5.11 Cosmic Microwave Background Observables
129
s Y lm ( ˜
ϑ) = e
isα(ϑ)
l
m =−l
¯
D
l
mm ( ˆ
θ) s Y lm (ϑ) ,
(5.11.35)
where D
l is the Wigner matrix associated to the Euler angles in ε i jk ˆ
θ
jk , yielding
with Eq. (5.11.26) the transformation rule
˜ ˆ
T lm ( ˆ
ω) :=
d ˜
s Y
∗
lm ( ˜
ϑ) ˜ ˆ
T ( ˜
ϑ, ˆ
ω) =
l
m =−l
¯
D
l
mm ( ˆ
θ)
d s Y
∗
lm (ϑ) ˆ
T (ϑ, ˆ
ω)
≡
l
m =−l
¯
D
l
mm ( ˆ
θ) ˆ
T lm ( ˆ
ω) .
(5.11.36)
Under local boosts, however, the transformation is more complicated, even for the
scalar quantities ˆ
I lm ( ˆ
ω) and ˆ
V lm ( ˆ
ω). First, we express the map (5.11.26) in an active
form, i.e. in terms of a unique ˆ
ω parametrization
˜ ˆ
T ( ˜
ϑ, ˆ
ω) = e
isα(ϑ) ˆ
T (ϑ, ˆ
−1
(ϑ) ˆ
ω) ,
(5.11.37)
because the two frequencies ˆ
ω and ˜ ˆ
ω are related by an angle-dependent factor and
here we are in harmonic space. Since we can rotate at will using Wigner matrices,
without loss of generality, we can focus on the case where the boost is along the i = 3
direction ˆ
θ
0i
= δ
i
3 η. Moreover, we will work at linear order in η for simplicity. The
LLT-induced coordinate transformation given in Eqs. (4.3.5) and (4.3.6) becomes
˜
ϑ = ϑ − η sin ϑ + O(η
2
) ,
˜
ϕ = ϕ ,
˜ ˆ
ω = (1 + η cos ϑ) ˆ
ω + O(η
2
) .
(5.11.38)
In particular, with this choice of direction we have a diagonal Jacobian matrix, meaning that there is no compensating local rotation in the transformation of the dyad
(4.3.18), i.e. α(ϑ) = 0 in (5.11.37). Thus, expanding to linear order in η we find
˜ ˆ
T ( ˜
ϑ, ˆ
ω) =
1 − η cos ϑ ˆ
ω∂ ˆ
ω + O(η
2
)
ˆ
T (ϑ, ˆ
ω) ,
(5.11.39)
and
d ˜
s Y
∗
lm ( ˜
ϑ) = d
1 − η (2 cos ϑ + sin ϑ ∂ ϑ ) + O(η
2
)
s Y
∗
lm (ϑ) .
(5.11.40)
Using the following identities [24]
sin ϑ ∂ ϑ s Y lm ≡ l s C l+1,m s Y l+1,m +
sm
l(l + 1)
s Y lm − (l + 1) s C lm s Y l−1,m ,
(5.11.41)
cos ϑ s Y lm ≡ s C l+1,m s Y l+1,m −
sm
l(l + 1)
s Y lm + s C lm s Y l−1,m ,