re-
Contrary to table I we could not compute the numbers
analytically and have estimated them from 104 random covered and the theoretical matrix, we can think of the
two-point correlation matrix as the covariance matrix of
realisations each of maps with the trivial topology and
the a . Then we may ask the question, what is the
the topology. ℓm
B probability of a covariance matrix given the measured
C
a . This can be answered using Bayesianstatistics.
topology ℓmax λ∞ λB S/N [σ] ℓm
Inafirststepweneedtoconstructthelikelihoodfunc-
T[2,2,2] 60 0±0.015 0.973±0.030 29.0
tion. TheprobabilitydistributionforaGaussianrandom
T[4,4,4] 60 0±0.051 0.976±0.044 14.5 variablexwithvarianceσ2 andzeroexpectationvalueis
T[2,2,2] 16 0±0.032 0.924±0.100 8.8
T[4,4,4] 16 0±0.091 0.948±0.100 7.0 p(xσ)= 1 e−2xσ22 (16)
| √2πσ
T[6,6,1] 16 0±0.083 0.894±0.200 4.1
T[15,15,6] 16 0±0.534 0.971±0.553 1.3 Ifweassumethatwemeasurexandwanttoknowσ,then
the likelihood function for finding a certainx is givenby
TABLE II: Comparison of the mean and standard deviation (σ) p(xσ). We write the likelihood as a function of
of λ for different topologies and different ℓmax, normalised tLhe va≡riance|, as this is the model parameter that we are
with the power spectrum estimated independently for each
interested in.
realisation. As we see, the signal to noise ratio is improved
For many independent variables, the probability dis-
considerably.
tribution is the product, which leads to a sum in the
exponent. In the case of the a , the random variables
ℓm
For an infinite universe Cs is independent of m and are not independent but are distributed according to a
it does not matter whether we divide by Cs or Cℓ. For multivariateGaussiandistributionwithacovariancema-
non-trivial topologies this is not the case as additional trix . The likelihood function then is
correlations are induced in different m modes. For this C
troeassoomn,ewthheadtisvtirsoionngebrycothnestmra-ianvtse.raOgfedcoCuℓrsteenwdeslotoseletahde p(aℓm|C)=L(C)∝ 1 exp−21 a∗sCs−s1′as′,
informationencodedinthepowerspectrum,likethesup- |C|  Xs,s′ 
pressionoffluctuations withwavelengthslargerthanthe p (17)
size of the universe. However, we feel that the improved where isthedeterminantofthematrix . Thecovari-
|C| C
stability to mis-estimatesofthe powerspectrumandthe ancematrixisgivenbythetwo-pointcorrelationmatrix,
reduced dependence on the cosmological parameters is and as = 0. Any further model assumptions are im-
h i
worth the trade-off. plicitly included in the choice of . Using Bayes law we
C
The numerical evaluation of Eq. (4) requires a double can invert the probability to find
sum over s = ℓ (ℓ +2) matrix coefficients. It
max max max p(a )p( )
scales therefore as ℓ4max. But the correlation matrix of p(C|aℓm)= pℓm(a|C ) C . (18)
an infinite universe is diagonal, so that we only need to ℓm
perform a single sum. It should therefore be possible The probability in the denominator is a normalisation
to reduce the work for matrices that are close to being constant, while p( ) is the prior probability of a given
C
5
topology encoded by . We will assume that we have no topology ℓmax χ2∞ χ2B S/N [σ]
C
prior information about the topology of the universe, so T[2,2,2] 60 37168±2373 3720±86 14.1
that this is a constant as well. In this case p( a )
ℓm T[4,4,4] 60 14656±1517 3720±86 7.2
C| ∝
( ), ie. we can use the likelihood function to estimate
L C T[2,2,2] 16 5608±738 288±24 7.2
the probability of a topology given a set of a . For
ℓm T[4,4,4] 16 1802±300 288±24 5.0
our purpose, the covariance matrix is just given by the
T[6,6,1] 16 20781±7103 288±24 2.9
correlation matrix . In general, one may have to add
noise to it, and mayBbe introduce a sky cut. T[15,15,6] 16 309±28 288±24 0.6
Generally it is preferable to consider the logarithm of
the likelihood, log( ) = 1/2(log( ) + χ2) + const. TABLE III: Sameas table I for χ2.
