3D Spectra and Angular Spectra

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

\[\begin{aligned} a(\theta,\phi) &= \sum_{\ell = 0}^\infty \sum_{m=-\ell}^\ell a_{\ell m}Y_{\ell m}(\theta,\phi), \end{aligned}\]

where the coefficients is

\[\begin{aligned} a_{\ell m} &= \int_0^\pi \sin\theta\,\mathrm{d}\theta\, \int_0^{2\pi}\,\mathrm{d}\phi\, a(\theta,\phi)Y_{\ell m}^*(\theta,\phi). \end{aligned}\]

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

\[\begin{aligned} f(\mathbf{x}) &= \int\,\mathrm{d}^3\mathbf{x}\, \mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}} f(\mathbf{k}), \end{aligned}\]

where the function is Fourier space is

\[\begin{aligned} f(\mathbf{k}) &= \int\,\dfrac{\mathrm{d}^3\mathbf{k}}{(2\pi)^3}\, \mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot\mathbf{x}} f(\mathbf{x}). \end{aligned}\]

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

\[\begin{aligned} \mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}} &= 4\pi \sum_{\ell =0}^\infty \sum_{m=-\ell}^\ell \mathrm{i}^{\ell} j_{\ell}(kx)Y^*_{\ell m}(\hat{k})Y_{\ell m}(\hat{x}). \end{aligned}\]

( \(k\) 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 \(W\) :

\[\begin{aligned} \kappa(\mathbf{\hat{n}}) &= \int_0^\infty \,\mathrm{d}\chi'\, W(\chi')\delta(\chi',\mathbf{\hat{n}}). \end{aligned}\]

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

\[\begin{aligned} \kappa_{\ell m} &= \int_\Omega \kappa(\mathbf{\hat{n}})Y^*_{\ell m}(\mathbf{\hat{n}}) \\ &= \int_\Omega \int_{\chi'}W(\chi')\delta(\chi',\mathbf{\hat{n}})Y^*_{\ell m} (\mathbf{\hat{n}}) \\ &= \int_\Omega\int_{\chi'}\int_{\mathbf{k}}W(\chi')\mathrm{e}^{-\mathrm{i}\mathbf{k}\chi'\mathbf{\hat{n}}}\delta(\mathbf{k}) Y^*_{\ell m}(\mathbf{\hat{n}}). \end{aligned}\]

Then the plan wave expansion told us

\[\begin{aligned} \int_\Omega \mathrm{e}^{-\mathrm{i}\mathbf{k}\chi'\mathbf{\hat{n}}}Y^*_{\ell m}(\mathbf{\hat{n}}) &= 4\pi \sum_{\ell' =0}^\infty\sum_{m'=-\ell'}^{\ell'} \mathrm{i}^{\ell'} j_{\ell'}(k\chi')Y^*_{\ell' m'}(\hat{k})\int_\Omega Y_{\ell' m'}(\mathbf{\hat{n}}) Y^*_{\ell m}(-\mathbf{\hat{n}}) \\ &= 4\pi\sum_{\ell' =0}^\infty\sum_{m'=-\ell'}^{\ell'} \mathrm{i}^{\ell'} j_{\ell'}(k\chi')Y^*_{\ell' m'}(\hat{k})\delta_{\ell \ell'}\delta_{m m'} \\ &= 4\pi \mathrm{i}^\ell j_\ell(k\chi')Y^*_{\ell m}(\hat{k}). \end{aligned}\]

So the final expression is

\[\boxed{ \begin{aligned} \kappa_{\ell m} &= 4\pi \mathrm{i}^\ell \int_{\chi'}\int_\mathbf{k} W(\chi')j_l(k\chi')Y^*_{\ell m}(\hat{k}) \delta(\mathbf{k}). \end{aligned} }\]

