Parity-Violating Primordial Non-Gaussianity

• Parity-violating primordial non-Gaussianity is defined by correlation functions that change sign under spatial inversion, highlighting unique pseudoscalar and chiral features in early-universe models.
• In the tensor sector, chiral bispectra from graviton interactions—affected by de Sitter symmetry and time-dependent couplings—quantify parity violation with observable CMB imprints when slow-roll effects break the cancellation.
• In the scalar sector, parity violation appears through an imaginary trispectrum constructed from pseudoscalar momentum products, with models such as U(1)-gauge and chiral scalar-tensor theories offering distinct collapsed and squeezed configuration signals.

Parity-violating primordial non-Gaussianity denotes connected correlation functions of primordial perturbations that change sign under spatial inversion, $\mathbf{x}\to-\mathbf{x}$ or equivalently $\mathbf{k}\to-\mathbf{k}$. In inflationary cosmology this phenomenon appears in two structurally different ways. In the tensor sector, parity acts by exchanging left- and right-handed graviton helicities, so parity violation is encoded directly in chiral bispectra of primordial gravitational waves. In the scalar sector, statistical homogeneity and isotropy forbid parity-odd two- and three-point functions, so the lowest-order parity-sensitive correlator is the trispectrum, whose parity-odd part is purely imaginary and carries pseudoscalar momentum dependence (Soda et al., 2011, Shiraishi, 2016, Kurita et al., 10 Sep 2025).

1. Symmetry structure and allowed correlators

In the tensor sector, parity flips helicities, $\lambda\to-\lambda$, together with $\mathbf{k}\to-\mathbf{k}$. A standard diagnostic therefore compares same-helicity bispectra,

$$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$

so that parity-even contributions are invariant under simultaneous helicity reversal, whereas parity-odd contributions change sign (Soda et al., 2011).

In the scalar sector, the situation is more restrictive. Under statistical homogeneity and isotropy, the leading and most prominent parity-odd signal appears in the four-point function rather than the bispectrum, because a parity-odd scalar trispectrum can be built from a pseudoscalar such as a scalar triple product, whereas the scalar bispectrum does not admit a parity-odd invariant under isotropy. A general decomposition writes

$$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$

and the parity-odd part takes the form

$$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$

with $\tau^{-}$ antisymmetric under permutations (Kurita et al., 10 Sep 2025).

These kinematic facts organize the subject. Tensor parity violation is naturally discussed in terms of helicity asymmetries, chiral dispersion, and parity-odd bispectra. Scalar parity violation is naturally discussed in terms of imaginary trispectra, collapsed limits, and pseudoscalar angular structures. A recurrent theme across both sectors is that the presence of a parity-odd operator in the action does not by itself guarantee an observable parity-odd correlator.

2. Parity-odd tensor PNG from graviton interactions

A canonical starting point is the Weyl-cubic parity-violating interaction studied by Soda, Kodama, and Nozawa,

$$S_{\rm PV}=-\,b \int d\eta\,d^3x\,\sqrt{-g}\; \epsilon^{\mu\nu\lambda\sigma}\, W_{\mu\nu}{}^{\alpha\beta}\, W_{\alpha\beta}{}^{\gamma\delta}\, W_{\gamma\delta\lambda\sigma},$$

with $b$ of mass dimension $\mathbf{k}\to-\mathbf{k}$0. In this setup the linear Weyl term vanishes on shell, while the quadratic term $\mathbf{k}\to-\mathbf{k}$1 is topological and does not contribute dynamically in pure gravity without a varying coupling. The central result is that, for exact de Sitter spacetime and constant $\mathbf{k}\to-\mathbf{k}$2, the observable parity-violating graviton bispectrum vanishes even though the interaction is parity odd. In particular,

$\mathbf{k}\to-\mathbf{k}$3

so $\mathbf{k}\to-\mathbf{k}$4 in pure de Sitter (Soda et al., 2011).

This cancellation is not a trivial statement about the action; it arises from the combination of de Sitter isometries, time-independent coupling, mode-function structure, and polarization identities. The result corrects a common misconception: parity-odd operators can yield nonzero intermediate chiral correlators while still producing no observable parity-odd graviton bispectrum in exact de Sitter.

Slow-roll inflation breaks the exact de Sitter cancellation. In that case the parity-odd bispectrum is generated at $\mathbf{k}\to-\mathbf{k}$5 from three sources: the change in the asymptotic mode function, the change in the interaction Hamiltonian when expressed in creation–annihilation operators, and altered time-integral kernels with noncancelling imaginary parts. The same-helicity bispectra become

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

$\mathbf{k}\to-\mathbf{k}$7

with

$\mathbf{k}\to-\mathbf{k}$8

Mixed-helicity channels remain parity even in this setup, with

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

The overall amplitude is therefore controlled by $\lambda\to-\lambda$0, which makes the signal slow-roll suppressed in the minimal scenario (Soda et al., 2011).