L − |B|
where we have defined
topology ℓmax χ2∞ χ2B S/N [σ]
χ2 = a∗sBs−s1′as′. (19) T[2,2,2] 60 37366±1123 4655±438 27.1
Xs,s′ T[4,4,4] 60 14932±1157 4027±162 9.3
T[2,2,2] 16 5690±477 474±131 10.5
We notice that there is a potential issue with the nor-
malisation of the input model: If a 0 then χ2 0 T[4,4,4] 16 1841±196 335±48 7.5
s
→ →
– generally any model whose a lead to a bad fit (high T[6,6,1] 16 21093±5645 786±557 3.6
s
χ2)couldberenormaliseduntilareasonablelikelihoodis T[15,15,6] 16 309±10 289±5 1.8
obtained. It is therefore required to fix the overall nor-
malisation,andwewilldothis byusingthewhiteneda , TABLEIV: Sameas table II for χ2.
s
in which case the normalisationis fixed by a 2 =1.
s| s|
Forthetwospecialcases,theinfinite universeanda
P ℓm numbers from 104 numerical realisations for each topol-
distributed according to , we can compute expectation
B ogy. Again the detection power increases considerably.
value and variance. For the general case we will write
asas′ = ss′. Then
h i A In appendix A we compare the structure of the corre-
lation estimator to the likelihood χ2. We find that for
hχ2i= ha∗sas′iBs−s1′ =tr(AB−1), (20) many cases the χ2 has minimal variance.
ss′
X
where we have used the hermeticity of the correlation
matrices. The two special cases are IV. ROTATING THE MAP INTO POSITION
χ2 = C −1 (21) Thesituationdiscussedsofarissomewhatmisleading:
h i∞ sBss
s Nature is rather unlikely to align the topology of the
X
χ2 = tr(1)=s (22) universe with our coordinate system. The correlation
B max
h i matrices are not invariant under rotations, as rotations
As the aℓm are Gaussian random variables, we expect mix aℓm with different m. To parametrise the rotations
to find that χ2 is distributed with a χ2-like distribution. we use the three Euler angles α, β and γ which describe
Thegeneralexpressionisrathercumbersome,butforthe threesubsequentrotationsaroundthez,they andagain
two special cases we find thez axis. Thefirstandlastrotationjustleadtoaphase
change. The rotationaroundthe y-axis couples different
σB2 ≡h(χ2)2iB−hχ2i2B =2smax (23) m and is given by Wigner rotation matrices dℓmm′,
and aℓm → e−i(mα+m′γ)dℓmm′(β)aℓm′. (25)
m′
σ∞2 =2 CsCs′|Bs−s1′|2. (24) Together, the thrXee rotations can describe any element
ss′
X of the rotation group of order ℓ. We use the relations
We list in table III some examples, together with the given in [15] to compute the rotation matrices. Figure 1
number of standard deviations that the two expectation showsanexamplewhereweplotλwhilerotatingthea
ℓm
values lie apart. azimuthally. The figure represents the case for ℓ =
max
Inthesecomputations,asinthecorrespondingonesfor 60, for lower values of ℓ the peaks are less sharp and
max
the correlationcoefficient,wehaveassumedthatwe nor- there is less sub-structure. The same is true for the χ2,
malise the observed a by the “true” power spectrum while the peaks for likelihood, which is proportional to
ℓm
(ordiagonal). However,wedonotknowwhatitis. Ifwe exp( χ2/2), are even much narrower.
