Bianchi Model CMB Polarization and its Implications for CMB Anomalies
aa r X i v : . [ a s t r o - ph ] A ug Mon. Not. R. Astron. Soc. , 000–000 (2007) Printed 9 November 2018 (MN L A TEX style file v2.2)
Bianchi Model CMB Polarization and its Implications forCMB Anomalies
Andrew Pontzen, ⋆ Anthony Challinor , † Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Accepted 2007 July 6. Received 2007 June 12.
ABSTRACT
We derive the CMB radiative transfer equation in the form of a multipole hierarchy inthe nearly-Friedmann-Robertson-Walker limit of homogeneous, but anisotropic, uni-verses classified via their Bianchi type. Compared with previous calculations, thisallows a more sophisticated treatment of recombination, produces predictions for thepolarization of the radiation, and allows for reionization. Our derivation is independentof any assumptions about the dynamical behaviour of the field equations, except thatit requires anisotropies to be small back to recombination; this is already demandedby observations.We calculate the polarization signal in the Bianchi VII h case, with the parametersrecently advocated to mimic the several large-angle anomalous features observed in theCMB. We find that the peak polarization signal is ∼ . µ K for the best-fit model to thetemperature anisotropies, and is mostly confined to multipoles l <
10. Remarkably, thepredicted large-angle EE and T E power spectra in the Bianchi model are consistentwith WMAP observations that are usually interpreted as evidence of early reionization.However, the power in B -mode polarization is predicted to be similar to the E -modepower and parity-violating correlations are also predicted by the model; the WMAPnon-detection of either of these signals casts further strong doubts on the veracity ofattempts to explain the large-angle anomalies with global anisotropy. On the otherhand, given that there exist further dynamical degrees of freedom in the VII h universesthat are yet to be compared with CMB observations, we cannot at this time definitivelyreject the anisotropy explanation. Key words: cosmic microwave background, cosmology: theory
There are a number of observed features in the large-angle temperature anisotropies of the cosmic microwavebackground (CMB) that are anomalous under the usualassumption of statistically-isotropic, Gaussian fluctuations(see Copi et al. 2007 for a recent summary). Recent analy-ses have suggested that a small global anisotropy in the formdescribed by Bianchi models could be to blame (Jaffe et al.2005). In classical general relativity, the dimension of ini-tial state space leading to models with such homogeneousanisotropies is always greater than the limitation of thesemodels to the exactly isotropic case; furthermore, nearlyisotropic models do not necessarily tend to late-time isotropy(Collins & Hawking 1973b). Although later work has sug-gested an inflationary epoch can be responsible for remov- ⋆ Email: [email protected] † Email: [email protected] ing anisotropies (Wald 1983), this analysis is not complete(Goliath & Ellis 1999). It is therefore worth investigatingfurther the observational predictions of such models; whilstto the skeptic they seem unlikely, the importance of a posi-tive detection would be great.Using the CMB models developed in Collins & Hawking(1973a) and Barrow, Juszkiewicz & Sonoda (1985), basedon a specific solution for a type-VII h universe with onlypressureless matter, Jaffe et al. (2005) showed that knownanomalies such as the low quadrupole amplitude, alignmentof low- l modes, large-scale power asymmetry and the ‘coldspot’ in the CMB could be mimicked. The work was ex-tended in Jaffe et al. (2006) to include the dynamical effectof dark energy, yielding a degeneracy in the Ω Λ - Ω M plane(which effectively arises through the angular diameter dis-tance relation, since much of the temperature anisotropy isgenerated at high redshift). The degenerate range of param-eters able to explain the large-angle anomalies was shown tobe inconsistent with the cosmological parameters required to c (cid:13) Andrew Pontzen, Anthony Challinor explain constraints such as supernovae observations and theCMB spectrum on smaller scales (i.e. the structure of theacoustic peaks). A more complete statistical analysis wasperformed by Bridges et al. (2007), in which the inconsis-tency of inferred parameters was confirmed.There are remaining dynamical freedoms in the VII h model which were not explored in the above papers. Asnoted in Jaffe et al. (2006), these will need investigationbefore any firm conclusions about the compatibility of themodel with observations can be made.A second important issue, hitherto not considered, iswhether the predicted CMB polarization pattern in theBianchi model that best fits the large-angle temperatureanisotropies is consistent with current observations. Thereason for its neglect seems to be that there are cur-rently no full predictions for polarization in Bianchi mod-els (Jaffe et al. 2006). In this paper, we address this is-sue by providing a complete and computationally conve-nient framework for calculating polarization in these mod-els, and make a first attempt at confronting the predictionswith the three-year data from the Wilkinson MicrowaveAnisotropy Probe ( WMAP ; Page et al. 2007). Polarizationof the CMB in anisotropic cosmologies was first discussedby Rees (1968), where a calculation for Bianchi I mod-els was outlined. Further consideration has been given tothe problem in Matzner & Tolman (1982), where Bianchitypes V and IX were considered. There are also a num-ber of papers considering the effect of homogeneous mag-netic fields on polarization (e.g. Milaneschi & Fabbri 1985;Fabbri & Tamburrano 1987), although we emphasize thatthe resulting Faraday rotation of the polarization is quitedistinct from the purely gravitational effect under consider-ation here.This paper is organized as follows. We briefly reviewBianchi models in Section 2, including a more intuitivederivation of their FRW limit than has previously been pub-lished. In Section 3 we derive the zero order evolution ofthe photon polarization direction along geodesics, and in-corporate these results into a multipole treatment of theBoltzmann equation that describes the radiative transfer inSection 4. Using this, we specialize our model to calculatethe temperature and polarization anisotropies expected ina VII h model with favoured parameters (Jaffe et al. 2006)in Section 5, and discuss the difficulties of reconciling theresults with existing observations. We defer a full statisticalreanalysis of Bianchi signatures, including reionization, tofuture work, where we will also explore more fully the extradynamical degrees of freedom in the models. In this section we give a brief review of the frameworkthat we use for our calculation. Anisotropic, homogeneouscosmological models can be classified according to the com-mutation relations of their spatial symmetry groups. Themost popular classification method is based on Bianchi’s(1897) classification of three-parameter Lie groups. For thedevelopment of these classifications in the cosmologicalcontext see Taub (1951); Heckmann & Schucking (1962);Estabrook, Wahlquist & Behr (1968); Ellis & MacCallum (1969), and for reviews see Wainwright (1997)and Ellis & van Elst (1998).We adopt a − +++ metric signature, and will use Greekspacetime indices (0 →
3) for tensor components in a generalbasis, early Latin indices ( a , b etc. running from 0 →
3) tolabel the vectors of any group-invariant tetrad, and middleLatin indices ( i , j etc. running from 1 →
