5.11 Cosmic Microwave Background Observables
125
where the 4-momenta here are null k a k
a
≡ 0. Let us next observe that, for a phase
space distribution evaluated at the observer position ˆ
P, the
k parameters play exactly
the same role as ˆ
ω and ˆ
n
i in the case of the geodesic associated with localized
sources, i.e. they parametrize the subspace of T ˆ
P M with respect to the spatial frame
ˆ
e i . We can therefore decompose the above
k-dependencies into the norm ˆ
ω := k and
direction ˆ
n :=
k/k and we subsequently express the latter using the standard angular
parameterization of Eq. (4.3.2). More generally, we can make use of the geometrical
machinery developed in Sect. 4 to describe the observer sky. First, note that what
is observed in practice is the two-point function of the complexified electric field
operator tangent to the observer sky
ˆ
I AB (ϑ, ˆ
ω) := ˆ
k
a
A
ˆ
k
b
B
ˆ
I 0a,0b (
k) ≡ ˆ
ω
2 ˆ
k
a
A
ˆ
k
b
B
ˆ
f ab (
k) ≡ ˆ
ω
2
ˆ
n
i
A (ϑ) ˆ
n
j
B (ϑ) ˆ
f i j (
k) , (5.11.5)
where we have used Eqs. (4.5.7), (4.5.25) and (5.8.7). Thus, we basically observe the
projection of the photon distribution f ab on the observed sky, up to a ˆ
ω
2 factor. From
the third expression in the series of equalities (5.11.5), we infer that the transformation
rule under LLTs of this hermitian 2 × 2 matrix is
˜ ˆ
I AB ( ˜
ϑ, ˜ ˆ
ω) = ˆ
2
(ϑ) R
C
A (ϑ) R
D
B (ϑ) ˆ
I C D (ϑ, ˆ
ω) ,
(5.11.6)
with R
AB being the LLT-compensating Sachs rotations (see again Sect. 4) and the ˆ
2
factor being due to the presence of the ˆ
ω
2 one in Eq. (5.11.5). These are now clearly
the components of a tensor field on the spectral observer sky S spec in the Sachs dyad
basis. Expressing f ab in terms of I , V and P ab through Eq. (5.8.16), we then find
ˆ
I AB (ϑ, ˆ
ω) ≡
1
2
ˆ
ω
2
δ AB ˆ
I + iε AB ˆ
V + ˆ
P AB
(ϑ, ˆ
ω) ,
(5.11.7)
where
ˆ
P AB (ϑ, ˆ
ω) := ˆ
k
a
A (ϑ) ˆ
k
b
B (ϑ) ˆ
P ab (
k) ≡ ˆ
n
i
A (ϑ) ˆ
n
i
B (ϑ) ˆ
P i j (
k) ,
ˆ
P AA ≡ 0 ,
ˆ
P AB ≡ ˆ
P B A .