−
instead normalise them by the estimated one (which is We can therefore not avoid probing all possible ro-
differentforeachrealisation),wechangethestatistics. It tations, either by computing the average or by taking
is now nolongerGaussian. Table IV reproducesthe pre- the maximum/minimum of our estimator over all orien-
vious one, but now for this scenario. We estimated the tations. Possiblythemoststraightforwardapproachisto
6
and searchfor a localextremum nearby. We then repeat
theprocedurefordifferentrandomstartinglocationsun-
til we have found a stable global maximum (for exam-
ple, eight times the same global maximum). This is the
safest method, and can be relatively fast depending on
the topology.
Computing the average is therefore quite difficult and
slow. We also found that using the maximum or mini-
mumresultsinamuchstrongerdetectionthanusingthe
average,atleastfortheλandχ2 estimator. Itispossible
to improve the average by using the likelihood which is
proportional to exp( χ2/2). This decreases the weight
−
of the “wrong” orientations exponentially. However, it
makestheaverageevenhardertocompute. Furthermore,
it lends itself readily to a Bayesian interpretation which
is quite different from the frequentist approach followed
so far. For this reason we will consider only the maxi-
mum/minimum approach here and defer the discussion
FIG. 1: Behaviour of the correlation coefficient λ under a of the average likelihood to section VII. We also note
rotation around the z-axis. The signal is maximal only for that it makes no difference if we consider the χ2 esti-
very well-defined alignments. We used a T[2,2,2] correlation mator or the likelihood when using the extremum over
matrix and aℓm derived from a T[2,2,2] topology. orientations. The exponential function is monotonic and
so the maximum or minimum point will not change un-
der it (except that the minimum of the χ2 will turn into
a maximum of the likelihood and vice versa). For the
try many random rotations [16]. This is simple to pro-
same reason,it does not change the statisticalweight. If
gram and uses automatically any symmetries present in
99 realisations of model A have a lower χ2 than any of
the template. But due to the precision needed to find
model B, then those 99 realisations will have a higher
the bestalignmentforsometemplates,wefoundthatwe
likelihood as well.
need in excess of 106 rotations to get correct results for
A drawback of using the extremum over all rotations
ℓ = 60. We can on the other hand probe systemati-
max
is that we do not know the resulting distribution func-
callyallorientations,forexamplewiththetotalconvolu-
tion. In general we have to compute a large number
tionmethod[17]. Inthisapproach,therotationswiththe
of test-cases to obtain the distribution, but this is very
three Euler angles are replaced by a three-dimensional
time-consuming and for high ℓ computing more than
FFT. This speeds the procedure up a by a large factor. max
a few hundred realisations becomes prohibitive, at least
However, we found that we may nonetheless miss the
on a single processor. Instead we can find a good ap-
best-fit peaks which can be very sharp (see Figs. 1 and
proximation to the new distribution by assuming that
2).
each rotation leads to a new independent Gaussian dis-
tribution. If there are N independent rotations then we
need to know the distribution of the maximal value of
N draws from a Gaussian distribution. This leads to an
extreme value distribution, and exact results are known
only for N < 6. However, for very large N, the dis-
tribution should converge to one of three limiting cases,
analogouslyto the centrallimit theorem(see eg.[18]). If
wefitthesedistributionstothenumericalresultsthenwe
canobtainconfidencelimitswithareasonableamountof
cpu-time. We discuss this in more detail in appendix B.
We compare in tables V and VI the minimal χ2 and
maximalλvaluesrespectively,takenoverallpossibleori-
entations. WealsoquotetheresultingS/Nvalue. Weno-
FIG. 2: The maximal correlation coefficient for the case of
tice that especially the χ2 estimator gains in sensitivity.
a universe with T[2,2,2] orientation. The sharp, high peaks
correspondtothecorrectorientationofthemapwithrespect Thisseemsrathersurprising,asthedistancebetweenthe
to thetemplate. estimator values of an infinite and a finite universe will
in general decrease when taking the extremum. How-
ever,we also notice that the variance is dramaticallyde-
Ifwelimitourselvestofindingthemaximum/minimum creased,which in turnleads to the evenhigher detection
efficiently,thenwecanalsostartwitharandomrotation power.