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

\[\begin{aligned} \langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\rangle _c &= (2\pi)^3 P(k_1)\delta^D(\mathbf{k}_{12}), \end{aligned}\] \[\begin{aligned} \langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\rangle _c &= (2\pi)^3\delta^D(\mathbf{k}_{123})B(k_1,k_2,k_3), \end{aligned}\] \[\begin{aligned} \langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\delta(\mathbf{k}_4)\rangle _c &= (2\pi)^3\delta^D(\mathbf{k}_{1234})T(k_1,k_2,k_3,k_4). \end{aligned}\] \[\begin{aligned} \left\langle \prod_i^n\delta(\mathbf{k}_i) \right\rangle _c &= (2\pi)^3 \delta^D(\mathbf{k}_{12\dots n}) C_n(k_1,k_2,\dots,k_n) \end{aligned}\]

while it’s also useful to remember the FT of Dirac delta \(\delta^D(\mathbf{k})\) since it can be related with plane wave expansion

\[\begin{aligned} (2\pi)^3\delta^D(\mathbf{k}_{12\dots n}) &= \int_{\mathbf{r}}\mathrm{e}^{-\mathrm{i}\mathbf{k}_{12\dots n}\cdot\mathbf{r}} \\ &= \int_{\mathbf{r}}\prod_i^n\mathrm{e}^{-\mathrm{i}\mathbf{k}_i\cdot\mathbf{r}} \\ &= \int_{\mathbf{r}}\prod_i^n 4\pi \sum_{\ell =0}^\infty \sum_{m=-\ell}^\ell (-\mathrm{i})^{\ell}Y_{\ell m}(\hat{k}_i) j_{\ell}(k_ir)Y^*_{\ell m}(\hat{r}). \end{aligned}\]

Remark: We did not distinguish connected part and disconnected part for the cumulant but we will do this when talk about 4-point correlator.

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

\[\begin{aligned} \langle \kappa_{1, \ell m}\kappa_{2,\ell' m'} \rangle = (4\pi)^2\mathrm{i}^{\ell+\ell'}\int_{\chi'}\int_{\chi''}\int_{\mathbf{k}_1}\int_{\mathbf{k}_2} & W_1(\chi')W_2(\chi'')j_{\ell}(k_1\chi')j_{\ell'} (k_2 \chi'')\\ &\times Y_{\ell m}(\hat{k}_1)Y_{\ell' m'}(\hat{k}_2)\langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\rangle. \end{aligned}\]

Substituting the definition of power spectrum and expansion of Dirac delta, we can obtain

\[\begin{aligned} C_{\kappa_1 \kappa_2} &= \langle \kappa_{1, \ell m}\kappa_{2,\ell' m'} \rangle \\ &= \dfrac{2}{\pi} \int_{\chi'}\int_{\chi''} W_1(\chi')W_2(\chi'')\int k^2\mathrm{d}k\, j_{\ell}(k\chi')j_{\ell}(k\chi'') P(k,\chi',\chi''). \end{aligned}\]

3-point Correlator

For 3-point correlator, we would start at a more general expression, like

\[\begin{aligned} \langle \kappa_{1,\ell_1m_1}\kappa_{2,\ell_2m_2}\kappa_{3,\ell_3m_3}\rangle &= \left[\prod_i^3 4\pi\mathrm{i}^{\ell_i}\int_{\chi_i}\int_{\mathbf{k}_i} W_i(\chi_i)j_{\ell_i}(k_i\chi_i)Y_{\ell_im_i}(\hat{k}_i) \right]\langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\rangle. \end{aligned}\]

With the same method like 2-point correlator, we can obtain

\[\begin{aligned} \langle \kappa_{1,\ell_1m_1}\kappa_{2,\ell_2m_2}\kappa_{3,\ell_3m_3}\rangle = & \dfrac{8}{\pi^3} \mathcal{G}_{m_1m_2m_3}^{\ell_1\ell_2\ell_3}\\ &\times \int r^2\mathrm{d}r\, \left[\prod_{i}^3\int_{\chi_i}\int k_i^2\mathrm{d}k_i\, W_i(\chi_i)j_{\ell_i}(k_i\chi_i)j_{\ell_i}(k_ir) \right]\\ & \quad \times B(k_1,k_2,k_3). \end{aligned}\]

