Large-Scale Spatial Geometry of the Universe  

In General > s.a. cosmological models; expansion rate [including vorticity and shear].
* Idea: The spatial geometry is approximately flat, homogeneous and isotropic, at least locally; The universe may not be "globally" homogeneous and isotropic.
* Distances: In cosmology, distances are determined based on standard candles (e.g., supernovae) or standard rulers (e.g., baryon oscillations); The two methods agree (duality) if some conditions on the physics are met.
* Radius of curvature: It is related to the Hubble parameter and the total mass-energy density by

R = (c/H0) (1 / |1−Ωtot|1/2) .

* Status: 2000, Evidence for zero overall spatial curvature (Boomerang); 2003, Evidence for small positive curvature (WMAP), with radius \(R = 3.0 \times 10^4\) Mpc using current parameter values; 2012, Evidence for small curvature \(\Omega_{\rm dm} = 0.002 \pm 0.009\), using the integrated Sachs-Wolfe effect; 2013, Evidence for small negative curvature from anomalies in the cmb fluctuation variance; 2020, Support for spatially flat universe.
@ II: Adams & Shapiro AS(01)sep [10 orientable Euclidean 3-geometries]; Levin 02.
@ General references: Fagundes GRG(92)-a0812 + GRG(98)gq [closed spaces, rev]; Manchak SHPMP(09) [global structure is underdetermined]; Stebbins IJMPD(12)-a1205-GRF [using observables as coordinates, without assumptions]; Bester et al MNRAS(15)-a1506 [algorithm using data].
> Related topics: see Flatness Problem; large-scale topology of the universe; quantum geometry [microscopic scale].

Distances and Observations in Cosmology > s.a. expansion rate; observation [cosmic parallax]; rotation.
@ Distances: Hogg ap/99 [pedagogical]; Jensen et al ap/03-in; Bassett & Kunz PRD(04), Kunz & Bassett ap/04-proc [distance duality, standard candles and rulers]; Lu & Hellaby CQG(07)-a0705 [determining the metric from observations]; Räsänen JCAP(09)-a0812 [redshift and areal distance, clumping effects]; de Grijs IAU(12)-a1209 [status]; Kaiser & Hudson MNRAS(15)-a1502 [kinematic bias]; Nikolaev & Chervon G&C(16)-a1604 [measuring angular diameter distances]; Holz et al PT(18)dec [gravitational waves, standard sirens]; Chassande-Mottin et al a1906 [gravitational waves]; Li & Zhang a2005 [novel method].
@ Luminosity distance: Caldwell & Kamionkowski JCAP(04)ap [effects of geometry and expansion history]; Malec et al GRG(04)gq [and areal distance]; Adachi & Kasai PTP(12)-a1111 [flat cosmologies with cosmological constant]; Flanagan et al PRD(13)-a1207 [fluctuations from lensing, Swiss-cheese model]; > s.a. effect of perturbations; light propagation.
@ Distance-redshift relation: Flanagan et al PRD(09)-a0810 [in modified gravity theories]; Lasky & Bolejko CQG(10)-a1001 [effect of pressure gradients]; Clarkson et al JCAP(14)-a1405 [shift in the distance-redshift relation, and consequences]; Hada & Futamase JCAP(14)-a1410 [magnitude-redshift relation in an inhomogeneous universe]; Wei et al MNRAS(15)-a1411 [galaxy-cluster angular size versus redshift]; Amirkhanyan AB(14)-a1501 [angular size and redshift]; Veropalumbo et al MNRAS(16)-a1510 [and baryon acoustic oscillations]; Shchigolev G&C(17)-a1511 [calculation via homotopy perturbation method]; Oguri PRD(16)-a1603 [from the cross-correlation of gravitational wave standard sirens and galaxies]; More et al a1612 [general Etherington distance duality relation].
@ Light propagation and geometry: Samushia et al PLB(10) [lookback time vs redshift, and dark energy]; López-Corredoira IJMPD(10)-a1002 [galaxy angular sizes]; Räsänen PRD(12)-a1107 [redshift, angular sizes]; Christiansen & Siver AJP(12)may-a1204 [brightness, angular sizes and expansion]; Melia CQG(13)-a1207 [proper size of the visible universe].

