Primordial Gravitational Wave Birefringence in a de Sitter Background with Chern-Simons Coupling

TL;DR

This work analyzes tensor perturbations in a de Sitter background within CS modified gravity and shows that parity-violating corrections arise from the Cotton tensor while the Pontryagin density vanishes at linear order. By diagonalizing the Cotton operator, the authors obtain decoupled equations for symmetric/antisymmetric tensor modes, revealing chiral corrections to dispersion relations and a birefringence that combines amplitude and phase effects, modulated by the scalar CS field. They derive sub- and super-horizon behaviors, a sourced particular solution, and compute the resulting phase difference and energy flux, highlighting a quadratic growth of phase difference inside the horizon and a frozen imprint outside. The analysis is extended to a massive CS field as dark matter, linking birefringence to axion-like phenomenology and showing late-time consistency with cosmology while preserving a detectable early-universe parity-violating signature.

Abstract

In this work, we investigate tensor perturbations in a de Sitter background within the framework of Chern-Simons modified gravity. We introduce transverse-traceless perturbations and analyze how the Chern-Simons Cotton tensor induces parity-violating modifications to gravitational wave propagation, while the Pontryagin density vanishes at linear order. Using a mode decomposition of the scalar background field, we derive the sub- and super-horizon limits of the wave equations and uncover chiral corrections in the dispersion relations of tensor modes. The resulting birefringence exhibits both amplitude and velocity components, alternating with the phase of the scalar field. Particular solutions sourced by the scalar background show helicity-dependent amplification and a characteristic scaling of the radiated flux that reduces smoothly to the Minkowski limit. The accumulated phase difference between right- and left-handed modes grows quadratically inside the horizon and becomes frozen outside, leaving a permanent parity-violating imprint in the primordial tensor spectrum. Finally, by promoting the Chern-Simons field to a massive dark matter candidate, we demonstrate how its mass-dependent dynamics connect gravitational birefringence to axion-like dark matter phenomenology.

Figures (4)

• Figure 1: The birefringence effect ($\Delta\omega$) versus conformal time ($\eta$) in the sub-horizon limit. The red line shows the imaginary part, and the blue line shows the real part for the calculated birefringence. The dashed line is the absolute value for $\Delta\omega$, which envelops the real and imaginary parts.
• Figure 2: The amplitude birefringence $\Delta\omega$ versus conformal time in the super-horizon limit. The red line again shows the imaginary part and the blue the real part, with the dashed line indicating the constant total magnitude.
• Figure 3: The leading order phase differences ($\Delta\phi$) versus conformal time for both the sub- and super-horizon limits (in arbitrary units). The plot on left for the sub-horizon limit shows the quadratic dependence ($\propto \eta^2$). The plot on right for the super-horizon limit shows the decaying parts as conformal time reaches the far future ($\eta \rightarrow 0^{-}$).
• Figure 4: The ratio of the log of amplitudes in the leading order, $\ln\left(A_+/A_-\right)$ versus conformal time($\eta$) for both the sub- and super-horizon limits. The plot on left for the sub-horizon limit shows the quadratic dependence ($\propto \eta^2$) and enveloped by an oscillatory term. The plot on right for the super-horizon limit shows the constant frozen-in amplitude difference that remains etched in the distant future($\eta \rightarrow 0^{-}$)