Constraining the polarization content of gravitational waves with astrometry

Gravitational waves perturb the paths of photons, impacting both the time of flight and the arrival direction of light from stars. Pulsar timing arrays can detect gravitational waves by measuring the variations in the time of flight of radio pulses, while astrometry missions such as Gaia can detect gravitational waves from the time-varying changes in the apparent position of a field of stars. Just as gravitational waves impart a characteristic correlation pattern in the arrival times of pulses from pulsars at different sky locations, the deflection of starlight is similarly correlated across the sky. Here, we compute the astrometric correlation patterns for the full range of polarization states found in alternative theories of gravity and decompose the sky-averaged correlation patterns into vector spherical harmonics. We find that the tensor and vector polarization states produce equal power in the electric- and magnetic-type vector spherical harmonics, while the scalar modes produce only electric-type correlations. Any difference in the measured electric and magnetic-type correlations would represent a clear violation of Einstein gravity. The angular correlation functions for the vector and scalar longitudinal modes show the same enhanced response at small angular separations that is familiar from pulsar timing.

Figure 1

The analytic correlation functions $\alpha (\mathrm{\Theta })$ and $\sigma (\mathrm{\Theta })$ for the TT, ST and VL modes.

Figure 2

The correlation functions $\alpha (\mathrm{\Theta })$ and $\sigma (\mathrm{\Theta })$ for the SL mode. The solid lines show the analytic approximation, valid for $\mathrm{\Theta }\gg 1/(fL)$. The dotted lines are found by numerical integration for the case $fL=f{L}^{\prime }=10$.

Figure 3

The upper plot shows EE and BB angular correlation functions for the longitudinal modes. The solid lines are the analytic approximations from Eq. (11) and the dotted lines are from a numerical evaluation with $fL=10$.

Figure 4

The EE angular correlation functions are compared to the corresponding correlation functions $\mathrm{\Gamma }$ for pulsar timing. Aside from overall scalings and offsets, the correlation functions for the transverse modes are identical. The vector longitudinal correlation functions are identical at large angles, but differ slightly at small angles, though both share a similar $\ln (fL)$ scaling at $\mathrm{\Theta }=0$. The correlation functions for the scalar longitudinal modes are qualitatively similar, but differ in their detailed form. They share a similar $fL$ dependent scaling at $\mathrm{\Theta }=0$. The plots are shown for $fL=10$.

Figure 5

Comparing the analytic expressions without the star term for ${\alpha }^{\mathrm{VL}}(\mathrm{\Theta })$, ${\alpha }^{\prime \mathrm{VL}}(\mathrm{\Theta })$, ${\alpha }^{\prime \prime \mathrm{VL}}(\mathrm{\Theta })$ to a numerical evaluation with the star terms illustrates the differences that occur at small angles, which are especially significant for the ${\alpha }^{\prime \prime \mathrm{VL}}(\mathrm{\Theta })$ term. The star terms play an important role in determining the behavior of the $\mathrm{EE}(\mathrm{\Theta })$ and $\mathrm{BB}(\mathrm{\Theta })$ correlation functions for small angles. The $\alpha$ terms for $fL=10$ and $fL=100$ are compared to the expression without the star term ($fL\to \infty$). As $fL$ becomes large, the star term is only important in a small region near $\mathrm{\Theta }=0$.

Figure 6

The EE and BB correlation functions for the TT mode at small angular separations. The star terms break the equality between EE and BB. The plots are shown for $fL=10$.