4.1 Fundamental Observables
53
and from now on it is understood that space-time fields, such as
a
b (x), are implicitly
evaluated on the light-like geodesic γ. Another combination that will appear often is
(( ⊥ )
i
j :=
i
j +
−1
i
k n
k
−
i
0
0
j ,
(4.1.10)
which maps vectors that are normal to n
i to vectors that are normal to the new one ˜
n
i
given in (4.1.8). For any two vectors X
i and Y
i that are normal to n
i , we then have
the identity
(( ⊥ )
i
j X
j
(( ⊥ )
i
k Y
k
≡ X
i Y
i
,
(4.1.11)
i.e. ⊥ preserves the Euclidean norm, because the Lorentz contraction effect is
absorbed in the transformation of n
i . It will also be convenient to have the corresponding variations in terms of the generators
δ θ ω = ω
n
i
θ
0i
+ O(θ
2
)
,
(4.1.12)
δ θ ζ = n
i
θ
0i
− ˆ
n
i ˆ
θ
0i
+ O(θ
2
) ,
(4.1.13)
δ θ n
i
= n
i j
θ
0 j
− θ
i j n
j
+ O(θ
2
) ,
(4.1.14)
where
n
i j
:= δ
i j
− n
i n
j
,
n
i n
i j
≡ 0 ,
(4.1.15)
is the projector to the normal subspace to n
i . Note, in particular, that the transformation rules of ζ and n
i conform to the ones of a cosmological observable, i.e. these
quantities are MD-invariant and only depend on
a
b ( ˆ
P) and
a
b (P) under LLTs.
4.2 Redshift Parametrization
Let us now write down the evolution Eqs. (3.3.22) and (3.3.23) for the case of interest
q = 0 in terms of ζ and n
i
∂ λ γ
μ
= ω
e
μ
0 − n
i e
μ
i
,
(4.2.1)
∂ λ ζ = ω 0 ,
(4.2.2)
∂ λ n
i
= ωn
i j
j ,
(4.2.3)
and we will also use the notation
X := ω
−1 k
a X a ≡
e
μ
0 − n
i e
μ
i
X μ .
(4.2.4)
Given Eq. (4.2.2), it is convenient to consider λ as dimensionless, so that now has
dimensions of area, and we now fix the LD gauge by the following condition
53
and from now on it is understood that space-time fields, such as
a
b (x), are implicitly
evaluated on the light-like geodesic γ. Another combination that will appear often is
(( ⊥ )
i
j :=
i
j +
−1
i
k n
k
−
i
0
0
j ,
(4.1.10)
which maps vectors that are normal to n
i to vectors that are normal to the new one ˜
n
i
given in (4.1.8). For any two vectors X
i and Y
i that are normal to n
i , we then have
the identity
(( ⊥ )
i
j X
j
(( ⊥ )
i
k Y
k
≡ X
i Y
i
,
(4.1.11)
i.e. ⊥ preserves the Euclidean norm, because the Lorentz contraction effect is
absorbed in the transformation of n
i . It will also be convenient to have the corresponding variations in terms of the generators
δ θ ω = ω
n
i
θ
0i
+ O(θ
2
)
,
(4.1.12)
δ θ ζ = n
i
θ
0i
− ˆ
n
i ˆ
θ
0i
+ O(θ
2
) ,
(4.1.13)
δ θ n
i
= n
i j
θ
0 j
− θ
i j n
j
+ O(θ
2
) ,
(4.1.14)
where
n
i j
:= δ
i j
− n
i n
j
,
n
i n
i j
≡ 0 ,
(4.1.15)
is the projector to the normal subspace to n
i . Note, in particular, that the transformation rules of ζ and n
i conform to the ones of a cosmological observable, i.e. these
quantities are MD-invariant and only depend on
a
b ( ˆ
P) and
a
b (P) under LLTs.
4.2 Redshift Parametrization
Let us now write down the evolution Eqs. (3.3.22) and (3.3.23) for the case of interest
q = 0 in terms of ζ and n
i
∂ λ γ
μ
= ω
e
μ
0 − n
i e
μ
i
,
(4.2.1)
∂ λ ζ = ω 0 ,
(4.2.2)
∂ λ n
i
= ωn
i j
j ,
(4.2.3)
and we will also use the notation
X := ω
−1 k
a X a ≡
e
μ
0 − n
i e
μ
i
X μ .
(4.2.4)
Given Eq. (4.2.2), it is convenient to consider λ as dimensionless, so that now has
dimensions of area, and we now fix the LD gauge by the following condition