Parity-Odd Kurto Spectra

• Parity-odd kurto spectra are one-dimensional observables that compress the scalar trispectrum’s parity-odd information via composite-field constructions.
• They are constructed by cross-correlating fields with opposite parity properties using quadratic–quadratic and cubic–linear estimators.
• Their reduced dimensionality offers computational advantages and robust statistical tests for detecting cosmic parity violation in galaxy surveys.

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 (Jamieson et al., 2024, Gao et al., 16 Sep 2025, Gao et al., 7 Apr 2026).

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 $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$ (Gao et al., 7 Apr 2026). 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 (Gao et al., 7 Apr 2026).

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 (Jamieson et al., 2024). 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 $s$0, while reducing the statistic to a function of a single wavenumber $s$1. This makes the observable formally analogous to a power spectrum, but targeted to the parity-odd part of the four-point function (Jamieson et al., 2024, Gao et al., 16 Sep 2025).

2. Composite-field construction and the two standard estimators

The general composite-field construction begins from local products of filtered fields,

$s$2

with Fourier transform given by the corresponding convolution, and composite-field spectra defined by

$s$3

This compresses the $s$4-point function into a two-point statistic (Gao et al., 7 Apr 2026).

Two parity-odd kurto spectra are standard in the recent literature. The first is the quadratic–quadratic, or vector–pseudo-vector, estimator,

$s$5

with quadratic fields

$s$6

Since $s$7 is a true vector and $s$8 a pseudo-vector, their cross-power flips sign under parity (Gao et al., 7 Apr 2026). In related notation, this estimator appears as the quadratic–quadratic kurto spectrum $s$9 or the vector POP spectrum (Gao et al., 16 Sep 2025, Jamieson et al., 2024).

The second is the cubic–linear, or scalar–pseudo-scalar, estimator,

$s(\mathbf{x})\to s(-\mathbf{x})$0

where

$s(\mathbf{x})\to s(-\mathbf{x})$1

Here $s(\mathbf{x})\to s(-\mathbf{x})$2 is a pseudo-scalar and $s(\mathbf{x})\to s(-\mathbf{x})$3 is a scalar, so the expectation value again vanishes unless parity is violated (Gao et al., 7 Apr 2026). In the theoretical literature this appears as the linear–cubic kurto spectrum $s(\mathbf{x})\to s(-\mathbf{x})$4 or the scalar POP spectrum (Gao et al., 16 Sep 2025, Jamieson et al., 2024).

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 $s(\mathbf{x})\to s(-\mathbf{x})$5 and $s(\mathbf{x})\to s(-\mathbf{x})$6, to avoid cancellation of longitudinal pieces or vanishing of the pseudo-vector (Gao et al., 16 Sep 2025, Jamieson et al., 2024). The survey analysis adopts

$s(\mathbf{x})\to s(-\mathbf{x})$7

for the filtered overdensity fields $s(\mathbf{x})\to s(-\mathbf{x})$8 (Gao et al., 7 Apr 2026).

3. Relation to the parity-odd trispectrum

The parity-odd scalar trispectrum is the object being compressed. In one formulation,

$s(\mathbf{x})\to s(-\mathbf{x})$9

with the imaginary part $p$0 proportional to

$p$1

where the triple product determines the handedness and $p$2 is totally antisymmetric under permutations (Jamieson et al., 2024). 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,

$p$3

and

$p$4

where the kernels are constructed from the operators, filters, and smoothing window used in the estimator (Gao et al., 7 Apr 2026).

The presence of $p$5 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 (Gao et al., 7 Apr 2026). 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 $p$6 and $p$7 appearing in the compression (Jamieson et al., 2024).

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 $p$8 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 (Gao et al., 7 Apr 2026, Gao et al., 16 Sep 2025).

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

$p$9

with a Heaviside-like tanh filter

$p(\mathbf{x})\to -p(-\mathbf{x})$0

filtering scales smaller than $p(\mathbf{x})\to -p(-\mathbf{x})$1 (Gao et al., 7 Apr 2026). Closely related work also advocates smooth tanh windows to mimic sharp cuts while avoiding ringing (Gao et al., 16 Sep 2025).

