68
4 Observer Space-Time Formalism
˜
γ
μ
( ˜
ζ, ˜
ϑ) = γ
μ
(ζ, ϑ) ,
(4.6.1)
˜
n
i
( ˜
ζ, ˜
ϑ) =
−1
i
j n
j
−
i
0
(ζ, ϑ) ,
(4.6.2)
˜
n A ( ˜
ζ, ˜
ϑ) = R
B
A (ϑ)
n B −
0
i − ˆ
0
i
n
i
B
(ζ, ϑ) ,
(4.6.3)
˜
n
i
A ( ˜
ζ, ˜
ϑ) = R
B
A (ϑ) (( ⊥ )
i
j (ζ, ϑ) n
j
B (ζ, ϑ) ,
(4.6.4)
In the ALD-compensated case, which is expressed in terms of variations, we must
be careful because each side is evaluated at different angles, thus bringing two possibilities, as in the case of the MD symmetry for instance (see Appendix 6.2). In the
case of the ζ parameter, this is a genuine coordinate system on the manifold L, so
an active diffeomorphism corresponds to a variation at fixed ζ, as we have implicitly
done so far. The ϑ
ˆ
A parameters, however, are rigidly related to the boundary values
of the n
i field, i.e. we have ˜ ˆ
n
i
( ˜
ϑ) = ˆ
n
i
( ˜
ϑ), instead of a scalar field transformation
˜ ˆ
n
i
( ˜
ϑ) = ˆ
n
i
(ϑ). It therefore makes no sense to compute the variation at fixed ϑ
ˆ
A ,
since we are precisely looking at how n
i varies. We must thus use
δ θ X (ζ, ϑ) := ˜
X (ζ, ˜
ϑ) − X (ζ, ϑ) ,
not
δ θ X (ζ, ϑ) := ˜
X (ζ, ϑ) − X (ζ, ϑ) ,
(4.6.5)
since the latter would give zero for n
i at ζ = 0. With this definition of δ θ we thus get
δ θ γ
μ
= −κ ∂ ζ γ
μ
+ O(θ
2
) ,
(4.6.6)
δ θ n
i
= −κ ∂ ζ n
i
+ n
i j
θ
0 j
− θ
i j n
j
+ O(θ
2
) ,
(4.6.7)
δ θ n A = −κ ∂ ζ n A − α ε AB n B + n
i
θ
0i n A +
θ
0i
− ˆ
θ
0i
n
i
A + O(θ
2
) , (4.6.8)
δ θ n
i
A = −κ ∂ ζ n
i
A − α ε AB n
i
B −
n
i
θ
0 j
+ θ
i j
n
j
A + O(θ
2
) .
(4.6.9)
4.7 Observables from Localized Sources
We can now consider the cosmological observables that are associated to localized
sources. These are typically frequency-independent and are therefore defined on the
observer space C.
4.7.1 Distances and Weak Lensing
We start by noting that ∂ ˆ
A γ
μ
(ζ, ϑ) is the linear map relating the difference in spacetime position between two infinitesimally close geodesics at (ζ, ϑ) to the corresponding observed angular deviation dϑ
ˆ
A
d ϑ γ
μ
:= dγ
μ
| dζ=0 ≡ ∂ ˆ
A γ
μ dϑ
ˆ
A
.
(4.7.1)
4 Observer Space-Time Formalism
˜
γ
μ
( ˜
ζ, ˜
ϑ) = γ
μ
(ζ, ϑ) ,
(4.6.1)
˜
n
i
( ˜
ζ, ˜
ϑ) =
−1
i
j n
j
−
i
0
(ζ, ϑ) ,
(4.6.2)
˜
n A ( ˜
ζ, ˜
ϑ) = R
B
A (ϑ)
n B −
0
i − ˆ
0
i
n
i
B
(ζ, ϑ) ,
(4.6.3)
˜
n
i
A ( ˜
ζ, ˜
ϑ) = R
B
A (ϑ) (( ⊥ )
i
j (ζ, ϑ) n
j
B (ζ, ϑ) ,
(4.6.4)
In the ALD-compensated case, which is expressed in terms of variations, we must
be careful because each side is evaluated at different angles, thus bringing two possibilities, as in the case of the MD symmetry for instance (see Appendix 6.2). In the
case of the ζ parameter, this is a genuine coordinate system on the manifold L, so
an active diffeomorphism corresponds to a variation at fixed ζ, as we have implicitly
done so far. The ϑ
ˆ
A parameters, however, are rigidly related to the boundary values
of the n
i field, i.e. we have ˜ ˆ
n
i
( ˜
ϑ) = ˆ
n
i
( ˜
ϑ), instead of a scalar field transformation
˜ ˆ
n
i
( ˜
ϑ) = ˆ
n
i
(ϑ). It therefore makes no sense to compute the variation at fixed ϑ
ˆ
A ,
since we are precisely looking at how n
i varies. We must thus use
δ θ X (ζ, ϑ) := ˜
X (ζ, ˜
ϑ) − X (ζ, ϑ) ,
not
δ θ X (ζ, ϑ) := ˜
X (ζ, ϑ) − X (ζ, ϑ) ,
(4.6.5)
since the latter would give zero for n
i at ζ = 0. With this definition of δ θ we thus get
δ θ γ
μ
= −κ ∂ ζ γ
μ
+ O(θ
2
) ,
(4.6.6)
δ θ n
i
= −κ ∂ ζ n
i
+ n
i j
θ
0 j
− θ
i j n
j
+ O(θ
2
) ,
(4.6.7)
δ θ n A = −κ ∂ ζ n A − α ε AB n B + n
i
θ
0i n A +
θ
0i
− ˆ
θ
0i
n
i
A + O(θ
2
) , (4.6.8)
δ θ n
i
A = −κ ∂ ζ n
i
A − α ε AB n
i
B −
n
i
θ
0 j
+ θ
i j
n
j
A + O(θ
2
) .
(4.6.9)
4.7 Observables from Localized Sources
We can now consider the cosmological observables that are associated to localized
sources. These are typically frequency-independent and are therefore defined on the
observer space C.
4.7.1 Distances and Weak Lensing
We start by noting that ∂ ˆ
A γ
μ
(ζ, ϑ) is the linear map relating the difference in spacetime position between two infinitesimally close geodesics at (ζ, ϑ) to the corresponding observed angular deviation dϑ
ˆ
A
d ϑ γ
μ
:= dγ
μ
| dζ=0 ≡ ∂ ˆ
A γ
μ dϑ
ˆ
A
.
(4.7.1)