Bipolar Harmonic encoding of CMB correlation patterns
aa r X i v : . [ a s t r o - ph . C O ] M a r Bipolar Harmonic encoding of CMB correlation patterns
Nidhi Joshi ∗ , S. Jhingan † , Tarun Souradeep ‡ , and Amir Hajian § Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi-110025, India IUCAA, Post Bag 4, Ganeshkhind, Pune-411007, India Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544
Deviations from statistical isotropy can be modeled in various ways, for instance, anisotropiccosmological models (Bianchi models), compact topologies and presence of primordial magneticfield. Signature of anisotropy manifests itself in CMB correlation patterns. Here we explore thesymmetries of the correlation function and its implications on the observable measures constructedwithin the Bipolar harmonic formalism for these variety of models. Different quantifiers within theBipolar harmonic representation are used to distinguish between plausible models of breakdown ofstatistical isotropy and as a spectroscopic tool for discriminating between distinct cosmic topology.
PACS numbers: 98.70.Vc, 98.80.Es
I. INTRODUCTION
The fluctuations in Cosmic Microwave Background(CMB) contain an amazing amount of information aboutour universe. Detailed measurements of anisotropy in theCMB reveal global properties, constituents and history ofthe universe. In standard cosmology, CMB anisotropy isassumed to be statistically isotropic and Gaussian. Gaus-sianity implies that the statistical properties of the tem-perature field can be completely characterized in termsof its mean < ∆ T > = 0, and auto-correlation function C (ˆ n , ˆ n ) = < ∆ T (ˆ n )∆ T (ˆ n ) > , where ˆ n = ( θ, φ ), is aunit vector on the sphere. The angular brackets < .. > denote ensemble expectation values, i.e, averages aboveare for all possible realizations of the field over a sphere.Since we have one CMB sky, that is just one out ofall possible realizations, the ensemble expectation value C (ˆ n , ˆ n ) can be estimated in terms of sky averages onlyto a limited extent, depending on underlying symmetriesin C (ˆ n , ˆ n ). Under the usual assumption of StatisticalIsotropy (SI), implying essentially Einstein’s cosmologi-cal principle for cosmological perturbations, the correla-tion function is invariant under rotations. It implies thecorrelation function C (ˆ n , ˆ n ) = C (ˆ n . ˆ n ) ≡ C ( θ ), canbe readily estimated by averaging over all pairs of skydirections separated by an angle θ .Spherical harmonics form a basis of the vector spaceof complex functions on a sphere, making them a naturalchoice for expanding the temperature anisotropy field,∆ T (ˆ n ) = X lm a lm Y lm (ˆ n ) . (1)Here ∆ T is the temperature fluctuation around some av-erage temperature T . The complex quantities a lm are ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] drawn from a Gaussian distribution, related to the Gaus-sian temperature anisotropy as a lm = Z d Ω ˆ n Y ∗ lm (ˆ n )∆ T (ˆ n ) . (2)The condition for SI now takes the form of a diagonalcovariance matrix, < a l m a ∗ l m > = C l δ l l δ m m . (3)Here C l is the well known angular power spectrum. Inthe SI case, the angular power spectrum carries completeinformation about the Gaussian field, and the statisticalexpectation values of the temperature fluctuations arepreserved under rotations in the sky. This property ofCMB has been under scrutiny since the release of the firstyear of WMAP data. Tantalizing evidence for statisticalisotropy violation in the WMAP data using a variety ofstatistical measures has also been claimed in recent lit-erature [1–3]. However, the origin of these ‘deviations’from SI remains to be modeled adequately. These devia-tions could be either genuinely cosmological, or statisticalcoincidence, or residual foreground contamination, or, asystematic error in the experiment and the data process-ing. Hence, it is important to carry out a systematicstudy of SI violations using statistical measures within aunified, mathematically complete, framework. Moreover,it is important to develop several independent statisticalmeasures to study SI violations that can capture differ-ent aspects of any measured violation and provide hintstoward its origin.While testing a fundamental assumption, such as SI,is in itself a justifiable end, there are also strong theo-retical motivation to hunt for SI violations in CMB onlarge scales. Topologically compact spaces [4–10] andanisotropic cosmological models [11–13], Cosmologicalmagnetic fields generated during an early epoch of in-flation [14] can also lead to violation of SI [15], are but afew examples.This paper focuses on linking measures of SI viola-tion to the reduced symmetries of the underlying cor-relation patterns in the CMB map or the correlationfunction. While we present illustrative examples of thesymmetries from various mechanisms of SI violation, thispaper does not concern itself with a study of specificmechanisms. We define, within the framework of Bipo-lar harmonic representation of CMB sky maps, a num-ber of observables that can be used to quantitatively testSI. We present a study of the properties of bipolar mea-sures as one systematically reduces the rotational symme-tries of the CMB correlations, as is expected in differenttheoretical scenarios. We recapitulate the bipolar har-monic representation and the definitions of a set of mea-surable quantities representing SI violation in section II.In Section III, these observable measures are computedfor different levels of residual rotational symmetries ofCMB correlations. This provides a clear understandingof the underlying symmetries revealed through the differ-ent bipolar measures. Section IV deals with bipolar for-malism measurables using Bianchi template. Section Vsummarizing conclusions and discussions is followed byappendices where details of the calculations leading toresults are presented. II. BIPOLAR FORMALISM AND THEOBSERVABLE MEASURES
Any deviations from SI introduces off-diagonal termsin the covariance matrix Eq. (3), thereby making C l aninadequate quantity to characterize the statistical prop-erties of the temperature field [16]. Under such a sit-uation Bipolar spherical harmonic expansion, proposedby Hajian and Souradeep [17, 19–23], proves to be themost general representation of the two point correlationfunction, where the angular power spectrum C l is a sub-set of Bipolar spherical harmonic coefficients (BipoSH).Two point correlation function of CMB anisotropies canbe expanded as C (ˆ n , ˆ n ) = X l ,l ,L,M A LMl l { Y l (ˆ n ) ⊗ Y l (ˆ n ) } LM , (4)here A LMl l are Bipolar Spherical Harmonic coefficients(BipoSH), | l − l | ≤ L ≤ ( l + l ), m + m = M ,and { Y l (ˆ n ) ⊗ Y l (ˆ n ) } LM are Bipolar spherical harmon-ics [24]. Bipolar spherical harmonics form an orthonor-mal basis on S × S , with transformation properties un-der rotations similar to spherical harmonics. The tensorproduct in harmonic space can be explicitly written usingClebsch-Gordan coefficients C LMl m l m as, { Y l (ˆ n ) ⊗ Y l (ˆ n ) } LM = X m m C LMl m l m Y l m (ˆ n ) Y l m (ˆ n ) . (5) In this paper, we use the term ‘correlation patterns’ to inter-changeably refer to SI violation
A. Bipolar spherical harmonic coefficients -BipoSH
