4.3 Observer Sky
57
˜ ˆ
ω = ˆ
(ϑ) ˆ
ω .
(4.3.6)
The Jacobian matrix of the {ϑ, ϕ, ˆ
ω} coordinate transformation is therefore not in
block-diagonal form and the partial derivatives on S spec transform as follows
˜
∂ ˆ
A =
∂ϑ
ˆ
B
∂ ˜
ϑ
ˆ
A
∂ ˆ
B + ˆ
−1
(ϑ) ˆ
0
i ∂ ˆ
B ˆ
n
i
(ϑ) ˆ
ω∂ ˆ
ω
,
˜
∂ ˆ
ω = ˆ
−1
(ϑ) ∂ ˆ
ω . (4.3.7)
The fact that the transformation of ∂ ˆ
ω does not depend on ∂ ˆ
A means that fields on
S spec with no ˆ
ω dependence have that property for all observers. Thus, if one is not
interested in spectral distributions on the sky f (ϑ, ˆ
ω), but rather fields on S alone
f (ϑ), then one simply sets ∂ ˆ
ω f = 0 and this condition is conserved under LLTs.
Now given this {ϑ, ϕ, ˆ
ω} parametrization and Eq. (4.1.1), the LLT-invariant lineelement on S spec is
d
2 l S spec := η ab d ˆ
k
a d ˆ
k
b
= η ab
∂ ˆ
A
ˆ
k
a
∂ ˆ
B
ˆ
k
b dϑ
ˆ
A dϑ
ˆ
B
+ 2∂ ˆ
ω
ˆ
k
a
∂ ˆ
A
ˆ
k
b d ˆ
ω dϑ
ˆ
A
+ ∂ ˆ
ω
ˆ
k
a
∂ ˆ
ω
ˆ
k
b d ˆ
ω
2
= η ab ∂ ˆ
A
ˆ
k
a
∂ ˆ
B
ˆ
k
b dϑ
ˆ
A dϑ
ˆ
B
= ˆ
ω
2
∂ ˆ
A ˆ
n
i
∂ ˆ
B ˆ
n
i dϑ
ˆ
A dϑ
ˆ
B
= ˆ
ω
2 dl
2
S ,
(4.3.8)
where
dl
2
S := dϑ
2
+ sin
2
ϑ dϕ
2
,
(4.3.9)
is the line-element of the observer sky S. In particular, we can express it in terms of
a metric
d
2 l S ≡ S ˆ
A ˆ
B (ϑ) dϑ
ˆ
A dϑ
ˆ
B
,
S ˆ
A ˆ
B (ϑ) := ∂ ˆ
A ˆ
n
i
∂ ˆ
B ˆ
n
i
=
1 0
0 sin
2
ϑ
ˆ
A ˆ
B
,
(4.3.10)
which is the one that is used in actual observations. The fact that there is no physical distance associated to a d ˆ
ω displacement in (4.3.8) means that, geometrically
speaking, S spec is a stack of superimposed regular spheres S parametrized by ˆ
ω.
Under LLTs, the invariance of d
2 l S spec , along with Eqs. (4.3.6) and (4.3.8), implies
the following transformation for the metric
˜
S ˆ
A ˆ
B ( ˜
ϑ) = ˆ
−2
(ϑ)
∂ϑ
ˆ
C
∂ ˜
ϑ
ˆ
A
( ˜
ϑ(ϑ))
∂ϑ
ˆ
D
∂ ˜
ϑ
ˆ
B
( ˜
ϑ(ϑ)) S ˆ
C ˆ
D (ϑ) .
(4.3.11)
Thus, one needs to work with the full S spec in order to interpret the effect of LLTs as
a coordinate transformation, given by (4.3.5) and (4.3.6). If instead one restricts to
the subspace S, then LLTs induce both a coordinate and a conformal transformation
57
˜ ˆ
ω = ˆ
(ϑ) ˆ
ω .
(4.3.6)
The Jacobian matrix of the {ϑ, ϕ, ˆ
ω} coordinate transformation is therefore not in
block-diagonal form and the partial derivatives on S spec transform as follows
˜
∂ ˆ
A =
∂ϑ
ˆ
B
∂ ˜
ϑ
ˆ
A
∂ ˆ
B + ˆ
−1
(ϑ) ˆ
0
i ∂ ˆ
B ˆ
n
i
(ϑ) ˆ
ω∂ ˆ
ω
,
˜
∂ ˆ
ω = ˆ
−1
(ϑ) ∂ ˆ
ω . (4.3.7)
The fact that the transformation of ∂ ˆ
ω does not depend on ∂ ˆ
A means that fields on
S spec with no ˆ
ω dependence have that property for all observers. Thus, if one is not
interested in spectral distributions on the sky f (ϑ, ˆ
ω), but rather fields on S alone
f (ϑ), then one simply sets ∂ ˆ
ω f = 0 and this condition is conserved under LLTs.
Now given this {ϑ, ϕ, ˆ
ω} parametrization and Eq. (4.1.1), the LLT-invariant lineelement on S spec is
d
2 l S spec := η ab d ˆ
k
a d ˆ
k
b
= η ab
∂ ˆ
A
ˆ
k
a
∂ ˆ
B
ˆ
k
b dϑ
ˆ
A dϑ
ˆ
B
+ 2∂ ˆ
ω
ˆ
k
a
∂ ˆ
A
ˆ
k
b d ˆ
ω dϑ
ˆ
A
+ ∂ ˆ
ω
ˆ
k
a
∂ ˆ
ω
ˆ
k
b d ˆ
ω
2
= η ab ∂ ˆ
A
ˆ
k
a
∂ ˆ
B
ˆ
k
b dϑ
ˆ
A dϑ
ˆ
B
= ˆ
ω
2
∂ ˆ
A ˆ
n
i
∂ ˆ
B ˆ
n
i dϑ
ˆ
A dϑ
ˆ
B
= ˆ
ω
2 dl
2
S ,
(4.3.8)
where
dl
2
S := dϑ
2
+ sin
2
ϑ dϕ
2
,
(4.3.9)
is the line-element of the observer sky S. In particular, we can express it in terms of
a metric
d
2 l S ≡ S ˆ
A ˆ
B (ϑ) dϑ
ˆ
A dϑ
ˆ
B
,
S ˆ
A ˆ
B (ϑ) := ∂ ˆ
A ˆ
n
i
∂ ˆ
B ˆ
n
i
=
1 0
0 sin
2
ϑ
ˆ
A ˆ
B
,
(4.3.10)
which is the one that is used in actual observations. The fact that there is no physical distance associated to a d ˆ
ω displacement in (4.3.8) means that, geometrically
speaking, S spec is a stack of superimposed regular spheres S parametrized by ˆ
ω.
Under LLTs, the invariance of d
2 l S spec , along with Eqs. (4.3.6) and (4.3.8), implies
the following transformation for the metric
˜
S ˆ
A ˆ
B ( ˜
ϑ) = ˆ
−2
(ϑ)
∂ϑ
ˆ
C
∂ ˜
ϑ
ˆ
A
( ˜
ϑ(ϑ))
∂ϑ
ˆ
D
∂ ˜
ϑ
ˆ
B
( ˜
ϑ(ϑ)) S ˆ
C ˆ
D (ϑ) .
(4.3.11)
Thus, one needs to work with the full S spec in order to interpret the effect of LLTs as
a coordinate transformation, given by (4.3.5) and (4.3.6). If instead one restricts to
the subspace S, then LLTs induce both a coordinate and a conformal transformation