7
topology ℓmax χ2∞ χ2B S/N [σ] ageoverdirections. Todothatweonlyneedtoinvertthe
T[2,2,2] 60 33237±586 4588±382 41 matrix once at the start, not for every evaluation. But
we need to evaluate the likelihood for each orientation,
T[4,4,4] 60 11146±438 4057±204 14
and the number of the required rotations scales roughly
T[2,2,2] 16 4062±172 469±172 17
like ℓ2 . We therefore end up with a ℓ6 scaling at
T[4,4,4] 16 1180±73 350±47 10 max max
any rate. Secondly, the most time consuming procedure
T[6,6,1] 16 7719±1125 675±370 6
is the estimation of the variance using simulated maps,
T[15,15,6] 16 287±2.1 285±2.5 0.6 and again we only need to invert the matrix once as it
stays the same. ℓ6 is a rather steep growth, and it
TABLEV:Comparisonofthemeanandstandarddeviationof max
is certainly preferable to use the smallest matrices that
the χ2 for different topologies and different ℓmax, normalised
guarantee a detection.
with thepower spectrum and minimised overrotations.
Onthe other hand, does the detection alwaysimprove
with growing ℓ ? Let us have a look at the correla-
max
tion estimator, in the case of a whitened map. Clearly
Thereductionofthevariance,especiallyfortheinfinite
σ2 = 2/U can only decrease as long as there are any
universe case, is easy to understand. In table II and IV ∞
off-diagonalelements in the correlationmatrix. But this
weusethebest-fitalignmentforthemapsofafiniteuni-
is not the dominant error. However, we expect that the
verse. Butthe maps withthe trivialtopologyarealways
maincontributiontoEq.(11)isdue totheremainingdi-
randomly aligned (being statistically isotropic). The
agonalentriess =s ands =s . Thistermofthesum
varianceforthe infinite universemapscontainstherefore 2 3 1 4
is equalto the auto-correlationU and so contributes the
an effective “random orientation” contribution. Taking
same error as σ2 . As the signature of the topology be-
the extremum over all orientations eliminates this con- ∞
comes very weak, we expect that the two errors become
tribution. As the infinite universe variance dominates
strongly in the case of the χ2 estimator, we find that comparable, but are still decreasing functions of ℓmax.
this estimator benefits more from the reduction of the
variance.
topology ℓmax λ∞ λB S/N [σ]
T[2,2,2] 60 0.08±0.01 0.98±0.03 28
T[4,4,4] 60 0.21±0.02 0.98±0.05 14
T[2,2,2] 16 0.16±0.02 0.95±0.08 10
T[4,4,4] 16 0.38±0.05 0.98±0.09 6
T[6,6,1] 16 0.35±0.05 0.94±0.19 3
T[15,15,6] 16 1.84±0.25 1.86±0.27 0
TABLEVI: Same as table V for λ.
Asafinalpoint,wenoticethatthemaximisedvalueof
λforthe T[15,15,6]topologyintableVI is largerthan1.
This is a sign that we cannot detect that topology. The
fluctuationsaresolargethatthey completelyoverwhelm
thesignal. Aftermaximisingoverorientationsweendup
with λ>1. FIG. 3: Detection significance assuming that we know the
correct orientation. The topologies were T[2,2,2] (solid black
anddottedredline)andT[4,4,4](dashedblueanddot-dashed
V. DISCUSSION OF GENERAL RESULTS magenta line). The estimators were respectively the correla-
tion amplitude λ (dotted red and dot-dashed magenta line)
and thelikelihood χ2 (solid black and dashed blue line).
