Primordial Parity-Odd Trispectrum

• Primordial parity-odd trispectrum is the imaginary component of the curvature four-point function, signaling chiral interactions and mirror asymmetry during inflation.
• It emerges in non-coplanar momentum configurations through Levi-Civita tensor structures and can be sourced by mechanisms like Chern–Simons gravity and axion couplings.
• Observational strategies using CMB and large-scale structure data aim to detect these parity-violating signals despite tree-level null results and complex nonlinear evolution.

Searching arXiv for the most relevant papers on primordial parity-odd trispectra and closely related observational/theoretical developments. arxiv_search(query="primordial parity-odd trispectrum inflation Chern-Simons axion scalar trispectrum", max_results=10) The primordial parity-odd trispectrum is the parity-odd component of the connected scalar four-point function of the primordial curvature perturbation $\zeta$. In statistically homogeneous and isotropic cosmologies, it is the lowest-order scalar correlator that can encode mirror asymmetry: two- and three-point scalar correlators are parity-even, whereas the scalar trispectrum can carry Levi–Civita structures and is therefore sensitive to parity-violating dynamics during or before inflation (Cabass et al., 2022, Shiraishi, 2016). For a real field, the odd part is purely imaginary, and its existence signals chiral interactions, chiral propagation, or pseudoscalar sourcing in the primordial sector (Fujita et al., 2023, Creque-Sarbinowski et al., 2023).

1. Definition and symmetry structure

The momentum-space definition of the curvature trispectrum is

$$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$

with the parity decomposition

$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$

Under $\mathbf{k}_i\to -\mathbf{k}_i$, the odd part changes sign while the even part is invariant (Creque-Sarbinowski et al., 2023). Equivalently, for real $\zeta$,

$$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$

so the imaginary part is the parity-odd signal (Fujita et al., 2023).

The kinematic origin of parity oddness is the Levi–Civita tensor. A generic parity-odd scalar trispectrum contains pseudoscalar factors such as

$\epsilon_{abc}\,k_1^a k_2^b k_3^c$

or, after normalization by magnitudes, $$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$ (Lee et al., 2023, Fujita et al., 2023). These structures vanish in planar configurations. This is why several explicit models predict that the parity-odd signal is absent for coplanar momentum tetrahedra and is largest for genuinely three-dimensional configurations (Niu et al., 2022, Yura et al., 22 May 2025).

A widely used isotropic parity-odd template is the Legendre-expanded form

$$t_{k_3 k_4 k_1 k_2}(K) = i \sum_n d_n^{\rm odd} \Big[ P_n(\hat{\mathbf{k}}_1\!\cdot\!\hat{\mathbf{k}}_3) + P_n(\hat{\mathbf{k}}_1\!\cdot\!\hat{\mathbf{K}}) + (-1)^n P_n(\hat{\mathbf{k}}_3\!\cdot\!\hat{\mathbf{K}}) \Big] \Big[(\hat{\mathbf{k}}_1\times \hat{\mathbf{k}}_3)\cdot \hat{\mathbf{K}}\Big] P_\zeta(k_1)P_\zeta(k_3)P_\zeta(K),$$

which makes explicit that the odd sector is imaginary and odd under mirror reflection (Shiraishi, 2016).

2. No-go theorems, reality, and loop-leading behavior

A central theoretical result is that the parity-odd scalar trispectrum obeys strong null theorems under standard inflationary assumptions. At tree level, in the decoupling limit of the Effective Field Theory of Inflation, with exact scale invariance and a Bunch–Davies vacuum, the parity-odd scalar trispectrum vanishes for any number of scalar fields with arbitrary mass (Cabass et al., 2022). Closely related analyses show that, for massless external scalars in the exact scale-invariant limit, the tree-level parity-odd trispectrum also vanishes under unitary time evolution and Bunch–Davies initial conditions (Lee et al., 2023).

