--- title: Parity-Odd Kurto Spectra url: https://www.emergentmind.com/topics/parity-odd-kurto-spectra type: topic --- # Parity-Odd Kurto Spectra Searching arXiv for the cited papers and closely related work on parity-odd kurto/composite-field spectra. Parity-odd kurto spectra are composite-field two-point statistics that compress parity-odd information in the scalar trispectrum into one-dimensional, power-spectrum-like observables. In large-scale structure, parity symmetry forbids leading parity-odd signatures in the two- and three-point functions of scalar observables such as the galaxy overdensity, making the four-point function the first parity-sensitive statistic. Parity-odd kurto spectra address the resulting high dimensionality of the trispectrum by correlating composite fields with opposite parity transformation properties, so that the expectation value vanishes unless parity is violated. This framework was introduced as “Parity-Odd Power (POP) spectra” and later developed as survey-ready parity-odd kurto spectra for galaxy clustering, culminating in measurements on BOSS DR12 and DESI DR1 luminous red galaxies [2406.15683; 2509.13207; 2604.06021]. ## 1. Concept and parity structure Parity acts as $\mathbf{x}\to-\mathbf{x}$. The relevant field transformations are: a scalar field $s$ transforms as $s(\mathbf{x})\to s(-\mathbf{x})$; a pseudo-scalar $p$ as $p(\mathbf{x})\to -p(-\mathbf{x})$; a vector $v_i$ as $v_i(\mathbf{x})\to -v_i(-\mathbf{x})$; and a pseudo-vector (axial vector) $a_i$ as $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$ [2604.06021]. A parity-odd spectrum is therefore obtained by cross-correlating one pseudo object with an even number of non-pseudo objects, or vice versa. In particular, a vector–pseudo-vector cross-power and a scalar–pseudo-scalar cross-power flip sign under parity and must vanish in a parity-symmetric theory [2604.06021]. For scalar observables, the four-point function is the leading parity-sensitive correlator. In Fourier space, the parity-odd part of the trispectrum is imaginary and must be proportional to a pseudoscalar triple product such as $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$, since isotropy allows no other pseudoscalar built from three independent momenta [2406.15683]. This establishes the trispectrum as the natural carrier of scalar-sector parity violation and explains why any practical parity test must either estimate the full trispectrum or compress it without discarding the parity-odd structures. The central idea of parity-odd kurto spectra is to preserve those antisymmetric structures through composite operators containing gradients, curls, and the Levi–Civita tensor $\epsilon_{ijk}$, while reducing the statistic to a function of a single wavenumber $k$. This makes the observable formally analogous to a power spectrum, but targeted to the parity-odd part of the four-point function [2406.15683; 2509.13207]. ## 2. Composite-field construction and the two standard estimators The general composite-field construction begins from local products of filtered fields, \[ {\mathcal D}_n[\delta](\mathbf{x})=\prod_{i=1}^{n}{\mathcal D}_i(\mathbf{x})\,\delta(\mathbf{x})\,, \] with Fourier transform given by the corresponding convolution, and composite-field spectra defined by \[ \langle D_n[\delta](\mathbf{k})D_m[\delta](\mathbf{k}')\rangle=(2\pi)^3\delta_D(\mathbf{k}+\mathbf{k}')\,\mathcal{P}_{D_n[\delta],D_m[\delta]}(\mathbf{k})\,. \] This compresses the $(n+m)$-point function into a two-point statistic [2604.06021]. Two parity-odd kurto spectra are standard in the recent literature. The first is the quadratic–quadratic, or vector–pseudo-vector, estimator, \[ \mathcal{P}_{2\times 2}^{\rm PO}(k)=\int \frac{d\Omega_k}{4\pi}\,\langle {\bf V}^{ab}(\mathbf{k}) \cdot {\bf A}^{cd}(-\mathbf{k}) \rangle\,, \] with quadratic fields \[ V_i^{ab}(\mathbf{x})=\delta^a(\mathbf{x})\,\partial_i \delta^b(\mathbf{x})\,, \qquad A_i^{cd}(\mathbf{x})=\epsilon_{ijk}\,\partial_j \partial^{-2} V_k^{cd}(\mathbf{x})\,. \] Since ${\bf V}$ is a true vector and ${\bf A}$ a pseudo-vector, their cross-power flips sign under parity [2604.06021]. In related notation, this estimator appears as the quadratic–quadratic kurto spectrum $K_{\rm QQ}(k)$ or the vector POP spectrum [2509.13207; 2406.15683]. The second is the cubic–linear, or scalar–pseudo-scalar, estimator, \[ \mathcal{P}^{\rm PO}_{3\times1}(k)=\int \frac{d\Omega_k}{4\pi}\,\langle \delta(\mathbf{k}) \,\Psi^{abc}(-\mathbf{k})\rangle\,, \] where \[ \Psi^{abc}(\mathbf{x})=\nabla \delta^a(\mathbf{x}) \cdot \mathbf{B}^{bc}(\mathbf{x})\,, \qquad \mathbf{B}^{bc}(\mathbf{x})=\nabla \delta^b(\mathbf{x}) \times \nabla \delta^c(\mathbf{x})\,. \] Here $\Psi$ is a pseudo-scalar and $\delta$ is a scalar, so the expectation value again vanishes unless parity is violated [2604.06021]. In the theoretical literature this appears as the linear–cubic kurto spectrum $K_{\rm LC}(k)$ or the scalar POP spectrum [2509.13207; 2406.15683]. A crucial operator-level requirement is that the filters entering the convolutions differ appropriately. For the vector construction, the filters must differ within each convolution, such as $f_a\neq f_b$ and $f_c\neq f_d$, to avoid cancellation of longitudinal pieces or vanishing of the pseudo-vector [2509.13207; 2406.15683]. The survey analysis adopts \[ f_a(k)=k^2,\qquad f_b(k)=f_d(k)=1,\qquad f_c(k)=k^{-2}\,, \] for the filtered overdensity fields $\delta^a$ [2604.06021]. ## 3. Relation to the parity-odd trispectrum The parity-odd scalar trispectrum is the object being compressed. In one formulation, \[ T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)=T_{+}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)+i\,T_{-}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3)\,, \] with the imaginary part $T_-$ proportional to \[ \mathbf{k}_1\cdot\big(\mathbf{k}_2\times\mathbf{k}_3\big)\,\tau_-(k_1,k_2,k_3,k_4,K,\tilde K)\,, \] where the triple product determines the handedness and $\tau_-$ is totally antisymmetric under permutations [2406.15683]. This form makes clear that parity-odd information is encoded in Levi–Civita-type structures. Substituting the composite operators into the two-point definitions gives explicit trispectrum compressions. For the survey-ready formalism, \[ \mathcal{P}_{2\times 2}(k)=\int\!\frac{d\Omega_k}{4\pi} \int_{\mathbf{q}_1\mathbf{q}_2\mathbf{q}_3} K^{(2\times2)}(\mathbf{k};\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3)\; T(\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3,\mathbf{k}-\mathbf{q}_1-\mathbf{q}_2-\mathbf{q}_3)\,, \] and \[ \mathcal{P}_{3\times 1}(k)=\int\!\frac{d\Omega_k}{4\pi} \int_{\mathbf{q}_1\mathbf{q}_2\mathbf{q}_3} K^{(3\times1)}(\mathbf{k};\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3)\; T(\mathbf{q}_1,\mathbf{q}_2,\mathbf{q}_3,\mathbf{k}-\mathbf{q}_1-\mathbf{q}_2-\mathbf{q}_3)\,, \] where the kernels are constructed from the operators, filters, and smoothing window used in the estimator [2604.06021]. The presence of $\epsilon_{ijk}$ in the pseudo-vector and pseudo-scalar operators ensures that these kernels isolate parity-odd components of the trispectrum, while parity-even contributions cancel in expectation [2604.06021]. In the earlier POP formalism, the corresponding explicit integral representations show that the resulting one-dimensional observables are weighted slices of the full trispectrum over triangle or tetrahedron configurations, with angular factors such as $\big[\mathbf{k}\cdot(\mathbf{k}_1\times\mathbf{k}_3)\big]^2$ and $\big[\mathbf{Q}\cdot(\mathbf{k}_2\times\mathbf{k}_4)\big]^2$ appearing in the compression [2406.15683]. This compression has a direct statistical consequence. The full trispectrum depends on three independent wavevectors and their relative angles; the kurto spectra depend only on $k$ after angle averaging. The data vector therefore has power-spectrum-like dimensionality rather than full four-point dimensionality, which is the basis for the method’s computational and covariance advantages [2604.06021; 2509.13207]. ## 4. Estimation, filtering, and fast prediction To suppress small-scale modelling uncertainties and potential systematics, the survey implementation smooths the density field in Fourier space using \[ \delta_W(\mathbf{x})=\int \frac{d^3 k}{(2\pi)^3}\,e^{-i\,\mathbf{k}\cdot \mathbf{x}}\,W(k)\,\delta(\mathbf{k})\,, \] with a Heaviside-like tanh filter \[ W(k)=\frac{1}{2}\,\left[1 - \tanh\!\left(\alpha\,\ln\frac{k}{k_{\rm max}}\right)\right]\,, \quad \alpha=50,\quad k_{\rm max}=0.45\,h\,\mathrm{Mpc}^{-1}\,, \] filtering scales smaller than $R\approx7\,h^{-1}\,\mathrm{Mpc}$ [2604.06021]. Closely related work also advocates smooth tanh windows to mimic sharp cuts while avoiding ringing [2509.13207]. In survey data, the fluctuation field is constructed from galaxies and random catalogues through an FKP-like weighted field, \[ F(\mathbf{x})=\frac{w_{\mathrm{FKP}}(\mathbf{x})}{I_{44}^{1/4}} \Big[w_{\mathrm{g}}(\mathbf{x})\,n_{\mathrm{g}}(\mathbf{x})-\alpha_{\mathrm{r}}\,w_{\mathrm{r}}(\mathbf{x})\,n_{\mathrm{r}}(\mathbf{x})\Big]\,, \] with \[ w_{\mathrm{FKP}}(\mathbf{x})=\frac{1}{1+\bar n(z)\,P_0}\,,\qquad