A. What angular resolution is required?
Isitbettertotestthemapstoarbitrarilyhighℓ ,or We compare in Figs. 3 and 4 the scaling of
max
tousealowerresolution? Oneimportantconsiderationis S/N(T[2,2,2]) and S/N(T[4,4,4]) respectively, for the
theamountofwork(andthusoftime)neededtoevaluate correlationestimator(reddotted/magentadash-dotted)
the estimator. For both estimators we need to sum over and the likelihood method (black solid / blue dashed).
s ands′. This meansthatthe requirednumber ofopera- In all cases we used 100 realisations to compute the av-
tions scales like ℓ4 . The matrix inversion required for erage and standard deviation, which explains the noisy
max
the likelihood evaluation scales like ℓ6 . However, for curves. As discussed earlier, we find that taking the ex-
max
tworeasonsitisnormallynotthelimitingfactor. Firstly, tremum over rotations can increase the detection power,
asdiscussedintheprevioussection,westillneedtoaver- especially for the χ2 estimator.
8
constrain it with one of our methods? Tables VI and
V show that we can indeed distinguish a universe with
T[4,4,4] topology from an infinite one at over 10 σ.
As in the previous section we plot in Figs. 5 and 6 the
detection significance both before and after maximising
over directions. This time we study two families of slab
spaces. Thefirstone,T[X,X,1],hasoneverysmalldirec-
tionofsize1/H andwevarytheothertwo. Wefindthat
0
we can clearly detect this kind of topology at ℓ =16
max
for any size of the larger dimensions. For this example-
topologyitis verystrikinghowthe correlationestimator
is better if we use the “correct” alignment, while the χ2
becomesmorepowerfulasweextremiseoverorientations.
FIG.4: Detection significancewhenmaximising overallori-
entations. The topologies were T[2,2,2] (solid black and dot-
ted red line) and T[4,4,4] (dashed blue and dot-dashed ma-
genta line). The estimators were respectively the correlation
amplitude λ (dotted red and dot-dashed magenta line) and
thelikelihood χ2 (solid black and dashed blueline).
We alsosee thatfor the T[4,4,4]topologyandthe cor-
rect orientation,the correlationmethod eventually over-
takes the likelihood method. This is most likely because
the T[4,4,4] correlation matrix is closer to being diago-
nal than the T[2,2,2] correlation matrix. At high ℓ the
diagonalelements startto dominate the contributions to
the χ2. The correlator method is not sensitive to this
contribution as it does not sum over the diagonal ele-
FIG. 5: Detection significance assuming that we know the
ments. After maximising over orientations, on the other
correctorientation. ThetopologieswereT[X,X,1](solidblack
hand, the likelihoodis alwayssuperior to the correlation and dotted red line) and T[15,15,X] (dashed blue and dot-
method, except maybe for the highest ℓmax. dashed magenta line). The estimators were respectively the
We further notice that the detection power keeps in- correlation amplitudeλ(dottedred anddot-dashedmagenta
creasingwithincreasingℓ ,eventhoughthingstendto line)andthelikelihoodχ2 (solidblackanddashedblueline).
max
slowdownbeyondℓ 40. Thismeansthatitisusefulto Weused ℓmax =16.
≈
consider the largest ℓ for which we have the correlation
matrixandwhichwecananalyseinareasonableamount
of time. Unfortunately, it is also the case (and hardly
surprising) that the smallest universes profit the most
from analysing smaller scales. The traces from large but Thesecondfamily,T[15,15,X]isconsiderablyharderto
finite universesbecome rapidly weakeras ℓmax increases. detectasheretwodirectionsareverylargeandeffectively
As there is little practical difference between a 20 σ de- infinite. Forlargevalues ofXwecannotfindadifference
tection and a 50 σ detection, it seems in general quite toaninfiniteuniverse. Asthethirddirectionshrinks,we
sufficient to consider scales up to ℓmax = 40 to 60. The start to see differences, but only for X 3/H0 can we
higher ℓ may become more important when we also con- detect the non-trivial topology at over 2≤σ. In this case
sider the ISW effect. thecorrelationmethodisalwaysinferiortotheχ2. Inap-
pendix C we considera morefundamental distance mea-
sure between correlationmatrices, namely the Kullback-
B. What size of the universe can be detected? Leiblerdivergence. Weconfirmthatwewillneverbeable
todistinguishT[15,15,X]withX >6/H fromaninfinite
0
From the suppression of the low-ℓ modes in the angu- universe, see also Fig. 15. This is not very surprising, as
lar power spectrum, the T[4,4,4] topology seems a good in this case the universe is in all directions larger than
candidate for the global shape of the universe. Can we the particle horizon today.
