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arXiv:gr-qc/0205009v1 2 May 2002
Eigenmodes of 3-dimensional spherical spaces and
their application to cosmology
Roland Lehoucq1,6, Jeffrey Weeks2, Jean-Philippe Uzan3,4
Evelise Gausmann5, and Jean-Pierre Luminet6
(1) CE-Saclay, DSM/DAPNIA/Service d’Astrophysique, F-91191 Gif sur Yvette
Cedex (France)
(2) 15 Farmer St., Canton NY 13617-1120 (USA).
(3) Institut d’Astrophysique de Paris, GReCO, CNRS-FRE 2435,
98 bis, Bd Arago, 75014 Paris (France).
(4) Laboratoire de Physique Th´eorique, CNRS-UMR 8627, Bˆat. 210, Universit´e
Paris XI, F–91405 Orsay Cedex (France),
(5) Instituto de F´ısica Te´orica, Rua Pamplona, 145 Bela Vista - S˜ao Paulo - SP,
CEP 01405-900 (Brasil)
(6) Laboratoire Univers et Th´eories, CNRS-FRE 2462, Observatoire de Paris,
F-92195 Meudon Cedex (France).
Abstract. This article investigates the computation of the eigenmodes of
the Laplacian operator in multi-connected three-dimensional spherical spaces.
General mathematical results and analytical solutions for lens and prism spaces
are presented. Three complementary numerical methods are developed and
compared with our analytic results and previous investigations. The cosmological
applications of these results are discussed, focusing on the cosmic microwave
background (CMB) anisotropies. In particular, whereas in the Euclidean case
too small universes are excluded by present CMB data, in the spherical case there
will always exist candidate topologies even if the total energy density parameter
of the universe is very close to unity.
PACS numbers: 98.80.-q, 04.20.-q, 02.040.Pc
1. Introduction
The search for the topology of our universe has made tremendous progress in the past
years and several methods have been designed using either galaxy catalogs or the CMB
(see e.g. [1, 2] for general reviews). The most promising dataset that can eventually
contain a topological signature is the cosmic microwave background (CMB) in the
form of pattern correlations (such as homologous circles in the sky [3], or anomalously
large temperature correlations in a set of directions [4], see [5] for a recent review on
the CMB methods) or non Gaussianity [6].
The detectability of the topology in datasets such as those that will be made
available by the MAP [7] and Planck [8] satellite missions requires to simulate maps
with the topological signature for a large set of topologies. These maps will have
mainly two uses: first, they will allow us to test the detection method and for instance
estimate its running time and second, once all sources of noises are added, it will help
us investigating to which extent a given method is well suited to detect the topological
signal and indeed if it is not blurred (in the same spirit as the investigation of the
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 2
“crystallographic” methods based on galaxy catalogs [9]). A prerequisite for any
further study is thus to simulate CMB maps with a topological signal.
At present, the status of the constraint on the topology of the universe is sparse.
Concerning locally Euclidean spaces, it was shown on the basis of the COBE data that
the size of the fundamental domain of a 3-torus has to be larger than L≥4800 h−1Mpc
[10, 11, 12, 13]. This constraint does not exclude a toroidal universe since there can
be up to N= 8 copies of the fundamental cell within our horizon. This constraint
relies mainly on the fact that the smallest wavenumber is 2π/L, which induces a
suppression of fluctuations on scales beyond the size of the fundamental domain. This
result holds only for the case of a vanishing cosmological constant and was generalized
to all Euclidean manifolds [14]. A non-vanishing cosmological constant induces larger
scale cosmological perturbations, via the integrated Sachs-Wolfe effect. For instance
if ΩΛ= 0.9 and Ωm= 0.1, the former is relaxed to allow for N= 49 copies of the
fundamental cell within our horizon. This constraint is also milder in the case of
compact hyperbolic manifolds and it was shown [15, 16, 17] that the angular power
spectrum was consistent with the COBE data on multipoles ranging from 2 to 20 for
the Weeks and Thurston manifolds. Another approach was developed in [18, 19, 20]
and is based on the method of images. Only one spherical space using this method of
images was considered in the literature, namely the case of the pro jective space [21].
Note that multiconnectedness breaks global homogeneity and isotropy (except
for the particular case of the projective space). It follows that the temperature
angular correlation function Cwill depend on the position of the observer and on
the orientation of the manifold, which is at odd with the standard lore. In a simply
connected universe the angular correlation function depends only on the angle between
the two directions whereas in a multi-connected universe, it will depend on the
two directions. It follows that the coefficients Cℓof the decomposition of Cinto
Legendre polynomials, obtained by averaging over the sky, loose much of topological
information. As clearly explained in [5], Ccan be decomposed into an isotropic and
an anisotropic part and the Cℓdepend solely on the former. The Cℓalone are a poor
indicator of the topology, despite the fact that they can help constraining the topology,
which backs up the necessity to study the full sky map.
In standard relativistic cosmology, the universe is described by a Friedmann-
Lemaˆıtre spacetime with locally isotropic and homogeneous spatial sections. These
spatial sections can be defined as the constant density or time hypersurfaces. With
such a splitting, the equations of evolution of the matter and geometry perturbations
that will give birth to the large scale structures of the universe reduce to a set of
coupled differential equations involving a Laplacian (see e.g. [22]). This system is
conveniently (numerically) solved in Fourier space but this requires to determine
the eigenmodes and eigenvalues of the Laplacian through the generalized Helmoltz
equation
∆Ψq=−q2Ψq.(1)
The Laplacian in Eq. (1) is defined as ∆ ≡DiDi,Dibeing the covariant derivative
associated with the metric γij of the spatial sections (i, j = 1..3). The eigenmodes on
which any function can be developed encode the boundary conditions imposed by the
topology, and any function developed on this basis will satisfy the required boundary
conditions.
Concerning the computation of the eigenmodes, the case of Euclidean manifolds
can be solved analytically and many numerical investigations of compact hyperbolic
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 3
manifolds have been performed [16, 23, 24, 25, 26]. The case of spherical manifolds
has been disgarded during a long time. The recent measurements of the density
parameters let the room for our universe to be slightly positively curved since they
are estimated to lie in the range Ω0≡Ωm0+ ΩΛ0= 1.11+0.13
−0.12 to 95% confidence [27].
The goal of this article is to focus on the computation of the eigenmodes of the
Laplacian in spherical spaces, having in mind their use to simulate CMB maps. In
Section 2, we recall the main mathematical results on the classification of spherical
spaces [28] and some analytical results on the eigenmodes of the Laplacian such as
the determination of the smallest wavenumber and its multiplicity. In section 3, we
describe the three numerical methods that are used to compute the eigenvalues and
eigenmodes and we discuss their precision and efficiency. We then describe briefly,
in Section 4, a method to obtain these eigenfunctions analytically in the particular
cases of lens and prism spaces. Such a result is of importance to compare with the
output of numerical computations. After describing some statistical properties of the
eigenmodes in Section 5, we present, in Section 6, some cosmological consequences
of our result. We estimate the effect of the topology on the Sachs-Wolfe plateau.
Appendix A gathers most of the results on the eigenfunctions of the Laplacian in
homogeneous and isotropic 3-dimensional simply connected spaces.
Notations: The local geometry of the universe is described by a Friedmann–Lemaˆıtre
metric
ds2=−dt2+a2(t)dχ2+f2(χ)dΩ2(2)
with f(χ) = (sinhχ, χ, sin χ) respectively for hyperbolic, Euclidean and spherical
spatial sections. a(t) is the scale factor, tthe cosmic time and dΩ2≡dθ2+ sin2θdϕ2
the infinitesimal solid angle. χis the (dimensionless) comoving radial distance in units
of the curvature radius RC.
