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 \(n=4\) .
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.
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 \(T^{\rm full}(k_1,k_2,k_3,k_4, k_{12},k_{34})\) .
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
\[\begin{aligned} Q_{\ell_3\ell_4}^{\ell_1\ell_2}(L) &= G_{\ell_3\ell_4}^{\ell_1\ell_2}(L)+T_{\ell_3\ell_4}^{\ell_1\ell_2}(L). \end{aligned}\]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
\[\begin{aligned} \delta_{\rm L}(\mathbf{q}) &= \int \mathrm{d}^3 r_{\rm L}\,\delta_{\rm L}(\mathbf{r}_{\rm L})\mathrm{e}^{-\mathrm{i}\mathbf{r}_{\rm L}\cdot \mathbf{q}} \\ &= \dfrac{1}{V_{\rm L}}\delta(\mathbf{q})W(-\mathbf{q}). \end{aligned}\]To deal with the annoying \(\mathbf{q}_1\) integral, we can make an approximation under the assumption \(q\ll k\) :
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 \(\mathcal{O}(q/k)\) .
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 \(P(k), P(q)\) .
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 \(\Phi_{\rm L}(q) = \delta(q)/\alpha(q)\) , \(\alpha(q) = \dfrac{2q^2 T^2(q)D(z)}{3\Omega_{\rm m}H_0^2}\) .
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.
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:
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.
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.
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.
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
\[\begin{aligned} C_{\kappa_1 \kappa_2} &= \langle \kappa_{1, \ell m}\kappa_{2,\ell' m'} \rangle \\ &= \dfrac{2}{\pi} \int_{\chi'}\int_{\chi''} W_1(\chi')W_2(\chi'')\int k^2\mathrm{d}k\, j_{\ell}(k\chi')j_{\ell}(k\chi'') P(k,\chi',\chi''). \end{aligned}\]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.
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:
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 \(\mathcal I(4)_{m_1m_2m_3m_4}^{\ell_1\ell_2\ell_3\ell_4} = \sum_{LM}(-1)^M \mathcal G^{\ell_1\ell_2L}_{m_1m_2,-M} \mathcal G^{\ell_3\ell_4L}_{m_3m_4,M}\).
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.
]]>每年按时写个总结，不知道十年后我会不会回看我的来时路.
已更新.
年度关键词：冒进，懒惰，寻路
研究/学习关键词：宇宙学，大尺度结构，暗能量，有效场论
2025真是烂完了，但最烂的还在一月份，所以占个坑和去年一样春节前写😋
2026.1.25凌晨更新于上海李政道研究所
结束了不紧不慢的期末周，随即来TDLI的“天问”论坛放松一下心情，也顺便回忆一下这一年的收获和感悟.
关键词：冒进，懒惰，寻路
这一年的记录开始于2024年回答结束后.
2025春季学期，我计划提前选课来为我大三腾出时间科研，不知道是出于“虚荣心”还是急躁的本性，我在大二下学期，也就是25春，修读了巨量的课程，计划在大二结束后完成四大力学与广义相对论的修读. 这句话从结果上来说实现了，但是我得到了大学前四个学期最低的gpa，当然，一方面确实课程太多，有些手忙脚乱；另一方面，我确实陷入了一种疲惫期，怠惰、不知道为什么做事、怎么做事，最后抄作业、迷迷糊糊应对完了一学期的课程，实际上这个学期我除了广义相对论的皮毛以外什么都没学到.
除了课程学期，我的科研训练也陷入了长达三个月的停滞期，四月开始几乎没有推动我的进度，“课程多、任务重”也仿佛成了我的挡箭牌. 最后综合我进度慢、成绩差、不愿意保研到对方组里等种种原因（均为臆想），在2025年暑假，我被当时的导师“劝退”了.
实际上在回顾2025年上半年的时候我是十分无力的，就像我从未亲身经历那段时间，我也不知道我既不做科研也没有学习的时间在做什么，时间如白驹过隙，甚至没有留下脚印，而我也四顾茫然.
