Trispectrum and Position-dependent Power Sepctrum

Ref: The Angular Trispectrum of the CMB, Weak Lensing Trispectrum and Kurt-Spectra, Galaxy Survey Cosmology by Hannu, Position-dependent power spectrum

Correlation function is the most useful statistics in cosmology data analysis. With the increasing quality of survey data in the further, the demanding of higher-order statistics, \(n\) -point correlation function and the corresponding spectra, increases too. Here we will review the complex but powerful cases of \(n=4\) .

Four-point correlation

We started in the most common situation that a four point correlation function in three Cartesian coordinate. For any field \(\delta(\mathbf{x})\) defined in \(\mathbb{R}^3\) , we can define their 4-point correlator with their Fourier mode

\[\begin{aligned} \langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\delta(\mathbf{k}_4)\rangle. \end{aligned}\]

And using cumulant expansion, we can decompose it into two part: disconnected (Gaussian) part and connected (non-Gaussian) part

\[\begin{aligned} \langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\delta(\mathbf{k}_4)\rangle &= \langle\delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\rangle\langle\delta(\mathbf{k}_3)\delta(\mathbf{k}_4)\rangle + \langle\delta(\mathbf{k}_1)\delta(\mathbf{k}_3)\rangle\langle\delta(\mathbf{k}_2)\delta(\mathbf{k}_4)\rangle \\ &\quad + \langle\delta(\mathbf{k}_1)\delta(\mathbf{k}_4)\rangle\langle\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\rangle + \langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\delta(\mathbf{k}_4)\rangle_c. \end{aligned}\]

The first three terms are the disconnected part, also known as the Wick pairs. The last term is the connected part and the exact additional information of 4-point correlator comparing with 2-point.

Trispectrum

3D Space

The trispectrum is the connected contribution of the 4-point correlation, as

\[\begin{aligned} (2\pi)^3T(k_1,k_2,k_3,k_4)\delta^D(\mathbf{k}_{1234}) &= \langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\delta(\mathbf{k}_4)\rangle_c. \end{aligned}\]

But for any tetrahedron in \(\mathbb{R}^3\) , we need \(4\times 3 - 3 -3 =6\) degree of freedom (dof) to parameterize it (reducing 3 rotation dof and 3 translation dof), i.e. we need to know the length of two diagonal, so a full trispectrum is usually written as \(T^{\rm full}(k_1,k_2,k_3,k_4, k_{12},k_{34})\) .

Sphere

We also usually use the trispectrum on the sphere. With the field \(\kappa(\hat{\mathbf{n}})\) defined in \(\mathbb{S}^2\) , we can define a 4-point correlator with its spherical harmonic modes

\[\begin{aligned} \langle \kappa_{\ell_1m_2}\kappa_{\ell_2 m_2}\kappa_{\ell_3m_3}\kappa_{\ell_4m_4}\rangle. \end{aligned}\]

We discuss this away from 3D since they are much different then power spectrum’s and bispectrum’s difference in space and sphere.

Using rotation symmetry for the correlator, a 4-point correlator in sphere can be written as

\[\begin{aligned} \langle \kappa_{\ell_1m_2}\kappa_{\ell_2 m_2}\kappa_{\ell_3m_3}\kappa_{\ell_4m_4}\rangle &= \sum_{LM}(-1)^M \begin{pmatrix}\ell_1 & \ell_2 & L \\ m_1 & m_2 & -M\end{pmatrix} \begin{pmatrix}\ell_3 & \ell_4 & L \\ m_3 & m_4 & M\end{pmatrix} Q^{\ell_1\ell_2}_{\ell_3\ell_4}(L). \end{aligned}\]

where the \(L\) can be seen as a length of diagonal and for spherical 4-point correlator, 5 dof is complete to parameterize it ( \(4\times 2 - 3 = 5\) ). Geometrically speaking, the \(Q^{\ell_1\ell_2}_{\ell_3\ell_4}(L)\) is a quadrilateral component determined by two triangular ( \(\ell_1,\ell_2,L\) ) and ( \(\ell_3,\ell_4,L\) ) . We call ( \(\ell_1,\ell_2,\ell_3,\ell_4,L\) ) a configuration.

