Parity-odd Primordial Trispectrum

• Parity-odd primordial trispectrum is the imaginary component of the four-point function of scalar perturbations that encodes parity violation via pseudoscalar momentum invariants.
• It is characterized by a distinctive nonplanar momentum dependence, vanishing in planar limits and revealing unique tetrahedral configurations.
• Observational probes from the CMB, large-scale structure, and gravitational waves rigorously test its amplitude and help constrain departures from standard inflationary models.

Searching arXiv for recent and foundational papers on parity-odd primordial trispectra. Parity-odd primordial trispectrum denotes the parity-violating component of the connected four-point function of primordial scalar perturbations, usually the curvature perturbation $\zeta$. In momentum space it is the lowest-order scalar statistic that can carry intrinsic parity information, because two- and three-point scalar correlators are parity-even under the usual assumptions, while a generic tetrahedral four-point configuration is not superimposable on its mirror image by a rotation. For a real scalar field, the Fourier-space parity-odd part is necessarily purely imaginary and built from pseudoscalar momentum invariants such as $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$ (Cabass et al., 2022).

1. Definition and symmetry structure

The primordial trispectrum is defined by

$$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$

Under parity, $\mathbf{k}_i\to -\mathbf{k}_i$. Because $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$ for a real field, the trispectrum obeys $$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$. It is therefore natural to decompose

$$T = T^{\rm even} + T^{\rm odd},\qquad T^{\rm even}=\Re T,\qquad T^{\rm odd}=i\,\Im T,$$

so the parity-odd component is the imaginary part in momentum space (Lee et al., 2023).

Rotational invariance forces the odd-parity sector to contain a Levi-Civita contraction. A generic form is

$$T^{\rm odd}\propto i\,\epsilon_{ijk}k_1^ik_2^jk_3^k\,f(k_a,\mathbf{k}_i\!\cdot\!\mathbf{k}_j,\ldots),$$

or equivalently $$i\,\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$$ times a parity-even scalar shape function. This immediately implies two kinematic facts. First, the parity-odd scalar trispectrum is the first scalar correlator sensitive to parity violation. Second, it vanishes for planar momentum configurations, because the scalar triple product vanishes when all momenta lie in a plane (Cabass et al., 2022).

The same logic underlies later phenomenological parameterizations. For example, one template writes

$$\mathcal{T}_{\rm odd} = i\,g\, \beta(\widehat{\mathbf{k}_1+\mathbf{k}_2},\mathbf{k}_1,\mathbf{k}_3)\, P_\zeta(k_1)P_\zeta(k_3)P_\zeta(|\mathbf{k}_1+\mathbf{k}_2|) +\text{23 permutations},$$

with $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$0 and dimensionless amplitude $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$1 (Ragavendra et al., 3 Jul 2025).

2. No-go theorems and the special role of loops

A central result of recent theory is that parity-odd scalar trispectra are strongly constrained in minimal inflationary settings. At tree level, in the decoupling limit of the Effective Field Theory of Inflation, with a Bunch-Davies vacuum, exact scale invariance, and IR-finite local interactions, the parity-odd scalar trispectrum vanishes for any number of scalar fields of arbitrary mass. The same conclusion holds for parity-odd scalar correlators in the presence of spinning fields whose mode functions are massless or conformally coupled in de Sitter (Cabass et al., 2022).

This no-go statement has several complementary formulations. One is geometric: with full de Sitter conformal symmetry, scalar four-point functions can be mapped into planar configurations, eliminating pseudoscalar tensor structures. Another is analytic: Hermitian analyticity, scale invariance, and the Cosmological Optical Theorem force the relevant tree-level wavefunction coefficients to be real, while a parity-odd scalar correlator requires an imaginary contribution (Cabass et al., 2022).

The immediate implication is that any nonzero tree-level parity-odd scalar trispectrum diagnoses a departure from at least one minimal assumption. The explicit yes-go examples are correspondingly organized: violations of scale invariance in single-clock inflation, modified dispersion relations such as Ghost Inflation, and interactions with massive spinning fields all produce nonvanishing parity-odd trispectra (Cabass et al., 2022).

Loops alter the conclusion. Because the tree-level parity-odd trispectrum vanishes in the exact scale-invariant Bunch-Davies setup, the leading contribution can appear at one loop. In that case the parity-odd trispectrum is nonzero and, for massless or conformally coupled fields, remarkably simplifies to a purely rational function of the external kinematics with only a total-energy pole and no partial-energy poles or branch cuts. Although this one-loop contribution is leading in the odd-parity sector, its signal-to-noise is typically bounded from above by that of an associated tree-level parity-even trispectrum in a perturbatively controlled EFT (Lee et al., 2023).

