A consistency relation for induced gravitational wave anisotropies

We show that the anisotropies in the spectrum of gravitational waves induced by scalar modes after the end of inflation in canonical, single-field models are completely determined by the tilt of the scalar and tensor power spectra. The latter contains information about anisotropies produced due to the propagation of the tensor modes in an inhomogeneous Universe, whereas the former represents the anisotropies generated at the time of production and arise only when non-Gaussian corrections to the angular power spectrum are considered. Our proof takes into account all scalar interactions in the cubic inflaton Lagrangian.