These null results are tied to analyticity and reality properties of the late-time wavefunction. For massless external scalars and gravitons, the maximally-connected parts of the wavefunction coefficients that generate total-energy singularities are purely real at tree level, so parity-odd correlators are factorized and do not diverge when total energy is conserved (Stefanyszyn et al., 2023). A non-perturbative conformal argument further implies that if full de Sitter isometries hold, parity-odd scalar four-point functions vanish because four boundary points can be conformally mapped to a plane, leaving no nontrivial parity-odd scalar structure (Cabass et al., 2022).

Relaxing these assumptions produces “yes-go” cases. The explicit mechanisms identified in the literature are: violations of scale invariance in single-clock inflation, modified dispersion relations such as the ghost condensate or ghost inflation, and interactions involving massive spinning fields (Cabass et al., 2022). In addition, one-loop effects can become the leading source of parity oddness. In the exact scale-invariant limit, the one-loop parity-odd trispectrum of a massless scalar is non-vanishing, UV finite, purely imaginary, and remarkably simple: it is a rational function of momenta with only total-energy poles and no branch cuts or partial-energy poles (Lee et al., 2023). This loop-leading structure is regular in the collapsed limit and suppressed in collinear limits by the antisymmetric Levi–Civita factor (Lee et al., 2023).

3. Microscopic mechanisms

Several concrete primordial mechanisms generate parity-odd scalar trispectra.

In dynamical Chern–Simons gravity, the inflaton couples to the gravitational Pontryagin density,

$$S = \int d^4x\,\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}R - \frac{1}{2}\,(\partial \phi)^2 - V(\phi) + \frac{1}{4f}\,\phi\, R\tilde{R}\right],$$

and the graviton helicities satisfy

$$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$0

One helicity undergoes a tachyonic instability, producing amplitude birefringence and a nonzero gravitational circular polarization

$$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$1

Graviton-mediated exchange between scalar pairs then yields a parity-odd scalar trispectrum linear in $$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$2 (Creque-Sarbinowski et al., 2023).

A distinct route uses a rolling spectator axion $$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$3 coupled to a $$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$4 gauge field through a Chern–Simons term. In conformal time the gauge-field helicities obey

$$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$5

and for $$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$6 only the $$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$7 mode is amplified (Fujita et al., 2023). The amplified gauge field sources axion fluctuations through $$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$8, which then source inflaton fluctuations and hence $$\big\langle \zeta_{\mathbf{k}_1}\,\zeta_{\mathbf{k}_2}\,\zeta_{\mathbf{k}_3}\,\zeta_{\mathbf{k}_4}\big\rangle = (2\pi)^3\,\delta^{(3)}(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4)\; T_\zeta(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4),$$9. The resulting sourced trispectrum is complex; its imaginary part is parity odd and is generated by helicity-projector contractions carrying Levi–Civita structure (Fujita et al., 2023).

Single-field axion inflation with $T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$0 coupling also produces a parity-odd scalar trispectrum through one-loop gauge-field exchange. In that case the transverse gauge modes obey

$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$1

with one helicity exponentially amplified. The parity-odd part requires non-coplanar momentum configurations and is smaller than the parity-even part: in the approximately massless case it is typically one to two orders of magnitude smaller, while for $T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$2 it is suppressed by about three orders of magnitude (Niu et al., 2022).

Chiral scalar-tensor theories of gravity extend dCS by including parity-violating operators built from first and second derivatives of the inflaton. In the parameter choice emphasized in the paper, chirality arises from the PV2 sector, producing unequal tensor power spectra

$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$3

while PV1 operators generate $T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$4 interaction vertices that mediate a parity-odd scalar trispectrum (Moretti et al., 2024).

Helical primordial magnetic fields provide a non-inflaton mechanism. Their two-point function contains

$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$5

so the helical part $T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$6 directly sources the imaginary, parity-odd component of the scalar trispectrum through the passive scalar mode $T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$7 (Yura et al., 22 May 2025).

4. Shapes, limits, and spin diagnostics

The parity-odd trispectrum is highly shape dependent. In the dCS graviton-exchange model, the collapsed limit yields analytic templates,

$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$8

$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)}.$9

with

$\mathbf{k}_i\to -\mathbf{k}_i$0

The $\mathbf{k}_i\to -\mathbf{k}_i$1 dependence is a direct imprint of spin-2 exchange and distinguishes graviton-mediated parity oddness from spin-0 or spin-1 exchange (Creque-Sarbinowski et al., 2023).

