Ground-state expectation value of the metric in the $\frac{1}{N}$ or semiclassical approximation to quantum gravity

We consider gravity coupled to $N$ conformally invariant scalar fields and compute the effective action to leading order in $\frac{1}{N}$. This approximation turns out to be closely related to semiclassical relativity, where one treats quantum matter interacting with classical gravity. In this approximation we consider two tests of the conjecture that the ground-state expectation value of the metric is flat space. We find evidence that this conjecture is false.