Renormalization of the Quantum Stress Tensor Fluctuations and the Limits of Semiclassical Gravity

We analyze the expectation value of the energy-momentum tensor and its fluctuations in quantum field theory on curved spacetimes $\langle T_{ab} \rangle$. A generally accepeted condition for the conceptual consistency of semiclassical gravity, where $\langle T_{ab} \rangle$ represent the sources of the Einstein equations, is that the fluctuations of the energy momentum tensor remain small compared to its expectation value. We study the renormalization of both the energy-momentum tensor $\langle T_{ab}(x)\rangle_{\rm ren}$ and the fluctuation tensor $\langle T_{ab}(x) T_{cd}(x) \rangle_{\rm ren}$ for suitable Hadamard states, using the operator product expansion for a free scalar field on a fixed curved background. We show that states (usually referred to as `squeezed vacua') -- arising naturally in black hole evaporation and in inflationary cosmology -- fail to satisfy the natural semiclassicality criterion.