2025年秋季学期是一个慢下来探索方向的过程，前导师只留给了我一点点宇宙学基础，使我对专业方向的未来毫无了解. 在不想面对理论物理巨大的压力下，我在大二下学期结束后的分流上选择了天文学系，我觉得天文学的竞争与氛围会让我松口气，事实也基本如此，除了一些烦人的专业必修课.
按照惯例先从学业讲起，大二的我给大三的我留了充足的时间，于是我在2025秋季学期选修了量子场论、物理宇宙学、宇宙大尺度结构三门研究生课程. 并由于上学期的广义相对论在期中均分50的情况下总评没有调分，导致我对成绩不满意，选择重新修读了（天文学系开设的）广义相对论课程，我认为天文学系开设的课程肯定会比理论物理专业的广义相对论简单，可事实我无法评价，因为我最终成绩还不如重修前. 而量子场论因为我觉得我学习的motivation不够，最后选择了退课，实际上我十分后悔这一决定…至于另外两门专业方向的研究生课程，我认为我学的挺开心的.
至于科研，我先后找了我校两位不同的宇宙学老师，了解了不同方向，最后决定都试一下，于是就有了手头的大尺度结构（galaxy clustering）相关的大研项目以及暗能量constrain相关的一个小课题，这一个学期我也去听了由东南大学丘成桐中心举行的引力波物理讨论班，了解了一些引力波的基础知识，再慢慢的了解下，我觉得我对宇宙学这个大的领域有了一点点初步的认知. 也算是我2025年最重要的收获. 整理思绪的同时，我也在这里写下我的兴趣，欢迎有人交流与提出建议：
但站在终点回顾2025秋季学期的经历，我觉得我依旧是十分贪玩、懒惰的，我在很长一段时间频繁的去网吧打游戏，逃避不知道为什么逃避的现实，一拖再拖我的科研进度. 我知道我自己总是在回顾的时候去茫然自己为何不抓紧时间，我希望未来的我能够更加充满干劲.
【实际上在写完所有内容后我惊醒，我的懒惰与摆烂似乎与过度依赖AI脱不开关系，2025年是我开始逐渐依赖AI的过程，我很遗憾我似乎没有做好self-thinking–AI-assisting balance，希望新的一年我能逐渐适应过来.】
最后是我2026年1月的一点进展，我“意外地“陶瓷到了UW-Madison的一位做AI和cosmology的老师，他给了我一份暑研的offer，希望我能做一些有意义的工作.
回顾全年，睽诸当下，我仍然要感谢这一年与我交流、陪伴我的所有人，我觉得我是一个渴望社交的人，偶尔的一些畅聊能让我很愉悦；以及无法忽视的我的女朋友，在我如此低沉、混乱的一年仍然能够包容我的各种诡异的性格与举动. 谢谢这个世界.
总的而言，我的学业在稳步推进，本科的bg也初具雏形，对科研、对世界的认知也再进一步，希望我的2026:
我是Uphi.，一名做宇宙学大尺度结构和暗能量相关的大三本科生，很高兴遇见你.
]]>2025.1.23 趁着农历2024还没结束占个坑，后天忙完了这一年最后一点事后写一写
2025.1.26 看了一眼日历才发现还有两天就过年了，遂跑来速速更新，最是一直到今天才算是放寒假闲下来（并没有）了
2024可以说是我正式开始物理学习的一年，经历了人生前20年的最低谷然后趋于平缓稳步上升
个人情况与回答摘要 23级的本科生，某酒吧舞技校在读，水平一般，技不如人…
本回答大篇碎碎念，很多东西都是说给我自己听的，毕竟作为一个普普通通的本科生能说出些什么呢？内容值得一看的大概有：课程学习/教材推荐，初入科研的感悟
24春（碎碎念） 24春可能是我人生前20年最低谷的时间了吧（其实还有大半年才满20岁），因为大一第一学期比较放纵，又还没有跳出高中的思维定势，总觉得自己没有学过物理竞赛导致在专业课学习的时候比别人菜很多，加之一段时间琐事很多，人际关系也不顺，最后整个人进入了极度焦虑抑郁的状态.
