Planck 2013 results. XXVI. Background geometry and topology of the Universe
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Warsaw University Observatory, Aleje Ujazdowskie 4, 00478
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Received:
22
March
2013
Accepted:
23
February
2014
The new cosmic microwave background (CMB) temperature maps from Planck provide the highestquality fullsky view of the surface of last scattering available to date. This allows us to detect possible departures from the standard model of a globally homogeneous and isotropic cosmology on the largest scales. We search for correlations induced by a possible nontrivial topology with a fundamental domain intersecting, or nearly intersecting, the last scattering surface (at comoving distance χ_{rec}), both via a direct search for matched circular patterns at the intersections and by an optimal likelihood search for specific topologies. For the latter we consider flat spaces with cubic toroidal (T3), equalsided chimney (T2) and slab (T1) topologies, three multiconnected spaces of constant positive curvature (dodecahedral, truncated cube and octahedral) and two compact negativecurvature spaces. These searches yield no detection of the compact topology with the scale below the diameter of the last scattering surface. For most compact topologies studied the likelihood maximized over the orientation of the space relative to the observed map shows some preference for multiconnected models just larger than the diameter of the last scattering surface. Since this effect is also present in simulated realizations of isotropic maps, we interpret it as the inevitable alignment of mild anisotropic correlations with chance features in a single sky realization; such a feature can also be present, in milder form, when the likelihood is marginalized over orientations. Thus marginalized, the limits on the radius ℛ_{i} of the largest sphere inscribed in topological domain (at loglikelihoodratio Δln ℒ > −5 relative to a simplyconnected flat Planck bestfit model) are: in a flat Universe, ℛ_{i}> 0.92χ_{rec} for the T3 cubic torus; ℛ_{i}> 0.71χ_{rec} for the T2 chimney; ℛ_{i}> 0.50χ_{rec} for the T1 slab; and in a positively curved Universe, ℛ_{i}> 1.03χ_{rec} for the dodecahedral space; ℛ_{i}> 1.0χ_{rec} for the truncated cube; and ℛ_{i}> 0.89χ_{rec} for the octahedral space. The limit for a wider class of topologies, i.e., those predicting matching pairs of backtoback circles, among them tori and the three spherical cases listed above, coming from the matchedcircles search, is ℛ_{i}> 0.94χ_{rec} at 99% confidence level. Similar limits apply to a wide, although not exhaustive, range of topologies. We also perform a Bayesian search for an anisotropic global Bianchi VII_{h} geometry. In the nonphysical setting where the Bianchi cosmology is decoupled from the standard cosmology, Planck data favour the inclusion of a Bianchi component with a Bayes factor of at least 1.5 units of logevidence. Indeed, the Bianchi pattern is quite efficient at accounting for some of the largescale anomalies found in Planck data. However, the cosmological parameters that generate this pattern are in strong disagreement with those found from CMB anisotropy data alone. In the physically motivated setting where the Bianchi parameters are coupled and fitted simultaneously with the standard cosmological parameters, we find no evidence for a Bianchi VII_{h} cosmology and constrain the vorticity of such models to (ω/H)_{0}< 8.1 × 10^{10} (95% confidence level).
Key words: cosmology: observations / cosmic background radiation / cosmological parameters / gravitation / methods: data analysis / methods: statistical
© ESO, 2014
1. Introduction
This paper, one of a set of papers associated with the 2013 release of data from the Planck^{1} mission (Planck Collaboration I 2014), describes the use of Planck data to limit departures from the global isotropy and homogeneity of spacetime. We will use Planck’s measurements of the cosmic microwave background (CMB) to assess the properties of anisotropic geometries (i.e., Bianchi models) and nontrivial topologies (e.g., the torus). The simplest models of spacetime are globally isotropic and simply connected. Although both are supported by both local observations and previous CMB observations, without a fundamental theory of the birth of the Universe, observational constraints on departures from global isotropy are necessary. General Relativity itself places no restrictions upon the topology of the Universe, as was recognised very early on (e.g., De Sitter 1917); most proposed theories of quantum gravity predict topologychange in the early Universe which could be visible at large scales today.
The Einstein field equations relate local properties of the curvature to the matter content in spacetime. By themselves they do not restrict the global properties of the space, allowing a universe with a given local geometry to have various global topologies. FriedmannRobertsonWalker (FRW) models of the universe observed to have the same average local properties everywhere still have freedom to describe quite different spaces at large scales. Perhaps the most remarkable possibility is that a vanishing or negative local curvature (Ω_{K} ≡ 1 − Ω_{tot} ≥ 0) does not necessarily mean that our Universe is infinite. Indeed we can still be living in a universe of finite volume due to the global topological multiconnectivity of space, even if described by the flat or hyperbolic FRW solutions. In particular, quantum fluctuations can produce compact spaces of constant curvature, both flat (e.g., Zeldovich & Starobinskii 1984) and curved (e.g., Coule & Martin 2000; Linde 2004), within the inflationary scenario.
The primary CMB anisotropy alone is incapable of constraining curvature due to the wellknown geometrical degeneracy which produces identical smallscale fluctuations when the recombination sound speed, initial fluctuations, and comoving distance to the last scattering surface are kept constant (e.g., Bond et al. 1997; Zaldarriaga & Seljak 1997; Stompor & Efstathiou 1999). The present results from Planck (Planck Collaboration XVI 2014) can therefore place restrictive constraints on the curvature of the Universe only when considering secondary anisotropies or nonCMB data: at 95%, considering CMB primary anisotropy and lensing from Planck (in the natural units with c = 1 we use throughout). This is equivalent to constraints on the radius of curvature R_{0}H_{0}> 19 for positive curvature (K = + 1) and R_{0}H_{0}> 33 for negative curvature (K = −1). CMB primary anisotropy alone gives limits on R_{0}H_{0} roughly a factor of two less restrictive (and strongly dependent on priors).
Thus, the global nature of the Universe we live in is still an open question and studying the observational effects of a possible finite universe is one way to address it. With topology not affecting local mean properties that are found to be well described by FRW parameters, its main observational effect is in setting boundary conditions on perturbation modes that can be excited and developed into the structure that we observe. Studying structure on the last scattering surface is the bestknown way to probe the global organisation of our Universe and the CMB provides the most detailed and best understood dataset for this purpose.
We can also relax assumptions about the global structure of spacetime by allowing anisotropy about each point in the Universe. This yields more general solutions to Einstein’s field equations, leading to the socalled Bianchi cosmologies. For small anisotropy, as demanded by current observations, linear perturbation about the standard FRW model may be applied. A universal shear and rotation induce a characteristic subdominant, deterministic signature in the CMB, which is embedded in the usual stochastic anisotropies. The deterministic CMB temperature fluctuations that result in the homogenous Bianchi models were first examined by Collins & Hawking (1973) and Barrow et al. (1985) (and subsequently Barrow 1986), however no dark energy component was included as it was not considered plausible at the time. More recently, Jaffe et al. (2006c), and independently Bridges et al. (2007), extended these solutions for the open and flat Bianchi VII_{h} models to include cosmologies with dark energy. It is these solutions to Bianchi VII_{h} models that we study in the current article. More accurate solutions were since derived by Pontzen & Challinor (2007), Pontzen (2009) and Pontzen & Challinor (2011), where recombination is treated in a more sophisticated manner and reionisation is supported. Furthermore, we note that in these works (Pontzen & Challinor 2007, 2011; Pontzen 2009) the induced CMB polarisation contributions that arise in Bianchi models have also been derived, although here focus is given to temperature contributions.
In this paper, we will explicitly consider models of global topology and anisotropy. In a chaotic inflation scenario, however, our postinflationary patch might exhibit largescale local topological features (“handles” and “holes”) the can mimic a global multiplyconnected topology in our observable volume. Similarly, it might also have residual shear or rotation which could mimic the properties of a global Bianchi spacetime.
Planck’s ability to discriminate and remove largescale astrophysical foregrounds (Planck Collaboration XII 2014) reduces the systematic error budget associated with measurements of the CMB sky significantly. Planck data therefore allow refined limits on the scale of the topology and the presence of anisotropy. Moreover, previous work in this field has been done by a wide variety of authors using a wide variety of data (e.g., COBE, WMAP 1year, 3year, 5year, etc.) and in this work we perform a coherent analysis.
In Sect. 2, we discuss previous attempts to limit the topology and global isotropy of the Universe. In Sect. 3 we discuss the signals induced in topologically nontrivial and Bianchi universes. In Sect. 4 the Planck data we use in the analysis are presented, and in Sect. 5 the methods we have developed to detect those signals are discussed. We apply those methods in Sect. 6 and discuss the results in Sect. 7.
2. Previous results
The first searches for nontrivial topology on cosmic scales looked for repeated patterns or individual objects in the distribution of galaxies (Sokolov & Shvartsman 1974; Fang & Sato 1983; Fagundes & Wichoski 1987; Lehoucq et al. 1996; Roukema 1996; Weatherley et al. 2003; Fujii & Yoshii 2011). The last scattering surface from which the CMB is released represents the most distant source of photons in the Universe, and hence the largest scales with which we could probe the topology of the Universe. This first became possible with the DMR instrument on the COBE satellite (Bennett et al. 1996): various searches found no evidence for nontrivial topologies (e.g., Starobinskij 1993; Sokolov 1993; Stevens et al. 1993; De OliveiraCosta & Smoot 1995; Levin et al. 1998; Bond et al. 1998, 2000b; Rocha et al. 2004; but see also Roukema 2000b,a), but sparked the creation of robust statistical tools, along with greater care in the enumeration of the possible topologies for a given geometry (see, for example, LachiezeRey & Luminet 1995 and Levin 2002 for reviews). With data from the WMAP satellite (Jarosik et al. 2011), these theoretical and observational tools were applied to a highquality dataset for the first time. Luminet et al. (2003) and Caillerie et al. (2007) claimed the low value of the low multipoles (compared to standard ΛCDM cosmology) as evidence for missing largescale power as predicted in a closed universe with a small fundamental domain (see also Aurich 1999; Aurich et al. 2004, 2005, 2006, 2008; Aurich & Lustig 2013; Lew & Roukema 2008; Roukema et al. 2008). However, searches in pixel space (Cornish et al. 2004; Key et al. 2007; Niarchou et al. 2004; Bielewicz & Riazuelo 2009; Dineen et al. 2005) and in harmonic space (Kunz et al. 2006) determined that this was an unlikely explanation for the low power. Bond et al. (1998, 2000a) and Riazuelo et al. (2004a,b) presented some of the mathematical formalism for the computation of the correlations induced by topology in a form suitable for use in cosmological calculations. Phillips & Kogut (2006) presented efficient algorithms for the computation of the correlation structure of the flat torus and applied it via a Bayesian formalism to the WMAP data; similar computations for a wider range of geometries were performed by Niarchou & Jaffe (2007).
These calculations used a variety of different vintages of the COBE and WMAP data, as well as a variety of different sky cuts (including the unmasked internal linear combination (ILC) map, not originally intended for cosmological studies). Nonetheless, none of the pixelspace calculations which took advantage of the full correlation structure induced by the topology found evidence for a multiplyconnected topology with a fundamental domain within or intersecting the last scattering surface. Hence in this paper we will attempt to corroborate this earlier work and put the calculations on a consistent footing.
