How would gravitons couple to the Stress-Energy tensor? How would gravitons couple to the Stress-Energy tensor $T^{\mu\nu}$? How did physicists arrive at this result? I've read that it follows from the analysis of irreducible representations of the 4-dimensional Poincaré group, but is this accurate?
 A: The stress-energy tensor, is up to multiplicative factors, can be defined by $\frac{\delta S}{\delta g^{\mu\nu}}$, where $S$ is the action and $g_{\mu\nu}$ is the metric. When people talk about the graviton, they talk about quantizing the metric around it's classical solution, so we consider field values $g_{\mu\nu} = g^{(c)}_{\mu\nu} + h_{\mu\nu}$, where $h$ is considered a small perturbation (there are a lot of gauge fixings left out here). In order to evaluate the action for this new field $h$, we would simply plug $g$ into the action and collect terms involving $h$, as a starting point. To lowest non-trivial order in $h$, we can Taylor expand:
$$S(g) = S(g^{(c)}) + \int\!\frac{\delta S}{\delta g^{\mu\nu}} h^{\mu\nu} + \int\!\frac{1}{2} h^{\alpha\beta} \frac{\delta^2 S}{\delta g^{\alpha\beta} \delta g^{\mu\nu}} h^{\mu\nu} + O(h^3).$$
Notice the second term is just $\int\!dx\, T_{\mu\nu} h^{\mu\nu}$, as advertised.  The third term is the kinetic term for the $h$ field, and gives a wave equation. Now, these gravitons are essentially free spin-2 particles moving in a classical GR background, with no interactions, because we have truncated the expansion at order 2. Once we try to add higher orders, however, quantum corrections require ever increasing powers of $h$ with no unique prescription to render their coefficients finite. The theory is said to be non-renormalizable.
A: It is not known yet. 
Gravitons are from quantum mechanics model, while stress-energy tensor is from General relativity (GR) model.
Two models are not connected until quantum gravity created.
Also, gravitons were never observed, so they are pretty hypothetical. 
Simultaneously, it is known, that metric tensor is "generated" by stress-energy tensor. Metric tensor is from GR model. Also GR model contains gravitation waves.
Gravitation waves were never observed too. (Gravitation waves were never observed directly, i.e. so that they affect matter on Earth, although they were confirmed indirectly, by predicting of energy loss in rotating heavy (neutron) star systems).
If gravitons exist, they should be a quantum representation of gravitation waves. And, it is known from it, that gravitons should have spin of 2.
This is the sequence: GR -> gravitation waves -> spin of 2.
Two last parts are hypothetical.
