In what regime is the semiclassical approximation of gravity valid? I am currently working on the regime of validity of Semiclassical Gravity, an approximation defined by the following equation: $G_{\mu\nu}= 8 \pi G_N \langle T_{\mu\nu} \rangle$, where the expectation value is taken in some state $\lvert \psi \rangle$.
For simplicity, let's consider a free, massless scalar field theory of a single real scalar. Then, assuming that $\lvert \psi \rangle$ is an eigenstate of $\hat{T}_{\mu\nu}$, the above equation reduces to $G_{\mu\nu}= 8 \pi G_N T_{\mu\nu}$, where now $T_{\mu\nu}$ is the eigenvalue.
Then in principle, I can use $T_{\mu\nu}$ to source a spacetime (at least one). Does that hold for all eigenvalues of $\hat{T}_{\mu\nu}$ in a given theory? Is there some further restriction? Is my reasoning flawed in some way? 
 A: Edit 1: Let me distinguist between two levels of approximations to deal with quantum field theory on curved spacetimes and the classical limit of quantum gravity.
First, you can of course consider a quantum field theory on a curved background spacetime, where you choose to ignore the backreaction (the expectation value of) the energy-momentum tensor itself might have on the dynamics of the spacetime. This is as valid as considering a quantum mechanical electron $\psi$ that couples to a classical electromagnetic field $A_\mu$. 
Second, you can try to take the next step and use the 'semiclassical approximation', where you consider a quantum field theory on a classical background spacetime and now take $\langle T^{\mu\nu} \rangle$ for the energy-momentum tensor in the Einstein equations. When we compare this with electromagnetism, this approach is similar to taking into account that the expectation value of the electron current $\langle J_\mu \rangle = \langle \bar{\psi} \gamma_mu \psi \rangle$ should create an additional electromagnetic field of its own.
Edit 2: I would say that both of these are a valid approximation as long as $\langle T^{\mu\nu} \rangle$ remains small (which effectively means that you can ignore the backreaction of $\langle T^{\mu\nu} \rangle$ on your spacetime and restrict yourself to the first case). As soon as $\langle T^{\mu\nu} \rangle$ is locally of order one (in Planck units), your quantum field theory (which is in a superposition of different field configurations) would (in the full quantum gravity theory) give you a superposition of different corresponding spacetimes. When we say that $T^{\mu\nu} \approx \langle T^{\mu\nu} \rangle$, we want our superposition of different spacetimes on which our scalar field moves to be roughly approximated by its 'average'.
Excitations of lengthscales $\lambda$, will carry an energy $\lambda^{-1}$. If $\lambda \approx \ell_{\text{Planck}}$ their wavelength would be smaller than their Schwarzschild horizon, causing such exciations to form small black holes of mass $M_{\text{Planck}}$ with an event horizon. Taking the expectation vlaue of such a distribution of spacetimes where we have event horizons and singularities popping into existence, is clearly bad and is where our approximation breaks down completely. 

Nevertheless you might still wonder, if we take the expectation value of $T^{\mu\nu}$ and consider the corresponding spacetime, does this give rise to a spacetime that we would consider as 'physical' in the sence that geodesics within that spacetime seem to behave they way they do in the world we are used to. I am afraid the question is still out on which spacetimes are exactly the physical ones and what would be the best constraint to impose on $T^{\mu\nu}$  to ensure they will be.
There are various constraints that people impose in cases like this to obtain, what they think, must be physically meaningful spacetimes. These constraints (or energy conditions) are largely based on what they people assume are reasonable assumptions for the behavior of a field theory (for example, having a positive energy density and not having its energy density flowing faster than light). Note that many inflation models or other models that consider a scalar field with a large potential violate all of these conditions. 
null energy condition for every future-pointing null vector field $\rho = T_{ab} \, X^a \, X^b \ge 0$
weak energy condition for every future-pointing causal vector field $\rho = T_{ab} \, X^a \, X^b \ge 0$
dominant energy condition for every future-pointing causal vector field $-{T^{a}}_{b}\,Y^{b}$ should also be a future-pointing causal vector (so $\rho$ can never be observed to be flowing faster than light).
strong energy condition for every future-pointing timelike vector field $\left(T_{{ab}}-{\frac  {1}{2}}\,T\,g_{{ab}}\right)\,X^{a}\,X^{b}\geq 0$ 
In addition to locally satisfying energy conditions such as these, you also might want your spacetime to satisfy various global conditions, for example, not having any closed time-like loops. This can still be the case for an arbitrary spacetime that does satisfy these energy conditions since timelike loops are a global property that depends on the topology of your spacetime.