BipoSH can be extracted by inverse transformationof Eq. (4), i.e., multiplying both sides of this equationby { Y l ′ (ˆ n ) ⊗ Y l ′ (ˆ n ) } ∗ L ′ M ′ , and using orthonormality ofBipolar spherical harmonics. Hence, given a real spacecorrelation pattern BipoSH coefficients can be found us-ing A LMl l = Z d Ω ˆ n d Ω ˆ n C (ˆ n , ˆ n ) { Y l (ˆ n ) ⊗ Y l (ˆ n ) } ∗ LM . Since C (ˆ n , ˆ n ) is symmetric under the exchange of ˆ n and ˆ n , this gives rise to following symmetry propertiesof BipoSH: A LMl l = ( − l + l − L A LMl l A LMll = A LMll δ L, k . k = 0 , , , , . . . . (6)Hence, A LMll exists for even L and vanishes otherwise. Itwas shown in [17] that the Bipolar Spherical Harmonic(BipoSH) coefficients A LMl l are a linear combination ofelements of the harmonic space covariance matrix includ-ing the off-diagonal elements that encode SI violation, A LMl l = X m m < a l m a ∗ l m > ( − m C LMl m l − m . (7)When SI holds the covariance matrix is diagonal, Eq. (3)and Clebsch property (H6), therefore A LMl l = ( − l C l (2 l + 1) / δ l l δ L δ M , (8)implying that A ll contains all the information on thediagonal harmonic space covariance matrix given by C l .The well known power spectrum C l thus forms a sub-space of BipoSH [19]. Under SI, the only non-zero Bipo-lar spherical harmonic coefficient will be A ll (equivalentof C l ), all the rest must vanish. The violation of SI thusimplies A ll are not sufficient to describe the field. Hence,BipoSH proves to be a better tool to test SI, as non-zero A LMl l , other then L = 0 and M = 0, terms should confirmits violation.It is impossible to measure all A LMl l from just one CMBmap because of cosmic variance. Thus we need to com-bine them in different ways to diagnose different aspectsof SI violations. B. Bipolar power spectrum- BiPS
The
Bipolar Power Spectrum (BiPS) is a rotationallyinvariant, quadratic measure that can be constructed outof BipoSH coefficients [17]. BiPS involves averaging overBipoSH that reduces cosmic variance in comparison to asingle CMB map, however this does not erase all the SIsignatures. BiPS is defined as κ L = X l ,l ,M | A LMl l | . (9)For statistically isotropic models κ L = κ δ L , i.e., κ L =0 ∀ L >
0. Thus a breakdown of SI will imply non-zero components of BiPS. In real space, κ L can be ex-pressed as κ L = (cid:18) L + 18 π (cid:19) Z d Ω ˆ n Z d Ω ˆ n (cid:20)Z dR χ L ( R ) C ( R ˆ n , R ˆ n ) (cid:21) , (10)where C ( R ˆ n , R ˆ n ) is the correlation function after ro-tating the coordinate system through an angle ω (0 ≤ ω ≤ π ), about an axis n (Θ , Φ). R ˆ n and R ˆ n are thecoordinates of the pixels ˆ n and ˆ n in the rotated coordi-nate system. The rotation axis n , is characterized by twoparameters Θ (0 ≤ Θ ≤ π ), and Φ (0 ≤ Φ ≤ π ). χ L , isthe trace of finite rotation matrix in LM -representationcalled the characteristic function , and it is invariant un-der rotation of coordinate system, χ L ( R ) = X M D LMM ( R ) . (11)Here dR is the volume element of the three-dimensionalrotation group given by dR = 4 sin (cid:16) ω (cid:17) dω sin Θ d Θ d Φ . (12)A simplified expression for BiPS in real space is κ L = (2 L + 1)8 π Z d Ω ˆ n Z d Ω ˆ n C (ˆ n , ˆ n ) Z dR χ L ( R ) C ( R ˆ n , R ˆ n ) . (13)For statistical isotropic model condition κ L = κ δ L canbe recovered using orthonormality of χ L ( R ), Z π χ L ( R ) χ L ′ ( R ) sin ω dω = π δ LL ′ . The BiPS of CMB anisotropy computed from the mapsmeasured by WMAP are consistent with SI, rulings outits radical violation [20]. An advantage of BiPS is that itsrotational invariance allows for constraints to be placedon the presence of specific forms of CMB correlation pat-terns independent of the overall orientation in the sky.
C. Reduced Bipolar coefficients- rBipoSH
In order to extract information on the orientation of SIviolation, or to detect correlation patterns in a specificdirection in the sky, the
Reduced Bipolar coefficients [23],obtained as A LM = ∞ X l =0 L + l X l = | L − l | A LMl l , (14) provide another set of measures. The summation of Bi-poSH over spherical wave-numbers l and l , reduces thecosmic variance rendering these measurable from the sin-gle CMB sky map available.Note that the summation involves both the terms A LMl l , and A LMl l , that are related via symmetry prop-erties Eq. (6). Thus for any such combination where l + l − L is odd, these two terms will cancel each otherleaving no contribution to the summation. The reducedBipolar coefficients A LM , by definition have the followingsymmetry A LM = ( − M A ∗ L − M , (15)which indicates A L are always real. When SI conditionis valid, the ensemble average of A LM vanishes for allnon-zero values of L , < A LM > = 0 , ∀ L = 0 . (16)These A LM coefficients fluctuate about zero in any givenCMB anisotropy map. Therefore, a statistically signifi-cant deviation from zero would confirm violation of SI.Unlike BiPS, reduced Bipolar coefficients are sensitive toorientation, hence they can assign directions to correla-tion patterns of the map. D. Bipolar map
It is possible to visualize correlation patterns using the
Bipolar map constructed from the reduced Bipolar coef-ficients A LM as [23],Θ(ˆ n ) = X LM A LM Y LM (ˆ n ) . (17)The Bipolar map from A LM is computed in the sameway as the temperature anisotropy map from a givenset of spherical harmonic coefficients, a lm . Bipolar mapcan also be represented in terms of Tripolar SphericalHarmonics of zero angular momentum (see appendix Bfor details),Θ(ˆ n ) = X L,l ,l Z d Ω ˆ n d Ω ˆ n C (ˆ n , ˆ n )( − l + l p (2 L + 1) δ λL { Y L (ˆ n ) ⊗ { Y l (ˆ n ) ⊗ Y l (ˆ n ) } λ } . (18)The tripolar spherical harmonics are expressed as [24] { Y l (ˆ n ) ⊗ { Y l (ˆ n ) ⊗ Y l (ˆ n ) } l } LM = X C LMl m l m C l m l m l m Y l m (ˆ n ) Y l m (ˆ n ) Y l m (ˆ n ) , where the summation is carried over m , m , m , and m . The transformations under rotations of tripolarspherical harmonics are identical to spherical harmonics.In particular, the tripolar scalar harmonics, which areinvariant under rotations, can be expressed as follows, { Y l (ˆ n ) ⊗ { Y l (ˆ n ) ⊗ Y l (ˆ n ) } λ } = ( − l + l + l δ λ l X m m m (cid:18) l l l m m m (cid:19) Y l m (ˆ n ) Y l m (ˆ n ) Y l m (ˆ n ) . Orthogonality and normalization relation is as follows,
Z Z Z d Ω ˆ n d Ω ˆ n d Ω ˆ n { Y l (ˆ n ) ⊗ { Y l (ˆ n ) ⊗ Y l (ˆ n ) } λ } LM { Y l ′ (ˆ n ) ⊗ { Y l ′ (ˆ n ) ⊗ Y l ′ (ˆ n ) } λ ′ } ∗ L ′ M ′ = δ l l ′ δ l l ′ δ l l ′ δ λλ ′ δ LL ′ δ MM ′ . From Eq. (18) its evident that under SI the Bipolarmap is invariant under the rotations, since tripolar scalarharmonics are rotationally invariant and C (ˆ n , ˆ n ) = C ( R ˆ n , R ˆ n ). Hence, the map gets contribution onlyfrom the monopole term A ,Θ = 12 X l ( − l r (2 l + 1) π C l . (19)Also, if the temperature map is rotated by a elementof rotation group, “ R ” then Bipolar map also rotatesidentically (see Appendix B). For example, if you rotatethe temperature map about the z-axis by some angle “ α ”,∆ T ( R ( θ, φ )) = X lm a lm Y lm ( θ, φ − α ) , the Bipolar map will also be rotated about z-axis throughsame angle “ α ”Θ( R ( θ, φ )) = X LM A LM Y LM ( θ, φ − α ) . However, the Wigner-D matrices in the two cases will bedifferent because of different m (or M ) values. III. BIPOLAR REPRESENTATION OF CMBCORRELATION SYMMETRIES