3) to label spa-tial tetrad vectors in expressions only involving the spatialvectors. For the latter, we also use upper-case early Latin( A , B etc.) when the tetrad is time-invariant (see below). Asusual, round brackets denote symmetrisation on the enclosedindices and square brackets denote anti-symmetrisation.A spacetime is said to be homogeneous if it can befoliated into space sections each admitting at least threelinearly-independent Killing vector fields (KVFs) { ξ } (sothat L ξ g = 0, where L denotes the Lie derivative and g is the metric tensor) with at least one subgroup that actssimply transitively in the space sections. We denote the ele-ments of the three-dimensional subgroup by ξ i where i = 1,2, 3. The commutator of two KVFs is also Killing and, sincethe ξ i form a subgroup, we must have[ ξ i , ξ j ] = C kij ξ k . (1)The C kij = C k [ ij ] are the structure constants of the Lie alge-bra of the group and are constant in space. (Through thefoliation construction below they may also be shown to beconstant throughout time.) The Jacobi identity,[ ξ i , [ ξ j , ξ k ]] + [ ξ j , [ ξ k , ξ i ]] + [ ξ k , [ ξ i , ξ j ]] = 0 , (2)restricts the structure constants to satisfy C mn [ i C njk ] = 0 . (3)Since a constant linear combination of KVFs will also be aKVF, one is permitted to perform global linear transforma-tions: ξ i → ξ ′ i = T ji ξ j , (4)under which the structure constants transform as a (mixed)3-tensor. Classifying all homogeneous spacetimes amountsto finding solutions to equation (3) that are inequivalentunder the linear transformations (4).The fiducial classification is achieved by decomposingthe structure constants into irreducible vector and pseudo-tensor parts: C mij = ǫ ijk n km + a i δ mj − a j δ mi ,n ij a i = 0 (Jacobi identities) , (5)where n ij is symmetric and ǫ ijk is the alternating tensor.Linear transformations can be used to diagonalise n ij and,since a i is an eigenvector of n ij one may take, without lossof generality: n ij = diag( n , n , n ) ,a i = ( a, , . (6)By rescaling (possibly reversing their directions) and rela-belling the KVFs, we may reduce the structure constantsto one of 10 distinct canonical forms that describe the 10possible group types (see, for example, Ellis & MacCallum1969). In these forms, all non-zero n i and a are either ± an n = 0 in which case thereis an additional parameter h ≡ a / ( n n ). c (cid:13) , 000–000 ianchi Model CMB Polarization In order to express homogeneous tensor fields in an in-variant way within this space, one must construct a suitablebasis tetrad { e a } that is invariant under the group action,i.e. satisfying[ e a , ξ i ] = 0 . (7)Any homogeneous tensor ( T where L ξ T = 0) has compo-nents relative to this tetrad that are constant throughoutspace. The unit normal n to the hypersurfaces of homo-geneity (satisfying n · ξ i = 0) is necessarily group-invariant([ n , ξ i ] = 0) via the Leibniz property of L , so we can al-ways take e = n . Because of the foliated construction ofthe spacetime, one may label the hypersurfaces with a co-ordinate time t such that n α = − t ,α and hence e can beshown to be geodesic. In any homogeneous hypersurface, wecan always construct the spatial part of the group-invarianttetrad by making an arbitrary choice for e i at a point, andthen dragging this frame out across the hypersurface withthe ξ i (see, for example, Ellis & MacCallum 1969).In general the elements of the tetrad will not commuteso one has[ e a , e b ] = γ cab e c , (8)where the γ cab = γ c [ ab ] are constant in the hypersurfaces of ho-mogeneity by the Jacobi identities. The rotation coefficientsencode the covariant derivatives of the tetrad:Γ cab ≡ e αc e βa ∇ β e bα , (9)where e αa are the components of the tetrad vectors in anarbitrary basis. The rotation coefficients are related to the γ cab by Γ abc = 12 ( − ∂ a g bc + ∂ b g ca + ∂ c g ab + γ abc + γ cab − γ bca ) , (10)where g ab ≡ g ( e a , e b ) are the tetrad components of the met-ric, γ abc ≡ g ad γ dbc and ∂ a is the ordinary derivative alongthe direction of e a . By taking e = n , we have g = − g i = 0 and, since n is normalized, geodesic and irrota-tional, γ ab = 0. The group-invariance of the tetrad restrictsthe form of the spatial components γ kij : they are related tothe group structure constants by a linear transformation andcan be classified in the same way (and, necessarily, fall intothe same Bianchi type: see MacCallum 1973).It is always possible to construct the tetrad so thatit is orthonormal; see Ellis & MacCallum (1969) for de-tails. However, a convenient alternative is the time-invariantframe used by Collins & Hawking (1973a). In this case,the tetrad is constructed so that [ n , e a ] = 0. The Jacobiidentities applied to n , e a and ξ i show that this is stillconsistent with the tetrad being group invariant. A spe-cific construction is to start with a group-invariant tetradover a hypersurface and to drag the tetrad along the nor-mal n to cover spacetime. An important consequence isthat the γ cab are constant throughout spacetime for thegroup-invariant tetrad. Moreover, γ i j = 0 and the onlynon-vanishing components are spatial, i.e. γ kij . Because theBianchi classification of the Killing group and tetrad com-mutators is the same, it is always possible to perform aglobal (three-dimensional) linear transformation of the time-invariant tetrad to bring the γ kij equal to the canonical struc-ture constants, i.e. γ kij = C kij . We shall use upper-case early Latin indices ( A , B etc.) to denote the spatial elements ofthis specific time-invariant tetrad.The spacetime metric can then be written in terms ofthe time-invariant tetrad as g = − n ⊗ n + g AB ( t ) e A ⊗ e B , (11)where the basis one-forms e Aα = g AB g αβ e βB satisfy e Aα e αB = δ AB . Here, as usual, g AB denotes the inverse of g AB . The e A are also group- and time-invariant. We decompose thespatial part of the metric following Misner (1968): g AB = e α ( t ) (cid:0) e β ( t ) (cid:1) AB , (12)where β is a matrix with zero trace and the matrix expo-nential is defined as usual via Taylor expansion, yieldingdet e β = 1. Thus α represents shape-preserving expansionwhilst β represents volume-preserving shape deformation.The expansion rate of the n congruence is ∇ α n α = 3 ˙ α andits shear – i.e. the trace-free, symmetric spatial projectionof ∇ α n β – has components σ AB = 12 e α (cid:0) e β (cid:1) · AB . (13)Here, and throughout, overdots denote derivatives with re-spect to t . We shall make use of conformal time, η , definedby d t = e α d η , and we denote derivatives with respect to η with primes.The field equations are generally more naturally ex-pressed in a (group-invariant) orthonormal frame, since themetric derivatives then vanish and one obtains first orderequations for their commutation functions. The spatial vec-tors of such a frame will be represented with lower-case mid-dle Latin indices, i , j etc. One can always define a specificorthonormal frame by (Hawking 1969) e i = e − α (cid:0) e − β (cid:1) iA e A . (14)The orthonormal tetrad is completed with n . The compo-nents of the shear in the orthonormal frame are σ ij = ( e β ) ˙ k ( i ( e − β ) j ) k . (15)Note that this corresponds to the matter shear tensor onlyin the case that the fluid flow is not tilted relative to thehypersurfaces; i.e. the case n = v where v is the fluid 4-velocity. In this paper, we do not specialize in this way. It is instructive to sketch the derivation of the FRW limitof spacetimes in the time-invariant frame (the calculation inthe orthonormal frame is described in Ellis & MacCallum1969 and yields identical conditions).A homogeneous spacetime can only be FRW if the ex-pansion and 3-curvature are isotropic. Vanishing shear re-quires β AB = const. and then we can always set β AB = 0(i.e. diagonalise the spatial metric) with a constant lineartransformation of the e A . After the transformation, the γ kij will no longer be equal to the canonical group structure con-stants (so we denote them with lower-case indices) but wecan perform a further (constant) orthogonal transformationof the e i (hence preserving β = 0) to bring the γ kij to thecanonical form up to positive scalings of a , n , n and n . c (cid:13)000
3) to label spa-tial tetrad vectors in expressions only involving the spatialvectors. For the latter, we also use upper-case early Latin( A , B etc.) when the tetrad is time-invariant (see below). Asusual, round brackets denote symmetrisation on the enclosedindices and square brackets denote anti-symmetrisation.A spacetime is said to be homogeneous if it can befoliated into space sections each admitting at least threelinearly-independent Killing vector fields (KVFs) { ξ } (sothat L ξ g = 0, where L denotes the Lie derivative and g is the metric tensor) with at least one subgroup that actssimply transitively in the space sections. We denote the ele-ments of the three-dimensional subgroup by ξ i where i = 1,2, 3. The commutator of two KVFs is also Killing and, sincethe ξ i form a subgroup, we must have[ ξ i , ξ j ] = C kij ξ k . (1)The C kij = C k [ ij ] are the structure constants of the Lie alge-bra of the group and are constant in space. (Through thefoliation construction below they may also be shown to beconstant throughout time.) The Jacobi identity,[ ξ i , [ ξ j , ξ k ]] + [ ξ j , [ ξ k , ξ i ]] + [ ξ k , [ ξ i , ξ j ]] = 0 , (2)restricts the structure constants to satisfy C mn [ i C njk ] = 0 . (3)Since a constant linear combination of KVFs will also be aKVF, one is permitted to perform global linear transforma-tions: ξ i → ξ ′ i = T ji ξ j , (4)under which the structure constants transform as a (mixed)3-tensor. Classifying all homogeneous spacetimes amountsto finding solutions to equation (3) that are inequivalentunder the linear transformations (4).The fiducial classification is achieved by decomposingthe structure constants into irreducible vector and pseudo-tensor parts: C mij = ǫ ijk n km + a i δ mj − a j δ mi ,n ij a i = 0 (Jacobi identities) , (5)where n ij is symmetric and ǫ ijk is the alternating tensor.Linear transformations can be used to diagonalise n ij and,since a i is an eigenvector of n ij one may take, without lossof generality: n ij = diag( n , n , n ) ,a i = ( a, , . (6)By rescaling (possibly reversing their directions) and rela-belling the KVFs, we may reduce the structure constantsto one of 10 distinct canonical forms that describe the 10possible group types (see, for example, Ellis & MacCallum1969). In these forms, all non-zero n i and a are either ± an n = 0 in which case thereis an additional parameter h ≡ a / ( n n ). c (cid:13) , 000–000 ianchi Model CMB Polarization In order to express homogeneous tensor fields in an in-variant way within this space, one must construct a suitablebasis tetrad { e a } that is invariant under the group action,i.e. satisfying[ e a , ξ i ] = 0 . (7)Any homogeneous tensor ( T where L ξ T = 0) has compo-nents relative to this tetrad that are constant throughoutspace. The unit normal n to the hypersurfaces of homo-geneity (satisfying n · ξ i = 0) is necessarily group-invariant([ n , ξ i ] = 0) via the Leibniz property of L , so we can al-ways take e = n . Because of the foliated construction ofthe spacetime, one may label the hypersurfaces with a co-ordinate time t such that n α = − t ,α and hence e can beshown to be geodesic. In any homogeneous hypersurface, wecan always construct the spatial part of the group-invarianttetrad by making an arbitrary choice for e i at a point, andthen dragging this frame out across the hypersurface withthe ξ i (see, for example, Ellis & MacCallum 1969).In general the elements of the tetrad will not commuteso one has[ e a , e b ] = γ cab e c , (8)where the γ cab = γ c [ ab ] are constant in the hypersurfaces of ho-mogeneity by the Jacobi identities. The rotation coefficientsencode the covariant derivatives of the tetrad:Γ cab ≡ e αc e βa ∇ β e bα , (9)where e αa are the components of the tetrad vectors in anarbitrary basis. The rotation coefficients are related to the γ cab by Γ abc = 12 ( − ∂ a g bc + ∂ b g ca + ∂ c g ab + γ abc + γ cab − γ bca ) , (10)where g ab ≡ g ( e a , e b ) are the tetrad components of the met-ric, γ abc ≡ g ad γ dbc and ∂ a is the ordinary derivative alongthe direction of e a . By taking e = n , we have g = − g i = 0 and, since n is normalized, geodesic and irrota-tional, γ ab = 0. The group-invariance of the tetrad restrictsthe form of the spatial components γ kij : they are related tothe group structure constants by a linear transformation andcan be classified in the same way (and, necessarily, fall intothe same Bianchi type: see MacCallum 1973).It is always possible to construct the tetrad so thatit is orthonormal; see Ellis & MacCallum (1969) for de-tails. However, a convenient alternative is the time-invariantframe used by Collins & Hawking (1973a). In this case,the tetrad is constructed so that [ n , e a ] = 0. The Jacobiidentities applied to n , e a and ξ i show that this is stillconsistent with the tetrad being group invariant. A spe-cific construction is to start with a group-invariant tetradover a hypersurface and to drag the tetrad along the nor-mal n to cover spacetime. An important consequence isthat the γ cab are constant throughout spacetime for thegroup-invariant tetrad. Moreover, γ i j = 0 and the onlynon-vanishing components are spatial, i.e. γ kij . Because theBianchi classification of the Killing group and tetrad com-mutators is the same, it is always possible to perform aglobal (three-dimensional) linear transformation of the time-invariant tetrad to bring the γ kij equal to the canonical struc-ture constants, i.e. γ kij = C kij . We shall use upper-case early Latin indices ( A , B etc.) to denote the spatial elements ofthis specific time-invariant tetrad.The spacetime metric can then be written in terms ofthe time-invariant tetrad as g = − n ⊗ n + g AB ( t ) e A ⊗ e B , (11)where the basis one-forms e Aα = g AB g αβ e βB satisfy e Aα e αB = δ AB . Here, as usual, g AB denotes the inverse of g AB . The e A are also group- and time-invariant. We decompose thespatial part of the metric following Misner (1968): g AB = e α ( t ) (cid:0) e β ( t ) (cid:1) AB , (12)where β is a matrix with zero trace and the matrix expo-nential is defined as usual via Taylor expansion, yieldingdet e β = 1. Thus α represents shape-preserving expansionwhilst β represents volume-preserving shape deformation.The expansion rate of the n congruence is ∇ α n α = 3 ˙ α andits shear – i.e. the trace-free, symmetric spatial projectionof ∇ α n β – has components σ AB = 12 e α (cid:0) e β (cid:1) · AB . (13)Here, and throughout, overdots denote derivatives with re-spect to t . We shall make use of conformal time, η , definedby d t = e α d η , and we denote derivatives with respect to η with primes.The field equations are generally more naturally ex-pressed in a (group-invariant) orthonormal frame, since themetric derivatives then vanish and one obtains first orderequations for their commutation functions. The spatial vec-tors of such a frame will be represented with lower-case mid-dle Latin indices, i , j etc. One can always define a specificorthonormal frame by (Hawking 1969) e i = e − α (cid:0) e − β (cid:1) iA e A . (14)The orthonormal tetrad is completed with n . The compo-nents of the shear in the orthonormal frame are σ ij = ( e β ) ˙ k ( i ( e − β ) j ) k . (15)Note that this corresponds to the matter shear tensor onlyin the case that the fluid flow is not tilted relative to thehypersurfaces; i.e. the case n = v where v is the fluid 4-velocity. In this paper, we do not specialize in this way. It is instructive to sketch the derivation of the FRW limitof spacetimes in the time-invariant frame (the calculation inthe orthonormal frame is described in Ellis & MacCallum1969 and yields identical conditions).A homogeneous spacetime can only be FRW if the ex-pansion and 3-curvature are isotropic. Vanishing shear re-quires β AB = const. and then we can always set β AB = 0(i.e. diagonalise the spatial metric) with a constant lineartransformation of the e A . After the transformation, the γ kij will no longer be equal to the canonical group structure con-stants (so we denote them with lower-case indices) but wecan perform a further (constant) orthogonal transformationof the e i (hence preserving β = 0) to bring the γ kij to thecanonical form up to positive scalings of a , n , n and n . c (cid:13)000 , 000–000 Andrew Pontzen, Anthony Challinor
Type a n n n Re α I 0 0 0 0 0V − − h −√ h − h IX 0 1 1 1 3 / Table 1.
Bianchi groups with FRW limit and their structure con-stants in canonical form. In the case VII h , we take a = −√ hn n for compatibility with the notation of Collins & Hawking (1973a)and Barrow et al. (1985). Note that a can also be positive, butthis case will be related by a rotation of the sky (constructedby e → − e followed by e → − e ) and need not be consid-ered separately. However, the overall parity must be considered;one way is to consider the transformation e → − e which re-verses the sign of a and all n i . The final column is the comoving3-curvature scalar in the β = 0 FRW limit. With β = 0, the 3-curvature is R (3) ij = 13 R (3) δ ij e α = − (cid:2) γ kil γ ljk + 2 γ lki γ lkj − γ ikl γ jkl + 2 γ kkl ( γ ijl + γ jil ) (cid:3) . (16)Zero- and first-order expressions (such as the one above) willoften contain surprising index placement; note that metricfactors are always included where necessary to account forthis.In terms of the decomposition of γ kij , one obtains the off-diagonal element R (3) 23 = a ( n − n ), yielding the condition a = 0 or n = n (17)if the curvature is to be isotropic. The vanishing of the re-mainder of the trace-free part further requires n + n n = n + n n = n + n n , (18)which implies that at least one of the n i are zero and theremaining two are equal. By examining a complete list ofdistinct group types, these conditions yield the spaces withan FRW limit (Table 1). In each case we can use a volumerescaling (change of α ) to set the non-zero n i to unity or,if the n i vanish, a to −
1. This is enough freedom to setthe γ kij of the time-invariant frame, with β = 0, equal tothe corresponding canonical group structure constants. Fora universe that is close to FRW, we can therefore treat β as a small perturbation while working in the time-invariantframe with γ CAB still in canonical form.
