Chiral gravitational waves from domain walls in Nieh-Yan gravity

We study the scattering of gravitational waves by axion domain walls in teleparallel gravity with the Nieh-Yan term. Since a domain wall causes the parity violation, the transmitted gravitational waves also exhibit the parity violation. We calculate the degree of circular polarization of gravitational waves. It turns out that gravitational waves after going through the domain wall could be chiral. Remarkably, the degree of circular polarization does not depend on the tension of the domain wall.

💡 Research Summary

This paper investigates the generation of chiral gravitational waves (GWs) through their scattering off axion domain walls within the framework of teleparallel gravity augmented by the Nieh-Yan term. The primary motivation is to explore parity violation in gravity using a theoretically healthy model, as the more familiar Chern-Simons modification of General Relativity suffers from ghost instabilities. The Nieh-Yan term in teleparallel gravity provides a ghost-free alternative to violate parity.

The authors begin by establishing the theoretical setup. Teleparallel gravity is formulated using tetrad fields and a flat, metric-compatible connection with torsion. The standard teleparallel action is equivalent to the Einstein-Hilbert action. The parity-violating Nieh-Yan term is introduced, which becomes physically non-trivial when coupled to an axion-like field ϕ. The total action considered includes the Einstein-Hilbert term (from the teleparallel equivalence), the axion-coupled Nieh-Yan term, and the axion’s kinetic and potential terms (Eq. 18). The axion potential is of the double-well type, leading to domain wall solutions. A static, planar domain wall solution oriented perpendicular to the z-axis is adopted as the background (Eq. 22).

The propagation of linear tensor metric perturbations (GWs) on this domain wall background is then analyzed. Assuming GW propagation along the z-axis, the linearized equation of motion is derived (Eq. 25). Decomposing the GWs into right-handed (R) and left-handed (L) circular polarization modes reveals that their evolution equations differ in sign (Eq. 29). Using a Fourier ansatz, the problem reduces to two independent 1D Schrödinger-type equations (Eq. 30) for the spatial profiles of the R and L modes. The effective potential in these equations is proportional to the derivative of the axion field ϕ′(z), which is localized at the domain wall, and crucially has opposite signs for the two polarizations (Eq. 31). This sign difference is the origin of the birefringence effect.

The scattering problem is solved analytically. By introducing dimensionless variables, the equation is transformed into a form solvable in terms of hypergeometric functions. The exact solution satisfying outgoing boundary conditions at infinity is found (Eq. 37). Analyzing its asymptotic behavior at negative infinity allows for the extraction of reflection and transmission coefficients. An analytic expression for the transmission probability |T|² is obtained (Eq. 41).

This general result is then evaluated separately for the R and L modes, which correspond to different parameter ranges. For the L-mode (always a potential well), the transmission probability is given by Eq. (45). For the R-mode, the expression depends on the frequency: Eq. (46) applies when the effective potential is a well, and Eq. (48) applies when it is a barrier. The degree of circular polarization Π is defined as the normalized difference between the squared transmission amplitudes of the two modes (Eq. 49). Analytic formulas for Π are provided for both parameter regimes (Eqs. 50, 51).

The key findings are: 1) GWs transmitted through the axion domain wall become chiral (Π ≠ 0). 2) Remarkably, the degree of circular polarization Π is essentially independent of the domain wall tension σ, which is set by the symmetry-breaking scale η and the coupling λ of the axion potential. Numerical plots (Fig. 2) confirm that Π depends mainly on the product of the GW frequency ω and the Nieh-Yan coupling length ℓ. A strong chiral signal (Π ≈ 1) is achieved when ωℓ is sufficiently large (Eq. 52), a condition unrelated to the wall’s tension.

In conclusion, the study demonstrates that the Nieh-Yan term in teleparallel gravity provides a viable mechanism for generating chiral gravitational waves via interaction with cosmic axion domain walls. The effect is robust and universal, as its strength does not depend on the specific energy scale of the domain wall, suggesting it could be a relevant source of parity violation in the stochastic gravitational wave background across different cosmological epochs.