The open and flat Bianchi type VII_{h} models have been compared previously to both the COBE (Bunn et al. 1996; Kogut et al. 1997) and WMAP (Jaffe et al. 2005, 2006b) data, albeit ignoring dark energy, in order to place limits on the global rotation and shear of the Universe. A statistically significant correlation between one of the Bianchi VII_{h} models and the WMAP ILC map (Bennett et al. 2003) was first detected by Jaffe et al. (2005). However, it was noted that the parameters of this model are inconsistent with standard constraints. Nevertheless, when the WMAP ILC map was “corrected” for the bestfit Bianchi template, some of the socalled “anomalies” reported in WMAP data disappear (Jaffe et al. 2005, 2006b; Cayón et al. 2006; McEwen et al. 2006). A modified template fitting technique was performed by Land & Magueijo (2006) and, although a statistically significant template fit was not reported, the corresponding “corrected” WMAP data were again free of many large scale “anomalies”. Subsequently, Ghosh et al. (2007) used the bipolar power spectrum of WMAP data to constrain the amplitude of any Bianchi component in the CMB. Due to the renewed interest in Bianchi models, solutions to the CMB temperature fluctuations induced in Bianchi VII_{h} models when incorporating dark energy were since derived by Jaffe et al. (2006c) and Bridges et al. (2007). Nevertheless, the cosmological parameters of the Bianchi template embedded in WMAP data in this setting remain inconsistent with constraints from the CMB alone (Jaffe et al. 2006a,c). Furthermore, Pontzen & Challinor (2007) compared the polarisation power spectra of the bestfit Bianchi VII_{h} model found by Jaffe et al. (2006a) with the WMAP 3year data (Page et al. 2007) and also concluded that the model could be ruled out since it produced greater polarization than observed in the WMAP data. A Bayesian analysis of Bianchi VII_{h} models was performed by Bridges et al. (2007) using WMAP ILC data to explore the joint cosmological and Bianchi parameter space via Markov chain Monte Carlo sampling, where it was again determined that the parameters of the resulting Bianchi cosmology were inconsistent with standard constraints. In a following study by Bridges et al. (2008) it was suggested that the CMB “cold spot” (Vielva et al. 2004; Cruz et al. 2006; Vielva 2010) could be driving evidence for a Bianchi component. Recently, this Bayesian analysis has been revisited by McEwen et al. (2013) to handle partialsky observations and to use nested sampling methods (Skilling 2004; Feroz & Hobson 2008; Feroz et al. 2009). McEwen et al. (2013) conclude that WMAP 9year temperature data do not favour Bianchi VII_{h} cosmologies over ΛCDM.
3. CMB correlations in anisotropic and multiplyconnected universes
3.1. Topology
All FRW models can describe multiconnected universes. In the case of flat space, there are a finite number of compactifications, the simplest of which are those of the torus. All of them have continuous parameters that describe the length of periodicity in some or all directions (e.g., Riazuelo et al. 2004b). In a space of constant nonzero curvature the situation is notably different – the presence of a length scale (the curvature radius R_{0}) precludes topological compactification at an arbitrary scale. The size of the space must now reflect its curvature, linking topological properties to Ω_{tot} = 1 − Ω_{K}. In the case of hyperbolic spacetimes, the list of possible compact spaces of constant negative curvature is still infinite, but discrete (Thurston 1982), while in the positive curvature spherical space there is only a finite set of wellproportioned possibilities (i.e., those with roughly comparable sizes in all directions; there are also the countably infinite lens and prism topologies) for a multiconnected space (e.g., Gausmann et al. 2001; Riazuelo et al. 2004a).
The effect of topology is equivalent to considering the full simplyconnected threedimensional spatial slice of the spacetime (known as the covering space) as being filled with repetitions of a shape which is finite in some or all directions (the fundamental domain) – by analogy with the twodimensional case, we say that the fundamental domain tiles the covering space. For the flat and hyperbolic geometries, there are infinite copies of the fundamental domain; for the spherical geometry, with a finite volume, there is a finite number of tiles. Physical fields repeat their configuration in every tile, and thus can be viewed as defined on the covering space but subject to periodic boundary conditions. Topological compactification always break isotropy, and for some topologies also the global homogeneity of physical fields. Positively curved and flat spaces studied in this paper are homogeneous, however hyperbolic multiconnected spaces are never homogeneous.
Parameters of analysed curved spaces.
The primary observable effect of a multiconnected universe is the existence of directions in which light could circumnavigate the space in cosmological time more than once, i.e., the radial distance χ_{rec} to the surface of last scattering exceeds the size of the universe. In these cases, the surface of last scattering can intersect the (notional) edge of a fundamental domain. At this intersection, we can view the same spacetime event from multiple directions – conversely, it appears in different directions when observed from a single point.
Thus, temperature perturbations in one direction, , become correlated with those in another direction by an amount that differs from the usual isotropic correlation function C(θ), where θ denotes the angle between and . Considering a pixelized map, this induces a correlation matrix C_{pp′} which depends on quantities other than the angular distance between pixels p and p′. This break from statistical isotropy can therefore be used to constrain topological models. Hence, we need to calculate the pixelspace correlation matrix or its equivalent in harmonic space.
In this paper we consider the following topologies using the likelihood method: a) toroidal flat models with equallength compactification size L in three directions, denoted T [ L,L,L ]^{2}; b) toroidal flat models with different compactification lengths, parametrized by L_{x},L_{y},L_{z}, denoted T [ L_{x},L_{y},L_{z} ]; c) three major types of singleaction positively curved spherical manifolds with dodecahedral, truncated cubical and octahedral fundamental domains (I^{∗}, O^{∗}, T^{∗} compactification groups correspondingly, see Gausmann et al. 2001); and d) two sample negative curvature hyperbolic spaces, m004(−5,1) being one of the smallest known compact hyperbolic spaces as well as the relatively large v3543(2,3)^{3}. Scales of fundamental domains of compactified curved spaces are fixed in the units of curvature and are summarised in Table 1, where we quote the volume , radius of the largest sphere that can be inscribed in the domain ℛ_{i} (equal to the distance to the nearest face from the origin of the domain), the smallest sphere in which the domain can be inscribed ℛ_{u} (equal to the distance to the farthest vertex), and the intermediate scale ℛ_{m} that is taken to be the distance to the edges for spherical spaces and the “spine” distance for hyperbolic topologies. For the cubic torus with edge length L, these lengths are ℛ_{i} = L/ 2, and . The ratio ℛ_{u}/ ℛ_{i} is a good indicator of the shape of the fundamental domain. Note that when χ_{rec} is less than ℛ_{i}, multiple images on large scales are not present, although the C_{pp′} correlation matrix is still modified versus the singlyconnected limit. The effects of topology usually become strong when χ_{rec} exceeds the intermediate ℛ_{m}; conversely, for flat and nearlyflat geometries, there are limits to the allowed topologies (Mota et al. 2011).
A much wider class of topologies is explicitly constrained using the matched circles method. As discussed in Sect. 5.1, because of computational limitations we restrict our analysis to pairs of circles centered around antipodal points, so called backtoback circles. Thus, we can constrain all topologies predicting pairs of such circles.The strongest constraints are imposed on topologies predicting backtoback circles in all directions i.e., all the single action manifolds, among them tori of any shape and the three spherical cases considered explicitly in the likelihood analysis. Weaker constraints are imposed on topologies with all backtoback circles centred on a great circle of the celestial sphere such as halfturn, quarterturn, thirdturn and sixthturn spaces, as well as Klein and chimney spaces. The statistic can also constrain the multiconnected spaces predicting one pair of antipodal matching circles such as Klein or chimney spaces with horizontal flip, vertical flip or halfturn and slab space translated without screw motion. Other topologies catalogued in Riazuelo et al. (2004b) are not constrained by this analysis: the HantzscheWendt space; the chimney space with halfturn and flip; the generic slab space; the slab space with flip; spherical manifolds with double and linked action; and all the hyperbolic topologies including those two cases considered using the likelihood method.
3.1.1. Computing correlation matrices
The CMB temperature pixelpixel correlation matrix is defined as the ensembleaverage product of the temperature at two different pixels: (1)It can be calculated as a double radial integral of the ensemble average of the product of the source functions that describe the transport of photons through the universe from the last scattering surface to the observer: (2)where and are unit vectors that point at pixels p and p′ on the sky, and χ and χ′ are proper distances along radial rays pointing towards the last scattering surface.
Two techniques have been developed to compute the CMB correlation function for multiplyconnected universes. In one approach, one constructs the orthonormal set of basis functions that satisfy the boundary conditions imposed by compactification (eigenfunctions of the Laplacian operator furnish such a basis), and assembles the spatial correlation function of the source from such a basis (Cornish & Spergel 1999; Lehoucq et al. 2002). In the other approach, one applies the method of images to create the compactified version of from the one computed on the universal covering space by resumming the latter over the images of the 3D spatial positions (Bond et al. 1998, 2000a,b): (3)where the superscripts c and u refer to the quantity in the multiplyconnected space and its universal cover, respectively. The tilde refers to the need for sum regularization in the models with an infinite set of images, e.g., hyperbolic and flat toroidal ones. Γ is the discrete subgroup of motions which defines the multiplyconnected space and γ [ x ] is the spatial point on the universal cover obtained by the action of the motion γ ∈ Γ on the point x. Note that we can consider the location of one of the pixels as fixed and consider the action of γ on the other due to symmetry. This equation defines the action of γ on the source function itself, needed unless all the terms in the source function are scalar quantities (which is the case if one limits consideration to SachsWolfe terms) when the action is trivial.
Both methods are general, but have practical considerations to take into account when one increases the pixel resolution. For computing C_{pp′} up to the resolution corresponding to harmonic mode ℓ ≈ 40 both methods have been tested and were found to work equally well. In this paper we employ both approaches.
The main effect of the compactification is that C_{pp′} is no longer a function of the angular separation between the pixels p and p′ only, due to the lack of global isotropy. In harmonic space the twopoint correlation function of the CMB is given by (4)where δ_{ℓℓ′} is the Kronecker delta symbol and a_{ℓm} are the spherical harmonic coefficients of the temperature on the sky when decomposed into the spherical harmonics by (5)Note that the twopoint correlation function is no longer diagonal, nor is it mindependent, as in an isotropic universe.
A flat universe provides an example when the eigenfunctions of the Laplacian are readily available in a set of plane waves. The topological compactification in the flat space discretizes the spectrum of the wavevector magnitudes k^{2} and selects the subset of allowed directions. For example, for a toroidal universe the length of the fundamental cell needs to be an integer multiple of the wavelength of the modes. We therefore recover a discrete sum over modes k_{n} = (2π/L)n for n = (n_{x},n_{y},n_{z}) a triplet of integers, instead of an integral over k, (6)where Δ_{ℓ}(k,Δη) is the radiation transfer function (e.g., Bond & Efstathiou 1987; Seljak & Zaldarriaga 1996). We refer to the cubic torus with three equal sides as the T3 topology; it is also possible for the fundamental domain to be compact in only two spatial dimensions (e.g., the socalled T2 “chimney” space) or one (the T1 “slab”, similar to the “lens” spaces available in manifolds with constant positive curvature) in which case the sum is replaced by an integral in those directions. These models serve as approximations to modifications to the local topology of the global manifold (albeit on cosmological scales): for example, the chimney space can mimic a “handle” connecting different regions of an approximately flat manifold.
In Fig. 1 we show rows of the pixelspace correlation matrix for a number of multiplyconnected topologies as maps, showing the magnitude of the correlation within a particular pixel. For the simplyconnected case, the map simply shows the same information as the correlation function C(θ); for the topologically nontrivial cases, we see the correlations depend on distance and direction and differ from pixel to pixel (i.e., from row to row of the matrix). In Fig. 2 we show example maps of CMB anisotropies in universes with these topologies, created by direct realisations of Gaussian fields with the correlation matrices of Fig. 1.
Fig. 1
Top row: correlation structure (i.e., a single row of the correlation matrix) of a simplyconnected universe with isotropic correlations. For subsequent rows, the left and middle column show positively curved multiplyconnected spaces (left: dedocahedral, middle: octahedral) and the right column shows equal sided tori. The upper row of three maps corresponds to the case when the size of the fundamental domain is of the size of the diameter to the last scattering surface and hence the first evidence for large angle excess correlation appears. Subsequent rows correspond to decreasing fundamental domain size with respect to the last scattering diameter, with parameters roughly chosen to maintain the same ratio between the models. 

