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Abstract

Recent observations seem to indicate that we live in a universe whose spatial sections are nearly or exactly flat. Motivated by this we study the problem of observational detection of the topology of universes with flat spatial sections. We first give a complete description of the diffeomorphic classification of compact flat 3-manifolds, and derive the expressions for the injectivity radii, and for the volume of each class of Euclidean 3-manifolds. There emerges from our calculations the undetectability conditions for each (topological) class of flat universes. To illustrate the detectability of flat topologies we construct toy models by using an assumption by Bernshteı̌n and Shvartsman which permits to establish a relation between topological typical lengths to the dynamics of flat models.

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Chapter
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Whether we live in a spatially finite universe, and what its shape and size may be, are among the fundamental long-standing questions in cosmology. These questions of topological nature have become particularly topical, given the wealth of increasingly accurate astro-cosmological observations, especially the recent observations of the cosmic microwave background radiation. An overview of the basic context of cosmic topology, the detectability constraints from recent observations, as well as the main methods for its detection and some recent results are briefly presented. Comment: 14 pages, 5 figures. Short review of the topics addressed with details in the lectures. To appear in the proc. of the XIth Brazilian School of Cosmology and Gravitation, eds. M.Novelo and S.E. Perez Bergliaffa, American Institute of Physics Conference Proceedings (2005)
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The observed values of density parameters inevitably involve uncertainties. We study the conditions for detectability and undetectability of cosmic topology in presence of such uncertainties. We present closed analytical forms of (un)detectability conditions for infinite redshift, which are important because: (i) they allow the examine of the detectability of cosmic topology not only for individual manifolds (topologies), but also in whole classes of manifolds; (ii) they are, to a very good approximation, (un)detectibility conditions for z=1100.
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Questions such as whether we live in a spatially finite universe, and what its shape and size may be, are among the fundamental open problems that high precision modern cosmology needs to resolve. These questions go beyond the scope of general relativity (GR), since as a (local) metrical theory GR leaves the global topology of the universe undetermined. Despite our present-day inability to predict the topology of the universe, given the wealth of increasingly accurate astro-cosmological observations it is expected that we should be able to detect it. An overview of basic features of cosmic topology, the main methods for its detection, and observational constraints on detectability are briefly presented. Recent theoretical and observational results related to cosmic topology are also discussed.
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It is shown here how prior estimates on the local shape of the universe can be used to reduce, to a small region, the full parameter space for the search of circles in the sky. This is the first step towards the development of efficient estrategies to look for these matched circles in order to detect a possible nontrivial topology of our Universe. It is shown how to calculate the unique point, in the parameter space, representing a pair of matched circles corresponding to a given isometry g (and its inverse). As a consequence, (i) given some fine estimates of the covering group Γ\Gamma of the spatial section of our universe, it is possible to confine, in a very effective way, the region of the parameter space in which to perform the searches for matched circles, and reciprocally (ii) once identified such pairs of matched circles, one could determine with greater precision the topology of our Universe and our location within it.
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Given the wealth of increasingly accurate cosmological observations, especially the recent results from the WMAP, and the development of methods and strategies in the search for cosmic topology, it is reasonable to expect that we should be able to detect the spatial topology of the Universe in the near future. Motivated by this, we examine to what extent a possible detection of a nontrivial topology of positively curved universe may be used to place constraints on the matter content of the Universe. We show through concrete examples that the knowledge of the spatial topology allows to place constraints on the density parameters associated to dark matter (Ωm\Omega_m) and dark energy (ΩΛ\Omega_{\Lambda})
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We propose an alternative formalism to simulate CMB temperature maps in Λ\LambdaCDM universes with nontrivial spatial topologies. This formalism avoids the need to explicitly compute the eigenmodes of the Laplacian operator in the spatial sections. Instead, the covariance matrix of the coefficients of the spherical harmonic decomposition of the temperature anisotropies is expressed in terms of the elements of the covering group of the space. We obtain a decomposition of the correlation matrix that isolates the topological contribution to the CMB temperature anisotropies out of the simply connected contribution. A further decomposition of the topological signature of the correlation matrix for an arbitrary topology allows us to compute it in terms of correlation matrices corresponding to simpler topologies, for which closed quadrature formulae might be derived. We also use this decomposition to show that CMB temperature maps of (not too large) multiply connected universes must show ``patterns of alignment'', and propose a method to look for these patterns, thus opening the door to the development of new methods for detecting the topology of our Universe even when the injectivity radius of space is slightly larger than the radius of the last scattering surface. We illustrate all these features with the simplest examples, those of flat homogeneous manifolds, i.e., tori, with special attention given to the cylinder, i.e., T1T^1 topology.