We parameterize the photon propagation direction vectoraccording to the convention in Barrow et al. (1985); al-though this unusually places the azimuthal direction along e , it more conveniently reflects the symmetries of the fidu-cial Bianchi classification, which ultimately yields much sim-pler equations. Thus in the orthonormal frame, the photondirection has components p i = (cos θ, sin θ cos φ, sin θ sin φ ) (19)and the photon 4-momentum is K = E ( n + p i e i ) , (20)where E is the photon energy measured by an observer on ahypersurface-orthogonal path. It is convenient to introducethe comoving energy, ǫ ≡ Ee α . In terms of this, the spatialcomponents of K on the time-invariant tetrad (in which thegeodesic equations take their simplest form) are K A = ǫ (cid:0) e β (cid:1) iA p i . (21)The energy change along a geodesic follows from differ-entiating E = − K · n . The geodesic equation then gives theexact result ǫ ′ = − ǫe α p i p j σ ij , (22)where ǫ ′ ≡ d ǫ/ d η . This is equivalent to the integral resultobtained by Hawking (1969). For the evolution of θ and φ ,we make use of the (exact) geodesic equation in the time-invariant frame (Barrow et al. 1985): K ′ A = (cid:0) e − β (cid:1) BD K C K D ǫ C CBA . (23)We only require the evolution of θ and φ to zero order in β since the radiation is necessarily isotropic in the FRW limit.Setting β = 0 in equation (23), we find θ ′ = [ a + ( n − n ) cos φ sin φ ] sin θ (24) φ ′ = [ n − n + ( n − n ) cos φ ] cos θ. (25)We denote the (complex) polarization 4-vector by P .It is normalized so that P · P ∗ = 1 and, in the Lorentzgauge, is orthogonal to K . The polarization P is paralleltransported along the photon geodesic. We are more inter-ested in the observed polarization ˜ P (i.e. the electric fielddirection for radiation in a pure state) relative to n . Thisis given by screen-projecting P perpendicular to n and thephoton direction p :˜ P α = H αβ P β , (26)where the screen-projection tensor H αβ is defined by H αβ ≡ g αβ + n α n β − p α p β . (27)The evolution of ˜ P follows from the parallel transport of P itself (Challinor 2000): H αβ (cid:0) K ρ ∇ ρ ˜ P β (cid:1) = 0 . (28)We find for the tetrad components of ˜ P in the time-invariantbasis the exact equation˙˜ P A − ˜ P B σ AB − p A ˜ P B p C σ BC − ˙ α ˜ P A + p B ˜ P C Γ ABC = 0 , (29)where Γ ABC are the rotation coefficients given by the spatialcomponents of equation (10). There is only one degree offreedom in the projected polarization vector, described bythe angle between it and the ˆ e θ direction: ˜ P = ˆ e θ cos ψ + ˆ e φ sin ψ (30)where ˆ e θ and ˆ e φ are constructed from the orthonormal-frame vectors. The components on the orthonormal frameare therefore The metric derivatives vanish in this case since g IJ is constantin each hypersurface. c (cid:13) , 000–000 ianchi Model CMB Polarization ˜ P i = − sin θ cos ψ cos θ cos φ cos ψ − sin φ sin ψ cos θ sin φ cos ψ + cos φ sin ψ ! . (31)The polarization is first-order in β so we only require theevolution of ψ at zero order. In this FRW limit, equation (29)reduces to2 ψ ′ = − n − ( n − n ) cos 2 φ. (32)Since we are considering models close to FRW, n = n (see Table 1) and our evolution equations simplify to θ ′ = −√ h sin θφ ′ = ( n − n ) cos θ ψ ′ = − n , (33)where, for type V universes, one takes √ h = 1. We see thatthe polarization only rotates relative to ˆ e θ and ˆ e φ in type-IXuniverses. These general equations for θ and φ agree with thespecific cases given in Appendix B of Barrow et al. (1985);the general ψ equation agrees with the previous analysis oftype-IX universes given in Matzner & Tolman (1982). In this section, we derive the first-order Boltzmann equationdescribing polarized radiative transfer in any Bianchi modelwith an FRW limit.CMB polarization is generated by Thomson scatteringand is therefore expected to be only linearly polarized. Wecan describe the radiation distribution function in terms ofStokes parameters f , q and u where f gives the expectednumber density of photons per proper phase-space volumeirrespective of their polarization state. The parameters q and u describe linear polarization relative to a basis that we taketo be ˆ e θ and ˆ e φ for propagation along p i . Then ( f + q ) / e θ linear-polarization state, and ( f + u ) / e φ ). In the absence of scattering, f is conserved along thephoton path in phase space and q and u are conserved ifreferred to bases that rotate like ˜ P in equation (28). Weparameterize f , q and u by the photon-direction angles θ and φ and by comoving energy ǫ . Since these parameters aredefined relative to a group-invariant tetrad, f is independentof position in the hypersurfaces of homogeneity. The sameis true for the Stokes parameters since the basis on whichthey are defined is constructed in an invariant manner.The Lagrangian derivative of f in phase space isD f D η = ∂f∂η + ∂f∂θ θ ′ + ∂f∂φ φ ′ + ∂f∂ǫ ǫ ′ , (34)and is only non-zero because of scattering. The quanti-ties ǫ ′ , θ ′ and φ ′ are given by equations (22) and (33).Anisotropies are formed through the last term in equa-tion (34) and by Thomson scattering off electrons with anon-zero peculiar velocity; both effects are first order. Theremaining terms ‘advect’ the resulting pattern on the sphereusing the zero-order transport equations. The energy de-pendence of the anisotropies is therefore proportional to ǫ∂ ¯ f/∂ǫ where ¯ f is the Planck distribution function in theFRW background. We define the dimensionless temperatureanisotropies Θ( θ, φ ; η ) by f ( K ; η ) = ¯ f ( ǫ ) (cid:18) − d ln ¯ f d ln ǫ Θ (cid:19) . (35)Polarization is generated by scattering the anisotropies and,since Thomson scattering is achromatic, the polarization hasthe same ǫ spectrum as the anisotropies. We can thereforeintroduce dimensionless ‘thermodynamic-equivalent’ Stokesparameters Q ( θ, φ ; η ) and U ( θ, φ ; η ) as( q ± iu )( K ; η ) = − d ¯ f d ln ǫ ( Q ± iU ) . (36)The quantities Q ± iU have spin-weight 2, i.e. under a changeof basisˆ e θ + i ˆ e φ → e iχ (ˆ e θ + i ˆ e φ ) ⇒ Q ± iU → e ± iχ ( Q ± iU ) . (37)Expressed in this way, the time dependence of the tem-perature anisotropies obeys: ∂ Θ ∂η = DΘD η − ∂ Θ ∂θ θ ′ − ∂ Θ ∂θ φ ′ − e α p i p j σ ij , (38)where the first term on the right describes the Thomsonscattering kernel (Section 4.3), the next two describe ad-vection of the patterns on the sphere, and the final term isgravitational redshifting in the anisotropic expansion due tothe shear. The advection terms are more transparent wheninterpreted as being due to the spatial dependence (relativeto a parallel-propagated rather than time-invariant basis)transforming to angular dependence through free streaming.For polarization, we have ∂ ( Q ± iU ) ∂η = D( Q ± iU )D η − ∂ ( Q ± iU ) ∂θ θ ′ − ∂ ( Q ± iU ) ∂φ φ ′ ± i ( Q ± iU ) ψ ′ , (39)where the last term arises from polarization rotation. Again,the first term on the right describes Thomson scattering (seeSection 4.3).We expand the temperature anisotropies in terms ofspherical harmonics about the propagation direction:Θ( p ) = X lm Θ ml Y ml ( p ) . (40)For polarization, we expand in spin-weighted spherical har-monics as( Q ± iU )( p ) = X lm ( E ml ± iB ml ) Y ± ml ( p ) . (41)Note that the Bianchi models are not parity invariant; themirror universe can be obtained using the standard trans-formationsΘ ml → ( − l Θ ml E ml → ( − l E ml B ml → ( − ( l +1) B ml , (42)or, equivalently, flipping the signs of a and all n i (whichinverts the direction of the e axis). For a brief review, see Appendices A-Cof Lewis, Challinor & Turok (2002).c (cid:13) , 000–000