P_0=10^{4}\,h^{-3}\,\mathrm{Mpc}^{3}\,, \] and a trispectrum-appropriate normalization $I_{44}$ [2604.06021]. For BOSS DR12, $I_{44}$ is approximated via Monte Carlo sums over catalogue objects; for DESI DR1 it is approximated on the FFT mesh with a $\lambda$-dependent expression and $\lambda=3$ chosen for stability [2604.06021]. The practical estimator is FFT-based. The field $F(\mathbf{x})$ is painted to a mesh using `nbodykit`, composite fields are built in configuration space, and cross-power spectra are then measured after FFT, angle averaging, and $k$-binning [2604.06021]. The POP work described a closely related real-space/Fourier-space pipeline: filtered fields are transformed, gradients are obtained by multiplying by $i\mathbf{k}$ in Fourier space, real-space products are formed, and the pseudo-vector or pseudo-scalar is constructed before shell averaging the final cross-power [2406.15683]. For theory prediction, one development is an FFTLog pipeline for efficient evaluation of the observables for separable parity-odd templates. In that approach, the induced non-Gaussian field and the QQ/LC correlators are reduced to products of radial correlation functions computed through Hankel transforms and FFTLog decompositions, implemented with `pyfftlog`. Reported computational costs are $\sim0.87\,\mathrm{s}$ for $P_{2\times2}$ and $\sim0.79\,\mathrm{s}$ for $P_{3\times1}$ per $k$-range on a laptop [2509.13207]. An earlier POP implementation reported 15 and 17 3D FFTs, with wall-times of approximately $6\,\mathrm{s}$ and $8\,\mathrm{s}$ on 128 cores for a $512^3$ grid [2406.15683]. ## 5. Noise, covariance, and comparison with parity-odd 4PCF methods Parity-odd kurto spectra vanish in mean for parity-even fields, but their variance does not. The dominant variance sources identified in the theoretical and simulation literature are the parity-even trispectrum, disconnected Gaussian terms, and tracer shot noise or stochasticity [2509.13207]. For discrete tracers, Poisson-like $1/\bar n$ contributions enter the composite spectra and can dominate the variance in halos [2509.13207]. In the survey analysis, residual parity-even noise contributes to the variance rather than to a bias, and is reduced by higher number density and smoothing with $k_{\rm max}=0.45\,h\,\mathrm{Mpc}^{-1}$ [2604.06021]. Using the stochasticity-dominated scaling, \[ \mathrm{Var}\propto \bar n^{-4}V^{-1} \] and, for fixed bin width, \[ \mathrm{Var}\propto \Delta k^{-1}\,, \] the DESI-to-BOSS error reduction is predicted and observed at approximately the expected level [2604.06021]. Covariances are estimated numerically from mocks and corrected for finite sample size by the Hartlap correction, \[ \hat{\mathrm{C}}_{ij}=\frac{N_{\mathrm{s}}-1}{N_{\mathrm{s}}-N_{\mathrm{b}}-2}\,\mathrm{C}^{\mathrm{N}}_{ij}\,, \] with $N_{\mathrm{b}}=70$ for BOSS and $N_{\mathrm{b}}=32$ for DESI [2604.06021]. The resulting correlation matrices are close to diagonal in both surveys, with weak off-diagonals largely irrelevant for $\chi^2$ tests [2604.06021]. Earlier POP simulations likewise found covariance matrices diagonally dominated with off-diagonal correlations below $10\%$ [2406.15683]. This near-diagonal covariance is a major methodological distinction relative to parity-odd four-point correlation function analyses. Recent parity-odd 4PCF studies employed data vectors with $N_{\mathrm{b}}\approx2760$ for 10 radial bins up to $18{,}768$ for 18 bins, far exceeding the numbers of available mocks for BOSS and DESI and forcing Gaussian analytic covariance models with imperfect treatment of redshift-space distortions, non-Gaussianities, and survey window effects. By contrast, kurto spectra retain $N_{\mathrm{b}}=70$ or $32$, enabling direct numerical covariance estimation from mocks and reducing sensitivity to covariance-modelling systematics [2604.06021]. A common misconception is that parity-odd compression trivially removes all noise because the mean must vanish in a parity-symmetric universe. The literature does not support that view. The expectation value vanishes, but parity-even Gaussian and nonlinear pieces remain in the variance, and halo stochasticity can be the dominant limitation unless mitigated by weighting or cross-tracer constructions [2509.13207]. ## 6. Measurements in BOSS DR12 and DESI DR1 The first measurement of parity-odd kurto spectra in spectroscopic galaxy survey data was performed on BOSS DR12 and DESI DR1 luminous red galaxies [2604.06021]. For BOSS DR12, the analysis used NGC ($7429\,\mathrm{deg}^2$) and SGC ($2823\,\mathrm{deg}^2$), the redshift range $0.43