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Fig. 2
Random realisations of temperature maps for the models in Fig. 1. The maps are smoothed with a Gaussian filter with fullwidthhalfmaximum FWHM = 640′. 

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3.2. Bianchi
Bianchi cosmologies include the class of homogeneous but anisotropic cosmologies, where the assumption of isotropy about each point in the Universe is relaxed. For small anisotropy, as demanded by current observations, linear perturbation theory about the standard FRW model may be applied, leading to a subdominant, deterministic contribution to the CMB fluctuations. In this setting CMB fluctuations may be viewed as the sum of a deterministic Bianchi contribution and the usual stochastic contribution that arises in the ΛCDM model. The deterministic CMB temperature fluctuations that result in the Bianchi models were derived by Barrow et al. (1985), although no dark energy component was included. More recently, Jaffe et al. (2006c), and independently Bridges et al. (2007), extended these solutions for the open and flat Bianchi VII_{h} models to include cosmologies with dark energy. We defer the details of the CMB temperature fluctuations induced in Bianchi models to these works and give only a brief description here.
Bianchi VII_{h} models describe a universe with overall rotation, parameterized by an angular velocity, ω, and a threedimensional rate of shear, parameterized by the tensor σ_{ij}; we take these to be relative to the z axis. The model has a free parameter, first identified by Collins & Hawking (1973), describing the comoving lengthscale over which the principal axes of shear and rotation change orientation. The ratio of this length scale to the present Hubble radius is typically denoted x, which defines the h parameter of type VII models through (Barrow et al. 1985) (7)where the total energy density Ω_{tot} = Ω_{m} + Ω_{Λ}. The parameter x acts to change the “tightness” of the spiraltype CMB temperature contributions that arise due to the geodesic focusing of Bianchi VII_{h} cosmologies. The shear modes σ_{ij} of combinations of orthogonal coordinate axes are also required to describe a Bianchi cosmology. The present dimensionless vorticity (ω/H)_{0} may be related to the dimensionless shear modes (σ_{ij}/H)_{0} by (Barrow et al. 1985) (8)where H is the Hubble parameter. The spherical harmonic coefficients of the Bianchi VII_{h} induced temperature component are proportional to [ (σ_{12} ± iσ_{13}) /H ] _{0} and are nonzero for azimuthal modes m = ∓ 1 only (Barrow et al. 1985; McEwen et al. 2006; Pontzen & Challinor 2007). Hence, varying the phase of σ_{12} + iσ_{13} corresponds to an azimuthal rotation, i.e., a change of coordinates, while the rotationally invariant part depends on , and we are thus free to choose equality of shear modes σ = σ_{12} = σ_{13} (Pontzen & Challinor 2007), which we do for consistency with previous studies (e.g. Jaffe et al. 2005). The amplitude of the deterministic CMB temperature fluctuations induced in Bianchi VII_{h} cosmologies may be characterised by either (σ/H)_{0} or (ω/H)_{0} since these parameters influence the amplitude of the induced temperature contribution only and not its morphology. The handedness of the coordinate system is also free in Bianchi VII_{h} models, hence both left and righthanded models arise. Since the Bianchiinduced temperature fluctuations are anisotropic on the sky the orientation of the resulting map may vary also, introducing three additional degreesoffreedom. The orientation of the map is described by the Euler angles^{4}(α,β,γ), where for (α,β,γ) = (0°,0°,0°) the swirl pattern typical of Bianchi templates is centred on the South pole.
Examples of simulated Bianchi VII_{h} CMB temperature maps are illustrated in Fig. 3 for a range of parameters. In the analysis performed herein the BIANCHI2^{5} (McEwen et al. 2013) code is used to simulate the temperature fluctuations induced in Bianchi VII_{h} models. Bianchi VII_{h} models induce only large scale temperature fluctuations in the CMB and consequently Bianchi maps have a particularly low bandlimit, both globally and azimuthally (i.e., in both ℓ and m in spherical harmonic space; indeed, as mentioned only those harmonic coefficients with m = ± 1 are nonzero).
Fig. 3
Simulated deterministic CMB temperature contributions in Bianchi VII_{h} cosmologies for varying x and Ω_{tot} (lefttoright Ω_{tot} ∈ { 0.10,0.50,0.95 }; toptobottom x ∈ { 0.1,0.3,0.7,1.5,6.0 }). In these maps the swirl pattern typical of Bianchiinduced temperature fluctuations is rotated from the South pole to the Galactic centre for illustrational purposes. 