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If the universe has a nontrivial shape (topology) the sky may show multiple correlated images of cosmic objects. These correlations can be couched in terms of distance correlations. We propose a statistical quantity which can be used to reveal the topological signature of any Robertson–Walker (RW) spacetime with nontrivial topology. We also show through computer-aided simulations how one can extract the topological signatures of flat, elliptic, and hyperbolic RW universes with nontrivial topology.
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A statistical quantity suitable for distinguishing simply-connected Robertson–Walker (RW) universes is introduced, and its explicit expressions for the three possible classes of simply-connected RW universes with an uniform distribution of matter are determined. Graphs of the distinguishing mark for each class of RW universes are presented and analyzed. There sprout from our results an improvement on the procedure to extract the topological signature of multiply-connected RW universes, and a refined understanding of that topological signature of these universes studied in previous works.
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The 1997 Cleveland Workshop on Topology and Cosmology was a remarkable opportunity for mathematicians and cosmologist with overlapping interests in the topology of 3-space to discuss the state of knowledge in their fields. This special issue reflects the wide range of talks given at the conference including both introductions/reviews of the relevant fields of mathematics and cosmology and new contributions to the scholarship in these fields. In this introduction to the special issue, a brief history is given of the study of the topology of the universe. Some attempt is also made to introduce and discuss each of the papers in the collection and to relate them to each other and to the wider conversation on the global topology of the universe.
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Using recent observational constraints on cosmological density parameters, together with recent mathematical results concerning small volume hyperbolic manifolds, we argue that, by employing pattern repetitions, the topology of nearly flat small hyperbolic universes can be observationally undetectable. This is important in view of the fact that quantum cosmology may favour hyperbolic universes with small volumes, and from the expectation, coming from inflationary scenarios, that Omega0 is likely to be very close to one.
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The anisotropy of the cosmic microwave background radiation contains information about the contents and history of the universe. We report new limits on cosmological parameters derived from the angular power spectrum measured in the first Antarctic flight of the Boomerang experiment. Within the framework of models with adiabatic perturbations, and using only weakly restrictive prior probabilities on the age of the universe and the Hubble expansion parameter h, we find that the curvature is consistent with flat and that the primordial fluctuation spectrum is consistent with scale invariant, in agreement with the basic inflation paradigm. We find that the data prefer a baryon density Ωbh2 above, though similar to, the estimates from light element abundances and big bang nucleosynthesis. When combined with large scale structure observations, the Boomerang data provide clear detections of both dark matter and dark energy contributions to the total energy density Ωtot, independent of data from high-redshift supernovae.
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WMAP precision data enable accurate testing of cosmological models. We find that the emerging standard model of cosmology, a flat Λ-dominated universe seeded by a nearly scale-invariant adiabatic Gaussian fluctuations, fits the WMAP data. For the WMAP data only, the best-fit parameters are h = 0.72 ± 0.05, Ωbh2 = 0.024 ± 0.001, Ωmh2 = 0.14 ± 0.02, τ = 0.166, ns = 0.99 ± 0.04, and σ8 = 0.9 ± 0.1. With parameters fixed only by WMAP data, we can fit finer scale cosmic microwave background (CMB) measurements and measurements of large-scale structure (galaxy surveys and the Lyα forest). This simple model is also consistent with a host of other astronomical measurements: its inferred age of the universe is consistent with stellar ages, the baryon/photon ratio is consistent with measurements of the [D/H] ratio, and the inferred Hubble constant is consistent with local observations of the expansion rate. We then fit the model parameters to a combination of WMAP data with other finer scale CMB experiments (ACBAR and CBI), 2dFGRS measurements, and Lyα forest data to find the model's best-fit cosmological parameters: h = 0.71, Ωbh2 = 0.0224 ± 0.0009, Ωmh2 = 0.135, τ = 0.17 ± 0.06, ns(0.05 Mpc-1) = 0.93 ± 0.03, and σ8 = 0.84 ± 0.04. WMAP's best determination of τ = 0.17 ± 0.04 arises directly from the temperature-polarization (TE) data and not from this model fit, but they are consistent. These parameters imply that the age of the universe is 13.7 ± 0.2 Gyr. With the Lyα forest data, the model favors but does not require a slowly varying spectral index. The significance of this running index is sensitive to the uncertainties in the Lyα forest.