Spatial Curvature > s.a. cosmological parameters.
@ General references: Roos & Harun-or-Rashid ap/00, ap/00 [flatness]; Knox PRD(06)ap/05 [proposal]; Vardanyan et al MNRAS(09)-a0901 [model comparison perspective]; Dossett & Ishak PRD(12)-a1205 [and cosmological tests of gravity]; Kleban & Schillo JCAP(12) [and falsification of eternal inflation]; Caldwell & Gubser PRD(13)-a1302 [scalar curvature in the early universe]; Bull & Kamionkowski PRD(13) [consequences of possible non-zero value]; Bayin a1309 [argument for slight negative curvature based on the concept of gravitational temperature]; Carr & Harada PRD(15)-a1405 [the separate universe problem]; Planck Collaboration A&A(16)-a1502; Cai et al PRD(16)-a1509 [model-independent null test]; Li et al ApJ(16)-a1611; Coley a1905 [current situation]; Efstathiou & Gratton a2002, Vagnozzi et al a2010 [evidence for flatness]; Tian et al PRD(21)-a2010 [in an inhomogeneous universe].
@ And perturbations, structure formation: Lyth et al JCAP(05)ap/04; Buchert et al GRG(09)-a0906; Bolejko a1707 [emergence of spatial curvature in a spatially flat model].
@ Measurement methods: Gu & Khlopov gq/07; Bozek et al PRD(09) [from causal entropic principle]; Mortonson PRD(09)-a0908 [model-independent tests]; Albrecht PRL(11)-a1104 [from de Sitter equilibrium cosmology]; Adler AJP(12)may [exact spatial flatness from the tipping pencil analogy and the uncertainty principle]; Wei & Wu ApJ(17)-a1611 [model-independent method, from H(z) and supernova data]; Park & Ratra a1809 [and Hubble constant]; Arjona & Nesseris a2103 [null tests, and machine learning].
@ Measurements, contraints: Lieu & Mittaz ApJ(05)ap/03 [exactly 0]; Mersini-Houghton & Kafexhiu APP(08)-a0705 [role of cosmic priors in bounds], comment Räsänen APP(08)-a0705 [analysis]; Okouma et al PLB(13)-a1207 [evidence for Ωdm = 0.002 ± 0.009]; news nbc(13)sep [possible evidence for negative curvature]; Leonard et al PRD(16)-a1604 [future data and accuracy]; Farooq et al ApJ(17)-a1607 [from Hubble parameter measurements].
@ From the cmb: Hu & White ap/96-proc; Uzan et al MNRAS(03)ap [and WMAP].
@ From supernova data: Wang et al PLB(05)ht/04; Araújo & Stoeger MNRAS(09)-a0705 [supernova luminosity distances]; Gong et al MNRAS(11)-a1008; Mörtsell & Jönsson a1102 [model-independent measure, from type-Ia supernovae]; Cai et al PRD(16)-a1509 [model-independent null test]; Jesus et al a1907; Wang et al a1910.
@ From lensing: Soucail et al A&A(04)ap [Abell 2218]; Bernstein ApJ(06)ap/05.

Other Aspects
@ Homogeneity, isotropy: Roukema ASR(03)ap/01 [local vs global]; Campanelli et al PRL(06) + pn(06)oct [ellipsoidal shape]; Räsänen PRD(09)-a0903 [and cmb isotropy]; Yu et al RAA(11)-a1008 [smoothness parameter]; Zunckel et al PRD(11)-a1009 [multipole vectors]; Koivisto et al PRD(11) [anisotropic curvature]; > s.a. cosmological principle; Homogeneity.
@ Singularities: Szydłowski et al PRD(05)ap [big bang vs bounce, observational tests]; > s.a. types of singularities.
@ Related topics: Johnson a1505 [black holes and the curvature of space]; Gaztañaga & Fosalba a2104-GRF [universe trapped inside an event horizon].

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