3. Time-dependent couplings and modified-gravity realizations

Time dependence changes the tensor-sector picture qualitatively. For Weyl-cubic interactions with

$\lambda\to-\lambda$1

the in-in time integral takes the form

$\lambda\to-\lambda$2

The parity-even $\lambda\to-\lambda$3 contribution is proportional to $\lambda\to-\lambda$4, whereas the parity-odd $\lambda\to-\lambda$5 contribution is proportional to $\lambda\to-\lambda$6. Consequently, $\lambda\to-\lambda$7 reproduces the de Sitter cancellation, but $\lambda\to-\lambda$8 generates a nonzero parity-violating graviton bispectrum even in exact de Sitter, without slow-roll suppression. The projected CMB bispectra occupy complementary multipole domains: the parity-conserving and parity-violating terms appear in completely different configurations of multipoles. For the diagonal large-scale CMB bispectrum,

$\lambda\to-\lambda$9

and comparison with $\mathbf{k}\to-\mathbf{k}$0 yields

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

(Shiraishi et al., 2011).

Chiral scalar-tensor theories extend Chern–Simons gravity by including parity-violating operators with first and second derivatives of the inflaton. In these models the tensor power spectra are chiral but their chirality parameter is suppressed in the EFT regime,

$\mathbf{k}\to-\mathbf{k}$2

At the bispectrum level, no new contributions arise if the couplings are constant in a pure de Sitter phase; all in-in time integrals are real, so the bispectrum vanishes. If the couplings are time dependent, however, the bispectra become nonzero even in exact de Sitter. The parity-odd component

$\mathbf{k}\to-\mathbf{k}$3

inherits scale dependence $\mathbf{k}\to-\mathbf{k}$4, with PV1 shapes peaking in the squeezed and equilateral limits and PV2 shapes strongly equilateral for same helicities. In CMB language, the parity-odd selection rule is

$\mathbf{k}\to-\mathbf{k}$5

which is a smoking-gun indicator because such configurations vanish in parity-conserving theories (Bartolo et al., 2020).

Hořava–Lifshitz gravity supplies another realization. There the leading tensor bispectrum comes from the usual second-order derivative terms and coincides with the general-relativistic result, while the contribution of an $\mathbf{k}\to-\mathbf{k}$6-th order spatial derivative operator is suppressed by

$\mathbf{k}\to-\mathbf{k}$7

The next-to-leading parity-violating contribution is then the 3-dimensional gravitational Chern–Simons term, and the analysis argues that this is the only parity-violating operator in the theory with non-vanishing contribution to the tensor bispectrum in the de Sitter, adiabatic setup (Zhu et al., 2013).

4. Scalar-sector parity-odd trispectra

A parity-odd scalar trispectrum is the lowest-order scalar correlator compatible with statistical homogeneity, isotropy, and parity violation. One convenient parameterization uses Legendre polynomials,

$\mathbf{k}\to-\mathbf{k}$8

so that parity violation is carried by the triple product and the reduced trispectrum is purely imaginary (Shiraishi, 2016).

Axially coupled gauge fields provide a concrete source. In the $\mathbf{k}\to-\mathbf{k}$9-gauge model with

$$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$0

the collapsed-type parity-odd trispectrum template can be written in $$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$1-notation as

$$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$2

with

$$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$3

In the equivalent Legendre basis, the $$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$4 case corresponds to

$$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$5

(Kurita et al., 10 Sep 2025).

Chiral scalar-tensor gravity provides a different scalar-sector mechanism. There the parity-odd scalar trispectrum is graviton mediated: PV1 operators generate intrinsically parity-odd $$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$6 vertices, while PV2 operators generate chiral tensor propagation. The resulting four-point function separates into real and imaginary parts, with the imaginary part proportional to $$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$7 and therefore odd under parity. The analysis estimates that, for a set of parameters of the theory, it is possible to produce a signal-to-noise ratio for the parity-violating part of the trispectrum of order one without introducing modifications to the single-field slow-roll setup (Moretti et al., 2024).

A useful structural distinction follows. Gauge-field models typically produce scalar parity violation through collapsed-type trispectra with strong diagonal dependence. Chiral scalar-tensor models produce it through graviton exchange, with angular dependence controlled by polarization tensors, azimuthal phases, and $$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$8 contractions. Both routes feed late-time parity-odd observables, but their configuration dependence is different.

