Large-Scale Spatial Topology of the Universe

Theory > s.a. 3D manifolds; general-relativistic cosmology; large-scale geometry of the universe.
* Idea: Spacetime appears to be topologically \(\mathbb R\)⁴, and is usually assumed to be simply connected, but it need not be, there is no specific prediction about it, and we don't have good evidence either way; Schwarzschild mentioned the possibility in 1900.
* Possibilities: Numbers of different spatially homogeneous topologies (* = used in the standard model)
     closed: ∞ with k = +1 (*), 10 with k = 0, ∞ with k = −1;
     open: 0 with k = +1, 8 with k = 0 (*), ? with k = −1 (*).
* Remark: The notion of topological defects is unrelated to this, since it does not refer to spatial topology.
@ General references: Høg a1408 [astronomers' opinions on whether the universe is infinite]; Galloway et al a2010 [general cosmological models].
@ Non-trivial: Ellis GRG(71); Galloway PLA(80); Fang & Liu MPLA(88); Fagundes GRG(92)-a0812 + GRG(98)gq; Lachièze-Rey & Luminet PRP(95)gq/96; Rebouças et al GRG(98)gq/97 [and fragility]; Roukema & Luminet A&A(99)ap [and curvature]; Yasuno et al CQG(01)gq/00 [3-manifold gluing]; Luminet a0802-proc; Roukema & Blanlœil CQG(10)-a0912 [measure on the set of compact FLRW models]; Asselmeyer-Maluga & Król MPLA(12)-a1206 [restrictions from differential topology]; Moncrief & Mondal a1903 [with Λ > 0]; > s.a. perturbations.
@ And brane-world gravity: McInnes NPB(05)ht/04 [5D anti-de Sitter spacetime]; Bento et al PRD(06)ap.
> Related topics: see Baby Universes; cosmological acceleration; initial-value formulation of general relativity; multiverse; modified newtonian gravity; perturbations; spacetime topology [including small-scale spatial topology].

Observations > s.a. fractals in physics [bounds on fractal dimensionality]; laplace equation [spectrum].
* Consequences: If the topology is not simply connected (e.g., T\(^3\)), one (1) Gets finite-size flat universes (this helps for probabilities, etc); (2) Explains possible periodicities in quasar redshifts and absorption lines (obs?); (3) Explains the quasars in close pairs with very different redshifts; (4) Predicts a much smaller microvawe anisotropy, and a maximum length scale of about 200 or 600 Mpc (present horizon is about 4000 Mpc).
* Methods: Global topology is only weakly related to local observations, but a simple approach is to look for multiple images of distant galaxies; More sophisticated ones are the crystallographic method, or the search for patterns present in the cosmic microwave background, such as "circles-in-the-sky" (pairs of matching or correlated circles of temperature fluctuations in maps of the cmb).
@ Reviews: Lachièze-Rey & Luminet PRP(95)gq/96; Luminet et al SA(99)apr; Rebouças & Gomero BJP(04)ap-proc; Luminet BJP(06)ap/05-proc; Rebouças AIP(05)ap, IJMPD(07)ap/06; Luminet pw(05)sep, a0704-proc; Luminet G&C(14)-a1310, Univ(16)-a1601 [status].
@ General references: Oldershaw Nat(90)aug; Kamionkowski & Toumbas ap/96-proc; Levin et al PRD(98)ap, CQG(98)gq, PRD(98)ap; Cornish & Weeks NAMS-ap/98; Gott CQG(98); Luminet gq/98-proc, & Roukema ap/99-proc; Uzan et al A&A(99)ap, gq/00-proc; Roukema Pra(99)ap-proc, MNRAS(00)ap/99, BASI(00)ap/00, ap/02-proc; Blanloeil & Roukema ap/00-ed; Levin PRP(02)gq/01; Gomero & Rebouças PLA(03)gq/02 [spatially flat]; Bernui & Villela A&A(06)ap/05 [angular distributions of objects]; Souradeep IJP(06)gq [spectroscopy]; Bielewicz et al a1303-proc [constraints from WMAP-7]; Di Valentino et al nAstr(19)oct-a1911 [evidence for a closed universe].
@ Detectability: Mota et al CQG(03)gq; Kunz et al PRD(08)-a0704; Mota et al a1007-MG12; Fujii & Yoshii A&A(11)-a1105 [celestial distribution of astronomical objects].
@ Search for periodicities: Hawkins et al MNRAS(02)ap [none seen]; Weatherley et al MNRAS(03)ap; Planck Collaboration A&A(16)-a1502.
@ Nearly flat universe: Gomero et al IJMPD(00)gq/01, CQG(01)gq; Weeks MPLA(03), et al CQG(03); Mota et al CQG(04)ap/03.
@ Small universe: Cornish & Spergel PRD(00)ap/99; Gomero et al PLA(00)gq/99, CQG(01)gq; Gomero CQG(03)ap [strategy]; Piechocki gq/99-conf [quantum]; Barrow & Kodama IJMPD(01)gq; Barrow & Levin MNRAS(03)gq [and the Copernican principle].
@ And dark-energy equation of state: Rebouças IJMPA(09)-a0902; Rebouças & Teixeira a1005-MG12; Vitenti et al in(10)-a1008; Espiro & Le Delliou a1506.
@ Topological acceleration: Ostrowski et al CQG(12)-a1109; Roukema AIP(10)-a0911; Roukema AIP(13)-a1212 [example in multiply-connected exact solution]; Hernández et al a1901; Lorca & Le Delliou a1910.
@ Related topics: Goncharov & Nesteruk EPL(91) [and density perturbations]; Lehoucq et al A&A(00)ap [crystallographic, robustness]; Gausmann et al CQG(01)gq [topological lensing]; Opher ap/04 [new method, ??]; Rebouças et al A&A(06)ap/05 [and supernova observations]; Rebouças & Alcaniz BJP(05)ap/06, MNRAS(06)ap, ap/07-MG11 [and cosmological parameters]; Roukema & Blanloeil PRD(13)-a1201 [implications of inhomogeneity].