We use the embedding of the 3-sphere S3in 4-dimensional Euclidean space
by introducing the set of coordinates (xµ)µ=0..3related to the intrinsic comoving
coordinates (χ, θ, ϕ) through (see e.g. Ref. [29])
x0= cos χ
x1= sin χsin θsin ϕ
x2= sin χsin θcos ϕ
x3= sin χcos θ, (3)
with 0 ≤χ≤π, 0 ≤θ≤πand 0 ≤ϕ≤2π. The 3–sphere is then the submanifold of
equation
xµxµ≡x2
0+x2
1+x2
2+x2
3= +1,(4)
where xµ=δµν xν. The comoving spatial distance dbetween any two points xand y
on S3can be computed using the inner product xµyµand is given by
d[x, y] = arc cos [xµyµ].(5)
The volume enclosed by a sphere of radius χis, in units of the curvature radius,
Vol(χ) = π(2χ−sin 2χ).(6)
We consider 3-dimensional multi-connected manifolds of the form M3=S3/Γ
where the holonomy group‡Γ is a discrete subgroup of SO(4) that acts without fixed
‡We use this term to mean the geometric version of the group of covering transformations, that is,
elements of the holonomy group are isometries, while covering transformations are homeomorphisms,
see [30].
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 4
point on the 3-dimensional covering space S3. It is isomorphic to the first fondamental
group π1(M3). The elements g∈Γ are isometries and can be expressed as 4 ×4
matrices acting on the elements of the 4-dimensional embedding Euclidean space.
The transformation (3) between the intrinsic coordinates (χ, θ, φ) and xµis
bijective. In the following of the article, if fis a function on S3, we will loosely
write f(x) for f(χ, θ, φ) and for instance f(gx) will refer to f(χ′, θ′, φ′) with (χ′, θ′, φ′)
being the intrinsic coordinates of the image x′=gx obtained from Eq. (3) [see e.g.
Eq. (A.14) for an example].
To finish, let us remark that any eigenmode of a multi-connected manifold S3/Γ
lifts to a Γ-invariant§eigenmode of S3, and conversely each Γ-invariant eigenmode of
S3projects down to an eigenmode of S3/Γ. Thus, with a slight abuse of terminology,
we may say that the eigenmodes of S3/Γ are the Γ-invariant eigenmodes of S3.
2. Mathematical results on the Laplacian operator in spherical spaces
2.1. Classification of spherical manifolds
In our preceding article [28], we presented in a pedestrian way the complete
classification of 3-dimensional spherical topologies and we described how to compute
their holonomy transformations. We briefly outline the main results of this study,
mainly to set our notations. The classification of constant curvature spherical
manifolds was first presented in [31, 29].
The isometry group of the 3-sphere is G=SO(4) and one needs to determine
all the finite subgroups of S O(4) acting fixed point freely. Every isometry of SO(4)
can be uniquely decomposed as the product of a right-handed (R) and a left-handed
(L) Clifford translationk, up to a multiplication by -1 of both factors. Besides, S3
enjoys a group structure as the set S3of unit length quaternions. It can then be
shown that each right-handed (resp. left-handed) Clifford translation corresponds to
a left (resp. right) multiplication of S3[q→xq (resp. q→qx)] so that the two
groups of right-handed and left-handed Clifford translations are isomorphic to S3. It
follows that SO(4) is isomorphic to S3× S 3/{±(1,1)}so that the classification of the
subgroups of SO(4) can be deduced from the classification of all subgroups of S3. It
can then be shown that there exists a two-to-one homomorphism from S3to SO(3)
the subgroups of which are known. It follows that the finite subgroups of S3are:
•the cyclic groups Znof order n,
•the binary dihedral groups D∗
mof order 4m,m≥2,
•the binary tetrahedral group T∗of order 24,
•the binary octahedral group O∗of order 48,
•the binary icosahedral group I∗of order 120,
where a binary group is the two-fold cover of the corresponding group.
¿From this classification, it can be shown that there are three categories of
spherical 3-manifolds.
•The single action manifolds are those for which a subgroup Rof S3acts as pure
right-handed Clifford translations. They are thus the simplest spherical manifolds
and can all be written as S3/Γ with Γ = Zn, D ∗
m, T ∗, O∗, I ∗.
§A Γ-invariant function fis a function satisfying f(x) = f(gx) for all g∈Γ and for all x.
kClifford translations are isometries that translate all points the same distance.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 5
•The double-action manifolds are those for which subgroups Rand Lof S3act
simultaneously as right- and left-handed Clifford translations, and every element
of Roccurs with every element of L. There are obtained for the groups Γ = Γ1×Γ2
with (Γ1,Γ2) = (Zm, Zn),(D∗
m, Zn),(T∗, Zn),(O∗, Zn),(I∗, Zn) respectively with
gcd(m, n) = 1, gcd(4m, n) = 1, gcd(24, n) = 1, gcd(48, n) = 1, gcd(120, n) = 1.
•The linked-action manifolds are similar to the double action manifolds, except
that each element of Roccurs with only some of the elements of L.
The classification of these manifolds is summarized in the figure 8 of [28].
We also define a lens space L(p, q) by identifying the lower surface of a lens-shaped
solid to the upper surface with a 2πq/p rotation for relatively prime integers pand q
with 0 < q < p. Furthermore, we may restrict our attention to 0 < q ≤p/2 because for
values of qin the range p/2< q < p the twist 2πq/p is the same as −2π(p−q)/p, thus
L(p, q) is the mirror image of L(p, p −q). Lens spaces can be of any of the category
described above and their classification is detailed in figure 9 of [28].
2.2. Spectrum of spherical manifolds
The Helmoltz equation (1) is usually rewritten in terms of the quantity βdefined as
β2≡q2+ 1 (7)
and takes the form
∆Ψk,s =−k(k+ 2)Ψk,s,(8)
after using the change of variable β=k+ 1 and where sis an integer labelling the
modes of same eigenvalue (see Ref. [32] for the properties of the Laplacian operator).
It follows that the eigenvalues of the Laplacian on S3are k(k+ 2), kbeing an integer,
and that the multiplicity of each eigenvalue is (k+ 1)2[see Appendix A for details].
The two modes k= 0 and k= 1 are gauge modes since they respectively correspond to
a change in the curvature radius (homogeneous deformation) and in a displacement of
the center of the 3-sphere and are thus physically not relevant [33]. It is thus clear that
the eigenvalues of the Laplacian on a multi-connected spherical manifold M3=S3/Γ
will be a subset of the eigenvalues k(k+ 2) of the Laplacian on S3.
For any spherical manifold M3, we can introduce its (discrete) spectrum as the
set of all eigenvalues of the Laplacian
Sp(M3) = {0 = λ0< λ1≤λ2≤...≤λi≤...}(9)
and two Riemannian manifolds M3and M′3are said to be isospectral if Sp(M3) =
Sp(M′3).
Ikeda and Yamamoto [34] studied the spectra of 3-dimensional lens spaces and
demonstrated that if two 3-dimensional lens spaces with fundamental group of order q
are isospectral to each other, then they are isometric to each other. Ikeda [35] extended
this result to show that if two 3-dimensional spherical manifolds are isospectral
then they are isometric. Combining this with previous results, it follows that a 3-
dimensional spherical manifold is completely determined, as a Riemannian manifold,
by its spectrum. Higher-dimensional generalizations appear in [36].
More related to our purpose is the work by Ikeda [37] in which the spectra of
single-action manifolds are determined. This result is of great interest for comparison
with our numerical computation. Unfortunately, it will give us only the wavenumbers
with their multiplicity for a restricted set of topologies and it does not determine the
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 6
Manifold Eigenvalue k(k+ 2) multiplicity
S3/Z2pk > 0 even (k+ 1) Pk
l=0 nkl
S3/Z2p+1 k≥2p+ 1 odd (k+ 1) Pk
l=0 nkl
k > 0 even
S3/D∗
m¯
keven (2¯
k+ 1)([¯
k/m] + 1)
¯
k > m odd (2¯
k+ 1)[¯
k/m]
S3/T ∗¯
k6= 1,2,5 (2¯
k+ 1)(1 + 2[¯
k/3] + [¯
k/2] −¯
k)
S3/O∗¯
k6= 1,2,3,5,7,11 (2¯
k+ 1)(1 + [¯
k/4] + [¯
k/3] + [¯
k/2] −¯
k)
S3/I∗¯
k6= 1,2,3,4,5,7,8,(2¯
k+ 1)(1 + [¯
k/5] + [¯
k/3] + [¯
k/2] −¯
k)
9,11,13,14,17,19,23,29
Table 1. Spectra of single-action manifolds. For spaces other than S3/Znwe
have set k= 2¯
kwith ¯
kbeing an integer. [p] refers to the integer part of pand nkl
is defined by nkl = 1 if k≡2l(mod n) and 0 otherwise. Adapted from Ref. [37].