The homogeneity and isotropy of cosmic mi-crowave background points to the Friedmann-Robertson-Walker(FRW) model of universe. Flat FRW model ad-equately describes the observed local properties of theuniverse, but the fact that universe with same local ge-ometry can admit different global topology has been ap-preciated since the advent of post GR modern cosmology.This is because Einstein’s equations describe local prop-erties of the spacetime and can only constrain, but notdetermine, the global topological structure.Symmetries of the space are preserved in the corre-lation function and global topology modifies correlationfunction. The simply connected (topologically trivial)hyperbolic 3-space H , and the flat Euclidean 3-space E ,are non-compact and have infinite volume. There are nu-merous theoretical motivations, however, to favor a spa-tially compact universe [4–7]. Compact topologies (more,generally, multiply connected space) break the statisticalisotropy of CMB in characteristic patterns and inducea cutoff in the power spectrum because of finite spatialsize [16, 25–27]. Theoretical possibilities include com-pact Euclidean and Hyperbolic 3-spaces which require the space to be multiply connected. The compact hyper-bolic manifolds are not globally homogeneous and theyturn out to be not of much interest for the class of sym-metries considered under the scope of this paper.Simply connected universes are statistically isotropic,i.e. C (ˆ n , ˆ n ) = C (ˆ n . ˆ n ). In contrast, all compactuniverse models with Euclidean or hyperbolic geometry C (ˆ n , ˆ n ) are statistically anisotropic. The isotropy ofspace is broken in multi-connected models; this break-ing of symmetry may be apparent through the presenceof some principal directions. In a cylinder, for instance,which is compact in one dimension and infinite in theother two, the metric tensor is exactly the same at ev-ery point hence it preserves local homogeneity. However,it is not globally isotropic and does not have the maxi-mal symmetry. It is noteworthy that globally anisotropicmodels do not contradict observations, since the homo-geneity of space and the local isotropy can ensure theobserved isotropy of the CMB, however can influencethe spectrum of density fluctuations. Multiply-connectedmodels with zero or negative curvature can be compactin some, or all their dimensions. For instance a toroidaluniverse, despite its zero spatial curvature, has a finitevolume which may in principle be measured. It containsa finite amount of matter. A cylindrical universe (in thesense that the spatial sections are cylinders), on the otherhand, is noncompact in one dimension only and has aninfinite volume, although with a finite circumference inthe principal direction.Homogeneity and isotropy are experimentally con-firmed in the observations of distribution of luminousred galaxies [28], and the isotropy of CMB back-ground [29, 30]. Most of the studies in CMB as-sume statistical isotropy of the universe (FRW model).However, indications for a preferred direction in CMB,have motivated the study of departures from statisticalisotropy [1]. These deviations can arise from non-trivialspatial topologies [4, 5, 8–10], or departures from thebackground FRW metric [11, 31]. Alternatively, statis-tical anisotropies might also arise from coherent mag-netic fields in the universe [14, 15, 32]. Anisotropic Cos-mological models have been considered in the past andthey lead to characteristic patterns in the CMB sky [13].The Bianchi template is an example of SI violation dueto departure from background FRW geometry. Here wewill discuss the signature of anisotropy due to existenceof preferred axis (axes) on BipoSH. Such SI violationscan arise due to non-trivial topologies as well as coherentmagnetic fields.Since Bipolar formalism is sensitive to structures andpatterns in the underlying two point correlation function,particularly the real space correlations, it is a novel toolto characterize statistical anisotropies [17, 19–23]. Ro-tational symmetry about a preferred axis (say ˆ z ) is thesimplest way to break SI.In general, the correlation function may be decom-posed into isotropic and anisotropic parts [26], C (ˆ n , ˆ n ) = C ( I ) (ˆ n , ˆ n ) + C ( A ) (ˆ n , ˆ n ) . (20)where C ( I ) (ˆ n , ˆ n ) = C ( n · n ) = X l l + 14 π C l P l (ˆ n · ˆ n ) , (21)and the anisotropic part C ( A ) is orthogonal to the Leg-endre polynomials Z d Ω ˆ n Z d Ω ˆ n C ( A ) (ˆ n , ˆ n ) P l (ˆ n · ˆ n ) = 0 . (22)This decomposition is useful in our study of the symme-tries of the CMB correlation patterns/structure that areexplicit in real space. A. Statistical Isotropy (Rotational symmetry)
Under SI, the correlation function is a function onlyof θ , the angle between the two directions, say, ˆ n andˆ n . Hence, C (ˆ n , ˆ n ) ≡ C (ˆ n · ˆ n ) = C ( θ ), and the cor-relation function can be expanded in terms of Legendrepolynomials C ( I ) ( θ ) = 14 π ∞ X l =2 (2 l + 1) C l P l (cos θ ) , (23)where C l is the angular power spectrum. The summa-tion starts from l = 2, since l = 0 and l = 1, respectively, monopole and dipole, are usually subtracted out. For SIthe angular power spectrum C l contains all the informa-tion.In Bipolar representation, the condition of SI for var-ious observables, described in the previous section, canbe summarized as follows [17, 23]: • BipoSH : A LMl l = ( − l C l (2 l + 1) / δ l l δ L δ M , • BiPS : κ L = κ δ L , • rBipoSH : A LM = P l ( − l C l (2 l +1) / δ L δ M , • Bipolar map: Θ = P l ( − l q (2 l +1) π C l . Therefore, to test a CMB map for statistical isotropy, oneshould compute the BipoSH coefficients for the maps andlook for non-zero BipoSH coefficients. Cosmic variancecalculated for BipoSH under statistical isotropy is (seeAppendix C) , σ SI ( ˜ A LMl l ) = C l C l [1 + ( − L δ l l ] (24)for rBipoSH is, σ SI ( ˜ A LM ) = X l l C l C l [1 + ( − l + l − L ] (25)and for BiPS [17, 18], σ SI ( κ L ) = X l :2 l ≥ L C l [2 (2 L + 1) l + 1 + ( − L (2 L + 1) + (1 + 2( − L ) F Lll ] + X l L + l X l = | L − l | C l C l [(2 L + 1) + F Ll l ]+8 X l (2 L + 1) l + 1 C l [ L + l X l = | L − l | C l ] + 16( − L X l :2 l ≥ L (2 L + 1) l + 1 L + l X l = | L − l | C l C l (26)where F Ll l = l X m m = − l l X m m = − l L X M,M ′ = − L C LMl − m l − m C LMl m l m × C LM ′ l m l m C LM ′ l − m l − m (27)and L is even. Statistically significant deviations fromzero would mean violation of statistical isotropy. B. Cylindrical symmetry
The correlation function must satisfy the symmetriesof the underlying theory. In Friedman models the sym-metry group is SO(3), hence the correlation function is invariant under rotations; any breakdown of SI will re-duce this symmetry group. The simplest way to break SIis to introduce a favored direction in the sky, in such acase the reduced symmetry group is SO(2) or cylindricalsymmetry. Assuming the favored axis to be z-axis, therotational symmetry about z-axis for any arbitrary ∆ φ will require, C ( A ) ( θ , φ , θ , φ ) = C ( A ) ( θ , φ + ∆ φ, θ , φ + ∆ φ ) . where n ≡ ( θ , φ ) and n ≡ ( θ , φ ). The most generalform of the correlation function in such a case is (seeAppendix D) C ( A ) ( θ , φ , θ , φ ) = X m f m ( θ , θ ) cos m ( φ − φ ) . (28)Further, if the correlation function is invariant under thereflection, i.e., looking at a correlation pattern in thesky one cannot distinguish whether we are looking upor down the preferred direction, then C ( π − θ , φ , π − θ , φ ) = C ( θ , φ , θ , φ ) , (29)which leads to a condition that l + l is even. BipoSHin such a case would be, or equivalently the covariancematrix will be [33], < a l m a ∗ l m > = δ m m C l l | m | , (30)where diagonal terms C lm of C l l | m | are called the cylindri-cal power spectrum, and | m | > l . There may be correlations be-tween various scales called connectivity of fluctuations.The expression for C l l | m | in terms of the correlation func-tion is C l l | m | = 18 π s (2 l + 1)(2 l + 1)( l − m )!