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4. Data description
We use Planck maps that have been processed by the various componentseparation pipelines described in Planck Collaboration XII (2014). The methods produce largely consistent maps of the sky, with detailed differences in pixel intensity, noise properties, and masks. Here, we consider maps produced by the CommanderRuler, NILC, SMICA and SEVEM methods. Each provides its own mask and we also consider the conservative common mask.
We note that because our methods rely on rather intensive pixel or harmonicspace calculations, in particular considering a full set of threedimensional orientations and, for the likelihood methods, manipulation of an anisotropic correlation matrix, computational efficiency requires the use of data degraded from the native HEALPix (Górski et al. 2005) N_{side} = 2048 resolution of the Planck maps. Because the signatures of either a multiplyconnected topology or a Bianchi model are most prominent on large angular scales, this does not result in a significant loss of ability to detect and discriminate amongst the models (see Sect. 5.3). However, it is worth pointing out that the likelihood and matched circles methods are sensitive to different angular scales as applied to Planck data here. The likelihood method explicitly retains only lowℓ (largescale) information in its correlation matrix, whereas the matchedcircles method considers anisotropies at angular scales down to tens of arcminutes (still large in comparison to the native resolution of the Planck maps). Of course, the matchedcircles method exploits the correlation of the smallscale patterns along matched circles potentially separated by large angles; this effect is not generated by intrinsically large angular scale anisotropies but by the boundary conditions of the fundamental domain imposed by the multiply connected topology. As described in Sect. 5.1, the matched circles statistic used here damps the anisotropies at the largest angular scales relative to those at smaller scales so sensitivity of the method does not rely on the former.
The topology analyses both rely on degraded maps and masks. The matchedcircles method smooths with a 30′ Gaussian filter and degrades the maps to N_{side} = 512, and uses a mask derived from the SEVEM component separation method. Because the performance of the matchedcircles statistic depends on anisotropies on smaller angular scales, it can be significantly degraded by the point source cut. As there are more point sources detected in the Planck maps than in the WMAP maps, the problem of point source masking is more severe in the present case. We mask only those point sources from the fullresolution f_{sky} = 0.73 SEVEM mask with amplitude, after smoothing and extrapolation to the 143 or 217 GHz channels, greater than the faintest source originally detected at those frequencies. The mask derived in this way retains f_{sky} = 0.76 of the sky.
The likelihood method smooths the maps and masks with an 11° Gaussian filter and then degrades them to N_{side} = 16 and conservatively masks out any pixel with more than 10% of its original subpixels masked. At full resolution, the common mask retains a fraction f_{sky} = 0.73 of the sky, and f_{sky} = 0.78 when degraded to N_{side} = 16 (the highresolution pointsource masks are largely filled in the degraded masks). The Bianchi analysis is performed in harmonic space, and so does not require explicit degradation in pixel space. Rather, a noisy mask is added in pixel space to effectively marginalise the pixel values in the masked region (as described in more detail below; see also McEwen et al. 2013), before the data are transformed at full resolution into harmonic space and considered only up to a specified maximum harmonic ℓ, where correlations due to the mask are taken into account.
Different combinations of these maps and masks are used to discriminate between the topological and anisotropic models described in Sect. 3.
5. Methods
5.1. Topology: circles in the sky
The first set of methods, exemplified by the circlesinthesky of Cornish et al. (1998), involves a frequentist analysis using a statistic which is expected to differ between the models examined. For the circles, this uses the fact that the intersection of the topological fundamental domain with the surface of last scattering is a circle, which one potentially views from two different directions in a multiplyconnected universe. Of course, the matches are not exact due to noise, foregrounds, the integrated SachsWolfe (ISW) and Doppler effects along the different lines of sight.
By creating a statistic based on the matching of different such circles, we can compare Monte Carlo simulations of both a simplyconnected, isotropic null model with specific anisotropic or topological models. We may then calibrate detections and nondetections using Monte Carlo simulations. In principle, these simulations should take into account the complications of noise, foreground contributions, systematics, the ISW and Doppler effects. However, they do not include gravitational lensing of the CMB as the lensing deflection angle is small compared to the minimal angular scale taken into account in our analysis. Note that the null test is generic (i.e., not tied to a specific topology) but any detection must be calibrated with specific simulations for a chosen topology or anisotropic model. A very similar technique can be used for polarisation by taking into account the fact that the polarisation pattern itself is now not directly repeated, but rather that the underlying quadrupole radiation field around each point on the sky is now seen from different directions (Bielewicz et al. 2012). These methods have been applied successfully to COBE DMR and WMAP data, and have recently been shown to be feasible for application to Planck data (Bielewicz et al. 2012).
The idea of using the matched circles to study topology is due to Cornish et al. (1998). In that work, a statistical tool was developed to detect correlated circles in all sky maps of the CMB anisotropy – the circle comparison statistic. In our studies we will use version of this statistic optimised for the smallscale anisotropies as defined by Cornish et al. (2004): (9)where ΔT_{i,m} and ΔT_{j,m} denote the Fourier coefficients of the temperature fluctuations around two circles of angular radius α centered at different points on the sky, i and j, respectively, with relative phase φ_{∗}. The mth harmonic of the temperature anisotropies around the circle is weighted by the factor  m , taking into account the number of degrees of freedom per mode. Such weighting enhances the contribution of smallscale structure relative to largescale fluctuations and is especially important since the largescale fluctuations are dominated by the ISW effect. This can obscure the image of the last scattering surface and reduce the ability to recognise possible matched patterns on it.
The above S^{+} statistic corresponds to pair of circles with the points ordered in a clockwise direction (phased). For alternative ordering, when along one of the circles the points are ordered in an anticlockwise direction (antiphased), the Fourier coefficients ΔT_{i,m} are complex conjugated, defining the S^{−} statistic. This allows the detection of both orientable and nonorientable topologies. For orientable topologies the matched circles have antiphased correlations while for nonorientable topologies they have a mixture of antiphased and phased correlations.
The statistic has a range over the interval [ −1,1 ]. Circles that are perfectly matched have S = 1, while uncorrelated circles will have a mean value of S = 0. Although the statistic can also take negative values for the temperature anisotropy generated by the Doppler term (Bielewicz et al. 2012), anticorrelated circles are not expected for the total temperature anisotropy considered in this work. To find matched circles for each radius α, the maximum value is determined.
Because general searches for matched circles are computationally very intensive, we restrict our analysis to a search for pairs of circles centered around antipodal points, so called backtoback circles. As described above, the maps were also downgraded to N_{side} = 512, which greatly speeds up the computations required, but with no significant loss of discriminatory power, as seen in Sect. 5.3.1. More details on the numerical implementation of the algorithm can be found in the paper by Bielewicz & Banday (2011).
As mentioned in Sect. 3.1, the constraints we will derive concern topologies that predict matching pairs of backtoback circles. However, the constraints do not apply to those universes for which the orientation of the matched circles is impossible to detect due to partial masking on the sky. Because of the larger sky fraction removed by the Planck common mask than for WMAP this probability is larger for the analysis of the Planck maps. Moreover, the smaller fraction of the sky used in the search of matched circles results in a false detection level larger with our f_{sky} = 0.76 mask than for the f_{sky} = 0.78 7year KQ85 WMAP mask. As a result we obtain weaker – but more conservative – constraints on topology than for similar analyses of WMAP data (Bielewicz & Banday 2011).
To draw any conclusions from an analysis based on the statistic , it is very important to correctly estimate the threshold for a statistically significant match of circle pairs. We used 300 Monte Carlo simulations of CMB maps, described in detail in Sect. 5.3.1, to establish the threshold such that fewer than 1% of simulations would yield a false event.
5.2. Bayesian analyses
The second set of methods take advantage of the fact that the underlying smallscale physics is unchanged in both anisotropic and topological models compared to the standard cosmology, and thus a Gaussian likelihood function will still describe the statistics of temperature and polarization on the sky, albeit no longer with isotropic correlations. When considering specific topologies, these likelihood methods instead calculate the pixelpixel correlation matrix. This has been done for various torus topologies (which are a continuous family of possibilities) in the flat Universe as well as for locally hyperbolic and spherical geometries (which have a discrete set of possibilities for a given value of the curvature). More general likelihoodbased techniques have been developed for generic mild anisotropies in the initial power spectrum (Hanson & Lewis 2009), which may have extension to other models. For the Bianchi setting, an isotropic zeromean Gaussian likelihood is recovered by subtracting a deterministic Bianchi component from the data, where the cosmological covariance matrix remains diagonal in harmonic space but masking introduces nondiagonal structure that must be taken into account.
Because these methods use the likelihood function directly, they can take advantage of any detailed noise correlation information that is available, including any correlations induced by the foregroundremoval process. We denote the data by the vector d, which may be in the form of harmonic coefficients d_{ℓm} or pixel temperatures d_{p} or, in general, coefficients of the temperature expansion in any set of basis functions. We denote the model under examination by the discrete parameter M, which can take on the appropriate value denoting the usual isotropic case, or the Bianchi case, or one of the possible multiplyconnected universes. The continuous parameters of model M are given by the vector Θ, which for this case we can partition into Θ_{C} for the cosmological parameters shared with the usual isotropic and simplyconnected case, and Θ_{A} which denotes the parameters for the appropriate anisotropic case, be it a topologically nontrivial universe or a Bianchi model. Note that all of the anisotropic cases contain “nuisance parameters” which give the orientation of either the fundamental domain or the Bianchi template which we can marginalize over as appropriate.
Given this notation, the posterior distribution for the parameters of a particular model, M, is given by Bayes’ theorem: (10)Here, P(Θ  M) = P(Θ_{C},Θ_{A}  M) is the joint prior probability of the standard cosmological parameters Θ_{C} and those describing the anisotropic universe Θ_{A}, P(d  Θ,M) ≡ ℒ is the likelihood, and the normalizing constant P(d  M) is the Bayesian evidence, which can be used to compare the models to one another.
We will usually take the priors to be simple “noninformative” distributions (e.g., uniform over the sphere for orientations, uniform in length for topology scales, etc.) as appropriate. The form of the likelihood function will depend on the anisotropic model: for multiplyconnected models, the topology induces anisotropic correlations, whereas for the Bianchi model, there is a deterministic template, which depends on the Bianchi parameters, in addition to the standard isotropic cosmological perturbations. We will assume that any other nonGaussian signal (either from noise or cosmology) is negligible (Planck Collaboration XXIII 2014; Planck Collaboration XXIV 2014) and use an appropriate multivariate Gaussian likelihood.
Given the signal and noise correlations, and a possible Bianchi template, the procedure is similar to that used in standard cosmologicalparameter estimation, with a few complications. Firstly, the evaluation of the likelihood function is computationally expensive and usually limited to large angular scales. This means that in practice the effect of the topology on the likelihood is usually only calculated on those large scales. Secondly, the orientation of the fundamental domain or Bianchi template requires searching (or marginalizing) over three additional parameters, the Euler angles.
5.2.1. Topology
In topological studies, the parameters of the model consist of Θ_{C}, the set of cosmological parameters for the fiducial bestfit flat cosmological model, and Θ_{T}, the topological parameters which include the set of compactification lengths L_{x},L_{y},L_{z} for flat toroidal model or the curvature parameter Ω_{K} for curved spaces, and a choice of compactification T. In our studies we keep Θ_{C} fixed, and vary Θ_{T} for a select choice of compactifications listed in Sect. 3.1. These parameters define the predicted twopoint signal correlation matrix C_{pp′} for each model, which are precomputed. Additional internal parameters, including the amplitude of the signal A and the angles of orientation of the fundamental domain of the compact space relative to the sky ϕ (e.g., parameterized by a vector of the three Euler angles), are maximized and/or marginalized over during likelihood evaluation.
The likelihood, i.e., the probability to find a temperature data map d with associated noise matrix N given a certain topological model is then given by (11)Working with a cutsky, it is often easier to start the analysis with data and a correlation matrix given in pixel space. However, especially in the realistic case of negligible noise on large scales, the matrix C + N is poorly conditioned in pixel space, and pixel space evaluation of the likelihood is, as a rule, not robust. Indeed, there are typically more pixels than independent modes that carry information about the signal (e.g., even in the standard isotropic case, subarcminute pixels would not be useful due to beamsmoothing; with anisotropic correlations and masked regions of the sky, more complicated linear combinations of pixels even on large scales may have very little signal content). Therefore in general we expand the temperature map d_{p}, the theoretical correlation matrix C_{pp′} and the noise covariance matrix N_{pp′} in a discrete set of mode functions ψ_{n}(p), orthonormal over the pixelized sphere, possibly with weights w(p), , obtaining the coefficients of expansion (12)Next we select N_{m} such modes for comparison and consider the likelihood marginalized over the remainder of the modes (13)where C and N are restricted to the N_{m} × N_{m} block of chosen modes. Flexibility in choosing mode functions and their number N_{m} is used to achieve the compromise between the robust invertibility of the projected C + N matrix on the one hand, and the amount of discriminating information retained in the data on the other. The weights w(p) can be used to improve the accuracy of transforms on a pixelized sky.
For fullsky analysis the natural choice of the mode functions is the set of ordinary spherical harmonics Y_{lm}(p) which leads to standard harmonic analysis with modes limited to a suitably chosen ℓ_{max}. Here, where we focus on masked data, we have made a somewhat different choice. As a mode set for comparison we use the N_{m} = 837 largest eigenvectors of the C_{pp′} matrix, restricted to the masked sky, for the fiducial flat isotropic model with bestfit parameters Θ_{C}. We emphasize that the correlation matrix computed for this reduced dataset has fewer modes, but contains no additional assumptions beyond those of the original C_{pp′}.
Since computation of C_{pp′} matrices for a range of topological models is expensive, we do not aim to determine the full Bayesian evidence P(d  T) which would require marginalization over all parameters Θ_{C}, Θ_{T}, an overall amplitude of the correlation matrix A (proportional to the physical amplitude σ_{8} or the scalar amplitude A_{s}), and orientation (Euler angles) ϕ, and would in addition be sensitive to the prior probabilities assumed for the size of the fundamental domain. Instead we directly compare the likelihood along the changing set of Θ_{T} that has as its limit the flat fiducial model defined by Θ_{C}. In case of toroidal topology such a limit is achieved by taking compactification lengths to infinity, while for curved models we vary Ω_{K} in comparison to the flat limit Ω_{K} = 0. In the latter case, for the spherical spaces we change Ω_{Λ} and H_{0} together with Ω_{K} to track the CMB geometrical degeneracy line in which the recombination sound speed, initial fluctuations, and comoving distance to the last scattering surface are kept constant (e.g., Bond et al. 1997; Zaldarriaga & Seljak 1997; Stompor & Efstathiou 1999), and for hyperbolic spaces we vary Ω_{K} while keeping H_{0} and Ω_{Λ} − Ω_{m} fixed to fiducial values. Note that hyperbolic multiconnected spaces, in contrast to tori and the singleaction positive curvature manifolds considered in this paper, are not only anisotropic but also inhomogeneous. Therefore, the likelihood is expected to be dependent on the position of the observer. We do not study this dependence here.
For each parameter choice, we find the likelihood at the best orientation ϕ of the topology with respect to the sky after marginalizing over the amplitude A of the signal (hence, this can be considered a profile likelihood with respect to the orientation parameters). This likelihood is compared both with the fiducial model applied to the observed temperature map and with the likelihood of the topological model applied to the simulated realization of the isotropic map drawn from the fiducial model. Such a strategy is optimized for the detection of topological signatures. For nondetections, the marginalized likelihood can be a better probe of the overall power of the data to reject a nontrivial topology, and so for real data below, we also show the likelihood marginalized over the orientations ϕ. We estimate the marginalized likelihood from the random sample of 10 000 orientations, drawn statistically uniformly on the S^{3} sphere of unit quaternions representing rotations of the fundamental domain relative to the observed sky.
5.2.2. Bianchi
For the Bianchi analysis the posterior distribution of the parameters of model M is given by Bayes’ Theorem, specified in Eq. (10), similar to the topological setting. The approach of McEwen et al. (2013) is followed, where the likelihood is made explicit in the context of fitting a deterministic Bianchi template embedded in a stochastic CMB background, defined by the power spectrum C_{ℓ}(Θ_{C}) for a given cosmological model with parameters Θ_{C}. The Bianchi VII_{h} parameters are denoted Θ_{B}. The corresponding likelihood is given by (14)where (15)and d = { d_{ℓm} } and b(Θ_{B}) = { b_{ℓm}(Θ_{B}) } are the spherical harmonic coefficients of the data and Bianchi template, respectively, considered up to the harmonic bandlimit ℓ_{max}. A bandlimit of ℓ_{max} = 32 is considered in the subsequent analysis for computational tractibility and since this is sufficient to capture the structure of the CMB temperature fluctuations induced in Bianchi VII_{h} models in the vacinity of the bestfit model found in WMAP data (see, e.g., McEwen et al. 2006). The likelihood is computed in harmonic space where rotations of the Bianchi template can be performed efficiently (McEwen et al. 2006).
The covariance matrix X(Θ_{C}) depends on whether the fullsky or partialsky masked setting is considered. In the fullsky setting X(Θ_{C}) = C(Θ_{C}) as first considered by Bridges et al. (2007), where C(Θ_{C}) is the diagonal CMB covariance matrix with entries C_{ℓ}(Θ_{C}) on the diagonal. In the case of a zero Bianchi component, Eq. (14) then reduces to the likelihood function used commonly to compute parameter estimates from the power spectrum estimated from CMB data (e.g., Verde et al. 2003). In the masked setting considered subsequently, the situation is a little more involved.
In order to handle a mask in the harmonic space analysis of Bianchi models we follow the approach of McEwen et al. (2013), where masking noise is added to the data to effectively marginalise over the pixel values of the data in the masked region. The masking noise m is chosen to be zeromean and large in the masked region of the data, and zero elsewhere. Consequently, the masking noise is anisotropic over the sky but may be chosen to be uncorrelated, and may thus be defined by its covariance (16)where δ_{ij} is Kronecker delta symbol, ω_{i} denotes the angular coordinate of pixel i, and the variance of the noise for pixel i is given by a constant value in the masked regions and zero elsewhere. By synthetically adding masking noise that is much larger than the original data in the masked region of the sky, we effectively marginalise over the pixel values of the data in this region. The noisy mask introduces coupling in harmonic space that must be accounted for in the analysis. The covariance matrix of the resultant data is given by X(Θ_{C}) = C(Θ_{C}) + M, where M is the nondiagonal mask covariance matrix: (17)and Ω_{i} is the area of pixel i (see McEwen et al. 2013 for further details).
The χ^{2} of the likelihood for the Bianchi case hence differs from the topology case by the nonzero Bianchi template b and the use of a correlation matrix M to account for the presence of the mask.
In the most physically motivated scenario, the Bianchi and cosmological parameters are coupled (e.g., the total density of the Bianchi and standard cosmological model are identical). However, it is also interesting to consider Bianchi templates as phenomenological models with parameters decoupled from the standard cosmological parameters, particularly for comparison with previous studies. Both scenarios are considered in the subsequent analysis. In the decoupled scenario a flat cosmological model is considered, whereas in the coupled scenario an open cosmological model is considered to be consistent with the Bianchi VII_{h} model; we label these models the flatdecoupledBianchi model and the opencoupledBianchi model, respectively.
To determine whether the inclusion of a Bianchi component better describes the data the Bayesian evidence is examined, as given by (18)Using the Bayesian evidence to distinguish between models naturally incorporates Occam’s razor, trading off model simplicity and accuracy. In the absence of any prior information on the preferred model, the Bayes factor given by the ratio of Bayesian evidences (i.e., E_{1}/E_{2}) is identical to the ratio of the model probabilities given the data. The Bayes factor is thus used to distinguish models. The Jeffreys scale (Jeffreys 1961) is often used as a ruleofthumb when comparing models via their Bayes factor. The logBayes factor Δln E = ln(E_{1}/E_{2}) (also called the logevidence difference) represents the degree by which the model corresponding to E_{1} is favoured over the model corresponding to E_{2}, where: 0 ≤ Δln E< 1 is regarded as inconclusive; 1 ≤ Δln E< 2.5 as significant; 2.5 ≤ Δln E< 5 as strong; and Δln E ≥ 5 as conclusive (without loss of generality we have assumed E_{1} ≥ E_{2}). For reference, a logBayes factor of 2.5 corresponds to odds of 1 in 12, approximately, while a factor of 5 corresponds to odds of 1 in 150, approximately.
The ANICOSMO^{6} code (McEwen et al. 2013) is used to perform a Bayesian analysis of Bianchi VII_{h} models, which in turn uses the public MultiNest^{7} code (Feroz & Hobson 2008; Feroz et al. 2009) to sample the posterior distribution and compute evidence values by nested sampling (Skilling 2004). We sample the parameters describing the Bianchi VII_{h} model and those describing the standard cosmology simultaneously.
5.3. Simulations and Validation
5.3.1. Topology
CirclesintheSky
Before beginning the search for pairs of matched circles in the Planck data, we validate our algorithm using simulations of the CMB sky for a universe with 3torus topology for which the dimension of the cubic fundamental domain is , and with cosmological parameters corresponding to the ΛCDM model (see Komatsu et al. 2011, Table 1) determined from the 7year WMAP results combined with the measurements of the distance from the baryon acoustic oscillations and the Hubble constant. We performed simulations computing directly the a_{ℓm} coefficients up to the multipole of order ℓ_{max} = 500 as described in Bielewicz & Banday (2011) and convolving them with the same smoothing beam profile as used for the data, i.e., a Gaussian beam with 30′ FWHM. In particular, we verified that our code is able to find all pairs of matched circles in such a map. The map with marked pairs of matched circles with radius α ≃ 24° and the statistic for the map are shown in Figs. 4 and 5, respectively. Note that the peak amplitudes in the statistic, corresponding to the temperature correlation for matched circles, decrease with radius of the circles. Cornish et al. (2004) noted that this is primarily caused by the Doppler term, which becomes increasingly anticorrelated for circles with radius smaller than 45°.
Fig. 4
A simulated map of the CMB sky in a universe with a T [ 2,2,2 ] toroidal topology. The dark circles show the locations of the same slice through the last scattering surface seen on opposite sides of the sky. They correspond to matched circles with radius α ≃ 24°. 