Andrew Pontzen, Anthony Challinor
The effect of the shear on the observed temperature patternreads (cid:18) ∂ Θ ∂η (cid:19) shear = − e α p i p j σ ij , (43)which is a pure quadrupole. In terms of spherical harmonics, (cid:18) ∂ Θ ∂η (cid:19) shear = r π
15 ( σ − σ + 2 iσ ) e α (cid:18) ∂ Θ ∂η (cid:19) shear = r π
15 ( σ − iσ ) e α (cid:18) ∂ Θ ∂η (cid:19) shear = − r π σ e α (cid:18) ∂ Θ − m ∂η (cid:19) shear = ( − m (cid:18) ∂ Θ m ∂η (cid:19) ∗ shear , (44)where σ ij are the components of the shear in the orthonor-mal frame. We see that shear injects power at l = 2, butsubsequent advection generally transports this to higher l . The advection part of the Boltzmann equation for the tem-perature is (cid:18) ∂ Θ( θ, φ ) ∂η (cid:19) advec ≡ − ∂ Θ( θ, φ ) ∂θ θ ′ − ∂ Θ( θ, φ ) ∂φ φ ′ , (45)with θ ′ and φ ′ given by equations (33). Making use of theresultssin θ ∂ s Y ml ∂θ = ll + 1 r [( l + 1) − m ][( l + 1) − s ](2 l + 1)(2 l + 3) s Y ml +1 + msl ( l + 1) s Y ml − l + 1 l r ( l − m )( l − s )(2 l − l + 1) s Y ml − cos θ s Y ml = 1 l + 1 r [( l + 1) − m ][( l + 1) − s ](2 l + 1)(2 l + 3) s Y ml +1 − msl ( l + 1) s Y ml + 1 l r ( l − m )( l − s )(2 l − l + 1) s Y ml − (46)which follow from standard recursion relationsfor the related Wigner functions D l − ms ( φ, θ, (cid:18) ∂ Θ ml ∂η (cid:19) advec = l +1 X l ′ = l − f mll ′ Θ ml ′ . (47)Here, f ml,l − = r l − m (2 l − l + 1) (cid:2) im ∆ n + ( l − √ h (cid:3) f ml,l = 0 f ml,l +1 = r ( l + 1) − m (2 l + 1)(2 l + 3) (cid:2) im ∆ n − ( l + 2) √ h (cid:3) (48)with ∆ n ≡ n − n . Note that ± f − mll ′ = ∓ f mll ′ ∗ , as requiredby the reality of Θ( θ, φ ) in equation (47). For polarization, we have (cid:18) ∂ ( Q ± iU ) ∂η (cid:19) advec ≡ − ∂ ( Q ± iU ) ∂θ θ ′ − ∂ ( Q ± iU ) ∂φ φ ′ ± iψ ′ ( Q ± iU ) . (49)We write (cid:18) ∂ ( E ml ± iB ml ) ∂η (cid:19) advec = l +1 X l ′ = l − g ± mll ′ ( E ml ′ ± iB ml ′ ) , (50)which may be expressed (cid:18) ∂E ml ∂η (cid:19) advec = 12 l +1 X l ′ = l − [( g + mll ′ + g − mll ′ ) E ml ′ + i ( g + mll ′ − g − mll ′ ) B ml ′ ] , (cid:18) ∂B ml ∂η (cid:19) advec = 12 l +1 X l ′ = l − [( g + mll ′ + g − mll ′ ) B ml ′ − i ( g + mll ′ − g − mll ′ ) E ml ′ ] . (51)We see that there are two modes of propagation; one trans-fers power amongst l and the other mixes E and B -modes.The mixing terms are g + ml,l − g − ml,l = − in + 4 (cid:0) √ h − im ∆ n (cid:1) ml ( l + 1) ,g + ml,l +1 − g − ml,l +1 = g + ml,l − − g − ml,l − = 0 . (52)Polarization is generated from Thomson scattering as a pureelectric quadrupole (see Section 4.3) but B -modes can subse-quently be produced through advection. This happens in allnearly-FRW Bianchi models except type I. The n term inequations (52) arises from the polarization rotation ψ ′ ; theremaining terms are from the evolution of the photon direc-tion relative to the invariant frame. For the power transfer,we find g + ml,l + g − ml,l = 0 ,g + ml,l − + g − ml,l − = 2 r ( l − m )( l − l (2 l − l + 1) × (cid:2) ( l − √ h + im ∆ n (cid:3) ,g + ml,l +1 + g − ml,l +1 = 2 r [( l + 1) − m ] [( l + 1) − l + 1) (2 l + 1)(2 l + 3) × (cid:2) − ( l + 2) √ h + im ∆ n (cid:3) , (53)which are not affected by polarization rotation. Note that ± g − mll ′ = ∓ g mll ′ ∗ , (54)as required by the reality of E ( θ, φ ) and B ( θ, φ ) in equa-tion (50). We use a standard Thomson scattering kernel in the formderived by Hu & White (1997) (see also Dautcourt & Rose1978): (cid:18) D ( E ml ± iB ml ) Dη (cid:19) = τ ′ (cid:18) − ( E ml ± iB ml )+ 35 δ l ( E m − √ m ) (cid:19) , c (cid:13) , 000–000 ianchi Model CMB Polarization (cid:18) D Θ ml Dη (cid:19) = τ ′ (cid:18) − Θ ml (1 − δ l )+ 110 δ l (Θ m − √ E m ) + δ l ˜ u m (cid:19) , (55)where τ ′ = n e σ t e α gives the scattering rate in conformaltime, and˜ u − = r π u + iu )˜ u = r π u ˜ u = r π − u + iu ) (56)are the dipole moments of the electron peculiar velocity inthe orthonormal frame. It follows from equation (55) thatThomson scattering of the temperature quadrupole gener-ates polarization that is an E -mode quadrupole. H UNIVERSES5.1 Field equations
The complete set of field equations are available from, for ex-ample, Wainwright (1997). Naturally, these reduce at zeroth-order to the standard Friedmann and acceleration equations,so that e α d η d z = − H ( z )= − H − (cid:0) Ω Λ , + Ω K, (1 + z ) +Ω M , (1 + z ) (cid:1) − / , (57)where Ω M , , Ω Λ , and Ω K, have their usual meanings,(1 + z ) − = e α − α and H = ˙ α . The Friedmann constraintequation relates the group parameter h to the curvature (seeTable 1): h = H e α Ω K, .The evolution of the shear is required at first order andis provided by the trace-free part of the spatial evolutionequations,˙ σ ij = − Hσ ij − S (3) ij (58)in the orthonormal frame. Here, (3) S ij ≡ R (3) ij − Rδ (3) ij / h case, (3) S and (3) S are zero to this accuracy, and, further-more, no coefficient of β or β enters into the expressionfor (3) S ij , so that one may study a simple model in which σ ij = 0 except for σ , σ ∝ e − α . The linear constraintequations show that the matter in such a model containsvorticity, i.e. the separation of neighbouring particles rotatesrelative to inertial gyroscopes.Whilst it is not prohibitively difficult to implement anumerical solution for the most general case, we defer sucha treatment to a later paper. Instead, we take advantage of the simplified solutions to derive the polarization in thefavoured models of Jaffe et al. (2006).We shall adopt the standard assumption that the CMBsignal from global anisotropy adds linearly to that from inho-mogeneities. This is clearly correct insofar as the linear-orderperturbations are concerned; however, given that genericanisotropy modes grow towards the initial singularity, thereis no guarantee that standard inflationary mechanisms forgenerating inhomogeneities can be invoked. Ignoring this po-tential inconsistency is pragmatic, but investigation wouldcertainly be necessary if the resulting models gain any sig-nificant observational support. To be consistent with our assumption of small departuresfrom FRW symmetry, we assume that all peculiar velocitiesare small. If we write the total momentum density of allmatter and radiation as P (tot) = P n ( ρ ( n ) + p ( n ) ) u ( n ) , thelinear constraint equation relates the spatial components tothe shear:8 πP (tot) i = e − α ( σ jk C jki − σ ij C kkj ) (60)to first order in the orthonormal frame. Here, the C ijk arethe structure constants in canonical form. In the restrictedsolution σ , σ ∝ e − α , so, assuming the total momentumdensity is dominated by a barotropic fluid with equation ofstate p = wρ for constant w , we have ρ ∝ e − w ) α and | u | = ( u i u i ) / ∝ e (3 w − α ∝ (cid:26) constant w = 1 / e − α w = 0 . (61)This behaviour of | u | is consistent with momentum con-servation. To see this, consider the Euler equation for a non-interacting ideal fluid in the time-invariant frame. The fluidpressure is constant on surfaces of homogeneity but gra-dients proportional to ˙ pu A appear in the fluid rest frame.These accelerate the fluid so that˙ pu A + ( ρ + p ) ˙ u A = 0 . (62)Solving gives u A ∝ e wα and, recalling the zero-order metric g AB = e − α , we recover equation (61).The above introduces a complication in multi-fluidmodels, which appears to have been overlooked in recentwork. For two components, say, the tilt velocities u (1) , u (2) need not be the same (except in the case of strong cou-pling). Only the total momentum density, ( ρ + p ) u (1) +( ρ + p ) u (2) is constrained by the shear so there is addi-tional freedom in the solution. Although the dark matterdensity will be dominant around the time of recombination, z ∼ z LSS , the tilt velocity of relevance for the CMB is mani-festly that of the baryons. Given that the baryons are tightlycoupled to the photons until the last scattering surface, theyexperience a significant pressure and their tilt decay will behalted; this will not be the case for the dark matter. Thus, We note that, with this effect in mind, the application of theterm ‘universal vorticity’ to describe VII h cosmologies is an over-simplification, since the vorticity of the dark matter need bear noresemblance to that of the baryons.c (cid:13) , 000–000 Andrew Pontzen, Anthony Challinor one needs to consider with care how to estimate the electronvelocity in the Thomson scattering terms (56).For most of cosmic history before recombination, thebaryon–photon plasma has an equation of state parameterclose to w = 1 /