There’re also other two pairing method for \(\ell_1\ell_3\|\ell_2\ell_4\) and \(\ell_1\ell_4\|\ell_2\ell_3\) and the relation between each \(Q\) under those configuration can be related using Wigner-6j symbols. But here, we discuss all derivation with \(\ell_1\ell_2\|\ell_3\ell_4\) .

\(Q_{\ell_3\ell_4}^{\ell_1\ell_2}(L)\) is also contributed by Gaussian part and non-Gaussian part, written as

\[\begin{aligned} Q_{\ell_3\ell_4}^{\ell_1\ell_2}(L) &= G_{\ell_3\ell_4}^{\ell_1\ell_2}(L)+T_{\ell_3\ell_4}^{\ell_1\ell_2}(L). \end{aligned}\]

Local Power Spectrum

Convention

Another useful observable is local power spectrum (or position dependent power spectrum). To define it, we can consider a finite survey region \(V_{\rm s}\) and a local small region \(V_{\rm L}\) at \(\mathbf{r}_{\rm L}\) , in the local region we have

\[\begin{aligned} \delta(\mathbf{k}; \mathbf{r}_{\rm L}) &= \int_{V_{\rm L}}\mathrm{d}^3r\, \delta(\mathbf{r})\mathrm{e}^{-\mathrm{i}\mathbf{r}\cdot\mathbf{k}} \\ &= \int \mathrm{d}^3r\,\delta(\mathbf{r})W_{\rm L}(\mathbf{r}-\mathbf{r}_{\rm L})\mathrm{e}^{-\mathrm{i}\mathbf{r}\cdot\mathbf{k}} \\ &= \int \dfrac{\mathrm{d}^3 q}{(2\pi)^3}\, \delta(\mathbf{k}-\mathbf{q})W_{\rm L}(\mathbf{q})\mathrm{e}^{-\mathrm{i}\mathbf{r}_{\rm L}\cdot\mathbf{q}}. \end{aligned}\]

Here the last step use Plancherel theorem :

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

and \(W_{\rm L}\) is the rectangle window function selecting the region \(V_{\rm L}\) .

Under this convention, the local power spectrum is defined as

\[\begin{aligned} P(k;\mathbf{r}_{\rm L}) &= \dfrac{1}{V_{\rm L}}\|\delta(\mathbf{k};\mathbf{r}_{\rm L})\|^2 \\ &= \dfrac{1}{V_{\rm L}} \int \dfrac{\mathrm{d}^3 q_1}{(2\pi)^3}\int \dfrac{\mathrm{d}^3 q_2}{(2\pi)^3} \, \delta(-\mathbf{k}-\mathbf{q}_1)\delta(\mathbf{k}-\mathbf{q}_2)W_{\rm L}(\mathbf{q}_1)W_{\rm L}(\mathbf{q}_2) \mathrm{e}^{-\mathrm{i}\mathbf{r}_{\rm L}\cdot(\mathbf{q}_1+\mathbf{q}_2)}. \end{aligned}\]

In the Fourier space for the observable, we have

\[\begin{aligned} P(k;q) &= \int \mathrm{d}^3r_{\rm L}\, P(k;\mathbf{r}_{\rm L}) \mathrm{e}^{-\mathrm{i}\mathbf{r}_{\rm L}\cdot \mathbf{q}} \\ &= \dfrac{1}{V_{\rm L}}\int \dfrac{\mathrm{d}^3 q_1}{(2\pi)^3}\delta(-\mathbf{k}-\mathbf{q}_1)\delta(\mathbf{k}+\mathbf{q}+\mathbf{q}_1) W_{\rm L}(\mathbf{q}_1)W_{\rm L}(-\mathbf{q}_1-\mathbf{q}). \end{aligned}\]