Here the integral of \(\mathbf{r}\) is from the expansion of Dirac delta, and the direction of \(\hat{r}\) went into Gaunt symbol which defined as

\[\begin{aligned} \mathcal G_{m_1m_2m_3}^{\ell_1\ell_2\ell_3} &\equiv \int_{\Omega_{\hat r}}\, Y_{\ell_1m_1} Y_{\ell_2m_2} Y_{\ell_3m_3}. \end{aligned}\]

The direction component of \(\mathbf{k}_i\) is integrated by the orthonormal of \(Y_{\ell_i m_i}(\hat{k}_i)\) from \(\kappa_{i,\ell_i m_i}\) and \(Y_{\ell_i m_i}^*(\hat{k}_i)\) from expansion of Dirac delta. More details will be discussed below.

4-point Correlator

Now we can give some discussion about the 4-point correlator which might be more and more useful is the further cosmology and carefully treat the connected and disconnected part.

For a 4-point correlator we have the expansion ( \(\delta_i\) refers to \(\delta(\mathbf{k}_i)\))

\[\begin{aligned} \langle \delta_1\delta_2\delta_3\delta_4\rangle &= \langle \delta_1\delta_2\rangle\langle \delta_3\delta_4\rangle + \langle \delta_1\delta_3\rangle \langle \delta_2\delta_4\rangle + \langle \delta_1\delta_4\rangle \langle \delta_2\delta_3\rangle +\langle \delta_1\delta_2\delta_3\delta_4\rangle _c. \end{aligned}\]

the first 3 terms are disconnected part and the final term is the connected part which is exactly the definition of trispectrum, or more precisely speaking, connected matter trispectrum.

So what we have is exactly (\(P_{ij}\) refers to \(P\) in \((2\pi)^3\delta^D(\mathbf{k}_{ij})P\))

\[\begin{aligned} \langle \delta_1\delta_2\delta_3\delta_4\rangle = & (2\pi)^6[\delta^D(\mathbf{k}_{12})\delta^D(\mathbf{k}_{34})P_{12}P_{34}+2 \,\mathrm{perm.}]\\ & +(2\pi)^3 \delta^D(\mathbf{k}_{1234})T(k_1,k_2,k_3,k_4, k_{12},k_{34}). \end{aligned}\]

where \(\mathrm{perm}.\) refers to those permutation terms. We can treated connected terms and disconnected term separately, for the former (12 and 34 for example)

\[\begin{aligned} &\quad\left[\prod_i^44\pi\mathrm{i}^{\ell_i}\int_{\chi_i}\int_{\mathbf{k}_i} W_i(\chi_i)j_{\ell_i}(k_i\chi_i)Y_{\ell_im_i}(\hat{k}_i) \right]\langle \delta_1 \delta_2 \rangle \langle \delta_3 \delta_4 \rangle\\ &= (-1)^{m_1+m_3} \delta^K_{\ell_1\ell_2}\delta^K_{m_1,-m_2}\delta^K_{\ell_3\ell_4}\delta^K_{m_3,-m_4} C_{\kappa_1\kappa_2}C_{\kappa_3\kappa_4} \\ &\equiv D_{12|34}. \end{aligned}\]

For the connected correaltor, we give a general expression

\[\begin{aligned} \left\langle \prod_{i}^n \kappa_{i, \ell_i m_i} \right\rangle _c &= \left[\prod_i^n4\pi\mathrm{i}^{\ell_i}\int_{\chi_i}\int_{\mathbf{k}_i} W_i(\chi_i)j_{\ell_i}(k_i\chi_i)Y_{\ell_im_i}(\hat{k}_i) \right]\left\langle \prod_i^n\delta(\mathbf{k}_i) \right\rangle _c. \end{aligned}\]

and substitute the expansion of Dirac delta, with operations below:

• Coefficient is \((4\pi)^n \cdot (4\pi)^n \cdot (2\pi)^{-3n} = \left(\dfrac{2}{\pi}\right)^n\) , which are from Dirac delta plane wave expansion, relation between \(\kappa_{\ell m}\) and \(\delta(\mathbf{k})\), and Fourier measurement \(\dfrac{\mathrm{d}^3\mathbf{k}_i}{(2\pi)^3}\).
• Each \(Y_{\ell_i m_i}(\hat{k}_i)\) are integrated by \(Y_{\ell_i m_i}^*(\hat{k}_i)\) comes from Dirac delta plane wave expansion, and the orthonormality selected \(\ell_i\) from the summation of plane wave expansion.
• \((-\mathrm{i})^{\ell_i}\) from Dirac delta plane wave expansion and \(\mathrm{i}^{\ell_i}\) from relation between \(\kappa_{\ell_i m_i}\) and \(\delta(\mathbf{k})\) canceled by each other.

And finally we left these terms

\[\boxed{ \begin{aligned} \left\langle \prod_{i}^n \kappa_{i, \ell_i m_i} \right\rangle _c = & \left(\dfrac{2}{\pi}\right)^n \mathcal{I}_{m_1m_2\dots m_n}^{\ell_1\ell_2\dots \ell_n} \\ & \times \int r^2\mathrm{d}r\, \left[\prod_{i}^n\int_{\chi_i}\int k_i^2\mathrm{d}k_i\, W_i(\chi_i)j_{\ell_i}(k_i\chi_i)j_{\ell_i}(k_ir) \right]\\ & \quad \times C_n(k_1,k_2,\dots,k_n). \end{aligned} }\]

where we additionally define the general integral

\[\begin{aligned} \mathcal{I}(n)_{m_1m_2\dots m_n}^{\ell_1\ell_2\dots\ell_n} &\equiv \int_{\Omega_{\hat r}}\prod_i^n Y_{\ell_i m_i}(\hat{r}). \end{aligned}\]

and \(\mathcal{I}(3) = \mathcal{G}\), \(\mathcal I_2= (-1)^{m_1} \delta_{\ell_1\ell_2}^{K} \delta_{m_1,-m_2}^{K}\).

We can easily verify this expression is consistent with 2- and 3-point correlator which the disconnected part is \(0\).

Remark: To obtain 2-point correlator we need extra equation of Bessel function:

\[\begin{aligned} \int_0^\infty dr\,r^2 j_\ell(k_1r)j_\ell(k_2r) &= \frac{\pi}{2k_1^2} \delta_{\rm D}(k_1-k_2) \end{aligned}\]

To sum up, the 4-point correlator should be

\[\begin{aligned} \left\langle \prod_{i}^4 \kappa_{i, \ell_i m_i} \right\rangle &= \left\langle \prod_{i}^4 \kappa_{i, \ell_i m_i} \right\rangle_c +\left(D_{12|34}+2\,\mathrm{perm.}\right). \end{aligned}\]

and \(\mathcal I(4)_{m_1m_2m_3m_4}^{\ell_1\ell_2\ell_3\ell_4} = \sum_{LM}(-1)^M \mathcal G^{\ell_1\ell_2L}_{m_1m_2,-M} \mathcal G^{\ell_3\ell_4L}_{m_3m_4,M}\).

Higher-order Correlator

After all, we can combine those conclusion above to a general result, as we suppose a \(n\)-point correlator

\[\begin{aligned} \left\langle \prod_i^n\delta(\mathbf{k}_i) \right\rangle &= \sum_{\mathscr{P}}\prod_{B\in \mathscr{P}}\left\langle \prod_{i\in B}\delta(\mathbf{k}_i)\right\rangle \end{aligned}\]

where \(\mathscr{P}\) is a set partition of \(\{1,2,\dots, n\}\) and \(B\in \mathscr{P}\) is a block. In cosmology, we usually set \(\langle \delta\rangle = 0\), i.e. the 1-point correlator vanished by reduce the average, the each order correlator can be decomposed as disconnected part and connected part. For the former lower-order correlator can be calculated by continuing decomposed, the later just use the connected result we derived above.