Bianchi
Type IX Models| Bianchi
Type IX Models |
In General > s.a. gravitational
instanton; minisuperspace quantum
cosmology.
$ Def: nAB = (+1, +1, +1), vA =
0; G is (the simply
connected covering group of) SO(3).
* History: The Russian
school (BKL) started studying Bianchi IX in 1962, hoping to understand the
behavior of the metric near a generic singularity
(introduced map for u); Misner started in 1966, but using
earlier work on Taub-NUT, hoping to understand anisotropy dissipation
(introduced
Hamiltonian and potential).
* Special cases: l1 = l2 = l3 =
R,
FRW models; Kasner solution; Taub-NUT solution; Diagonal models, the metric
has gab =
diag(l1, l2, l3),
with
li functions of time; The
diagonal vacuum model is also called Mixmaster universe.
* Geometry: 3V = 16
2 l1 l2 l3,
1 = cos
d
+
sin sin d
, 2 = sin d – cos sin d , 3 = d +
cos d .
* Evolution: Well approximated by a sequence of Bianchi I (Kasner) epochs; Each one is characterized by the value of a parameter u, which gives rise to an approximate discrete dynamics, the Gauss map
un+1 = (un – [un])–1 ;
In the Hamiltonian approach, each epoch is the time between two bounces off
the potential; At each bounce, two scale factors switch between expansion and
contraction,
while the third one keeps contracting; An era is a set of epochs with
the same two factors switching behavior, i.e., bouncing off the same
pair
of walls.
* Better approximation: & Garfinkle.
References > s.a. types of singularities;
Taub-NUT [early work].
@ General: Montani et al IJMPA(08)-a0712 [classical
and quantum, review].
@ ADM approach: Misner ApJ(68), PR(69), PRL(69);
in Misner et al 73; in Ryan & Shepley
75;
Imponente & Montani gq/02-in,
IJMPD(02).
@ BKL approach: Belinskii et al AiP(70), JETP(71), AiP(82); Manojlovic & Mikovic
JMP(00)mp [Painlevé III]; JKPS(03)gq/02.
@ Other approach: Creighton & Hobill in(94) [Ellis-MacCallum-Wainwright];
Gogilidze et al
G&C(97)
[Hamiltonian, non-diagonal].
@ Other dynamics: Llibre & Valls JMP(05), JMP(06)
[Darboux first integrals]; Buzzi et al JPA(07); > s.a.
chaos in bianchi models.
@ Self-dual: Tod PLA(94);
Chakravarty & Ablowitz PRL(96); Maszczyk CQG(96).
@ Self-similar: Apostolopoulos & Tsamparlis GRG(03)gq.
@ Isotropization: Guzman IJTP(96);
Bergamini et al PRD(97)gq/96 [inflation];
Cervantes-Cota & Chauvet PRD(99)gq/98 [induced
gravity]; Kirillov & Montani
PRD(02)gq [and
inflation]; Battisti et al a0903-in [semiclassical mechanism].
@ With matter:
Waller PRD(84)
[electromagnetism];
Banerjee et al ASS(90)
[viscous fluid]; Toporensky & Ustiansky gq/99,
Fay
& Lehner GRG(05)gq [massive
scalar].
@ In Horava-Lifshitz gravity: Myung et al a0911 [chaotic
and non-chaotic
solutions]; Bakas et al a0911 [and chaos].
@ Other theories: Belinskii et al PLB(78),
in Cotsakis 90 [Euclidean];
Barrow & Dabrowski
PRD(98)ht/97 [low-energy
string theory]; Garcia de Andrade & Monerat ap/01/C&G
[with torsion]; Halpern GRG(03)gq/02 [5D];
van den Hoogen et al PRD(03)gq/02 [brane];
Bergshoeff
et al CQG(03)ht [supergravity];
Maceda et al PRD(08)
[non-commutative]; > s.a. modified
uncertainty
relations.
@ Related topics: Chitre PRD(72);
Hu PRD(73);
Lin & Wald PRD(90)
[recollapse];
King PRD(91);
in Misner in(94) [as geodesic motion]; Berger et
al CQG(97)gq/96,
gq/97-in [other
algorithms]; Cotsakis et al PRD(98)gq/97 [adiabatic
invariants and catastrophes];
Barguine
et al PRD(01)
[with cosmological constant,
homoclinic structure]; Battisti & Montani a0903-in [gup approach].
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send feedback and suggestions to bombelli at olemiss.edu – modified 16
nov
2009