Astrometric effects of a stochastic gravitational wave background

A stochastic gravitational wave background causes the apparent positions of distant sources to fluctuate, with angular deflections of order the characteristic strain amplitude of the gravitational waves. These fluctuations may be detectable with high precision astrometry, as first suggested by Braginsky et al. in 1990. Several researchers have made order of magnitude estimates of the upper limits obtainable on the gravitational wave spectrum ${\Omega }_{\mathrm{gw}}(f)$, at frequencies of order $f\sim 1{\mathrm{yr}}^{-1}$, both for the future space-based optical interferometry missions GAIA and SIM, and for very long baseline interferometry in radio wavelengths with the SKA. For GAIA, tracking $N\sim {10}^6$ quasars over a time of $T\sim 1\mathrm{yr}$ with an angular accuracy of $\Delta \theta \sim 10\mu$ as would yield a sensitivity level of ${\Omega }_{\mathrm{gw}}\sim (\Delta \theta {)}^2/(N{T}^2{H}_0^2)\sim {10}^{-6}$, which would be comparable with pulsar timing. In this paper we take a first step toward firming up these estimates by computing in detail the statistical properties of the angular deflections caused by a stochastic background. We compute analytically the two-point correlation function of the deflections on the sphere, and the spectrum as a function of frequency and angular scale. The fluctuations are concentrated at low frequencies (for a scale invariant stochastic background), and at large angular scales, starting with the quadrupole. The magnetic-type and electric-type pieces of the fluctuations have equal amounts of power.

Figure 1

Here we plot the coefficients ${\alpha }_l^{EE}$ as defined in Eq. (87) vs multipole $l$.

Figure 2

Here we plot the function $\alpha (\Theta )$, the coefficient of $H_{ij}(\mathbf{n},{\mathbf{n}}^{\prime })$ as shown in Eq. (82), as a function of the angle $\Theta$ between $\mathbf{n}$ and ${\mathbf{n}}^{\prime }$.