Axion–gauge models instead favor equilateral-like shapes because particle production is localized near horizon crossing. In the spectator-axion model, the parity-odd fraction reaches $\mathbf{k}_i\to -\mathbf{k}_i$2 in exact equilateral configurations and can reach or surpass the parity-even part in quasi-equilateral shapes where one side is slightly longer than the others (Fujita et al., 2023). In single-field axion inflation with gauge production, the odd component vanishes at $\mathbf{k}_i\to -\mathbf{k}_i$3 and is largest for intermediate non-planar angles; in the massless case $\mathbf{k}_i\to -\mathbf{k}_i$4 in the representative $\mathbf{k}_i\to -\mathbf{k}_i$5 scan, while the massive case is much more strongly suppressed (Niu et al., 2022).

The loop-leading parity-odd trispectrum derived in the exact scale-invariant limit is unusual in that it has only total-energy poles, no partial-energy poles, and a regular collapsed limit (Lee et al., 2023). By contrast, the helical-PMF trispectrum can display local-type divergences in collapsed channels through pole enhancement, yet its odd part still vanishes in planar limits because the scalar triple product disappears there (Yura et al., 22 May 2025).

These shape statements matter observationally because different templates project onto different experimental configurations. Collapsed graviton exchange is naturally targeted by large-scale-structure mode-coupling estimators, whereas equilateral and quasi-equilateral shapes are better matched to tetrahedral 4-point estimators and orientation-dependent analyses (Creque-Sarbinowski et al., 2023, Fujita et al., 2023).

5. Projection to the CMB and large-scale structure

In harmonic space, a primordial parity-odd scalar trispectrum projects onto a CMB trispectrum with support only in the domain

$\mathbf{k}_i\to -\mathbf{k}_i$6

whereas parity-even scalar trispectra live only in the even-sum sector (Shiraishi, 2016). This selection rule follows from the parity transformation $\mathbf{k}_i\to -\mathbf{k}_i$7 together with the fact that the primordial odd trispectrum is imaginary (Shiraishi, 2016). For an isotropic parity-odd dipolar template, a Fisher analysis for a full-sky, cosmic-variance-limited temperature survey up to $\mathbf{k}_i\to -\mathbf{k}_i$8 gives a minimum detectable coefficient $\mathbf{k}_i\to -\mathbf{k}_i$9, with $\zeta$0 (Shiraishi, 2016).

A direct CMB temperature search on Planck 2018 SMICA data over $\zeta$1 found no evidence for parity violation. The parity-odd trispectrum is consistent with parity conservation at roughly $\zeta$2, and among eight primordial models the maximum apparent detection significance is $\zeta$3 (Philcox, 2023). That study also argues that the CMB null result disfavors a primordial interpretation of recent parity-odd signals reported in galaxy four-point statistics, because the CMB contains roughly $\zeta$4 more primordial modes and is easier to interpret owing to linear physics and near-Gaussian statistics (Philcox, 2023).

In large-scale structure, the primordial curvature trispectrum transfers to galaxy and matter four-point functions. For dCS gravity, the parity-odd galaxy trispectrum in the collapsed limit leads to the mode-counting sensitivity estimate

$\zeta$5

and the same analysis quotes $\zeta$6 as a representative enhancement threshold for detectability in extended models (Creque-Sarbinowski et al., 2023). In the CST framework, the authors estimate that a parameter choice exists for which the signal-to-noise ratio of the parity-violating part of the trispectrum is of order one without modifying the single-field slow-roll background (Moretti et al., 2024).

At the same time, the observed galaxy trispectrum is not a pristine primordial observable. Even if the primordial trispectrum is parity-even, relativistic projection effects in redshift space generate a parity-odd observed trispectrum. In the leading-order relativistic analysis, $\zeta$7 is typically $\zeta$8 near equality scales and can reach $\zeta$9–$$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$0 for particular viewing-angle choices, depending strongly on the evolution and magnification biases $$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$1 (Paul et al., 2024). This establishes a substantive late-time contaminant that must be modeled when interpreting any galaxy parity-odd signal as primordial (Paul et al., 2024).

