Bianchi Classification of Cosmological Models| Bianchi Classification of Cosmological Models |
In General
> s.a. Homogeneity; Isotropic Metric.
* Idea: It is a classification
of most spatially homogeneous anisotropic cosmologies; What Bianchi really
did was to classify the three-dimensional Lie algebras, but the Lie algebra
can be taken to be that of the isometry group, and the two things become
equivalent; The original scheme has been later modified.
$ Def: Spacetimes with a 3D
group G of isometries acting freely and (simply) transitively
on spacelike hypersurfaces.
* Motivation: The vacuum or
perfect fluid models give simple equations, which however are non-linear
enough to study anisotropy, singularities, chaotic dynamics, horizon
structure, gravitational waves, non-Newtonian Weyl curvature, etc.
* Classification: If the
Lie algebra is generated by Killing vector fields ξi
(i = 1, 2, 3), with commutators [ξi,
ξj]
= Cijk
ξk, choose a triad of left-invariant
vector fields Li such that
Cijk = εijl nlk + δjk vi − δik vj ,
where nij
= diag(n1,
n2,
n3),
ni
= ±1, 0, and vi
= (0,1; 0,1; 0,1), with vi
nij
= 0; Types are classified by the combinations of values
of ni
and vi
(class A has vi = 0,
class B has vi ≠ 0);
The dual forms χi
are also left-invariant.
* Killing vector fields:
They generate the isometries, so they are right-invariant and satisfy
[ξi, ξj]a = Ckij ξka , [ξ, η]a = Cabc ξb ηc .
* Left-invariant forms: There are three
1-forms \(\sigma^i{}_a\), such that qab
= gij
σia
σ jb
and ∇[a
σib]
= −\(1\over2\)Ccab
σic.
* Metric: It is often parametrized in
an ADM-like form by a volume element and a shear matrix β, such that
ds2 = −N2(t) dt2 + gij(t) σiσj , gij = exp{−2Ω(t)} (exp{2β(t)})ij ;
the simplest case is for types I, II, VIII, IX, in which g or β are diagonalizable; With some loss of generality,
gij = e−2Ω(t) diag(a(t), b(t), 1/ab) , βij = diag(β+ + 31/2 β−, β+ − 31/2 β−, − 2β+) .
Special Cases and Models
> s.a. bianchi I, bianchi IX, and
other models; scalar-tensor theories.
@ Class / type A:
Schücking et al GRG(03) [diagram];
Apostolopoulos CQG(03)gq [self-similar];
Ferragut et al JGP(12) [with k = 1, analytic integrability].
@ Class / type B: Apostolopoulos CQG(05)gq/04 [self-similar];
Yamamoto PRD(12)-a1108 [with electromagnetic fields].
@ Compact: Koike et al JMP(94) [topologies];
Tanimoto et al JMP(97)gq/96,
JMP(97)gq;
Coley & Goliath PRD(00)gq [fluid + scalar];
Kodama PTP(02)gq/01 [fluid];
Torre CQG(04)gq [weakly locally homogeneous].
Specific models: see bianchi I models;
bianchi II-V models; bianchi VI-VIII models;
bianchi IX models.
References
> s.a. geodesics; minisuperspace quantum
cosmology; neural networks; quasilocal energy;
types of singularities.
@ Early references: Bianchi MMSI(1897),
translation GRG(01);
Bianchi 18;
Misner PR(69) [ADM].
@ Reviews: in Misner et al 73;
MacCallum in(73)-a2001;
Ryan & Shepley 75;
MacCallum in(79);
Jantzen in(84);
Ellis GRG(06).
@ EMW variables: Ellis & MacCallum CMP(69);
MacCallum & Ellis CMP(70);
MacCallum CMP(71);
Wainwright in(88);
Wainwright & Hsu CQG(89);
Wainwright & Ma in(92);
Hewitt & Wainwright CQG(93).
@ Ashtekar variables: Kodama PTP(88);
Ashtekar & Pullin in(90);
Ashtekar & Samuel CQG(91);
Manojlović & Miković CQG(93);
Obregón et al PRD(93)gq;
González & Tate CQG(95)gq/94 [I and II];
Calzetta & Thibeault gq/97/CQG [I, II, IX].
@ Symmetries and reduction: Coussaert & Henneaux CQG(93)gq;
Schirmer CQG(95);
Capozziello et al IJMPD(97) [Noether symmetries and general relativity];
Christodoulakis et al CMP(02)gq/01 [invariant characterization];
Apostolopoulos CQG(05) [new approach].
@ Hamiltonian / Lagrangian:
Ryan & Waller gq/97 [class B];
Tanimoto et al JMP(97)gq [compact];
Pons & Shepley PRD(98)gq.
@ Dynamics, general: Bonilla CQG(98);
Byland & Scialom PRD(98)gq [I, II, Kantowski-Sachs];
Billyard PhD(99)gq;
Szydłowski & Krawiec IJMPA(00) [constraint solved];
Aguiar & Crawford PRD(00)gq [I + III, dust and Λ];
Barrow & Hervik CQG(02)gq [Weyl curvature invariant];
Gambini & Pullin CQG(03) [discretized];
Fay gq/05-in [+ massive scalar, ADM];
Goheer et al CQG(07)-a0710 [in f(R) theories].
@ Stability:
Barrow & Sonoda PRP(86) [several V, VI, VII models];
Zotov gq/99.
@ Isotropization:
Cervantes-Cota & Chauvet PRD(99)gq/98 [I-V-IX, induced gravity];
Fay CQG(03) [class A],
gq/05-in [scalar-tensor];
Barzegar et al a1904 [massless Einstein-Vlasov system];
> s.a. types I, VII, IX.
@ In Einstein-Yang-Mills theory: Donets et al PRD(99) [N = 2 supersymmetric].
@ In higher-order theories:
Querella PhD(98)gq/99;
Barrow & Hervik PRD(06)gq [I + II, quadratic theories];
Middleton CQG(10) [I, VIII, IX, anisotropy and approach to singularity].
@ Other theories: Savaria gq/97 [non-symmetric gravity];
Bonneau CQG(98) [Einstein-Weyl spaces];
Cognola & Zerbini IJTP(08)-a0802 [generalized gravity theories];
> s.a. modified electrodynamics; non-commutative physics.
@ Kinds of matter: Rendall & Uggla CQG(00)gq [Einstein-Vlasov];
Calogero & Heinzle PhyD(11)-a0911 [anisotropic matter];
Normann et al CQG(18)-a1712 [p-form gauge fields];
Thorsrud CQG(19)-a1905 [free massless scalar field].
@ Phenomenology:
Coley & Lim CQG(07) [bounds on shear];
> s.a. cmb anisotropy.
@ Related topics: Estabrook et al JMP(68);
Siklos CMP(78) [horizons and whimper singularities];
Rosquist & Jantzen PRP(88);
Fujiwara et al CQG(93)gq;
Di Pietro & Demaret IJMPD(99)gq [duality];
De Ritis et al NCB(01) [horizons];
Doliwa et al JGP(04) [discretization];
Hervik ap/05,
Palle ap/05
[re vorticity, > s.a. cmb];
Heinzle & Uggla CQG(10) [monotonic functions];
Avetisyan & Verch CQG(13)-a1212 [harmonic and spectral analysis].
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