Cosmic Topology of Polyhedral Double-Action Manifolds

A special class of non-trivial topologies of the spherical space S^3 is investigated with respect to their cosmic microwave background (CMB) anisotropies. The observed correlations of the anisotropies on the CMB sky possess on large separation angles surprising low amplitudes which might be naturally be explained by models of the Universe having a multiconnected spatial space. We analysed in CQG 29(2012)215005 the CMB properties of prism double-action manifolds that are generated by a binary dihedral group D^*_p and a cyclic group Z_n up to a group order of 180. Here we extend the CMB analysis to polyhedral double-action manifolds which are generated by the three binary polyhedral groups (T^*, O^*, I^*) and a cyclic group Z_n up to a group order of 1000. There are 20 such polyhedral double-action manifolds. Some of them turn out to have even lower CMB correlations on large angles than the Poincare dodecahedron

Cosmic microwave anisotropies in an inhomogeneous compact flat universe

The anisotropies of the cosmic microwave background (CMB) are computed for the half-turn space E_2 which represents a compact flat model of the Universe, i.e. one with finite volume. This model is inhomogeneous in the sense that the statistical properties of the CMB depend on the position of the observer within the fundamental cell. It is shown that the half-turn space describes the observed CMB anisotropies on large scales better than the concordance model with infinite volume. For most observer positions it matches the temperature correlation function even slightly better than the well studied 3-torus topology

How well-proportioned are lens and prism spaces?

The CMB anisotropies in spherical 3-spaces with a non-trivial topology are analysed with a focus on lens and prism shaped fundamental cells. The conjecture is tested that well proportioned spaces lead to a suppression of large-scale anisotropies according to the observed cosmic microwave background (CMB). The focus is put on lens spaces L(p,q) which are supposed to be oddly proportioned. However, there are inhomogeneous lens spaces whose shape of the Voronoi domain depends on the position of the observer within the manifold. Such manifolds possess no fixed measure of well-proportioned and allow a predestined test of the well-proportioned conjecture. Topologies having the same Voronoi domain are shown to possess distinct CMB statistics which thus provide a counter-example to the well-proportioned conjecture. The CMB properties are analysed in terms of cyclic subgroups Z_p, and new point of view for the superior behaviour of the Poincar\'e dodecahedron is found

The optimal phase of the generalised Poincare dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps

Several studies have proposed that the shape of the Universe may be a Poincare dodecahedral space (PDS) rather than an infinite, simply connected, flat space. Both models assume a close to flat FLRW metric of about 30% matter density. We study two predictions of the PDS model. (i) For the correct model, the spatial two-point cross-correlation function, \ximc, of temperature fluctuations in the covering space, where the two points in any pair are on different copies of the surface of last scattering (SLS), should be of a similar order of magnitude to the auto-correlation function, \xisc, on a single copy of the SLS. (ii) The optimal orientation and identified circle radius for a "generalised" PDS model of arbitrary twist $\phi$, found by maximising \ximc relative to \xisc in the WMAP maps, should yield $\phi \in \{\pm 36\deg\}$. We optimise the cross-correlation at scales < 4.0 h^-1 Gpc using a Markov chain Monte Carlo (MCMC) method over orientation, circle size and $\phi$. Both predictions were satisfied: (i) an optimal "generalised" PDS solution was found, with a strong cross-correlation between points which would be distant and only weakly correlated according to the simply connected hypothesis, for two different foreground-reduced versions of the WMAP 3-year all-sky map, both with and without the kp2 Galaxy mask: the face centres are $(l,b)_{i=1,6}\approx (184d, 62d), (305d, 44d), (46d, 49d), (117d, 20d), (176d, -4d), (240d, 13d) to within ~2d, and their antipodes; (ii) this solution has twist \phi= (+39 \pm 2.5)d, in agreement with the PDS model. The chance of this occurring in the simply connected model, assuming a uniform distribution$\phi \in [0,2\pi]$, is about 6-9%.Comment: 20 pages, 22 figures, accepted in Astronomy & Astrophysics, software available at http://adjani.astro.umk.pl/GPLdownload/dodec/ and MCMCs at http://adjani.astro.umk.pl/GPLdownload/MCM

