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Particle production in models with helicity-0 graviton ghost in de Sitter spacetime

We revisit the problem of the helicity-0 ghost mode of massive graviton in the de Sitter background. In general, the presence of a ghost particle, which has negative energy, drives the vacuum to be unstable through pair production of ghost particles and ordinary particles. In the case that the vacuum state preserves the de Sitter invariance, the number density created by the pair production inevitably diverges due to unsuppressed ultra-violet(UV) contributions. In such cases one can immediately conclude that the model is not viable. However, in the massive gravity theory we cannot construct a vacuum state which respects the de Sitter invariance. Therefore the presence of a ghost does not immediately mean the breakdown of the model. Explicitly estimating the number density and the energy density of particles created by the pair production of two conformal scalar particles and one helicity-0 ghost graviton, we find that these densities both diverge. However, since models with helicity-0 ghost graviton have no de Sitter invariant vacuum state, it is rather natural to consider a UV cutoff scale in the three-dimensional momentum space. Then, even if we take the cutoff scale as large as the Planck scale, the created number density and energy density are well suppressed. In many models the cutoff scale is smaller than the Planck scale. In such models the created number density and the energy density are negligiblly small as long as only the physics below the cutoff scale is concerned.