(5.11.8)
Thus, the dimensionless CMB observables, that are the intensity spectrum map
ˆ
I (ϑ, ˆ
ω), circular polarization spectrum map ˆ
V (ϑ, ˆ
ω) and linear polarization spectrum map ˆ
P ab (ϑ, ˆ
ω), are simply related to the phase space fields I (x,
k), V (x,
k) and
P ab (x,
k) by evaluation at the observer position ˆ
P and projection on the observer
sky for the P ab . The dimensionful (spectral radiance) observables are then simply
obtained by multiplying by ˆ
ω
3 . The advantage of the dimensionless ones is that they
transform tensorially on S spec under LLTs
˜ ˆ
I ( ˜
ϑ, ˜ ˆ
ω) = ˆ
I (ϑ, ˆ
ω) ,
(5.11.9)
˜ ˆ
V ( ˜
ϑ, ˜ ˆ
ω) = ˆ
V (ϑ, ˆ
ω) ,
(5.11.10)
˜ ˆ
P
AB
( ˜
ϑ, ˜ ˆ
ω) = R
A
C (ϑ) R
B
D (ϑ) ˆ
P
C D
(ϑ, ˆ
ω) ,
(5.11.11)
125
where the 4-momenta here are null k a k
a
≡ 0. Let us next observe that, for a phase
space distribution evaluated at the observer position ˆ
P, the
k parameters play exactly
the same role as ˆ
ω and ˆ
n
i in the case of the geodesic associated with localized
sources, i.e. they parametrize the subspace of T ˆ
P M with respect to the spatial frame
ˆ
e i . We can therefore decompose the above
k-dependencies into the norm ˆ
ω := k and
direction ˆ
n :=
k/k and we subsequently express the latter using the standard angular
parameterization of Eq. (4.3.2). More generally, we can make use of the geometrical
machinery developed in Sect. 4 to describe the observer sky. First, note that what
is observed in practice is the two-point function of the complexified electric field
operator tangent to the observer sky
ˆ
I AB (ϑ, ˆ
ω) := ˆ
k
a
A
ˆ
k
b
B
ˆ
I 0a,0b (
k) ≡ ˆ
ω
2 ˆ
k
a
A
ˆ
k
b
B
ˆ
f ab (
k) ≡ ˆ
ω
2
ˆ
n
i
A (ϑ) ˆ
n
j
B (ϑ) ˆ
f i j (
k) , (5.11.5)
where we have used Eqs. (4.5.7), (4.5.25) and (5.8.7). Thus, we basically observe the
projection of the photon distribution f ab on the observed sky, up to a ˆ
ω
2 factor. From
the third expression in the series of equalities (5.11.5), we infer that the transformation
rule under LLTs of this hermitian 2 × 2 matrix is
˜ ˆ
I AB ( ˜
ϑ, ˜ ˆ
ω) = ˆ
2
(ϑ) R
C
A (ϑ) R
D
B (ϑ) ˆ
I C D (ϑ, ˆ
ω) ,
(5.11.6)
with R
AB being the LLT-compensating Sachs rotations (see again Sect. 4) and the ˆ
2
factor being due to the presence of the ˆ
ω
2 one in Eq. (5.11.5). These are now clearly
the components of a tensor field on the spectral observer sky S spec in the Sachs dyad
basis. Expressing f ab in terms of I , V and P ab through Eq. (5.8.16), we then find
ˆ
I AB (ϑ, ˆ
ω) ≡
1
2
ˆ
ω
2
δ AB ˆ
I + iε AB ˆ
V + ˆ
P AB
(ϑ, ˆ
ω) ,
(5.11.7)
where
ˆ
P AB (ϑ, ˆ
ω) := ˆ
k
a
A (ϑ) ˆ
k
b
B (ϑ) ˆ
P ab (
k) ≡ ˆ
n
i
A (ϑ) ˆ
n
i
B (ϑ) ˆ
P i j (
k) ,
ˆ
P AA ≡ 0 ,
ˆ
P AB ≡ ˆ
P B A .
(5.11.8)
Thus, the dimensionless CMB observables, that are the intensity spectrum map
ˆ
I (ϑ, ˆ
ω), circular polarization spectrum map ˆ
V (ϑ, ˆ
ω) and linear polarization spectrum map ˆ
P ab (ϑ, ˆ
ω), are simply related to the phase space fields I (x,
k), V (x,
k) and
P ab (x,
k) by evaluation at the observer position ˆ
P and projection on the observer
sky for the P ab . The dimensionful (spectral radiance) observables are then simply
obtained by multiplying by ˆ
ω
3 . The advantage of the dimensionless ones is that they
transform tensorially on S spec under LLTs
˜ ˆ
I ( ˜
ϑ, ˜ ˆ
ω) = ˆ
I (ϑ, ˆ
ω) ,
(5.11.9)
˜ ˆ
V ( ˜
ϑ, ˜ ˆ
ω) = ˆ
V (ϑ, ˆ
ω) ,
(5.11.10)
˜ ˆ
P
AB
( ˜
ϑ, ˜ ˆ
ω) = R
A
C (ϑ) R
B
D (ϑ) ˆ
P
C D
(ϑ, ˆ
ω) ,
(5.11.11)