What are some experimental verification of the non-linearity of gravitation?

According to my (limited) knowledge, all experiments to date probe only situations which can be understood using the linearized version of general relativity. For example, measuring gravitational waves do not require knowledge of non-linear corrections to the metric.

Friedmann cosmology and the Schwarzschild metric are solutions of the full Einstein's equation. But has there been experimental confirmation of their predictions which differ from predictions due to linearized versions of them?

Any other experimental confirmation of the non-linearity of gravitation?

• While measuring gravitational waves only requires linearized GR, their production from black hole inspirals requires fully-fledged strong-field GR. Since the waves we have observed agree very well with our numerical models of that, I'd argue that this is probing strong-field GR.
– user107153
Commented Feb 24, 2017 at 10:11
• You are missing some key points. In GR the gravitational field can affect other gravitational fields when they "cross" path (intersecting geodesics) and because of this fact we can't simply write a linear sum of the fields of all the bodies (a simple field superposition) like we do in Newtonian approximation. For example gravitational lensing also affects gravitational fields. Scwarzschild metric is only the metric of a star when no other stars are present, when we disregard other bodies. It is also only valid at the exterior of the object not inside it. Another example: 2 black holes merging. Commented Apr 6, 2017 at 8:55
• To conclude: gravitational lensing, 2 black holes merging, precession of the Mercury Commented Apr 6, 2017 at 8:56

To solve this problem we use Schwarzschild metric which is the solution to full (non-linearized) Einstein equations. By using killing fields ($k_1=\partial_t$ and $k2=\partial_{\phi}$) and normalization condition ($u^2=1$) for time-like geodesic we get the equation.
$$E=\dot{r}^2+\left(1-\frac{2m}{r}\right)\left(1+\frac{L^2}{r^2} \right)$$
If we treat the term with brackets as a potential we can get $$V=1-\frac{2m}{r}+\frac{L^2}{r^2}-\frac{2mL^2}{r^3}$$ where first therm is a constant, so doesn't contribute to equations of motion, the second and third therm are just potential of Keplers problem (Newtonian gravitational potential plus centrifugal term).