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Fig. 5
An example of the statistic as a function of circle radius α for a simulated CMB map (shown in Fig. 4) of a universe with the topology of a cubic 3torus with dimensions (solid line). The dashdotted line show the false detection level established such that fewer than 1% out of 300 Monte Carlo simulations of the CMB map, smoothed and masked in the same way as the data, would yield a false event. 

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The intersection of the peaks in the matching statistic with the false detection level estimated for the CMB map corresponding to the simplyconnected universe defines the minimum radius of the correlated circles which can be detected for this map. The height of the peak with the smallest radius seen in Fig. 5 indicates that the minimum radius is about α_{min} ≈ 20°.
For the Monte Carlo simulations of the CMB maps for the simplyconnected universe we used the same cosmological parameters as for the multiconnected universe, i.e., corresponding to the ΛCDM model determined from the 7year WMAP results. The maps were also convolved with the same beam profile as for the simulated map for the 3torus universe and data, as well as masked with the same cut used for the analysis of data. The false detection threshold was established such that fewer than 1% of 300 Monte Carlo simulations would yield a false event.
Bayesian Analysis
Because of the expense of the calculation of the correlation matrix, we wish to limit the number of threedimensional wavevectors k we consider, as well as the number of spherical harmonic modes ℓ, and finally the number of different correlation matrices as a whole. We need to ensure that the full set of matrices that we calculate contains all of the available information on the correlations induced by the topology in a sufficiently finegrained grid. For this purpose, we consider the KullbackLeibler (KL) divergence as a diagnostic (see, e.g., Kunz et al. 2006, 2008, for applications of the KL divergence to topology). The KL divergence between two probability distributions p_{1}(x) and p_{2}(x) is given by (19)If the two distributions are Gaussian with correlation matrices C_{1} and C_{2}, this expression simplifies to (20)and is thus a measure of the discrepancy between the correlation matrices. The KL divergence can be interpreted as the ensemble average of the loglikelihoodratio Δln ℒ between realizations of the two distributions. Hence, they enable us to probe the ability to tell if, on average, we can distinguish realizations of p_{1} from a fixed p_{2} without having to perform a bruteforce Monte Carlo integration. Thus, the KL divergence is related to ensemble averages of the likelihoodratio plots that we present for simulations (Fig. 10) and real data (Sect. 6), but does not depend on simulated or real data.
We first use the KL divergence to determine the size of the fundamental domain which we can consider to be equivalent to the simplyconnected case (i.e., the limit in which all dimensions of the fundamental domain go to infinity). We note that in our standard ΛCDM model, the distance to the surface of last scattering is χ_{rec} ≈ 3.1416(H_{0})^{1}. We would naively expect that as long as the sphere enclosing the last scattering surface can be enclosed by the fundamental domain (L = 2χ_{rec}), we would no longer see the effects of nontrivial topology. However, because the correlation matrix includes the full threedimensional correlation information (not merely the purely geometrical effects of completely correlated points) we would see some longscale correlation effects even for larger fundamental domains. In Fig. 6 we show the KL divergence (as a function of (LH_{0})^{1} so that the simplyconnected limit L → ∞ is at a finite position) for the T [ L,L,L ] (cubic), T [ L,L,7 ] (chimney) and T [ L,7,7 ] (slab) spaces and show that it begins to level off for (LH_{0})^{1}^{<~}1 / 5, although these topologies are still distinguishable from the T [ 7,7,7 ] torus which is yet closer to the value for a simplyconnected universe d_{KL} [ 7,7,7 ] ≃ 1.1. These figures indicate that a length of is an acceptable proxy for the simplyconnected infinite Universe. The figures, as well as the likelihoods computed on simulations and data, show steps and other structures on a variety of scales corresponding to the crossing of the different length scales of the fundamental domain ℛ_{u}, ℛ_{m}, and ℛ_{i} crossing the last scattering surface; smaller fundamental domains with longer intersections with the last scattering surface are easier to detect.
Fig. 6
KL divergence computed for torus models as a function of the (inverse) length of a side of the cube. T [ L_{1},L_{2},L_{3} ] refers to a torus with edge lengths L_{i}. 

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Fig. 7
KL divergence between a supposed correct model and other models. We show differences of cubic tori with respect to models with (LH_{0})^{1} = 1 / 4.5 ≃ 0.22 (aligned with our grid of models), (LH_{0})^{1} = 1 / 5.25 ≃ 0.19 (in between the gridpoints) and and a T [ 5,5,7 ] chimney model with (LH_{0})^{1} = 1 / 5 in two directions and (LH_{0})^{1} = 1 / 7 ≃ 0.14 in the third. 

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Computational limitations further prevent us from calculating the likelihood at arbitrary values of the fundamental domain size parameters. We must therefore ensure that our coarsegrained correlation matrices are sufficient to detect a topology even if it lies between our gridpoints. In Fig. 7 we show the KL divergence as a function of the size of the fundamental domain, relative to various models, both aligned with our grid (LH_{0} = 4.5) and in between our grid points (LH_{0} = 5.25). We see that the peak is wide enough that we can detect a peak within δLH_{0} ~ 0.1 of the correct value. We also show that we can detect anisotropic fundamental domains even when scanning through cubic tori: we show a case which approximates a “chimney” universe with one direction much larger than the distance to the last scattering surface.
Because our topological analyses do not simultaneously vary the background cosmological parameters along with those describing the topology, we also probe the sensitivity to the cosmology. In Fig. 8 we show the effect of varying the fiducial cosmology from the Planck Collaboration XVI (2014) bestfit values to those reported by WMAP (Komatsu et al. 2011)^{8}. We see that this induces a small bias of δLH_{0} ≃ 0.2 but does not hinder the ability to detect a nontrivial topology. This indicates that small deviations from the correct background cosmology do not hinder our ability to detect (or rule out) topological signals.
We have also directly validated the topological Bayesian techniques with simulations. In Fig. 9 we show the loglikelihood for the above T [ 2,2,2 ] simulations as a function of two of the Euler angles, maximized over the third. We find a strong peak at the correct orientation, with a multiplicity due to the degenerate orientations corresponding to the faces of the cube (there are peaks at the North and South poles, which are difficult to see in this projection). Note that the peaks correspond to ratios of more than exp(700) compared to the relatively smooth minima elsewhere.
In Fig. 10 we also test the ability of the Bayesian likelihood technique to detect the compactification of the space in the simulated temperature realizations drawn from the dodecahedral closed model. For curved geometries, the size of the fundamental domain is fixed with respect to the varying curvature scale (R_{0}), whereas the distance to the last scattering χ_{rec} is constant. Hence we plot the likelihood as a function of χ_{rec}/R_{0}, inversely proportional to the scale of the fundamental domain.
Two mulitplyconnected realizations of the sky were tested: one corresponding to the space in which the last scattering sphere can be just inscribed into the fundamental domain, χ_{rec} = ℛ_{i}, when just the first large angle correlations appear, and the second drawn from a somewhat smaller space for which χ_{rec} = R_{e}. We see detections in both cases, stronger as the fundamental domain shrinks relative to χ_{rec}. We also calculate the likelihood for a model known to be simplyconnected.
Fig. 8
KL divergence between a model generated with the WMAP bestfit cosmological parameters as a background cosmology and a T [ 5,5,5 ] cubic torus topology with respect to a Planck bestfit cosmology and a varying cubic topology. 

