5.11 Cosmic Microwave Background Observables
127
˜
∇ A = R
B
A
ˆ
∇ B + ˆ
0
i ˆ
n
i
B ˆ
ω∂ ω
,
(5.11.19)
where we have used Eqs. (4.3.18) and (4.5.17). Therefore, this a covariant derivative
only for fields on S, i.e. with no ˆ
ω dependence, or only under purely rotational LLTs
for fields on S spec . In what follows we will work with a real fully symmetric traceless
tensor of arbitrary rank ˆ
T A 1 ...A s ( ˜
ϑ, ˜ ˆ
ω), thus transforming as the generalization of
(5.11.11) under LLTs
˜ ˆ
T
A 1 ...A s ( ˜
ϑ, ˜ ˆ
ω) = R
A 1
B 1
(ϑ) . . . R
A s
B s
(ϑ) ˆ
T
B 1 ...B s (ϑ, ˆ
ω) .
(5.11.20)
This way we will be treating the three cases ˆ
I , ˆ
V and ˆ
P
AB simultaneously. In two
dimensions such a tensor has only two independent components and these can be
expressed in terms of two scalars under local rotations: the “electric” and “magnetic”
fields
11
ˆ
T A 1 ...A s ≡ ∇ A 1 ... A s ˆ
E + ε
B
(A 1
∇ A 2 ... A s )B ˆ
B ,
∇ A 1 ...A s := ∇ A 1 . . . ∇ A s ,
(5.11.21)
where . . . denotes full symmetrization and removal of traces. In terms of the complex operator
D A :=
δ
B
A + iε
B
A
∇ B ,
(5.11.22)
which satisfies the convenient identities
D A D
A
≡ 0 ,
[D A , D B ] ≡ 0 ,
(5.11.23)
we obtain a much simpler result
ˆ
T A 1 ...A s ≡ ˆ
T
C
A 1 ...A s
+ c.c. ,
ˆ
T
C
A 1 ...A s
:= D A 1 . . . D A s
ˆ
E + i ˆ
B
, (5.11.24)
as one can show by induction. The full information of the tensor can now be stored
in the first component of the complexified field
ˆ
T := ˆ
T
C
1...1 ≡ D
s
1
ˆ
E + i ˆ
B
.
(5.11.25)
Under LLTs at the observer, Eqs. (5.11.20) and (5.11.25) lead to the generalization
of Eq. (5.11.14), i.e. the helicity s representation of the local rotations on S
˜ ˆ
T ( ˜
ϑ, ˜ ˆ
ω) = e
isα(ϑ) ˆ
T (ϑ, ˆ
ω) .
(5.11.26)
11 This expression can be obtained straightforwardly by performing a harmonic decomposition
ˆ
T A1...As ≡
s
k=0 ∇ A1...A k h A k+1 ...As , where all the h A1...A k are totally symmetric, traceless and
transverse. In two dimensions these conditions are more than the number of independent components
for h A1...A k>1 , so these fields are zero, while the condition ∇ A h A ≡ 0 implies h A ≡ AB ∇ B ˜
h, thus
yielding the form (5.11.21).
127
˜
∇ A = R
B
A
ˆ
∇ B + ˆ
0
i ˆ
n
i
B ˆ
ω∂ ω
,
(5.11.19)
where we have used Eqs. (4.3.18) and (4.5.17). Therefore, this a covariant derivative
only for fields on S, i.e. with no ˆ
ω dependence, or only under purely rotational LLTs
for fields on S spec . In what follows we will work with a real fully symmetric traceless
tensor of arbitrary rank ˆ
T A 1 ...A s ( ˜
ϑ, ˜ ˆ
ω), thus transforming as the generalization of
(5.11.11) under LLTs
˜ ˆ
T
A 1 ...A s ( ˜
ϑ, ˜ ˆ
ω) = R
A 1
B 1
(ϑ) . . . R
A s
B s
(ϑ) ˆ
T
B 1 ...B s (ϑ, ˆ
ω) .
(5.11.20)
This way we will be treating the three cases ˆ
I , ˆ
V and ˆ
P
AB simultaneously. In two
dimensions such a tensor has only two independent components and these can be
expressed in terms of two scalars under local rotations: the “electric” and “magnetic”
fields
11
ˆ
T A 1 ...A s ≡ ∇ A 1 ... A s ˆ
E + ε
B
(A 1
∇ A 2 ... A s )B ˆ
B ,
∇ A 1 ...A s := ∇ A 1 . . . ∇ A s ,
(5.11.21)
where . . . denotes full symmetrization and removal of traces. In terms of the complex operator
D A :=
δ
B
A + iε
B
A
∇ B ,
(5.11.22)
which satisfies the convenient identities
D A D
A
≡ 0 ,
[D A , D B ] ≡ 0 ,
(5.11.23)
we obtain a much simpler result
ˆ
T A 1 ...A s ≡ ˆ
T
C
A 1 ...A s
+ c.c. ,
ˆ
T
C
A 1 ...A s
:= D A 1 . . . D A s
ˆ
E + i ˆ
B
, (5.11.24)
as one can show by induction. The full information of the tensor can now be stored
in the first component of the complexified field
ˆ
T := ˆ
T
C
1...1 ≡ D
s
1
ˆ
E + i ˆ
B
.
(5.11.25)
Under LLTs at the observer, Eqs. (5.11.20) and (5.11.25) lead to the generalization
of Eq. (5.11.14), i.e. the helicity s representation of the local rotations on S
˜ ˆ
T ( ˜
ϑ, ˜ ˆ
ω) = e
isα(ϑ) ˆ
T (ϑ, ˆ
ω) .
(5.11.26)
11 This expression can be obtained straightforwardly by performing a harmonic decomposition
ˆ
T A1...As ≡
s
k=0 ∇ A1...A k h A k+1 ...As , where all the h A1...A k are totally symmetric, traceless and
transverse. In two dimensions these conditions are more than the number of independent components
for h A1...A k>1 , so these fields are zero, while the condition ∇ A h A ≡ 0 implies h A ≡ AB ∇ B ˜
h, thus
yielding the form (5.11.21).