And we define the background overdensity

\[\begin{aligned} \delta_{\rm L}(\mathbf{r}_{\rm L}) &= \dfrac{ \delta(\mathbf{k}=0;\mathbf{r}_{\rm L})}{V_{\rm L}}. \end{aligned}\]

and its Fourier transform

\[\begin{aligned} \delta_{\rm L}(\mathbf{q}) &= \int \mathrm{d}^3 r_{\rm L}\,\delta_{\rm L}(\mathbf{r}_{\rm L})\mathrm{e}^{-\mathrm{i}\mathbf{r}_{\rm L}\cdot \mathbf{q}} \\ &= \dfrac{1}{V_{\rm L}}\delta(\mathbf{q})W(-\mathbf{q}). \end{aligned}\]

Approximation

To deal with the annoying \(\mathbf{q}_1\) integral, we can make an approximation under the assumption \(q\ll k\) :

• observables \(\delta, P\) and spectrum \(B,T\) (will be discussed later) just keep zero order approximation, i.e. \(k\pm q = k\) .
• Window functions keep the first order term and are integrated:

\[\begin{aligned} \int \dfrac{\mathrm{d}^3q_1}{(2\pi)^3} W(\mathbf{q}_1)W(-\mathbf{q}-\mathbf{q}_1) &= \mathscr{F}[W^2(\mathbf{x})](-\mathbf{q}) \\ &= W(-\mathbf{q}). \end{aligned}\]

Here \(\mathscr{F}\) denotes the Fourier transform.

In this way, the local power spectrum in Fourier space are

\[\begin{aligned} P(k;q) &\simeq \dfrac{W(-\mathbf{q})}{V_{\rm L}}\delta(-\mathbf{k})\delta(\mathbf{k}+\mathbf{q}). \end{aligned}\]

This approximation are accurate before \(\mathcal{O}(q/k)\) .

Response

After squeezed approximation \(q\ll k\) , we can expand local power spectrum as the response to local density and gravitational potential

\[\begin{aligned} P(k;\mathbf{r}_{\rm L}) &= P(k)[1+R_1\delta_{\rm L}(\mathbf{r}_{\rm L}) + b_\phi \Phi_{\rm L}(\mathbf{\rm L})+\mathcal{O}(\delta_{\rm L}^2)] \\ &\quad + \epsilon_{\rm L}(\mathbf{r}_{\rm L}). \end{aligned}\]

Here \(R_1\) is the gravitational response, \(b_{\phi} = 4f_{\rm NL}\) is the response to local-type non-Gaussianity and \(\epsilon_{\rm L}\) is the noise term. We will show how the response bridge the relation between higher order statistics and \(P(k), P(q)\) .

Squeezed Bispectrum

Since local power spectrum is a observable at \(\mathbf{r}_{\rm L}\) , we can calculate its cross-correlation with the background overdensity \(\delta_{\rm L}(\mathbf{r}_{\rm L})\) . So we can get

