Slowdown of gravitational waves measurement

As far as I can understand, the gravitational waves also slow down on interaction with matter similar to light rays. I am trying to understand if we can measure this?

Suppose we put two detectors on 2 opposite side of the globe and measure the time difference with the help of LIGO detectors. Then, we follow up for the location of the black holes with the help of some other em-observations. Now, we know the predicted time difference that should occur if the gravitational waves are travelling at the speed of light and the observed time difference. The difference of these 2 should be because of the slowdown of gravitational waves caused by the earth. Has such a calculation been done before?

• The Shapiro effect due to Earth's spacetime curvature is likely going to dominate. From my cursory glance at the literature the matter interaction effect is going to be pretty minor because matter has a microscopic index of refraction relative to the waves. The abstract of journals.aps.org/prd/abstract/10.1103/PhysRevD.9.2207 gives $n-1=2\pi G\rho/\omega^2$, which for LIGO is about 0.00348. Maybe measurable (it would give a phase shift between detectors), but much smaller than the delay in arrival. – Anders Sandberg Jun 19 '18 at 10:33
• @AndersSandberg I did the calculation. Assuming the frequency of 50 Hz and density of earth as 5510 kg/m^3, the change in refractive index is of the order of 10^-11 and the time difference is of the order of 10^-12 sec if the wave travels through the entire earth. I am not sure if we can achieve this accuracy. – Rishabh Jain Jun 19 '18 at 15:15