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We present full-sky microwave maps in five frequency bands (23-94 GHz) from the Wilkinson Microwave Anisotropy Probe (WMAP) first-year sky survey. Calibration errors are less than 0.5%, and the low systematic error level is well specified. The cosmic microwave background (CMB) is separated from the foregrounds using multifrequency data. The sky maps are consistent with the 7° FWHM Cosmic Background Explorer (COBE) maps. We report more precise, but consistent, dipole and quadrupole values. The CMB anisotropy obeys Gaussian statistics with -58 < fNL < 134 (95% confidence level [CL]). The 2 ≤ ℓ ≤ 900 anisotropy power spectrum is cosmic-variance-limited for ℓ < 354, with a signal-to-noise ratio greater than 1 per mode to ℓ = 658. The temperature-polarization cross-power spectrum reveals both acoustic features and a large-angle correlation from reionization. The optical depth of reionization is τ = 0.17 ± 0.04, which implies a reionization epoch of tr = 180 Myr (95% CL) after the big bang at a redshift of zr = 20 (95% CL) for a range of ionization scenarios. This early reionization is incompatible with the presence of a significant warm dark matter density.
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General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi- rather than simply-connected. We review the main mathematical properties of multi-connected spaces, and the different tools to classify them and to analyse their properties. Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. We briefly discuss some implications of multi-connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. We consider in details the properties of the cosmological models where space is multi-connected, with emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible signature of multi-connectedness, or to constrain the models. They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,…, or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them.
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The fluctuations of the cosmic microwave background (CMB) are investigated for a hyperbolic universe with finite volume. Four-component models with radiation, matter, vacuum energy, and an extra spatially constant dark energy X-component are considered. The general solution of the Friedmann equation for the cosmic scale factor a(eta) is given for the four-component models in terms of the Weierstrass P-function. The lower part of the angular power spectra C_l of the CMB anisotropy is computed for nearly flat models with Omega_tot <= 0.95. It is shown that the particular compact fundamental cell, which is considered in this paper, leads to a suppression in C_l for l < 10 and Omega_tot <= 0.9.
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We study the effect of global topology of the spatial geometry on the cosmic microwave background (CMB) for closed flat and closed hyperbolic models in which the spatial hypersurface is multiply connected. If the CMB temperature fluctuations were entirely produced at the last scattering, then the large-angle fluctuations would be much suppressed in comparison with the simply connected counterparts which is at variance with the observational data. However, as we shall show in this thesis, for low matter density models the observational constraints are less stringent since a large amount of large-angle fluctuations could be produced at late times. On the other hand, a slight suppression in large-angle temperature correlations in such models explains rather naturally the observed anomalously low quadrupole which is incompatible with the prediction of the "standard" Friedmann-Robertson-Walker-Lemaitre models. Interestingly, moreover, the development in the astronomical observation technology has made it possible to directly explore the imprint of the non-trivial topology by looking for identical objects so called "ghosts" in wide separated directions. For the CMB temperature fluctuations identical patterns would appear on a pair of circles in the sky. Another interesting feature is the non-Gaussianity in the temperature fluctuations. Inhomogeneous and anisotropic Gaussian fluctuations for a particular choice of position and orientation are regarded as non-Gaussian fluctuations for a homogeneous and isotropic ensemble.
