Rømer time-delay determination of the gravitational-wave propagation speed

In 1676 Olaus Rømer presented the first observational evidence for a finite light velocity $c_{\mathrm{em}}$. He formed his estimate by attributing the periodically varying discrepancy between the observed and expected occultation times of the Galilean satellite Io by its planetary host Jupiter to the time it takes light to cross Earth’s orbital diameter. Given a stable celestial clock that can be observed in gravitational waves the same principle can be used to measure the propagation speed $c_{\mathrm{gw}}$ of gravitational radiation. Space-based “LISA”-like detectors will, and terrestrial LIGO-like detectors may, observe such clocks and thus be capable of directly measuring the propagation velocity of gravitational waves. In the case of space-based detectors the clocks will be galactic close white-dwarf binary systems; in the case of terrestrial detectors, the most likely candidate clock is the periodic gravitational radiation from a rapidly rotating nonaxisymmetric neutron star. Here we evaluate the accuracy that may be expected of such a Rømer-type measurement of $c_{\mathrm{gw}}$ by foreseeable future space-based and terrestrial detectors. For space-based, LISA-like detectors, periodic sources are plentiful: by the end of the first year of scientific operations a LISA-like detector will have measured $c_{\mathrm{gw}}$ to better than a part in a thousand. Periodic sources may not be accessible in terrestrial detectors available to us in the foreseeable future; however, if such a source is detected then with a year of observations we could measure $c_{\mathrm{gw}}$ to better than a part in a million.

Figure 1

The relevant geometric quantities used in the calculation: $\widehat{k}$ is the unit vector pointing in the direction of wave propagation; $\theta$ is the ecliptic latitude (i.e., the angle that $-\widehat{k}$ makes with the plane of the ecliptic); $\varphi$ is the azimuthal angle of the source with respect to the Earth’s orbital position at $t=0$. The detector antenna pattern functions $F_{+}$ and $F_{\times }$ from Eq. (2) are defined with respect to the polarization tensors constructed from $\widehat{u}$ and $\widehat{v}$, which are proportional to the unit vectors $\widehat{\varphi }$ and $\widehat{\theta }$, respectively.