Manifold First eigenvalue k(k+ 2) kmin multiplicity
S3/Z28 2 9
S3/Zn,n > 2 8 2 3
S3/D∗
224 4 10
S3/D∗
m,m > 2 24 4 5
S3/T ∗48 6 7
S3/O∗80 8 9
S3/I∗168 12 13
Table 2. Value and multiplicity of the first non zero eigenvalue for the single-
action manifolds. From Ref. [37]. Note that we have corrected a mistake on the
multiplicity of the first eigenvalue of S3/D∗
m.
eigenfunctions. In tables 1 and 2, we sum up the main results on the wavenumbers
of single-action manifolds. In figure 1, we compare the spectra of the projective space
P3=S3/Z2and of the 3-sphere. Due to the Z2symmetry, half of the modes are
lost because they are not invariant under the antipodal map. In figure 2 we compare
the spectra of the single-action manifolds obtained from the groups Z7,Z8,Z9,D∗
2,
D∗
3, and D∗
4and figure 3 compares the ones obtained from the binary tetrahedral,
octahedral and icosahedral groups.
3. Numerical determination of the eigenmodes
To determine the eigenvalues and eigenmodes of the Laplacian, one has to find a way
to take into account the boundary conditions imposed by the topology. Different
routes have been investigated. The case of locally Euclidean manifolds is somehow
trivial since the problem can be solved analytically (see e.g. [38]). The hyperbolic case
was first addressed using the boundary element method first developed by Aurich and
Steiner [23] for the study of 2-dimensional hyperbolic surfaces. Inoue [24] developed
the direct boundary element method and was the first to determine precise eigenmodes
of 3-dimensional compact hyperbolic manifolds and get the 36 first eigenmodes of
Thurston space for k≤10 [24] and then for k≤13 [16] (see also Ref. [25] for the
first computation of the eigenmodes of a cusp manifold). Recently a new method was
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 7
0 50 100 150 200 250 300 350 400 450
0
50
100
150
200
250
300
350
400
450
S3/Z2
multiplicity
eigenvalue k(k+2)
Figure 1. Comparison of the spectra of the 3-sphere and of the projective space.
Half of the modes are lost due to the reflection symmetry (since antipodal points
are identified, the eigenfunction must vanish on the equator).
0 50 100 150 200 250 300 350 400 450
0
20
40
60
80 S3/Z7
0 50 100 150 200 250 300 350 400 450
0
50
100
150
multiplicity
S3/Z8
0 50 100 150 200 250 300 350 400 450
0
20
40
60
80 S3/Z9
eigenvalue k(k+2)
0 50 100 150 200 250 300 350 400 450
0
50
100
150
S3/D2
*
0 50 100 150 200 250 300 350 400 450
0
50
100
multiplicity
S3/D3
*
0 50 100 150 200 250 300 350 400 450
0
20
40
60
80 S3/D4
*
eigenvalue k(k+2)
Figure 2. Spectra of single-action manifolds generated from [left] the cyclic
groups Z7,Z8and Z9and from [right] the binary dihedral groups D∗
2,D∗
3and
D∗
4.
proposed by Cornish and Spergel [26] in the framework of hyperbolic spaces; such a
method can be adapted to the case of spherical spaces, and thus will be described in
more details below as the “ghosts method”.
In this section, we present three independent methods to compute the eigenmodes.
3.1. Ghosts method
The ghosts method is based on the idea that any square integrable function in
L2(X/Γ), Xbeing the universal covering space, satisfies
Ψ(x) = Ψ(gx) (10)
for all g∈Γ. Any function of L2(X/Γ) can be lifted to a (Γ-invariant) function of
L2(X) and, reciprocally, any Γ-invariant function projects down to a function on X/Γ.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 8
0 50 100 150 200 250 300 350 400 450
0
20
40
60
S3/T*
0 50 100 150 200 250 300 350 400 450
0
10
20
30
multiplicity
S3/O*
0 50 100 150 200 250 300 350 400 450
0
10
20
30
S3/I*
eigenvalue k(k+2)
Figure 3. Spectra of single-action manifolds generated from the binary
tetrahedral, octahedral and icosahedral groups.
It follows that the eigenmodes of the Laplacian can be decomposed as
Ψ[Γ]
β,s(x) = ∞
X
ℓ=0
ℓ
X
m=−ℓ
ξβ,sℓmYβℓm (x) (11)
0 where there is no summation on βbecause the eigenfunctions of the universal
covering space, Yβℓm =Rβ ℓ(χ)Yℓm (θ, φ) (see Appendix A) form a complete and
linearly independent family. Note that the coefficients ξβ,sℓm are obtained once the
holonomies are known and they will depend on the base point.
Now, choose randomly dpoints, xi, in the fundamental domain and consider the
niimages of each point up to a distance ρmax . We also need to truncate the sum (11)
to a maximum value Lof ℓ. Each point generates ni(ni+ 1)/2 constraints of the form
(10)
Ψ[Γ]
β,s(gaxi)−Ψ[Γ]
β,s (gbxi) = 0 (12)
for a6=band a, b =...|Γ|. With the decomposition (11) the set of constraints (12)
for all the dpoints xitakes the form
Aβvβ,s = 0 (13)
where the (L+ 1)2-components vector vβ,s is defined by
vβa ≡
ξβ,s00
.
.
.
ξβ,sLL
(14)
The matrix Aβwith M=Pd
j=1 nj(nj+ 1)/2 rows and N= (L+ 1)2columns is
defined by¶
Aβ≡
Yβ00(g1x1)− Yβ00(g2x1)... Yβ LL(g1x1)− Yβ LL(g2x1)
.
.
.....
.
.
Yβ00(gaxd)− Yβ00(gbxd)... Yβ LL(gaxd)− Yβ LL(gbxd)
(15)
¶Note that there is a typo in the equation (2.3) of [26].
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 9
If M > N the system (13) is over constrained and has a solution if and only if q
is an eigenvalue of the compact space. We named this method as the ghosts method
because of its close analogy with the simulation of galaxy catalogs in a multi-connected
universe [9, 39, 40, 41].
Numerically, one uses a single value decomposition (SVD) method to extract the
eigenmodes, i.e. which form a basis of Ker(Aβ). This decomposition is based on the
theorem stating that any M×Nmatrix Awith M≥Ncan be written as the product
A=UDtV(16)
where Uis a M×Nunitary matrix, D= diag(w1,...,wN) is a N×Ndiagonal
matrix, the wibeing non negative and Vis a N×Nunitary matrix. The columns
of Uassociated to non-zero wiform an orthonormal basis spanning the range of Aβ.
The columns of Vcorresponding to the vanishing wiform a basis of the nullspace of
Aβsince it is solution of the equation (13), i.e. of the subspace E[Γ]
βof the eigenmodes
of X/Γ with eigenvalue q.
This method was first implemented [26] to compute the lowest eigenvalues and
eigenfunctions for 12 hyperbolic manifolds. We have extended the method to include
any topology. In hyperbolic spaces, we recover the results of [26] and in the Euclidean
case we compared our result with the analytic solutions. When the universal covering
space is not compact, one has to specify the value of the degree of constraint c=M/N,
of the cut-off Land of the maximum radius ρmax up to which the images are considered.
Our main interest in this article is the case of spherical spaces and it enjoys a number
of simplifications. First, we can set ρmax =πsince the group Γ is finite. It follows
that any of the dpoints has exactly |Γ| − 1 images so that M=d|Γ|(|Γ| − 1)/2 and
we end with only two free parameters (L, d) or equivalently (L, c). This simplifies the
discussion concerning the choice of the different cut-offs L.
