Energy Conditions in Eternal Inflation
We show that the eternal inflation scenario requires violations of the Null Energy Condition (NEC) in de Sitter space. Within the context of semiclassical gravity, NEC violations are absent if the quantum state of fields is assumed to be de Sitter invariant since the expectation value of the “quantum NEC operator” vanishes. We then discuss the necessity of defining a smeared NEC operator. We find that smeared energy-momentum fluctuations in de Sitter space violate the NEC with probability 1/2 and the magnitude of the violations depends on the spatial and temporal smearing scales, with the largest violations occurring on the smallest scales. In the case when both smearing scales are equal to the de Sitter horizon H, the magnitude of the NEC violation is given by ∼ H/(2π). We also discuss NEC violations during slow roll inflation. Our results are consistent with what is obtained in the literature based on fluctuations of the scalar field except in the diffusive regime where quantum jumps are thought to be important. We discuss the need for a scheme to calculate the backreaction of such NEC violating quantum fluctuations on the metric.