And the Cosmic Microwave Background
* Results: 2007, Analysis of WMAP3 data puts a lower bound of \(5 \times 10^3\) Gpc\(^3\) on the volume of a flat space with 3-torus topology.
@ General references: Cornish et al PRD(98)ap/97, PNAS(98)gq/97, CQG(98)ap; Bond et al CQG(98)ap-proc; Olson & Starkman CQG(00)gq; Inoue PhD(01)ap; Bowen & Ferreira PRD(02); Inoue & Sugiyama PRD(03)ap/02 [size]; Cornish et al PRL(04)ap/03 [WMAP, topology scale > 24 Gpc]; Phillips & Kogut ApJ(06)ap/04, Aurich et al PRL(05)ap/04 [spectrum, WMAP]; Hipólito-Ricaldi & Gomero PRD(05)ap; Riazuelo et al ap/06/PRD; Weeks & Gundermann CQG(07)ap/06 [quadrupole-octupole alignment]; Niarchou & Jaffe PRL(07)ap [no evidence from WMAP3]; Aurich et al CQG(07) [alignment]; Aurich et al CQG(08)-a0708 [bound on volume]; Gurzadyan MPLA(07)-a0709 [alleged evidence]; Kramer AIP(10)-a1009; Aslanyan & Manohar JCAP(12)-a1104 [using WMAP-7 data]; Aslanyan et al JCAP(13)-a1304; Fabre et al PRD(15)-a1311 [topological signature in the temperature and polarization anisotropies].
@ Circles: Levin & Heard ap/99-proc; Levin PRD(04)ap [displacements]; Gausmann & Opher ap/04-wd [problems]; Dineen et al MNRAS(05)ap/04; Mota et al PRD(08)-a0808; Mota et al PRD(10)-a1002 [in a flat universe]; Wehus & Eriksen ApJL(11)-a1011, Moss et al JCAP(11)-a1012, Hajian ApJ(11)-a1012, Bielewicz & Banday MNRAS(11)-a1012 [no convincing evidence]; Mota et al PRD(11)-a1108 [single-circle detection scenario]; Vaudrevange et al PRD(12)-a1206 [extension to general geometries]; Aurich & Lustig MNRAS(13)-a1303 [search in WMAP-9 data]; Gomero et al PRD(16)-a1604 [limitations of the searches].
@ Poincaré dodecahedral space: Roukema et al A&A(04)ap [WMAP, hint]; Roukema AIP(06)ap, IJMPD(09)-a0905-GRF; Caillerie et al A&A(07)-a0705; Roukema & Kazimierczak A&A(11)-a1106.
@ Other topologies: Aurich & Lustig CQG(12)-a1207 [polyhedral double-action manifolds].

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