The parameter cis adjusted numerically. It needs to increase fast with the order
|Γ|of the group, but for each order cneeds to increase slowly with the increase of β.
For example for |Γ|= 8, c=|Γ|+βworks very well until β= 15. For β > 15, we
need either a smaller cor a cthat increases slower with β. The constraint determines
the dpoints that we need to generate randomly in the fundamental domain for each
β. Apparently the only reason for choosing a specific value for dis that the method
does not work without a good balance between M(number of rows) and N(number
of columns) for each |Γ|and β. A wrong choice of ccould cause the failure of the
method. The results of this method have been compared with Ikeda’s results and
agree with them.
3.2. Averaging method
When focusing on spherical spaces, one can take into account the fact that the number
of images of any point is exactly given by the order of the group, and is thus finite,
to develop another numerical method for computing the eigenfunctions, completely
independent from the previous one.
This method is based on the two remarks that
•if Ψ is an eigenmode, then Ψ ◦gis also an eigenmode
•for any function fon S3, then ¯
fdefined by
¯
f=1
|Γ|X
g∈Γ
f◦g(17)
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 10
is a Γ-invariant function of L2
Γ(S3), the indice recalling the Γ-invariance, which
can thus be identified to a function of L2(S3/Γ). The operation
avΓ:L2(S3)→L2
Γ(S3)
f7→ ¯
f(18)
is a projection on the subspace of Γ-invariant functions (since av2
Γ= 1 and
avΓ(f) = fiff fis Γ-invariant). This induces an equivalence relation g∼f
iff avΓ(g) = avΓ(f) so that L2(S3/Γ) is just given as the set of the functions of
L2(S3) modulo the equivalence relation, i.e. {avΓ(f), f ∈L2(S3)}
It follows that the eigenmodes of the Laplacian on S3/Γ are explicitly given in terms
of the eigenmodes of the Laplacian on the universal covering space (see Appendix A)
by
Ψ[Γ]
β,s =1
|Γ|X
g∈Γ
Yβℓm ◦g. (19)
Indeed, the (k+ 1)2linearly independent eigenmodes Ykℓm project down to (k+ 1)2
Γ-invariant eigenmodes Ψ[Γ]
k. This set of functions is a generator of the eigenspace
E[Γ]
kbut it is not a free family because we expect dim(E[Γ]
k)<(k+ 1)2. We will thus
have to pick up the dim(E[Γ]
k) independent functions of the family (19). This can
be performed by using an orthonormalisation procedure (the classical Gram-Schmidt
method itself being numerically disastrous).
Numerically, we compute the average family (19) and then decompose it on the
basis Ykℓm as
Ψ[Γ]
β,s =X
ℓ,m
ξβ,sℓmYβℓm (20)
and use the SVD method (as described in the previous section) to perform the
orthonormalisation. For that purpose we have to choose d= (k+ 1)2and L=β
so that there is no free parameter to choose and
c=1
2k+ 1
k+ 2 2
|Γ|(|Γ| − 1).(21)
3.3. Projection method
The ghosts method and the averaging method, presented in the two previous sections,
both compute an orthonormal basis Ψ[Γ]
βℓm for the space of eigenmodes of S3/Γ, so that
we may later compute a random eigenmode Ψ[Γ]
βof S3/Γ as a linear combination (11).
The projection method, by contrast, computes the random eigenmode Ψ[Γ]
βdirectly,
without explicitly computing a basis for the eigenspace. Throughout this section
we assume the wavenumber kis fixed, thus fixing the eigenvalue k(k+ 2) and the
parameter β=k+ 1 as well.
3.3.1. The algorithm The idea is to first compute a random eigenmode of S3, and
then project that random eigenmode down to an eigenmode of S3/Γ. More precisely,
one proceeds as follows:
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 11
Step 1: Construct a random eigenmode on S3.
Construct a random eigenmode Ψβon S3as a linear combination
Ψβ=X
ℓ,m
ζβℓm Yβℓm .(22)
Choose the coefficients ζβℓm relative to a Gaussian distribution with mean 0 and
standard deviation 1. The resulting point (ζβℓm )∈Rβ2will, in effect, be chosen
relative to a spherically symmetric distribution, because the product measure
Y
ℓ,m
exp(−ζ2
βℓm/2) = exp(−r2/2)
depends only on the radial distance r2=Pℓ,m ζ2
βℓm in Rβ2. Note that the
expected value E(ζβℓm) of each coefficient ζβℓm is 1, so the expected value of the
squared radius r2is simply the dimension β2of the space:
E(r2) = E(X
ℓ,m
ζ2
βℓm ) = X
ℓ,m
E(ζ2
βℓm ) = X
ℓ,m
1 = β2.
In the infinitely unlikely event that r= 0, discard the randomly chosen ζβℓm and
choose a new set.
Step 2. Construct the average avΓ(Ψβ).
Given the eigenmode Ψβof S3, define the Γ-invariant eigenmode avΓ(Ψβ) of S3/Γ
via the averaging formula (17). As mentioned earlier, a Γ-invariant eigenmode of
S3corresponds to an eigenmode of the quotient space S3/Γ. In principle we can
evaluate formula (17) for any x∈S3, but in practice we need evaluate it only for
those xlying on the last scattering surface.
Computational note: We begin with xin rectangular coordinates (x1, x2, x3, x4),
so that g(x) may be computed quickly and easily as a 4 ×4 matrix times a 4-
element vector. We then convert the result to spherical or toroidal coordinates
(χ, θ, ϕ) for efficient evaluation of Eq. (22).
3.3.2. Proof of orthogonality Let n=β2be the dimension of the full β-eigenspace
of S3, and let mbe the (unknown) dimension of the Γ-invariant subspace, i.e. m
is the dimension of the β-eigenspace of S3/Γ. Thus the averaging operator avΓ() in
Step 2 projects the eigenspace Rnof S3down onto the subspace Rmof Γ-invariant
eigenmodes. The question is, relative to the standard inner product in function space,
is this an orthogonal projection from Rnto Rm? If so, then a spherically symmetric
distribution of points in Rn(corresponding to random eigenmodes of S3) will project
to a spherically symmetric distribution of points in Rm(corresponding to random
eigenmodes of S3/Γ). However, if the projection is not orthogonal – for example if
it involves a shearing motion – then a spherically symmetric distribution of points
in Rnwill not project to a spherically symmetric distribution in Rm, but rather to
a distribution that is slanted to one side or another. Fortunately the projection is
orthogonal, as shown in Proposition I.
Proposition I. Let Hbe the space of all β-eigenmodes on S3, and HΓbe the space of
β-eigenmodes that are invariant under the action of a finite group Γ⊂SO(4). Then
the averaging function avΓdefined in Eq. (18) maps Horthogonally onto HΓ.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 12
Proof. Let H⊥be the orthogonal complement of HΓin H. That is, let H⊥consist of
those eigenmodes in Hthat are orthogonal to all eigenmodes in HΓ. Thus H≃Rn,
HΓ≃Rm, and H⊥≃Rn−mfor some nand m. Our goal is to show that the averaging
function avΓmaps H⊥to 0.
Let Ψ⊥be an element of H⊥. This means that hΨ⊥,ΨΓi= 0 for all ΨΓ∈HΓ.
The inner product h,iis invariant under the action of SO(4), so for every g∈Γ,
hΨ⊥◦g, ΨΓ◦gi= 0. But ΨΓis, by definition, invariant under Γ, so ΨΓ◦g= ΨΓ,
hence hΨ⊥◦g, ΨΓi= 0 for all ΨΓ∈HΓ. In other words, Ψ⊥◦g∈H⊥for all g∈Γ.
This implies that avΓ(Ψ⊥)∈H⊥. But avΓ(Ψ⊥) is also an element of HΓ. Because
HΓ∩H⊥= 0, this implies avΓ(Ψ⊥) = 0, as required. QED
3.3.3. Interpreting the norm The squared norm |avΓ(Ψβ)|2provides a statistical
estimate of the dimension of the Γ-invariant eigenspace.