\[\begin{aligned} \xi_{P\delta}(\mathbf{r}\|k) &= \langle P(k;\mathbf{r}_{\rm L}) \delta_{\rm L}(\mathbf{r}_{\rm L}+\mathbf{r})\rangle \\ &= \dfrac{1}{V_{\rm L}^2} \int \dfrac{\mathrm{d}^3 q_1}{(2\pi)^3}\int \dfrac{\mathrm{d}^3 q_2}{(2\pi)^3}\int \dfrac{\mathrm{d}^3 q_3}{(2\pi)^3} \, \langle\delta(-\mathbf{k}-\mathbf{q}_1)\delta(\mathbf{k}-\mathbf{q}_2)\delta(-\mathbf{q}_3)\rangle \\ &\quad \times W_{\rm L}(\mathbf{q}_1)W_{\rm L}(\mathbf{q}_2)W_{\rm L}(\mathbf{q}_3) \mathrm{e}^{-\mathrm{i}\mathbf{r}_{\rm L}\cdot(\mathbf{q}_1+\mathbf{q}_2+\mathbf{q}_3)}\mathrm{e}^{-\mathrm{i}\mathbf{r}\cdot\mathbf{q}_3} \\ &= \dfrac{1}{V_{\rm L}^2} \int \dfrac{\mathrm{d}^3 q_1}{(2\pi)^3}\int \dfrac{\mathrm{d}^3 q_3}{(2\pi)^3} \, B(-\mathbf{k}-\mathbf{q}_1, \mathbf{k}+\mathbf{q}_1+\mathbf{q}_3, -\mathbf{q}_3) \\ &\quad \times W_{\rm L}(\mathbf{q}_1)W_{\rm L}(-\mathbf{q}_1-\mathbf{q}_3)W_{\rm L}(\mathbf{q}_3)\mathrm{e}^{-\mathrm{i}\mathbf{r}\cdot\mathbf{q}_3}. \end{aligned}\]

The bispectrum is defined as \(B(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)(2\pi)^3 \delta^{\rm D}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3) = \langle \delta(\mathbf{k}_1)\delta(\mathbf{k}_2)\delta(\mathbf{k}_3)\rangle\) .

So we can define the cross power spectrum between local power spectrum and background overdensity

\[\begin{aligned} P_{P\delta}(q\|k) &= \int \mathrm{d}^3 r \,\xi_{P\delta}(\mathbf{r}\|k)\mathrm{e}^{-\mathrm{i}\mathbf{r}\cdot\mathbf{q}} \\ &= \dfrac{1}{V_{\rm L}^2} \int \dfrac{\mathrm{d}^3 q_1}{(2\pi)^3}\int \dfrac{\mathrm{d}^3 q_3}{(2\pi)^3}\int \mathrm{d}^3 r\, B(-\mathbf{k}-\mathbf{q}_1, \mathbf{k}+\mathbf{q}_1+\mathbf{q}_3, -\mathbf{q}_3) \\ &\quad \times W_{\rm L}(\mathbf{q}_1)W_{\rm L}(-\mathbf{q}_1-\mathbf{q}_3)W_{\rm L}(\mathbf{q}_3)\mathrm{e}^{-\mathrm{i}\mathbf{r}\cdot(\mathbf{q}_3+\mathbf{q})} \\ &=\dfrac{1}{V_{\rm L}^2} \int \dfrac{\mathrm{d}^3 q_1}{(2\pi)^3} B(-\mathbf{k}-\mathbf{q}_1, \mathbf{k}+\mathbf{q}_1-\mathbf{q}, \mathbf{q}) \\ &\quad \times W_{\rm L}(\mathbf{q}_1)W_{\rm L}(\mathbf{q}-\mathbf{q}_1)W_{\rm L}(-\mathbf{q}). \end{aligned}\]

Use the squeezed limit \(q\ll k\) , we have

\[\begin{aligned} P_{P\delta }(q\|k) &\simeq \dfrac{\|W(q)\|^2}{V_{\rm L}^2}B(-\mathbf{k},\mathbf{k}-\mathbf{q},\mathbf{q}) \\ &:= \dfrac{\|W(q)\|^2}{V_{\rm L}^2}B_{\rm sq}(k,q). \end{aligned}\]

This can also be derived from \(P_{P\delta}(q\|k) = \langle P(k;-q)\delta_{\rm L}(\mathbf{q})\rangle\) where \(-q\) making the momentum conserved. In this definition, the response model gives

\[\begin{aligned} B_{\rm sq}(k,q) &= \left[ R_1(k) + \dfrac{4f_{\rm NL}}{\alpha(q)}\right]P(k)P(q). \end{aligned}\]

where \(\alpha (q)\) comes from the Poisson equation \(\Phi_{\rm L}(q) = \delta(q)/\alpha(q)\) , \(\alpha(q) = \dfrac{2q^2 T^2(q)D(z)}{3\Omega_{\rm m}H_0^2}\) .

