Birefringent gravitational waves and the consistency check of inflation

In this work we show that the gravitational Chern-Simons term, aside from being a key ingredient in inflationary baryogenesis, modifies superhorizon gravitational waves produced during inflation. We compute the super-Hubble gravitational power spectrum in the slow-roll approximation and show that its overall amplitude is modified while its spectral index remains unchanged (at leading order in the slow-roll parameters). Then, we calculate the correction to the tensor to scalar ratio, $T/S$. We find a correction of $T/S$ which is dependent on $\mathcal{N}$ (more precisely quadratic in $\mathcal{N}$), the parameter characterizing the amplitude of the Chern-Simons terms. In a stringy embedding of the leptogenesis mechanism, $\mathcal{N}$ is the ratio between the Planck scale and the fundamental string scale. Thus, in principle, we provide a direct probe of leptogenesis due to stringy dynamics in the cosmic microwave background. However, we demonstrate that the corresponding correction of $T/S$ is in fact very small and not observable in the regime where our calculations are valid. To obtain a sizable effect, we argue that a nonlinear calculation is necessary.

Figure 1

Effective potential for the two states of polarization (solid line for the right polarization state and dashed line for the left polarization state). At $x=-1$ or $\eta =-8/(k\Theta )$, the effective potential $f_{\mathrm{L}}(x)$ blows up. For $x>-1$, the slight difference between $f_{\mathrm{L}}(x)$ and $f_{\mathrm{R}}(x)$ mathematically originates from the term ${\lambda }^s/[x(1-{\lambda }^{s}x)]$ in $z_s^{\prime \prime }/z_s$ and, physically, from the phenomenon of birefringence. As $x\to 0$, the standard term $(2+3ϵ)/x^2$ dominates. Since this term does not depend on the polarization state, one has $f_{\mathrm{R}}(x)\to f_{\mathrm{L}}(x)$.