Proposition II. For each β, the expected value of the squared norm havΓ(Ψβ),avΓ(Ψβ)i
is the dimension of the Γ-invariant eigenspace HΓ.
Proof. The comments in Step 1 of the algorithm in Section 3.3.1 show that the
expected squared norm of the original Ψβ(before averaging) tells the dimension of
the full eigenspace:
E(hΨβ,Ψβi) = E(Xζ2
βℓm ) = β2.
Choose an orthonormal basis {Yi}for the eigenspace such that the basis vectors
{Y1,...,Ym}span the Γ-invariant eigenspace HΓwhile the remaining basis vectors
{Ym+1,...,Yn}lie orthogonal to HΓ. Because the basis {Yi}is orthonormal, the
distribution exp(−r2/2) factors as a product
exp(−r2/2) =
n
Y
i=1
exp(−ζ2
i/2).
Proposition I implies that, relative to this basis, the averaging operator avΓ() preserves
the first mcoordinates while collapsing the remaining n−mcoordinates to zero. Thus
the distribution of avΓ(Ψβ) is given by the restricted product
m
Y
i=1
exp(−ζ2
i/2) = exp(−r2
Γ/2)
where r2
Γ=Pm
i=1 ζ2
i, and the above reasoning now implies that
E(havΓ(Ψβ),avΓ(Ψβ)i) = m,
as required. QED
Proposition II remains valid even if we do not explicitly construct the basis {Yi}.
We may instead compute the squared norm havΓ(Ψβ),avΓ(Ψβ)iby sampling points.
Taking 32 random eigenmodes Ψβand for each one evaluating havΓ(Ψβ),avΓ(Ψβ)i
at 32 random points yields the following rough estimates for the dimension of the
eigenspace (PDS designates the Poincar´e Dodecahedral Space of order 120):
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 13
k2 3 4 5 6 7 8 9 10 11 12
S39.1 17.1 22.2 40.0 44.1 60.7 83.9 106.4 129.5 144.6 163.9
P38.5 0.0 23.9 0.0 47.2 0.0 80.5 0.0 120.2 0.0 165.8
L(5,2) 1.4 3.6 3.9 6.6 7.9 12.7 16.5 19.0 26.9 25.1 31.9
PDS 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 11.8
The above computations took only a few minutes on a 300 MHz desktop computer.
Increasing the number of random eigenmodes and the number of sampling points
would increase the accuracy of the results at the expense of a longer computation
time.
3.3.4. Computational complexity The projection method is reasonably fast. The
choice of the ζβlm in Step 1 requires only O(β2) time, and is completed almost
instantaneously on a desktop PC. Thereafter the evaluation of avΓ(Ψβ)(x) for each
point x∈S3takes |Γ|times as long as evaluating the underlying random eigenmode
Ψβ(x) of S3at the same point x. In other words, using the projection method, an
eigenmode of S3/Γ is |Γ|times as expensive to compute as an eigenmode of S3. More
precisely, the time to evaluate Ψβ(x) grows as |Γ|β3,because for a fixed value of β
the eigenspace has dimension β2, meaning that there are β2terms to evaluate, each
of which requires O(β) steps. In practice we evaluated the eigenmodes of S3using
the toroidal coordinates method of [42], but in principle the same runtime could be
obtained using spherical coordinates.
4. Analytical solutions for lens and prism spaces
Besides the numerical methods presented above, there are special cases for which the
eigenfunctions can obtained analytically [42]. The results are indeed of importance
to test the accuracy of our numerical computations. The method is based on the use
of torus coordinates and applies to lens L(p, q) and prism S3/D∗
mspaces. We briefly
recall the main points and results of [42].
4.1. Summary of the general method
The key of the method is to choose a coordinate system that respects the holonomy
group Γ. We introduce the coordinates in R4, (x, y, z, w) by
x= cos χ′cos θ′
y= cos χ′sin θ′
z= sin χ′cos ϕ′
w= sin χ′sin ϕ′(23)
so that the equation of the 3-sphere is simply x2+y2+z2+w2= 1. Note that they
are different from the 4-dimensional coordinates (3) introduced previously and that
now the intrinsic coordinates have to range as
0≤χ′≤π/2,0≤θ′≤2π0≤ϕ′≤2π. (24)
For each fixed value of χ′∈[0, π/2], the θ′and ϕ′coordinates sweep out a torus. Taken
together, these tori almost fill S3. The exceptions occur at the endpoints χ′= 0 and
χ′=π/2, where the stack of tori collapses to the circles x2+y2= 1 and z2+w2= 1,
respectively.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 14
Identifying the eigenmodes of S3/Γ and the Γ-invariant eigenmodes of S3, as
explained in our introductory remark, the eigenmodes of a lens or prism space are
given by Zp-invariant or D∗
m-invariant eigenmodes of S3.
An elementary construction [42] shows that for each wavenumber k, with
eigenvalue k(k+ 2), the corresponding eigenspace of S3is spanned by the basis
Bk={ Qkℓm | |ℓ|+|m| ≤ kand |ℓ|+|m| ≡ k(mod 2) }(25)
where
Qkℓm = cos|ℓ|χ′sin|m|χ′P|m|,|ℓ|
d(cos 2χ′)
×(cos |ℓ|θ′or sin |ℓ|θ′)×(cos |m|ϕ′or sin |m|ϕ′) (26)
P|m|,|ℓ|
dbeing the Jacobi polynomial
P|m|,|ℓ|
d(x) = 1
2d
d
X
i=0 |m|+d
i|ℓ|+d
d−i(x+ 1)i(x−1)d−i
and cos |ℓ|θ′(resp. sin |ℓ|θ′) being used when ℓ≥0 (resp. ℓ < 0), and similarly for the
choice of cos |m|ϕ′or sin |m|ϕ′.
It is straightforward to see how the generating isometry of a lens space L(p, q ),
given in rectangular coordinates by
cos 2π/p −sin 2π/p 0 0
sin 2π/p cos 2π/p 0 0
0 0 cos 2πq/p −sin 2πq/p
0 0 sin 2πq/p cos 2πq/p
(27)
or in toroidal coordinates by
χ′→χ′
θ′→θ′+ 2π/p
ϕ′→ϕ′+ 2πq/p, (28)
acts on the Qkℓm . The eigenmodes of L(p, q) comprise the fixed point set of this
action. A set of simple numerical conditions tells how to select an orthogonal basis for
this fixed point set, essentially as a subset of the basis Bk. Specifically, the eigenbasis
for L(p, q) includes
Ψ[L(p,q)]
kℓm =
Qk00 always
Qk±ℓ0iff ℓ≡0(modp)
Qk0±miff qm ≡0(modp)
Qkℓm+Qk−ℓ−m
√2,Qk−ℓm−Qkℓ−m
√2iff ℓ≡qm(modp)
Qkℓm−Qk−ℓ−m
√2,Qk−ℓm+Qkℓ−m
√2iff ℓ≡ −qm(modp)
(29)
For details as well as for the explicit form of the eigenmodes of prism spaces, please
see Ref. [42]. A similar analysis yields an explicit eigenbasis for a prism space. The
Qkℓm are already mutually orthogonal, so after normalizing them to unit length we
may use the above basis to construct unbiased random eigenmodes of a lens or prism
space, with wavenumber k.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 15
4.2. Extracting the coefficients ξβaℓm
The previous analysis gives the decomposition of the eigenmodes on the basis Qβℓm
as
Ψβ,s =Xηβ,sℓmQβ ℓm(χ′, θ′, ϕ′) (30)
but what we need are the coefficients ξβ,sℓm of the decomposition on the basis Yβ ℓm
as given in Eq. (11).