3. A simple approximation is thus obtainedby assuming the baryon tilt velocity remains constant be-fore recombination, after which it decays as the inverse scalefactor. If dark matter decouples at z DM , the ratio of baryonto dark-matter peculiar velocities at last scattering will be | u (b) || u (c) | (cid:12)(cid:12)(cid:12)(cid:12) LSS ≈ z DM z LSS . (63)We take dark-matter decoupling to be at redshift (see e.g.Loeb & Zaldarriaga 2005) z DM ∼
10 MeV T CMB k B (cid:16) M σ
100 GeV (cid:17) (cid:16) M
100 GeV (cid:17) / ≃ , (64)where k B is Boltzmann’s constant, M σ is the coupling mass, M is the particle mass, and the chosen values assume asuper-symmetric origin of the CDM particle.Unfortunately, the linearisation will break down at highredshift as the expansion-normalized scales as e − α , so thevalue by which the dark matter tilt is suppressed rela-tive to the baryon tilt is unclear. However, equation (63)strongly suggests that the baryon–photon plasma dominatesthe momentum density at last scattering and that its tiltshould be properly determined at z LSS by equation (60) with P = ( ρ b + 4 ρ γ / u (b) on the left-hand side. After this u (b) decays as 1 + z as determined by (62). The usual procedureof assuming that all components have the same tilt under-estimates | u (b) | at last scattering by the ratio of the baryon–photon enthalpy to the total enthalpy. For the majority ofour results, we follow the usual procedure for consistencywith previous work. We consider the effect of the improvedvelocity analysis in Section 5.7, where we show that it willhave a significant impact on statistical studies, but does notchange our qualitative results. For ΛCDM, the background model may be specified fullyby the physical densities in CDM ( ω c , ≡ Ω c , h with H = 100 h km s − Mpc − ) and baryons ( ω b , ) with Ω Λ , and Ω K, . The Hubble constant and Ω M , are then derivedquantities. Models with fixed ω c , and ω b , have the sameearly-universe history and reproduce the same acoustic peakstructure in the CMB spectra if the angular-diameter dis-tance to last-scattering and the primordial power spectraare additionally held fixed (Efstathiou & Bond 1999). TheBianchi representation with structure constants in canonicalform further requires us to specify e α , although this is of nophysical consequence in the background. In the perturbedmodel, the current scale factor e α determines the physi-cal size over which the shear eigenvectors rotate in spaceon a parallel-propagated triad. For the simplified perturbedmodel, we must additionally specify initial values for σ and σ . Due to the rotational symmetry of the VII h structureconstants about e , only m = ± σ ∓ iσ ) /H ] respectively. Varying the phase of σ + iσ amounts to rotating the sky about e (reflecting the residual freedom in the choice of e and e ), while the rotationally-invariant content depends on σ + σ . We can, therefore,always choose σ = σ which we do for compatibility withprevious studies.The morphology of the CMB anisotropy and polariza-tion patterns in the Bianchi model is determined largely bythe parameters Ω M , , Ω Λ , and the conformal Hubble pa-rameter e α H . The expansion-normalized shear ( σ /H ) and ( σ /H ) determine the amplitude. Collins & Hawking(1973a) denote the conformal Hubble parameter by x , i.e. x = ˙ α e α = r h Ω K, , (65)where the latter relation arises directly from the FRW def-inition of Ω K, , with K = − h in our case (Section 2.1).With x , Ω M , and Ω Λ , fixed, variations in e α change phys-ical scales in the model (e.g. the age) but do not affect theconformal properties. There is an approximate degeneracyamongst Ω M , , Ω Λ , and x that preserves the morphology ofthe Bianchi patterns (Jaffe et al. 2006; Bridges et al. 2007).In our results, we follow Jaffe et al. (2006) by fixing ω c , and ω b , and use an ionization history consistent with thesechoices. Further specifying Ω M , and Ω Λ , determines H ;the current scale factor is then fixed by x . We assemble a hierarchy of multipole equations for Θ lm , E lm and B lm using the results of Section 4. The Thomson scat-tering rate ˙ τ requires a model for the recombination (and po-tentially reionization) history, for which we use RECFAST (Seager, Sasselov & Scott 1999).Starting at z ≃ z ≃ l >
15, we truncate the hierarchy at l = 60 with-out any special boundary conditions. Performing the calcu-lation with a higher truncation ( l = 120) made no differenceto the results for l <
30. During the numerical integrationwe ensure at each timestep δη ≪ σδη ≪ τδη ≪ x, Ω Λ , , Ω M , ) = (0 . , , . . , . , .
2) respectively, both with “right-handed”parity. The latter model is as close to a concordance value as The more refined treatment of tilt velocities requires one tospecify also the fraction of baryons to dark matter and the physi-cal Hubble parameter. However, the Doppler terms are generallyonly a small correction to the anisotropy accumulated throughthe shear. The details of recombination introduce further depen-dencies on the physical densities of baryons and dark matter.c (cid:13) , 000–000 ianchi Model CMB Polarization Figure 1.
Temperature (top), E -mode (middle) and B -modepolarization (bottom) maps for the Bianchi VII h model with( x, Ω Λ , , Ω M , )=(0 . , , .
5) and a consistent recombination his-tory and no reionization. The maps have been transformed to theobservational basis ( − p , ˆ e θ , ˆ e φ ), which involves a parity changeof the form (42), and rotated to match the orientation of the tem-plate given in Jaffe et al. (2006). The masks used in the WMAP polarization analysis (Page et al. 2007) are overlaid on the polar-ization maps. the Bianchi fitting allows (see Fig. 7 in Bridges et al. 2007).In both cases, we take ω b , = 0 .
022 and ω c , = 0 .
110 and theconsistent recombination history with no reionization. Thesemodels produce almost identical polarization patterns, forreasons outlined below. We briefly discuss the effects of al-tering the ionization history in various ways (including reion-ization) in Section 5.6.In each case, we normalize such that the maximumtemperature anisotropy corresponds to ∆ T = ± µ K.Note that the amplitude of the polarization anisotropy sim-ply scales linearly with the magnitude of the temperatureanisotropy.
Figure 2.
Auto- and cross-correlation power spectra for theBianchi models ( x, Ω Λ , , Ω M , ) = (0 . , , .
5) (solid lines) and( x, Ω Λ , , Ω M , ) = (1 . , . , .
