If space can expand faster than light why can't gravitational wave? I heard that gravitational wave is the measure of stretchiness of space time, so since there is no limit to how fast space can stretch what about gravitational wave?
 A: There is an existing mathematical model of astrophysical observations which used in the Big Bang model fits the observations well. This model is based on general relativity and in General Relativity the velocity of light as c is inherent in the equations and thus also in the derivation of gravitational waves.

The speed of gravitational waves in the general theory of relativity is equal to the speed of light in vacuum, c.1 Within the theory of special relativity, the constant c is not exclusively about light; instead it is the highest possible speed for any interaction in nature. Formally, c is a conversion factor for changing the unit of time to the unit of space.2 This makes it the only speed which does not depend either on the motion of an observer or a source of light and/or gravity. Thus, the speed of "light" is also the speed of gravitational waves and any other massless particle.

This is so by construction of the mathematics.
Until the recent LIGO experimental findings gravitational waves were just a hypothesis, and now one sees data that fit this hypothesis. At present the expansion of space is modeled within General Relativity, and this expansion does not affect   the velocity of propagation in the solutions of the wave equations, by construction. What it does affect is the wavelength, as happens also with the electromagnetic radiation which by the redshift effect has given us the data for the expansion of the universe.
A gravitational wave will get a larger and larger wavelength as it propagates on an expanding space, itself with velocity c.
The gravitational wave is an oscillatory disturbance on the stress energy tensor that propagates from where energy and mass are concentrated as a distortion of space time. The expansion of space is an independent change of the metric of space due to the cosmological constant or , for accelerating expansion, other mechanisms under research. Both the electromagnetic waves and the gravitational waves propagate on the space time but are not generating it, (except as tiny perturbations due to the energy they carry).
A: Assuming space really stretches and say it does so at Faster Than Light - 
The gravitational wave in an FTL-stretching space would lose its amplitude as the wave itself will stretch with space. Wave is space, it is not an object (bound) so it will stretch with space. Its wavelength will also increase at the same time.
Therefore, part wave will travel/stretch at FTL, but not due to the wave nature, but only due to stretching space. Soon it would lose all the amplitude and spread too thin to be called a wave anymore.
Now, can we use presence of gravitational wave as an evidence that space is not stretching at FTL? Because, if it were, then the GW in a stretching space would die soon.