Yβℓm and Qβℓm are two basis of dimension (k+ 1)2for each β=k+ 1, so up to
a change of coordinates between toroidal and spherical coordinates, we can write
Qβℓm =Xαβℓℓ′mm′Yβℓ′m′.(31)
Note that (ℓ, m) and (ℓ′, m′) do not vary in the same range since 0 ≤ℓ′≤β−1,
|m′| ≤ ℓ′and |ℓ|+|m|< β,|ℓ|+|m| ≡ β−1 (mod2). The coefficients αβℓℓ′mm′are
explicitly given by
αβℓℓ′mm′=ZQβℓm(χ′, θ′, ϕ′)Y∗
βℓ′m′(χ, θ, ϕ) sin2χsin θdχdθdϕ(32)
where (χ′, θ′, ϕ′) are functions of (χ, θ, ϕ). It can be checked from (3) and (23) that
χ′= a rccos [cos χcos θ] (33)
θ′= a rccos "cos χsin θ
p1−cos2χcos2θ#(34)
ϕ′=ϕ. (35)
It can be checked that the two basis differ by more than just a change of coordinates.
It follows that for the particular case of lens and prism spaces, the computation goes
as follows:
(i) Determine the coefficients ηβ,sℓm as given in Ref. [42]; thus mainly a tedious
book-keeping operation.
(ii) Compute the coefficients of the change of basis (31); this has to be performed
once for all for all spaces.
(iii) The required coefficients are obtained by a matrix multiplication
ξβ,sℓm =X
ℓ′m′
αβℓℓ′mm′ηβ,sℓ′m′.(36)
4.3. The simplest example
For the projective space, S3/Z2, there is only one generator which brings any point
to its antipodal point. It follows that the eigenfunctions are just the average of the
spherical harmonics evaluated with their antipodal equivalent. This can be sorted out
analytically and one easily obtains that
ξ[Z2]
β,sℓm =1
21−(−1)βδs,(ℓm)(37)
where the function δs,(ℓm)= 1 if the label scan be identified with (ℓ, m) and is zero
otherwise. We recover the results from figure 1, that is that there is no modes for β
even and that the dimension of the eigenspace for βodd is equal to the one of S3,
that is (k+ 1)2.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 16
5. Numerical results
An interesting property is the distribution of the number of modes per wavenumber
interval. In hyperbolic manifolds, the number N(≤q) of modes smaller than qis well
described by the Weyl asymptotic formula
N(≤q)∼Vol(H3/Γ)
6π2(q2−1)3/2(38)
for q≫1. In the case of the 3-sphere, q2=k(k+2) with multiplicity mult(q) = (k+1)2
so that
N[S3](≤q) =
k
X
p=2
(p+ 1)2=(k+ 1)(k+ 2)(2k+ 3) −30
6(39)
and the Weyl formula, which applies also to the spherical manifolds, tells us that
N[Γ] ∼N[S3]/|Γ|.(40)
We computed the eigenmodes and eigenfunctions of some spherical spaces with
the different methods described above. First, it was checked that the spectra agree
with the theoretical ones in the cases described in table 1. In the particular case of
lens and prism spaces, the eigenmodes and eigenfunctions agree with the ones obtained
analytically in Ref. [42].
The computational time of the averaging method is experimentally found to be
proportional to |Γ|β4.68 , the typical running time being less than 10 seconds on a
desktop computer for β < 15.
In figures 4 and 5, we present the two examples of the lowest modes of S3/D∗
2
and S3/Z8. In figure 6 we depict one of the ten eigenmodes of S3/D∗
2with k= 4 for
different values of the radial coordinate χ.
6. Some cosmological implications
The goal of this section is not to compute the CMB anisotropies in details (this task
will be delt with in a follow-up article [43]), but to give estimate of the expected effects
on large angular scales.
The evolution of the scale factor, a, of the universe is dictated by the Friedmann
equation that can be recast under the form
H
H02
= Ωr0x−2+ Ωm0x−1+ ΩΛ0x2+ (1 −Ωr0−Ωm0−ΩΛ0)(41)
where H ≡ a′/a, a prime denoting a derivative with respect to the conformal time η.
We have introduce x≡a/a0≡(1 + z)−1,zbeing the redshift and a0the value of the
scale factor today. The density parameters are defined by
Ωr0≡κρr0
3H2
0
,Ωm0≡κρm0
3H2
0
,ΩΛ0≡κρΛ0
3H2
0
(42)
where κ≡8πG and H0=H0/a0. In terms of these quantities, the physical curvature
radius today is given by
Rphys
C0≡a0RC0=c
H0
1
p|ΩΛ0+ Ωm0−1|(43)
We can choose a0to be the physical curvature radius today, i.e. a0=Rphys
C0, which
amounts to choosing the units on the comoving sphere such that RC0= 1, hence
determining the value of the constant a0=c|ΩΛ0+ Ωm0−1|−1/2/H0.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 17
Figure 4. The ten eigenmodes of S3/D∗
2for k= 4 in Hammer-Aitoff pro jection
for χ=π/2. Note that some are identical but only on the equator and it can be
checked that these modes are indeed linearly independent.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 18
Figure 5. The three eigenmodes of S3/Z8for k= 2 in Hammer-Aitoff projection
for χ=π/2.
6.1. Generalities
When studying the CMB anisotropies, one has to go beyond the homogeneous and
isotropic description of our universe and need to condider a perturbed spacetime with
metric
ds2=a2(η)−(1 + 2Φ)dη2+ (1 −2Ψ)γij dxidxj(44)
where we consider only scalar modes and working in longitudinal gauge. Vector and
tensor modes would have to be added for a complete description (including for instance
gravitational waves, but are negligible on large angular scales. On these scales, we
can neglect the effect of the anisotropic pressure so that Ψ = Φ and the effect of the
radiation between the last scattering surface and today.
Under these assumptions, the temperature fluctuation in a direction ncan be
related [44, 45] to the gravitational potential Φ by, again in the particular case of
adiabatic initial perturbations,
δT
T(n) = 1
3Φ[ηLSS ,(η0−ηLSS )n] + 2 Zη0
ηLSS
∂Φ[η, (η0−η)n]
∂η dη(45)
where ηLSS and η0are the value of the conformal cosmic time at the emission of
the photon (last scattering surface) and at the reception (observer). According to the
standard nomenclature, we will refer to the first term as the ordinary Sachs-Wolfe term
(OSW) and to the second as the integrated Sachs-Wolfe term (ISW). The temperature
angular correlation function is then defined by
C(θ)≡δT
T(n1)δT
T(n2)cos θ=n1.n2
.(46)
Decomposing the temperature fluctuation on the spherical harmonics as
δT
T(n) = X
ℓ
ℓ
X
m=−ℓ
aℓmYℓm (θ, φ) (47)
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 19
Figure 6. A mode of S3/D∗
2with k= 4 for 10 different values of χ=iπ/20
(i= 1..10). It must be read left to right, top to bottom.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 20
the coefficients Cℓof the development of C(θ) on Legendre polynomials are given by
(2ℓ+ 1)Cℓ=
ℓ
X
m=−ℓ
haℓma∗
ℓmi.(48)
To compute these quantities, one needs to determine the gravitational potential
Φ. Its evolution is dictated [22] by the equation
Φ′′ + 3H(1 + c2
s)Φ′−c2
s∆2Φ + [2H′+ (1 + 3c2
s)(H2−K)]Φ = 0.(49)
This relation strictly holds only for initial adiabatic perturbations as predicted by
most of the inflationary scenarios. c2
s=P′/ρ′is the sound speed and is given by
c2
s=ρr/(ρm+ 4/3ρr)/3. After decomposing Φ on the eigenmodes as
Φ(η, x) = X
β,s
Φβ,s Ψ[Γ]
β,s (x),(50)
one can easily show that if the universe is matter dominated between the last scattering
epoch and today then
Φβ,s (η) = F(η)Φ0(q) (51)
where we use the notation that q= (β , s). Φ0is the value of the gravitational potential
at the beginning of the matter era, but since in the early universe the curvature
term is negligible and the dynamics is dominated by the radiation (so that a∝η)
it can be shown that, for long wavelengths (kη ≪1), the non decaying mode of the
gravitational potential is constant so that Φ0is in fact the primordial gravitational
potential. Inflationary theories predict that it is a Gaussian field, and that all modes
are independent, with power spectrum [46]
hΦ0(q)Φ∗
0(q′)i=2π2
β(β2−K)PΦ(β)δ(β−β′)δss′.(52)
where K= 0,±1 is the sign of the curvature. In the case of a scale invariant Harrison-
Zel’dovich spectrum, PΦ∝β0.