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Fig. 9
Loglikelihood with respect to the peak as a function of the orientation of the fundamental T [ 2,2,2 ] torus domain for the simulations. The third Euler angle is marginalized over. We see peaks at the orientations corresponding to the six faces of the cubic fundamental domain (there are peaks at the North and South poles, which are difficult to see in this projection). 

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Fig. 10
Test for likelihood detectability of compactified space for the example of a dodecahedral (I^{∗}) closed universe. The vertical axis shows the loglikelihood relative to the largest model considered. Values are given for the orientations of the models which maximize the likelihood (top) and marginalized over the orientations (bottom). Different size models are tested against two HEALPix N_{side} = 16 temperature realizations drawn from the model with χ_{rec}/R_{0} = 0.314 = ℛ_{i} (blue) and χ_{rec}/R_{0} = 0.361 (black). No noise is added and the common mask has been applied. Dots mark the positions of the models for which the likelihoods were computed. The vertical lines show characteristic scales of the fundamental domain of the models in the units of curvature, from smaller to larger, ℛ_{i}/R_{0}, ℛ_{m}/R_{0} and ℛ_{u}/R_{0}. The variable χ_{rec}/R_{0} gives the size of the last scattering surface in the same units. The R_{0} → ∞ limit corresponds to the flat simplyconnected space. Both maximized and marginalized likelihoods show a detection relative to the isotropic sky realization drawn from the fiducial flat infinite universe (red) with the detection stronger for smaller spaces. However only the maximized likelihood unambigously distinguishes the correct compact model from spaces that exceed the lastscattering diameter, which shows that the likelihood for small models is narrowly peaked at the correct orientation and suppressed otherwise. 

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Note that the likelihood taken at the best orientations of the compact models generically shows a slight increase relative to that for the limiting simply connected space as one brings the size of the fundamental domain down to the size of the last scattering surface (χ_{rec} ≈ ℛ_{i}), followed, in the absence of signal in the map, by a rapid drop as soon as the models smaller than χ_{rec} are applied. This small increase is also present in the fiducial exactly isotropic sky, a single realization of which is shown in the figure, but is a generic feature irrespective of the topology being tested (occurring also in models with R< ℛ_{i}), and thus should not be taken as an indication for compact topology. The reason for the increase is the possibility of aligning the model with a weak anisotropic correlation feature with chance patterns of a single sky realization. However the fit drastically worsens as soon as the correlation features in a model become pronounced. Moreover, the feature becomes considerably less significant when the likelihood is marginalized over the orientation (Euler angles) of the fundamental domain.
All of these results (KL divergences and likelihoods) were computed with ℓ_{max} = 40, corresponding approximately to N_{side} = 16, indicating that this is more than adequate for detecting even relatively small fundamental domains such as the T [ 2,2,2 ] case simulated above. We also calculate d_{KL} between the correlation matrices for the T [ 7,7,7 ] torus (as a proxy for the simplyconnected case) and the T [ 5,5,5 ] torus, as a function of the maximum multipole ℓ_{max} used in the calculation of the correlation matrix: we find that d_{KL} continues to increase beyond ℓ_{max} = 60. Thus, higherresolution maps (as used by the matchedcircles methods) contain more information, but with the very low level of noise in the Planck CMB maps, ℓ_{max} = 40 would nonetheless give a robust detection of a multiplyconnected topology, even with the conservative foreground masking we apply.
We note that it is difficult to compress the content of these likelihood figures down to limits upon the size of the fundamental domain. This arises because it is difficult to provide a physicallymotivated prior distribution for quantities related to the size of the fundamental domain. Most naive priors would diverge towards arbitrarily large fundamental domain sizes or would otherwise depend on arbitrary limits to the topological parameters.
5.3.2. Bianchi
The ANICOSMO code (McEwen et al. 2013) is used to perform a Bayesian analysis of Bianchi VII_{h} models, which has been extensively validated by McEwen et al. (2013) already; we briefly summarise the validation performed for the masked analysis. In McEwen et al. (2013) a CMB map is simulated, in which a simulated Bianchi temperature map with a large vorticity (i.e., amplitude) is embedded, before applying a beam, adding isotropic noise and applying a mask. Both the underlying cosmological and Bianchi parameters used to generate the simulations are well recovered. For this simulation the coupled Bianchi model is favoured over ΛCDM, with a logBayes factor of ΔlnE ~ 50. As expected, one finds that the logBayes factor favours ΛCDM in simulations where no Bianchi component is added. For further details see McEwen et al. (2013).
6. Results
We now discuss the results of applying the circlesinthesky and likelihood methods to Planck data to study topology and Bianchi VII_{h} cosmologies.
Fig. 11
The (upper) and (lower) statistics as a function of circle radius α for the Planck CMB maps estimated using CommanderRuler (shortdashed green line), NILC (blue long dashed line), SEVEM (dotdashed red line) and SMICA (orange three dotsdashed line). Dotted line shows the false detection level established such that fewer than 1% out of 300 Monte Carlo simulations of the CMB map, smoothed and masked in the same way as the data, would yield a false event. The peak at 90° corresponds to a match between two copies of the same circle of radius 90° centered around two antipodal points. 

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Fig. 12
Top: likelihood as a function of the length of an edge of the fundamental domain L for a cubictorus topology. In this figure, χ_{rec} gives the distance to the surface of recombination. The data are componentseparated CMB temperature maps degraded to HEALPix N_{side} = 16 resolution and smoothed with an FWHM = 660′ Gaussian filter. The common mask of f_{sky} = 0.78 is used. The likelihood is marginalized over the amplitude of fluctuations, but maximized over the orientation of the fundamental domain. Lines for different estimates of the CMB temperature from Planck data are black: SMICA; magenta: SEVEM; green: CommanderRuler; blue: NILC. The red line is for a simulated isotropic sky from a fiducial flat simplyconnected model. Noise has been accounted for but is negligible at N_{side} = 16. The likelihoods are normalized to match the likelihood obtained with the common mask in the R_{0} → ∞ isotropic flat limit. The vertical lines mark the positions where χ_{rec} is equal to the characteristic sizes of the fundamental domain, from left to right, ℛ_{i} = L/ 2, and . Dots, superimposed onto the SMICA curve, designate the discrete set of models studied. Bottom: zoom into the transitional region near χ_{rec} ≈ R_{i}. Black PlanckSMICA and red fiducial curves are the same as in the top panel. The grey curve (open circles) is the likelihood marginalized over the orientations for the PlanckSMICA map. Only ℛ_{i} and ℛ_{m} are within the scale range shown. 

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Fig. 13
Same as Fig. 12, but for a toroidal space with one large dimension fixed at and two short dimensions of equal size L (approximating the “chimney” space). ℛ_{i} and ℛ_{m} are marked while ℛ_{u} = ∞ 

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Fig. 14
Same as Fig. 12, but for a toroidal space with two large dimensions fixed at and one short dimension of variable L (approximating the “slab” space). ℛ_{i} is marked while ℛ_{m} = ℛ_{u} = ∞. 

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6.1. Topology
Neither the circlesinthesky search nor the likelihood method find evidence for a multiplyconnected topology. We show the matched circle statistic in Fig. 11. We do not find any statistically significant correlation of circle pairs in any map. As seen in Fig. 5, the minimum radius at which the peaks expected for the matching statistic are larger than the false detection level is α_{min} ≈ 20°. Thus, we can exclude at the confidence level of 99% any topology that predicts matching pairs of backtoback circles larger than this radius, assuming that relative orientation of the fundamental domain and mask allows its detection. This implies that in a flat universe described otherwise by the Planck fiducial ΛCDM model, a 99% confidencelimit lower bound on the size of the fundamental domain is . This is better than the limits from the marginalized likelihood ratios below for the tori and octahedron topologies and slightly worse than the limits for the dodecahedron and truncated cube. However, this constraint is not limited only to these few topologies. The frequentist analysis provides constraints upon a much wider class of topologies than those explicitly considered in the Bayesian likelihood approach; it concerns all topologies listed in Sect. 3.1.
Fig. 15
Top: likelihood as a function of the distance to last scattering surface in curvature units for a locally spherical multiplyconnected universe with a dodecahedral (I^{∗}) fundamental domain with ℛ_{i} = 0.31R_{0}. Lines are for different estimates of the CMB temperature from Planck data as in Fig. 12. In this figure, the χ_{rec}/R_{0} parameterizes the position of the model on the geometrical degeneracy line which links H_{0} and Ω_{Λ} with Ω_{K}. The degeneracy relations are approximated as Ω_{Λ} = 0.691 + 2.705Ω_{K} and . The red reference curve is for the random isotropic realization from a fiducial flat model. Vertical lines mark when χ_{rec} equals each of ℛ_{i},ℛ_{m}, and ℛ_{u}, the characteristic scales of the fundamental domain. Bottom: zoom into the transitional region near χ_{rec} ≈ R_{i}. Both the likelihood at the best orientation of the domain versus the sky (black for the PlanckSMICA CMB map and red for the fiducial realization, as in the top panel) and the likelihood marginalized over the orientations for PlanckSMICA map (gray curve, open circles) are shown. 

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Fig. 16
Likelihood for a constant positive curvature multiplyconnected universe with a truncated cube (O^{∗}) fundamental domain with R_{i} = 0.39R_{0}. Notation is the same as in Fig. 15. 

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The likelihood method also show no evidence of a multiply connected universe. We present the likelihood for various models. In Fig. 12 we show the likelihood (marginalized over amplitude and maximized over orientation of the fundamental domain) for the cubic torus, fixing the background cosmology to the bestfit flat Universe Planck model (Planck Collaboration XVI 2014). We see that this is maximized for L> 2χ_{rec}, i.e., showing no evidence for nontrivial topology. Note that the likelihood shows mild features as the size goes through the other scales associated with the topology, in particular a small increase in the likelihood when the scale of the inscribed sphere ℛ_{i} is crossed. However, the same increase is found when the toroidal model is compared to a single realization of a strictly isotropic fiducial sky, and thus, should not be interpreted as a detection of multiconnected topology. The origin of this likelihood behaviour at best fit angles is that the freedom of orientation can be used to align small enhancements in largeangle correlations in the anisotropic L ≈ 2ℛ_{i} model with random features in the given single realization of the sky. When marginalized over all possible orientations the effect is significantly reduced; the slight rise is Δln ℒ ≃ 1.9 from a likelihood of P = 650, which is comparable to the numerical noise inherent in our stochastic integration. For even smaller spaces, more extensive correlations of the temperature can no longer be accommodated and for L< 2ℛ_{m} the likelihood of the T3 cubic toroidal model drops quickly, although not strictly monotonically.
Fig. 17
Likelihood for a constant positive curvature multiplyconnected universe with an octahedral (T^{∗}) fundamental domain with R_{i} = 0.45R_{0}. Notation is the same as in Fig. 15. 

