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 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].
@ 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]; > 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., T3), one (1) Gets finite-size flat universes (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 too 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].
@ 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 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 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 a1109/MNRAS; Roukema AIP(10)-a0911; Roukema AIP(13)-a1212 [example in multiply-connected exact solution].
@ 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 × 103 Gpc3 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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