Now, inserting the decomposition (50) with the solution (51) in Eq. (45) and
decomposing the eigenmodes as in Eq. (11), one finally gets that the coefficients of
the development (47) are given by
aℓm =X
β,s
Φ0(β, s)ξβ,sℓmGβ ℓ (53)
where Gβℓ is defined by
Gβℓ ≡1
3F(ηLSS )Rβℓ(η0−ηLSS ) + 2 ZηLSS
η0
F′(η)Rβℓ (η0−η)dη. (54)
The correlation
haℓmaℓ′m′i=X
β,s
2π2
β(β2−K)PΦ(β)Gβℓ Gβℓ′ξβ ,sℓmξ∗
β,sℓ′m′(55)
has non-zero off-diagonal terms which reflects the fact that there is a global anisotropy
due to the non–trivial topology. In a simply–connected homogeneous and isotropic
universe haℓmaℓ′m′i=Cℓδℓℓ′δmm′. These off-diagonal terms are characteristic of the
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 21
non-Gaussianity induced by the topology. From the expression (55), we can extract
the Cℓwhich characterise only the isotropic part of the temperature distribution, as
(2ℓ+ 1)C[Γ]
ℓ=X
β,s,m
2π2
β(β2−K)PΦ(β)|ξβ,sℓm|2|Gβℓ|2.(56)
Indeed, in the Euclidean and hyperbolic cases, the sum over βhas to be replaced by
an integral; in the spherical case, β≥max(3, ℓ + 1). This result has to be compared
to its covering space analog, obtained by setting ξβ,sℓm = 1 when the index scan be
assigned the value (ℓm) and zero otherwise, so that the sum over mgives (2ℓ+ 1),
C[U]
ℓ=X
β
2π2
β(β2−K)PΦ(β)|Gβℓ|2,(57)
so that there is an average effect with the ponderation |ξβ ,sℓm|2/(2ℓ+ 1).
The effect of the topology on large scales can thus be investigated, in a first step,
by considering the index
Υ[Γ]
ℓ≡ |Γ|C[Γ]
ℓ
C[U]
ℓ
.(58)
At large angular scales, Υℓwill oscillate due to the missing eigenmodes while it
will converge to unity, mainly because of Eq. (40), on small angular scales where
the topology becomes irrelevant. Indeed, this will allow to put constraints on some
topologies but an unambiguous detection will have to use a full-sky CMB map.
6.2. Example of the three-torus
As a first example, let us consider a cubic 3-torus of comoving size L. In the simplest
case in which ΩΛ= 0, Fis constant and
C[T3]
ℓ=X
k
PΦ(k)
k3L3j2
ℓ(k(η0−ηLSS )),(59)
with k= 2π(n1, n2, n3)/L, and for the universal covering space we have
C[U]
ℓ∝Zdu
uj2
ℓ(u)∝1
ℓ(ℓ+ 1) (60)
known as the Sachs-Wolfe plateau.
On this simple example, we can determine the cut-off ℓcut below which there is a
suppression of the power spectrum. The first approach is to remind that the Bessel
functions peak at k(η0−ηLSS )∼ℓ. Hence a mode kcontributes maximally to the
angle subtended by its corresponding scale at last scattering. In flat models, η0is
given [47] by
η0≃2ca0
H0pΩm01 + ln Ω0.085
m0.(61)
It follows that the OSW term has a cut-off round
ℓcut ∼4πc
L0H0pΩm01 + ln Ω0.085
m0.(62)
This is analogous to the estimate by Inoue [48].
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 22
Another approach, first introduced in [49, 50], is to compute the angle θcut under
which the maximum comoving wavelength λmax =Lat last scattering. It is given by
θcut =a0L
(1 + z)dA(z)(63)
where dA= dL/(1 + z)2is the angular distance and where the luminosity distance is
given by
dL(z) = (1 + z)ca0
H0
fZz
0
du
(1 + u)E(u).(64)
In the case where ΩΛ= 0, dL(z) = 2a0(1+z)(1−(1+z)−1/2)/H0so that θcut ∼H0L/2.
We have introduced E≡ H/H0. Now, the ℓth Legendre polynomial is a polynomial
of degree ℓin cos θand has ℓzeros in [−1,1] (or 2ℓzeros in [−π, π ] if working in θ)
with approximatively the same spacing. We can estimate that θ∼π/ℓ and thus that
the cut in the OSW contribution is expected to be round
ℓcut ∼2πc
H0LfZzLSS
0
du
(1 + u)E(u),(65)
the factor 2 arising from the fact that an oscillation corresponds to 2 zero. This
corresponds exactly to the previous estimate (62).
6.3. Spherical spaces
In the spherical case we expect the same kind of effects, i.e. a suppression of the large
scale ISW effect due of the existence of a maximal wavelength. The OSW term will
be approximatively the same as the one computed in the flat case since our universe,
even if closed, is still very flat.
Let us first remark a crucial difference between Euclidean manifolds and spherical
manifolds. For the former, as we have seen in the previous section, the smallest
multipole is directly related to the size Lof the fundamental polyhedron so that one
cannot consider too small universes. For spherical universes, the situation is a priori
different. As seen on the example of single-action manifolds, the value of the first
non-zero eigenvalue does not depend on the order of the group, at least for cyclic and
binary dihedral groups.
Let us take the example of lens spaces, the first nonzero eigenvalue is indeed the
same for all S3/Zm(m > 1), namely k= 2 and eigenvalue 2(2+2) = 8. However, the
multiplicity is 3 for homogeneous lens spaces L(p, 1), but only 1 for nonhomogeneous
lens spaces L(p, q), as shown in Table 1 of [42]. This can be understood by the fact
that the space is becoming smaller and smaller in only one direction. In perpendicular
directions the space remains large. The waves do not have distinct peaks (like
mountaintops found in nature) but rather have ridges (perfect horizontal ridges, which
are never found in nature). First imagine a wave in S3, as follows: the set of maxima
is a great circle, while the set of minima is a complementary great circle. As time
passes, the wave goes up and down, so what is the top at a time t= 0 becomes the
bottom at time t= 1/2 say, and vice versa, returning to its original position at time
t= 1. Midway between the “top ridge” and the “bottom ridge” is a torus which
remains at height 0 for all times t. In toroidal coordinates (23), one ridge is the circle
at χ= 0, the other ridge is the circle at χ′=π/2, and fixed torus lies at χ′=π/4.
This wave is preserved by all corkscrew motions along the natural axes, so it pro jects
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 23
down to an eigenmode of all lens spaces L(p, q). In the notations of Section 4, it is
given in toroidal coordinates as
Ψ200(χ′, θ′, φ′) = sin2χ′−cos2χ′=−cos 2χ′
thus verifying that it is a wave as described above, and in particular with no
dependence on θ′or φ′. In the special case of a homogeneous lens space L(p, 1),
we get two more eigenmodes
sin(2χ′) cos(θ′−φ′) and sin(2χ′) sin(θ′−φ′).
These modes are constant along helices (where θ′−φ′is constant), so they are
eigenmodes of all homogeneous lens spaces, whose holonomies are Clifford translations,
but not eigenmodes of nonhomogeneous lens spaces.
We thus expect the effect of the cut-off in the Sachs-Wolfe plateau to be milder
than for Euclidean spaces. But, on the other hand we expect to have a more irregular
Sachs-Wolfe plateau due of the fact that some wavelengths are missing from the
spectrum. Another observational consequences missed by the angular power spectrum
is a global large scale anisotropy that, at least, is expected for non-homogeneous lens
spaces.
To illustrate this, consider the simplest example S3/Z2for which half of the
modes have disappeared so that k= 2p,pbeing an integer. It follows that for ℓ∼2p,
C[Z2]
ℓ∼C[S3]
ℓand for ℓ∼(2p+ 1), C[Z2]
ℓ∼0. We thus expect that the Cℓcurve
oscillates around C[S3]
ℓ/2 with a frequency of order ℓ∼η0.