2) (dotted lines), normalized suchthat the maximum ∆ T = ± µ K. (The units of the vertical axisare µ K.) The main difference between the models is a shift ofpower to larger scales (lower l ) in the model with Λ; this is wellunderstood in terms of the reduced focusing given lower Ω K, (seetext), and causes no difference to our conclusions. Note that the T B correlation is negative for l < l <
The resulting temperature and polarization E - and B -modemaps for the Ω Λ , = 0 case are illustrated in Fig. 1. The levelof the polarization is very high, approximately 1 µ K. Heuris-tically, this is because the shear modes considered here de-cay as (1 + z ) , so that a substantial portion of the finaltemperature anisotropy can be built up between individualscattering events at high redshift. Because of the efficientconversion of E -modes to B , (equation 52), the B -mode con-tribution is of similar magnitude to the E -mode.Although computing the power spectra, C XYl = 12 l + 1 X m a X ∗ lm a Ylm , (66)does throw away useful information in these models, it pro-vides a fast and efficient way to compare our results withknown, and robust, polarization constraints. Since the mul-tipole hierarchy does not transfer power between different m values, and the implemented cosmology only generatesanisotropies with m = ±
1, in forming the power spectrumwe are throwing away only phase information.Given the position of the Bianchi-like features on thesky given in Jaffe et al. (2006), we may be confident thatthe P06, and even the P02, mask of the
WMAP polarizationanalysis (Page et al. 2007) could not hide the polarizationsignal to a great extent (Fig. 1). Although the relation be-tween the Stokes parameters and E and B is non-local, wenote that maps of the Stokes parameters have their powerlocalized in a similar way to E and B (and T ). We therefore c (cid:13)000
WMAP polarizationanalysis (Page et al. 2007) could not hide the polarizationsignal to a great extent (Fig. 1). Although the relation be-tween the Stokes parameters and E and B is non-local, wenote that maps of the Stokes parameters have their powerlocalized in a similar way to E and B (and T ). We therefore c (cid:13)000 , 000–000 Andrew Pontzen, Anthony Challinor
Figure 3.
Growth of observable r.m.s. signal in the T and E -and B -mode polarization plotted against a = e α − α = (1 + z ) − for the models ( x, Ω Λ , , Ω M , )=(0 . , , .
5) (solid lines) and( x, Ω Λ , , Ω M , )=(1 . , . , .
2) (dotted lines). Note that the powergrows rapidly at high redshift while the shear is still significant,then remains constant (although it is transferred to higher l ,which cannot be seen in this diagram). It is for this reason thatthe polarization is remarkably strong and relatively insensitiveto the cosmology along the line for which the VII h temperaturepatterns are degenerate. calculate the full-sky power spectra without any considera-tion of the effect of masking nor the weighting with the in-verse of the (non-diagonal) pixel-pixel noise covariance ma-trix that were employed by the WMAP team. Given that ther.m.s. Bianchi signal inside the masks is lower than outside,we expect the effects of masking would increase the esti-mated Bianchi power spectra over the full-sky values plottedin Fig. 2.The major difference between the two parameter setsconsidered is that, for the Ω Λ , = 0 . K, to 0 . . l (see Fig. 2). The existingstatistical studies show that distinguishing these cases ob-servationally is currently not possible (Bridges et al. 2007).There is no significant difference in the overall polariza-tion power. This follows because the majority of the poweris built up rapidly at high redshifts as the universe becomesoptically thin and the shear term has not decayed: at thispoint, the model is insensitive to the values of Ω Λ , andΩ K, (Fig. 3). Allowing ω b , to vary introduces much moresubstantial variations in the relative level of polarization;however, this introduces a further degree of freedom and isbeyond the scope of our current analysis.In Fig. 4, we compare the power spectra in the BianchiΩ Λ , = 0 model with the power expected in a ‘concordance’model with standard, statistically-isotropic and homoge-neous perturbations. The latter spectra are computed usingCAMB (Lewis, Challinor & Lasenby 2000) for two models, Figure 4.
Bianchi VII h induced power in the CMB (solidlines) for ( x, Ω Λ , , Ω M , )=(0 . , , .
5) and no reionization,compared with Gaussian power from inhomogeneities for( ω c , , ω b , , σ , r ) = (0 . , . , . , .
3) with reionization op-tical depth τ = 0 (dash-dotted lines) and τ = 0 .
10 (dashedlines). The polarization data are from the
WMAP three-yearrelease (Page et al. 2007). From the
T E and EE power spec-tra alone, the Bianchi-induced polarization can mimic the ef-fect of early reionization in the standard scenario (the conven-tional interpretation of the large-scale polarization power seen by WMAP ). However, the best-fit Bianchi model to the tempera-ture map clearly over-produces B -mode power compared to the WMAP upper limit (plotted) ruling out the simple model imme-diately. one with no reionization (dot-dashed lines) and a favouredreionization model (dashed lines; τ = 0 . T E and EE power spectra areconcerned, the Bianchi model can mimic the observed large-angle power that is conventionally attributed to reioniza-tion. Of course, the ‘corrected’ power in such a model wouldprobably lead to an unfeasibly low estimate for τ in lightof other data such as the Gunn-Peterson constraints (e.g.Fan, Carilli & Keating 2006). So, at least with the fiducialsimplified dynamics outlined in Section 5.1, this already pro-vides strong evidence against the VII h model.More challenging for the Bianchi model is the B -modepolarization, which is at a similar level to the E -mode. InPage et al. (2007), the B -modes for l <
10 are found to beconsistent with zero with errors better than σ ∼ . µ K at each multipole. At this level, the signal-to-noise on the B -mode spectrum in the Bianchi model should be at leastunity for each < l <
8, and would have produced a highlysignificant detection of large-angle B -modes overall.Finally, the Bianchi models are not parity-invariantand one therefore obtains a T B and EB cross-correlation c (cid:13) , 000–000 ianchi Model CMB Polarization (Fig. 2). To get a rough estimate of the current statis-tical power of these spectra in constraining the Bianchimodel, we compute the χ between the model predictionand the WMAP estimates of C TEl and C TBl available onthe LAMBDA website . We use the spectra from l = 2–16and, since only the diagonal errors are publically available,we ignore correlations between the estimates and compli-cations due to the shape of the low- l likelihood. As notedearlier, we also ignore the effects of foreground masking andnoise-weighting. We find reduced χ values of 0 . T B and 4 . EB for 15 degrees of freedom. The correspondingfigures for null C TBl and C EBl are 0 . .
6. Note that,although the Bianchi power is typically two orders of magni-tude smaller in EB than T B , the EB estimates have smallererrors as C Tl exceeds the variance of the polarization noiseon these scales. The interpretation of these χ values is thatthe data are too noisy to distinguish the Bianchi model fromthe null case for T B (both are perfectly consistent) but the EB spectra disfavour the Bianchi model over the null case. Since the shear decays rapidly, σ ∝ (1 + z ) , our inclusionof a more detailed recombination calculation will affect thetemperature maps somewhat. We take the Ω Λ , = 0 modeland run the Boltzmann hierarchy twice; first with instan-taneous recombination at z = 1100 and then with the full RECFAST history (with no reionization). There is no qual-itative difference in the temperature maps produced, butthere is an approximately 15 percent decrease in the temper-ature amplitude in the latter case. Of course, this is simplyreflected in a slightly different estimate of ( σ/H ) and has nosignificant impact on previous probes of Bianchi signatures.However, the detailed recombination history does havea significant impact on the amplitude of polarization. Withthe detailed model, the amplitude is approximately fivetimes larger than that derived from the instantaneous model.Note that this puts the amplitude of polarization in the in-stantaneous model in agreement with the estimation in Rees(1968). It is unsurprising that the polarization is so sensi-tive to the recombination model, given that it arises throughthe detailed interplay of the rapidly decreasing shear andsharply peaked visibility function ˙ τ e − τ .The effect of adding reionization is not as dramatic asfor standard FRW perturbations; this is because of the highlevel of the primordial polarization relative to the tempera-ture signal ( ∼ /
25) on large scales. Adding reionization asearly as z = 15 produces only a ∼
50 percent increase in the EE power. The numerical results presented so far have derived thebaryon tilt assuming the same tilt for all particle species, http://lambda.gsfc.nasa.gov/ We checked that the additional variance in the power spec-trum estimates (when averaging over statistically-isotropic CMBfluctuations and noise) due to products between the two-pointfunctions of the Bianchi signal and fluctuations is only a smallcorrection to the errors computed by the
WMAP team.
Figure 5. (Top) Normalized temperature map for BianchiVII h model ( x, Ω Λ , , Ω M , )=(0 . , , .
5) with improved tilt con-straint; (bottom) residuals in this map compared to the standardconstraint map (Fig. 1). The centre of the maps are here orienteddown the e axis. Figure 6.