( l − m )!( l + m )!( l + m )! × Z π P l m (cos θ ) P l m (cos θ ) f m ( θ , θ ) d (cos θ ) d (cos θ ) . Using Eq. (D11) in the appendix, we obtain A LMl l = [1 + ( − l + l − L ] X m ( − m C l l | m | C LMl ml − m δ M . When m is even, the functions P ml will be odd or evenfunctions of their arguments, depending on whether l isodd or even respectively. Similarly, for the odd m ’s. Inboth the cases when only one of l and l is odd, theintegral vanishes. Therefore we have to consider caseswhen both of them are either odd or even. In such a case l + l is even and hence A LMl l vanishes for L = odd, A LMl l = A LMl l δ L, k δ M , where k = 0 , , , , .... (31) A LMl l = A LMl l δ L δ M δ l l + A LMl l δ M . (32)Therefore, the BipoSH present under cylindrical symme-try are A ll and A L l l with even L . Using symmetry prop-erty of BipoSH (6), under cylindrical symmetry we have A LMl l = A LMl l , i.e., the BipoSH are symmetric under theexchange of l and l . There is another possibility that a lm ’s have a gaussian distribution with different variancefor each m mode corresponding to a particular l . Thisimplies breakdown of SI, as power in each m mode is dif-ferent, C l l | m | = δ l l C l | m | , and the corresponding Bipolarcoefficients are, A LMl l = A LMl l δ L δ M δ l l + A LMl l δ M δ l l . (33)In such a case non-zero BipoSH are A ll and A L ll , wheremultipole moment is even and ( L ≥ C l | m | = C l and the rotational symmetry SO(3) of co-variance matrix is restored. The rBipoSH for cylindricalsymmetry are, A LM = X l l A LMl l = A LM δ L, k δ M where k = 0 , , , , .... (34)Hence the Bipolar map for such a symmetry will be,Θ(ˆ n ) = X LM A LM Y LM (ˆ n ) = X L A L δ L, k Y L (ˆ n )= X L r L + 14 π A L δ L, k P L (cos θ ) (35)Thus the map here looks like a sphere which is dividedinto latitude bands, or zones, without any longitudinalvariation.A realistic example of cylindrical symmetry is a pri-mordial homogeneous magnetic field which breaks sta-tistical isotropy by inducing a preferred direction ( e ).Therefore, the correlation function between two points( n and n ′ ) depends not only on the angular separationbetween two points ( n . n ′ ) but also on their orientationwith respect to the magnetic field. This dependence ofcorrelation function on angles between n and e (as well as n ′ and e ) leads to correlation between l and l ± D l ( m ) = < a ∗ l − m a l +1 m > ≡ < a ∗ l +1 m a l − m > . (36)Here D l is the power spectrum of the off-diagonal ele-ments of the covariance matrix, and the correlation func-tion shows up as, < a l m a ∗ l m > = δ m m δ l l C l + (37) δ m m ( δ l +1 ,l − + δ l − ,l +1 ) D l . The BipoSH corresponding to this covariance matrix are[18], A LMl l = ( − l (2 l + 1) / C l δ l l δ L δ M (38)+ D l δ l l ± δ M X m ( − m C L l ml − m . The non-zero BipoSH in this case are A ll and A L l l ± .The reduced Bipolar spherical harmonic coefficients(rBipoSH) for this case are A LM = X l δ M δ L (2 l + 1) / C l +2 X lm ( − m D l C L l − ml +1 m δ M . (39)These coefficients are non-zero only for l + l − L = even,thus L can take only even values. Finally, the Bipolarmap is,Θ(ˆ n ) = X L A L δ L, k Y L (ˆ n ) , k = 0 , , ,
3= 12 √ π A + X L A L δ L, a P L (cos θ ) , a = 1 , .. where A and A L are given by Eq.(39). C. n-fold discrete Cylindrical symmetry
Violation of SI also manifests itself in compact uni-verses with flat universal cover, which exhibits a n-foldrotational symmetry about an axis. There are six pos-sible compact models of the universe having a flat uni-versal cover (UC) [5]. These are visualized by identifyingopposite sides of the fundamental polyhedra. The fun-damental polyhedron (FP) may be a parallelepiped. Thepossible identifications then are (figure 1)
FIG. 1:
The locally Euclidean, closed, oriented 3-spaces.
1) - opposite faces by translations.2) - opposite faces, one pair being rotated by angle π .3) - opposite faces, one pair being rotated by π/ π .The fundamental polyhedron can also be the interior ofan hexagonal prism, with two possible identifications (fig-ure 2) : FIG. 2:
The locally Euclidean, closed, oriented 3-spaces.
1) - opposite faces, the top face being rotated by anangle 2 π/ π/ C ( A ) ( θ , φ , θ , φ ) = C ( A ) ( θ , φ + 2 πn , θ , φ + 2 πn ) . (40)This symmetry enforces m + m = nk , where n canbe odd or even, depending upon the symmetry of thecompact universe and k = 0 , , , ... . Thus the generalform of correlation function is (see Appendix E), C ( A ) ( θ , φ , θ , φ ) = X m ,m f m ,m ( θ , θ ) × e i ( m φ + m φ ) δ m + m ,nk , k = 0 , ± , ± ... (41)Corresponding Bipolar spherical harmonic coefficientsunder the symmetry eq.(40) are, A LMl l = 14 π X m m s (2 l + 1)(2 l + 1)( l − m )!( l − m )!( l + m )!( l + m )! δ m + m ,nk C LMl m l m Z π Z π Z π Z π × C ( θ , φ , θ , φ ) e − i ( m φ + m φ ) P l m (cos θ ) P l m (cos θ ) d (cos θ ) d (cos θ ) dφ dφ . (42)All possible Euclidean models of compact universe ex-hibit reflection symmetry about the xy-plane. The cor-relation function under reflection symmetry is, C ( θ , φ , θ , φ ) = C ( π − θ , φ , π − θ , φ ) , (43)also under reflection of the coordinate system about x-yplane the spherical harmonics transform as, Y lm ( π − θ, φ ) = ( − l + m Y lm ( θ, φ ) . (44) Therefore, reflection symmetry demands, P m l (cos θ ) = P m l (cos( π − θ )) = P m l ( − cos θ ) . (45)This implies l + m is even and similarly l + m . Herewe have used the symmetry property of Legendre poly-nomials, P lm ( − x ) = ( − l + m P lm ( x ). Interestingly, fromsymmetries of spherical harmonics, one can show that n-fold symmetries are ruled out for odd n (see AppendixF).Topologically compact universes exhibits even foldsymmetry, but the emergence of this fact from the sym-metry of two-point correlation pattern itself is neverthe-less instructive. Therefore, we need to look at the caseswhen n is even. D. Even-fold Cylindrical symmetry
Even fold symmetry refers to the case when n is even.For compact topologies this is always the case, for in-stance, Dirichlet domain (DD) of a T toroidal universe[34], and a T have a 4-fold symmetry, that of a hexago-nal prism has a 6-fold symmetry and a squeezed torus has 2 fold symmetry. This symmetry puts another restrictionon correlation function, C ( θ , φ , θ , φ ) = C ( θ , − φ , θ , − φ ) . (46)Hence most general correlation function under even-foldsymmetry is (see Appendix E), C ( A ) ( θ , φ , θ , φ ) = X m ,m f m ,m ( θ , θ ) δ m + m ,nk cos( m φ + m φ ) . Therefore Bipolar spherical harmonic coefficients are, A LMl l = [1 + ( − l + l − L ] X m m s (2 l + 1)(2 l + 1)( l − m )!( l − m )!( l + m )!( l + m )! C LMl m l m δ m + m ,nk Z π Z π Z π Z π C ( θ , φ , θ , φ ) cos( m φ + m φ ) P m l (cos θ ) P m l (cos θ ) dφ dφ d (cos θ ) d (cos θ ) . (47)Reflection symmetry allows A LMl l only for even values of L . For odd indices coefficients vanish A LMl l = A LMl l δ Mnk δ L a , k = 0 , , ..., a = 0 , , , ... (48)Here nk is even, therefore for an even-fold symmetry i.e.,( n = 2 , , , ... ) M is even and multipole moment L ≥ M = 2 , , , .... , similarly for4-fold symmetry M = 4 , , , ... and so on. Anothersymmetry property of BipoSH is Eq. (6), so under even-fold rotational symmetry l + l + m + m = even whichimplies l + l is even hence A LMl l = A LMl l i.e., BipoSH aresymmetric under the exchange of l and l . Note that forall possible compact flat spaces BipoSH vanish for oddindices, and the fact that two-point correlation functionis invariant under reflection about xy-plane plays a piv-otal role in restricting non-zero BipoSH only to even L ’s.The rBiposh for even universe with even fold symmetryare, A LM = A LM δ Mnk δ L a . (49)Bipolar map will be,Θ(ˆ n ) = X LM A LM Y LM (ˆ n ) δ Mnk δ L a , (50) L will take only even values and M will run from − L to L , subsequently picking up even values. IV. BIPOLAR MAP: EXAMPLE OF THEBIANCHI TEMPLATE