9
distributions agree quite well, to within their own vari-
ance. The variance observed between the different re-
constructedskymaps(WMAP,TC,TWandLILC)isof
thesameorderofmagnitude. ThisexampleisforT[2,2,2]
andℓ =16,butitisrepresentativeoftheothercases.
max
FIG.6: Detection significancewhenmaximising overallori-
entations. ThetopologieswereT[X,X,1](solidblackanddot-
tedredline)andT[15,15,X](dashedblueanddot-dashedma-
genta line). The estimators were respectively the correlation
amplitude λ (dotted red and dot-dashed magenta line) and
the likelihood χ2 (solid black and dashed blue line). Again
ℓmax =16. FIG.7: Thedistributionofthehχ2i∞ estimatorvalueswhen
testing for a T[2,2,2] universe with ℓmax = 16. The black
solid histogram is computed from 10000 noiseless full-sky re-
VI. A SIMPLIFIED ANALYSIS OF WMAP alisations used throughout this paper, while the red dashed
DATA histogram used 1000 simulated LILC maps (see text). The
verticallines show the χ2 valuesof themeasured maps, from
theleft LILC, TW and WMAP (coincident) and TC.
Toillustratetheapplicationoftheseteststorealdata,
weperformasimplifiedanalysisoftheWMAP[19]data.
Simplified in the sense that we do not deal with issues
For our standard example, the T[4,4,4] template, we
like map noise and sky cuts. In general, one has to sim-
find a maximal value for the 1st year WMAP ILC map
ulate a largenumber ofmaps where both ofthese effects
of λ = 0.20. This is about expected for an infinite
max
areincluded, andwhicharethenanalysedwiththe same
universe. A universe exhibiting a genuine T[4,4,4] topol-
pipeline as the actual data map. However, as an illus-
ogy should lead to roughly λ =1.
max
tration we will analyse reconstructed full-sky maps. We
use the internal linear combination (ILC) map created
bythe WMAP team,whichwewillcallthe WMAP map topology ℓmax χ2 P∞ PB λ P∞ PB
fromnowon. Wealsousetwomapreconstructionsbyby T[2,2,2] 60 33130 0.39 0 0.087 0.20 0
Tegmark,aWienerfilteredmap(TW)andaforeground- T[4,4,4] 60 11020 0.40 0 0.20 0.64 0
cleaned map (TC) [20]. All of these maps are publicly T[6,6,1] 16 8805 0.85 10−6 0.37 0.29 10−5
available in HEALPix format [21] with a resolution of T[15,15,6] 16 290 0.95 0.01 1.6 0.16 0.84
N = 512. We use this software package to read the
side
map files and to convert them into a . TABLEVII:The valueof χ2 and λ obtained for theWMAP
ℓm
To get some idea of the systematic errors in this anal- map,togetherwiththeprobabilityofmeasuringsuchavalue
ysis, we additionally analyse the ILC map reconstructed if theuniverseis infinite(P∞) and if theuniversehasindeed
by Eriksen et al. (LILC). They also produced a set of thetopology that we test for (PB).
simulated LILC maps (for the trivial topology) with the
same pipeline [22, 23]. It is a necessary (but not suffi- We give in table VII the values of χ2 and λ for the
cient) condition to trust our simplified analysis that the WMAP map. The values for the other maps are not
results from these maps are consistent with our results very different. We also give two probabilities for both
for an infinite universe. As an illustration we plot in estimators, P and P . The first one is the probability
∞ B
Fig. 7 the distribution of χ2 for our simple infinite uni- of measuring a larger value of λ (or a smaller value of
versemaps(blacksolidhistogram)andforthe simulated χ2) if the universe is infinite. P on the other hand is
B
ILC maps which contain noise and foreground contri- the probability of measuring a smaller value of λ (or a
butions (red dashed histogram). We see that the two largervalueofχ2 ifthe universehasindeedthe topology
10
that we tested for. For a non-detection of any topology
we require P to be not too small. A positive detection
∞
of a topology on the other hand requires a larger P . If
B
both probabilities are large then we cannot detect that
topology (as exemplified eg. for the case of T[15,15,6]).