Collapsed Trispectrum

Alternative trispectrum is usually hard to be measured since they have 6 free parameters. But similar to bispectrum, we can focus on special configuration of it. In this section, we take space trispectrum as example to discuss a configuration called collapsed trispectrum, which have \(\|\mathbf{k}_1+\mathbf{k}_2\| = \|\mathbf{k}_3 +\mathbf{k}_4\| \ll \min(k_1,k_2,k_3,k_4)\) .

Similar to bispectrum, we can construct the trispectrum by calculating the local power spectrum auto-correlation function

\[\begin{aligned} \xi_{PP}(\mathbf{r}\|k,k') &= \langle P(k;\mathbf{r}_{\rm L}) P(k';\mathbf{r}_{\rm L}+\mathbf{r})\rangle \\ &= \dfrac{1}{V_{\rm L}^2} \int \dfrac{\mathrm{d}^3 q_1}{(2\pi)^3}\int \dfrac{\mathrm{d}^3 q_2}{(2\pi)^3} \int \dfrac{\mathrm{d}^3 q_3}{(2\pi)^3}\int \dfrac{\mathrm{d}^3 q_4}{(2\pi)^3} \\ &\quad \times \langle\delta(-\mathbf{k}-\mathbf{q}_1)\delta(\mathbf{k}-\mathbf{q}_2) \delta(-\mathbf{k}'-\mathbf{q}_3)\delta(\mathbf{k}'-\mathbf{q}_4)\rangle \\ &\quad \times W_{\rm L}(\mathbf{q}_1)W_{\rm L}(\mathbf{q}_2)W_{\rm L}(\mathbf{q}_3)W_{\rm L}(\mathbf{q}_4) \mathrm{e}^{-\mathrm{i}\mathbf{r}_{\rm L}\cdot(\mathbf{q}_1+\mathbf{q}_2+\mathbf{q}_3+\mathbf{q}_4)} \\ &\quad \times \mathrm{e}^{-\mathrm{i}\mathbf{r}\cdot(\mathbf{q}_3+\mathbf{q}_4)}. \end{aligned}\]

To avoid heavy calculation, we can just consider it in Fourier space where we have

\[\begin{aligned} P'_{PP}(q\|k,k') &= \langle P(k;q)P(k';-q)\rangle' \\ &\simeq \dfrac{\|W(q)\|^2}{V_{\rm L}^2}T (-\mathbf{k},\mathbf{k}+\mathbf{q},-\mathbf{k}',\mathbf{k}'-\mathbf{q}) \\ &:= \dfrac{\|W(q)\|^2}{V_{\rm L}^2}T_{\rm coll}(k,k',q). \end{aligned}\]

Here we use prime to denote the connected statistics.

Also the response gives

\[\begin{aligned} T_{\rm coll}(k,k',q) &= \left[R_1(k)R_1(k') +\dfrac{4f_{\rm NL}(R_1(k)+R_1(k'))}{\alpha(q)} +\dfrac{(100/9)\tau_{\rm NL}}{\alpha^2(q)} \right]P(k)P(k')P(q). \end{aligned}\]

\(\tau_{\rm NL}\) are defined in inflation theory that \(T_{\zeta} \supset \tau_{\rm NL}(P_{\zeta}P_{\zeta}P_{\zeta}(k_{12}) + 12{\rm perm.})\) and in single field inflation it satisfies \(\tau_{\rm NL} = \left(\dfrac{6}{5} f_{\rm NL}\right)^2\) .

Disconnected term should be treated carefully when doing research. But we won’t discuss it now.