Now we will consider a Bianchi template as an exam-ple to show how a Bipolar map looks like for a giventemperature map. The choice of Friedmann-Robertson-Walker (FRW) model as a model of our universe wasinitially due to its simplicity, and later because of ob-servational evidence which strongly suggests universe tobe homogeneous and isotropic at large scales. However,the presently observed isotropy may not necessarily holdin the past and the universe may have been anisotropicin its early stages and tends to FRW only later as itevolves. Bianchi models are the simplest examples whichhave the property to isotropise as they evolve in future.Bianchi classification contains 10 equivalent classes giv-ing generic description of a homogeneous and anisotropiccosmology [12]. The most general Bianchi type which ad-mits FRW at late time are
V II h and IX . However, thetype IX re-collapses after a finite time hence do not comearbitrarily close to isotropy. Spiral pattern are charac-teristic signatures of V II and V II h models [12, 35, 36].Jaffe et. al. proposed Bianchi V II h models as an ex-planation of WMAP anomalies. Since class V II h modelsresembles a universe with vorticity and hence can lead tobounds on the universal rotation in cosmological (CMB)data [37]. They proposed correction for some anomaliesin the first year maps from WMAP, however, introduc-ing such corrections induces other features like preferreddirection and violation of SI. Pontzen et al. calculatedvarious temperature and polarisation anisotropy patternswhich may be formed in Bianchi cosmologies [38]. Ghoshet. al. analyzed the temperature map for Bianchi V II h template [13]. Given the temperature map for Bianchi V II h template, here we see how a Bipolar map actuallylooks like.The temperature map for Bianchi V II h template is ofthe form∆ T B ( θ, φ ) = f ( θ ) sin φ + f ( θ ) cos φ, (51) where super-script B signifies Bianchi, and f ( θ ) and f ( θ ) are parameters of the model which should be cal-culated numerically [12].BipoSH for Bianchi template are, A LMl l = Z π Z π { W l l ( θ , θ ) C LMl − l δ M + X l l ( θ , θ ) C LMl − l − δ M − + Y l l ( θ , θ ) C LMl l − δ M + Z l l ( θ , θ ) C LMl l δ M } d (cos θ ) d (cos θ ) . (52)Therefore, rBiposh are, A LM = X l l Z π Z π { W l l ( θ , θ ) C LMl − l δ M + X l l ( θ , θ ) C LMl − l − δ M − + Y l l ( θ , θ ) C LMl l − δ M + Z l l ( θ , θ ) C LMl l δ M } d (cos θ ) d (cos θ ) , (53)where W l l = π s (2 l + 1)(2 l + 1)( l + 1)!( l − π ) ( l − l + 1)! { f ( θ ) f ( θ ) + i ( f ( θ ) f ( θ ) − f ( θ ) f ( θ )) + f ( θ ) f ( θ ) } P − l (cos θ ) P l (cos θ ) , X l l = π s (2 l + 1)(2 l + 1)( l + 1)!( l + 1)!(4 π ) ( l − l − {− f ( θ ) f ( θ ) + i ( f ( θ ) f ( θ ) + f ( θ ) f ( θ )) + f ( θ ) f ( θ ) } P − l (cos θ ) P − l (cos θ ) , Y l l = π s (2 l + 1)(2 l + 1)( l − l + 1)!(4 π ) ( l + 1)!( l − { f ( θ ) f ( θ ) + i ( − f ( θ ) f ( θ ) + f ( θ ) f ( θ )) + f ( θ ) f ( θ ) } P l (cos θ ) P − l (cos θ ) , Z l l = π s (2 l + 1)(2 l + 1)( l − l − π ) ( l + 1)!( l + 1)! {− f ( θ ) f ( θ ) − i ( f ( θ ) f ( θ ) + f ( θ ) f ( θ )) + f ( θ ) f ( θ ) } P l (cos θ ) P l (cos θ ) . Hence, A LM = A LM δ M,k k = 0 , ± . (54)Possible values of M are 0 , ±
2. Note that A LM existsonly for l + l − L = even , and vanishes otherwise. Keep-ing in mind the reality of two-point correlation function,i.e., A LM = ( − M A ∗ L − M , here we have A L = A ∗ L − .Since A LM coefficients are complex numbers we can de-fine, X LM = ℜ ( A LM ) and Z LM = ℑ ( A LM ). Therefore, Bipolar map for a Bianchi template looks like (see ap-pendix G),Θ(ˆ n ) = X L A L Y L ( θ, φ ) + 2 X L X L − G L ( θ ) cos 2 φ − X L Z L − G L ( θ ) sin 2 φ, (55)0 FIG. 3:
Temperature map for Bianchi VII h . where G L ( θ ) = 1(sin θ ) s ( L − L ( L + 1)( L + 2)4 π (2 L + 1) (56) h P L − (cos θ )2 L − − L + 1) P L (cos θ )(2 L − L + 3) + P L +2 (cos θ )2 L + 3 i . FIG. 4:
Bipolar map for Bianchi VII h . Thus, a spiral pattern in temperature map will showup as double spiral pattern in Bipolar map.
V. CONCLUSION AND DISCUSSION
Representation of correlation function of CMBanisotropy in terms of Bipolar spherical harmonics pro-vides a novel approach to study violations of SI. Very re-cently the Bipolar representation has been used to quan-tify anomalies in the analysis of WMAP seven - yeardata [42]. These anisotropies can arise due to departurefrom FRW metric (eg. Bianchi models), non-trivial spa-tial topologies (compact spaces) or from primordial mag-netic fields, among others. Here we have studied variousmeasurable quantities of Bipolar formalism to quantifybreakdown of SI.We studied anisotropic homogeneous cosmologieswhich leave a characteristic pattern on CMB. LikeBianchi
V II h temperature map which has a spiral pat-tern of a pair of cold and hot spots with a dipole in azimuthal space. We found that the corresponding pat-tern in Bipolar space becomes a double spiral having aquadrupole in azimuthal space of the Bipolar map.Another application is in case of homogeneous isotropicmodels where an anisotropic topological identificationhas been imposed. As an example, if the space is com-pact in one (or more) direction(s), the statistical isotropyis broken due to introduction of preferred direction(s).We calculate BipoSH when this preferred direction isintroduced. We have shown here that for compact topolo-gies, symmetry requirements can restrict BipoSH to evenmultipole moments, i.e., BipoSH vanish for odd indicesfor all kind of physically plausible models of flat multi-connected universe. Hyperbolic manifolds do not havethe desired symmetry and hence we expect odd multi-poles to be non-zero in these manifolds. Hence, we havea tool to distinguish different topologies. In case of ho-mogeneous magnetic fields we have shown that BipoSH’sare restricted to even L and M = 0.A new representation of Bipolar map has been pro-posed. Further work needs to be done in this directionto extract new information from this representation.This technique can be applied to polarization mapsand it may prove to be a powerful method to decipherthe topology of the universe, something on which generalrelativity is completely silent. The Bipolar formalismcan also be applied to various anisotropic universes andcan be used as a tool to distinguish various types of SIbreakdown. Acknowledgments
The authors deeply regret the untimely demise of theircolleague, Himan Mukhopadhyay, and acknowledge hervaluable contribution in this work that originates andbuilds upon the excellent work carried out during a grad-uate school project at IUCAA [39]. We also thank TuhinGhosh for discussions at several stages of this work. SJand NJ would like to thank IUCAA for its hospitalityand computational facilities.