现在回想起来那段时候确实是比较可悲的，不过也是人生一定要经历一下的过程吧，感觉那段“至暗时刻”给现在的我确实带来了大心脏和更加平和的心态，也让我对物理专业的学习产生了新的感觉——物理学习不止有做题，还需要更加活跃的思路和扎实而广阔的知识架构
（当然我承认也有我实在是卷不过而找的借口（），做题同样尤其重要性和必要性，一方面是GPA作为客观条件不可或缺，另一方面做题也是对知识运用的深度刻画）
极度焦虑抑郁的情况持续到了五月初，也是在五月初我开始更加活跃的水群（？），从而在网上认识了现在的女朋友（逃）. 期末周的复习让我逐渐找回了一些学习的感觉，最后也在24春学期取得了还看得过去的成绩（对我来说已经很好了）
24夏（机缘巧合） 24夏我先试报名了学校的项目去中科院物理所参观，当时对超导比较感兴趣（当时也是没有太大的主见，打算投身凝聚态洪流），了解了一下前置课程（当时的个人思路，完全不具有参考价值）：
线性代数->量子力学->热力学与统计物理->固体物理->超导概论
看起来还是十分劝退的，但我还是打算试一试，尽管学习周期可能长达两到三个学期.
再后来就是和女朋友（刚放假的时候面基，随后就确定了关系）交流，她推荐我和她一起看梁灿彬老师的《微分几何与广义相对论》，我想着反正也没什么事，就看了一点，可能就注定了我和偏理论方向的物理（我并不敢说是理论物理，实际上我对“理论物理”一词始终抱有十足的敬畏之心）奠定了机缘.
最大的转折发生在八月初，女朋友给我推荐了国家天文台的大研计划，我抱着试一试的心态和她挑了一个看起来十分高端大气上档次的课题名称联系了导师（是后话了），发了人生的第一封“套瓷信”（我女朋友如是说，对我来说并不是，第一我感觉这算不上陶瓷，只是想咨询一下大研的招生；第二是我前段时间和物理所的一位做超导的老师邮件联系过几次，请教了一些基本问题），并获得了面试机会，而非常惊喜意外的是，我们两个都从二十多人的面试中被老师选中，与其余五人组成了一个七人大研课题组. 我与宇宙学的故事也就此开始.
实际上我暑假也试图简单入门了一下量子力学这门课程，对于小白入手量子力学有一定的思路心得，放在这里仅供参考（希望我的回答能让别人学有所得）：
基本路径：线性代数基础（矩阵与特征根、特征向量）->《量子力学导论》(Introduction to Quantum Mechanics, David J. Griffiths) +BiliBili 网课 姚珩《量子力学》 （小众但个人感觉还不错）
另一参考路径：《量子力学 第一卷》(Quantum Mechanics, Vol.1, Cohen Tannoudji)->《现代量子力学》(Modern Quantum Mechanics, J.J. Sakurai)
避雷：个人不建议无任何基础的人直接读Sakurai 的书，第一章感觉略有一些劝退（或许学了近代物理（普物）会好一些），Cohen 的书确实是百科全书，拿来入手学习数学符号之类的十分合适
24秋（学业部分） 考虑到大一玩了一年，了解了做科研需要的一众前置知识后本人在这学期开始狂暴选课，直接提前修读了量子力学和电动力学两门课（原本是固体物理，后来被劝退了），造就了2门数学4门物理1门导论课的盛世，万幸的是期末考试除了光学一言难尽其余的都还看得过去.
首先说一下课内部分吧，理论力学的期中分数4开头的逆天分数给了我当头一棒，我不得不痛定思痛，好在期末成绩还行+潘爷妙手神捞，最后成绩十分意外. 这门课程教材的一些建议：
首先我个人愿意评价高显老师的《经典力学》（科学出版社）为国内理论力学教材的巅峰（刘川老师书略薄，是郎道的风格），其次潘海俊老师的《理论力学导论》（中国科学技术大学出版社）也是意外之喜，这本书网上没有电子版，但是作为潘爷讲义的合集修订本，整本书内容可圈可点，值得一读
量子力学是我这学期精力精力投入最大的一门课，尽管最后分数略有些不尽人意，但我还是收获了很多. 十分可惜的是由于课时问题量子力学中的一些板块（比如散射（？））没有提及，不过给我后续学习奠定了基本的物理图像，我也会在后续继续深入学习. 由于老师上课风格的问题，量子力学我在学期中几乎没有读新的教材，全靠老师的Beamer学习，这里便不再多提
光学一学期只学了考前的两天，分数的惨烈只能说罪有应得.