Using the coefficients (37) for the projective space, we plot in figure 7, the function
Υℓfor ℓ≤20 for different curvature radius. We do not include the cosmic variance.
This confirms the previous semi-analytical analysis and is in agreement with the
numerical computations performed for S3/Z2[21]. When |Ω−1| ≪ 1, C[Z2]
ℓ∼C[S3]
ℓ
because χLSS < π/2, the topological scale. When χLSS ∼π/2, both antipodal points
on the last scattering surface are close to the equator so that the angular correlation
function in opposite directions is expected to be higher. The case in which χLSS > π/2
is even more intricate because the geodesics are warping around the universe more
than once.
Such features are general to all spherical spaces, but the higher the order of the
group, the larger the minimal curvature radius to get a topological signal. Since the
size of the manifold decreases with the order of the group, there will always exist
potentially detectable topologies even for spaces very close to flatness.
Note also that in the case of the projective space,
haℓmaℓ′m′i ∝ δℓℓ′δmm′
which can be understood if one remembers that there is no breaking of global isotropy
and homogeneity for the projective plane. This is the only exception for which one
can have a non-trivial topology and no preferred direction.
7. Conclusion
In this article, we have investigated in details the structure of the eigenmodes of
the Laplacian operator in spherical spaces. A series of analytical and mathematical
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 24
0,4
0,6
0,8
1
1,2
1,4
1,6
0 5 10 15 20
0,4
0,6
0,8
1
1,2
1,4
1,6
0 5 10 15 20
0,4
0,6
0,8
1
1,2
1,4
1,6
0 5 10 15 20
0,4
0,6
0,8
1
1,2
1,4
1,6
0 5 10 15 20
0,4
0,6
0,8
1
1,2
1,4
1,6
0 5 10 15 20
0,4
0,6
0,8
1
1,2
1,4
1,6
0 5 10 15 20
Figure 7. The Υℓfor ℓ < 20 for S3/Z2. From top to bottom and left to right the
curvature increases as Ω = 1.1,1.25,1.5,1.7,1.8,1.9 with Λ = 0. This corresponds
respectively to χLSS = 0.5939,0.8990,1.1945,1.3528,1.4173,1.4746. For higher
curvature, the topologicaal scale has more and more importance. The growth of
the amplitude of the oscillations at small multipole is dur to the growth of the Cℓ
due to the integrated Sachs-Wolfe effect.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 25
results have been either reviewed or obtained and we have introduced various efficient
numerical methods to compute them. These methods were compared together and to
some analytical results.
We have also investigated some cosmological consequences of this work par-
ticularly concerning the large angular scales of the cosmic microwave background
anisotropies. The effect of the topology in CMB calculation has been described and
these results are now being included to a full Boltzmann code to simulate CMB maps
with a topological signal [43]. As an example, we considered the simplest of all cases,
that is the pro jective space.
Appendix A. Eigenmodes of the Laplacian of homogeneous and isotropic
simply connected three-dimensional spaces
This appendix follows the work by Abbott and Schaeffer [51] and Harrison [52]. Its
goal is to summarize the derivation and explicit forms of the scalar harmonic functions
solutions of the Helmoltz equations (1).
We rewrite the Friedmann metric (2) as
ds2=−dt2+a2(t)
σ2(r)dr2+r2dΩ2(A.1)
with σ(r) = 1 + Kr2/4. By splitting radial and angular variables, it can be shown
that the eigenmodes decompose as
Yβℓm(r, θ, φ) = Rβℓ (r)Yℓm(θ, φ).(A.2)
Yℓm(θ, φ) are the spherical harmonics, related to associated Legendre polynomials Pm
ℓ
by
Yℓm(θ, φ)≡(2ℓ+ 1)(ℓ−m)!
4π(ℓ+m)! 1/2
Pm
ℓ(cos θ)eimφ (A.3)
and satisfy the relation
Yℓm(θ, φ) = (−1)mY∗
ℓm(θ, φ).(A.4)
The radial eigenfunctions Rβ ℓ(r) are solutions of the radial harmonic equation
σ3(r)
r2
d
drr2
σ(r)
dRβℓ
dr+q2−σ2(r)ℓ(ℓ+ 1)
r2Rβℓ = 0.(A.5)
In flat space (K= 0) the radial eigenfunctions are simply the spherical Bessel
functions Rβℓ =jℓ(βr).
The cases K=±1 can be treated simultaneously by using the variable ξdefined
by
sin ξ=K1/2r
σ(r).(A.6)
Note that ξis related to the dimensionless comoving radial distance in units of the
curvature radius by ξ=K1/2χ. In terms of these variables the radial equation (A.5)
takes the form
1
sin2ξ
d
dξsin2ξdRβℓ
dξ+Kq2−ℓ(ℓ+ 1)
sin2ξRβℓ = 0.(A.7)
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 26
Introducing the function Πβℓ (ξ) = Rβℓ (ξ) sin1/2(ξ) allows to solve the radial equation
in terms of associated Legendre functions Pµ
ν(cos ξ).
For K= +1 (i.e. β2=q2+ 1)+, the radial eigenfunctions are given by
Rβℓ (χ)∝1
sin2χP−1/2−ℓ
−1/2+β(cos χ) (A.8)
with β≥max(3, ℓ + 1) being an integer∗.
For K=−1 (i.e. β2=q2−1) the radial eigenfunctions are given by
Rβℓ (χ)∝1
sinh2χP−1/2−ℓ
−1/2+iβ (cosh χ) (A.9)
and βcan now take on any positive real value since there are no periodic boundary
conditions to satisfy.
We use the normalisation condition
ZY∗
βℓm Yβ′ℓ′m′
r2drdΩ
σ3=δ(β−β′)δℓℓ′δmm′,(A.10)
so that the properly normalized functions take the following form
Rβℓ (χ) =
Nβℓ
sinh χ1/2
P−1/2−ℓ
−1/2+iβ (cosh χ)K=−1
2β2/π1/2jℓ(βχ)K= 0
Mβℓ
sin χ1/2
P−1/2−ℓ
−1/2+β(cos χ)K= +1
(A.11)
with the two coefficients
Nβℓ ≡
ℓ
Y
n=0
(β2+n2)Mβℓ ≡
ℓ
Y
n=0
(β2−n2).(A.12)
The radial eigenfunctions (A.11) differ from those determined by Abbott and
Schaeffer [51] by an overall factor (2β2/π)−1/2due to the fact that they used the
normalisation
ZY∗
βℓm Yβ′ℓ′m′
r2drdΩ
σ3(r)=π
2β2δ(β−β′)δℓℓ′δmm′.(A.13)
To finish, in the case of spherical space, the harmonic functions can be expressed in
terms of the 4-dimensional coordinates (3) as
Ykℓm(x) = (2ℓ+ 1)(ℓ−m)!
4π(ℓ+m)! 1/2 Mβℓ
p1−x2
0!1/2
P−1/2−ℓ
−1/2+β(x0)Pm
ℓ x2
p1−x2
0!x1+ix2
px2
1+x2
2
.(A.14)
Those expressions are of little value for numerical computation. There are two
routes to compute numerically the eigenmodes. First, and as explained in Abbott
and Schaeffer [51], one can use a recursive relation between Rβℓ,Rβ ,ℓ−1and Rβ,ℓ−2.
Another efficient method [53] makes use of a WKB approximation. These two methods
are complementary.
+We recall that β2=q2+K.
∗It is well known that homogeneous harmonic polynomials of degree kon R4restricted to S3are
eigenmodes of the Laplacian with eigenvalues k(k+ 2). It follows that β=k−1 is necessarily an
integer.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology 27
Acknowledgments
We thank S. Helgason for discussions on the Laplacian operator during the
Williamstown meeting of the American Mathematical Society, and Alain Riazuelo and
Simon Prunet for discussions on the numerical CMB computations. JW thanks the
MacArthur Foundation for its support. EG thanks FAPESP-Brazil (Proc 01/10328-6)
for financial support.
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