Appendix A: A review of topologically compactspaces
Topologically compact spaces break the statisticalisotropy, thereby introducing signatures in CMB corre-lation patterns. A compact cosmological model, M , isa Quotient space , constructed by identifying points ofstandard FRW space under the action of suitable dis-crete subgroup of motions Γ , of the full isometry group G of the FRW space. The isometry group G is the groupof motions which preserves the distance between points.The simply connected infinite FRW spatial hypersurfacewith same constant curvature geometry is the universalcover (UC), M u , tiled by the copies of the compact space,1 M . It can be spherical ( S ), Euclidean ( E ) or hyper-bolic ( H ). The compact space for a given location ofobserver is represented as Dirichlet domain (DD), withthe observer at its basepoint. Any point x of the com-pact space has an image, x i = γ i x , in each copy of DDon the universal cover, where γ i ε Γ . By constructionDD represents the compact space as a convex polyhedron with even number of faces identified pairwise under Γ . Incosmology, DD around the observer represents the uni-verse as seen by the observer and the symmetries of thecorrelation function are nothing but the symmetries ofthe corresponding DD.Correlation function of a scalar field, Φ , on a compactmanifold, M , can be expressed as [40], ξ CΦ ( x , x ′ ) = X i m i X j =1 P Φ ( k i )Ψ ij ( x )Ψ ∗ ij ( x ′ ) , (A1)where ( ∇ + k i )Ψ ij = 0 . (A2)Ψ i are orthonormal set of eigenfunctions of Laplacianon the hypersurface, having positive and discrete set ofeigenvalues { k i } ( k o = 0 and k i < k i +1 ) with multiplic-ities m i . On a compact manifold, M , the set of eigen-functions and eigenvalues are not always easy to obtainin closed form (even numerically, for compact hyperbolicspaces). On the other hand, eigenfunctions Ψ uj ( k, x ) ofthe universal cover (UC), M u , of a compact manifoldusually known because of their simplicity (e.g., H , S and E ), hence they can be used to compute the corre-lation functions ξ CΦ ( x , x ′ ) on the UC. For flat and hy-perbolic UC’s the set of eigenvalues are continuous. Thefunction P Φ ( k i ) is the rms amplitude of the eigenmodeexpansion of the field Φ , whose information lies in thephysical mechanism responsible for the generation of Φ .The regularized method of images [25], describes how cor-relation function on the compact manifold can be calcu-lated once the correlation function on the universal coveris known [16, 25, 26], which is expressed as, ξ CΦ ( x , x ′ ) = ] X γεΓ ξ u Φ ( x , x ′ ) . (A3)This implies that correlation function on a compactspace, M , can be expressed as sum over the correlationfunction on its universal covering space, M u , calculatedbetween x and the images γ x ′ ( γ ε Γ ) of x ′ . The localhomogeneity and isotropy demands that the correlationfunction on the UC is only a function of the distancebetween two points x and x ′ i.e r ≡ d ( x , x ′ ). The corre-lation function on a compact universe with flat UC is, ξ CΦ ( x , x ′ ) = X i Z dkk P Φ ( k ) sin kd i kd i , (A4)Here, P Φ ( k ) can be determined from the early universephysical mechanism and d i is the distance between the images of x and x ′ ( d is the distance between originalpoints). Summation implies summing over all images.Hence, correlation function depends not only on the dis-tance between two points and the distance of their im-ages but symmetry defines both the pair to have identicaldistance from their images i.e., take any two points onthe last scattering surface and their corresponding imagesabout xy-plane, correlation function will turn out to beinvariant under this reflection. Figure (5) illustrates this FIG. 5:
Images of two-pairs of point which are mirror imagesof each other about the XY plane in a T space (one dimensionsuppressed). point for a T universe. The DD of a squeezed torus isshown in figure (6). The choice of the axes here is a littlebit more non-trivial. The xy-plane is not parallel to anyof the faces of the DD or the FP, but still it would cut theLSS into two halves in such a way that there will be sym-metry under reflection about the xy-plane and on the xy-plane there will be 2-fold rotational symmetry. However, FIG. 6:
Dirichlet domain of a squeezed torus. here we point out that reflection symmetry does not hold2good for the compact spaces for which the opposite facesare glued together with a twist [39].Topology of the universe leaves characteristic signa-tures on CMB. If the universe is finite and smaller thanthe distance to the last scattering surface(LSS), then thesignature of the topology of the universe is imprinted onthe CMB. For such a small universe LSS can wrap aroundthe universe and will self-intersect. The intersection ofthe LSS, which is a 2-sphere with itself is a circle thatwill appear twice in the cosmic microwave background.Hence, there might exist pairs of circles which share cor-related patterns of temperature fluctuations. This circlesin the sky [41] method is a powerful and direct probefor detecting non-trivial spatial topology. The correlatedpatterns would be matching perfectly if the temperaturefluctuation did not depend on the direction of observationand if the patterns were not distorted. However, the ob-served temperature fluctuations has direction dependentcomponents, i.e. the
Doppler effect and the integratedSachs-Wolfe effect . Also observationally, galaxy cut andforeground removals can also distort the matching. How-ever, one can search for such patterns in CMB correlationfunction statistically. In a multi-connected space, thereexist preferred direction(s) so that global isotropy is bro-ken. The angular correlation will then depend on twodirections of observations and can also depend on the po-sition of the observer. This induces correlations between a lm ’s of different l and m . Thus, another indirect probeis to search such patterns or signatures in the statisticsof CMB temperature fluctuations. Appendix B: Bipolar map representation in terms oftripolar spherical harmonics