电动力学的教材是郭硕鸿老师的《电动力学》（高等教育出版社），我不是很喜欢郭书，首先就是万恶的 度规！！！其次就是感觉讲的深度不够，像是电动力学pro，书中也没有讲张量/经典场论的内容（提了一句），令我失望（或许是我对这门课的理解有些问题），我导师推荐去读郎道的《经典场论》(Classical Field Theory, Landau)，后面有时间我会仔细品鉴.
两门数学课概率论与数理统计、复变函数，最后分数平平无奇，学习过程也很平淡，但是很多概念确实需要理解记忆，在后续做科研的某一天肯定会用到.
一门导论课是天文学导论，是薛永泉老师上的，上课比较有意思，但是期末的难度还是比较令人破防.
24秋与24冬（科研部分） 首先十分感谢我的导师，尽管是一个大研项目，但我真真切切的感受到了导师从0培养本科生的思想，这半年来受益匪浅.
我们做的相关课题是宇宙学（早期宇宙）方向的，组里7个人只有两个人有一点相关的科研经历（顺带一提，我们组是来自天南海北的4个大三和3个大二学生，很可惜的是有一位大三的学长中途退出了），于是导师找来了两位他的合作者，是两位在美国读博士（一位今年申请博后）的学长（后文称为小导），给我们进行了为期半年的基础知识能力的教学，这里我不得不提的是
理论最小值 (Theoretical Minimum) 的形式是我见过最快的接触、上手科研必备能力的形式，整套 任务19题，从宇宙学基本概念（参考书《现代宇宙学》(Modern Cosmology, Scott Dodelson) 到我们做的相关内容（此处省略展开），这半年感觉自己快速入门，感觉十分良好（虽然但是，对于成体系的理解还是略有欠缺，毕竟时间有限我也没有完整的读过一本宇宙学的书，甚至没有广义相对论的基础）
非常感谢两位小导的付出！
这个回答拖到现在才发也是因为前一周我们组5位同学（有一位没来）一起来到了国家天文台进行线下的讨论，也是让我感受到了我导的魅力，从他的话中，我感觉国内的宇宙学领域还是较为贫瘠的土壤，希望我能出一些力. 在国台我们领到了自己的子课题，虽然感觉我的课题是比较trivial的（？），但这也为我后续学习课内留了一些时间吧（感觉后续会给新的课题）
作为普通的本科生，一开始接触科研感觉导师的带领是十分重要的，我并不是很了解周围的同学初次接触科研的感受，但我希望“系统地培养科研能力”和“提供快速上手科研的途径”是当代高校、课题组较为缺少的，很多导师一开始就让本科生看文章、讲想法，固然有许多天才可以做到（物理学不缺天才，我知道），但我也希望我们这些普通的本科生能有一条自己走的下去的路，从基础能力到兴趣，或许只需要一个激发的契机.
我时常感叹物理学海的广袤与深厚，我希望能有一盏灯指引我前行，如果不够，我希望未来我也能成为一盏灯. 自己学会的知识就像是火，火应该照亮前路，点燃别人的火，而不是拿来炫技展示. 全文完.
最后的最后还是感谢一下我的女朋友，是她带来了文章中那么多的机缘巧合，也是她在我沉沦的时候拉了我一把，让我逐渐走上正轨，我爱她.
后记碎碎念：这篇文章写的时候已经是深夜了，很多东西有感而发而丧失了一名物理学习者的理性与框架性，作为我一整年独特的经历，希望能给读文章的你带来一点点感悟与提升.
与君共勉.
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