For a continuous and unbounded field, Fourier Transform (FT) can be applied:
f ( x ) = ∫ Ω d p k F ( k ) e i 2 π k ⋅ x . f(\bm{x}) = \int_\Omega d^p\bm{k} F(\bm{k}) e^{i2\pi\bm{k}\cdot\bm{x}}. f ( x ) = ∫ Ω d p k F ( k ) e i 2 π k ⋅ x . θ ≡ x l ≡ 2 π k . \begin{align}
\bm{\theta} &\equiv \bm{x}\\
\bm{l} &\equiv 2\pi\bm{k}.
\end{align} θ l ≡ x ≡ 2 π k . We obtain,
f ( θ ) = ∫ Ω d p ( l 2 π ) F ( l ) e i l ⋅ θ = 1 ( 2 π ) p ∫ Ω d p l F ( l ) e i l ⋅ θ . \begin{align}
f(\bm{\theta}) &= \int_\Omega d^p \left(\frac{\bm{l}}{2\pi}\right) F(\bm{l}) e^{i\bm{l}\cdot\bm{\theta}}\\
& = \frac{1}{(2\pi)^p}\int_\Omega d^p\bm{l} F(\bm{l}) e^{i\bm{l}\cdot\bm{\theta}}.
\end{align} f ( θ ) = ∫ Ω d p ( 2 π l ) F ( l ) e i l ⋅ θ = ( 2 π ) p 1 ∫ Ω d p l F ( l ) e i l ⋅ θ . In fact, in the flat sky approximation we can treat C ( l ) C(l) C ( l ) Dodelson & Schmidt (2021)
w ( θ ) = 1 4 π 2 ∫ Ω d 2 l C ( l ) e i l ⋅ θ w(\theta)=\frac{1}{4\pi^2}\int_\Omega d^2\bm{l} C(l) e^{i\bm{l}\cdot\bm{\theta}} w ( θ ) = 4 π 2 1 ∫ Ω d 2 l C ( l ) e i l ⋅ θ Full vs Flat ¶ Full sky
w ( X , Y ) = ⟨ δ ^ ( X ) , δ ^ ( Y ) ⟩ = ∑ l l ′ ∑ m m ′ Y l m Y ‾ l ′ m ′ ⟨ δ ^ m m ′ δ ^ l l ′ ⟩ = ∑ l l ′ ∑ m m ′ Y l m Y ‾ l ′ m ′ δ m m ′ δ l l ′ C l = ∑ l C l ∑ m Y l m Y ‾ l m = \begin{align}
w(X,Y) &=\langle \hat{\delta}(X), \hat{\delta}(Y)\rangle \nonumber \\
&= \sum_{ll'}\sum_{mm'} Y^m_l \overline{Y}^{m'}_{l'} \langle \hat{\delta}_{mm'}\hat{\delta}_{ll'} \rangle \nonumber \\
&= \sum_{ll'}\sum_{mm'} Y^m_l \overline{Y}^{m'}_{l'}\delta_{mm'}\delta_{ll'}C_l \nonumber \\
&= \sum_{l}C_l\sum_{m} Y^m_l \overline{Y}^{m}_{l} \nonumber \\
&=
\end{align} w ( X , Y ) = ⟨ δ ^ ( X ) , δ ^ ( Y )⟩ = l l ′ ∑ m m ′ ∑ Y l m Y l ′ m ′ ⟨ δ ^ m m ′ δ ^ l l ′ ⟩ = l l ′ ∑ m m ′ ∑ Y l m Y l ′ m ′ δ m m ′ δ l l ′ C l = l ∑ C l m ∑ Y l m Y l m = w ( μ ) = ∑ l 2 l + 1 4 π C l P l ( μ ) ⟹ ∫ μ w ( μ ) P l ′ ( μ ) = ∑ l 2 l + 1 4 π C l ∫ μ P l ′ ( μ ) P l ( μ ) = 1 2 π C l ′ C l = 2 π ∫ μ w ( μ ) P l ( μ ) \begin{align}
w(\mu) &=\sum_{l}\frac{2l+1}{4\pi}C_lP_l(\mu)\\