In survey data, the fluctuation field is constructed from galaxies and random catalogues through an FKP-like weighted field,

$p(\mathbf{x})\to -p(-\mathbf{x})$2

with

$p(\mathbf{x})\to -p(-\mathbf{x})$3

and a trispectrum-appropriate normalization $p(\mathbf{x})\to -p(-\mathbf{x})$4 (Gao et al., 7 Apr 2026). For BOSS DR12, $p(\mathbf{x})\to -p(-\mathbf{x})$5 is approximated via Monte Carlo sums over catalogue objects; for DESI DR1 it is approximated on the FFT mesh with a $p(\mathbf{x})\to -p(-\mathbf{x})$6-dependent expression and $p(\mathbf{x})\to -p(-\mathbf{x})$7 chosen for stability (Gao et al., 7 Apr 2026).

The practical estimator is FFT-based. The field $p(\mathbf{x})\to -p(-\mathbf{x})$8 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 $p(\mathbf{x})\to -p(-\mathbf{x})$9-binning (Gao et al., 7 Apr 2026). The POP work described a closely related real-space/Fourier-space pipeline: filtered fields are transformed, gradients are obtained by multiplying by $v_i$0 in Fourier space, real-space products are formed, and the pseudo-vector or pseudo-scalar is constructed before shell averaging the final cross-power (Jamieson et al., 2024).

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 $v_i$1 for $v_i$2 and $v_i$3 for $v_i$4 per $v_i$5-range on a laptop (Gao et al., 16 Sep 2025). An earlier POP implementation reported 15 and 17 3D FFTs, with wall-times of approximately $v_i$6 and $v_i$7 on 128 cores for a $v_i$8 grid (Jamieson et al., 2024).

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 (Gao et al., 16 Sep 2025). For discrete tracers, Poisson-like $v_i$9 contributions enter the composite spectra and can dominate the variance in halos (Gao et al., 16 Sep 2025).

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 $v_i(\mathbf{x})\to -v_i(-\mathbf{x})$0 (Gao et al., 7 Apr 2026). Using the stochasticity-dominated scaling,

$v_i(\mathbf{x})\to -v_i(-\mathbf{x})$1

and, for fixed bin width,

$v_i(\mathbf{x})\to -v_i(-\mathbf{x})$2

the DESI-to-BOSS error reduction is predicted and observed at approximately the expected level (Gao et al., 7 Apr 2026).

Covariances are estimated numerically from mocks and corrected for finite sample size by the Hartlap correction,

$v_i(\mathbf{x})\to -v_i(-\mathbf{x})$3

with $v_i(\mathbf{x})\to -v_i(-\mathbf{x})$4 for BOSS and $v_i(\mathbf{x})\to -v_i(-\mathbf{x})$5 for DESI (Gao et al., 7 Apr 2026). The resulting correlation matrices are close to diagonal in both surveys, with weak off-diagonals largely irrelevant for $v_i(\mathbf{x})\to -v_i(-\mathbf{x})$6 tests (Gao et al., 7 Apr 2026). Earlier POP simulations likewise found covariance matrices diagonally dominated with off-diagonal correlations below $v_i(\mathbf{x})\to -v_i(-\mathbf{x})$7 (Jamieson et al., 2024).

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 $v_i(\mathbf{x})\to -v_i(-\mathbf{x})$8 for 10 radial bins up to $v_i(\mathbf{x})\to -v_i(-\mathbf{x})$9 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 $a_i$0 or $a_i$1, enabling direct numerical covariance estimation from mocks and reducing sensitivity to covariance-modelling systematics (Gao et al., 7 Apr 2026).

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 (Gao et al., 16 Sep 2025).

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 (Gao et al., 7 Apr 2026). For BOSS DR12, the analysis used NGC ($a_i$2) and SGC ($a_i$3), the redshift range $a_i$4, approximately $a_i$5 galaxies in NGC and $a_i$6 in SGC, mean number density approximately $a_i$7, and 70 bins with $a_i$8 (Gao et al., 7 Apr 2026). For DESI DR1, the analysis used NGC ($a_i$9), SGC ($a_i(\mathbf{x})\to +a_i(-\mathbf{x})$0), and sub-patches NGC-1, NGC-2, and SGC-3, over $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$1, with $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$2 up to $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$3, and 32 bins with $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$4 (Gao et al., 7 Apr 2026).

Mock suites were central to the analysis. BOSS used 2048 MultiDark-Patchy realizations and 2000 GLAM-Uchuu realizations; DESI used 1000 EZmocks plus 25 AbacusSummit realizations each for FFA and altMTL fibre assignment (Gao et al., 7 Apr 2026). In BOSS, Patchy variances were approximately $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$5 larger than Uchuu variances across $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$6; in DESI, EZmocks exhibited small off-diagonals, while Abacus showed larger off-diagonals due to limited $a_i(\mathbf{x})\to +a_i(-\mathbf{x})$7 (Gao et al., 7 Apr 2026).