CMB Anisotropy of Spherical Spaces

The first-year WMAP data taken at their face value hint that the Universe might be slightly positively curved and therefore necessarily finite, since all spherical (Clifford-Klein) space forms M^3 = S^3/Gamma, given by the quotient of S^3 by a group Gamma of covering transformations, possess this property. We examine the anisotropy of the cosmic microwave background (CMB) for all typical groups Gamma corresponding to homogeneous universes. The CMB angular power spectrum and the temperature correlation function are computed for the homogeneous spaces as a function of the total energy density parameter Omega_tot in the large range [1.01, 1.20] and are compared with the WMAP data. We find that out of the infinitely many homogeneous spaces only the three corresponding to the binary dihedral group T*, the binary octahedral group O*, and the binary icosahedral group I* are in agreement with the WMAP observations. Furthermore, if Omega_tot is restricted to the interval [1.00, 1.04], the space described by T* is excluded since it requires a value of Omega_tot which is probably too large being in the range [1.06, 1.07]. We thus conclude that there remain only the two homogeneous spherical spaces S^3/O* and S^3/I* with Omega_tot of about 1.038 and 1.018, respectively, as possible topologies for our Universe.Comment: A version with high resolution sky maps can be obtained at http://www.physik.uni-ulm.de/theo/qc

Hot pixel contamination in the CMB correlation function?

Recently, it was suggested that the map-making procedure, which is applied to the time-ordered CMB data by the WMAP team, might be flawed by hot pixels. This could lead to a bias in the pixels having an angular distance of about 141 degrees from hot pixels due to the differential measuring process of the satellite WMAP. Here, the bias is confirmed, and the temperature two-point correlation function C(theta) is reevaluated by excluding the affected pixels. It is shown that the most significant effect occurs in C(theta) at the largest angles near theta = 180 degrees. Furthermore, the corrected correlation function C(theta) is applied to the cubic topology of the Universe, and it is found that such a multi-connected universe matches the temperature correlation better than the LCDM concordance model, provided the cubic length scale is close to L=4 measured in units of the Hubble length

Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds

Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes for all globally homogeneous 3-manifolds: the Poincare dodecahedral space S3/I*, the binary octahedral space S3/O*, the binary tetrahedral space S3/T*, the prism manifolds S3/D_m* and the lens spaces L(p,1).Comment: v3. Final published version. 27 pages, 1 figur

The Topology and Size of the Universe from the Cosmic Microwave Background

We study the possibility that the universe has compact topologies T^3, T^2 x R^1, or S^1 x R^2 using the seven-year WMAP data. The maximum likelihood 95% confidence intervals for the size L of the compact direction are 1.7 < L/L_0 < 2.1, 1.8 < L/L_0 < 2.0, 1.2 < L/L_0 < 2.1 for the three cases, respectively, where L_0=14.4 Gpc is the distance to the last scattering surface. An infinite universe is compatible with the data at 4.3 sigma. We find using a Bayesian analysis that the most probable universe has topology T^2 x R^1, with L/L_0=1.9.Comment: Additional checks, Monte-Carlo skies, and study of dipole contamination added. References added. 13 pages, 11 figure

Cosmic Topology of Prism Double-Action Manifolds

The cosmic microwave background (CMB) anisotropies in spherical 3-spaces with a non-trivial topology are studied. This paper discusses the special class of the so-called double-action manifolds, which are for the first time analysed with respect to their CMB anisotropies. The CMB anisotropies are computed for all prism double-action manifolds generated by a binary dihedral and a cyclic group with a group order of up to 180 leading to 33 different topologies. Several spaces are found which show a suppression of the CMB anisotropies on large angular distances as it is found on the real CMB sky. It turns out that two of these spaces possess Dirichlet domains which are not very far from highly symmetric polyhedra like Platonic or Archimedean ones

A measure on the set of compact Friedmann-Lemaitre-Robertson-Walker models

Compact, flat Friedmann-Lemaitre-Robertson-Walker (FLRW) models have recently regained interest as a good fit to the observed cosmic microwave background temperature fluctuations. However, it is generally thought that a globally, exactly-flat FLRW model is theoretically improbable. Here, in order to obtain a probability space on the set F of compact, comoving, 3-spatial sections of FLRW models, a physically motivated hypothesis is proposed, using the density parameter Omega as a derived rather than fundamental parameter. We assume that the processes that select the 3-manifold also select a global mass-energy and a Hubble parameter. The inferred range in Omega consists of a single real value for any 3-manifold. Thus, the obvious measure over F is the discrete measure. Hence, if the global mass-energy and Hubble parameter are a function of 3-manifold choice among compact FLRW models, then probability spaces parametrised by Omega do not, in general, give a zero probability of a flat model. Alternatively, parametrisation by the injectivity radius r_inj ("size") suggests the Lebesgue measure. In this case, the probability space over the injectivity radius implies that flat models occur almost surely (a.s.), in the sense of probability theory, and non-flat models a.s. do not occur.Comment: 19 pages, 4 figures; v2: minor language improvements; v3: generalisation: m, H functions of