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REVIEW 3 major objections 7 minor 1 cited by

The Odd-Parity Part of the Observed Galaxy Trispectrum

T0 review · 3 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Even a parity-symmetric universe produces a parity-odd galaxy trispectrum.

desk verdict Solid follow-up with a robust qualitative claim, but the headline 10–80% amplitudes rest on a truncated third-order kernel and are not yet reproducible from the paper alone. read the letter →

arxiv 2411.10897 v1 pith:LRY2WSF2 submitted 2024-11-16 astro-ph.CO gr-qc

Pritha Paul , Chris Clarkson , Roy Maartens This is my paper · ORCID
classification astro-ph.COgr-qc PACS 98.80.-k98.80.Es
keywords trispectrumparityviolationrelativisticprojectioneffectsredshift-spacedistortionsgalaxynumbercountsfour-pointcorrelationfunctiontetrahedronconfigurationlarge-scalestructure
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that a nonzero parity-odd part of the galaxy trispectrum is not evidence for intrinsic parity violation, because relativistic projection effects generate it on their own. The authors compute the observed galaxy number counts to third order in perturbation theory, keeping the leading Doppler-type terms that are suppressed by $H/k$ and odd under $k \to -k$, and assemble the Fourier-space four-point function (trispectrum) for a tetrahedron of wavevectors. For two representative surveys, at equality scales the odd part is $10\%$ or more of the even Newtonian trispectrum, rising to $60$–$80\%$ for some viewing angles. The consequence is that a parity-odd four-point signal is a standard observational artefact of redshift-space galaxy data, and intrinsic-parity claims must survive its subtraction.

What carries the argument

The load-bearing object is the odd-parity trispectrum $T_{\rm odd} = \frac{1}{2}[T_g(k_1,k_2,k_3,k_4)-T_g(-k_1,-k_2,-k_3,-k_4)]$, which isolates the imaginary part of the four-point spectrum. The calculation is carried by the relativistic kernels $K_{\rm GR}^{(1)}, K_{\rm GR}^{(2)}, K_{\rm GR}^{(3)}$, which contain only odd powers of $\mu_i = \hat{k}_i \cdot \hat{n}$ (the cosine between each wavevector and the line of sight), whereas the Newtonian kernels contain only even powers. These odd powers encode the Doppler, evolution-bias, and magnification-bias terms that break symmetry under $k \to -k$ once a line of sight is fixed. Inserted into the tree-level trispectrum with the tetrahedron geometry parameterized by viewing angles $\theta,\phi$ and configuration/folding angles $\Theta,\Phi,\Psi$, they produce the nonzero $T_{\rm odd}$ that the paper computes.

What would settle it

Recompute the trispectrum with the dropped lensing, integrated Sachs–Wolfe, and gravitational-potential terms included, or replace the plane-parallel single-line-of-sight approximation with a wide-angle light-cone computation; if the dropped terms contribute at the same order as the kept Doppler terms, or if wide-angle effects significantly alter the angular dependence, the numerical $10$–$80\%$ claim fails even though a nonzero odd part survives. A simulation-based test is also decisive: generate a parity-symmetric field, apply the full relativistic mapping, and check whether the measured odd trispectrum matches the predicted dependence on viewing angles.

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Extended reading notes

Core claim

The paper's central claim is that even when the underlying matter density field is parity symmetric, the observed galaxy trispectrum in redshift space has a nonzero odd-parity part, because the relativistic corrections to the number counts are not invariant under reversing every wavevector. Working at tree level, the authors split the trispectrum into $T_{\rm even}$ and $T_{\rm odd} = \frac{1}{2}[T_g(k_1,k_2,k_3,k_4)-T_g(-k_1,-k_2,-k_3,-k_4)]$, derive the first-, second-, and third-order Newtonian and relativistic kernels, and evaluate the ratio $|T_{\rm odd}/T_{\rm even}|$ numerically over the tetrahedron's configuration and viewing angles. For the two representative surveys (a spectroscopic galaxy survey and a 21cm intensity-mapping survey), the ratio is $10\%$ or larger at $k \simeq 0.01\,h\,{\rm Mpc}^{-1}$ and reaches $60$–$80\%$ for particular viewing angles. The authors conclude that the apparent parity violation in the four-point function is a relativistic projection effect whose size is set by $H/k$ and by the survey's evolution and magnification biases.

