---
title: Parity-Violating Primordial Non-Gaussianity
url: https://www.emergentmind.com/topics/parity-violating-primordial-non-gaussianity-png
type: topic
---

# Parity-Violating Primordial Non-Gaussianity

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 [1106.3228] [1608.00368] [2509.08787].

## 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 [1106.3228].

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 [2509.08787].

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 $[\mathrm{length}]^2$. In this setup the linear Weyl term vanishes on shell, while the quadratic term $\int\sqrt{-g}\,W\tilde W$ 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 $b$, the observable parity-violating graviton bispectrum vanishes even though the interaction is parity odd. In particular,
\[
\langle h_R h_R h_R\rangle' = 0,\qquad
\langle h_L h_L h_L\rangle' = 0,
\]
so $B^{\rm PV}=0$ in pure de Sitter [1106.3228].

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 ${\cal O}(\epsilon)$ 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
\[
\langle h_R(\mathbf{k}_1)h_R(\mathbf{k}_2)h_R(\mathbf{k}_3)\rangle'
= -\,{\cal A}_{\rm PV}\,\frac{F(k_1,k_2,k_3)}{(k_1+k_2+k_3)^6},
\]
\[
\langle h_L(\mathbf{k}_1)h_L(\mathbf{k}_2)h_L(\mathbf{k}_3)\rangle'
= +\,{\cal A}_{\rm PV}\,\frac{F(k_1,k_2,k_3)}{(k_1+k_2+k_3)^6},
\]
with
\[
{\cal A}_{\rm PV}=64(2\pi)^4\,\epsilon\,b\,\kappa^6\,H^8.
\]
Mixed-helicity channels remain parity even in this setup, with
\[
\langle h_R h_R h_L\rangle'=\langle h_L h_L h_R\rangle'=0
\quad\text{(no parity-odd contribution)}.
\]
The overall amplitude is therefore controlled by $\epsilon\,b\,\kappa^6 H^8$, which makes the signal slow-roll suppressed in the minimal scenario [1106.3228].

## 3. Time-dependent couplings and modified-gravity realizations

Time dependence changes the tensor-sector picture qualitatively. For Weyl-cubic interactions with
\[
f(\tau)=\left(\frac{\tau}{\tau_*}\right)^A,
\]
the in-in time integral takes the form
\[
I_A(k_t)\propto
\left[\cos\!\left(\frac{\pi A}{2}\right)+i\,\sin\!\left(\frac{\pi A}{2}\right)\right]\times(\cdots).
\]
The parity-even $W^3$ contribution is proportional to $\mathrm{Re}(I_A)$, whereas the parity-odd $\widetilde W W^2$ contribution is proportional to $\mathrm{Im}(I_A)$. Consequently, $A=0$ reproduces the de Sitter cancellation, but $A\neq0$ 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,
\[
|b_{\ell\ell\ell}|\sim \ell^{-4}\times 3.2\times 10^{-2}
\left(\frac{\mathrm{GeV}}{\Lambda}\right)^2
\left(\frac{r}{0.1}\right)^4,
\]
and comparison with $f_{\rm NL}^{\rm eq}<300$ yields
\[
\Lambda \gtrsim 3\times 10^6\ {\rm GeV}
\quad\text{for } r=0.1
\]
[1108.0175].

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,
\[
\chi\equiv\frac{P_T^R-P_T^L}{P_T^R+P_T^L}
= - \frac{\pi}{16}\frac{H}{M_{\rm PV}}\ll 1.
\]
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
\[
B^{\rm odd}(k_i)\equiv \frac{1}{2}\big[B_{+++}(k_i)-B_{---}(k_i)\big]
\]
inherits scale dependence $\propto (k_T\tau_*)^A$, 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
\[
\langle a^{(t)X_1}_{\ell_1 m_1}a^{(t)X_2}_{\ell_2 m_2}a^{(t)X_3}_{\ell_3 m_3}\rangle\neq 0
\Longrightarrow
x_1+x_2+x_3+\ell_1+\ell_2+\ell_3=\text{odd},
\]
which is a smoking-gun indicator because such configurations vanish in parity-conserving theories [2008.01715].

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 $n$-th order spatial derivative operator is suppressed by
\[
\left(\frac{H}{M_*}\right)^{n-2},\qquad n\ge 3.
\]
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 [1305.0600].

## 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,
\[
t^{\,\mathbf{k}_1\mathbf{k}_2}_{\,\mathbf{k}_3\mathbf{k}_4}(\mathbf{K})
= i\sum_n d_n^{\rm odd}
\Big[
P_n(\hat{\mathbf{k}}_1\!\cdot\!\hat{\mathbf{k}}_3)
+P_n(\hat{\mathbf{k}}_1\!\cdot\!\hat{\mathbf{K}})
+(-1)^n P_n(\hat{\mathbf{k}}_3\!\cdot\!\hat{\mathbf{K}})
\Big]
\big[(\hat{\mathbf{k}}_1\times\hat{\mathbf{k}}_3)\cdot\hat{\mathbf{K}}\big]\,
P_\zeta(k_1)P_\zeta(k_3)P_\zeta(K),
\]
so that parity violation is carried by the triple product and the reduced trispectrum is purely imaginary [1608.00368].