Bipolar map is defined asΘ(ˆ n ) = X LM A LM Y LM (ˆ n ) , (B1)where A LM = P l l A LMl l , thereforeΘ(ˆ n ) = X LM X l l A LMl l Y LM (ˆ n ) (B2)Now using the expansion of A LMl l we getΘ(ˆ n ) = X LM X l l Z d Ω ˆ n Z d Ω ˆ n C (ˆ n , ˆ n ) ×{ Y l (ˆ n ) ⊗ Y l (ˆ n ) } ∗ LM Y LM (ˆ n )tripolar scalar spherical harmonics are defined as { Y l (ˆ n ) ⊗ { Y L (ˆ n ) ⊗ Y l (ˆ n ) } λ } = ( − l + l + L (B3) δ λl X m m M (cid:18) l L l m M m (cid:19) Y l m (ˆ n ) Y LM (ˆ n ) Y l m (ˆ n ) where (cid:18) l L l m M m (cid:19) are Wigner-3j symbols and are re-lated to Clebsch Gordan coefficients in the following way, C l m l m l m = ( − l − l + m p l + 1 (cid:18) l l l m m − m (cid:19) Hence Bipolar map can be represented in terms of tripo-lar scalar spherical harmonics,Θ(ˆ n ) = X L,l ,l Z d Ω ˆ n d Ω ˆ n C (ˆ n , ˆ n )( − l + l √ L + 1 δ λL { Y l (ˆ n ) ⊗ { Y L (ˆ n ) ⊗ Y l (ˆ n ) } λ } (B4)The representation of the Bipolar map in terms of tripo-lar harmonic function makes the transformation proper-ties of the bipolar map under rotations explicit.In a rotated sky map,Θ ′ (ˆ n ) = X LMl l Z [ X l ′ ,l ′ ,L ′ ,M ′ A L ′ M ′ l ′ l ′ X M ′′ D L ′ M ′′ M ′ ( R ) { Y l ′ (ˆ n ) ⊗ Y l ′ (ˆ n ) } L ′ M ′′ ] ×{ Y l (ˆ n ) ⊗ Y l (ˆ n ) } ∗ LM Y LM (ˆ n ) d Ω ˆ n d Ω ˆ n (B5)Using orthogonality of Bipolar spherical harmonicsΘ ′ (ˆ n ) = X LMl l X l ′ ,l ′ ,L ′ ,M ′ A L ′ M ′ l ′ l ′ X M ′′ D L ′ M ′′ M ′ ( R ) Y LM δ l l ′ δ l l ′ δ LL ′ δ MM ′ (B6)Θ ′ (ˆ n ) = X LM ′ A LM ′ X M D LMM ′ ( R ) Y LM (ˆ n ) = Θ( R ˆ n )(B7)Thus when correlation pattern is rotated by “ R ”, Bipolarmap also rotates by “ R ”. Appendix C: Cosmic Variance of Bipolar Quantities
Cosmic variance is defined as the variance of estimatorof an observable constructed from a single sky map. Inparticular for BipoSH σ ( ˜ A LMl l ) = < ( ˜ A LMl l ) > − < ˜ A LMl l > . (C1)Using Gaussianity of ∆ T , one can analytically computethe variance of ˜ A LMl l .˜ A LMl l = X m m a l m a ∗ l m ( − m C LMl m l − m (C2)therefore < ˜ A LMl l ˜ A ∗ LMl l > = X m m X m ′ m ′ < a l m a ∗ l m a ∗ l m ′ a l m ′ > ( − m + m ′ C LMl m l − m C LMl m ′ l − m ′ (C3)3Considering temperature field to be a Gaussian randomfield, one can expand the four-point correlation functionin terms of two-point correlation function. Further, con-sidering the fact that under statistical isotropy the covari-ance matrix is diagonal eq.(3) the above equation reducesto < ˜ A LMl l ˜ A ∗ LMl l > = C l C l δ l l (2 l + 1) δ L δ M + C l C l [1 + ( − L δ l l ] (C4)Also, we have < ˜ A LMl l > = (2 l + 1) / C l δ l l δ L δ M (C5)Hence the cosmic variance is σ SI ( ˜ A LMl l ) = C l C l [1 + ( − L δ l l ] (C6)Similarly for rBipoSH, σ SI ( ˜ A LM ) = X l l C l C l [1 + ( − l + l − L ] (C7) Appendix D: Correlation function for Cylindricalsymmetry
Expansion of correlation function in terms of Bipolarspherical harmonics is, C ( A ) (ˆ n , ˆ n ) = X l ,l ,L,M A LMl l X m m C LMl m l m Y l m (ˆ n ) Y l m (ˆ n ) (D1)Now rotational symmetry about z-axis for any arbitrary∆ φ implies, C ( A ) ( θ , φ , θ , φ ) = C A ( θ , φ + ∆ φ, θ , φ + ∆ φ )(D2)Therefore X l ,l ,L,M,m ,m A LMl l C LMl m l m Y l m ( θ , φ ) Y l m ( θ , φ )= X l ,l ,L,M,m ,m A LMl l C LMl m l m Y l m ( θ , φ + ∆ φ ) Y l m ( θ , φ + ∆ φ )which means e i ( m + m )∆ φ = 1 (D3) therefore m + m = 2 kπ ∆ φ , k = 0 , ± , ± .... (D4)for zero fold symmetry m + m = 0 which means m = − m , hence C ( A ) (ˆ n , ˆ n ) = X l ,l ,L,M,m ,m A LMl l C LMl m l m Y l m (ˆ n ) Y l m (ˆ n ) δ m , − m (D5)using the expansion of spherical harmonics in terms ofassociated Legendre polynomials, Y lm ( θ, φ ) = e imφ s (2 l + 1)( l − m )!4 π ( l + m )! P ml (cos θ ) (D6)therefore correlation function will be, C ( A ) (ˆ n , ˆ n ) = X m f m ( θ , θ ) e im ( φ − φ ) (D7)where f m ( θ , θ ) = 14 π X l ,l ,L A LMl l C LMl ml − m δ M s (2 l + 1)(2 l + 1)( l − m )!( l + m )! l + m )!( l − m )! P ml (cos θ ) P − ml (cos θ ) (D8)Symmetry ensures, C ( A ) ( θ , φ , θ , φ ) = C A ( θ , − φ , θ , − φ ) (D9)Imposing this symmetry we get, C ( A ) ( θ , φ , θ , φ ) = X m f m ( θ , θ ) cos m ( φ − φ )(D10)This is the most general correlation function under zerofold rotational symmetry. Using A LMl l = Z d Ω ˆ n d Ω ˆ n C (ˆ n , ˆ n ) { Y l (ˆ n ) ⊗ Y l (ˆ n ) } ∗ LM and eq.(D10) we get, A LMl l = [1 + ( − l + l − L ] X m ( − m p (2 l + 1)(2 l + 1)( l − m )!( l − m )!(4 π ) ( l + m )!( l + m )! C LMl ml − m δ M Z π Z π P l m (cos θ ) P l m (cos θ ) f m ( θ , θ ) d (cos θ ) d (cos θ ) . (D11)4 Appendix E: n-fold cylindrical symmetry
Correlation function in such a case is, C ( A ) ( θ , φ , θ , φ ) = C ( A ) ( θ , φ + 2 πn , θ , φ + 2 πn ) . (E1)This implies, X l l m m LM A LMl l C LMl m l m Y l m ( θ , φ ) Y l m ( θ , φ ) = X l l m m LM A LMl l C LMl m l m Y l m ( θ , φ + 2 πn ) Y l m ( θ , φ + 2 πn )Hence, e i ( m + m ) πn = 1, which implies m + m = nk, k = 0 , ± , ± , ± ..... .Most general form of correlation function will be, C ( A ) ( θ , φ , θ , φ ) = X m m f m m ( θ , θ ) e i ( m φ + m φ ) δ m + m ,nk where f m m ( θ , θ ) = 14 π X l l LM s (2 l + 1)(2 l + 1)( l − m )!( l − m )!( l + m )!( l + m )! A LMl l C LMl m l m P m l (cos θ ) P m l (cos θ )Demanding explicitly the two fold symmetry that holdsfor all even-fold symmetry, C ( θ , φ , θ , φ ) = C ( θ , − φ , θ , − φ ) (E2)this symmetry rules out the presence of sine terms incorrelation function. Hence for even-fold symmetry cor-relation function reduces to, C ( A ) ( θ , φ , θ , φ ) = X m ,m f m ,m ( θ , θ ) δ m + m ,nk cos( m φ + m φ ) (E3) Appendix F: Absence of odd-fold symmetries forcompact spaces
Let us take a compact space with reflection symmetry.It would demand that C ( π − θ , φ , π − θ , φ ) = C ( θ , φ , θ , φ ) . (F1)It can be shown that C ( θ , φ , θ , φ ) = X l l m m LM ( − m < a l m a ∗ l m > × C LMl m l − m { Y l (ˆ n ) ⊗ Y l (ˆ n ) } LM . (F2)The symmetry of spherical harmonics would ensure that { Y l ( π − θ , φ ) ⊗ Y l ( π − θ , φ ) } LM = ( − m + m { Y l ( θ , φ ) ⊗ Y l ( θ , φ ) } LM . (F3)This put together with equations (F1) and (F2) indicatesthat m must be an even number. Since they are dummyindices, m would be even too. Now let us consider sucha space with n fold symmetry. Evidently C ( θ , φ + 2 πn , θ , φ + 2 πn ) = C ( θ , φ , θ , φ ) (F4) since we know { Y l ( θ , φ + 2 πn ) ⊗ Y l ( θ , φ + 2 πn ) } LM =exp( 2 π ( m + m ) n ) { Y l ( θ , φ ) ⊗ Y l ( θ , φ ) } LM . (F5)For Eq. (F4) to hold ( m + m ) /n must be even. Since m and m are even, n has to be even too. Appendix G: Bianchi template