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If the universe is finite and smaller than the distance to the surface of last scatter, then the signature of the topology of the universe is writ large on the microwave background sky. We show that the microwave background will be identified at the intersections of the surface of last scattering as seen by different ``copies'' of the observer. Since the surface of last scattering is a two-sphere, these intersections will be circles, regardless of the background geometry or topology. We therefore propose a statistic that is sensitive to all small, locally homogeneous topologies. Here, small means that the distance to the surface of last scatter is smaller than the ``topology scale'' of the universe. Comment: 14 pages, 10 figures, IOP format. This paper is a direct descendant of gr-qc/9602039. To appear in a special proceedings issue of Class. Quant. Grav. covering the Cleveland Topology & Cosmology Workshop
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The past, present and future of cosmic microwave background (CMB) anisotropy research is discussed, with emphasis on the Boomerang and Maxima balloon experiments. These data are combined with large scale structure (LSS) information and high redshift supernova (SN1) observations to explore the inflation-based cosmic structure formation paradigm. Here we primarily focus on a simplified inflation parameter set, {omega_b,omega_{cdm},Omega_{tot}, Omega_Q,w_Q, n_s,tau_C, sigma_8}. After marginalizing over the other cosmic and experimental variables, we find the current CMB+LSS+SN1 data gives Omega_{tot}=1.04\pm 0.05, consistent with (non-baroque) inflation theory. Restricting to Omega_{tot}=1, we find a nearly scale invariant spectrum, n_s =1.03 \pm 0.07. The CDM density, omega_{cdm}=0.17\pm 0.02, is in the expected range, but the baryon density, omega_b=0.030\pm 0.004, is slightly larger than the current nucleosynthesis estimate. Substantial dark energy is inferred, Omega_Q\approx 0.68\pm 0.05, and CMB+LSS Omega_Q values are compatible with the independent SN1 estimates. The dark energy equation of state, parameterized by a quintessence-field pressure-to-density ratio w_Q, is not well determined by CMB+LSS (w_Q<-0.3 at 95%CL), but when combined with SN1 the resulting w_Q<-0.7 limit is quite consistent with the w_Q=-1 cosmological constant case. Though forecasts of statistical errors on parameters for current and future experiments are rosy, rooting out systematic errors will define the true progress. Comment: 14 pages, 3 figs., in Proc. CAPP-2000 (AIP), CITA-2000-64
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CMB anisotropy measurements have brought the issue of global topology of the universe from the realm of theoretical possibility to within the grasp of observations. The global topology of the universe modifies the correlation properties of cosmic fields. In particular, strong correlations are predicted in CMB anisotropy patterns on the largest observable scales if the size of the Universe is comparable to the distance to the CMB last scattering surface. We describe in detail our completely general scheme using a regularized method of images for calculating such correlation functions in models with nontrivial topology, and apply it to the computationally challenging compact hyperbolic spaces. Our procedure directly sums over images within a specified radius, ideally many times the diameter of the space, effectively treats more distant images in a continuous approximation, and uses Cesaro resummation to further sharpen the results. At all levels of approximation the symmetries of the space are preserved in the correlation function. This new technique eliminates the need for the difficult task of spatial eigenmode decomposition on these spaces. Although the eigenspectrum can be obtained by this method if desired, at a given level of approximation the correlation functions are more accurately determined. We use the 3-torus example to demonstrate that the method works very well. We apply it to power spectrum as well as correlation function evaluations in a number of compact hyperbolic (CH) spaces. Application to the computation of CMB anisotropy correlations on CH spaces, and the observational constraints following from them, are given in a companion paper. Comment: 27 pages, Latex, 11 figures, submitted to Phys. Rev. D, March 11, 1999
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A statistical quantity suitable for distinguishing simply-connected Robertson-Walker (RW) universes is introduced, and its explicit expressions for the three possible classes of simply-connected RW universes with an uniform distribution of matter are determined. Graphs of the distinguishing mark for each class of RW universes are presented and analyzed. There sprout from our results an improvement on the procedure to extract the topological signature of multiply-connected RW universes, and a refined understanding of that topological signature of these universes studied in previous works.