Axially coupled gauge fields provide a concrete source. In the $U(1)$-gauge model with
\[
{\cal L}\supset -\frac14 f(\phi)F_{\mu\nu}F^{\mu\nu}
+\frac{\gamma}{4} f(\phi)F_{\mu\nu}\tilde F^{\mu\nu},
\]
the collapsed-type parity-odd trispectrum template can be written in $\Phi$-notation as
\[
f^{-}(k_1,k_3,k_{12})
= -\frac{25}{3}A_{\rm gauge}
\,F(\mu_{13},\mu_1,\mu_3)\,
\frac{P_\Phi(k_1)}{k_1}
\frac{P_\Phi(k_3)}{k_3}
\frac{P_\Phi(k_{12})}{k_{12}},
\]
with
\[
F(\mu_{13},\mu_1,\mu_3)=1-\mu_{13}+\mu_1-\mu_3.
\]
In the equivalent Legendre basis, the $U(1)$ case corresponds to
\[
d_0^{\rm odd}=-A_{\rm gauge},\qquad
d_1^{\rm odd}=+3A_{\rm gauge},\qquad
d_{n\ge2}^{\rm odd}=0
\]
[2509.08787].

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 $\zeta\zeta\gamma$ 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 $T_R-T_L$ 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 [2410.11801].

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 $\varepsilon^{ijk}$ 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
\[
\ell_1+\ell_2+\ell_3+x_1+x_2+x_3
=
\begin{cases}
\text{even} & \text{for parity-even } B_h^{+},\\
\text{odd} & \text{for parity-odd } B_h^{-},
\end{cases}
\]
with $x=0$ for $T,E$ and $x=1$ for $B$. 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
\[
f_{\rm NL}^{ttt}=900\pm 700,\qquad
f_{\rm NL}^{ttt,+}=1100\pm 800,\qquad
f_{\rm NL}^{ttt,-}=0\pm 1400,
\]
with no detection of non-Gaussianity in either parity sector. B-modes strengthen constraints on the parity-even sector by $\simeq 30\%$ and dominate the parity-odd bounds [2312.12498].

For scalar parity violation, the hallmark CMB signature is instead a trispectrum in the otherwise forbidden domain
\[
\ell_1+\ell_2+\ell_3+\ell_4=\text{odd}.
\]
In the Legendre-dipolar parameterization, the squeezed configurations dominate the signal-to-noise ratio, especially the $J=2$ channel, and a Fisher analysis gives a minimum detectable dipolar coefficient
\[
d_1^{\rm odd}=640
\]
for a cosmic-variance-limited-level temperature survey [1608.00368].

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
\[
\Pi_k\equiv \frac{P_V(k,\eta)}{P_I(k,\eta)}
=\frac{P_{LL}-P_{RR}}{P_{LL}+P_{RR}}.
\]
Gaussian scalar perturbations give $P_V=0$, so any nonzero $P_V$ selects the parity-odd scalar trispectrum. For a scale-invariant scalar spectrum with $A_s=2\times10^{-9}$ and $g=1$, the chirality is tiny,
\[
\Pi_k\simeq -2\times10^{-9}.
\]
For a peaked scalar spectrum with $A_p=10^{-2}$ and $g=\chi=1$, however, the chirality can reach
\[
\Pi_k \simeq 0.1-1
\quad\text{near } k\simeq 3k_p,
\]
and may directly measure the ratio of odd to even trispectrum amplitudes, $\Pi_k\simeq g/\chi$, in the clean regime where Gaussian contributions do not dominate [2507.02733].

## 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 $U(1)$-gauge model the renormalized large-scale result is especially simple:
\[
P_-^{(\lambda)}(k)=A_{\rm gauge}[b_-^{(\lambda)}]^2 P_\Phi(k)
= A_{\rm gauge}[b_-^{(\lambda)}]^2 \frac{9}{25}P_\zeta(k),
\]
with $P_\zeta(k)\propto k^{-3}$. The signal is purely imaginary and odd under $\hat{\mathbf{k}}\to-\hat{\mathbf{k}}$, and in the projected-shape basis it appears through an imaginary EB power spectrum,
\[
P_{EB}(k,\mu)=
-\frac{i}{4}\mu(1-\mu^2)P_-^{(1)}(k)
-\frac{i}{8}\mu(1+\mu^2)P_-^{(2)}(k),
\]
which is odd in $\mu$ [2509.08787].

Forecasts show that this channel is competitive. For DESI, the intrinsic-alignment analysis yields
\[
\sigma(A_{\rm gauge})\simeq 0.73
\quad\text{for fiducial }k_{\min},
\qquad
\sigma(A_{\rm gauge})\simeq 0.14
\quad\text{for optimistic }k_{\min}.
\]
For LSST, the corresponding forecasts are
\[
\sigma(A_{\rm gauge})\simeq 0.088
\quad\text{for fiducial }\ell_{\min},
\qquad
\sigma(A_{\rm gauge})\simeq 0.013
\quad\text{for optimistic }\ell_{\min}.
\]
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 $d_0^{\rm odd}$, whereas intrinsic alignments are most sensitive to the collapsed limit and hence to $d_1^{\rm odd}$ [2509.08787].

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 [2410.11801].

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.

Source: https://www.emergentmind.com/topics/parity-violating-primordial-non-gaussianity-png