3. Inflationary and early-universe generation mechanisms

Several concrete mechanisms now exist for generating parity-odd primordial trispectra.

A particularly explicit tree-level construction introduces a massive spin-1 field $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$2 during inflation together with a helical chemical potential $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$3. In the reduced-sound-speed regime $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$4, integrating out the vector yields an effective single-field description that is local in time but non-local in space. The parity-odd scalar trispectrum originates in the parity-odd non-local kernel

$\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$5

and the resulting template can reach $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$6 while remaining under perturbative control for representative small-$\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$7 parameters (Jazayeri et al., 2023).

Axion–gauge systems provide a second major class. In axion inflation with a $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$8 field coupled through $\mathbf{k}_1\cdot(\mathbf{k}_2\times\mathbf{k}_3)$9, one transverse helicity is predominantly produced, and one-loop gauge-field diagrams generate both parity-even and parity-odd scalar trispectra. In the numerical study with approximately massless and massive vectors, the parity-odd contribution is generically smaller than the even one, typically by one to two orders of magnitude in the massless case and by one to three orders of magnitude in the massive case (Niu et al., 2022). A related but distinct spectator-axion model with a rolling axion and chiral gauge production produces much larger odd-to-even ratios: the parity-odd part can reach $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$0 in exact equilateral configurations and can match or exceed the parity-even part in quasi-equilateral configurations (Fujita et al., 2023).

Parity-violating gravity offers a third route. In dynamical Chern-Simons gravity, one graviton helicity is amplified during inflation, and graviton exchange between two scalar pairs generates a parity-odd scalar trispectrum. In the collapsed limit, the even and odd parts take the form

$$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$1

$$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$2

so that

$$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$3

The odd component is therefore of the same structural form as the even one, scaled by the degree of gravitational circular polarization $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$4 (Creque-Sarbinowski et al., 2023).

The phenomenon is not confined to inflationary vacuum fluctuations. Helical primordial magnetic fields source curvature perturbations through the passive scalar mode, $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$5, and the helical part of the magnetic two-point function induces an imaginary, parity-odd curvature trispectrum. For nearly scale-invariant PMF spectra and $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$6, the analysis gives a rough upper bound $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$7 on the helical-to-non-helical power ratio (Yura et al., 22 May 2025).

4. Characteristic kinematics and shape phenomenology

The most distinctive physics of parity-odd primordial trispectra lies in their momentum dependence. The universal feature is nonplanarity: odd-parity contributions vanish in planar limits and are maximized in genuinely three-dimensional tetrahedral configurations (Yura et al., 22 May 2025).

In the emergent non-locality scenario, the leading template factorizes into an overall amplitude, a parity-odd polarization structure, and a dynamical shape function. In the $$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$8-channel one has

$$\langle \zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\zeta_{\mathbf{k}_4}\rangle_c = (2\pi)^3\delta^{(3)}\!\Big(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}_4\Big)\, T(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4).$$9

so the trispectrum vanishes for planar configurations by construction. In the internal soft limit $\mathbf{k}_i\to -\mathbf{k}_i$0, the same model predicts a “low-speed collider resonance” at

$\mathbf{k}_i\to -\mathbf{k}_i$1

together with oscillations periodic in the momentum ratio $\mathbf{k}_i\to -\mathbf{k}_i$2, not in $\mathbf{k}_i\to -\mathbf{k}_i$3. Their frequency is controlled by $\mathbf{k}_i\to -\mathbf{k}_i$4 rather than by the mass $\mathbf{k}_i\to -\mathbf{k}_i$5 (Jazayeri et al., 2023).

In rolling-axion models, angular dependence in equilateral and quasi-equilateral quadrilaterals is equally diagnostic. In the exact equilateral family, the parity-even part is symmetric under $\mathbf{k}_i\to -\mathbf{k}_i$6 and the parity-odd part is antisymmetric. In quasi-equilateral configurations with one side slightly longer than the other three, this simple symmetry is broken and the parity-odd piece can surpass the parity-even one over substantial regions of angle space (Fujita et al., 2023).