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In Figs. 13 and 14 we show the likelihood for the T [ L,L,7 ] chimney and T [ L,7,7 ] slab topologies, which are also maximized in the simplyconnected limit. The T2 chimney, with only two compact dimensions, is less constrained than the T3 cube, and the T1 slab, with one compact dimension, even less so.
We find similar limits for the topologies allowed in a closed universe with a locally spherical geometry. In Fig. 15 we show the likelihood for the dodecahedral fundamental domain, in Fig. 16 for the truncated cube, and in Fig. 17 for the octahedron. In this case, we do not fix the background cosmological model, but rather account for the geometrical degeneracy line which links H_{0} and Ω_{Λ} with Ω_{K}. The degeneracy relations are approximated as Ω_{Λ} = 0.691 + 2.705Ω_{K} and . As in the toroidal case, there is no detection of a small space at the level expected from the simulations of Sect. 5. Fundamental domains larger than the last scattering diameter are preferred for the dodecahedral and truncated cube spaces with somewhat weaker restriction for the octahedral case. Note that an observationally motivated prior on H_{0} or Ω_{K} would be yet more restrictive on the fundamental domain size. For all three topologies, again as in the toroidal case, the maximum of the likelihood at best fit orientation is detected for the finite volume spaces with χ_{rec} ≈ ℛ_{i} at the level Δln ℒ ≈ + 4 relative to the fiducial flat simplyconnected model. Since this feature is seen in the isotropic fiducial sky as well, we cannot take it as an indication of a detection of a multiconnected space. In the case of curved spaces we see that this mild increase disappears when we consider the likelihood marginalized over orientations.
We present numerical limits for these flat and positively curved spaces in Table 2. Because of the onesided nature of these limits, we characterize the shape of the likelihood by the steepness of its fall from the value as the scale of the fundamental domain goes to infinity (i.e., the simplyconnected limit). Hence, we show limits for Δln ℒ < −5, (roughly equivalent to a 3σ – 99% confidence limit – fall for a Gaussian; because of the very steep gradient, the 2σ limits are very similar) and Δln ℒ < −12.5 (5σ). Note that the limits differ depending on whether we marginalize or maximize the likelihood over the orientation angles. We show lower limits on the quantity ℛ_{i} (L/ 2 for a torus with edge length L) in units of the last scattering distance χ_{rec} (in conventional units, χ_{rec} ≈ 14 Gpc for the fiducial Planck parameters; Planck Collaboration XVI 2014). In most cases, the limits are roughly – the scale of the fundamental domain must be greater than that of the last scattering surface. We place the most restrictive limits on the dodecahedron with ℛ_{i}> 1.03χ_{rec} using marginalized values for Δln ℒ < −5. Conversely, the chimney and slab spaces are less constrained as the expected correlations are weaker in one or two directions; for the slab space, we only constrain ℛ_{i} = L/ 2^{>~}0.5χ_{rec}.
In Fig. 18 we show the likelihood for the two hyperbolic models listed in Table 1, which also show no detection of the multiconnected topology. In the hyperbolic case we space the range of space sizes by varying Ω_{K} while keeping Ω_{Λ} − Ω_{m} as well as H_{0} constant at fiducial values.
All of these results show at least some increase in the likelihood for certain orientations when one of the characteristic scales of the fundamental domain (ℛ_{u}, ℛ_{m}, or ℛ_{i}) just exceed the surface of last scattering, and so no longer produces matched patterns, but induces extra correlations at large angular separations. Chance patterns can then mimic these correlations, and this is exacerbated by our conservative sky masks, which allow arbitrary patterns in the masked regions.
Lower limits on the size of the fundamental domain for different multiplyconnected spaces, in units of the distance to the last scattering surface, χ_{rec}.
Fig. 18
Likelihood for two constant negative curvature multiplyconnected universe, top: m004(−5,1); bottom: v3543(2,3). Notation is as in Fig. 15 except that only ℛ_{i}/R_{0} is shown by vertical lines. 

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6.2. Bianchi
Masked Planck data are analysed for evidence of a Bianchi VII_{h} component, where the prior parameter ranges adopted are the same as those specified by McEwen et al. (2013). The analysis is performed on the SMICA componentseparated map, using the mask defined for this method, and is repeated on the SEVEM componentseparated map for validation purposes (using the mask defined for the SEVEM method). The Bayes factors for the various Bianchi VII_{h} models and the equivalent standard cosmological models are shown in Table 3.
LogBayes factor relative to equivalent ΛCDM model (positive favours Bianchi model).
For the phenomenological flatdecoupledBianchi model, evidence in support of a lefthanded Bianchi template is found. On the Jeffreys scale (Jeffreys 1961), evidence for this model would be referred to as strong for the SMICA map and significant for the SEVEM map. For both SMICA and SEVEM componentseparated data, recovered posterior distributions for the flatdecoupledBianchi model are shown in Fig. 19a, where similar posterior distributions are recovered for both component separation methods. Recall that the Bianchi parameters are decoupled from the standard cosmology in the flatdecoupledBianchi model, hence for this model and are specific to the Bianchi model and should not be compared with standard values. The maximum a posteriori (MAP) bestfit template found for SMICA componentseparated data is shown in Fig. 20b, with the difference between this template and the template found in WMAP 9year data (McEwen et al. 2013) shown in Fig. 21. Note that the template found in Planck data is very similar to the template found in WMAP 9year data (McEwen et al. 2013), which in turn is similar to the template first found by Jaffe et al. (2005). However, the template found in WMAP 9year data (McEwen et al. 2013) is only significant in fullsky data, but not when the 9year KQ75 WMAP mask (Bennett et al. 2013) is applied. Since the PlanckSMICA and SEVEM masks are less conservative than the KQ75 mask, these findings suggest data near the Galactic plane may be playing a considerable role in supporting a Bianchi component in Planck data. The SMICA CMB map and a Bianchisubtracted version of this map are also shown in Fig. 20. The bestfit parameters of the templates found in PlanckSMICA and SEVEM componentseparated data are displayed in Table 4, for both the MAP and meanposterior estimates. The analysis was also performed on a SMICA componentseparated Gaussian simulation, yielding a null detection (i.e., no evidence for a Bianchi component), as expected.
For the most physically motivated opencoupledBianchi model where the Bianchi VII_{h} model is coupled to the standard cosmology, there is no evidence in support of a Bianchi contribution. Recovered posterior distributions for the opencoupledBianchi model are shown in Fig. 19b for both SMICA and SEVEM componentseparated data. Although the cosmological Bianchi parameters agree reasonably well between these different componentseparated data, the posterior distributions recovered for the Euler angles differ. For SEVEM data, an additional mode of the posterior distribution is found; the mode found with SMICA data is still present in SEVEM data but is not dominant. Consequently, the bestfit estimates for the Euler angles differ between the SMICA and SEVEM componentseparated data. Note that the additional mode found in SEVEM data is also present in WMAP 9year data (McEwen et al. 2013). The resulting bestfit parameters for the opencoupledBianchi model are displayed in Table 5, while the corresponding MAP bestfit maps are shown in Fig. 22. Nevertheless, for both SMICA and SEVEM data the Bayes factors computed (Table 3) do not favour the inclusion of any Bianchi component for the opencoupledBianchi model. Planck data thus do not provide evidence in support of Bianchi VII_{h} cosmologies. However, neither is it possible to conclusively discount Bianchi VII_{h} cosmologies in favour of ΛCDM cosmologies. The constraints (ω/H)_{0}< 7.6 × 10^{10} (95% confidence level) on the vorticity of the physical coupled Bianchi VII_{h} lefthanded models and (ω/H)_{0}< 8.1 × 10^{10} (95% confidence level) for righthanded models are recovered from SMICA componentseparated data.
Fig. 19
Posterior distributions of Bianchi parameters recovered from PlanckSMICA (solid curves) and SEVEM (dashed curves) componentseparated data for lefthanded models. Planck data provide evidence in support of a Bianchi component in the phenomenological flatdecoupledBianchi model (panel a)) but not in the physical opencoupledBianchi model (panel b)). 

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Fig. 20
Bestfit template of lefthanded flatdecoupledBianchi VII_{h} model subtracted from PlanckSMICA componentseparated data. Before subtraction, the peaktopeak variation is ± 594 μK, reduced to ± 564 μK after subtraction. 

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Fig. 21
Difference between bestfit template of flatdecoupledBianchi VII_{h} model recovered from WMAP 9year data and from PlanckSMICA componentseparated data. 

Open with DEXTER 
Parameters recovered for lefthanded flatdecoupledBianchi model.
Parameters recovered for lefthanded opencoupledBianchi model.
Fig. 22
Bestfit templates of lefthanded opencoupledBianchi VII_{h} model recovered from PlanckSMICA and SEVEM componentseparated data. The Bayes factors for this model indicate that Planck data do not favour the inclusion of these Bianchi maps. 

Open with DEXTER 
7. Discussion
We have used the Planck temperature anisotropy maps to probe the largescale structure of spacetime. We have calculated the Bayesian likelihood for specific topological models in universes with locally flat, hyperbolic and spherical geometries, all of which find no evidence for a multiplyconnected topology with a fundamental domain within the last scattering surface. After calibration on simulations, direct searches for matching circles resulting from the intersection of the fundamental topological domain with the surface of last scattering also give a null result at high confidence. These results use conservative masks of the sky, unlike previous WMAP results, which used fullsky internal linear combination maps (not originally intended for cosmological studies) or less conservative foreground masks. Hence, the results presented here, while corroborating the previous nondetections, use a single, selfconsistent, and conservative dataset. The masked sky also increases the possibility of chance patterns in the actual sky mimicking the correlations expected for topologies with a characteristic scale near that of the last scattering surface.
Depending on the shape of the fundamental domain, we find (Table 2) with detailed 99% confidence limits (considering the likelihood marginalized over the orientation of the fundamental domain) varying from 0.9χ_{rec} for the cubic torus in a flat universe to 1.03χ_{rec} for the dodecahedron in a positively curved universe, with somewhat weaker constraints for poorlyproportioned spaces that are considerably larger along some directions. In the case of the torus and octahedron topologies, a tighter constraint of 0.94χ_{rec} comes from the matched circles method (albeit with a somewhat different interpretation of frequentist and Bayesian limits). The constraint derived using this method applies to a wide class of topologies, listed in Sect. 3.1, predicting matching pairs of backtoback circles.
Note that the results derived using the likelihood method make use of the expected pixelspace correlations as a unique signal of nontrivial topology. Hence, although a small fundamental domain will suppress power on the largest scales of the CMB, observation of such low power on large scales as observed by COBE (Bond et al. 2000c), and confirmed by WMAP (Luminet et al. 2003), is not sufficient for the detection of topology. Conversely, because our methods search directly for these correlations (and indeed marginalize over the amplitude of fluctuations), a slight modification of the background FRW cosmology by lowering power in some or all multipoles (Planck Collaboration XV 2014) will not affect the ability to detect the correlations induced by such topologies.
Similarly, using a Bayesian analysis we find no evidence for a physical, anisotropic Bianchi VII_{h} universe. However, Planck data do provide evidence supporting a phenomenological Bianchi VII_{h} component, where the parameters of the Bianchi component are decoupled from standard cosmology. The resulting bestfit Bianchi VII_{h} template found in Planck data is similar to that found in WMAP data previously (Jaffe et al. 2005; McEwen et al. 2013). However, although this Bianchi component can produce some of the (possibly anisotropic) temperature patterns seen on the largest angular scales (see also Planck Collaboration XXIII 2014), there is no set of cosmological parameters which can simultaneously produce these patterns and the observed anisotropies on other scales. Moreover, the parameters of the bestfit Bianchi VII_{h} template in the decoupled setting are in strong disagreement with other measurements of the cosmological parameters.
These results are expected from previous measurements from COBE and WMAP, but Planck’s higher sensitivity and lower level of foreground contamination provides further confirmation. We have shown that the results are insensitive to the details of the preparation of the temperature maps (in particular, the method by which the cosmological signal is separated from astrophysical foreground contamination). Future Planck measurement of CMB polarization will allow us to further test models of anisotropic geometries and nontrivial topologies and may provide more definitive conclusions, for example allowing us to moderately extend the sensitivity to largescale topology (Bielewicz et al. 2012).
Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark.
The nomenclature for hyperbolic spaces follows J. Weeks’ census, as incorporated in the freely available SnapPea software, http://www.geometrygames.org/SnapPea; see also Thurston & Levy (1997).
We use the wmap7+bao+h0 results from http://lambda.gsfc.nasa.gov.
Acknowledgments
The development of Planck has been supported by: ESA; CNES and CNRS/INSUIN2P3INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration. The authors thank the anonymous referee for helpful comments and acknowledge the use of the UCL Legion High Performance Computing Facility (Legion@UCL), and associated support services, in the completion of this work. Part of the computations were performed on the Andromeda cluster of the University of Geneve, the Hopper Cray XE6 at NERSC and on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund – Research Excellence; and the University of Toronto.
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All Tables
Lower limits on the size of the fundamental domain for different multiplyconnected spaces, in units of the distance to the last scattering surface, χ_{rec}.
LogBayes factor relative to equivalent ΛCDM model (positive favours Bianchi model).
All Figures
Fig. 1
Top row: correlation structure (i.e., a single row of the correlation matrix) of a simplyconnected universe with isotropic correlations. For subsequent rows, the left and middle column show positively curved multiplyconnected spaces (left: dedocahedral, middle: octahedral) and the right column shows equal sided tori. The upper row of three maps corresponds to the case when the size of the fundamental domain is of the size of the diameter to the last scattering surface and hence the first evidence for large angle excess correlation appears. Subsequent rows correspond to decreasing fundamental domain size with respect to the last scattering diameter, with parameters roughly chosen to maintain the same ratio between the models. 