Load-bearing premise

The numerical amplitudes assume that the only relativistic corrections that matter are the Doppler-type line-of-sight terms kept in the kernels, with lensing, integrated Sachs–Wolfe, and gravitational-potential terms negligible, and that a single global line of sight is accurate.

Editorial extensions

If this is right

  • Any measurement of the galaxy four-point function in redshift space must subtract or model the relativistic odd-parity contribution before an intrinsic parity-violation claim can be made.
  • At scales near equality ($k \simeq 0.01\,h\,{\rm Mpc}^{-1}$) the odd part is at least $10\%$ of the even Newtonian trispectrum, so the effect is not negligible in wide-angle surveys.
  • The contamination is survey-specific: it depends on the evolution and magnification biases, so different galaxy surveys will see different amplitudes for the same geometry.
  • Strong cancellations occur between the $T_{1113}$ and $T_{1122}$ contributions, so the total odd signal can be much smaller than the individual terms—relevant for forecasting detectability.
  • Because the effect grows with $H/k$, it becomes more important at larger scales and higher redshifts, exactly where future surveys gain new sky coverage.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because published parity-odd 4PCF detections average over lines of sight, an orientation-resolved measurement would directly separate this relativistic template from any intrinsic signal.
  • Wide-angle corrections beyond the single-line-of-sight plane-parallel approximation could redistribute the odd signal across multipoles or change the 10–80% amplitudes, an extension the paper leaves for future work.
  • The same third-order machinery could jointly model the parity-even part of the trispectrum, where relativistic corrections also enter, potentially yielding cleaner constraints on $H/k$ effects and on primordial non-Gaussianity.
  • A signal-to-noise forecast for upcoming surveys would determine whether the predicted odd part is actually detectable rather than merely present—a step the paper does not take.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The paper computes the tree-level galaxy trispectrum in redshift space, including leading relativistic corrections to the observed number counts up to third order in perturbation theory. It separates the trispectrum into parity-even and parity-odd parts under the transformation k_i -> -k_i with a fixed line-of-sight direction, and numerically evaluates the ratio |T_odd/T_even| for Euclid-like and SKA-like survey parameters as a function of configuration angles, viewing angles, scale, folding angle, and redshift. The central quantitative claim is that relativistic projection effects produce a parity-odd trispectrum at the 10% or more level, reaching about 80% in parts of parameter space at k ~ 0.01 h/Mpc.

Significance. If the quantitative claim is correct, the paper identifies an important astrophysical systematic for the recent and future searches for parity violation in the galaxy 4-point function: a parity-odd trispectrum can arise purely from relativistic projection effects even when the underlying density field is parity symmetric. The paper has clear strengths: it correctly identifies the trispectrum as the lowest-order parity-sensitive statistic for a scalar field, assembles the tree-level trispectrum in a standard way, provides explicit first- and second-order kernels and a long third-order GR kernel, and explores a broad parameter space with realistic Euclid- and SKA-like bias parameters using CAMB power spectra. The qualitative conclusion, that a nonzero parity-odd part exists, is robust. However, the headline amplitudes are not fully verifiable from the manuscript as written because the third-order relativistic kernel is presented only through 'dominant' terms, and the treatment of neglected lensing, ISW, and gravitational-potential contributions is not quantified.