Comparison of power spectra with standard constraint(solid lines) and improved constraint (dashed lines) after renor-malizing the maximum temperature anisotropy to ± µ K.c (cid:13)000
Comparison of power spectra with standard constraint(solid lines) and improved constraint (dashed lines) after renor-malizing the maximum temperature anisotropy to ± µ K.c (cid:13)000 , 000–000 Andrew Pontzen, Anthony Challinor consistent with previous work. In this section, we analyze theeffect of dropping this assumption and adopting the moresophisticated model of Section 5.2.One may see heuristically that the shear generallycontributes more to the temperature anisotropy than thedipole, because its integrated effect (for models with mat-ter domination over most of the line of sight) scales as ∼ Ω − / , ( σ/H ) (1 + z LSS ) / × O (1) whereas the dipole isimprinted instantaneously at scattering and scales in theimproved model as | u | ∼ √ h (1 + z LSS ) x Ω b , (cid:16) σH (cid:17) . (67)Using this new approach with the two sets of parametersconsidered above, the ratio of the shear to Doppler contri-butions is ∼
6. If instead we assume the same tilt for allspecies, as in the previous section, we should replace Ω b , by Ω M , in equation (67) and this ratio becomes ∼ T /T , and only the quadrupole at high redshiftis responsible for producing polarization.The revised temperature map and residual map for the( x, Ω Λ , , Ω M , ) = (0 . , , .
5) model are shown in Fig. 5.The power spectra are plotted in Fig. 6. As expected, thepolarization strength is somewhat lower after renormaliza-tion. The effects are in accordance with our expectations:the difference in the temperature maps amounts to a 15%effect, whilst the polarization level is reduced by approxi-mately 10%.It is clear that the details of how the tilt is treated willimpact on a detailed statistical comparison of the modelswith the
WMAP data, but a full study is beyond the scopeof the present work. However these effects are not sufficientlylarge to make the Bianchi B -mode polarization unobservableat the three-year WMAP sensitivity (cf. Fig. 4), or changeour overall conclusions.
We have derived the radiative-transfer equation for theCMB, including polarization, in all nearly-FRW Bianchi uni-verses in the form of a hierarchy of multipole equationswhich can be easily integrated numerically. These can becoupled with the dynamical (i.e. Einstein) equations to com-pute maps of the CMB temperature anisotropies and polar-ization in any such model. B -mode polarization is generic,being produced in all Bianchi types except I. We appliedthese equations to the Bianchi VII h case, with parameterstuned to address the anomalous features observed in theCMB temperature on large scales (Jaffe et al. 2005, 2006).Our treatment includes a more physical treatment of thetilt velocity in CDM models with sub-dominant baryons.Whilst this does not make a qualitative difference to our re-sults, more detailed statistical studies could well be affectedby its ∼
20% corrections.Our temperature maps are similar to those derived fromearlier studies (Collins & Hawking 1973a; Barrow et al. 1985), although the amplitude is modified somewhat dueto the better treatment of the ionization history. Polariza-tion maps, with the generality presented here, do not appearto have been computed before. Note also that for these, adetailed treatment of recombination is required for accu-rate results. The power spectra of our type-VII h polariza-tion maps apparently put these models in contradiction ofthe large-scale polarization results from WMAP (Page et al.2007).During the drafting of this paper, an analysis of uni-verses equivalent to Bianchi I models, tuned to accountfor the low CMB quadrupole (Campanelli, Cea & Tedesco2006), was shown to give a similar level of polarization tothat computed here (Cea 2007). This is not surprising giventhat the anisotropies are tuned to address some of the sameproblems, and that the added complications induced by theVII h geometry do not substantially alter the amplitude ofthe effect (in the simplified dynamical model). In the type-Imodel, the temperature anisotropy and E -mode polariza-tion are simply quadrupoles, and no B -modes are produced(see equation 52). Although the type-I model does not sufferthe same observational constraints as type-VII h in polariza-tion, the latter has the virtues in temperature of resolvingessentially all of the large-angle anomalies.Our cursory glance over the available data appears torule out the VII h models employed in recent papers on thebasis that they over-produce B -mode power. This is espe-cially significant given that the results hold for all modelson the Bianchi degeneracy line given in Jaffe et al. (2006).Our polarization results, combined with the failure of theBianchi degeneracy region to include well-established val-ues for the cosmological parameters, suggest that the simpleVII h model, as it stands, is unsuitable to describe the avail-able data. However, to reject completely the hypothesis thatour universe contains anisotropic perturbations that are ho-mogeneous under groups of motions with Bianchi type VII h requires a fuller treatment of the dynamics of the linearizedmodel (Section 5.1). We intend to address this problem, andto search for statistical correlations between the morphol-ogy of the generalized model’s polarization and the WMAP data, in future work.
ACKNOWLEDGMENTS
AP is supported by a STFC (formerly PPARC) studentshipand scholarship at St John’s College, Cambridge. AC ac-knowledges a Royal Society University Research Fellowship.We thank Kendrick Smith, Antony Lewis and John Barrowfor helpful discussions.
REFERENCES
Barrow J. D., Juszkiewicz R., Sonoda D. H., 1985, MNRAS,213, 917Bianchi, I. 1897, Mem. Soc. Ital. Sci. Ser IIIa, 11, 267Bridges M., McEwen J. D., Lasenby A. N., Hobson M. P.,2007, MNRAS, 377, 1473Campanelli L., Cea P., Tedesco L., 2006, Phys. Rev. Lett.,97, 209903Cea P., 2007, preprint (astro-ph/0702293) c (cid:13) , 000–000 ianchi Model CMB Polarization Challinor A., 2000, Phys. Rev. D, 62, 043004Collins C. B., Hawking S. W., 1973a, MNRAS, 162, 307Collins C. B., Hawking S. W., 1973b, ApJ, 180, 317Copi C. J., Huterer D., Schwarz D. J., Starkman G. D.,2007, Phys. Rev. D, 75, 023507Dautcourt G., Rose K., 1978, Astronomische Nachrichten,299, 13Efstathiou G., Bond J. R., 1999, MNRAS, 304, 75Ellis G., MacCallum M., 1969, Comm. Math. Phys., 12,108Ellis G., van Elst H., 1998, Arxiv preprint gr-qc/9812046Estabrook F., Wahlquist H., Behr C., 1968, J. Math. Phys.,9, 497Fabbri R., Tamburrano M., 1987, A&A, 179, 11Fan X., Carilli C. L., Keating B., 2006, ARA&A, 44, 415Goliath M., Ellis G. F. R., 1999, Phys. Rev. D, 60, 023502Hawking S., 1969, MNRAS, 142, 129Heckmann O., Schucking E., 1962, Gravitation: An Intro-duction to Current Research. Wiley New York, p. 438Hu W., White M., 1997, Phys. Rev. D, 56, 596Jaffe T. R., Banday A. J., Eriksen H. K., G´orski K. M.,Hansen F. K., 2005, ApJ, 629, L1Jaffe T. R., Hervik S., Banday A. J., G´orski K. M., 2006,ApJ, 644, 701Lewis A., Challinor A., Lasenby A., 2000, ApJ, 538, 473Lewis A., Challinor A., Turok N., 2002, Phys. Rev. D, 65,023505Loeb A., Zaldarriaga M., 2005, Phys. Rev. D, 71, 103520MacCallum M. A. H., 1973, in Schatzman E., ed., CargeseLectures in Physics, Vol. 6, p. 61Matzner R. A., Tolman B. W., 1982, Phys. Rev. D, 26,2951Milaneschi E., Fabbri R., 1985, A&A, 151, 7Misner C. W., 1968, ApJ, 151, 431Page L. e. a., 2007, ApJS, 170, 335Rees M. J., 1968, ApJ, 153, L1Seager S., Sasselov D. D., Scott D., 1999, ApJ, 523, L1Spergel D. N. e. a., 2007, ApJS, 170, 377Taub A., 1951, The Annals of Mathematics, 53, 472Varshalovich D. A., Moskalev A. N., Khersonskii V. K.,1998, Quantum theory of angular momentum. World Sci-entific: Singapore, 1988Wainwright J., 1997, Dynamical Systems in Cosmology.Cambridge University PressWald R. M., 1983, Phys. Rev. D, 28, 2118 c (cid:13)000