Open with DEXTER  
In the text 
Fig. 2
Random realisations of temperature maps for the models in Fig. 1. The maps are smoothed with a Gaussian filter with fullwidthhalfmaximum FWHM = 640′. 

Open with DEXTER  
In the text 
Fig. 3
Simulated deterministic CMB temperature contributions in Bianchi VII_{h} cosmologies for varying x and Ω_{tot} (lefttoright Ω_{tot} ∈ { 0.10,0.50,0.95 }; toptobottom x ∈ { 0.1,0.3,0.7,1.5,6.0 }). In these maps the swirl pattern typical of Bianchiinduced temperature fluctuations is rotated from the South pole to the Galactic centre for illustrational purposes. 

Open with DEXTER  
In the text 
Fig. 4
A simulated map of the CMB sky in a universe with a T [ 2,2,2 ] toroidal topology. The dark circles show the locations of the same slice through the last scattering surface seen on opposite sides of the sky. They correspond to matched circles with radius α ≃ 24°. 

Open with DEXTER  
In the text 
Fig. 5
An example of the statistic as a function of circle radius α for a simulated CMB map (shown in Fig. 4) of a universe with the topology of a cubic 3torus with dimensions (solid line). The dashdotted line show the false detection level established such that fewer than 1% out of 300 Monte Carlo simulations of the CMB map, smoothed and masked in the same way as the data, would yield a false event. 

Open with DEXTER  
In the text 
Fig. 6
KL divergence computed for torus models as a function of the (inverse) length of a side of the cube. T [ L_{1},L_{2},L_{3} ] refers to a torus with edge lengths L_{i}. 

Open with DEXTER  
In the text 
Fig. 7
KL divergence between a supposed correct model and other models. We show differences of cubic tori with respect to models with (LH_{0})^{1} = 1 / 4.5 ≃ 0.22 (aligned with our grid of models), (LH_{0})^{1} = 1 / 5.25 ≃ 0.19 (in between the gridpoints) and and a T [ 5,5,7 ] chimney model with (LH_{0})^{1} = 1 / 5 in two directions and (LH_{0})^{1} = 1 / 7 ≃ 0.14 in the third. 

Open with DEXTER  
In the text 
Fig. 8
KL divergence between a model generated with the WMAP bestfit cosmological parameters as a background cosmology and a T [ 5,5,5 ] cubic torus topology with respect to a Planck bestfit cosmology and a varying cubic topology. 

Open with DEXTER  
In the text 
Fig. 9
Loglikelihood with respect to the peak as a function of the orientation of the fundamental T [ 2,2,2 ] torus domain for the simulations. The third Euler angle is marginalized over. We see peaks at the orientations corresponding to the six faces of the cubic fundamental domain (there are peaks at the North and South poles, which are difficult to see in this projection). 

Open with DEXTER  
In the text 
Fig. 10
Test for likelihood detectability of compactified space for the example of a dodecahedral (I^{∗}) closed universe. The vertical axis shows the loglikelihood relative to the largest model considered. Values are given for the orientations of the models which maximize the likelihood (top) and marginalized over the orientations (bottom). Different size models are tested against two HEALPix N_{side} = 16 temperature realizations drawn from the model with χ_{rec}/R_{0} = 0.314 = ℛ_{i} (blue) and χ_{rec}/R_{0} = 0.361 (black). No noise is added and the common mask has been applied. Dots mark the positions of the models for which the likelihoods were computed. The vertical lines show characteristic scales of the fundamental domain of the models in the units of curvature, from smaller to larger, ℛ_{i}/R_{0}, ℛ_{m}/R_{0} and ℛ_{u}/R_{0}. The variable χ_{rec}/R_{0} gives the size of the last scattering surface in the same units. The R_{0} → ∞ limit corresponds to the flat simplyconnected space. Both maximized and marginalized likelihoods show a detection relative to the isotropic sky realization drawn from the fiducial flat infinite universe (red) with the detection stronger for smaller spaces. However only the maximized likelihood unambigously distinguishes the correct compact model from spaces that exceed the lastscattering diameter, which shows that the likelihood for small models is narrowly peaked at the correct orientation and suppressed otherwise. 

Open with DEXTER  
In the text 
Fig. 11
The (upper) and (lower) statistics as a function of circle radius α for the Planck CMB maps estimated using CommanderRuler (shortdashed green line), NILC (blue long dashed line), SEVEM (dotdashed red line) and SMICA (orange three dotsdashed line). Dotted line shows the false detection level established such that fewer than 1% out of 300 Monte Carlo simulations of the CMB map, smoothed and masked in the same way as the data, would yield a false event. The peak at 90° corresponds to a match between two copies of the same circle of radius 90° centered around two antipodal points. 

Open with DEXTER  
In the text 
Fig. 12
Top: likelihood as a function of the length of an edge of the fundamental domain L for a cubictorus topology. In this figure, χ_{rec} gives the distance to the surface of recombination. The data are componentseparated CMB temperature maps degraded to HEALPix N_{side} = 16 resolution and smoothed with an FWHM = 660′ Gaussian filter. The common mask of f_{sky} = 0.78 is used. The likelihood is marginalized over the amplitude of fluctuations, but maximized over the orientation of the fundamental domain. Lines for different estimates of the CMB temperature from Planck data are black: SMICA; magenta: SEVEM; green: CommanderRuler; blue: NILC. The red line is for a simulated isotropic sky from a fiducial flat simplyconnected model. Noise has been accounted for but is negligible at N_{side} = 16. The likelihoods are normalized to match the likelihood obtained with the common mask in the R_{0} → ∞ isotropic flat limit. The vertical lines mark the positions where χ_{rec} is equal to the characteristic sizes of the fundamental domain, from left to right, ℛ_{i} = L/ 2, and . Dots, superimposed onto the SMICA curve, designate the discrete set of models studied. Bottom: zoom into the transitional region near χ_{rec} ≈ R_{i}. Black PlanckSMICA and red fiducial curves are the same as in the top panel. The grey curve (open circles) is the likelihood marginalized over the orientations for the PlanckSMICA map. Only ℛ_{i} and ℛ_{m} are within the scale range shown. 

Open with DEXTER  
In the text 
Fig. 13
Same as Fig. 12, but for a toroidal space with one large dimension fixed at and two short dimensions of equal size L (approximating the “chimney” space). ℛ_{i} and ℛ_{m} are marked while ℛ_{u} = ∞ 

Open with DEXTER  
In the text 
Fig. 14
Same as Fig. 12, but for a toroidal space with two large dimensions fixed at and one short dimension of variable L (approximating the “slab” space). ℛ_{i} is marked while ℛ_{m} = ℛ_{u} = ∞. 

Open with DEXTER  
In the text 
Fig. 15
Top: likelihood as a function of the distance to last scattering surface in curvature units for a locally spherical multiplyconnected universe with a dodecahedral (I^{∗}) fundamental domain with ℛ_{i} = 0.31R_{0}. Lines are for different estimates of the CMB temperature from Planck data as in Fig. 12. In this figure, the χ_{rec}/R_{0} parameterizes the position of the model on the geometrical degeneracy line which links H_{0} and Ω_{Λ} with Ω_{K}. The degeneracy relations are approximated as Ω_{Λ} = 0.691 + 2.705Ω_{K} and . The red reference curve is for the random isotropic realization from a fiducial flat model. Vertical lines mark when χ_{rec} equals each of ℛ_{i},ℛ_{m}, and ℛ_{u}, the characteristic scales of the fundamental domain. Bottom: zoom into the transitional region near χ_{rec} ≈ R_{i}. Both the likelihood at the best orientation of the domain versus the sky (black for the PlanckSMICA CMB map and red for the fiducial realization, as in the top panel) and the likelihood marginalized over the orientations for PlanckSMICA map (gray curve, open circles) are shown. 

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In the text 
Fig. 16
Likelihood for a constant positive curvature multiplyconnected universe with a truncated cube (O^{∗}) fundamental domain with R_{i} = 0.39R_{0}. Notation is the same as in Fig. 15. 

Open with DEXTER  
In the text 
Fig. 17
Likelihood for a constant positive curvature multiplyconnected universe with an octahedral (T^{∗}) fundamental domain with R_{i} = 0.45R_{0}. Notation is the same as in Fig. 15. 

Open with DEXTER  
In the text 
Fig. 18
Likelihood for two constant negative curvature multiplyconnected universe, top: m004(−5,1); bottom: v3543(2,3). Notation is as in Fig. 15 except that only ℛ_{i}/R_{0} is shown by vertical lines. 

Open with DEXTER  
In the text 
Fig. 19
Posterior distributions of Bianchi parameters recovered from PlanckSMICA (solid curves) and SEVEM (dashed curves) componentseparated data for lefthanded models. Planck data provide evidence in support of a Bianchi component in the phenomenological flatdecoupledBianchi model (panel a)) but not in the physical opencoupledBianchi model (panel b)). 

Open with DEXTER  
In the text 
Fig. 20
Bestfit template of lefthanded flatdecoupledBianchi VII_{h} model subtracted from PlanckSMICA componentseparated data. Before subtraction, the peaktopeak variation is ± 594 μK, reduced to ± 564 μK after subtraction. 

Open with DEXTER  
In the text 
Fig. 21
Difference between bestfit template of flatdecoupledBianchi VII_{h} model recovered from WMAP 9year data and from PlanckSMICA componentseparated data. 

Open with DEXTER  
In the text 
Fig. 22
Bestfit templates of lefthanded opencoupledBianchi VII_{h} model recovered from PlanckSMICA and SEVEM componentseparated data. The Bayes factors for this model indicate that Planck data do not favour the inclusion of these Bianchi maps. 

Open with DEXTER  
In the text 