major comments (3)
  1. [§4.3, Eq. (4.26)] The text states: 'The full expression for the third-order relativistic terms is extremely long and we have given the terms that dominate the relativistic corrections.' The manuscript does not specify which terms of Eq. (4.24) are omitted, why they are subdominant, or provide the full expression. Since Eq. (4.26) enters directly into the trispectrum through Eq. (3.10), and since the paper's central claim is the 10-80% amplitude of |T_odd/T_even|, the amplitude claim is not reproducible or checkable from the paper as written. Please provide the complete K_GR^(3), or an explicit list of the dropped terms with scaling estimates, and reconcile this with Appendix C: if Appendix C is intended to be the full kernel, state that explicitly and remove the 'dominant terms' wording.
  2. [§4.1 vs §4.3 and Appendix C] Section 4.1 says 'we also neglect the terms that involve gravitational potentials, which are responsible for the effect arising from gravitational redshift... although we leave it for a future analysis.' This is inconsistent with Eq. (4.26), which contains many Omega_m terms originating from the psi couplings in Eq. (4.24) (for example the -9/2 Omega_m bracket), and with Appendix C, which lists these psi terms explicitly. If potential terms are included at third order, the text must say so; if they are meant to be dropped, then those Omega_m terms should be removed and the numerical results recomputed. This matters because the potential terms also carry odd powers of mu and contribute to T_odd.
  3. [§4.1 and §5.2] The calculation neglects the integrated lensing-magnification and ISW terms in Eq. (4.7). These are not estimated anywhere, even though they can contribute to parity-odd correlators at comparable order in H/k in some survey configurations, especially at high redshift where magnification bias is non-negligible. Please provide an order-of-magnitude estimate for their contribution to T_odd, or include the dominant integrated terms, or give a quantitative argument for why they are subdominant at the scales and redshifts used for the 10-80% claim.
minor comments (7)
  1. [Throughout] There are many typographical errors, including 'T rispectrum' in the title, 'bipsectrum', 'signifcantly', 'evergy', and 'asymmetries'; the manuscript needs a careful proofreading pass.
  2. [References] Reference [7] (Cahn, Slepian, Hou 2021) is missing publication details; please add the journal, arXiv identifier, or DOI.
  3. [§4.3, Eq. (4.21)] The bias expansion 'delta_g^(3) = b1 delta^(3) + 3b2 delta delta^2 + b3 delta^3' is notationally ambiguous; write the second-order field explicitly as delta^(2) and clarify the factorial/normalization conventions used for b2 and b3.
  4. [§5.2, Figs. 2-3] The color maps do not have color bars, and the text says the bright yellow regions are where T_even = 0, but the plotted ratio |T_odd/T_even| is then undefined; please add color bars and state how singular regions are treated in the plots.
  5. [§5.5] The bias relation in Eq. (5.9) and the redshift-dependent evolution and magnification bias values should be referenced explicitly to the relevant table in [36].
  6. [§5.1 and §5.2] The numerical results use a single global line-of-sight direction in Eq. (5.2); for the smallest k values shown (k ~ 10^-3 h/Mpc) the combination k r is only of order ten, so the plane-parallel approximation may break down at the same order as the H/k effects being computed. Please state the range of validity of this approximation.
  7. [§6] The conclusion refers to the 'monopole of the trispectrum' and its 'dipole' without having defined a multipole decomposition of the trispectrum earlier; please define these terms or rephrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the odd-parity trispectrum follows algebraically from standard relativistic number-count kernels, with no fitted parameter renamed as a prediction.

full rationale

The paper's central claim is that relativistic Doppler-type projection effects, which enter the galaxy number counts with odd powers of mu = k-hat·n, generate a parity-odd trispectrum even when the underlying density field is parity-symmetric. The derivation chain is: standard first- and second-order relativistic number counts from the literature, third-order number counts from Di Dio & Seljak [19], Fourier-space kernels obtained by explicitly transforming those expressions (Appendices A-C), then the tree-level trispectrum via Wick contractions and the odd/even split defined in Eqs. (5.7)-(5.8). The non-zero odd part is a direct algebraic consequence of the odd-mu structure of the input relativistic kernels, not an assumption equivalent to the conclusion. No parameter is fitted to the target trispectrum; the survey bias parameters are taken from an external forecast table [36], and the matter power spectrum is computed with CAMB. The self-citations [16, 25, 27, 36] are to earlier derivations, to a companion paper giving the same formalism, or to survey parameter tables; they are not the sole justification for the load-bearing input, which is independently supported by references [18, 32, 33, 34]. The explicit truncation of the full third-order relativistic kernel in Section 4.3 is a completeness and reproducibility concern, not a circularity: it affects the numerical amplitude of the 10-80% ratios, but the qualitative structure follows from the stated leading-order kernel. The paper therefore contains no circular step that reduces a prediction to its inputs by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new entities are introduced and no constants are fitted inside the paper. The numerical amplitude depends on survey bias parameters from [36] and on the standard perturbative framework. The main burdens are the assumed Gaussian initial conditions, the plane-parallel LOS, and the truncation of omitted relativistic effects.