The null hypothesis was that the expectation value of each parity-odd kurto spectrum is zero. Tests used Gaussian likelihoods with

$a_i(\mathbf{x})\to +a_i(-\mathbf{x})$8

and a signal-sensitive cross-patch statistic

$a_i(\mathbf{x})\to +a_i(-\mathbf{x})$9

following Krolewski et al. (Gao et al., 7 Apr 2026).

The principal result was non-detection in both surveys. In BOSS DR12, both $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$0 and $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$1 fluctuated about zero within mock scatters; apparent large-scale outliers in $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$2 were not shared across patches and were therefore not robust (Gao et al., 7 Apr 2026). The $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$3 values lay within mock distributions, and cross-patch $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$4 was consistent with zero across NGC$\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$5SGC (Gao et al., 7 Apr 2026).

In DESI DR1, both spectra again fluctuated about zero. The largest nominal deviations were $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$6 for SGC-Full in $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$7 and $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$8 for NGC-Full in $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$9, but these were sensitive to a few small-scale modes and not robust under $s$00-cuts; cross-patch $s$01 values were also consistent with the null, with the largest deviation of approximately $s$02 for NGC-1$s$03NGC-2, plausibly reflecting shared local systematics (Gao et al., 7 Apr 2026). The DESI DR1 scatter was smaller than that of BOSS DR12 by about a factor of four, consistent with the higher tracer number density and larger volume (Gao et al., 7 Apr 2026).

7. Theory templates, systematics, and outlook

The theoretical motivation for parity-odd kurto spectra includes primordial and late-time sources of parity violation. The literature explicitly mentions inflationary mechanisms such as axion-gauge couplings, massive spin exchange, and ghost inflation; late-time gravity such as chiral gravity; and astrophysical sources such as helical magnetic fields (Gao et al., 16 Sep 2025). One tested primordial template is a separable contact-type parity-odd trispectrum generated by a cubic correction to the primordial field,

$s$04

leading to

$s$05

This maps to matter through transfer functions and to galaxies approximately as $s$06 under linear bias (Gao et al., 16 Sep 2025).

Simulation studies validated the estimators on perturbative dark matter fields and on Quijote simulations with and without parity-odd initial conditions, in real and redshift space (Gao et al., 16 Sep 2025). For dark matter, phase-matched fiducial subtraction,

$s$07

substantially suppresses parity-even variance because the fiducial and odd simulations share phases; for halos, stochasticity differences make the subtraction much less effective (Gao et al., 16 Sep 2025). Optimal inverse-variance weights and halo–matter cross kurto spectra were found to mitigate halo stochasticity and improve detectability (Gao et al., 16 Sep 2025). This suggests that survey applications could benefit from template-matched weighting and cross-tracer constructions, although the BOSS/DESI analysis did not introduce a parametric amplitude model and remained a non-parametric null test (Gao et al., 7 Apr 2026).

Survey systematics are treated largely through forward realism rather than exact deconvolution. Redshift-space distortions and survey geometry are incorporated in the window-convolved estimators through random catalogues and FFTs, while mock-based covariances capture anisotropies without relying on Gaussian analytic approximations (Gao et al., 7 Apr 2026). In DESI, Abacus mocks with FFA and altMTL showed small differences in $s$08 distributions, suggesting weak sensitivity to fibre assignment at current precision, plausibly because the smoothing removes fluctuations below $s$09 (Gao et al., 7 Apr 2026). The pseudo constructions cancel parity-even contributions in expectation, leaving parity-even noise as variance rather than bias (Gao et al., 7 Apr 2026).

The current evidence therefore supports a conservative conclusion: parity-odd kurto spectra provide a physically motivated, low-dimensional, and computationally efficient compression of parity-odd trispectrum information, and the first survey measurements in BOSS DR12 and DESI DR1 find no evidence for cosmological parity violation in the scalar sector (Gao et al., 7 Apr 2026). Future DESI releases, with larger volume and number density, together with larger suites of high-fidelity mocks and template-based analyses, are expected to sharpen these tests substantially (Gao et al., 7 Apr 2026). A plausible implication is that the method’s long-term significance will depend less on the basic estimator design—which is already well defined—and more on improvements in mock fidelity, weighting, and the construction of template banks for specific parity-violating scenarios such as massive-spin exchange, axion–gauge models, and helical primordial magnetic fields (Gao et al., 16 Sep 2025, Gao et al., 7 Apr 2026).