6. Nonlinear evolution, alternative probes, and current directions

After horizon entry, primordial parity-odd trispectra are reshaped by nonlinear structure formation. In the EFT of large-scale structure, the one-loop parity-odd matter trispectrum can be written in terms of the primordial odd trispectrum and standard SPT kernels,

$$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$2

and purely gravitational evolution does not generate parity-odd trispectra de novo (Azyzy et al., 7 Oct 2025). The one-loop analysis shows explicit infrared cancellations required by the equivalence principle and finds that the only necessary EFT counterterm at this order is proportional to $$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$3 times the tree-level odd trispectrum, aside from a possible primordial five-point counterterm (Azyzy et al., 7 Oct 2025). For a representative squeezed-like configuration and a calibrated EFT sound speed, the resulting EFT prediction is reliable up to $$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$4 at $$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$5 and $$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$6 at $$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$7 accuracy for $$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$8 (Azyzy et al., 7 Oct 2025).

Several lower-dimensional observables have been proposed to compress trispectrum information. Galaxy intrinsic alignments provide a parity-odd $$T_\zeta^{(+)} = \mathrm{Re}[T_\zeta], \qquad T_\zeta^{(-)} = i\,\mathrm{Im}[T_\zeta],$$9 signal sensitive to the collapsed limit of the primordial parity-odd trispectrum; for a $\epsilon_{abc}\,k_1^a k_2^b k_3^c$0-gauge inflationary model, the IA power spectrum scales as

$\epsilon_{abc}\,k_1^a k_2^b k_3^c$1

with $\epsilon_{abc}\,k_1^a k_2^b k_3^c$2 (Kurita et al., 10 Sep 2025). Forecasts quoted for this channel give $\epsilon_{abc}\,k_1^a k_2^b k_3^c$3 for DESI Y5 3D $\epsilon_{abc}\,k_1^a k_2^b k_3^c$4 in a fiducial setup and $\epsilon_{abc}\,k_1^a k_2^b k_3^c$5 for LSST Y10 angular $\epsilon_{abc}\,k_1^a k_2^b k_3^c$6, with optimistic configurations reaching $\epsilon_{abc}\,k_1^a k_2^b k_3^c$7 and $\epsilon_{abc}\,k_1^a k_2^b k_3^c$8, respectively (Kurita et al., 10 Sep 2025).

Parity-odd “kurto spectra” compress the galaxy or matter trispectrum into one-dimensional composite-field power spectra. For the specific parity-odd primordial template implemented in Quijote-like simulations, a Euclid-like spectroscopic survey yields cumulative signal-to-noise ratios of order $\epsilon_{abc}\,k_1^a k_2^b k_3^c$9–$$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$0, depending on estimator and weighting (Gao et al., 16 Sep 2025). This approach is motivated by the observation that the full parity-odd scalar trispectrum is high-dimensional, whereas suitably weighted quadratic–quadratic or cubic–linear correlators isolate the odd sector more directly (Gao et al., 16 Sep 2025).

Finally, scalar-induced gravitational waves offer a complementary probe. A parity-odd primordial scalar trispectrum sources a chiral SIGW background with

$$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$1

For scale-invariant scalar power and the template analyzed in that work, $$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$2 for $$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$3, implying the theoretical bound $$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$4; for lognormal-peaked scalar power, the chirality can become much larger, reaching $$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$5–$$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$6 near $$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$7, where it approximately measures the ratio $$\hat{\mathbf{k}}_i\cdot(\hat{\mathbf{k}}_j\times \hat{\mathbf{k}}_\ell)$$8 of odd to even trispectrum amplitudes (Ragavendra et al., 3 Jul 2025).

Taken together, these results place the primordial parity-odd trispectrum at the intersection of symmetry theorems, chiral early-Universe dynamics, and late-time higher-order statistics. The subject is now defined as much by its null structure—tree-level vanishing in vanilla settings—as by the concrete parity-violating mechanisms that evade those null results and the increasingly diverse observables that can test them (Cabass et al., 2022, Lee et al., 2023).