Rotations of the polarization of a gravitational wave propagating in universe

Abstract

In this paper, we study the polarization of a gravitational wave (GW) emitted by an astrophysical source at a cosmic distance propagating through the Friedmann-Lemaître-Robertson-Walk (FLRW) universe. By considering the null geodesic deviations, we first provide a definition of the polarization of the GW in terms of the Weyl scalars with respect to a parallel-transported frame along the null geodesics, and then show explicitly that, due to different effects of the expansion of the universe on the two polarization modes, the so-called “+” and “×” modes, the polarization angle of the GW changes generically, when it is propagating through the curved background. More precisely, due to the presence of the matter field of the FLRW background, both of the “×” and “+” modes can get amplified or diluted, depending on their waveforms, so in principle the effects to the “×” modes are different from those to the “+” modes. As a result, the polarization angle will change along the wave path, regardless of which source it is. As a concrete example, we directly compute the polarization angle in a binary system, and show that different epochs, radiation-, matter- and Λ-dominated, have different effects on the polarization. In particular, for a GW emitted by a binary system, we find explicitly the relation between the change of the polarization angle |Δφ| and the redshift zs of the source in different epochs. In the ΛCDM model, we find that the order of |Δφ|η0F is typically O(10−3) to O(103), depending on the values of zs, where η0 is the (comoving) time of the current universe, and F≡(52561τobs)3/8(GNMc)−5/8 with τobs and Mc being, respectively, the time to coalescence in the observer's frame and the chirp mass of the binary system. The typical value of |Δφ| for LIGO-Virgo sources is 10−21. Hence, it may not be easily detected with current detectors.