The speed of gravitational waves which was epxerimentally found by LIGO at 2017 to be the same with the speed of light c in the vacuum.
But we know that for exmaple the speed of light varies with the type of a material medium. Is this also true for the gravitational waves. What our theories and experiments say about this?
Does gravity behave the same as light in this case or is different than light?
The mechanism for index of refraction in a dielectric is that the passing light wave's electric field polarized the material, which leads to a time-varying dipole which radiates in a manner such that the total wave moves slower.
So we should look for a similar mechanism in gravitational radiation. Here, the waves would induce a time-varying quadruple moment which then emits coherent waves such that the sum is slower.
LIGO had a 4 km sized quadrupole that changed length by $10^{-18}\,$m. That is not much of a change, so I am going with "no", in practice.
It's easier with EM because in an atom, say carbon in a clear plastic, you have a solid $+6e$ and you just need to separate the mean electron position ($-6e$) to create a dipole where there was none.
With gravity, you would need to make the material have a different quadrupole moment, but there is no negative mass, so you to completely deform it, and as shown above, 1 attometer doesn't cut it.
Note that the amplitude falls of linearly with distance for gravitation radiation, so if the two 50-ish solar mass blackhole had been at spin (8 light minutes) instead of 1 Bly, the amplitude would be:
$$ (10^{-18}\,{\rm m})\frac{1B\,{\rm year}}{8\,{\rm s}} \approx 60\,\mu{\rm m}$$
so that's not much.
No. Gravitational waves are fundamentally different in that they propagate through the gravitational field, i.e. a perturbation in the geometry of spacetime, not the electromagnetic field. The EM field has varying properties depending on the medium; spacetime is just spacetime.
By definition the weak-field limit where we deal with gravitational waves is such that spacetime is approximately flat everywhere; as such the speed of light (as in the velocity $\sum (v^i)^2$ of a null ray, not the constant $c$) is roughly constant everywhere, and as such you expect that a gravitational wave, not otherwise interacting gravitationally in a significant way (remember, weak-field limit) to maintain a constant speed.
However, although I do not have a definite answer at this instant, I would be interested to see the behavior of gravitational waves propagating near strong fields, as in near the event horizons of black holes. In those cases the speed of light (velocity of null ray) does actually change and I would expect the propagation velocity of gravitational waves to change accordingly.