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If the universe has a nontrivial shape (topology) the sky may show multiple correlated images of cosmic objects. These correlations can be couched in terms of distance correlations. We propose a statistical quantity which can be used to reveal the topological signature of any Robertson-Walker (RW) spacetime with nontrivial topology. We also show through computer-aided simulations how one can extract the topological signatures of flat, elliptic, and hyperbolic RW universes with nontrivial topology.
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This book lays out the logical structure and relationships of contemporary cosmology. Both the classical principles of cosmology and the latest problems in this field are considered. The general theory of relativity is presented in intelligible form together with the theories of elementary particles and physical statistics. Topics include: the geometry of space, the theory of the expanding hot isotropic universe, the problem of the initial singularity, physical processes in the early stages of cosmological expansion, gravitational instability and the formation of galaxies, propagation of light and neutrinos in the expanding universe, the cosmological redshift, gravitational theories, anisotropic cosmological models, and steady-state cosmology.
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We present a modified version of the cosmic crystallography method, especially useful for testing closed models of negative spatial curvature. The images of clusters of galaxies in simulated catalogs are `pulled back' to the fundamental domain before the set of distances is calculated.
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We study the topological signature of euclidean isometries in pair separations histograms (PSH) and elucidate some unsettled issues regarding distance correlations between cosmic sources in cosmic crystallography. Reducing the noise of individual PSHs using mean pair separations histograms we show how to distinguish between topological and statistical spikes. We report results of simulations that evince that topological spikes are not enough to distinguish between manifolds with the same set of Clifford translations in their covering groups, and that they are not the only signature of topology in PSHs corresponding to euclidean small universes. We also show how to evince the topological signature due to non-translational isometries.
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Nature abhors an infinity. The limits of general relativity are often signaled by infinities: infinite curvature as in the center of a black hole, the infinite energy of the singular big bang. We might be inclined to add an infinite universe to the list of intolerable infinities. Theories that move beyond general relativity naturally treat space as finite. In this review we discuss the mathematics of finite spaces and our aspirations to observe the finite extent of the universe in the cosmic background radiation. Comment: Hilarioulsy forgot to remove comments to myself in previous version. Reference added. Submitted to Physics Reports
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Cosmic microwave background data shows the observable universe to be nearly flat, but leaves open the question of whether it is simply or multiply connected. Several authors have investigated whether the topology of a multiply connect hyperbolic universe would be detectable when 0.9 < Omega < 1. However, the possibility of detecting a given topology varies depending on the location of the observer within the space. Recent studies have assumed the observer sits at a favorable location. The present paper extends that work to consider observers at all points in the space, and (for given values of Omega_m and Omega_Lambda and a given topology) computes the probability that a randomly placed observer could detect the topology. The computations show that when Omega = 0.98 a randomly placed observer has a reasonable chance (~50%) of detecting a hyperbolic topology, but when Omega = 0.99 the chances are low (<10%) and decrease still further as Omega approaches one. Comment: 9 pages, 5 figures
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Since the dawn of civilization, humanity has grappled with the big questions of existence and creation. Modern cosmology seeks to answer some of these questions using a combination of mathematics and measurement. The questions people hope to answer include ``how did the universe begin?''; ``how will the universe end?''; ``is space finite or infinite?''. After a century of remarkable progress, cosmologists may be on the verge of answering at least one of these questions -- is space finite? Using some simple geometry and a NASA satellite set for launch in the year 2000, the authors and their colleagues hope to measure the size and shape of space. This review article explains the mathematics behind the measurement, and the cosmology behind the observations.
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In recent years, the large angle COBE--DMR data have been used to place constraints on the size and shape of certain topologically compact models of the universe. Here we show that this approach does not work for generic compact models. In particular, we show that compact hyperbolic models do not suffer the same loss of large angle power seen in flat or spherical models. This follows from applying a topological theorem to show that generic hyperbolic three manifolds support long wavelength fluctuations, and by taking into account the dominant role played by the integrated Sachs-Wolfe effect in a hyperbolic universe.