The temperature map for Bianchi template is writtenas ∆ T ( θ, φ ) = f ( θ ) sin φ + f ( θ ) cos φ . (G1)Bipolar map can be expressed as,Θ(ˆ n ) = X LM X l l X m m Z d Ω ˆ n d Ω ˆ n < ∆ T (ˆ n )∆ T (ˆ n ) >C LMl m l m Y ∗ l m (ˆ n ) Y ∗ l m (ˆ n ) Y LM (ˆ n ) . The integrals over φ contribute only for m = ± m = ± m = ±
1, admits only M = 0 , ±
2. Reduced Bipolarcoefficient is then A LM = X l l X m m Z d Ω ˆ n d Ω ˆ n < ∆ T (ˆ n )∆ T (ˆ n ) > ( − m + m C LMl m l m Y l − m (ˆ n ) Y l − m (ˆ n ) .A LM exists only for l + l − L = even , and vanishesotherwise and the reality condition demands A LM , i.e., A LM = ( − M A ∗ L − M . Now Bipolar map isΘ(ˆ n ) = X LM A LM Y LM (ˆ n ) (G2)5but for Bianchi template it will beΘ(ˆ n ) = X L A L Y L (ˆ n ) + X L ≥ A L Y L (ˆ n )+ X L ≥ A L − Y L − (ˆ n )which can be written asΘ(ˆ n ) = X L A L Y L (ˆ n ) + X L ≥ A ∗ L − Y ∗ L − (ˆ n )+ X L ≥ A L − Y L − (ˆ n ) (G3)Since A ′ LM s are complex numbers, we define A LM = X LM + iZ LM and A ∗ LM = X LM − iZ LM (G4)and the Bipolar map (G3) can then be written as θ (ˆ n ) = X L A L Y L (ˆ n ) + X L X L − (cid:0) Y ∗ L − (ˆ n ) + Y L − (ˆ n ) (cid:1) + i X L Z L − (cid:0) Y L − (ˆ n ) − Y ∗ L − (ˆ n ) (cid:1) . Defining, G L ( θ ) = 1(sin θ ) s ( L − L ( L + 1)( L + 2)4 π (2 L + 1) h P L − (cos θ )2 L − − L + 1) P L (cos θ )(2 L − L + 3) + P L +2 (cos θ )2 L + 3 i the Bipolar map is represented asΘ( θ, φ ) = X L A L Y L ( θ, φ ) + X L X L − G L ( θ )2 cos 2 φ − X L Z L − G L ( θ )2 sin 2 φ (G5)where we have used expansion of spherical harmonics interms of associated Legendre polynomials Y lm ( θ, φ ) = e imφ s (2 l + 1)( l − m )!4 π ( l + m )! P ml (cos θ )and Y l ± ( θ, φ ) = e i ± φ (sin θ ) s ( L − L ( L + 1)( L + 2)4 π (2 L + 1) h P L − (cos θ )2 L − − L + 1) P L (cos θ )(2 L − L + 3) + P L +2 (cos θ )2 L + 3 i . Appendix H: Useful Mathematical Relations
Orthonormality of spherical harmonics Z d Ω ˆ n Y l m (ˆ n ) Y ∗ l m (ˆ n ) = δ l l δ m m (H1)Symmetry property of spherical harmonics Y ∗ lm (ˆ n ) = ( − m Y l − m (ˆ n ) . (H2) Spherical harmonic expansion of Legendre polynomials P l (ˆ n · ˆ n ′ ) = 4 π l + 1 l X m = − l Y ∗ lm (ˆ n ) Y lm (ˆ n ′ ) . (H3)Property of legendre polynomial P − ml = ( − m ( l − m )!( l + m )! P ml . (H4)Symmetry properties of Clebsch-Gordan coefficients C cγaαbβ = ( − a + b − c C cγbβaα , (H5) C cγaαbβ = ( − a + b − c C c − γa − αb − β . Summation rules of Clebsch-Gordan coefficients X αβ C cγaαbβ C c ′ γ ′ aαbβ = δ cc ′ δ γγ ′ { abc }{ abc ′ } X aγ C cγaαbβ C cγaαb ′ β ′ = 2 c + 12 b + 1 δ bb ′ δ ββ ′ { abc }{ ab ′ c } X cγ C cγaαbβ C cγaα ′ bβ ′ = δ αα ′ δ ββ ′ { abc } X b ( − a − b C c aba − b = Y a δ c (H6)where Y abc..... = [(2 a + 1)(2 b + 1) .... (2 c + 1)] / (H7)and where { abc } is defined by { abc } = (cid:26) a + b + c is an integer0 otherwise (H8)and where a, b and c satisfy triangle inequality | a − b | ≤ c ≤ ( a + b ).Tripolar spherical harmonics are expressed as, { Y l (ˆ n ) ⊗ { Y l (ˆ n ) ⊗ Y l (ˆ n ) } l } LM = X m m m m C LMl m l m C l m l m l m Y l m (ˆ n ) Y l m (ˆ n ) Y l m (ˆ n ) . Tripolar scalar spherical harmonics are defined as { Y l (ˆ n ) ⊗ { Y L (ˆ n ) ⊗ Y l (ˆ n ) } λ } = ( − l + l + L (H9) δ λl X m m M (cid:18) l L l m M m (cid:19) Y l m (ˆ n ) Y LM (ˆ n ) Y l m (ˆ n ) . where (cid:18) l L l m M m (cid:19) are Wigner-3j symbols.Orthogonality of tripolar spherical harmonics, is given as Z Z Z d Ω ˆ n d Ω ˆ n d Ω ˆ n { Y l (ˆ n ) ⊗ { Y l (ˆ n ) ⊗ Y l (ˆ n ) } λ } LM { Y l ′ (ˆ n ) ⊗ { Y l ′ (ˆ n ) ⊗ Y l ′ (ˆ n ) } λ ′ } ∗ L ′ M ′ = δ l l ′ δ l l ′ δ l l ′ δ λλ ′ δ LL ′ δ MM ′ . [1] A. de Oliveira-Costa, M. Tegmark, M. Zaldarriagaand A. Hamilton, Phys. Rev. D , 063516 (2004)[arXiv:astro-ph/0307282]; C. J. Copi, D. Huterer andG. D. Starkman, Phys. Rev. D , 043515 (2004)[arXiv:astro-ph/0310511]; D. J. Schwarz, G. D. Stark-man, D. Huterer and C. J. Copi, Phys. Rev. Lett. , 221301 (2004) [arXiv:astro-ph/0403353]; S. Prunet,J. P. Uzan, F. Bernardeau and T. Brunier, Phys. Rev. D , 083508 (2005) [arXiv:astro-ph/0406364]; H. K. Erik-sen, A. J. Banday, K. M. Gorski and P. B. Lilje, As-trophys. J. , 58 (2005) [arXiv:astro-ph/0407271];K. Land and J. Magueijo, Phys. Rev. Lett. ,071301 (2005) [arXiv:astro-ph/0502237]; T. R. Jaffe,A. J. Banday, H. K. Eriksen, K. M. Gorskiand F. K. Hansen, Astrophys. J. , L1 (2005)[arXiv:astro-ph/0503213]; C. J. Copi, D. Huterer,D. J. Schwarz and G. D. Starkman, Mon. Not. Roy.Astron. Soc. , 79 (2006) [arXiv:astro-ph/0508047];K. Land and J. Magueijo, Mon. Not. Roy. As-tron. Soc. , 1714 (2006) [arXiv:astro-ph/0509752];A. Bernui, T. Villela, C. A. Wuensche, R. Leonardiand I. Ferreira, Astron. Astrophys. , 409 (2006)[arXiv:astro-ph/0601593]; L. R. Abramo, A. Bernui,I. S. Ferreira, T. Villela and C. A. Wuensche, Phys.Rev. D , 063506 (2006) [arXiv:astro-ph/0604346];J. Magueijo and R. D. Sorkin, Mon. Not. Roy. Astron.Soc. Lett. , L39 (2007) [arXiv:astro-ph/0604410];C. G. Park, C. Park and J. R. I. Gott, Astrophys. J. ,959 (2007) [arXiv:astro-ph/0608129]; D. Huterer, NewAstron. Rev. , 868 (2006) [arXiv:astro-ph/0608318];P. Vielva, Y. Wiaux, E. Martinez-Gonzalez andP. Vandergheynst, New Astron. Rev. , 880 (2006)[arXiv:astro-ph/0609147]; K. Land and J. Magueijo,Mon. Not. Roy. Astron. Soc. , 153 (2007)[arXiv:astro-ph/0611518]; C. Gordon, W. Hu, D. Hutererand T. Crawford, Phys. Rev. D , 103002 (2005)[arXiv:astro-ph/0509301]; J. G. Cresswell, A. R. Lid-dle, P. Mukherjee and A. Riazuelo, Phys. Rev. D ,041302 (2006) [arXiv:astro-ph/0512017]. S. H. S. Alexan-der, arXiv:hep-th/0601034.[2] H. K. Eriksen, F. K. Hansen, A. J. Banday, K. M. Gorskiand P. B. Lilje, Astrophys. J. , 14 (2004) [Erratum-ibid. , 1198 (2004)] [arXiv:astro-ph/0307507].[3] F. K. Hansen, A. J. Banday and K. M. Gorski,arXiv:astro-ph/0404206.[4] G. F. R. Ellis, Gen. Rel. Grav. , 7 (1971).[5] M. Lachieze-Rey and J. P. Luminet, Phys. Rept. ,135 (1995) [arXiv:gr-qc/9605010].[6] J. R. Gott, Mon. Not. R. Astr. Soc. , 153 (1980).[7] N. J. Cornish, D. N. Spergel and G. D. Starkman, Phys.Rev. Lett. , 215 (1996).[8] J. Levin, Phys. Rept. , 251 (2002)[arXiv:gr-qc/0108043].[9] A. Linde, JCAP , 004 (2004)[arXiv:hep-th/0408164].[10] T. Souradeep, Spectroscopy of Cosmic Topology [arXiv:gr-qc/0609026].[11] G. Ellis and M. MacCallum, Commun. Math. Phys. ,108 (1969).[12] J. D. Barrow, R. Juszkiewicz and D. H. Sonoda, Mon.Not. R. astr. Soc.,
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