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 R⁴, and is usually assumed to be simply connected, but it need not be and we don't have good evidence either way because global topology is only weakly related to local observations; Schwarzschild mentioned the possibility in 1900; 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).
* 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 does not refer to spatial topology.
@ Non-trivial: Ellis GRG(71); Galloway PLA(80); Fang & Liu MPLA(88); Fagundes GRG(92) + 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-in; > s.a. perturbations.
@ And brane-world gravity: McInnes NPB(05)ht/04 [5D AdS]; Bento et al PRD(06)ap.
> Related topics: see initial value formulation of general relativity; multiverse; perturbations; spacetime topology [including small-scale spatial topology].

Observations > s.a. laplace [spectrum].
* Methods: 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 cmb, such as pairs of "circles-in-the-sky" around which the temperature fluctuations are correlated.
* Results: 2007, Analysis of WMAP3 data puts lower bound of 5 10³ Gpc³ on the volume of a flat space with 3-torus topology.
@ Reviews: Lachièze-Rey & Luminet PRP(95)gq/96; Luminet et al SA(99)apr; Rebouças & Gomero ap/04-in; Luminet ap/05-in; Rebouças ap/05-in, ap/06/IJMPD; Luminet pw(05)sep, a0704-in.
@ General references: Oldershaw Nat(90)aug; Kamionkowski & Toumbas ap/96-in; Levin et al PRD(98)ap, CQG(98)gq, PRD(98)ap; Cornish & Weeks ap/98/NAMS; Gott CQG(98); Luminet gq/98-in, & Roukema ap/99-in; Uzan et al A&A(99)ap, gq/00-in; Roukema Pra(99)ap-in, MNRAS(00)ap/99, BASI(00)ap/00, ap/02-in; Blanloeil & Roukema ap/00-ed; Levin PRP(02)gq/01; Gomero & Rebouças PLA(03)gq/02 [spatially flat]; Bernui & Villela ap/05/A&A [angular distributions of objects]; Souradeep IJP(06)gq [spectroscopy].
@ Detectability: Mota et al CQG(03)gq; Kunz et al PRD(08)-a0704.
@ Search for periodicities: Hawkins et al MNRAS(02)ap [none seen]; Weatherley et al MNRAS(03)ap.
@ 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-in [quantum]; Barrow & Kodama IJMPD(01)gq; Barrow & Levin MNRAS(03)gq [and Copernican principle].
@ And cmb: Cornish et al PRD(98)ap/97, PNAS(98)gq/97, CQG(98)ap; Bond et al CQG(98)ap-in; Olson & Starkman CQG(00)gq; Inoue ap/01-PhD; 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].
@ And cmb, circles: Levin & Heard ap/99-in; Levin PRD(04)ap [displacements]; Gausmann & Opher ap/04-wd [problems]; Dineen et al MNRAS(05)ap/04.
@ And cmb, Poincaré dodecahedral space: Roukema et al A&A(04)ap [WMAP, hint]; Roukema ap/06-in; Caillerie et al A&A-a0705.
@ 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 sn observations]; Rebouças & Alcaniz BJP(05)ap/06, MNRAS(06)ap, ap/07-in [and cosmological parameters].

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