Math Notes: Spherical Harmonic Transform and Fourier Transform

In cosmology, spherical harmonic transform (SHT) and Fourier transform (FT) are two of the most significant mathematics when we process data. In this notes, we will shortly review the definition of two transforms and then derive some useful equations about them.

SHT

For a field $a(\theta,\phi)$ in sphere $\mathbb{S}^2$, the SHT is defined as \(a(\theta,\phi) = \sum_{\ell = 0}^\infty \sum_{m=-\ell}^\ell a_{\ell m}Y_{\ell m}(\theta,\phi),\) where the coefficients is \(a_{\ell m} = \int_0^\pi \sin\theta\,\mathrm{d}\theta\, \int_0^{2\pi}\,\mathrm{d}\phi\, a(\theta,\phi)Y_{\ell m}^*(\theta,\phi).\) Here we use super-script star to denote the complex conjugate. The $Y_{\ell m}(\theta,\phi)$ is spherical harmonics with these important properties:

• Orthonormal: $\displaystyle\int_0^\pi \sin\theta\,\mathrm{d}\theta\,\int_0^{2\pi}\,\mathrm{d}\phi\, Y_{\ell m}(\theta,\phi)Y^*{\ell’ m’}(\theta,\phi) = \delta^K{\ell \ell’}\delta^K_{mm’}$. Here $\delta^K$ is Kerdonecker delta.
• Parity: $Y_{\ell m}(-\theta,-\phi) = (-1)^\ell Y_{\ell m}(\theta,\phi)$.

Hereafter we use $\displaystyle\int_\Omega = \int_0^\pi \sin\theta\,\mathrm{d}\theta\,\int_0^{2\pi}\,\mathrm{d}\phi$ and $\mathbf{\hat{n}} = (\theta,\phi)$ to simplify the notation.

FT

Taking 3D space for example, a field $f(\mathbf{x})$ in $\mathbb{R}^3$ can be expanded by FT \(f(\mathbf{x}) = \int\,\mathrm{d}^3\mathbf{x}\, \mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}} f(\mathbf{k}),\) where the function is Fourier space is \(f(\mathbf{k}) = \int\,\dfrac{\mathrm{d}^3\mathbf{k}}{(2\pi)^3}\, \mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot\mathbf{x}} f(\mathbf{x}).\) One should notice that in different field people have different convention for the $(2\pi)^3$ coefficients, and the convention above (maybe) is the most popular in cosmology.

Hereafter we use $\displaystyle\int_{\mathbf{x}} = \int \,\mathrm{d}^3\mathbf{x}$ and $\displaystyle\int_{\mathbf{k}} = \int \,\dfrac{\mathrm{d}^3\mathbf{k}}{(2\pi)^3}$ to simplify the notation.