Wecomputetheseprobabilitieswiththebest-fittingthe-
oretical PDF, as discussed in the appendix B.
FIG.9: ThesameasFig.8fortheT[15,15,X]topology. Again
allWMAPmapsareconsistent withan infiniteuniverse,but
we can only rule out the universes with X <3 at more than
95% CL.
the probability of that model, given the data. Using the
three Euler angles as parameters Θ, defining the model
FIG.8: Medianand95%confidencelimitsasmeasuredwith to be a given topology, and the data d the measured
the χ2 estimator for infinite universes (upper green limits) aM we can write the model evidence as
ℓm
anduniverseswithaT[X,X,1]topology(redlowerlimits),asa
functionofsizeX inunits1/H0. Wealsoplottheχ2valuesof ( ) p(d )= µ(Θ)π(Θ) (Θ), (26)
theWMAP map (red crosses), theTW map (cyan triangles) E M ∝ |M L
Z
andTCmap(bluecircles)andtheLILCmap(magentastars).
where π(Θ) is the prior on the orientation of the map,
Allskymapsare consistent with an infiniteuniverseandnot
see eg. [24]. The ratio of the evidence for two topologies
consistent with a T[X,X,1] topology for any X. We also plot
is a Bayesian measure of the relative probability. We
errorbars for theLILC map simulations.
canthink ofit as the relativeodds ofthe two topologies.
A similar method to constrain the topology was applied
Fig. 8 shows 95% confidence limits (estimated numer- previously to the COBE data, see [25].
ically from 104 samples) when testing for the presence The measure µ(Θ) on SO(3) needs to be independent
(red, lower band) or absence (green, upper band) of a of the orientation[30], which pretty much singles out the
T[X,X,1] topology. The WMAP data (points) are all Haar measure (up to an irrelevant constant). In terms
compatible with the infinite universe and rule out this of the Euler angles it is dαdβdγsin(β)/(8π2) with α and
kind of topology very strongly. The bounds from the γ going from 0 to 2π, and β from 0 to π. The volume
simulated LILC maps (black error bars) are consistent of SO(3) is then µ(Θ) = 1. A simple way to generate
withoursimulatedmapswithatrivialtopology,butsys- random orientations is to select α and γ uniformly in
R
tematically a bit lower. We plot the same in Fig. 9 for a [0,2π] and u in [ 1,1] and then set β =arccos(u).
−
T[15,15,X] topology. Again, WMAP is compatible with The advantage of using Bayesian evidence is that it
the infinite universe. But as discussed before, we cannot provides a natural probabilistic interpretation which de-
detect these universes for X >3/H . Overall,all results pends only on the actually observed data, but not on
0
are consistent with an infinite universe. simulated data sets. Because of this, there is no need to
run large comparison sets. This is a very different view
point from the frequentist approach followed so far.
VII. BAYESIAN MODEL SELECTION For an infinite universe the correlationmatrix is diag-
onal and rotationally invariant (due to isotropy). The
integral over the alignment becomes trivial in this case.
The likelihood can also be used in a purely Bayesian
If we use whitening then the correlation matrix is just
approach. Weareinterestedintheprobabilityofamodel
the unit matrix and we have
given the data, p( d). If all topologies are taken to
M|
be equally probable, then through Bayes theorem the χ2 = a 2 =s . (27)
s max
statistical evidence ( ) for a model is proportional to | |
E M Xs