5. CMB and gravitational-wave observables

The CMB provides the cleanest harmonic-space classification of parity-violating PNG. For tensor-sourced bispectra, the general selection rule is

$$B^{\rm PV}(k_1,k_2,k_3)\equiv \frac{1}{2}\big[B_{+++}(k_1,k_2,k_3)-B_{---}(k_1,k_2,k_3)\big],$$9

with $$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$0 for $$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$1 and $$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$2 for $$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$3. This implies that parity-odd tensor bispectra populate configurations forbidden to parity-conserving sources. Using Planck PR4 temperature and polarization data, including B-modes for the first time in a primordial non-Gaussianity analysis, the constrained amplitudes of a maximally chiral equilateral tensor template are

$$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$4

with no detection of non-Gaussianity in either parity sector. B-modes strengthen constraints on the parity-even sector by $$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$5 and dominate the parity-odd bounds (Philcox et al., 2023).

For scalar parity violation, the hallmark CMB signature is instead a trispectrum in the otherwise forbidden domain

$$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$6

In the Legendre-dipolar parameterization, the squeezed configurations dominate the signal-to-noise ratio, especially the $$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$7 channel, and a Fisher analysis gives a minimum detectable dipolar coefficient

$$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$8

for a cosmic-variance-limited-level temperature survey (Shiraishi, 2016).

Scalar-induced gravitational waves supply a distinct probe of parity-odd scalar trispectra. In this case the helicity asymmetry of the induced gravitational-wave background is quantified by

$$T_\zeta = T_\zeta^{(+)} + T_\zeta^{(-)},\qquad \Pi[T_\zeta^{(\pm)}]=\pm T_\zeta^{(\pm)},$$9

Gaussian scalar perturbations give $$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$0, so any nonzero $$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$1 selects the parity-odd scalar trispectrum. For a scale-invariant scalar spectrum with $$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$2 and $$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$3, the chirality is tiny,

$$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$4

For a peaked scalar spectrum with $$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$5 and $$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$6, however, the chirality can reach

$$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$7

and may directly measure the ratio of odd to even trispectrum amplitudes, $$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$8, in the clean regime where Gaussian contributions do not dominate (Ragavendra et al., 3 Jul 2025).

6. Large-scale-structure probes, forecasts, and open issues

Large-scale structure accesses parity-violating PNG primarily through four-point information and collapsed-limit responses. A recent development is the use of galaxy intrinsic alignments as a two-point probe of a primordial parity-odd trispectrum. In the EFT of intrinsic alignments, the parity-odd helicity power spectra are sourced at one loop by the parity-odd matter trispectrum, and for the $$T_\zeta^{(-)}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}_4) = i \,[\mathbf{k}_1\cdot(\mathbf{k}_2\times \mathbf{k}_3)]\,\tau^{-}(k_1,k_2,k_3,k_4;k_{12},k_{14}),$$9-gauge model the renormalized large-scale result is especially simple: $\tau^{-}$0 with $\tau^{-}$1. The signal is purely imaginary and odd under $\tau^{-}$2, and in the projected-shape basis it appears through an imaginary EB power spectrum,

$\tau^{-}$3

which is odd in $\tau^{-}$4 (Kurita et al., 10 Sep 2025).

Forecasts show that this channel is competitive. For DESI, the intrinsic-alignment analysis yields

$\tau^{-}$5

For LSST, the corresponding forecasts are

$\tau^{-}$6

These are complementary to current limits from galaxy four-point correlation and CMB trispectrum analyses, because galaxy shapes are sensitive to different scales and trispectrum configurations. In the same framework, galaxy four-point statistics are more sensitive to equilateral configurations and dominantly constrain $\tau^{-}$7, whereas intrinsic alignments are most sensitive to the collapsed limit and hence to $\tau^{-}$8 (Kurita et al., 10 Sep 2025).

This complementarity matters because the current phenomenology is not settled. Possible hints of parity violation have been reported in the connected galaxy four-point correlation function, but their physical origin has been debated. If such hints are physical, they would point to primordial non-Gaussianity and new parity-violating physics; if they are spurious, the same analyses still constrain the free parameters of chiral scalar-tensor theories and related models (Moretti et al., 2024).

Across the literature, three conclusions are robust. First, parity-odd interactions do not automatically produce parity-odd observables; exact de Sitter with constant couplings is often too symmetric. Second, time dependence, chiral propagation, and collapsed or squeezed kinematics are the principal mechanisms that convert parity-odd operators into measurable PNG. Third, the most diagnostic observables are those forbidden in parity-conserving cosmology: odd-sum CMB bispectra or trispectra, chiral scalar-induced gravitational waves, and imaginary parity-odd shape correlators in large-scale structure. These channels jointly define the present research program on parity-violating primordial non-Gaussianity.