Relation Between SHT and FT

When we deal with the data in the sky sphere especially gravitation lensing (GL), we usually try to find the relation between the angular spectra and spectra in 3D. The bridge between them is the plane wave expansion \(\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}} = 4\pi \sum_{\ell =0}^\infty \sum_{m=-\ell}^\ell \mathrm{i}^{\ell} j_{\ell}(kx)Y^*_{\ell m}(\hat{k})Y_{\ell m}(\hat{x}).\) ($k$ and $x$ are equivalent so you can choose which $Y_{\ell m}$ to conjugate) With the projection from a 3D field $\delta(\mathbf{x})$ to a sphere map $\kappa(\mathbf{\hat{n}})$ with kernel function $W$ : \(\kappa(\mathbf{\hat{n}}) = \int_0^\infty \,\mathrm{d}\chi'\, W(\chi')\delta(\chi',\mathbf{\hat{n}}).\) we can derive the relation between $\kappa_{\ell m}$ and $\delta(\mathbf{k})$ so that we can connect $C_{\kappa\kappa}(\ell)$ with $P_{\delta\delta}(k)$. Suppose $\mathbf{x} = \chi\mathbf{\hat{n}}$ and $\displaystyle \int_\chi = \int_{0}^\infty \,\mathrm{d}\chi$, we can write down \(\kappa_{\ell m} = \int_\Omega \kappa(\mathbf{\hat{n}})Y^*_{\ell m}(\mathbf{\hat{n}}) = \int_\Omega \int_{\chi'}W(\chi')\delta(\chi',\mathbf{\hat{n}})Y^*_{\ell m} (\mathbf{\hat{n}})= \int_\Omega\int_{\chi'}\int_{\mathbf{k}}W(\chi')\mathrm{e}^{-\mathrm{i}\mathbf{k}\chi'\mathbf{\hat{n}}}\delta(\mathbf{k}) Y^*_{\ell m}(\mathbf{\hat{n}}).\) Then the plan wave expansion told us \(\int_\Omega \mathrm{e}^{-\mathrm{i}\mathbf{k}\chi'\mathbf{\hat{n}}}Y^*_{\ell m}(\mathbf{\hat{n}})= 4\pi \sum_{\ell' =0}^\infty\sum_{m'=-\ell'}^{\ell'} \mathrm{i}^{\ell'} j_{\ell'}(k\chi')Y^*_{\ell' m'}(\hat{k})\int_\Omega Y_{\ell' m'}(\mathbf{\hat{n}}) Y^*_{\ell m}(-\mathbf{\hat{n}}) = 4\pi\sum_{\ell' =0}^\infty\sum_{m'=-\ell'}^{\ell'} \mathrm{i}^{\ell'} j_{\ell'}(k\chi')Y^*_{\ell' m'}(\hat{k})\delta_{\ell \ell'}\delta_{m m'} = 4\pi \mathrm{i}^\ell j_\ell(k\chi')Y^*_{\ell m}(\hat{k}).\) So the final expression is \(\kappa_{\ell m} = 4\pi \mathrm{i}^\ell \int_{\chi'}\int_\mathbf{k} W(\chi')j_l(k\chi')Y^*_{\ell m}(\hat{k}) \delta(\mathbf{k}).\) Denoting $\mathbf{k}_{12\dots n} = \mathbf{k}_1+\mathbf{k}_2 + \cdots + \mathbf{k}_3$, we firstly show the definition of power spectrum, bispectrum and trispectrum before we start \(\langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\rangle =(2\pi)^3 P(k_1)\delta^D(\mathbf{k}_{12}),\) \(\langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\rangle = (2\pi)^3\delta(\mathbf{k}_{123})^DB(k_1,k_2,k_3),\) \(\langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\delta(\mathbf{k}_4)\rangle = (2\pi)^3\delta(\mathbf{k}_{1234})^DT(k_1,k_2,k_3,k_4).\)

2-point Correlator

For 2-point correlator, we can start from the angular power spectrum and generally derive with $\kappa_i =\displaystyle \int_{\chi’} W_i(\chi’)\delta$ , i.e. different projected field \(C_{\kappa_1\kappa_2} =\langle \kappa_{1, \ell m}\kappa_{2,\ell m} \rangle = \int_{\chi'}\int_{\chi''}\int_{\mathbf{k}_1}\int_{\mathbf{k}_2}W_1(\chi')W_2(\chi'')j_{\ell}(k_1\chi')j_{\ell} (k_2 \chi'')Y_{\ell m}(\hat{k}_1)Y_{\ell m}(\hat{k}_2)\langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\rangle.\) The power spectrum is defined as $\langle \delta(\mathbf{k}1)\delta(\mathbf{k}_2)\rangle =(2\pi)^3 P(k_1)\delta^D(\mathbf{k}{12})$ . So after integral over $k_2$ getting $k=k_1=k_2$ , and using the orthonormal integral for $Y_{\ell m}$ , we finally get \(C_{\kappa_1 \kappa_2} = \int_{\chi'}\int_{\chi''} W_1(\chi')W_2(\chi'')\int k^2\mathrm{d}k\, j_{\ell}(k\chi')j_{\ell}(k\chi'') P(k,\chi',\chi'').\)

3-point Correlator

For 3-point correlator, we would start at a more general expression,like \(\langle \kappa_{1,\ell_1m_1}\kappa_{2,\ell_2m_2}\kappa_{3,\ell_3m_3}\rangle=\int_{\chi_1}\int_{\chi_2}\int_{\chi_3}\int_{\mathbf{k}_1}\int_{\mathbf{k}_2}\int_{\mathbf{k}_3}W_1(\chi_1)W_2(\chi_2)W_3(\chi_3)j_{\ell_1}(k_1\chi_1)j_{\ell_2} (k_2 \chi_2)j_{\ell_3}(k_3\chi_3)Y_{\ell_1 m_1}(\hat{k}_1)Y_{\ell_2 m_2}(\hat{k}_2)Y_{\ell_3m_3}(\hat{k}_3)\langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\rangle.\) With the bispectrum $\langle \delta(\mathbf{k}1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\rangle = (2\pi)^3\delta(\mathbf{k}{123})^DB(k_1,k_2,k_3)$

4-point Correlator