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B.1 Homogeneous models

In Bianchi models the transitive symmetry group acts freely on Σ, which implies that Σ can locally be identified with the group manifold S. The three generators of ℒS will be denoted as TI, 1 ≤ I ≤ 3, with relations [T ,T ] = CK T I J IJ K, where CK IJ are the structure constants of ℒS fulfilling J C IJ = 0 for class A models by definition. The Maurer–Cartan form on S is given by I θMC = ω TI with left-invariant 1-forms I ω on S, which fulfill the Maurer–Cartan equations
d ωI = − 1CI ωJ ∧ ωK . (86 ) 2 JK
Due to F = {1 }, all homomorphisms λ : F → G are given by 1 ↦→ 1, and we can use the embedding ι = id: S∕F `→ S. An invariant connection then takes the form A = φ&tidle;∘ θMC = &tidle;φiIτiωI = Aiaτidxa with matrices τi generating ℒSU (2). The scalar field is given by φ&tidle;: ℒS → ℒG, TI ↦→ &tidle;φ(TI ) =: &tidle;φiτi I already in its final form, because condition (84View Equation) is empty for a trivial isotropy group.

Using left-invariant vector fields XI obeying ωI (XJ ) = δI J and with Lie brackets [XI ,XJ ] = CK XK IJ the momenta canonically conjugate to Ai = &tidle;φiωI a I a can be written as Ea = √g--&tidle;pIXa i 0 i I with &tidle;pI i being canonically conjugate to i &tidle;φI. Here, I 2 g0 = det(ω a) is the determinant of the left-invariant metric ∑ (g0)ab := I ωIaωIb on Σ, which is used to provide the density weight of Eai. The symplectic structure can be derived from

∫ ∫ --1--- 3 ˙i a --1--- 3 √ --&tidle;˙i J I --V0--˙&tidle;i I 8πγG Σ d xA aE i = 8πγG Σ d x g0φIp&tidle;i ω (XJ ) = 8πγG φI&tidle;pi ,

to obtain

{&tidle;φi, &tidle;pJ} = 8πγGV0 δiδJ (87 ) I j j I
with the volume ∫ √ -- V0 := Σd3x g0 of Σ measured with the invariant metric g0.

It is convenient to absorb the coordinate volume V 0 into the fields by redefining φi := V 1∕3φ&tidle;i I 0 I and I 2∕3 I pi := V0 &tidle;pi. This makes the symplectic structure independent of V0 in accordance with background independence. These redefined variables automatically appear in holonomies and fluxes through coordinate integrations.

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