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B.2 Isotropic models

On Bianchi models, additional symmetries can be imposed, which corresponds to a further symmetry reduction and introduces non-trivial isotropy subgroups. These models with enhanced symmetry can be treated on an equal footing by writing the symmetry group as a semi-direct product S = N ⋊ F ρ, with the isotropy subgroup F and the translational subgroup N, which is one of the Bianchi groups. Composition in this group is defined as (n1,f1)(n2,f2) := (n1ρ (f1)(n2),f1f2), which depends on the group homomorphism ρ: F → AutN into the automorphism group of N (which will be denoted by the same letter as the representation on AutℒN used below). Inverse elements are given by (n,f )−1 = (ρ(f−1)(n− 1),f −1). To determine the form of the invariant connections we have to compute the Maurer–Cartan form on S (using the usual notation):
(S) θMC (n,f) = (n,f )−1d(n,f) = (ρ(f −1)(n−1),f− 1)(dn, df) = (ρ(f− 1)(n −1)ρ(f−1)(dn),f −1df) = (ρ(f −1)(n −1dn ),f −1df ) ( ) = ρ (f −1)(θ(NMC)(n)),θ(FMC)(f ) . (88 )
Here the Maurer–Cartan forms θ(MNC) on N and θ(FMC) on F appear. We then choose an embedding ι: S∕F = N `→ S, which can most easily be done as ι: n ↦→ (n,1). Thus, ∗ (S ) (N ) ι θMC = θMC, and a reconstructed connection takes the form (S) &tidle;φ ∘ ι∗θMC = &tidle;φiIωIτi, which is the same as for anisotropic models before (where now ωI are left-invariant 1-forms on the translation group N). However, here φ&tidle; is constrained by Equation (84View Equation) and we get only a subset as isotropic connections.

To solve Equation (84View Equation) we have to treat LRS (locally rotationally symmetric) models with a single rotational symmetry and isotropic models separately. In the first case we choose ℒF = ⟨τ3⟩, whereas in the second case we have ℒF = ⟨τ1, τ2,τ3⟩ (⟨⋅⟩ denotes the linear span). Equation (84View Equation) can be written infinitesimally as

φ&tidle;(adτi(TI)) = add λ(τi)&tidle;φ(TI ) = [dλ (τi), &tidle;φ(TI)]

(i = 3 for LRS, 1 ≤ i ≤ 3 for isotropy). The TI are generators of ℒN = ℒF ⊥, on which the isotropy subgroup F acts by rotation, ad τ(TI) = εiIK TK i. This is the derivative of the representation ρ defining the semi-direct product S: conjugation on the left-hand side of (84View Equation) is −1 Ad(1,f)(n,1) = (1,f )(n, 1)(1,f ) = (ρ(f)(n),1), which follows from the composition in S.

Next, we have to determine the possible conjugacy classes of homomorphisms λ : F → G. For LRS models their representatives are given by

λ : U (1) → SU (2), exp tτ ↦→ exp ktτ k 3 3

for k ∈ = {0, 1,...} 0 (as will be shown in detail below for spherically-symmetric connections). For the components &tidle;i φI of &tidle; φ defined by &tidle; &tidle;i φ (TI ) = φIτi, Equation (84View Equation) takes the form &tidle;j &tidle;l ε3IK φK = kε3ljφI. This has a non-trivial solution only for k = 1, in which case &tidle;φ can be written as

&tidle;φ1 = &tidle;aτ1 + &tidle;bτ2, &tidle;φ2 = − &tidle;bτ1 + &tidle;aτ2 , φ&tidle;3 = c&tidle;τ3

with arbitrary numbers &tidle;a, &tidle;b, &tidle;c (the factors of 1 2− 2 are introduced for the sake of normalization). Their conjugate momenta take the form

&tidle;p1 = 1(&tidle;paτ1 + &tidle;pbτ2), &tidle;p2 = 1(− &tidle;pbτ1 + &tidle;paτ2), &tidle;p3 = &tidle;pcτ3, 2 2

and the symplectic structure is given by

{&tidle;a, &tidle;p } = {&tidle;b,p&tidle;} = {&tidle;c, &tidle;p } = 8πγGV a b c 0

and vanishes in all other cases. There is remaining gauge freedom from the reduced structure group Z ∼= U (1 ) λ, which rotates the pairs (&tidle;a,&tidle;b) and (&tidle;p , &tidle;p ) a b. Then only ∘ ------- &tidle;a2 + &tidle;b2 and its momentum ∘ ------- (&tidle;a&tidle;pa + &tidle;b&tidle;pb)∕ &tidle;a2 + &tidle;b2 are gauge invariant.

In the case of isotropic models we have only two homomorphisms λ0 : SU (2) → SU (2),f ↦→ 1 and λ1 = id up to conjugation (to simplify notation we use the same letters for the homomorphisms as in the LRS case, which is justified by the fact that the LRS homomorphisms are restrictions of those appearing here). Equation (84View Equation) takes the form &tidle;j εiIK φK = 0 for λ0 without non-trivial solutions, and j εiIKφ&tidle;K = εilj&tidle;φlI for λ1. Each of the last equations has the same form as for LRS models with k = 1, and their solution is &tidle;φiI = &tidle;cδiI with an arbitrary &tidle;c. In this case the conjugate momenta can be written as &tidle;pI = &tidle;pδI i i and we have the symplectic structure {&tidle;c,p&tidle;} = 8πG γV 3 0.

Thus, in both cases there is a unique non-trivial sector, and no topological charge appears. The symplectic structure can again be made independent of V0 by redefining 1∕3 a := V 0 &tidle;a, 1∕3 b := V0 &tidle;b, c := V01∕3&tidle;c and pa := V02∕3p&tidle;a, pb := V20∕3&tidle;pb, pc := V02∕3&tidle;pc, p := V20∕3&tidle;p. If one computes the isotropic reduction of a Bianchi IX metric following from the left-invariant 1-forms of SU(2), one obtains a closed Friedmann–Robertson–Walker metric with scale factor ∘ --- a = 2&tidle;a = 2 |&tidle;p| (see [44] for the calculation). Thus, we obtain identification (19View Equation) used in isotropic loop cosmology. (Such a normalization can only be obtained in curved models.)

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