4.2 Redshift Parametrization
55
Simplifying the derivative term
k
b k
c
a
0
d
b ∂ c da = −k
b k
c
d
b da ∂ c
a
0 ≡ −k a k
c ∂ c
a
0
(3.3.22)
= −k a ∂
a
0
= −∂
a
0 k a
+
a
0 ∂ k a
(3.3.23)
= ∂
ω
−
a
0 abc k
b k
c (4.2.12)
and setting again ζ = λ, one gets
∂ ˜
λ
∂λ
= ∂ λ log
e
λ
,
(4.2.13)
which can be solved with the boundary condition ˆ ˜
λ = 0 to yield
˜
λ(λ) = λ + log
ˆ
,
(4.2.14)
and is indeed the transformation of ζ in Eq. (4.1.7). In the active compensation case,
one can look at the variation of ζ in Eq. (4.1.13) to get the generating vector κ to
linear order
κ = θ
0i n
i
− ˆ
θ
0i
ˆ
n
i
+ O(θ
2
) .
(4.2.15)
For the remaining variables, under a PLD-compensated LLT we have
˜
γ
μ
( ˜
ζ) = γ
μ
(ζ) ,
˜
n
i
( ˜
ζ) =
−1
i
j n
j
−
i
0
(ζ) ,
(4.2.16)
since these quantities are LD scalars. Under an ALD-compensated LLT, we have
δ θ γ
μ
= −κ ∂ ζ γ
μ
+ O(θ
2
) ,
δ θ n
i
= −κ ∂ ζ n
i
+ n
i j
θ
0 j
− θ
i j n
j
+ O(θ
2
) .
(4.2.17)
Now whether we use a PLD or a ALD compensation is irrelevant at the level of
the resulting equations of motion, they will be invariant under the combined transformation either way by construction. However, at the level of the cosmological
observables, which involve integrals over L, it is important that we use a PLD, otherwise we will have extra terms due to the boundary of the integral (see the end of
Appendix 6.2). Finally, now that we have fixed , it is more convenient to use ∇ ζ
instead of ∇ (see (3.3.24)), which therefore reads
∇ ζ X
a
≡ ∂ ζ X
a
+
a
bμ ∂ ζ γ
μ X
b
= ∂ ζ X
a
+
a
b
0
X
b
,
(4.2.18)
and we have used (4.2.8) in the last step. In particular, the second equation of (4.2.10)
reads ∇ ζ k
a
= 0.
55
Simplifying the derivative term
k
b k
c
a
0
d
b ∂ c da = −k
b k
c
d
b da ∂ c
a
0 ≡ −k a k
c ∂ c
a
0
(3.3.22)
= −k a ∂
a
0
= −∂
a
0 k a
+
a
0 ∂ k a
(3.3.23)
= ∂
ω
−
a
0 abc k
b k
c (4.2.12)
and setting again ζ = λ, one gets
∂ ˜
λ
∂λ
= ∂ λ log
e
λ
,
(4.2.13)
which can be solved with the boundary condition ˆ ˜
λ = 0 to yield
˜
λ(λ) = λ + log
ˆ
,
(4.2.14)
and is indeed the transformation of ζ in Eq. (4.1.7). In the active compensation case,
one can look at the variation of ζ in Eq. (4.1.13) to get the generating vector κ to
linear order
κ = θ
0i n
i
− ˆ
θ
0i
ˆ
n
i
+ O(θ
2
) .
(4.2.15)
For the remaining variables, under a PLD-compensated LLT we have
˜
γ
μ
( ˜
ζ) = γ
μ
(ζ) ,
˜
n
i
( ˜
ζ) =
−1
i
j n
j
−
i
0
(ζ) ,
(4.2.16)
since these quantities are LD scalars. Under an ALD-compensated LLT, we have
δ θ γ
μ
= −κ ∂ ζ γ
μ
+ O(θ
2
) ,
δ θ n
i
= −κ ∂ ζ n
i
+ n
i j
θ
0 j
− θ
i j n
j
+ O(θ
2
) .
(4.2.17)
Now whether we use a PLD or a ALD compensation is irrelevant at the level of
the resulting equations of motion, they will be invariant under the combined transformation either way by construction. However, at the level of the cosmological
observables, which involve integrals over L, it is important that we use a PLD, otherwise we will have extra terms due to the boundary of the integral (see the end of
Appendix 6.2). Finally, now that we have fixed , it is more convenient to use ∇ ζ
instead of ∇ (see (3.3.24)), which therefore reads
∇ ζ X
a
≡ ∂ ζ X
a
+
a
bμ ∂ ζ γ
μ X
b
= ∂ ζ X
a
+
a
b
0
X
b
,
(4.2.18)
and we have used (4.2.8) in the last step. In particular, the second equation of (4.2.10)
reads ∇ ζ k
a
= 0.