Collapsed limits are especially important because they connect primordial non-Gaussianity to late-time two-point observables. In the Chern-Simons-gravity case the collapsed ratio $\mathbf{k}_i\to -\mathbf{k}_i$7 directly encodes the spin-2 nature of the exchanged graviton through the harmonic dependence on $\mathbf{k}_i\to -\mathbf{k}_i$8 (Creque-Sarbinowski et al., 2023). In galaxy-shape EFT, the collapsed limit of the parity-odd primordial trispectrum induces a parity-odd intrinsic-alignment power spectrum

$\mathbf{k}_i\to -\mathbf{k}_i$9

which inherits the large-scale behavior $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$0 from the primordial spectrum (Kurita et al., 10 Sep 2025).

5. Observational probes, estimators, and current limits

The scalar trispectrum is the first place where primordial parity violation can be tested directly in scalar observables, and several complementary probes now exist.

For the CMB, the first dedicated measurement of the parity-odd scalar trispectrum used Planck temperature anisotropies on $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$1. New quasi-maximum-likelihood estimators for binned correlators account for mask convolution and leakage between even- and odd-parity components and achieve ideal variances within $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$2. The resulting null test found consistency with parity conservation at the $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$3 level. Eight primordial models were constrained, including Ghost Inflation, Cosmological Collider scenarios, and Chern-Simons gauge fields, with a maximal detection significance of $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$4 (Philcox, 2023).

For late-time large-scale structure, $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$5-body simulations with explicitly parity-violating initial conditions show that the realization-averaged power spectrum, bispectrum, halo mass function, and matter PDF are not affected by an imaginary primordial trispectrum, while the parity-odd matter trispectrum is proportional to the primordial amplitude parameter $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$6. The same simulations also reveal a nonzero correlation between halo angular momentum and smoothed velocity field, again proportional to $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$7 (Coulton et al., 2023).

Scalar-induced gravitational waves provide an indirect but clean probe. The SIGW circular-polarization spectrum

$\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$8

receives contributions only from the parity-odd trispectrum, whereas the intensity $\zeta(-\mathbf{k})=\zeta^*(\mathbf{k})$9 also contains Gaussian and parity-even pieces. The chirality parameter

$$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$0

therefore isolates odd-parity non-Gaussianity. In peaked small-scale scenarios, $$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$1 can reach $$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$2, and around $$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$3 it can approximately measure the ratio $$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$4 of odd- to even-trispectrum amplitudes (Ragavendra et al., 3 Jul 2025).

Galaxy intrinsic alignments extend this logic. The parity-odd IA power spectrum is sensitive to the collapsed limit of the primordial parity-odd trispectrum, and simulation-calibrated IA bias parameters permit forecasts for DESI and LSST. The resulting IA constraints are complementary to galaxy four-point and CMB trispectrum analyses because they probe different scales and trispectrum configurations (Kurita et al., 10 Sep 2025).

6. Interpretation, contaminants, and broader status

A nonzero odd-parity four-point function in observed galaxy data is not, by itself, evidence for primordial parity violation. Even though standard Newtonian redshift-space distortions are parity symmetric, the full relativistic number-count expression is not. Doppler-type terms proportional to $$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$5 generate purely imaginary, odd-in-$$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$6 kernels in Fourier space, so the observed redshift-space galaxy trispectrum acquires a parity-odd component from projection effects alone. This establishes a calculable relativistic contaminant that must be modeled before any observed odd-parity signal can be interpreted as primordial (Paul et al., 2024).

Within primordial theory itself, the current status is sharply bifurcated. In minimal single-clock settings with Bunch-Davies initial conditions and exact scale invariance, the tree-level parity-odd scalar trispectrum vanishes; if a signal is nevertheless observed, it points to broken scale invariance, modified dispersion relations or non-Bunch-Davies structure, emergent non-locality, or additional spinning fields with parity-violating couplings (Cabass et al., 2022). At the same time, concrete models now exist that generate amplitudes ranging from a percent-level odd-to-even ratio to $$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$7 parity violation in selected configurations, and in some cases $$T(-\mathbf{k}_1,-\mathbf{k}_2,-\mathbf{k}_3,-\mathbf{k}_4)=T^*(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4)$$8 dimensionless amplitudes within perturbative control (Jazayeri et al., 2023).

The parity-odd primordial trispectrum has therefore become a precision diagnostic rather than a generic expectation. Its mathematical signature is unambiguous—an imaginary four-point function built from pseudoscalar momentum invariants—and its physical interpretation is correspondingly selective. When present, it isolates symmetry breaking that is invisible in scalar two- and three-point functions; when absent, it enforces stringent constraints on parity-violating inflationary dynamics and on post-inflationary sources of helical scalar non-Gaussianity (Philcox, 2023).