free parameters (2)
  • Euclid-like bias set (b1, b2, b1', be, s) = b1=1.2, b2=-0.74, b1'=-1.6e-4 Mpc^-1, be=-4, s=-0.95 at z=1
    Taken from [36] to set the numerical amplitude; the parity-odd effect exists regardless of these values.
  • SKA-like bias set (b1, b2, b1', be, s) = b1=0.856, b2=-0.321, b1'=-0.5e-4 Mpc^-1, be=-0.5, s=1
    Same [36] input for the 21cm intensity mapping case.
assumptions (6)
  • standard math Wick's theorem and Gaussian initial conditions justify the tree-level contractions in Eqs. (3.8) and (3.9).
    Used in Section 3 to reduce the four-point function.
  • domain assumption Standard perturbation theory kernels F2, F3, G2, G3 from [34] describe the matter density and velocity at second and third order.
    Invoked in Eqs. (4.11)-(4.12) and (4.18)-(4.19).
  • domain assumption The relativistic galaxy number count expressions of [13-15, 19] are correct at first, second, and third order.
    Eqs. (4.7), (4.14)-(4.15), and (4.23)-(4.24) are taken from this literature, some co-authored by the present authors.
  • domain assumption A single global line-of-sight vector n applies to all four wavevectors (plane-parallel approximation).
    Eqs. (5.1)-(5.6) fix one n; wide-angle effects are not modeled.
  • domain assumption Integrated lensing, ISW, and gravitational potential terms are negligible at the orders considered.
    Stated in Section 4.1; these terms are left for future work.
  • ad hoc to paper The galaxy bias expansion is truncated at third order with b3=0 and no tidal bias.
    Section 5.2: 'We ignore tidal bias and set the third-order bias to 0 for simplicity.'

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Cite this review

Pith. "Pith review of The Odd-Parity Part of the Observed Galaxy Trispectrum." pith.science (2026). https://pith.science/paper/LRY2WSF2

@misc{pith2026241110897,
  author       = {Pith},
  title        = {Pith review of: The Odd-Parity Part of the Observed Galaxy Trispectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LRY2WSF2}},
  note         = {Machine review of arXiv:2411.10897}
}
read the original abstract

Recently the galaxy matter density 4-point correlation function has been looked at to investigate parity violation in large scale structure surveys. The 4-point correlation function is the lowest order statistic which is sensitive to parity violation, since a tetrahedron is the simplest shape that cannot be superimposed on its mirror image by a rotation. If the parity violation is intrinsic in nature, this could give us a window into inflationary physics. However, we need to exhaust all other contaminations before we consider them to be intrinsic. Even though the standard Newtonian redshift-space distortions are parity symmetric, the full relativistic picture is not. Therefore, we expect a parity-odd trispectrum when observing in redshift space. We calculate the trispectrum with the leading-order relativistic effects and investigate in detail the parameter space of the trispectrum and the effects of these relativistic corrections for different parameter values and configurations. We also look at different surveys and how the evolution and magnification biases can be affected by different parameter choices.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Measurement of Parity-Violating Modes of the Dark Energy Spectroscopic Instrument (DESI) Year 1 Luminous Red Galaxies' 4-Point Correlation Function

    astro-ph.CO 2025-08 conditional novelty 6.0 of 10

    DESI's first parity-violation search shows a strong auto-correlation signal that disappears when the sky is split into patches, pointing to underestimated mocks' variance rather than new physics.

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