\implies \int \mu w(\mu) P_{l'}(\mu) &= \sum_{l}\frac{2l+1}{4\pi}C_l\int \mu P_{l'}(\mu)P_l(\mu)=\frac{1}{2\pi}C_{l'} \nonumber \\
C_l &= 2\pi \int \mu w(\mu) P_{l}(\mu)
\end{align} w ( μ ) ⟹ ∫ μ w ( μ ) P l ′ ( μ ) C l = l ∑ 4 π 2 l + 1 C l P l ( μ ) = l ∑ 4 π 2 l + 1 C l ∫ μ P l ′ ( μ ) P l ( μ ) = 2 π 1 C l ′ = 2 π ∫ μ w ( μ ) P l ( μ ) Flat sky
w ( θ ) = F T − 1 C ( l ) = ∫ d 2 l ( 2 π ) 2 e i l ⋅ θ C ( l ) = ∫ d l ( 2 π ) 2 l C ( l ) ∫ d ϕ e i l θ c o s ( ϕ ) = ∫ d l 2 π l C ( l ) J 0 ( l θ ) \begin{align} w(\theta)
&=FT^{-1}C(\bm l)\\
&=\int \frac{d^2l}{(2\pi)^2} e^{i\bm l \cdot \bm \theta}C(\bm l) \nonumber\\
&=\int \frac{dl}{(2\pi)^2} l C(l)\int d\phi e^{il\theta cos(\phi)} \nonumber\\
&=\int \frac{dl}{2\pi} l C(l) J_0(l\theta)
\end{align} w ( θ ) = F T − 1 C ( l ) = ∫ ( 2 π ) 2 d 2 l e i l ⋅ θ C ( l ) = ∫ ( 2 π ) 2 d l lC ( l ) ∫ d ϕ e i lθ cos ( ϕ ) = ∫ 2 π d l lC ( l ) J 0 ( lθ ) Now to prove the equivalence between the flat sky and full sky relation, plug in the full sky Eq. 6
C l = 2 π ∫ μ ∫ d l ′ 2 π l ′ C ( l ′ ) J 0 ( l ′ θ ( μ ) ) P l ( μ ) = ∫ d l ′ l ′ C ( l ′ ) ∫ μ J 0 ( l ′ θ ( μ ) ) P l ( μ ) \begin{align*}
C_l &= 2\pi \int \mu \int \frac{dl'}{2\pi} l' C(l') J_0(l'\theta(\mu)) P_{l}(\mu) \nonumber\\
&= \int dl' l' C(l') \int \mu J_0(l'\theta(\mu)) P_{l}(\mu) \nonumber
\end{align*} C l = 2 π ∫ μ ∫ 2 π d l ′ l ′ C ( l ′ ) J 0 ( l ′ θ ( μ )) P l ( μ ) = ∫ d l ′ l ′ C ( l ′ ) ∫ μ J 0 ( l ′ θ ( μ )) P l ( μ ) for l ≫ 1 , J 0 ( l ′ θ ) → P l ′ ( μ ) l\gg1, J_0(l'\theta)\rightarrow P_{l'}(\mu) l ≫ 1 , J 0 ( l ′ θ ) → P l ′ ( μ )
∼ ∫ d l ′ l ′ C ( l ′ ) ∫ μ P l ′ ( μ ) P l ( μ ) ∼ ∫ d l ′ l ′ C ( l ′ ) δ l l ′ l = C ( l ) \begin{align}
&\sim \int dl' l' C(l') \int \mu P_{l'}(\mu) P_{l}(\mu) \nonumber\\
&\sim \int dl' l' C(l') \frac{\delta_{ll'}}{l} \nonumber\\
&= C(l)
\end{align} ∼ ∫ d l ′ l ′ C ( l ′ ) ∫ μ P l ′ ( μ ) P l ( μ ) ∼ ∫ d l ′ l ′ C ( l ′ ) l δ l l ′ = C ( l ) Moving to the discrete and bounded case, a Discrete Fuorier Transform (DFT) approach is needed:
f ( n ) = 1 N 1 N 2 . . . N p ∑ p = 0 N − 1 F ( p ) e i 2 π p ⋅ ( n ∘ N ∘ − 1 ) . f(\bm{n}) = \frac{1}{N_1N_2...N_p}\sum_{\bm{p}=0}^{\bm{N}-1} F(\bm{p}) e^{i2\pi\bm{p}\cdot \left( \bm{n}\circ \bm{N}^{\circ -1}\right)}. f ( n ) = N 1 N 2 ... N p 1 p = 0 ∑ N − 1 F ( p ) e i 2 π p ⋅ ( n ∘ N ∘− 1 ) . θ ≡ L ∘ n ∘ N ∘ − 1 l ≡ 2 π p ∘ L ∘ − 1 \begin{align}
\bm{\theta} &\equiv \bm{L}\circ \bm{n}\circ \bm{N}^{\circ -1}\\
\bm{l} &\equiv 2\pi\bm{p}\circ\bm{L^{\circ-1}}
\end{align} θ l ≡ L ∘ n ∘ N ∘− 1 ≡ 2 π p ∘ L ∘− 1 We obtain,
f ( θ ) = 1 N 1 N 2 . . . N p ∑ l = 0 2 π L ∘ − 1 ( N − 1 ) F ( l ) e i l ⋅ θ f(\bm{\bm{\theta}}) = \frac{1}{N_1N_2...N_p}\sum_{\bm{l}=0}^{2\pi\bm{L^{\circ-1}}(\bm{N}-1)} F(\bm{l}) e^{i\bm{l}\cdot\bm{\theta}} f ( θ ) = N 1 N 2 ... N p 1 l = 0 ∑ 2 π L ∘− 1 ( N − 1 ) F ( l ) e i l ⋅ θ where we have made use of the properties of the Hadamard Product ∘, a pair-wise product operation. We can additionally define
l m a x ≡ 2 π L ∘ − 1 ( N − 1 ) \bm{l_{max}} \equiv 2\pi\bm{L^{\circ-1}}(\bm{N}-1) l max ≡ 2 π L ∘− 1 ( N − 1 ) to obtain
f ( θ ) = 1 N 1 N 2 . . . N p ∑ l = 0 l m a x F ( l ) e i l ⋅ θ f(\bm{\bm{\theta}}) = \frac{1}{N_1N_2...N_p}\sum_{\bm{l}=0}^{\bm{l_{max}}} F(\bm{l}) e^{i\bm{l}\cdot\bm{\theta}} f ( θ ) = N 1 N 2 ... N p 1 l = 0 ∑ l max F ( l ) e i l ⋅ θ Applying this general equation to our 2D flat sky case of a field defined on a square box of size (L, L) and pixels (N, N).
θ ≡ n L N l ≡ 2 π p L l m a x ≡ 2 π N − 1 L \begin{align}
\bm{\theta} &\equiv \bm{n}\frac{L}{N}\\
\bm{l} &\equiv 2\pi\frac{\bm{p}}{L}\\
l_{max} &\equiv 2\pi\frac{N-1}{L}
\end{align} θ l l ma x ≡ n N L ≡ 2 π L p ≡ 2 π L N − 1 w ( θ ) = 1 N 2 ∑ l = 0 l m a x C ( l ) e i l ⋅ θ w(\theta)=\frac{1}{N^2}\sum_{\bm{l}=0}^{l_{max}} C(l) e^{i\bm{l}\cdot\bm{\theta}} w ( θ ) = N 2 1 l = 0 ∑ l ma x C ( l ) e i l ⋅ θ
Dodelson, S., & Schmidt, F. (2021). 13 - Probes of structure: lensing. In S. Dodelson & F. Schmidt (Eds.), Modern Cosmology (Second Edition) (Second Edition, pp. 373–399). Academic Press. https://doi.org/10.1016/B978-0-12-815948-4.00019-X